Nonlinear Aeroservoelasticity - Politecnico di Milano...Abstract This thesis presents an insight into the behavior of aeroelastic systems in presence of aerodynamic and structural

POLITECNICO DI MILANO

DEPARTMENT OF AEROSPACE SCIENCE AND TECHNOLOGY

DOCTORAL PROGRAM IN AEROSPACE ENGINEERING – XXVIII CYCLE

Nonlinear AeroservoelasticityReduced Order Modeling and Active Control

Ph.D. Candidate: Andrea Mannarino

Thesis Advisor: Prof. Paolo Mantegazza

Tutor: Prof. Pierangelo Masarati

Chair of the Ph.D. programme: Prof. Luigi Vigevano

Abstract

This thesis presents an insight into the behavior of aeroelastic systems in presence of aerodynamicand structural nonlinearities. It is shown that such effects, often not accounted in classical aeroelasticanalyses, might greatly affect the system stability and shape peculiar dynamic responses. Their earlyinclusion in the design process of aircraft components may therefore lead to substantial weight savingand to a paradigm shift in control law design.

Particular emphasis is given here to the mathematical modeling of aerodynamic nonlinearities,such as large shock wave motions that induce limit cycle oscillations in flexible wings. Reduced ordermodels based on neural networks are developed to extract information from computational fluiddynamics analyses and reconstruct unsteady nonlinear behaviors of the aerodynamic system in acompact way. Structural nonlinearities are instead studied through simpler static models, whichnonetheless permit a physically meaningful representation of the responses of interest. The coupledeffect of structural and aerodynamic nonlinearities on a typical aeroelastic test case is assessed, show-ing a very particular behavior that depends on the problem initial condition and on the disturbancetype and amplitude that is used to excite the system dynamics. Limit cycle oscillations are studied andtheir dependence on the flight speed and other key parameters is assessed. The results computed bythe reduced order model are then validated through computational fluid dynamics-based aeroelasticsimulations.

Having available a tool to describe compactly nonlinear unsteady aeroelastic responses, classicaland adaptive controllers are developed aiming at improving the system performance and eliminatingpossible instabilities. It is found that the suppression of aerodynamic nonlinearities well differs fromthe approaches used to compensate effectively the presence of structural nonlinearities such as free-play and friction in control surfaces actuation chains.At first these kind of nonlinearities and their related suppression techniques are studied separately,while at the end a systematic approach is presented to design a controller that compensates botheffects through an integrated approach. All the results are validated first designing and tuning thecontrollers on reduced order models and then verifying their behavior through the time integration ofa computational fluid dynamics-based aeroelastic code.

Finally, the design of a control system aiming at reconstructing the behavior of physically limitedactuators is considered. Particular focus is given to the choice of the control architecture required toreproduce position, rate and torque saturations, developing also an automatic tuning algorithm thatrequires the knowledge of a very little data, eventually refined through a frequency-based optimization.

Concluding remarks are given highlighting the most interesting findings of this work and indicatingpossible paths to be followed for extending the present results.

Contents1 Introduction 1

1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Nonlinearities in Aeroservoelastic Systems: Where They Come From . . . . . . . . . . . 2

1.2.1 Structural Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Aerodynamic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Control Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Reduced Order Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Aeronautical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Different approaches to nonlinear reduced order modeling . . . . . . . . . . . . . 13

1.4 Modern Control of Nonlinear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.1 Aeronautical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4.2 Different approaches to nonlinear control . . . . . . . . . . . . . . . . . . . . . . . 31

1.5 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 Aerodynamic Reduced Order Modeling through a Linear-Neural Model 47

2.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2 Aerodynamic Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.1 Reduced order model training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.2 Training signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.4 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.1 Two degree-of-freedom typical section . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4.2 Four degree-of-freedom typical section . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4.3 Transonic aeroelasticity in presence of free-plays . . . . . . . . . . . . . . . . . . . 68

3 Structural Nonlinearities Compensation 73

3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Dual-Loop PID Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.1 Frequency-based virtual reference tuning . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Adaptive Control through Nonlinearity Inversion . . . . . . . . . . . . . . . . . . . . . . . 79

3.4 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4.1 Two-mass with free-play benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4.2 Aileron actuation system with gear-box free-play . . . . . . . . . . . . . . . . . . . 90

3.4.3 Limit cycle oscillations of a vertical T-tail . . . . . . . . . . . . . . . . . . . . . . . . 101

3.5 Sliding Mode Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

i

ii Contents

4 Aerodynamic Nonlinearities Compensation 1154.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Aeroservoelastic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3 Flutter Suppression by Immersion and Invariance Control . . . . . . . . . . . . . . . . . . 118

4.3.1 Design of the target dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.3.2 Design of the parameters estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.3 Final control law and proof of stability . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.4.1 Typical section with trailing edge control surface . . . . . . . . . . . . . . . . . . . 1254.4.2 BACT wing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.3 Goland wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5 Integrated Compensation of Aerodynamic and Structural Nonlinearities 1355.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.2 Aeroservoelastic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.2.1 Design model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.2.2 A comment on the design of the flutter suppression system . . . . . . . . . . . . . 1385.2.3 Verification model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.3.1 Four degree-of-freedom typical section with free-play . . . . . . . . . . . . . . . . 1405.3.2 Goland wing with free-play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Conclusions and Final Recommendations 157

A Virtual Realization of Actuation System Nonlinearities 161A.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.2 Control Architecture and Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162A.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Bibliography 178

Introduction

CHAPTER 1

1.1 Motivations

During the last decades, the research on the active suppression of aerospace structures vibrationshas attempted to achieve a high level of performance and safety through improved design methods.Researchers and academia have focused their efforts on the synthesis aspects of control systems, inparticular by demonstrating the applicability and strengths of novel, robust, multivariable synthesistools.Such a development demands a comprehensive approach to appropriately deal with optimized de-signs, covering the whole spectrum of problems integrating flight mechanics and aeroservoelasticity,e.g.: flutter, control effectiveness and divergence, maneuver and gust loads, buffeting, flight perfor-mances [1, 2, 3, 4]. Until the more recent decades, a somewhat inadequate computational power hasrestricted the ordinary study of aeroservoelastic systems to linear(ized) subsonic and supersonic flightregimes [5].

Nowadays, advances in computers technology and computational fluid dynamics (CFD) allow toadequately evaluate nonlinear unsteady loads for inviscid and viscous flows. Therefore, the adoptionof CFD-based aeroservoelastic analyses is becoming more and more viable [6], thus allowing to betterdeal with transonic flows and strong oscillating shocks. The full control of these, possibly dangerous,nonlinear events is of utmost importance in avoiding unacceptable self-induced oscillations, instabili-ties, limit cycles, ride-quality deterioration and fatigue failures [1].Nevertheless, even with the computational power currently available, the costs of solving the relatednonlinear high order problems still impede their routine adoption in many repeated calculationsrequired in preliminary aircraft design phase, making them more viable for the detailed validationstypical of advanced design phases [7, 8, 9].Because of these reasons, if the objective of the study is the development of control laws able to copewith the previously mentioned aeroelastic problems, it is very important to develop models whichcaptures the main physical phenomena of interest while keeping the computational time as low aspossible. Such a result can be obtained in two ways.When the problem admits a clear physical interpretation, then models based on physical principlescan be used, giving to the engineer a great deal of comprehension of the situation under study andtherefore leading to solutions easy to understand and routinely employed in industry.Unfortunately, this is not always the case, especially when studying problems that have receivedattentions in the last years, for example the modeling of the dynamics of moving shock waves, un-steady flow separations or even the transition to turbulence [8, 10, 11]. In this case, the developmentof reduced order models (ROM) out of high fidelity numerical solvers permits the use of compactnonlinear models for a wide analysis and design spectrum, while maintaining the needed level ofaccuracy required by the applications of interest.It is therefore appealing to the designer to have available a computationally cheap tool to be usedfor designing control laws that then will be tested on the related expensive high-fidelity model and

1

2 Chapter 1

possibly experimented in laboratory. The aim of this work is to present some of the methods requiredto achieve such a goal and detail a general procedure to study, analyze and design advanced controllaws for multi-input, nonlinear aeroservoelastic system.

In what follows in this chapter, an introduction to the phenomena that characterize nonlinearaeroservoelastic problems will be presented, giving then a broad overview of methods that nowadaysare studied and employed in the fields of reduced order modeling and automatic control. After that,the original contribution of this research to the mentioned fields of interest will be detailed, and finally,a summary of what will be presented in this work will be provided.

1.2 Nonlinearities in Aeroservoelastic Systems: Where They Come From

Aero-servo-elasticity is a well known multi-disciplinary field that studies the structural behavior underthe interaction of inertial, structural, aerodynamic forces and controls. Traditionally, linear modelshave always been used in this field in stability analysis and control design, limiting the focus on systemnonlinearities only in the verification and validation phases through an Increased Order Modelingapproach [12, 13, 14].

Nonetheless, nonlinearities may be present in an aeroelastic system due to both structure, controllaw and aerodynamics, and their effect might be not negligible. The resulting aeroservoelastic behaviorcannot be accurately predicted in general by standard linear analysis methods. Such nonlinearitieshave been identified through mathematical models, wind-tunnel, and flight tests [4, 15, 16, 17, 18].Here below a description of the possible sources of nonlinearities in aeroservoelastic systems is given,summarizing also their effect on the stability characteristics and on some peculiar responses that theymay induce.

1.2.1 Structural Nonlinearities

Structural nonlinearities can be important and are the result of a given aerodynamic force on thestructure creating a response that is no longer linearly proportional to the applied force [3, 4, 19,20]. Free-play [18, 19, 21, 22, 23, 24, 25, 26] and geometrical nonlinearities [16, 27, 28, 29] are primeexamples. However, the internal damping forces in a structure may also have a nonlinear relationshipto structural motion [4], with dry friction being an example that has received limited attention to date.

Free-play is a concern with respect to control surface actuation, but it has also been suggestedas a possible source of flutter and limit cycle oscillations in wing/store attachments [30]. A sampleof an experimentally reproduced control surface free-play is depicted in Figure 1.1a. This is still anopen area of investigation, but recent progress for free-play in control surfaces offers an opportunityto enhance both analysis and design methods and may lead to a paradigm shift in design criteria [4].The main effect of such a nonlinearity is to shift back the flutter point, which is possibly reached atlower flight speeds, often inducing structural limit cycle oscillations (LCOs). Such responses are infact self-induced oscillations driven by the discontinuous motion of the control surface, induced bythe nonlinearity in the actuation torque as shown in Figure 1.1b. The related mathematical model foreach control surface can be described as follows:

mβ =

kβ

(∆β+βFP

)for ∆β<−βFP

0 for |∆β| <βFP

kβ

(∆β−βFP

)for ∆β>βFP

∆β=βext +βc −β

(1.1)

1.2. Nonlinearities in Aeroservoelastic Systems: Where They Come From 3

(a) Rendering of an experimentally reproduced free-play

!! !" # " !!$##

!%##

!&##

!!##

#

!##

&##

%##

$##

∆β [deg]

'()*+,-./+)0,12/3

,

,

45+67,8()*+,769

:;0<67,8()*+,769

-βFP

βFP

(b) Nonlinearity between surface deflection and actuationtorque

Figure 1.1: Control surface free-play

The variable ∆β is often referred as aileron dynamic response [31] and it can be decomposed into threecontributions: βext is the applied external command, βc the command computed by some control logicand β the effective aileron deflection. In addition, kβ is the equivalent stiffness connecting the pilotcommand to the control surface andβFP is the semi-width of the free-play, here assumed symmetric.Another unwanted effect is experienced at low flights speeds, since the command given by the pilotto the control flight system will not be followed correctly by the control surface, unless additionalcorrections are taken into consideration [24]. For all these reasons, strict limitations are still imposednowadays on the free-play maximum value, which has to be limited to small fraction of a degree [25].

Recent computational and experimental work [19, 22, 23, 25] has shed new light on this behavior.It is now understood that in fact, for an unloaded control surface, the flow velocity at which LCO beginsis independent of the degree of free-play.However, it is now known that the amplitude of the LCO and the amount of loading required to precludeLCO is strongly dependent on the degree of free-play. In fact the LCO amplitude scales in proportionto the degree of free-play and the amount of loading required to suppress flutter/LCO does as well. Forexample, the LCO amplitude will be of the order of the degree of free-play and if the loading is dueto placing the airfoil at an angle of attack, the angle of attack required to totally eliminate free-playis about five times the degree of free-play. Thus for a freeplay of 1/64 degrees the LCO amplitudewill be about 1/64 degrees and an angle of attack of 5/64 degrees is sufficient to suppress the LCOaltogether [18, 25]. These works have confirmed the effect of free-play and loading on LCO. Suchresults show that varying the degree of freeplay simply changes the LCO amplitude in proportion whilethe LCO frequency is unchanged.The work of Schlomach [32] for the F-35 program has provided independent verification of the aboveresults and extended them into the high subsonic/transonic flow regime. As expected the quantitativeagreement between theory and experiment is less satisfactory in the transonic regime because of thechallenging environment for modeling the aerodynamic forces. However, even so, the same scalinglaws for the effect of freeplay and loading were also found in the just mentioned study.

Nevertheless, the above cited works aimed mainly at the analysis of such peculiar nonlinearbehaviors, often without addressing the possibility of compensating it. It is well known that thepresence of free-play can significatively alter the stability characteristics of an aeroservoelastic system,both in open and closed loop [24, 31]. In particular, several studies have shown that the presence of

4 Chapter 1

such a nonlinearity can jeopardize the properties of a control law designed without accounting for it,thus leading to closed loop responses characterized by residual LCOs [24, 33, 34, 35].

Another not infrequently encountered and documented case is the limit cycle oscillation thatfollows the onset of flutter in plate-like structures. The structure has a nonlinear stiffening/softening asa result of the tension induced by mid-plane stretching of the plate that arises from its lateral bending.A sample of this behavior is depicted in Figure 1.2a. This is most commonly encountered in what isoften called panel flutter where a local element of a wing or fuselage skin encounters flutter and thenLCO. There have been many incidents reported in the literature dating back to the V-2 rocket of WorldWar II, the X-15, the Saturn Launch Vehicle of the Apollo program [36] and continuing on to the presentday. It has been recently recognized that low aspect ratio wings may behave as structural plates andthus the entire wing may undergo a form of plate-like flutter and LCO. This has been seen in both windtunnel models and computations [4].Such a stiffening effect is usually resembled through the following mathematical model [27]:

Felastic = kθθ+kNLθ3 (1.2)

in the case of a torsional nonlinearity, where θ is the rotation angle. As can be noted from Eq. 1.2, forsmall rotations the elastic force generated is the same provided by a linear spring. As the rotationincreases though, the nonlinear cubic effect becomes predominant, and LCOs may appear. As wellhighlighted in Figure 1.2a, when kNL is positive it induces a nonlinear stiffening, on the other hand,when negative, it produces a nonlinear softening. At this point, it is important to point out thatdifferently from the effect introduced by a free-play this nonlinearity does not affect the flutter point ofthe system. This is because linearized analyses will only take count of the first contribution, which isthe classical linear term traditionally employed in aeroelasticity. Therefore, the nonlinear stiffening (orsoftening) term will be only responsible of the shaping of possible LCOs beyond the flutter point.

O’Neill [27] provides an example of a theoretical and experimental study that explores nonlinearstiffening-induced aeroelastic LCOs. The study is based on a rigid wing section with a store mountedon a pitch-and- plunge system, the so-called Nonlinear Aeroelastic Test Apparatus (NATA), shown inFigure 1.2b. The analysis represents the structure by a rigid airfoil with 2 degree-of-freedom, namely,plunge and pitch. The airflow is assumed to conform to quasi-steady potential-flow strip theory. Thesystem is kinematically nonlinear as a result of the aerodynamic center, elastic axis, and center of massbeing vertically offset from each other, and is also constrained in pitch by a nonlinear spring. In thisstudy, the nonlinear pitch stiffness is identified as the dominant factor responsible for the occurrenceof LCOs.

Another cause of nonlinearity is dry friction. A control surface restrained only by solid frictionwill ideally have a Coulombian force-velocity characteristic like that roughly sketched in Figure 1.3a.In practice, it will probably behave more like a Stribeck friction model [37, 38, 39], resembled inFigure 1.3b. If both friction and backlash are present, the a hysteresis nonlinearity of the type shownin Figure 1.4 would be influencing the system response. This nonlinearity is characterized by a forceor moment which increases linearly with displacement until a value is reached at which a jumpoccurs, after which the system is again linear. On the return path, a corresponding jump occurs atanother value of the force or moment. The hysteresis introduces damping, and the time variations ofdisplacement for a one degree-of-freedom mechanical vibration system usually shows damping tovary with amplitude of oscillations for moderately large amplitudes. However, for small amplitudes thesystem oscillates on a line through the box with a linear spring constant and the hysteretic dampingbecomes zero. For very large amplitudes, the effect of the hysteresis is small and can be neglected inthe limit when the ratio of the width of the hysteresis box to the oscillation amplitude approacheszero [37].


(a) Nonlinear stiffening/softening effect (b) Nonlinear Aeroelastic Test Apparatus. From [27]

Figure 1.2: Nonlinear stiffening/softening phenomena in aeroelasticity

(a) Coulomb friction (b) Stribeck friction

Figure 1.3: Different friction models

Another interesting effect of hysteresis on the structural properties of a vibrating system is tointroduce an effectively weaker spring as in the case of a free-play. At amplitudes of oscillation forwhich the system passes completely through the hysteresis box, the frequency of oscillation is less thanthe frequency at low amplitudes where the system is linear. For very large amplitudes, the effectivestiffness is again approximately linear. Many structural systems are composed of ductile materials andare assembled in such a way that they exhibit a hardening hysteresis behavior under cyclic loading.An example of a hardening system can be found in riveted and bolted structures where slippingconnections provide a major contribution to the overall damping within the structure [37, 40].

Another type of structural nonlinearity which has been studied recently is the effect of underwingstores [16, 28, 30, 41]. These elements are typically attached to the wing and restrained via non-homogeneous elements such as rails, sliders, hooks, sway braces or crutches, all of which providescope for nonlinear response. Ground Vibration Tests on the F-16 have revealed wing-store attachmentnonlinearities of hardening as well as softening types [30]. In the case of the F/A-18 aircraft, after yearsof usage, it has been observed from flight tests that with light-to-intermediate weight stores and with

6 Chapter 1

Figure 1.4: Hysteresis effect.

wing tip missiles, the aircraft was subject to lightly damped low-frequency LCOs [37]. Thompson andStrganac [41] provide an example of a theoretical and experimental study that explores store-inducedaeroelastic LCOs, again based on the NATA model previously introduced.

As can be seen from this review, structural nonlinearities can be introduced by several sources intoan aeroelastic system. There is a slight but essential difference between them though. Nonlinearitiessuch as free-play, friction and hysteresis, are also called hard nonlinearities. Regarding an aeroelasticsystem, they modify not only its dynamic response, but they also change the related stability properties,usually moving backwards the flutter point considerably. The same cannot be said about the other typesof nonlinearity, such as the nonlinear stiffening/softening effect, which are called soft nonlinearity.This kind of behavior mainly influences the system response, without greatly affecting its stabilitycharacteristics.

1.2.2 Aerodynamic Nonlinearities

At high flight speeds, compressibility effects are important and they can have a pronounced influenceon the aeroelastic response compared to that using incompressible theory. Differently from thestructural nonlinearities seen before, all aerodynamic nonlinearities can be of the soft type. We shouldthen consider a few situations in which aerodynamic nonlinearities are important in their interactionwith the structural dynamics.

The first is associated with the presence of shock waves in transonic flows. The flow is assumed tobe inviscid and separation may be not so significant. In this situation, the unsteady forces generatedby the motion of the shock wave have been shown to destabilize the pitching motion of a singledegree-of-freedom airfoil and affect the bending-torsional flutter by lowering the flutter speed at theso-called transonic-dip regime, as shown by [42, 43]. In these works it was pointed out that the shockwaves located on the upper and lower wing surfaces move periodically with large phase lags with theoscillatory airfoil motion. In this case, the flutter boundary can be 25% lower than its conventionalshock-free counterpart.The distance traversed by the shocks on the airfoil surfaces depends on the frequency and can be quitelarge compared with the airfoil motion. A sample of this behavior is given in Figure 1.5. Consideringmore realistic systems, like flexible wings, LCOs may be induced by the interaction between themoving shock structure, which quite be quite complex, e.g. λ-shaped shocks, and the related structuralmotion [45], as shown in Figure 1.6. To capture the effects of aerodynamic nonlinearities on aeroelastic


(a) Flow field sketch around a moving shock (b) Shock motion cycle during an LCO

Figure 1.5: Effect of a moving shock on a flexibly mounted airfoil. From [44]

7.91e+03

4000

5000

6000

7000

p

3.16e+03

(a) LCO lower limit. The shock reaches its full strength onthe upper surface while it is absent on the lower surface

7.91e+03

4000

5000

6000

7000

p

3.16e+03

(b) LCO upper limit. The shock reaches its full strength onthe lower surface while it is absent on the upper surface

Figure 1.6: Numerical simulation of an LCO experienced by the AGARD 445.6 wing

behavior, time integration methods based on the transonic small disturbance method, Euler andfull Navier-Stokes equations have been developed, see [8] and references therein. These methodscan provide accurate approximations of the system behavior, but in general they require very largecomputation times owing to the large number of variables and the characteristic integration timesrequired to establish steady flow stability properties [8].

When viscous effects are considered, flow separation can occur due to shock-boundary layerinteraction. This can cause single degree-of-freedom flutter, such as control surface buzz [46, 47], andbuffeting [44, 48, 49, 50].

With the term buzz are referred the self-excited oscillations of a flap about its hinge. Such phe-nomenon can be associated with several regimes of flow and it is likely that the mechanism of excitationis different for each type [46], as shown in Figure 1.7. At least one form of instability can be relatedto negative aerodynamic damping predicted theoretically and does not depend on boundary-layereffects. Other forms of buzz depend on the occurrence of shock-induced separation ahead of the flaphinge, or on the presence of shock waves at the surface of the flap itself. In the experiments outlined

8 Chapter 1

in [46], it was found out that the lowest critical Mach number for buzz is a little above the critical Machnumber at which the flow becomes supersonic locally at the surface of the airfoil and is found to beassociated with a shock at the surface of the airfoil ahead of the hinge line causing separation of theboundary layer.As detailed in [46], for incidences very close to zero, the first onset of buzz occurs with shock-inducedseparation present at both upper and lower surfaces of the airfoil, as shown in Figure 1.7b(a). For higherincidences buzz occurs with shock-induced separation at the upper surface only, as in Figure 1.7b(b).Oscillation of the flap is coupled with backwards and forwards movement of each shock wave, but itsexcursion does not extend onto the flap itself.A second region of instability is associated with the regions of local supersonic flow extending rear-wards onto the surface of the flap, e.g. see Figure 1.7b(c), and also appears to involve separationsinduced by shock waves.With further increase of Mach number, the shock waves moves rearwards to the trailing edge, thuscorresponding to local supersonic flow everywhere over the flap. A further increase of Mach numberto a value greater than unity, e.g. Figure 1.7b(e), leads to an other behavior, in which the oscillation,for small amplitudes at least, does not appear to involve shock-wave or boundary-layer effects. Theresults present hysteresis effects. In fact, as the Mach number is lowered from condition (e), the flap isstable for small disturbances, but buzz will be initiated if the flap is given a disturbance sufficient tolead to a region of supersonic flow forming locally at the convex corner formed by the deflected flap,shown in Figure 1.7b(d).Numerical simulations were also carried out in [47] using an Euler flow model, attempting to generatea physics-based reduced order model to study this peculiar behavior. A sample of the obtained resultsis depicted in Figure 1.7a.

Shock buffet is the term used for the self-sustained, low-frequency, large-amplitude shock oscilla-tions that are observed for certain combinations of Mach number and steady mean angle of attack attransonic flows, even in the absence of airfoil motion. Currently, advances in viscous Navier–StokesCFD modeling and simulation methods enable the prediction of heavily separated and buffeting flowswith good correlation to wind-tunnel test experiments [50].The interaction of the shock buffet with the elastic structural motion may induce significant aeroelasticresponses that are highly undesirable from the point of view of structural integrity and flight handlingqualities.A recent studies by Raveh [51] presented Navier–Stokes simulations of airfoil responses to prescribedmotions (plunge, pitch, and trailing-edge flap rotation) about flow conditions that exhibit shock buffet.A lock-in phenomenon was discovered, like the one experienced by electric cables during galloppingconditions [3], in which the shock-buffet frequency synchronizes with the frequency of the prescribedairfoil motion. It was found that lock-in occurs when the shock-buffet frequency and the prescribedairfoil motion frequency are sufficiently close, and the amplitude of the prescribed airfoil motion isabove a certain threshold.Raveh and Dowell presented in [50] a study focused on the characterization of the aeroelastic responseof a spring suspended airfoil in transonic buffeting flows. Navier–Stokes turbulent simulations of anaeroelastic system composed of a NACA 0012 airfoil suspended on a single-degree-of-freedom pitchspring, as well as on a combination of 2-degree-of-freedom pitch-and-heave springs, were performedto compute the aerodynamic forces and elastic displacements. In such an effort, the natural frequen-cies of the aeroelastic system are varied from below to above the shock buffet frequency to study theireffects on the aeroelastic responses.


(a) Numerical simulation of a nonclassical buzz. From [47] (b) Buzz classification

Figure 1.7: Transonic buzz

(a) Lift coefficient vs. time in response to the step-upincrements to the mean flow angle of attack. M∞ = 0.72,Re = 107

(b) Moment coefficient vs. time in response to the step-down decrements to the mean flow angle of attack. M∞ =0.72, Re = 107

Figure 1.8: Buffeting flow solutions. Results from [4]

10 Chapter 1

1.2.3 Control Nonlinearities

Nonlinearities in a aeroservoelastic system may be introduced by control laws and actuators as well. Indesigning control laws, linear actuators models are usually taken into account as a first approximation.But, in reality, actuators are intrinsically nonlinear, because they presents limitations on the maximumachievable displacement (position saturation), on the speed with which they can reach a new position(rate saturation) and on the maximum force that they would be able to generate (force saturation) [52].In addition, electro-hydraulic actuators may present other internal nonlinearities, such as free-playand friction, usually experienced during the motion of the actuation valves [ 53] and turbulent fluxesacross valves and piston orifices [54]. Hysteresis is another phenomenon that can be found on this typeof actuators [55]. Actuator failures can also result in significant deviation from the nominal dynamicsand may cause departure in to highly nonlinear regimes [56].In certain cases, even when the actuation system can be considered linear in a first approximation,nonlinearities may be introduced by the control law employed. In fact, usually the plants to becontrolled are intrinsically nonlinear, and linear controllers can be employed only after the plantlinearization around an equilibrium point (a trimmed condition for an aircraft). If the designer wantsto use these type of controllers, then a well-known technique to link different working conditions andthe related control systems is the so-called gain scheduling [57, 58]. Nonetheless, in the last decadesanother family of controllers, more flexible than the linear ones, have been developed. These are theso-called nonlinear and adaptive controllers. Within the nonlinear class, back-stepping [59], feedbacklinearization [60] and sliding mode control [61], Lyapunov redesign and nonlinear damping [ 62] canbe cited. Gain scheduling can also be included in this class.In the context of adaptive control instead, a plethora of techniques are available to the designer. Modelreference adaptive control [63, 64, 65], neural networks-based controllers [66, 67], L -1 controllers [68],state-dependent Riccati equation methods [69], fuzzy-logic based controllers [70, 71], control lawsbased on nonlinearity inversion [72] and immersion and invariance control [73] belong to this class,for example.

Nonlinear control may be necessary to achieve the desired performance if large range and/or highspeed motions have to be controlled, since nonlinear effects can be significant in the system dynamics.On the other hand, adaptive control is an approach to dealing with uncertain systems or time-varyingsystems. Although the term adaptive can have broad meanings, current adaptive control designs applymainly to systems with known dynamic structure, but unknown constant or slowly-varying parameters.Adaptive controllers, whether developed for linear systems or for nonlinear systems, are inherentlynonlinear. Therefore, classical nonlinear controllers can be used when nonlinearities are modeled inthe system dynamics but no variation of the system properties is experienced. Instead, when suchproperties are subjected to large changes during the operational life of the system, adaptive controllermay be a better choice thanks to their ability of changing their behavior in the face of substantialchanges in the system response.The nonlinearities introduced by the control law may also lead to unwanted vibrations and LCOs whenthe closed loop system works in off-design conditions. For example, the discontinuous nature of asliding mode controller could lead to chattering [61] when the actuation system has not an adequatebandwidth to follow the control command. Other effects, like the delay introduced by digital/analogconversion, control computation and signals noise filtering may lead to nonlinear responses, as studiedin [74].

1.3. Reduced Order Modeling 11

1.3 Reduced Order Modeling

In recent years, the huge increase in computational power has allowed to solve problems once impossi-ble to approach. Nowadays, systems of Partial Differential Equations (PDE), representative of complexphysics, can be discretized and solved with numerical algorithms on massive computational systems.Simulations of systems with millions (or even tens of millions) of unknowns has become relativelycommonplace [8].However, such massive simulations require an adequate computational power to be afforded. More-over, a direct numerical simulation of the related system of PDE is often insufficient to understand aphysical phenomenon in depth. Reduced Order Models (ROMs) are a set of theoretical and numericalstructures that permits to reconstruct and represent a generic nonlinear system with a small numberof Degree-Of-Freedom (DOF). They thus allow a simpler formulation of a nonlinear problem, gainingboth a deeper understanding of the physical phenomenon and an uttermost saving of simulationtime. Of course, a simpler representation of a system allow also an effective formulation of the variousanalysis that have to be performed. In brief, a ROM permits to represent the nonlinear, dynamicbehavior of a system, once it is able to capture the main fundamentals dynamic embedded in thesystem itself.

Reduced Order Models have progressively gained space in the analysis of physical system, asnowadays numerical simulations present two main problems:

• Direct simulation can provide a detailed response history of the field variables, but such resultsmay not help the user in gaining an increased level of understanding of the essential physics ofthe problem;

• In the absence of massive computational resources, the simulation of large-scale problemsremains not practical to be used in various design phases, such as stability evaluation, controlsystem design and optimization.

Considering the aeronautical world, an example of direct numerical simulation of a PDE system is ofcourse the field of the CFD. It permits an adequately accurate simulation of a turbulent, viscous flowfield in the three-dimensional space and it is very useful to validate experimental results. Nevertheless,its computational cost is still too high to be used during a design phase of an aircraft, because in suchcase there is the need of many fast simulations. Moreover the analyses that have to be performed oftendo not require a point valued solution available in the whole domain. For example the evaluation ofthe loads acting on a body immersed in a fluid stream requires only the knowledge of the solution onthe body itself.

Reduced order modeling can be used both for linear and nonlinear systems. As an example, astraightforward order reduction in the linear case is of course the use of normal modes in structuralengineering [75]. Normal modes permit an accurate representation of the movement and the loadacting on the structure using a relatively small set of DOFs. An order reduction for a nonlinear systemis even more attractive, with the possibility of capturing physical phenomena characteristic of suchsystems, e.g. possible LCO behaviors [3], maintaining a low number of DOF.

A ROM should provide a sufficiently accurate description of the system dynamics but requiring acomputational effort much lower than the one requested by a direct numerical simulation, leading toan easier interpretation of the dynamic response. Such model order reduction can be interpreted as ageneralized projection of the full order system, characterized by a high number of DOFs, onto a muchsmaller space, which encapsulate most, if not all, of the system fundamental dynamics [8].Even if it inherits a lot of procedures from system identification theory, it should not be mistaken thatreduced order modeling is well different from system identification. In fact, while system identification

12 Chapter 1

Figure 1.9: Flow chart of a reduced order analysis

techniques are adopted when an unknown system has to be modeled, starting from an unknownparametrization or with a fixed structure but with unknown parameters (in this case it would be morecorrect to talk about parameter estimation), reduced order models start from a system which has beenalready modeled mathematically, typically through a system of PDE, and the successive semi-analyticaldiscretization has produced a very large system of Ordinary Differential Equations (ODE). Typicallysuch spatial discretization is performed considering the real geometry of the physical problem, withouttaking into account the efficiency of the basis on which the solution is approximated.Using reduced order models, the dynamical system is projected onto a subspace of the original physicalspace, and in this case the basis on which the solution is discretized is more efficient. In fact such asubspace is composed of basis functions which inherit the own characteristics of the overall solution.The basic idea of a ROM representation of a system is given in Figure 1.9.

1.3.1 Aeronautical problems

There are a lot of reduced order formulation in the aeronautical world, especially for linear systems.For example, the use of normal modes has just been cited as an effective technique to reduce thesize of a structural system. Regarding the aerodynamic representation, panel methods [ 76] has beenextensively used for load estimation of steady/unsteady, incompressible and linearized compressibleflows. These can be classified as reduced order models, since they capture the physical behavior of acontinuous system with a small number of unknowns.

In the field on aeroelasticity, example of reduced order models are those based on simple physicalprinciples, such the representation of a wing as a cantilevered elastic beam, represented by the firstbending and torsional normal modes coupled with a nonlinear aerodynamic model based on a look-uptable constructed by means of experimental data, theoretical formulas, panel methods or a few CFDsimulations [3]. These models permit fast simulations of the system, useful during a control law designfor example, with the accuracy of the results that depends on the structural and aerodynamic modeling.Nevertheless, they can be considered valuable only if adequately rebust controllers can be designed.


Modern airplanes that present their cruise at transonic flow regimes requires special attention,e.g. because potential based methods are not adequate when strong shocks are present in the flowfield. Therefore in such situations at least the adoption of an Euler flow model is required. In case ofvery strong shocks even this latter flow model should not be employed, since it neglects the effect ofthermal conductivity which can be important. In the viscous case, e.g. Navier Stokes flow model, thesituation is even more complex, with the possibility of large flow separation.Therefore, the flow behavior around an aircraft can be accurately predicted by high fidelity CFD codes.However, their direct coupling with a structural solver for the determination of aeroelastic responsescan be quite expensive from the computational point of view, due to the application of multi-steptime-domain scheme used in the integration of the flow dynamics equations. This integration implieslong calculation times, also with the modern high-performance computers.Moreover, many design and control applications in aeroelasticity may require relatively simple modelsto predict the system dynamics with as little computational (and experimental) effort as possible,especially in the conceptual and preliminary design phases of an aircraft. Reducing the complexityof the physical model is certainly a non-trivial task, in particular for the case of strongly nonlinear oreven chaotic systems.

The solution proposed in this work, aimed at reducing the overall design computational cost, is therealization of a ROM that can predict the unsteady, nonlinear behavior of an external flow. Then, bycoupling such compact aerodynamic model to the related mechanical system, fast simulations of theresulting nonlinear aeroelastic system will be possible.

1.3.2 Different approaches to nonlinear reduced order modeling

Different methods for modeling nonlinear systems with a few DOFs have been developed in recentyears. In the field of aeroelasticity in particular, a substantial effort has been made to reduce thecomputational cost relative to high-fidelity, nonlinear aeroelastic simulations. The ROMs proposed inthe literature are able to capture essential nonlinear flow characteristics with a greater computationalefficiency than full CFD simulations.Such models have been used to perform transonic flutter analysis [77], responses at high angle ofattack [78], gust response [79], integrated aeroservoelastic optimization [80] and control law design [81,82].However, most of the ROMs proposed for predicting aerodynamic responses are dynamically linear [3]:this means that the ROM is basically a linear system near the reference solution, with aerodynamicloads that varies linearly with respect to relatively small structural motions. Those models can beuseful for flutter investigations, under the assumption of small structural vibrations, but they lose theirreliability when the amplitude of the structural motion is large [8].

Aerodynamic reduced order modeling is usually approached from four distinct lanes. The firstcharacterizes the aerodynamic flow field in terms of a relatively small number of global modes, a modebeing intended here as a mean distribution of the variables characterizing some gross motion of theflow. This approach is the one followed by the Proper Orthogonal Decomposition (POD) [83], which

considers the aerodynamic model as characterized by modes describing the physical behavior of thesystem, similarly to the linear case. Considering such an information, a small set of dominant modescan be retained, constructing a model of compact dimensions. In this way, a typical CFD model, whichcan be described by 104 to 106 or more DOF, may be reduced to a model containing only a hundred, oreven tens of modes, remaining capable to accurately describe the main phenomena of interest.The second approach follows the one proposed by the system identification theory. It appeals to theidea that only a small set of input, i.e. structural modes, and a correspondingly small number of output,i.e. modal loads, are dominant in the system dynamic evolution. This approach is the one followed by

14 Chapter 1

Volterra series [84], Wiener-type models [85] and neural networks [86].The third approach instead does not consider a proper model order reduction. In this case the solutionshape is imposed a priori. For example, when considering periodic solutions, a Fourier series expansionbecomes the most natural choice. Then, substituting the imposed shape in the full order problem, anonlinear, algebraic system for the Fourier series coefficients is obtained and then solved with standardnonlinear solvers. This is the Harmonic Balance (HB) method [87], which has proved itself very efficientin the computation of periodic responses of nonlinear systems with respect to the brute force full ordersimulation.Eventually a fourth approach can be considered, but it is less employed in nonlinear reduced ordermodeling with respect to the other methods. Such a method is based on a accurate interpolation of thefull order system over various sampling points in the parameter space. The basis of this method arebriefly exposed in this chapter.

A comparison of the different reduced order modeling approaches available in the literature willbe given in the following paragraphs. It should be mentioned that all of the techniques presented herecan be efficiently used also to derive analytical models from experimental data.

Volterra theory

The idea of a time-domain, linear, unsteady aerodynamic model has been extensively used in theliterature regarding aircraft and helicopter aeroelasticity, by means of input-output models such asthe indicial response [88], or a state-space representation [89, 90, 91]. These models have proved tobe sufficiently accurate and robust in predicting aerodynamic loads related to small changes of theboundary conditions.

The modeling of nonlinear aerodynamic responses, such the ones coming from transonic flowshas always been a challenging problem [8]. Solutions as the transonic indicial response [92] havebeen proposed but more general approaches have been developed also. One of these is the Volterratheory of nonlinear systems. In practice this approach is a nonlinear Green’s function method thatprovides a natural and intuitive extension of well understood linear input-output formulation intothe nonlinear domain. In particular, Green functions can be interpreted as a linear subset of a muchbroader nonlinear Volterra functional space [8].

It is a nonlinear generalization of the concept of impulse response of a physical system: definingu(t ) a generic, scalar input function of the time t , for t ≥ 0, the response of the system w(t) is given bythe following linear combination of nonlinear terms:

w = w0 +∫ t

0h1(t −τ)u(τ)dτ+

∫ t

0

∫ t

0h2(t −τ1, t −τ2)u(τ1)u(τ2)dτ1dτ2 + . . . (1.3)

where w0 is the initial state of the system, meanwhile the functions hi are defined as kernels of theVolterra series. Their identification is based on measuring the response of the system to a set Diracdelta input at the time τi , i = 1,n, where n is the order of the series. For aeronautical applications,this identification is carried out from a small number of CFD simulations, where the input can bethe imposed motion of the body and the output are the aerodynamic loads. These input should becarefully chosen, because of the general nonlinear behavior of the system. Of course, for a linearsystem, only h1 has to be computed, and the other terms vanish. Some analytical expressions havebeen derived for 1st and 2nd order kernels, as shown in [93]. Also a discrete time version of this methodcan be designed for numerical applications [8].This formulation permits an analytical approach to the problem, but its accuracy has shown to be verysensitive to the choice of the input amplitude and to the computational time step used to discretize


Eq. 1.3, not only for nonlinear systems but even in the linear case [94]. In addition, Volterra series haveshown some difficulties in the identification of higher order kernels [8, 94, 95].

Sometimes, in order to reduce the computational cost of CFD-based simulations, linearized modelare considered.Such a linearization entails the evaluation of reference equilibrium solutions using a nonlinear model,but then the dynamic response varies linearly to the application of a generic input, which should be ofsmall amplitude in order to provide acceptable results. In the literature these models are also calleddynamically linear [3]. These linearized models can be then transformed to a state-space form. Theidentification of such models can be carried out through CFD analyses, computing the response ofthe aerodynamic system to an imposed input. Various strategies have been followed in the literature,for example a structural mode at a time is excited with an input chosen between a step, a ramp, asine function, a gaussian pulse or a blended step [93, 94]. These time domain responses are thentransformed into the frequency domain obtaining the well known generalized aerodynamic forcecoefficient matrix. A good feature of this approach is the ability of characterizing the aerodynamiclinearized system using a small number of CFD analyses. Once defined, the model of Eq. 1.3 canbe efficiently used to obtain a prediction of the aerodynamic response without costly repeated usesof a CFD solver. Resulting linear ROMs can be computed efficiently in a state-space form with theeigenvalue realization algorithm [8].

More recently, sparse Volterra reduced-order models [95] were efficiently used to model aerody-namically induced limit-cycle oscillations of the prototypical NACA 0012 benchmark model. In such awork a sparse representation of the Volterra series was explored for which the identification costs weresignificantly lower than the identification costs of the full Volterra series, with this fact permitting theidentification of high order kernels, usually associated with stronger nonlinearities.

Volterra series show some similarities with neural network modeling, and this fact will be high-lighted in the following sections.

Proper orthogonal decomposition

The proper orthogonal decomposition technique, also known as principal component analysis orKarhunen-Loéve expansion can be interpreted as a generalized nonlinear version of the modal repre-sentation of a linear system. It is also referred as an empirical spectral method [3, 8].This technique is normally used for determining efficient bases for approximating dynamic systemswith a few DOFs. It was first introduced in the study of turbulent coherent structures [96].As in the linear case, the system state x can be expressed as a modal expansion of the type:

x =Φq (1.4)

where Φ = [φ1φ2 · · ·φN

]is a modal matrix and q is the related amplitude vector. The difference is

that in this case the dynamic system is nonlinear, governed for example by the following autonomousdynamic relation:

x = R(x) (1.5)

The basis vectors are computed to maximize the following cost function:

J = maxϕ

< (x,ϕ)2 >(ϕ,ϕ)

= < (x,φ)2 >(φ,φ)

(1.6)

where the operator (·, ·) denotes the inner product in the related space and < · > is a time averageoperator.In practice, the POD basis is derived from a set of observations the system responses. Samples,

16 Chapter 1

or snapshots of the system behavior are used to compute an appropriate set of basis functions torepresent the system variables. It is important to remark that the basis functions so obtained are notonly appropriate, but optimal [8, 97]. The need of obtaining samples from the full order system can beviewed as both a strength and a weakness of the method. Its strength derives from the fact that thesampling of various system responses the ROM can be efficiently tuned so to obtain an high-fidelitybehavior with a small number of DOFs. Its weakness is of course related to the need of carrying out asignificant number of analyses of the full order model to construct an appropriate set of snapshots,combined to a possible lack of model robustness to the changes of the parameters governing thesystem behavior.

It has been shown in [3, 96], that taking a set of responses of the system at different time steps orfrom different input, let us say M << N samples, it is possible to construct the two-point correlationmatrix Z:

Zi , j = (xi ,x j ); (1.7)

being xi and x j the evaluation of the system state at ti and t j if the simulation is carried out in the timedomain. The eigenvalue problem associated to this matrix:

Zφ=λφ (1.8)

produces the optimal set of basis vectors, Φ= [φ1|φ2|...|φM ]. The eigenvalue λi represents some formof energy contribution of the mode φi to the system response. In practice, fewer than M modes areretained to simulate the system behavior, and these are selected considering the amplitude of themodule of the eigenvalue λi . However, it has been observed that the range of validity of the ROMdecreased as the number of retained mode is decreased [8, 83].This approach can also be applied in the frequency domain, where the response needed for construct-ing the two-point correlation matrix is retained only for a finite set of frequencies [97].It is worth remarking that the POD approach to reduced order modeling can be strictly related to thesingular value decomposition (SVD) of rectangular matrices [98].Recalling briefly what SVD means, consider a matrix X ∈Rn×m with rank d . It can be decomposed bymeans of the following relation:

UH XV =[Σd 00 0

]=Σ ∈Rn×m (1.9)

where U ∈Rn×n and V ∈Rm×m are two orthogonal matrices, composed by the left and right singularvectors of X and Σd = diagd (σ1,σ2, · · · ,σd ) ∈Rd×d , with σi the i-th singular value of X. The superscriptH is the transpose and conjugate operator, which reduces to the classical transposed operator T forreal-valued problems. Left and right singular vectors can be computed by:

XH ui =σ2i ui and Xvi =σ2

i vi i = 1, ...,d (1.10)

Such a connection between the two methods lies in the fact that the approximating POD basis shouldcontain as much information or energy as possible. With this consideration, the approximation ofthe snapshot vector xi through the single vector φ can be interpreted as the following constrainedoptimization problem:

maxφ

F =M∑

j=1

(xi ,φ

)2 such that:(φ,φ

)= 1 (1.11)

Using the Lagrange multipliers method, it can be derived that the solution of Eq. 1.11 is given by thefollowing eigenvalue problem:

XXHφ=σ2φ (1.12)


which is indeed equal to Eq. 1.8. The singular value analysis yields that the vector φ1 is a singulareigenvector of Eq. 1.12 and the relative singular value is equal to the functional value defined in Eq. 1.11.Iterating such a procedure, the vectors φi , i = 1, ...,d can be computed:

maxφ

F =M∑

j=1

(xi ,φ

)2 such that:(φ,φ

)= 1 and(φ,φ j

)= 0, j = 1, ..., i −1 (1.13)

with the corresponding functional value equal to σ2i . It is now clear that for every d ≤ M the approxi-

mation of the columns of X by the first d singular vectorsφi

di=1 is optimal in the least squares sense

among all rank d approximations to the column of X. Moreover, this approach leads to a practicaldetermination of a POD basis of rank d [98].

At this point the representation of Eq. 1.4 can be replaced in the original nonlinear system of Eq. 1.5,and solved by a weighted residual technique. This reads as:

F = x−R(x) (1.14)

where F the dynamic residual, which can be forced to vanish after weighting it with each of the Mmodes:

ΦT F =ΦT(

d(Φq

)d t

−R(Φq)

)= 0 (1.15)

Assuming that the modes are normalized such that ΦT Φ= I, Eq. 1.15 results in a subspace projectionof the full order system:

dq

d t=ΦT R(Φq) (1.16)

If one is interested to a steady-state solution of Eq. 1.16 the equilibrium solution satisfies the followingsystem of nonlinear, algebraic equations:

R =ΦT R(Φq) = 0 (1.17)

An extension of the subspace projection method is the so called direct projection [8], which utilizeshigher order terms of the function R (x), for example considering its Taylor series expansion:

R (x0 +∆x) = R (x0)+N∑

i=1

∂R

∂xi

∣∣∣∣x0

∆xi

+ 1

2

N∑i , j=1

∂2R

∂xi∂x j

∣∣∣∣x0

∆xi∆x j

+ 1

6

N∑i , j ,k=1

∂3R

∂xi∂x j∂xk

∣∣∣∣x0

∆xi∆x j∆xk

+O(|∆x|4)

(1.18)

The following pre-multiplication by ΦT and post-multiplication by Φ will produce a set of nonlinearODEs, with the linear portion in state-space form. However the numerical computation of the Jaco-bians can be burdensome if an inadequate technique is adopted to compute the needed derivatives.

Proper orthogonal decomposition has also proven very efficient in model sensitivity analysis.Assuming that the general model of Eq. 1.5 depends on a some set of parameters, here called θ ∈Rnθ ,this can be explicitly expressed as:

x = R (x;θ) (1.19)

18 Chapter 1

Having defined R in Eq. 1.17, the Jacobian matrices with respect of q and θ can be computed:

J = ∂R

∂qY = ∂R

∂θ(1.20)

Then, if Eq. 1.17 holds, the following relation is also true:

dR = Jdq+Ydθ = 0 (1.21)

and consequently:dq

dθ=−J−1Y (1.22)

remembering the transformation of Eq. 1.4:

dx

dθ=−ΦJ−1ΦT ∂R

∂θ(1.23)

This is the sensitivity dynamic equation, that can be formulated is one wishes to analyze the changingbehavior of the system with the modification of its parameters. Of course, as usual, the inverse ofthe Jacobian matrix J is not computed directly. It is first factorized and then repeatedly used in theevaluation of Eq. 1.23. Moreover, the computation of the Jacobian matrices can be quite a burdenfor the method proposed, but since the model has reduced dimensions, or if more sophisticatedtechniques are used [99], these terms can be efficiently obtained. Proper orthogonal decompositionhas been effectively used also for linear modeling. The reference equilibrium solution is computedwith a CFD solver with accelerating techniques, which are described in [100], toward the steady statevalue. Then the governing equations are linearized for periodic disturbances of small amplitude,placed in the frequency domain form, and solved with an another CFD run. This solution is thusgathered for a range of different frequencies to form the set of POD snapshots, computing then theresulting linear ROM model [8]. From this, a further order reduction can be obtained through the wellknown balanced reduction, described briefly in [3] and detailed in [97].

Finally, it should be noted that for linear systems, of the type: x = Ax+Bu, the POD methodpresents the same features of a subspace iteration scheme [75, 101] or the Arnoldi method [101] in thecomputation of the eigenvalues of a matrix. In fact it linearly projects the system states on a smallbasis characterized by a high precision, because of the subspace is composed by basis functions whichinherit already special characteristics of the overall solution. Of course also this method presents theneed of an accurate design of the input in order to represent the general behavior of the system. Ifno physical interpretation is considered during the input design, the results can be inaccurate [101].Similarly to the Volterra series, in this case the assembling of the correlation matrix needs the collectionof the samples derived from the numerical simulation of the full-order nonlinear problem, and this isof course the most expensive part of the ROM construction. Moreover, the collection of the samplesis not straightforward, because no standard rules are defined about how to choose them. The finalconsequence is a problem-dependent formulation, making the use of this method difficult for non-expert designers.

Harmonic balance method

Assuming that the solution of Eq. 1.5 is periodic in time, e.g. as in the case of a limit cycle, it can beexpressed in form of the Fourier series:

x =∞∑

m=−∞xm exp

(jω0m t

)(1.24)


where ω0 is the fundamental frequency, defined as ω0 = 2πT , with T the period of the harmonic response,

and j is the imaginary unit, such that j =p−1. Once known that Eq. 1.5 exhibit periodic solutions, it issufficient to compute such solution only in a time interval which corresponds to the period.

The Harmonic balance (HB) technique, also known as time-Galerkin method, consists in assumingthe solution of Eq. 1.5 in the form of a truncated Fourier series, with a predetermined number ofharmonics, here referred with NH . Then, this expression is substituted in the original ODE, andalgebraically manipulating the results to collect terms belonging to the same frequency. Any resultingterm with a frequency not in the Fourier expansion is dropped. Each harmonic is then balanced byrequiring that equal frequency terms on each side of the equation satisfy the equality independently.This balancing results in a system of coupled, generally nonlinear, algebraic equations which is thensolved for the Fourier coefficients x [102].

If one wants to remain the time domain for computing the solution of Eq. 1.5, the HB methodpermits to recast the problem in a steady-state form that accounts for the underlying time periodicityof the solution [103] and can be solved with a pseudo-time integration using convergence accelerationtechniques [100].

In practice, the truncated Fourier series results in:

x =NH∑

m=−NH

xm exp(

jω0m t)

(1.25)

Substituting the definition of Eq. 1.25 into Eq. 1.5 and integrating over a period, a system for the Fouriercoefficients is obtained:

AX−R(X) = 0 (1.26)

where:

A = diag(− j NH , ..., j NH ) X =

x−NH

...xNH

R =

R(x−NH

)...

R(xNH

) (1.27)

The problem represented by Eq. 1.26 can be solved with a nonlinear algorithm such as Netwon-Raphson technique. Typically, a reduced set of Fourier modes is required in order to capture thenonlinear behavior of the high-fidelity system. It has been found out that if the body motion is small, asingle harmonic is often sufficient in order to obtain accurate results [3].In the particular case of analyzing limit cycle oscillations, the periodT is unknown. This problem canbe solved just fixing a component of a single Fourier coefficient, for example x1NH

and then solving forthe period along with the remaining coefficients [102, 104, 105, 106].The evaluation of the so called harmonic fluxes R(X) of Eq. 1.26 is usually computationally expensive,in the order of O

(N N 3

H

)for Euler system in the fluid dynamics case, and are not easy to extend to

turbulent, viscous flows [102].Reference [102], which treats nonlinear aeroelastic problems, proposes also a simpler and moreefficient formulation. First the solution x(t ) of Eq. 1.5 is collocated at 2 NH +1 instant of times, evenlydistributed about the periodic orbit, and this terms are collected in the following array:

X∗ =(

xT (0),xT(

T

2 NH +1

), ... ,xT

(2 NH T

2 NH +1

))T

(1.28)

Then a Fouries transform operator E ∈RN NH×N NH relates X, defined in Eq. 1.27 to X∗:

X = EX∗ (1.29)

20 Chapter 1

the operator E has a blocked structure. The problem of Eq. 1.5 is then formulated in the frequencydomain:

j ωx = R (x) (1.30)

Defining the following elements:

N = diag(−NH , ... , NH ) R∗ =

R (x(0))

R(

x(

T

2 NH +1

))...

R(

x(

2 NH T

2 NH +1

))

(1.31)

The array R∗ can be related with the array R of Eq. 1.26 with the same previous operator E:

R = ER∗ (1.32)

Substituting these definitions in Eq. 1.30, the following nonlinear, algebraic problem is obtained:

j ωNEX∗ = ER∗ equivalently j ωE−1NEX∗ = R∗ (1.33)

Where ω= 2πT . In [102] a pseudo time τ has been introduced, by which Eq. 1.33 is integrated forward in

the pseudo time toward the fully developed solution:

∂X∗

∂τ+ j ωE−1NEX∗ = R∗ (

X∗)(1.34)

This approach allows the time dependent solution of Eq. 1.34 to be computed with the existingacceleration algorithms toward the steady-state [100], that needs far fewer iterations than a full ordertime-accurate integration method. In this way, the evaluation of the fluxes has a computational cost inthe order of O (N NH ) [102], bounding the cost of the numerical scheme.

However, it should be noted that such an approach was first developed for studying problemswhere the period T is known, e.g in the analysis of flows past the front stage rotor of a high-pressurecompressor. Turbomachinery flows are naturally unsteady mainly due to the relative motion of rotorsand stators and the natural flow instabilities present in tip gaps and secondary flows. It is noticedin [87] that full-scale time-dependent calculations of unsteady turbomachinery flows are still tooexpensive to be suitable for daily design purposes. One of the reasons for this large cost is the fact thatin practical turbomachinery configurations blade counts are chosen such that periodicity does notoccur, thus avoiding resonances, leading to mistuned configurations. In order to minimize the size ofthe problem that needs to be computed, various approximations were introduced, and nowadays HBis one of the most used. The effectiveness of harmonic balance in these applications is nonethelessacknowledged, so that it can be found directly implemented in industrial CFD softwares, such asSTAR-CCM+.The distinguishing feature of this reduced-order model is that only a specified set of frequencies,comprising combinations of the neighbor blade passing frequencies, is resolved in each blade row.Unlike single-stage problems, in multi-stage machinery, where each blade row has more than oneneighbor, this would amount to resolving frequencies that are not multiples of a single fundamentalfrequency. At the interface between blade rows, the flow variables are Fourier transformed once in timeand once in the circumferential direction. These Fourier coefficients are passed on to the neighboringblade row after non-reflecting boundary conditions are applied to those frequencies that are not


coupled, hence eliminating unwanted spurious frequencies, making this a time-domain/frequency-domain method. The Fourier coefficients are then transformed back to the physical domain in thereceiving blade row.

By definition, harmonic balance is not a reduced order modeling technique in a strict sense, sinceit solves the full-order nonlinear problem, but the computational time is greatly reduced for time-periodic problems.In fact it does not reduce the number of variable resulting from the spatial discretization and doesnot provide a model that is a compact representation of the full order system. Thus HB does notinvolve a compression of the spatial data, preserving the parametric relationships present in thesystem without loss of fidelity. However it should be noted that with respect to the methods presentedbefore, the HB method requires a re-formulation of the governing equations of the system, and thiscan be seen as one of its weaknesses. Nonetheless, it does yield an efficient representation of thetemporal variations of complex systems experiencing periodic behavior in time, presenting also a greataccuracy [8, 45, 87, 99, 102].

Thus, in conclusion, the Harmonic Balance method is an efficient tool for reduced order modelingof nonlinear systems, but it is limited to the study of time-periodic problems. So, its application toaeroelasticity is confined at the moment to the study of LCOs. No calculations of generic aeroelasticresponses are available in the literature yet.

High order interpolation of the high-fidelity model

In order to reduce the computational cost of simulations related to high-fidelity models, a high orderinterpolation of the solution at different sampling points may be exploited, obtaining smooth modelswhich can be employed in fast analyses.Different applications of such a method are available in the literature, both for large linear [ 107] andnonlinear [108, 109] systems. In particular, the latter references are related to aeroelastic problems,where the computation of the stability properties (in the specific, the flutter boundary) of the system iscarried out coupling the structural system, described in modal form, with the aerodynamics, describedby a CFD model. Such nonlinear dynamic system can be represented by:

x = R(x,µ

)(1.35)

where x is the system state and µ represents a set of parameters, typically the dynamic pressure, theMach number, the altitude, etc. The stability of the systems is evaluated through the computation ofthe eigenvalues of the linearized system near a trim condition:

R(x0,µ

)= 0 −→ A = ∂R

∂x

∣∣∣∣x0,µ

(1.36)

and checking the well known stability rule (asymptotic stability is related to a negative real part ofall the eigenvalues). Because of the presence of an aerodynamic model described through a CFDscheme, the resulting eigenvalue problem Ap =λp is very large, therefore very expensive. In [108], theJacobian matrix is decomposed in structural and aerodynamic part, and only the structural eigenvaluesub-problem is retained of interest, considering only the interaction with the aerodynamic system.

p =(

ps

pa

)A =

[Ass Asa

Aas Aaa

]ps ∈R2nstruct pa ∈Rna (1.37)

The resulting eigenvalue problem can be written as:

S(λ)ps =λps (1.38)

22 Chapter 1

Where S(λ) is the Schur complement of A, and it is equal to:

S(λ) = Ass −Asa (Aaa −λI)−1 Aas = Ass +Sc (λ) (1.39)

The size of the nonlinear eigenvalue problem written in Eq. 1.38 is equal to 2nstruct +1 (the structuralvariables plus the eigenvalue), instead of the na + 2nstruct + 1 unknowns of the original one. Thecomputational saving is considerable, because of na >> 2nstruct. Since the eigenvalue problem ofEq. 1.38 is nonlinear, it has to be solved with a iterative technique, i.e. Newton-Raphson. However, theevaluation of the coupling term Sc (λ) is computationally involved, since the aerodynamics is repre-sented by a very large system. The solution proposed in [108] avoids such large cost, approximatingthe Schur complement defined in Eq. 1.39. In the proposed procedure, the coupling term is evaluatedon different sampling points selected in the parameter space described by µ, then its extension overall the domain is performed interpolating the interaction matrix over such few samples, where it isevaluated through the high-fidelity model, obtaining the following nonlinear eigenvalue problem:(

Ass + Sc (λ))

ps =λps (1.40)

which is far away cheaper than the previous version, since the full interaction term has not to beevaluated at each iteration of the Newton-Raphson algorithm. In this particular case, the interactionterm Sc is interpolated through a Kriging technique, which is a response surface method. Withoutconsidering the details, that can be found extensively in [108, 109], such a predictor is composed by twomain terms: a low-order regression model and a random normally distributed signal, characterized bya co-variance that depends on the variance of the input samples and the two-point correlation matrixrelated to such samples. It is important to note that the mean square error of the prediction vanishesat a sampled location, interpolating the exact system response [110].An important issue that can be considered is how to choose the samples. In the literature thereare different efficient algorithms which permit to place the samples where some cost function ismaximized. We can cite, for example, the risk-based sampling, the latin hypercube and the expectedimprovement algorithm [108, 110].

Recently, Kriging-based ROMs have also been used in analyzing the aerodynamic response ofpitching/plunging airfoils subjected to fixed or time-varying freestream Mach number [111]. In sucha work, the reduced-order model uses Kriging surrogates to account for flow nonlinearities andrecurrence solutions to account for time-history effects associated with unsteadiness. The resultingsurrogate-based recurrence framework generates time-domain predictions of unsteady lift, moment,and drag that accurately approximate computational fluid dynamics solution.

Other types of interpolation strategies can be used. For example Radial Basis Functions (RBFs)are often adopted in different applications, an example can be found in [107], because of their highsmoothness and accuracy, permitting a reliable interpolation of the data computed by the high-fidelitymodel over all the parameter space of interest. Other candidates as high-dimensional interpolators arelow-order polynomials and moving least squares approaches.

Wiener-type models

A Wiener model is depicted in Figure 1.10. It consists of a linear dynamic system followed by a staticnonlinearity. The input u and the output y are measurable, possibly with noise, but the intermediatesignal z cannot be measured. Recently, Wiener models have received an increasing attention becausetheir identification algorithms are much simpler than those used for the Volterra series [112].

In [112], the presented discrete-time nonlinear aerodynamic ROM includes a finite sum of Wiener-type cascade models. In the studied case, as shown in Figure 1.10, us(k) is the vector of structural


Figure 1.10: General Wiener-type model, one of many paths in [112].

displacement, and ya(k) is the vector of aerodynamic output of the CFD-based aerodynamic system.At first, a nonlinear block-structured model, i.e., a linear dynamic element h1 (·) followed by a staticnonlinear term F1 (z1(k)), is fitted between the inputs and outputs. The outputs of the first systemy1(k) are computed and subtracted from the outputs computed by the direct CFD analysis. The secondblock-structured model is then fitted between the input and the output residuals y1(k). Such a processis repeated until the residuals of outputs contain only noise.Within such a framework, the outputs of the nonlinear ROM can be expressed as follows:

ya(k) =N∑

i=1yi (k) (1.41)

As shown in Figure 1.10, each path of the parallel cascades model is modeled by a state-space modelfollowed by a neural network model.Let (Aa , Ba , Ca , Da) be a state-space representation of the dynamic linear part of each path, that is:

xa(k +1) = Aaxa(k)+Baus(k)

z(k) = Caxa(k)+Daus(k)(1.42)

The static nonlinear part of each path, represented by a single-layer neural network model, can bewritten as:

yi ,l (k) =N∑

j=1w s

j ,l

[tanh

(w j ,l zi (k)+b j ,l

)]+b(N+1),1, l = 1,2, · · · , p (1.43)

where w j ,l and b j ,l are the weight coefficients and threshold values of the hidden layer, respectively.The elements w s

j ,l and b(N+1),1 are the weights and threshold values of the output layer, respectively.Given the input and output residual data of the generic i-th path, those parameters of the Wiener-typecascade model can be estimated by using the Levenberg–Marquadt algorithm [113].To demonstrate the performance of this nonlinear reduced-order model in reproducing the staticallynonlinear and dynamically linearized behavior of a nonlinear aerodynamic system, the unsteadytransonic compressible flow over a two-degree-of-freedom wing section with the NACA 64A010 airfoilwas presented in [85]. The numerical results indicated that the proposed nonlinear reduced-ordermodel can accurately identify the outputs of aerodynamic systems subject to a weak excitation.More recently, the same technique, improved through a more stable algorithm in the identificationof the linear sub-system [112] was used to investigate the nonlinear aeroservoelastic behavior of theBenchmark Active Control Technology (BACT) wing. The numerical results demonstrated that the

24 Chapter 1

−5 0 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Neuron potential, v

Activationfunction,φ(v)

(a) Squashing function (b) Simple neural network with one hidden layer

parallel cascade of Wiener-type models is capable of modeling open/closed-loop responses of such awing.

This kind of ROM has shown promising results, first of all because the related identification al-gorithm is much simpler than those used by the Volterra series [85, 112], providing a fair level ofrobustness, and then because it has already been applied to quite realistic aeroservoelastic applica-tions [112]. Nonetheless, in [112] as well, the authors have shown how the identification of aeroelasticLCOs could be difficult if the chosen training signal does not excite properly the system dynamics.

Recurrent neural networks

A neural network is a massively parallel distributed process based on simple processing units, theneurons. Using an analogy with our brain, a large number of interconnected neurons would havethe natural capability of learning new rules through experience, using them when needed. Thisknowledge is acquired through a learning process, and stored in the synaptic connections linking theneurons. Neural networks have been widely used in nonlinear system identifications because of theirself-learning abilities, adaptivity and nonlinear modeling.As demonstrated in the literature [66, 114, 115], neural models are a powerful tool for approximatingnonlinear dynamic systems, even when the system itself is unknown and only the input-output dataare available. Therefore, they permit a sort of black-box modeling of any nonlinear system, avoidingthe burden of formulating a structured parametrization of the equations describing the physical model.In fact, when using neural networks, the model structure is determined only by the layout of thenetwork connections, and the related parameters are determined either through experimental orcomputational models, thus requiring none or a very little prior knowledge at most [115]. A neuralnetwork is generally composed by an input layer, an arbitrary number of hidden layers and an outputlayer.The input layer receives the input data from the external environment and passes it to the computa-tional kernel represented by the neurons. These units receive a linear combination of the input, whosecoefficients are called synaptic weights. This signal is passed through the neurons, which are modeledby a nonlinear squashing map, e.g. a logistic or hyperbolic tangent function. If the network is used toapproximate a nonlinear process, then its computational layer should be hidden, meaning that theoutput of its neurons should not be the direct output of the network. Thus the real output is usuallya linear combination of the outcome of each neuron, presented to the external environment by theoutput layer. An example of such a framework is given in Figure 1.11b.


Figure 1.11: Partially recurrent neural network

Because of the universal approximation theorem, a neural network with one hidden layer is sufficientfor approximating a nonlinear function with arbitrary accuracy, provided that the number of neuronsis adequate [66].

The modeling of nonlinear dynamic systems is more interesting. In such a case the network shouldbe able to learn the evolutionary path of a general system from the input-output data pairs only. Anetwork memory is introduced in the process through a time delay applied to the hidden neurons,defining thus a state of the network.Basically, a network state is a hidden neuron which is connected to an other hidden neuron. Such anetwork is defined as recurrent. The previously mentioned input layer is now composed by the set ofthe input data, which is now time-dependent, and the set of the hidden neurons delayed output.In a fully connected recurrent neural network feedback all the hidden neurons output to the inputlayer, meanwhile a partially connected recurrent neural network feedback only a small fraction ofsuch output, as shown in Figure 1.11. The resulting internal dynamics realizes a nonlinear statespace model without information about the true system states. The internal states can be thereforeinterpreted as an artificial tool employed for the realization of the desired dynamic input/outputbehavior. Such models with internal dynamics are most frequently based on a multi-layer perceptornetwork architecture [116, 66]. Such structured recurrent scheme leads to faster training and fewerstability problems. Nevertheless, the number of states is related to the number of neurons in thehidden layer, and this fact restricts the flexibility of such a tool, leading to the well known problem ofthe curse of dimensionality [66].Such recurrent networks can be formalized through the following discrete state-space model:

xn+1 =φ(Waxn +Wbun)

yn = Wcxn(1.44)

or through a continuous model:

x(t ) =φ(Wax(t )+Wbu(t ))

y(t ) = Wcx(t )(1.45)

where x represent the state of the network, u the input, y the output, Wa, Wb the synaptic weightsbetween the hidden and the input layer, Wc the synaptic weights between the hidden and the output

26 Chapter 1

layer and φ the set of nonlinear activation function, defined as:

φ=

φ1(v)...

φnx (v)

(1.46)

if nx is the dimension of the state space. Several parametrizations of recurrent neural networks, forboth discrete and continuous formulation have been proposed in the literature [66, 117, 118, 119, 120]but no standard formulation is nowadays of general use for the representation of nonlinear dynamicsystems.

Like an human brain, in order to behave like the nonlinear dynamic system of interest, the neuralnetwork must gain its experience over a training experience, which should be able to excite the mainsystem dynamics.The choice of the model input is typically realized by a trial-and-error approach with the help of anyavailable prior knowledge. In the fields of structural mechanics, aerodynamics, and other branches ofphysics, the influence of the singular variables on the system output is usually quite clear, thus therelevant model input can be chosen by available insight into the physics of the process [116].The training of neural network can be performed in two main ways:

• Unsupervised training: only the system input are available. The principal tool employed for thiskind of training is the principal component analysis [66], which permits to discard non-relevantinput with a low computational demand;

• Supervised training: both the system input and output are available for the training of thenetwork. The resulting training process can be interpreted as an optimization problem wherethe error, defined as the difference between the real system output and the network output, isminimized varying the synaptic weights of the network.

After the training phase, if the training signal has excited all the nonlinearities, frequencies andamplitudes of interest, the resulting neural network should be able to generalize the knowledge learnedinto the behavior of the full order nonlinear system. Thus for generalization is meant the capacity ofthe neural network to reproduce the full system output for an input data sample not contained in thetraining set.

Different approaches to the system identification of nonlinear aeroelastic systems with discretetime recurrent neural network have been adopted: for example the utilization of a recurrent neuralnetwork with radial basis function (RBF) as activation functions has been able to predict the limitcycle oscillation in a transonic flow of the NACA 64A010A airfoil and the Benchmark Active ControlTechnology (BACT) wing [121]. Such recurrent RBF network can be interpreted as a nonlinear versionof the well known autoregressive model with exogenous input identification technique, in fact indifferent sources of the literature this kind of neural network is also defined as nonlinear autoregressivemodel with exogenous input (NARX). Instead, in [122] its has been presented an aerodynamic systemidentification technique based on the rigorous mathematical theory of the support-vector-machine,always related to a discrete time recurrent neural network, which has been able to predict differentlimit cycle oscillation conditions for various Mach numbers.

More recently, a combination of parameters reduction via proper orthogonal decompositionand system identification methods based on RBF networks was designed to reproduce compactlynonlinear aerodynamic effects such as viscosity as well as transient effects [123]. In such a work theROM is identified from a set of transient forced motion CFD analyses. After identification, the modelis used to predict the discrete surface force distribution of the NLR7301 airfoil in static as well as


Figure 1.12: Scheme of the ROM used in [123]

transient coupled analyses. The modeling framework used by the authors is shown in Figure 1.12.Static aeroelastic equilibrium and a limit cycle oscillation case are predicted by the surrogate modelwith sufficient accuracy, highlighting the effect of the viscosity in the studied problem. In [124] isdemonstrated the application of the same technique on a more realistic three-dimensional case, thehigh-Reynolds-number aerostructural dynamics (HIRE-NASD).However, the approaches just discussed above are usually quite involved from the computational pointof view. This is because to train the synaptic weights of the network, the RBF parameters, such that thefunction center and the function spread (or radius), must be trained as well. Different algorithms arepresent in the literature for predicting these function parameters from the input data, the most knownone being the K-means clustering algorithm [66].In the case of [121], a NARX model provides only a one-step ahead prediction of the response, ratherthan optimizing the simulation error. However, once the neuron centers and spreads have beencomputed, the resulting training algorithm is reduced to a linear least squares problem, as shownin [86, 121], the latter phase being characterized by a very small computational time. The approachstudied in [122] is well known for static neural networks, but a small number of applications have beenpresented so far in the contest of dynamics systems. A further consideration is that all the mentionedapproaches are formulated in a discrete-time domain, with no applications in the continuous-timedomain. This could be seen as a limitation if the design of control laws for suppressing the studiedunstable phenomena is of interest.

Of course neural networks present also some drawbacks. In fact there is no mathematical theoremthat guarantees the achievement of the global minimum of the error during the training phase, thusthere is the possibility of converging onto a local minimum, with the related poor performances.Moreover, too long training data sets may lead to over-fitting a specific training with an ensuing poorgeneralization capability. Thus, in order to obtain a functional recurrent neural network, a goodparametrization of the network itself and a careful design of the training signal is required, and so theywill be studied in depth in this work.

Under certain aspects, neural networks are similar to Volterra series. Each of them involves thecharacterization of a system through an input-output mapping, and there is a direct relationshipbetween the synaptic weights of the network and the kernels used in the Volterra representation [8].The major difference between the two approaches is the training effort. As previously mentioned, thefirst kernels of a Volterra series are known analytically with no need of training. The definition of akernel as a generalization of the impulse response of a system made the series approach adapt alsoto physical interpretation. Nevertheless, recurrent neural networks do not present the disadvantagesrelated to the input amplitude limitations required for convergence of the Volterra series, as citedin [94]. Moreover, the need of higher order terms, which can be computationally expensive, is nolonger required as shown in [8, 94, 95].Neural networks may be considered similar to Wiener-type models also, mainly because of the type of

28 Chapter 1

nonlinearities employed by these methodologies is mostly the same. In dynamic systems modeling,while the recurrent neural networks are somewhat compact because a small number of neurons isoften adequate to capture the main nonlinear behavior of interest, Wiener-type models are way moresimilar to the modern deep neural networks [125], because of the recurrent framework adopted duringtheir training, where the residual errors are minimized recursively, in the same way as in a multi-layer

neural network. The problem is that, using this analogy, as soon as the residual error requires to beminimized to very low values, the number of layers must be increased, leading to a deep network,i.e. a network built up by several hidden layers. It is well know that this kind of models may sufferof the vanishing gradient problem, because as the residual errors propagate from layer to layer, theyshrink exponentially with the number of layers. This fact is generally a problem in the training of thesemodels, because the related error is typically minimized through some versions of the gradient descenttechnique. Recurrent neural networks used in dynamic system modeling usually do not suffer thisproblem, thanks to the more limited number of neurons involved.Comparing recurrent neural networks to the others method presented, i.e. POD and HB, it is straight-forward to note that the only design parameter required to neural network are the number of states, i.e.neurons, used and the type of training signal. The choice of snapshots in POD design can be non-trivialand made the solution snapshot-dependent.If opportunely trained, a neural network can be a general approximator of a dynamic system, thuscan be used to predict the response to a general input. Instead, harmonic balance method is typicallyemployed to analyze the stability of time-periodic systems, with no development in the study of ageneric dynamic response [3, 102].

1.4 Modern Control of Nonlinear Dynamical Systems

The general objective of control design can be stated as follows: given a physical system to be con-trolled and the specifications of its desired behavior, construct a feedback (sometimes combinedwith a feedforward) control law to make the closed-loop system robustly display the desired behavior.Generally, the tasks of control systems can be divided into two categories: stabilization (or regulation)and tracking (or servo).As introduced in Section 1.2.3, even if all physical systems are intrinsically nonlinear, controllers maybe linear or nonlinear depending on the formulation of the problem under study. This thesis will mainlydeal with nonlinear systems, therefore a great effort is spent here in describing nonlinear controllers,also because a vast literature regarding linear control is already available to any reader [126, 127].

Nonlinear controllers present large differences with respect to linear ones, and a few of them aredetailed in this section. For example, in linear control, the desired behavior of a control system can besystematically specified, either in the time-domain (in terms of rise time, overshoot and settling timecorresponding to a step command) or in the frequency domain (in terms of regions in which the looptransfer function must lie at low frequencies and at high frequencies). In linear control design, onefirst lays down the quantitative specifications of the closed-loop control system, and then synthesizesa controller which meets these specifications.However, systematic specification for nonlinear systems is much less obvious because the responseof a nonlinear system to one command does not reflect its response to another command, and, inaddition to this, a frequency-domain description is not possible.As a result, for nonlinear systems, one often looks for some qualitative specifications of the desiredbehavior in the operating region of interest. Computer simulation is an important complement toanalytical tools in determining whether such specifications are met.Regarding the desired behavior of nonlinear control systems, a designer can consider the following

1.4. Modern Control of Nonlinear Dynamical Systems 29

characteristics:

• Stability must be guaranteed for the nominal model (the model used for design), either in alocal sense or in a global sense. The region of stability and convergence are also of interest

• Accuracy and speed of response may be considered for some "typical" motion trajectories inthe region of operation. For some classes of systems, appropriate controller design can actuallyguarantee consistent tracking accuracy independently of the desired trajectory

• Robustness is the sensitivity to effects which are not considered in the design, such as imprecisemodels, disturbances, measurement noise, unmodeled dynamics, etc. The system should beable to withstand these neglected effects when performing the tasks of interest.

• Cost of a control system is determined mainly by the number and type of actuators, sensors,and processing units necessary to implement it. The actuators, sensors and the controller com-plexity (affecting computing requirement) should be chosen consistently and suit the particularapplication.

Of course, the above qualities conflict to some extent, and a good control system can be obtained onlyon the base of effective trade-offs in terms of stability/robustness, stability/performance, cost/perfor-mance, and so on.Let us now focus on aeronautical applications.

1.4.1 Aeronautical problems

Even if nowadays classical linear control is still the most used method in aerospace industry thanks toits simplicity and intrinsic robustness, the variations in the dynamic response of an elastic aircraft withflight conditions, e.g. dynamic pressure, Mach number, angle of attack, must be taken into account incertain cases.For example, if a control law is supposed to cover a wide range of operating conditions, all the linearcontrol laws designed for each configuration must be scheduled to maintain an acceptable perfor-mance, leading to the so called gain scheduling [58, 128]. This approach is the most used by theindustry, because it is straightforward to be understood by engineers and it has proven to be robustin realistic applications, providing that the design points are adequately dense. However, this kind ofdesign may result in a very long tuning process, delaying the development of the control system. Thisfact may be undesirable in the early design stages of an aircraft, where a large number of configurationsmust be analyzed.

A possible solution to this drawback could be the adoption of self-adaptive control laws able tochange their behavior over various flight conditions, with adequate adaption rates to new configura-tions to avoid excessive loads while the transitioning conditions.Such a solution is provided by the so-called adaptive control laws [64], which have been developed inthe last decades to deal with large plant uncertainties and unmodeled dynamics, and that now seemsto be ready to be tested on realistic engineering problems [129, 130].

Adaptive controllers may be distinguished between direct and indirect methods. Direct methodsare ones wherein the estimated parameters are those directly used in the adaptive controller. Incontrast, indirect methods are those in which the estimated parameters are used to calculate requiredcontroller parameters [64].Therefore, an adaptive controller is a dynamic system with an on-line parameter estimation: it isinherently nonlinear and in general its analysis and design rely on Lyapunov Stability Theory.

30 Chapter 1

Another distinction should be made between certainty- and non-certainty-equivalence adaptivecontrollers. Certainty-Equivalence (CE) controllers permit to retain a controller structure that isidentical to that of the deterministic-case controller wherein no uncertainty is present, except forthe introduction of an additional carefully designed parameter estimation mechanism that ensuresstability with the adaptive controller and boundedness of all resulting closed-loop signals [131]. Intheoretical terms, the closed-loop error dynamics generated by CE-based adaptive control solutionsare exactly equivalent to the deterministic-case control error dynamics whenever the estimatedparameters coincide with their corresponding unknown true values. Of course, this happens onlywhen the underlying reference trajectory satisfies suitable persistence of excitation (PE) conditions. Asa result, typically, the control performance of CE-based adaptive control methods for either set-pointregulation or trajectory-tracking problems can, at best, match the performance of the deterministic-case controller, but only when the PE hypothesis ensures sufficiently fast convergence of parameterestimates to their true values. However, in practice, the closed-loop performance obtained fromCE-based adaptive controllers is often seen to be arbitrarily poor when compared with the idealdeterministic control case, due to the missing satisfaction of PE conditions and/or slow convergencerates for the parameter estimates. Such a performance degradation is mainly caused by the fact thatsearch/estimation efforts of the parameter-update law act like an additive disturbance imposed ontothe deterministic-case closed-loop dynamics. Another cause is the fact that parameter-estimationdynamics are driven by the state-regulation errors or tracking errors, which results in the undesirablefeature of parameter estimates being unable to get locked onto their corresponding true values, evenif, at any given instant during the estimation process, the estimates are equal to their true values [131].Non-Certainty-Equivalence (NCE) controllers on the other end permit to overcomes many of theperformance limitations arising from CE-based designs. Unlike their counterpart, the estimatedparameters of the NCE system includes not only the estimates generated by the adaptive law butalso include compensating nonlinear functions. Such additional nonlinear terms in the estimatedparameter vector provide improved controller performance, by eliminating, in a stable manner, thedisturbances arising from the estimation of uncertain parameters or by forcing the parameter estimatesto stay locked at their true parameters once they are attained during the estimation process.

This kind of control is different from robust control in that it does not need a priori informationabout the bounds on these uncertain or time-varying parameters; robust control guarantees that ifthe changes are within given bounds the control law need not be changed, while adaptive control isconcerned with control law changing themselves [64].However, there is a connection between adaptive and robust controllers. In fact, when adaptivecontrol laws have to be implemented and tested on the real physical system, modeling errors anddisturbances are inevitably present and can cause instability if it .does not have certain robustnessproperties. Robust adaptive control provides the solid foundations for adaptive control applications inwhich modeling errors and disturbances are inevitably present and can cause instability if an adaptivecontroller does not have certain robustness properties [132]. Among the commonly used algorithms,the σ-modification not only has the desired robustness properties, but also has some major advantages:its design parameters are practically easy to choose, and it does not prevent the restoration of theideal adaptive control system performance (that is, asymptotic tracking) when the modeling errorsand disturbances are absent [132]. Another practically useful robust adaptive control algorithm isparameter projection [132, 133], which has the advantage of ensuring the parameter estimates to stayin a specified region containing the true parameters to be estimated. Such a property is importantfor applications when some estimated parameters have to be within certain intervals to reflect thephysical meanings of the corresponding true parameters.

In the following review of modern nonlinear and adaptive controllers, the different laws will be


Figure 1.13: Input-Output linearization framework

classified according to the previous definitions.

1.4.2 Different approaches to nonlinear control

As anticipated in Section 1.2.3 a large number of techniques is available to control systems that presentnonlinear behaviors. Some of such methods are detailed here, providing an overview of the moststudied methods nowadays employed in the control research community.

Feedback linearization

The central idea of feedback linearization is to algebraically transform a nonlinear system dynamicsinto a (fully or partly) linear one, so that linear control techniques can be applied [60]. This differsentirely from conventional linearization, i.e., Jacobian linearization, because feedback linearizationis achieved by exact state transformations and feedback, rather than by linear approximations of thedynamics. Feedback linearization techniques can be viewed as ways of transforming original systemmodels into equivalent models of a simpler form. Thus, they can also be used in the development ofrobust or adaptive nonlinear controllers [134, 135]. Feedback linearization has been used successfullyto address some practical control problems. These include the control of helicopters [136], vehicledynamics [137] and electromechanical actuators [138].

Feedback linearization may be obtained through two ways: Input-State linearization and Input-Output linearization. Consider the problem of designing the control input u for a single-input nonlin-ear system of the form x = f (x,u). The technique of input-state linearization solves this problem in twosteps. First, one finds a state transformation z = z (x) and an input transformation u = u (x, v) so that

the nonlinear system dynamics is transformed into an equivalent linear time-invariant dynamics, inthe familiar form z = Az+bv . Second, one uses standard linear techniques (such as pole placement) todesign v . Such a procedure is summed up in Figure 1.13. As can be seen from Figure 1.13, in orderto implement the control law, all the state components must be available. If they are not physicallymeaningful or cannot be measured directly, a nonlinear observer must be used for state estimation.This drawback is common to almost all the adaptive control techniques available in the literature.

Consider now a tracking control problem, with the system:x = f (x,u)

y = h (x)(1.47)

and assume that the objective is to make the output y(t ) track a desired trajectory yd (t ) while keepingthe whole state bounded, where y(t ) and its time derivatives up to a sufficiently high order are assumedto be known and bounded. An apparent difficulty with this model is that the output y is only indirectlyrelated to the input u, through the state variable x and Eq. 1.47. Therefore, it is not easy to see how theinput can be designed to control the tracking behavior of the output. However, inspired by the results

32 Chapter 1

of the Input-State linearization technique, one might guess that the difficulty of the tracking controldesign can be reduced if a direct and simple relation between the system output and the controlinput can be found. Indeed, this idea constitutes the intuitive basis for the so-called Input-Outputlinearization approach to nonlinear control design [60].This is obtained in practice by differentiating the output of a system r times to generate an explicitrelationship between the output y and input u. The value of r defines the relative degree of the system.As shown in [60], such a procedure based on the time differentiation of the output as many times asrequired to obtain a direct relationship between the output itself and the system input produces areduced order controller. This means that the control law will be able to regulate the output and itsderivatives which are directly influenced by the input. As a result, part of the system state will not becontrolled, and if such internal dynamics is unstable, then the closed loop system will still be unstable.Please refer to [60] for a detailed analysis. Some application in the field of aeroelasticity can be foundin [139, 140, 141], while feedback linearization has been used extensively in the control of the rigiddynamics of aircraft [142, 143, 144, 145].

Nonetheless, the method suffers a number of important limitations:

• It cannot be used for all nonlinear systems. The applicability of input-state linearization is quan-tified by a set of somewhat stringent conditions [60], while input-output feedback linearizationcannot be applied when the relative degree is not defined and lacks systematic global results.

• The full state has to be measured

• No robustness is guaranteed in the presence of parameter uncertainty or unmodeled dynamics

Lyapunov redesign and nonlinear damping

Lyapunov’s method, introduced originally as an analysis tool, turns out to be a useful tool in feedbackdesign. Many feedback control techniques, e.g. back-stepping, sliding mode, are based on the ideaof designing the feedback control in such a way that a Lyapunov function, or more specifically thederivative of a Lyapunov function, has certain properties that guarantee bounded trajectories andconvergence to an equilibrium point or an equilibrium set.Lyapunov redesign uses a Lyapunov function of a nominal system to design an additional controlcomponent to robustify the design to a class of large uncertainties that satisfy the matching condition;that is, the uncertain terms enter the state equation at the same point as the control input. It can beused to achieve robust stabilization, and to introduce nonlinear damping that guarantees boundednessof trajectories even when no upper bound on the uncertainty is known.The choice of a Lyapunov function is often driven by the physics governing the problem at hand.However, the choice of a Lyapunov function is not unique, and a bad choice of it may lead closed loopsystems that do not exhibit good performance when compared to other control laws.

The related mathematical treatment is quite involved and therefore the reader is referred to Ref. [62]for a proper introduction. A few aeroelastic applications of Lyapunov-based control can be found inthe literature [146, 147, 148].

Back-stepping

Back-stepping is a systematic, Lyapunov-based method for nonlinear control design. Its name refersto the recursive nature of the design procedure. The design procedure starts at the scalar equationwhich is separated by the largest number of integrations from the control input and steps back towardthe control input. Each step an intermediate or virtual control law is calculated and in the last step the


real control law is found. Two comprehensive textbooks that deal with back-stepping and Lyapunovtheory are [59] and especially [149]. The origins of the back-stepping method are traced in the surveypaper by Kokotovìc [150].This technique presents some similarities to the feedback linearization previously presented, becauseit requires a sort of manipulation in order to control the state that is not directly affected by the input.With respect to the previous method, back-stepping provides the designer a more systematic way ofdesign and therefore a greater flexibility. However, as will be seen shortly, this technique is of easyimplementation for systems with a well defined structure and a limited number of states. In fact, assoon as the system size increases, the resulting control law design may result a bit cumbersome.

Such an approach provides a recursive method for stabilizing the origin of a system in strict-feedback form. That is, consider the following problem:

x = fx (x)+ gx (x)z1

z1 = f1(x, z1)+ g1(x, z1)z2

z2 = f2(x, z1, z2)+ g2(x, z1, z2)z3...

zi = fi (x, z1, z2, . . . , zi−1, zi )+ gi (x, z1, z2, . . . , zi−1, zi )zi+1 for 1 ≤ i < k −1...

zk−1 = fk−1(x, z1, z2, . . . , zk−1)+ gk−1(x, z1, z2, . . . , zk−1)zk

zk = fk (x, z1, z2, . . . , zk−1, zk )+ gk (x, z1, z2, . . . , zk−1, zk )u

(1.48)

where: x ∈ Rn with n ≥ 1, z1, . . . , zi , . . . , zk are scalars, u is a scalar input to the system, f1, . . . , fi , . . . , fk

vanish at the origin (i.e., fi (0,0, . . . ,0) = 0), g1, . . . , gi , . . . , gk are nonzero over the domain of interest (i.e.,gi (x, z1, . . . , zk ) 6= 0 for 1 ≤ i ≤ k). Also assume that the subsystem:

x = fx (x)+ gx (x)ux (x) (1.49)

is stabilized to the origin (i.e., x = 0) by some known control ux (x) such that ux (0) = 0. It is alsoassumed that a Lyapunov function Vx for this stable subsystem is known. That is, this x subsystem isstabilized by some other method and back-stepping extends its stability to the z shell around it.

In systems of this strict-feedback form around a stable x subsystem,

• The backstepping-designed control input u has its most immediate stabilizing impact on statezn .

• The state zn then acts like a stabilizing control on the state zn−1 before it.

• This process continues so that each state zi is stabilized by the fictitious "control" zi+1.

The backstepping approach determines how to stabilize thex subsystem using z1, and then proceedswith determining how to make the next state z2 drive z1 to the control required to stabilize x. Hence,the process steps backward from x out of the strict-feedback form system until the ultimate control uis designed.Because:

• fi vanish at the origin for 0 ≤ i ≤ k

• gi are nonzero for 1 ≤ i ≤ k,

34 Chapter 1

• the given control ux has ux (0) = 0,

then the resulting system has an equilibrium at the origin (i.e., where x = 0 , z1 = 0, z2 = 0, ..., zk−1 = 0,and zk = 0) that is globally asymptotically stable [59].

This technique has been also used in an adaptive way by coupling it with neural networks [151]or sliding mode controllers [152]. Adaptive back-stepping control has a more sophisticated way ofdealing with large uncertainties [153]. The related controllers do not only employ static state feedbacklike the controllers designed in the previous section, but it contains also a dynamic feedback part. Thisdynamic part of the control law is used as a parameter update law to continuously adapt the static partto new parameter estimates.

Despite such strong mathematical properties, the main limitation of back-stepping is its ap-plicability to strict-feedback form dynamical systems only, as shown in Eq. 1.48 and it is of easyimplementation in the case of small size systems. For this reason, it has been studied on a largenumber of benchmark cases [154, 155, 156], but it lacks of realistic applications, especially in the fieldof aeroelasticity, while it seems to be a well established technique in the flight dynamics control ofrigid aircraft [157, 158, 159].

Sliding mode control

Sliding mode control alters the dynamics of a nonlinear system by applying of a discontinuous controlsignal that forces the system to "slide" along a cross-section of the system normal behavior [ 61, 160].The state-feedback control law is not a continuous function of time. Instead, it can switch from onecontinuous structure to another based on the current position in the state space. Hence, slidingmode control is a variable structure control method. The multiple control structures are designed sothat trajectories always move toward an adjacent region with a different control structure, and so theultimate trajectory will not exist entirely within one control structure. Instead, it will slide along theboundaries of the control structures. The motion of the system as it slides along these boundariesis called a sliding mode and the geometrical locus consisting of the boundaries is called the sliding(hyper)surface [61].

Figure 1.14a shows an example trajectory of a system under sliding mode control. The slidingsurface is described by z = 0, and the sliding mode along the surface commences after the finite timewhen system trajectories have reached the surface. In the theoretical description of sliding modes,the system stays confined to the sliding surface and need only be viewed as sliding along the surface.However, real implementations of sliding mode control approximate this theoretical behavior witha high-frequency and generally non-deterministic switching control signal that causes the systemto "chatter" in a tight neighborhood of the sliding surface. This chattering behavior is evident inFigure 1.14b.

Intuitively, sliding mode control uses practically infinite gain to force the trajectories of a dynamicsystem to slide along the restricted sliding mode subspace. Trajectories from this reduced-ordersliding mode have desirable properties, e.g., the system naturally slides along it until it comes to restat a desired equilibrium. The main strength of sliding mode control is its robustness. Because thecontrol can be as simple as a switching between two states, it needs not to be precise and will beloosely sensitive to parameter variations entering the control law. Additionally, because it is in facta discontinuous control, the sliding mode can be reached in finite time, i.e. better than asymptoticbehavior.A sliding-mode control scheme always involves the following two steps, which are followed also by theimmersion and invariance technique described in the next sections:


(a) Typical phase portrait under sliding mode control (b) Chattering

Figure 1.14: Typical sliding modes

• Selection of a hyper-surface or a manifold, i.e. the sliding surface, such that the system trajectoryexhibits a desirable behavior when confined to this manifold

• Finding feedback gains so that the system trajectory intersects and stays on the manifold

Of course the complexity of these two points depends on the nonlinearity involved in the analysis andon the system order.

Since the sliding mode control is a powerful technique in the stabilization of nonlinear systemsbut it does not provide any degree of adaptiveness, it is usually coupled with other control techniquesto exploit its robustness in the face of system uncertainty and unmodeled dynamics, while includingthe ability of adapting to new configurations. Samples of such hybrid controllers in the aeroelasticcontrol literature can be found in [152, 161, 162, 163, 164].Sliding mode control theory will be resumed in the next chapters when a sliding mode observer will beemployed in nonlinear aeroservoelastic problems.

Control based on the state dependent Riccati equation

Techniques like feedback linearization and back-stepping have shown to have a very limited applicabil-ity because of the strong mathematical conditions imposed on the system. Control system designershave been striving for control algorithms that are systematic, simple, and yet optimize performances,providing trade-offs between control effort and state errors.Developed in the mid-90’s, the statedDependent Riccati equation (SDRE) strategy is now significantlywell-known and has become very popular within the control community over the last decade, provid-ing a very effective algorithm for synthesizing nonlinear feedback controls of nonlinear systems, whileadditionally offering great design flexibility through state-dependent weighting matrices [165, 166].This method, first proposed by Pearson [167] and later expanded by Wernli and Cook [168], wasindependently studied by Mracek and Cloutier [ 169] after having beed previously hinted at by Fried-land [170]. The method entails a particular factorization (that is, parameterization) of the nonlineardynamics into the state vector and the product of a matrix-valued function that depends on the stateitself. In doing so, the SDRE algorithm fully captures the nonlinearities of the system, bringing the

36 Chapter 1

nonlinear system to a (nonunique) linear structure having state-dependent coefficient (SDC) matrices,and minimizing a nonlinear performance index having a quadratic-like structure, as in a classicallinear optimal controller. Such a nonuniqueness of the parameterization creates extra degrees offreedom, which can be used to enhance controller performance. An algebraic Riccati equation usingthe SDC matrices is then solved on-line to give a suboptimum control law. Clearly, the coefficients ofthis equation vary with the given point in state space. The algorithm thus involves solving, at a givenpoint in state space, an algebraic state-dependent Riccati equation, or SDRE [171].

Mathematically speaking, considering a system which is is full-state observable, autonomous,nonlinear in the state, and affine in the input, represented in the form:

x(t ) = f(x)+B(x)u(t ), x(0) = x0 (1.50)

Without any loss of generality, the origin x = 0 is assumed to be an equilibrium point, such that f(0) = 0.In this context, the minimization of the infinite-time performance criterion:

J (x,u) =∫ ∞

−∞xTQ (x)x+uTR (x)udt (1.51)

is considered, which is nonquadratic in x but quadratic in u. As suggested by Eq. 1.51, the state andinput weighting matrices are assumed state-dependent. These design parameters satisfy Q (x) ≥ 0 andR (x) > 0 for all x.Under the specified conditions, a control law of the type:

u (x) = k (x) =−K(x)x, k(0) = 0 (1.52)

where k (x) is then sought that will minimize the cost function 1.51 subject to the input-affine nonlineardifferential constraint 1.52 while regulating the system to the origin [165].

As commented before, in order to apply the SDRE method, the nonlinear term of Eq. 1.50 f (x) isfactorized (also known as extended linearization [171] or apparent linearization [170] or SDC parame-terization [169]) in a linear-like structure composed by a matrix-vector multiplication: f (x) = A (x)x.This factorization is clearly non-unique when the order of the system is greater than 1.

The system of Eq. 1.50 now becomes x(t ) = A (x)x+B(x)u(t ). Motivated by the LQR problem [169],SDRE feedback control is an "extended linearization control method" that provides a similar approachto the nonlinear regulation problem for the input-affine system of Eq. 1.50 with the cost functional ofEq. 1.51. By mimicking the LQR formulation, the state-feedback controller is obtained in the form:

u (x) =−R−1 (x)BT(x)P(x)x (1.53)

where P (x) is the unique, symmetric, positive-definite solution of the algebraic State-DependentRiccati Equation:

P(x)A(x)+AT(x)P(x)−P(x)B(x)R−1(x)BT(x)P(x)+Q(x) = 0 (1.54)

hence the name SDRE control. The SDRE solution to the infinite-horizon autonomous nonlinearregulator problem is a true generalization of the infinite-horizon time-invariant LQR problem, whereall of the coefficient matrices are state-dependent. Therefore, at each time instant, the method treatsthe state-dependent coefficients matrices as being constant, and computes a control action by solvingan LQ optimal control problem.A full treatment of the mathematical conditions under which such an approach is applicable can befound in [171] and references therein.

The clearest benefit of the SDRE algorithm is its simplicity and its apparent effectiveness. Of course,the factorization of f (x) can be difficult to figure out, because it depends on the type of nonlinearities


Figure 1.15: MRAC framework

involved in the modeling. Computational implementation is another important practical considera-tion. Implementing the SDRE algorithm, at least for simulations, is relatively straightforward and canbe easily mechanized using commercially available software.On-line computation of SDRE feedback controls makes the technique ideal for real-time implementa-tion, so that the controller must perform all operations inreal-time. Depending on the design systemsize, a real-time implementation may be feasible or not. Moreover, because a full estimate of the stateis required at each sampling time to compute the control action of Eq. 1.53, a nonlinear observer willalways be required. This latter may be designed always following the SDRE theory [171, 172], and therelated computational effort will be added to the one related to the controller, therefore limiting itstheoretical efficacy.

Even if some authors have been criticizing the stability and convergence properties of this kind offormulation in the case of a generic nonlinear system [173, 172], there is a large number of applicationsof control laws based on the SDRE. These include advanced guidance law development [174], autopilotdesign [175], seismic isolators [176], satellite and spacecraft control [177, 178] and suppression ofaeroelastic instabilities [179, 180, 181, 182].

Model-reference adaptive control

Generally, a model-reference adaptive control (MRAC) system can be schematically represented byFigure 1.15. It is composed of four parts: a plant containing unknown parameters, a reference modelfor compactly specifying the desired output of the control system, a feedback control law contain-ing adjustable parameters, and an adaptation mechanism for updating the adjustable parameters.This control technique can be implemented as a either direct or indirect controller, whereas thedirect version can be more easily found in the literature, being one of the first adaptive controllersimplemented [64, 65, 183, 184]. Such a method can also be classified as a CE controller, because theestimation of the system uncertain parameters is directly applied in the control law.

The plant is assumed to have a known structure, although the parameters are unknown. Fornonlinear systems, this implies that the structure of the dynamic equations is known, but that someparameters are not.A reference model is used to specify the ideal response of the adaptive control system to the externalcommand. Intuitively, it provides the ideal plant response which the adaptation mechanism shouldseek in adjusting the parameters. The choice of the reference model is part of the adaptive controlsystem design. This choice has to satisfy two requirements. On one hand, it should reflect theperformance specification in the control tasks, such as rise time, settling time, overshoot or othercharacteristics. On the other hand, this ideal behavior should be achievable for the adaptive controlsystem to be designed, i.e., there are some inherent constraints on the structure of the reference model,e.g. its order and relative degree, given the assumed structure of the plant model.

The controller is usually parameterized by a number of adjustable parameters, i.e. implying thatone may obtain a family of controllers by assigning various values to the adjustable parameters. The

38 Chapter 1

controller should have a perfect tracking capacity to achieve tracking convergence. That is, whenthe plant parameters are exactly known, the corresponding controller parameters should make theplant output identical to that of the reference model. When the plant parameters are not known,the adaptation mechanism will adjust the controller parameters so that perfect tracking is asymp-totically achieved. If the control law is linear in terms of the adjustable parameters, it is said to belinearly parameterized. Existing adaptive control designs normally require linear parametrizationof the controller in order to obtain adaptation mechanisms with guaranteed stability and trackingconvergence.

The adaptation mechanism is used to adjust the parameters in the control law. In MRAC systems,the adaptation law searches for parameters such that the response of the plant under adaptive controlbecomes the same as that of the reference model, i.e., the objective of the adaptation is to makethe tracking error converge to zero. Clearly, the main difference from more conventional controlslies in the existence of this mechanism. The main issue in adaptation design is to synthesize anadaptation mechanism which will guarantee that the control system remains stable and the trackingerror converges to zero as the parameters are varied. Lyapunov theory is usually used in MRAC designto synthesize the adaptation mechanism.

The reader is referred to [64, 65] for a detailed introduction. Model Reference Adaptive Controlis usually applied to linear and linear time variant systems, where the model parameters may beunknown [185]. Applications on nonlinear systems can also be found in the literature [186, 187]. Alarge number of applications of this control technique to aeroelastic problems, regarding both fluttersuppression and gust alleviation, can be found in [188, 189, 190]. Also in this case, the full knowledgeof the system state is required. Therefore, when it is not available, an observer is required for stateestimation.

L -1 control

Model reference adaptive control was developed conventionally to control linear systems with un-known coefficients [64]. This architecture has been facilitated by the Lyapunov stability theory, whichgives sufficient conditions for stable performance without characterizing the frequency propertiesof the resulting controller. Application of adaptive controllers was therefore largely restricted dueto the fact that the system uncertainties during the transient have led to unpredictable/undesirablesituations, involving control signals of high-frequency or large amplitudes, large transient errors orslow convergence rate of tracking errors, to name a few.

Nevertheless, several important aspects of the transient performance analysis seem to be missingin the analysis framework of a MRAC. First, limit bounds are computed for the tracking error only, andnot for the control signals. Although the latter can be deduced from the former, it is straightforwardto verify that the ability to adjust the former may not extend to the latter in case of nonlinear controllaws [191, 192, 193]. Second, since the purpose of adaptive control is to ensure a stable performance inthe presence of modeling uncertainties, one needs to ensure that the changes in reference input andunknown parameters due to possible faults or unexpected uncertainties do not lead to unacceptabletransient deviations or oscillatory control signals, implying that a re-tuning of adaptive parametersis required. Finally, it is needed to ensure that whatever modifications or solutions are suggestedfor performance improvement of adaptive controllers, they are not achieved via high-gain feedback,otherwise the system high frequency modes would be excited, leading to unwanted vibrations, or eveninstability.The L -1 control framework consists, as for any other adaptive controller, of a state predictor, adapta-tion law and control law, as shown by Figure 1.16.

The main benefit of L -1 adaptive control is its fast and robust adaptation, which does not affect


Figure 1.16: L -1 control framework

significantly the needed trade–off between performance and robustness.In fact, the architectures of L -1 adaptive control theory have guaranteed transient performance andguaranteed robustness in the presence of fast adaptation, without introducing or enforcing persistenceof excitation, without any gain scheduling in the controller parameters, and without resorting tohigh–gain feedback, as could happen in MRAC design [ 192, 194]. In these architectures, the speed ofadaptation is limited only by the available hardware and actuators bandwidth, while robustness canbe achieved via the methods of robust control theory.

The philosophy of L -1 adaptive control architectures is to obtain the estimate of the uncertaintiesvia a fast estimation scheme, defining the control signal as the output of alow–pass linear filter, a factwhich not only guarantees that the control signal stays in the low-frequency range in the presenceof fast adaptation and large reference inputs, but also leads to separation between adaptation androbustness.In particular, proper filter design leads to desired transient performance for the system input andoutput signals simultaneously, in addition to steady-state tracking. In addition to this, L -1 adaptivecontrol theory provides a systematic design procedure that significantly reduces the tuning effortrequired to achieve desired closed-loop performance, particularly while operating in the presence ofuncertainties and failures. It is also important to mention that the fast adaptation capability of thiskind of control architectures allows for control of time-varying nonlinear systems by adapting twoparameters only, which is in contrast to the use of on–line neural networks which may have problemsof computational predictability and for which the only validation and verification techniques are basedon brute-force Monte-Carlo testing [68]. This methodology is usually implemented to result in anindirect adaptive controller, which is also CE, because of the strong similarities with the MRAC method.

However, recently different researchers [195, 196] pointed out that L -1 control for plants withmeasured states is simply a standard MRAC with a low pass filter included just in front of the controlinput. Such a claim is based on the fact that the analysis of the control scheme is almost identical tothat of MRAC as the same Lyapunov function is used to establish stability [195]. In such a work it isstated that the motivation for using the filter in L -1 control is the fact that for this class of adaptiveschemes, i.e. MRAC for plants with full state measurement, the tracking error can be made arbitrarilysmall during transient by increasing the adaptive gain.A high adaptive gain however makes the differential equation of the adaptive law very stiff and leadsto numerical problems that cause high oscillations in the estimated parameters leading to loss ofadaptivity and deviations from the theoretical behavior. The L -1 approach typically mistakes thesenumerical oscillations as properties of the adaptive scheme and employs an input low pass filter inorder to filtering them out. While such a filter helps in reducing the frequency of these oscillations inthe control law, the price paid is high, i.e. the loss of the theoretical predicted closed loop performance,as reported in [195].

A complete introduction to L -1 control can be found in [68]. Numerous applications of this

40 Chapter 1

Figure 1.17: Fuzzy controller architecture

promising technique are available in the literature [194, 197, 198, 199, 200]. Aeroelastic systems havebeen also analyzed by control specialists following the L -1 theory. Various applications, some of themwith experimental validation, can be found in [133, 201], regarding both flutter suppression and gustalleviation of relatively simple systems.

Fuzzy logic control

As first pointed out in Section 1.1, even if a relatively accurate model of a dynamic system can bedeveloped, it is often too complex to use it for a controller development. It is for this reason that inpractice conventional controllers are often developed via simple models of the system behavior thatsatisfy the necessary assumptions, and via an ad-hoc tuning of relatively simple linear or nonlinearcontrollers.Regardless of this, it is well understood that heuristics enter in the conventional control design processas long as one is concerned with the actual implementation of the control system. It must be acknowl-edged, moreover, that conventional control engineering approaches that use appropriate heuristics totune the design have been often relatively successful.

Fuzzy control provides a formal methodology for representing, manipulating, and implementing ahuman’s heuristic knowledge about how to control a system [202].A fuzzy controller block diagram is given in Figure 1.17, where fuzzy controller embedded in a closed-loop control system is shown. The fuzzy controller has four main components: (1) The “rule-base”holds the knowledge, in the form of a set of rules, of how best to control the system. (2) The inferencemechanism evaluates which control rules are relevant at the current time and then decides what theinput to the plant should be. (3) The fuzzification interface simply modifies the inputs so that theycan be interpreted and compared to the rules in the rule-base. And (4) the defuzzification interfaceconverts the conclusions reached by the inference mechanism into the inputs to the plant. Basically,the fuzzy controller is an artificial decision maker that operates in a closed-loop system in real time. Itgathers plant output data y(t ), compares it to the reference input r (t ), and then decides what the plantinput u(t ) should be to ensure that the performance objectives will be met.

To design the fuzzy controller, the control engineer must gather information on how the artificialdecision maker should act in closing the loop. Sometimes this information can come from a humandecision maker performing the control task, while at other times the control engineer can come tounderstand the plant dynamics and write down a set of rules about how to control the system withoutoutside help.Depending on the formulation of the control problem, fuzzy logic-based controllers can be classifiedas either direct or inverse adaptive controllers. On the other hand, they are usually CE.

Thanks to their intrinsic ability to adapt to new conditions, fuzzy controllers are often used inconjunction with other robust control techniques to develop control laws that are both adaptive and


Figure 1.18: A sample of state feedback inverse compensation control system, from [210]

robust to unmodeled dynamics [203, 204, 205, 206]. Within this realm of controllers, a large number ofapplications in typical aeroelastic problems can be found in the literature [207, 208, 209], regardingboth flutter suppression and gust alleviation.

Control based on nonlinearity inversion

Adaptive inverse design techniques provide readily implementable algorithms to cancel the effectsof unknown nonlinearities. Such an approach for handling uncertain nonlinearities has been welldeveloped for SISO and MIMO systems [72].The essence of such an approach is to use adaptive algorithms to estimate the presence of possiblenonlinearities such as free-play, friction, hysteresis, and other characteristics, and to compute controlsignals to compensate the related effects. A key component of this approach is an adaptive inverse ofthe functions representing these nonlinearities, which generates an input signal to adaptively cancelthe effects of the uncertain nonlinearity. A sample of the related closed-loop framework is shownin Figure 1.18. While the design for SISO systems is quite straightforward, the main technical issuein developing this approach for MIMO systems is caused by the dynamic couplings and parameteruncertainties in such systems [210]. Controllers based on nonlinearity inversion are intrinsicallyindirect and their classical implementation lead to a CE control framework.

Different applications of this technique can be found in the literature. For example, nonsmoothactuators nonlinearities can be compensated by this technique [210, 211, 212]. This approach hasbeen also used to compensate the nonlinearities of synthetic jet actuators for aircraft flight controlapplications [213, 214].Nevertheless it seems that there is a lack of studies focused on aeroservoelastic applications. One ofthe scopes of this work will be to provide a series of applications in the field of aeroelasticity that makesuse of this simple, yet effective technique.

Neural networks-based control

Neural networks are attracting increasing attention in applications where autonomy is an importantfeature or where it is virtually impossible to analyze, in advance, all possible environmental conditionsthat may arise during a mission. Control applications which need to react to catastrophic changes(e.g., a broken control surface of an aircraft) can also benefit from a neural network based system [67,144, 215, 216, 217, 218].

Neural controllers are typically composed by two networks of the type introduced in Section1.3.2. The first one is used to identify the dynamics of the plant under control and the information soobtained is fed to the second network which acts like a controller. Thanks to its learning capability,the system under control can be identified on-line and no model is required to design the controller.Such a framework in depicted in Figure 1.19a. This adaptive black-box approach permits to avoidscheduling procedures to cover a large range of operations and reduces the errors that may come from

42 Chapter 1

(a) Recurrent neural network-based control framework (b) Closed loop system with neural controller

Figure 1.19: Layout of a dynamic neural network controller. From [67]

linearizations and approximations used to describe the dynamic behavior of the system [67, 215].Let us assume to have an available model of a process able to produce any sort of output. These

are the known, measured, quantities of the model, indicated by yMn . Subjected to the same input of

the physical model, the identifier should be able to minimize the error between its output and themeasured one. The network input and output will be defined on a case by case basis. The training ofsuch a network can be interpreted as an optimization algorithm which minimizes the following costfunction:

E IDn = 1

2eT

nen , en = yIDn −yM

n (1.55)

The real-time recurrent learning algorithm [219] is usually employed to modify the network synapticweights and by doing so minimize the cost function of Eq. 1.55. Such an algorithm derives its namefrom the fact that adjustments are made to the synaptic weights of a fully connected recurrent networkin real time, i.e. while the network continues to perform its signal-processing function [220, 219]. Thiskind of approach fits perfectly the needs of an adaptive controller, potentially leading to an accuratedescription, in terms of input-output relation, of the plant under control at each sampling time. Noticealso that in this kind of approach the identifier acts like a reduced order observer for the system andtherefore a further state estimation is not required.

Starting with a basic interpretation, the controller network will be fed in input by the outcomes ofthe identifier network, and it will give in output a new value of the control input, which will minimizethe control cost function:

E COn = 1

2eT

nen + 1

2ρyCOT

n yCOn , en = yID

n −yrefn (1.56)

Where yrefn is a reference output. The parameter ρ is defined as the control penalization parameter and

is used to limit the control effort in a way similar to what is done in a classical LQG controller. In anon-minimal phase system, all the zeros in the right half of the complex plane may be transformedinto unstable poles by the controller through a straight system inversion. In this case, the penalty termis necessary to avoid the divergence of the control [67, 215]. The resultant closed loop system is shownin Figure 1.19b.

Neural networks lead to indirect adaptive controllers that belong to the CE family. These controllershave shown promising results in their application, and they are currently employed by the aeronauticalindustry in research and development projects [221]. Nevertheless, in safety-critical applications, noneural network-based system would be used nowadays unless verification and validation (V&V) candemonstrate its reliability in a cost effective manner. As pointed out in [216], the adaptive nature ofneural networks requires a significantly different approach to V&V methods. In order to perform all


required V&V activities, the software development process, e.g. as prescribed by IEEE standards, needto be adapted accordingly. Nevertheless, V&V of neural network-based control systems is consideredto be very difficult, because the underlying theory of machine learning usually does not providean enough mathematically rigorous help for verification. Therefore, V&V must be augmented byspecifically tailored validation and dynamic monitoring tools, and this is still an open problem in therelated research community [216, 218].

A large number of applications in aeroelasticity can be found in the related literature, regardingboth static [217, 222, 223] and recurrent [67, 215] neural networks, where experimental results are alsoreported. More recently, neural networks have been used to shape dynamic surface controllers [224,225] in supersonic and hypersonic aeroelastic problems, where structural nonlinearities were present.Basically, this approach uses a back-stepping control technique, where nonlinearities and systemuncertainties are approximated by radial basis function neural networks.

Immersion and invariance control

The immersion and invariance (I&I) principle is a relatively new method for designing nonlinear andadaptive controllers [73, 226, 227]. The method relies upon the notions of system immersion andmanifold invariance. The basic idea of this approach is to achieve the control objective by immersingthe plant dynamics into a (possibly lower-order) target system that captures the desired behavior [227].This is achieved by finding a manifold in the state-space that can be rendered invariant and attractive,with internal dynamics that reflect the desired closed-loop dynamics and by designing a control lawthat takes the state of the system towards the manifold. Immersion and invariance-based controllersare by definition NCE and indirect, as will be shown in the problems studied in this work.

The I&I method consists in four main ingredients. The target dynamics evolve over a separate,usually lower dimensional state space, a smooth mapping, called the immersion, maps this target statespace to a sub-manifold of the original state space, the control law renders this sub-manifold invariantand such that the dynamics on this sub-manifold are the image of the target dynamics through theimmersion, and the control law also renders this sub-manifold globally attractive while keeping allstates bounded [73]. It follows from continuity reasons that the trajectories of the closed-loop systemapproximately track the image of the target dynamics after an initial transient phase during which thestates converge to the sub-manifold.The design flexibility of this methodology include the choice of the target dynamics, the choice of theimmersion and thus of the resulting sub-manifold, and the choice of the off-manifold control action,which renders the sub-manifold globally attractive [73].

In addition, I&I design does not require a complete knowledge of the system dynamics or of aLyapunov function for a particular subsystem. This results in a simpler control law than other designs,with its performance easier to tune. In fact, its authors claim that I&I control does not require theconstruction of any Lyapunov function during the design of the control law, although using it to provethe asymptotic stability of the controller in closed-loop [73]. This kind of controller is usually designedin a continuous time domain, even if recently discrete time designs have been presented for certainclasses of nonlinear systems [228, 229].

As can be noticed from [73], the I&I reformulation of the stabilization problem is implicit in slidingmode control, where the target dynamics are the dynamics of the system on the sliding surface, which ismade attractive by a discontinuous control law. Therefore, sliding mode and I&I may be considered tobelong to the same class of controllers, with the difference that I&I is adaptive to system changes whilesliding mode control is more appropriate when large system uncertainties and unmodeled dynamicsare present. Nevertheless, it is worth to be pointed out that while I&I control leads to a classicaladaptive controller behavior, sliding mode control is not able to adapt to new system configurations. In

44 Chapter 1

addition, a I&I-based control is usually characterized by nonlinear, smooth laws in closed loop, whilethe implicit discontinuous nature of a sliding mode controller may lead to unwanted phenomena inmechanical systems, like chattering.

In spite of these strong stabilization properties, applications of I&I adaptive control to generalmulti-input nonlinear systems have been somewhat limited for a while because of certain restrictiverealizability and manifold attractivity conditions [73]. Seo and Akella [131] extended the I&I frameworkby circumventing such restrictions. With their proposed procedure, all the key beneficial features ofthe I&I adaptive control methodology are preserved by introducing stable linear filters so that theparameter-adaptation dynamics reside within a stable and attracting manifold. In particular, sucha design approach ensures that the additive disturbance type of term arising within the closed-loopdynamics due to the parameter estimation error decays to zero.

Because a I&I control law requires the knowledge of the full system state, an observer is usuallyrequired for a realistic implementation. In [230] a methodology based on I&I is presented but therelated design must respect strong mathematical conditions that usually are not satisfied by a genericnonlinear system. The number of applications of this technique are therefore more limited [231, 232,233].

Flexible-joint robots [226, 234] and magnetic levitation systems [226, 235] have been alreadycontrolled effectively by I&I-based laws. Applications on the attitude control of spacecraft [ 131, 236]and trajectory tracking of aircraft [237, 238] can be found as well. Relatively simple aeroelastic systemshave been stabilized by using this technique also [239, 240, 241]. Nonetheless, it seems that in thisfield there is a lack of applications where realistic configurations are taken into account.For example, nonlinear unsteady aerodynamics, nonlinear actuation systems, interaction with stateestimators, effects of sensors dynamics and control law discretization have never been studied so farin this field considering this kind of adaptive controller. One of the scopes of this work will be to studya series of applications of increasing complexity that involves such a control technique. Several proofswill be provided showing the efficacy of such a control strategy applied to aero-servo-elastic typicalproblems, highlighting its strengths and pointing out some limitations.

1.5 Thesis Contributions

Even if already sufficiently rich of real-world applications [15, 16, 18, 26, 45, 242, 243], nonlinearitiesin the field of aeroservoelasticity have not been studied systematically so far. This work has the aimof presenting a systematic framework to study the stability properties and various typical responsesof nonlinear aeroservoelastic systems, eventually designing active controllers to improve the systemperformance.

In trying to pursue such a scope, a novel aerodynamic reduced order model technique obtained asa mix between classical linear identification techniques and neural network will be presented, studyingcases of increasing structural complexity. The convergence properties of such methodology to thereference CFD results will be also studied, proving that, in the cases considered, nonlinear aeroelasticresponses can be computed with good accuracy also using dynamical systems of compact size.On the base of these results, the interaction between aerodynamics and structural nonlinearities, suchas free-play and friction is studied. It is found that their combination leads to a bi-stable behaviorthat was never mentioned before in the related literature. In such systems, stability properties andresponses depend not only on the flight speed, but also on the initial condition and the amplitude andtype of the input signal.Having available a tool for evaluating aeroelastic responses in a short time, active controllers are thendesigned to improve the system performance. Particular emphasis is given to adaptive controllers,

1.6. Thesis Outline 45

because aeroelastic systems have the feature of changing their stability properties as the flight speed isvaried. For this reason, adaptive control laws seem to be the perfect candidate to stabilize an aeroelasticsystem in a even wider range than the one covered by a classical linear controller.A systematic procedure to design immersion and invariance-based controllers is provided, also in thecase of multi-input, multi-output applications. This can be considered a quite original contribution,because all the applications found in the literature present ad-hoc procedures for the control lawdesign. In this work instead, an easy-to-use method is used to design the control framework andchoose the controller gains.

Usually, in the field of adaptive control, the researchers prefer to focus more on the mathematicalproperties of the controller rather then evaluate its performance when it is employed in realistic appli-cations [133, 154, 155, 156, 163, 164, 179, 180, 190, 201, 239, 240, 241, 244, 245, 246], where very largeuncertainties and important unmodeled dynamics may be present. Moreover, only a few applicationscan be found in the literature where the controller is tested on a verification model with nonlinearitiesnot included in the design.This work tries to fill such a gap systematically studying the effect of actuators and sensors dynamicsin the closed loop response. The influence of the actuators bandwidth on the controller performanceis studied as well.

The coupling between controller and nonlinear observer is also studied, providing interestinginsights in a problem that is not usually studied in adaptive control applications, especially when struc-tural and aerodynamics nonlinearities are both present. Finally, the effect of the time discretizationof the control law, e.g. the time delay related to analog-to-digital and digital-to-analog conversion isanalyzed, examining the differences with the related continuous time counterparts.

The effects of nonlinearities in the actuation system are also investigated, analyzing how thecontrol system reacts to them when they are not taken into account during the design, and eventuallydesigning additional control laws aiming at compensating them. Nonlinearities such as free-play andfriction has been already studied in depth in the aeroelastic community, but it appears that only a fewauthors in the literature have addressed the problem of eliminating such a detrimental effect in realisticapplications. In this work, a simple but naturally adapt approach based on the nonlinearity inversionmethod is studied. The designed control law provides and adaptive estimation of the nonlinearitieskey features, permitting an effective suppression of the related unwanted effects.

In summary, this work tries to provide the reader a systematic framework for modeling and controlclassical nonlinear aeroservoelastic systems. The proposed techniques can be applied to a broad rangeof physical problems, and they are not restrained to aeroelastic applications only. With particular focuson the aeronautical industry, in the next few decades such a techniques may permit the inclusion ofnonlinear phenomena in the conceptual and preliminary stages of aircraft design, eventually leadingto improved and less expensive products.

1.6 Thesis Outline

This work is organized in three interfaced parts. The first part details the implementation of a novelhybrid linear-neural numerical technique to represent compactly large order nonlinear dynamicsystems. Particular focus is given to aerodynamic and aeroelastic applications, studying test cases ofincreasing complexity.

The second part regards the development and implementation of active control techniques aimedat suppressing structural and aerodynamic nonlinearities separately. Therefore, at first cases with onlystructural nonlinearities will be considered, developing compensation laws based on both classical andadaptive controllers. It will be shown that advanced techniques will be required when high accuracy

46 Chapter 1

is requested and large disturbances coming from linear aerodynamic systems may influence thesystem stability. Then, and adaptive suppression technique is developed to deal with the instabilitiesintroduced by nonlinear aerodynamic models.

The third part puts everything together and presents an integrated approach for compensatingstructural and aerodynamic nonlinearities at the same time. All the presented approaches aim atbeing as systematic as possible to be applied to generic aeroelastic systems formulated in a state spaceframework.

Concluding remarks are given at the end enhancing the most interesting findings of this work andaddressing possible future extensions. In appendix, some details related to the design and realizationof the control surfaces actuation system of a flexible wind tunnel aircraft model are given, focusing onthe virtual realization of actuation system nonlinearities through linear controllers.

Aerodynamic Reduced Order Modelingthrough a Linear-Neural Model

CHAPTER 2

2.1 Problem Description

Recently, reduced order models have been gaining more and more relevance in different branches ofcomputational physics. The ability of capturing the main features of complex phenomena by makinguse of a small number of states is particularly appealing in time critical applications, such as thoserelated to aerospace analysis and design.Within this context, CFD is probably the most reliable tool for evaluating aerodynamic loads in thetransonic regime during preliminary, or even conceptual, design stages of an aircraft. Even withthe impressive improvements of modern computational methods, these computations are still toodemanding to be employed in routine industrial analyses, where sensitivity calculations and controllaw designs must be carried out over a large number of configurations and operational conditions. Thesituation is even more challenging when the interaction between structural flexibility and aerodynamicloads is taken into account, leading to aeroelastic models and dynamic response calculations. In thiscase, a wide spectrum of phenomena could be of interest, i.e. identification of flutter conditions andeventual limit cycle oscillations.

It is therefore of great importance to develop reduced order models (ROM) from high fidelitynumerical schemes to permit the use of nonlinear aerodynamics in a far wider analysis and designspectrum, while maintaining a reasonable level of accuracy. Reduced order models are not onlyencountered in aerodynamics, in fact they are of great interest also in electromechanics [247, 248]. Instructural dynamics such a concept is relatively old, used for many purposes ranging from the designof a test-analysis model to provide a basis for comparing computational and experimental results, tothe alleviation of the computational burden associated with large-scale finite element models [249].

In the last decade, a significant effort has been directed toward developing compact models oftransonic aerodynamics and computational aeroelastic problems [8, 83, 102]. Those ROMs must beable to capture the main features of the nonlinear problem under consideration, while maintaining alimited number of states. In the aeroelastic community, the most popular approaches include properorthogonal decomposition [83, 250], harmonic balance [102], generalized interpolation methods, e.g.radial basis function or Kriging interpolators [108, 109], Volterra theory [95, 84] and neural networks(NN), both static [251] and dynamic [252, 253]. These methods tackle the problem of model orderreduction in different ways, as pointed out in Chapter 1.

The method that will be presented in the following sections will follow an NN-based approach,though structuring the aerodynamic model equations as a combination of linear and nonlinearcontributions. In this way, the linear part will be responsible for the estimation of the system bifurcationpoint, while the nonlinear terms will shape possible limit cycle oscillations beyond it. Instead of usinga brute force identification of the aerodynamic loads, as done in [253, 121, 122, 86], the proposedapproach permits an improved physical understanding of the considered problem, as well as being ageneralized reduced order modeling technique, applicable to a broad range of unsteady problems incomputational physics. Also the training phase is different from the one discussed in [253], and the

47

48 Chapter 2

present approach leads to a faster generation of reduced order models than the cited technique, whilemaintaining a good level of accuracy.

The present chapter aims at presenting a novel, physics-based ROM technique, evaluating theimportance of a correct training signal in the nonlinear domain, and finally performing convergenceanalyses in both aerodynamic and aeroelastic applications to determine the ROM sensitivity to param-eter changes.

2.2 Aerodynamic Solver

In the next sections, and throughout this thesis, whenever a high fidelity model of the aerodynamicproblem is required, the in-house solver AeroFoam developed at Politecnico di Milano [254] is chosen.This application is supported by OpenFOAM libraries for the management of the mesh data, thecomputation of the numerical solution and the pre/post-processing phases. It is a Reynolds-averagedNavier Stokes (RANS) density-based solver for aero-servo-elastic applications, written exploiting thearbitrary-Lagrangian-Eulerian formulation for moving grids. It is a finite volume, cell-centered solver,that can treat both structured and unstructured grids. In the present computations, the Euler flowmodel is chosen, therefore the effects of viscosity and thermal conductivity will be neglected.AeroFoam is the first density-based RANS solver implemented within the framework of OpenFOAM,realized to overcome the limits of built-in pressure-based solvers in the transonic regime, e.g. sonic-

Foam, because their non-conservative formulations do not permit to solve accurately transonic andsupersonic regimes.Regarding the present inviscid application, the convective fluxes are discretized by the classical Roe’s

approximated Riemann solver, which is a first order, monotone scheme, blended by the centeredapproximation provided by the Lax-Wendroff scheme, resulting in a second order, high-resolutionscheme. The spatial discretization is completed by the entropy fix of Harten and Hyman and the fluxlimiter by van Leer [100].The time discretization is performed by an explicit five-stage Runge-Kutta scheme, which presents afirst order convergence. Dual time stepping and a full approximation storage multi-grid technique arecombined to speed up the convergence of time-accurate simulations.An extended illustration of the aeroelastic capabilities of AeroFoam can be found in [7, 254].

As will be shown in the next sections, this CFD solver will be used to generate the input-output timehistories required for tuning the neural model employed in this work. In the present applications, theinput will be represented by the structural motion, while the outputs will be the associated aerodynamicloads.

2.3 Model Order Reduction

The proposed identification technique is here directly applied to a generalized aerodynamic problem,however the same approach can be followed in several branches of physics. Furthermore, the datasource on which the model will be trained may be generated in different ways. In this case onlyCFD-based results will be analyzed, but experimental data could also be employed as a training signal.Using the following identification-like approach, it is possible to structure the dynamic equations ofthe reduced order model in any way the analyst would prefer. The following model is adopted here:

xa = Aaexa +Baexs +Eaeφ (xa)

fa/q∞ = Caexa +Daexs +Faeφ (xa)(2.1)

2.3. Model Order Reduction 49

where xa and xs are the aerodynamic and structural dynamics state, Aae,Bae,Cae,Dae,Eae,Fae are theROM parameters, fa are the aerodynamic loads and q∞ is the flight dynamic pressure. The vectorsφ (xa) are the nonlinear contributions to the ROM dynamics, where each component is a hyperbolictangent function of the input, i.e. φi

(xa,i

) = tanh(xa,i

)− xa,i . The dimension of the aerodynamicstate will be referred as Na . The derivative of this last contribution with respect to the aerodynamicstate presents a null value at the origin, i.e. xa,i = 0, and this fact will be exploited in the constructionof the reduced order model. The problem is formulated in the continuous time domain in order topermit variations in the integration time step depending on the analysis under consideration. Forexample, the time discretization can be modified during the design of a control law where the effectof different sampling times on the closed-loop system may be of interest. Such a variable-time-stepfeature could also be exploited in the search of limit cycle oscillations, as reported in [253]. Equation 2.1represents a nonlinear model in state space form, where the input is the structural dynamics statexs while the output is the generalized aerodynamic load fa. An aerodynamic state is introduced torepresent the intrinsic memory of the dynamic model, and in general it does not have any particularphysical meaning. Nonlinear functions like the one employed in this case have already proved theirpotential in system identification [219].

Note also the particular structure of Eq. 2.1, where the model is nonlinear in the aerodynamic stateonly, while it is linear with respect to the structural state. In this way we are not forcing the system todepend nonlinearly on the input, as a brute force NN approach would have done, instead we are takinginto consideration the physics of the problem. In fact, after running several CFD-based aeroelasticsimulations, it has been noticed that in the transonic regime large amplitude limit cycle oscillationsare induced by the large motion of the shock wave on the moving body. Therefore nonlinearities areintroduced in the model even when a linear structural system is considered. This means that the basicnonlinear behavior is mainly introduced by the aerodynamic system, and this is the reason of thelinear dependence on the structural state in Eq. 2.1.

The determination of the model parameters is obtained through a two-level training procedure,whereas in the first stage the matrices associated to the linear part of the model are computed bya robust subspace projection technique, while the remaining nonlinear terms are determined by anonlinear output error minimization procedure in a second stage.Once the linear sub-model is trained, the nonlinear optimization will change only the terms related tothe nonlinear part. In fact, thanks to the proposed nonlinear functions, i.e. φi (x) = tanh(x)− x, thenonlinear terms will not contribute to the system linearization around the origin.

Finally, we would also point out the versatility of the present reduced order model. Even if it will behere used for predicting nonlinear responses due to fluid-structure interaction, its parameterizationcould also allow to represent other nonlinear aerodynamic phenomena, such as vortex shedding [255].

2.3.1 Reduced order model training

The matrix entries of Eq. 2.1 must be determined through an optimization procedure. Because of thedynamic behavior of the model, and since the problem is formulated in the continuous time domain,a simple linear least square approache cannot be used.The training of the reduced order model is tackled in two main stages. The first identifies the linearcontribution, i.e. matrices Aae, Bae, Cae, Dae, with a classical linear subspace projection technique [256].The second stage of the training refines the reduced order model response by adding the nonlinearterms to the optimization procedure, i.e. matrices Eae and Fae. Because of the intrinsic nonlinearity ofthe model, a generalized optimization method, such as the Levenberg-Marquardt (LM) [257] methodis employed.

We remark that once the linear sub-model is trained, the nonlinear optimization will change

50 Chapter 2

only the terms related to the nonlinear part. Note that these terms will not contribute to the systemlinearization around the origin, thanks to the special type of nonlinearities employed, in fact φ′(0) =sech2(0)−1 = 1−1 = 0. Thus dynamic linearized analyses can be performed right after the first part ofthe training, while the nonlinear contribution will only shape possible nonlinear responses.

As a first step, the training signal is generated by means of a CFD code. The type of training signalchosen in our simulations will be detailed in the next section, but the proposed procedure can workwith signals generated by experimental data as well. Such a signal combines input-output data pairs,where the input is the structural motion while the output are the aerodynamic loads, and it is given asinput to the following linear model:

xa = Aaexa +Baexs

fa/q∞ = Caexa +Daexs(2.2)

This training phase makes use of a subspace projection technique to determine the matrices. Thealgorithm determines state sequences xa through the projection of input and output data, xs andfa/q∞ respectively. These state sequences are outputs of non-steady state Kalman filter banks appliedin parallel to the training data. From these results it is possible to determine the state space systemmatrices. Such an algorithm has proved to be always convergent (non-iterative) and numerically stablesince it uses only QR and singular value decompositions, as reported in [256].

The second stage of the training introduces the nonlinear terms. Because of the intrinsic nonlinearformulation of the model, a classical linear identification method cannot be employed. Thus, weemploy the LM method to determine the entries of the matrices multiplying the nonlinear terms, butanother optimization approach could have been used. Collecting the entries of the matrices Eae andFae in a single, unknown vector p, the following system of ordinary differential equations must besolved until the output error e(t ) = fCFD

a (t )− fa(t ;p

)is arbitrarily small:

xa = Aaexa +Baexs +Eaeφ (xa)

xa/p = (Aae +Eaeφ (xa)/xa

)xa/p +Ea/pφ (xa)

fa/q∞ = (Caexa +Daexs +Faeφ (xa)

)fa/p /q∞ = (

Cae +Faeφ (xa)/xa

)xa/p +Fa/pφ (xa)

(2.3)

Analytic expressions for the sensitivity terms xa/p and fa/p are now provided. Calling vec(·) theoperator which stacks the column of a matrix in a vector, we can define the optimization unknown as

p = vec(Eae)T ,vec(Fae)T

T. The sensitivity term xa/p is then computed as:

xa/Ei j = (Aae +EaeΦ (xa))xa/Ei j + Ii jφ (xa) i , j = 1, ... , Na (2.4a)

xa/Fi j = (Aae +EaeΦ (xa))xa/Fi j i = 1, ... , Nout, j = 1, ... , Na (2.4b)

xa/p = (Aae +EaeΦ (xa))xa/p +Ux (2.4c)

where Φ (xa) = Diag(φ′ (xa,1

),φ′ (xa,2

), ... ,φ′ (xa,Na

)), being φ′ (·) the first derivative of φ (·) with respect

to its argument, while matrix Ux is so defined:

Ux =[φ

(xa,1

)INa , φ

(xa,2

)INa , · · · |φ(

xa,Na

)INa , 0 , · · · , 0

](2.5)

being INa the identity matrix of size Na .In order to assemble the Jacobian matrix of the following Eq. 2.8, the sensitivity term fa/p has to be


computed as well. This can be performed similarly to Eqs. 2.4:

fa/Ei j /q∞ = (Cae +FaeΦ (xa))xa/Ei j i , j = 1, ... , Na (2.6a)

fa/Fi j /q∞ = (Cae +FaeΦ (xa))xa/Fi j + Ii jφ (xa) i = 1, ... , Nout, j = 1, ... , Na (2.6b)

fa/p /q∞ = (Cae +FaeΦ (xa))xa/p +Ufa (2.6c)

where matrix Ufa is so defined:

Ufa =[0 , · · · , 0 , φ

(xa,1

)INout , φ

(xa,2

)INout , · · · , φ

(xa,Na

)INout

](2.7)

At this point, stacking the values of the computed e and fa/p at each simulation time step ti in thetwo following quantities:

g =

e (t1)e (t2)

...e (ti )

...e (tN )

J =

e/p (t1)e/p (t2)

...e/p (ti )

...e/p (tN )

=−

fa/p (t1)fa/p (t2)

...fa/p (ti )

...fa/p (tN )

(2.8)

the vector p is updated at each iteration through the solution of the following least-square problem:(JTJ+λI

)(pnew −pold

)=−JTg (2.9)

being λ a regularization parameter, usually very small, used to keep the left hand side matrix alwaysnon-singular [257]. The same training signal is presented to Eq. 2.3 at each optimization iteration, thetime histories of xa and fa are computed, and the value of p is updated through Eq. 2.9. The iterationsare stopped when a sufficiently small euclidean norm of the output error ||e||2 is achieved or when thevariation between two iterations is smaller than a user-defined threshold.

2.3.2 Training signals

The design of training signals in nonlinear identification problems is not as simple as its linear coun-terpart. In fact, it is not sufficient to excite the desired frequency range to obtain physically meaningfulROMs, but it is also necessary to perturb the system with signals with suitable amplitudes. Further-more, the superposition principle cannot be applied in these cases, and, to overcome this obstacle, theproblem must be tackled from a different prospective. From the linearized analysis, often available inpreliminary design concepts, we may take advantage of knowledge of two very important parameters:the flutter frequency and the aeroelastic eigenmode at the flutter point.The first parameter will help us in determining the frequency range to be excited by the training signal.As a matter of fact, at the generalized force level, aerodynamic nonlinearities are mostly smooth, e.g.they change the system properties in a regular way. The same cannot be said about the so called ’hard’nonlinearities, e.g. structural friction and free-play [22], which are able to switch the system behaviorin a discontinuous manner. Assuming such an assumption as acceptable, the training signal is de-signed to excite a relatively broad frequency range, covering the critical frequencies of interest. Evenin presence of ’hard’ nonlinearities this approach would still be valid, taking care of broadening thesignal frequency content. Also, signals with a wide frequency content would be suitable in analyzing alarge spectrum of responses, thus being particularly useful in control law designs.

52 Chapter 2

(a) Maximum aft position of the shock on the airfoil uppersurface

(b) Maximum aft position of the shock on the airfoil lowersurface

Figure 2.1: Shock oscillation during an LCO

The aeroelastic mode at the flutter point leads us to the determination of proper signal amplitudes.After, fixing the amplitude of one degree of freedom, all the others will follow, since the eigenvectoris defined. So, even running a coarse CFD-based simulation, or with a trial-and-error approach, theamplitude range for the training signal can be easily determined.Finally, the type of training signal has to be chosen. Since the superposition principle is not valid, thesystem has to be simultaneously excited in all of its degrees of freedom. This has been verified throughextensive simulations. Furthermore, step sequences and frequency sweep signals have not proven tobe good excitations for the here presented applications. This is in contrast with the results obtainedin [84, 94, 258], where Volterra series are used to predict unsteady responses of nonlinear CFD-basedaerodynamic simulations. In the present case, random-like and noisy sweep signals have alwaysproduced physically meaningful ROMs, and this is in accordance with the results reported in [95]. It is

thought that to identify large amplitude limit cycle oscillations, a series of step and impulse functionsis not adequate. This kind of signal would be useful in the identification of weak nonlinearities. In thepresent case, the moving shock wave undergoes very large displacements, eventually disappearingand reappearing during a cycle, as shown in Figure 2.1 in the case of an airfoil experiencing an LCO.It is evident that when the shock on the upper surface reaches its maximum strength the one on thelower surface disappears, and vice versa. This kind of behavior can be better captured by the trainingsignals used in this work. An example of this kind of signal is given in Fig. 2.2. A detailed definitionof these signals is provided here. The so called random-like signal is generated by first reproducing afinite time white noise sequence with assigned root mean square and then a second shaping order

filter, of the type G(s) = ω20

s2 +2ξω0s +ω20

is applied to it, with ξ= 1 and ω0 dependent on the problem

under consideration. The resulting signal is a smooth function of time which persistently excites theamplitude and frequency range of interest.The noisy sweep signal is a classical frequency sweep signal disturbed in both amplitude and phase bya white noise with a small root mean square, defined by:

u(t ) = (A+wgnA

) · sin

(ωi +

ω f −ωi

Tt +φ+wgnφ

), t ≤ T (2.10)

Here A is the signal amplitude and wgnA is the related disturbance, ωi and ω f are the initial andfinal frequency of the signal, T is the simulation time, φ is the signal phase and wgnφ is the relateddisturbance.It might be thought that a signal similar to a limit cycle oscillation would help the ROM in identifying


0 2 4 6 8 10−0.1

−0.05

0

0.05

0.1

Time [s]

h/c

(a) Random-like signal

0 2 4 6 8 10

−0.1

−0.05

0

0.05

0.1

Time [s]

h/c

(b) Disturbed sweep signal

Figure 2.2: Sample of training signals capable of producing physically meaningful ROMs

t - τ [s]-20 -10 0 10 20

Cor

rela

tion

betw

een

h an

d θ

×10-4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 2.3: Cross correlation between plunge and pitch training signals in the two degree-of-freedomcase

this kind of behavior once coupled with a mechanical system. According to the results of this work,this is indeed the case, but only when such a signal contains a sufficient level of noise in amplitude andphase. In fact, it is found that a training signal which considers an input of harmonic oscillations fromthe aeroelastic LCOs leads to an erroneous identification of the system nonlinear behavior, alwaysresulting in unbounded, unstable aeroelastic responses, as if the identified system was linear. Thus arandom-like signal is selected in the following analyses. The excited frequencies and amplitudes canbe chosen a priori, selecting the lower and upper values of the frequency range and the root meansquare of the signal. Furthermore, using a white noise based signal we are assured that the time historyof the various input channels will not be correlated, as shown in Figure 2.3, increasing the chances ofobtaining meaningful identification results.

54 Chapter 2

Figure 2.4: Typical section

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

x/c

z/c

(a) Two degree-of-freedom model - mesh close-up

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

x/c

z/c

(b) Four degree-of-freedom model - mesh close-up

Figure 2.5: Different meshes used in CFD calculations

2.4 Test Cases

The previously presented technique is here applied to a plunging–pitching NACA 0012 airfoil, flyingat M∞ = 0.8 in air. The airfoil is equipped with leading and trailing edge control surfaces, their twohinges being placed at 15% and 75% chord respectively. A schematic representation of the system isdepicted in Fig. 2.4. The same structural model is employed in the following two examples, where atfirst the control surfaces are held fixed, reducing the number of structural degrees of freedom to two,while in the second case all of the airfoil parts are left free to move.The computational meshes used in CFD calculations are depicted in Fig. 2.5. After a convergenceanalysis based on static aerodynamic data, the C-type topology mesh around the airfoil is discretizedwith 30000 elements in the case of the two degree-of-freedom model, as shown in Fig. 2.5a. Regardingthe four degree-of-freedom case instead, the mesh around the airfoil is discretized with 32000 elements,increasing the cell density near the control surfaces hinges, in order to track the local surface motionwith higher accuracy, as shown in Fig. 2.5b. The system of ordinary differential equations governingthe dynamics of the aeroelastic system is given in Eq. 2.11, while the structural parameters are reportedin Table 2.1.

1 xθ xβLE xβTE

xθ r 2θ

j 2βLE

j 2βTE

xβLE j 2βLE

r 2βLE

0

xβTE j 2βTE

0 r 2βTE

hττ/bθττ

βLE,ττ

βTE,ττ

+

(ωh/ωθ)2 0 0 0

0 r 2θ

0 0

0 0(ωβ/ωθ

)2 0

0 0 0(ωβ/ωθ

)2

h/bθ

βLE

βTE

= Fext (2.11)

2.4. Test Cases 55

xθ r 2θ

ωh/ωθ µ xβLE xβTE jβLE jβTE r 2βLE

r 2βTE

ωβ/ωθ

0.25 0.75 0.5 75 0.0375 0.05 0 0 0.1125 0.15 3

Table 2.1: Structural parameters of the typical section

0.65 0.7 0.75 0.8 0.850

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

V

h LCO

/c

(a) Plunge trend

0.65 0.7 0.75 0.8 0.850

1

2

3

4

5

6

V

θLC

O [d

eg]

(b) Pitch trend

Figure 2.6: LCO trends computed by CFD-based simulations

with:

Fext = V 2

π

−CL(τ)2CM (τ)

2CMβLE(τ)

2CMβTE(τ)

+

0

r 2θ

00

θ0 (2.12)

The variable θ0 in Eq. 2.11 represents the static pre-twist in the torsional spring and it will be usedto test the robustness of the computed ROM outside the training region. The reduced velocityV = U∞/(ωθb

pµ) is chosen as the bifurcation parameter of the model. Beyond a critical value,

the linearized system will become unstable, and, because of the nonlinearities introduced by theaerodynamics, the response will eventually converge on stable limit cycle oscillation trajectories.

2.4.1 Two degree-of-freedom typical section

Before generating the signals used to train the ROM, several CFD-based aeroelastic analyses have beencarried out to better understand the physics behind the nonlinear behavior of this test case. It hasbeen determined that LCOs are brought into the system due to the large motion of strong shock wavesover the oscillating airfoil. The resulting oscillations increase in amplitude as the reduced velocity isincreased, as shown in Fig. 2.6. This growth in amplitude can be explained by analyzing the amplitudeof the shock oscillation, as reported in Fig. 2.7. It can be seen that the chord length swept by the shockis always larger than 13%c, indicating that the aerodynamics are introducing strong nonlinear effectsin the system. It is also clearly visible how the growth of the LCO amplitude depends almost linearly onthe length swept by the shock wave. However even though these two variables are in linear proportionto each other, the aerodynamic loads are nonlinear functions of the LCO amplitude as is expected for anonlinear system. Computing Fast Fourier Transforms of the obtained LCOs, it is then possible toanalyze the variation of frequency with the reduced velocity, as depicted in Fig. 2.8. As it can be noticed,

56 Chapter 2

10 15 20 25 300.1

0.15

0.2

0.25

0.3

0.35

h/c

Length swept by the shock wave [%c]

h/c = 0.0094*s

Linear regressionReal relationship

NOLCO

(a) Plunge dependence

10 15 20 25 300.03

0.04

0.05

0.06

0.07

0.08

θ [r

ad]

Length swept by the shock wave [%c]

θ = 0.0024*s

Linear regressionReal relationship

NO LCO

(b) Pitch dependence

Figure 2.7: Dependence of the LCO amplitude on the length swept by the shock wave

0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

k

FF

T a

mpl

itude

V = 0.70V = 0.725V = 0.75V = 0.775V = 0.80

(a) Plunge frequency trend

0.15 0.2 0.250

0.5

1

1.5

2

2.5

3

3.5

k

FF

T a

mpl

itude

[deg

]

V = 0.70V = 0.725V = 0.75V = 0.775V = 0.80

(b) Pitch frequency trend

Figure 2.8: LCO frequency trends computed by CFD-based simulations

the LCO frequency remains confined in a narrow range, e.g. k = 0.2−0.23, so it may be assumed that atraining signal designed to excite the aerodynamic system within such a frequency range will probablylead to a correct identification of this peculiar nonlinear behavior. The time required to fully developan LCO using standard CFD time marching methods is about 8 hours on 4 cores Intel r i5-3470 CPUunits running at 3.2 GHz on a desktop workstation. All the simulations are run with a physical timestep ∆tCFD = 10−3 seconds, using a 5th order accurate Runge-Kutta scheme. The convergence betweentime steps is accelerated by multigrid and dual time stepping methods.A training signal can now be designed based on the information collected through such preliminaryanalyses. The input-output pair used in the training stage is shown in Fig. 2.9. A zoom of the samesignal near its end is presented in Figure 2.10 to show its smooth behavior. It covers a broad frequencyspectrum, as reported in Fig. 2.11a. The structural motion used as training input has a ratio h/(cθ) = 3.5,and, as shown in Fig. 2.11b, this ratio remains almost constant in a wide range of flight speeds. Thetraining signal is generated by imposing the structural motion depicted in Fig. 2.9a to the CFD solver,thus providing the load history of Fig. 2.9b. The root mean square of the plunge and pitch degreesof freedom is set to 0.15 and 0.045 radians respectively, while the parameter ω0 of the smoothing

2.4. Test Cases 57

Time [s]0 5 10 15

Str

uctu

ral m

otio

n

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

h/cθ [rad]

(a) Input: structural motion

Time [s]0 5 10 15

Aer

odyn

amic

load

s-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

CL

CM

(b) Output: aerodynamic loads

Figure 2.9: Training signal

Time [s]19 19.2 19.4 19.6 19.8

Str

uctu

ral m

otio

n

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

h/cθ [rad]


Time [s]19 19.2 19.4 19.6 19.8

Aer

odyn

amic

load

s

-0.6

-0.4

-0.2

0

0.2

0.4

0.6C

LC

M


Figure 2.10: Close-up near the training signal end

58 Chapter 2

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

FF

T A

mpl

itude

k

(a) Fast Fourier transform of the plunge training signal

0.55 0.6 0.65 0.7 0.75 0.8 0.853

3.5

4

4.5

V

h/(c

θ)

(b) LCO amplitude ratio

Figure 2.11: Frequency and amplitude characteristics of the training signal

filter is set to a cut-off reduced frequency of k = 0.4. The resulting signal is quite long in time (20seconds) and it is used to perform a ROM convergence analysis. Several training periods are carriedout considering different time intervals of the same signal, requiring only one reference signal, andobtaining the required results in about 10 hours. Series of ROMs are then computed consideringan increasing number of aerodynamic states Na , from 5 to 12, and various training time intervals:5, 10, 15 and 20 seconds. Such a convergence analysis is carried out by assuming the bifurcationvelocity of the system as a target value, usually available from linearized analyses. In this case insteadsuch a value has been extrapolated from CFD analyses, resulting in Vbif = 0.64. Fig. 2.12a presentsthe convergence analysis performed on this test case. It is evident that as the length of the trainingsignal is increased, the convergence towards the reference value becomes smoother. For signals longerthan 5 seconds the results are very similar for a number of aerodynamic states greater than 10. InFig. 2.12b instead it is shown the time required to train a ROM on the different training signals. Noticethat the ROM training is performed on a single CPU of the previously mentioned workstation. Themaximum number of iterations allowed to the LM algorithm is limited to 50, with a converged costfunction threshold value set at 10−3. Both the input and the output have been normalized by theirmaximum value, to appropriately weigh the fitting errors. A reasonably good trade-off between theaccuracy and computational time required is achieved by ROMs trained on the 10 second long signal.The comparison between CFD and these ROM-based results is depicted in Fig. 2.13. It can be seenthat a reasonable match between the two methods is obtained for a number of states Na > 10. Thebest bifurcation point estimation is obtained by the ROM with Na = 12, which predicts Vbif = 0.635.Considering only the linear part of the same model, the computed bifurcation speed is Vbif = 0.629,therefore proving that the assumptions of Section 2.3.1 are valid in this case. After a convergenceanalysis, a number of states equal to nx = 4 and a number of hidden neurons equal to nh = 8 haveproven to be adequate in LCO predictions. For the details please see [ 253]. As can be seen, this ROMproduces accurate LCO amplitude trends. Finally, in Fig. 2.14 is shown the comparison betweenthe LCO frequency trends. Once again the ROM proves to be quite accurate in the prediction ofthe LCO main features. The results obtained by the ROM of [253] are almost identical. All ROM-based simulations are performed using an implicit, L-stable time integration method, with tunablenumerical dissipation [259], run with a time step ∆tROM = 10 ·∆tCFD, demonstrating the robustnessof the proposed approach against the time discretization. In conclusion to this analysis, Table 2.2

2.4. Test Cases 59

5 6 7 8 9 10 11 120.3

0.4

0.5

0.6

0.7

0.8

System order, Na

Bifu

rcat

ion

poin

t, V

bif

Ts = 5 [s]

Ts = 10 [s]

Ts = 15 [s]

Ts = 20 [s]

CFD value

(a) Convergence analysis based on the bifurcation point

0 5 10 15 20 250

1

2

3

4

5

6

7

Training signal length [s]

Ave

rage

trai

ning

tim

e [h

]

(b) Time required for the training

Figure 2.12: Convergence analysis for the two degree-of-freedom airfoil

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

V

h LCO

/c

CFD7 states9 states11 states12 statesROM, ref. 253

(a) Plunge trend

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

V

θLC

O [d

eg]

CFD7 states9 states11 states12 statesROM, ref. 253

(b) Pitch trend

Figure 2.13: Comparison of LCO amplitude trends. The training signal considered is 10-second long

reports a comparison of the time required to compute the LCO trends depicted in Fig. 2.13. Accordingto the obtained results, the time savings is substantial. What would take a CFD model an hour tocalculate, the ROM can do in 0.7 seconds, maintaining a reasonable accuracy. It can be noticed thatthe ROM of [253] requires a longer training time. This is due to the training approach used in that case,composed always by two stages but made up by a genetic algorithm and the LM method, because ofthe implicit nonlinear parameterization of the system [253]. Therefore, in comparison with the oneof [253], the ROM here presented is computed in a shorter time and permits the analyst to parameterizethe system in a much simpler way.In order to test the ROM robustness in aeroelastic applications, a non-null pre-twist pitch angle θ0

is now considered. The computation of LCO trends is performed at θ0 = 1 deg and θ0 = 2 deg. Giventhat the maximum LCO pitch angle experienced in the null pre-twist case was equal to 4 degrees, thecomputation of aeroelastic responses with these new pre-twist values means that we are consideringnew perturbed conditions, of 25% and 50% respectively, with respect to the nominal condition forwhich the ROM was originally computed. The results relative to θ0 = 1 deg are shown in Figure 2.15,while those obtained with θ0 = 2 deg are presented in Figure 2.16. Notice that the variables are now

60 Chapter 2

0.65 0.7 0.75 0.80.05

0.1

0.15

0.2

0.25

0.3

V

k

CFDROM − 12 statesROM, ref. 253

Figure 2.14: Comparison of LCO frequency trend. The training signal considered is 10-second long

CFD ROM ROM, ref. [253]

Training signal generation, hours - 10 10Training stage, hours - 2.5 4

Trend calculation, hours 8 × 8 points 0.0015 × 22 points 0.0012 × 22 points64 0.033 0.0264

Total, hours 64 12.533 14.0264

Table 2.2: Computational time required for computing the LCO trends, 2 degree-of-freedom typicalsection

∆h/c and ∆θ, computed as [max(LCO)−min(LCO)]/2, because the LCOs are no longer symmetric inthis case.Good results, comparable with the θ0 = 0 deg case, are obtained for θ0 = 1 deg. The amplitude trend

is tracked with a good accuracy by the ROM with Na = 12. Similar results are obtained with the ROMof [253]. Unfortunately, the same cannot be said for the case θ0 = 2 deg where the amplitude trendchanges dramatically in shape, resulting in an unstable LCO behavior [3], probably because of thestrong influence of the pre-twist angle. The ROM, being not able to change its behavior, producesan LCO trend similar to the previous cases. Qualitatively, the same results are obtained by the ROMof [253]. A possible solution to this problem could be a new training which takes into account a non-null mean pitch angle, but this is beyond the scope of the present analysis, and will not be pursued here.In conclusion, it can be stated that the ROM presents good robustness features around the nominalcondition for which it has been trained, producing accurate results at least up to a 25% perturbation inthe pre-twist angle.

In addition to these analyses, it is worth noting that, since a reduced order aeroelastic modelhas been tested so far, a reduced order aerodynamic model is available as well. So, if the analyst isnot only interested in evaluating nonlinear aeroelastic responses, but also in computing unsteadyaerodynamic loads given a prescribed motion, the proposed ROM is able to perform such a task. Atfirst, the aerodynamic ROM is excited by an harmonic input with small amplitude in both plungeand pitch degrees of freedom. The motion amplitude is set to h/c = 0.01 for the plunge and θ = 0.1deg for the pitch. The excitation reduced frequency is set to k = 0.6, which is not well excited by thetraining signal used before, as shown in Figure 2.11a. The computational time required for simulating1.5 seconds of physical time by the present CFD solver is in this case equal to 20 minutes. The ROM

2.4. Test Cases 61

V0.6 0.65 0.7 0.75 0.8 0.85

∆h/

c

0

0.05

0.1

0.15

0.2

0.25CFDROMROM, ref. 253

(a) Plunge trend

V0.6 0.65 0.7 0.75 0.8 0.85

∆θ [d

eg]

0

0.5

1

1.5

2

2.5

3

3.5

4CFDROMROM, ref. 253

(b) Pitch trend

Figure 2.15: Comparison of LCO amplitude trends. Pre-twist angle of 1 deg

V0.6 0.65 0.7 0.75 0.8 0.85

∆h/

c

0

0.05

0.1

0.15

0.2

0.25

0.3CFDROMROM, ref. 253

(a) Plunge trend

V0.6 0.65 0.7 0.75 0.8 0.85

∆θ [d

eg]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5CFDROMROM, ref. 253

(b) Pitch trend

Figure 2.16: Comparison of LCO amplitude trends. Pre-twist angle of 2 deg

62 Chapter 2

0.58 0.6 0.62 0.64 0.66 0.68 0.70

0.01

0.02

0.03

0.04

0.05

0.06

k

FF

T a

mpl

itude

CFDFirst stage − 6 statesFirst stage − 12 statesSecond stage − 6 statesSecond stage − 12 states

(a) Lift coefficient

0.58 0.6 0.62 0.64 0.66 0.68 0.70

2

4

6

8

10

x 10−3

k

FF

T a

mpl

itude

CFDFirst stage − 6 statesFirst stage − 12 statesSecond stage − 6 statesSecond stage − 12 states

(b) Moment coefficient

Figure 2.17: Aerodynamic loads FFT for small structural motion

is able to compute the same response in a few seconds. This test is performed to assess the ROMperformance for small input perturbations only, comparing the variation of the predicted resultsbetween the two training stages. The results in terms of lift and moment coefficients are presented inFigure 2.17, showing a close-up near the FFT peak. As it can be noticed from Figure 2.17a, very good,almost identical results are obtained both by linear (one training stage only) and nonlinear ROMsin the prediction of the lift coefficient. The same accuracy is obtained for the moment coefficient,as shown in Figure 2.17b. In this case however, some differences can be distinguished between thedifferent ROMs. Even though all of them present almost the same accuracy, the nonlinear ROM withNa = 12 is the one closest to the reference response, while the others present a convergent behaviortoward the reference CFD solution as the number of aerodynamic states is increased.Two conclusions can be drawn from these results. First, both linear and nonlinear ROMs presentgood accuracy when the input amplitude is small enough to assume the aerodynamic system as linear.Second, the nonlinear ROM always performs better than the linear ROM.The nonlinear performance of the ROM is evaluated by considering large amplitude structural input,always at the same frequency of the previous case. The motion amplitude is set to h/c = 0.3 for theplunge and θ = 5 deg for the pitch degree of freedom, both much greater than the maximum amplitudeexcited by the training signal shown in Figure 2.9. In this case the attention is focused on the momentcoefficient, because it is the one characterized by the strongest nonlinear behavior. The computationaltime required for simulating 1.5 seconds of physical time by the present CFD solver is about 3 hours.Thus, in this case the computational savings introduced by the ROM is substantial. The results arepresented in Figure 2.18, with a close-up near the FFT peaks. It can be noticed that, as the number ofstates is increased, the ROM response gets closer to the reference solution, even if small differencesstill appear. This is due to an insufficient excitation of these input amplitudes during the trainingphase. Notice however that the refined ROM is able to replicate accurately the different harmonics inthe response.The robustness of the proposed ROM in both linear and nonlinear regimes demonstrates that rea-sonably accurate results can be obtained by the linear ROM also, taking care of operating with smallstructural input.

2.4. Test Cases 63

0.6 0.65 0.70

0.1

0.2

0.3

0.4

0.5

k

FF

T a

mpl

itude

CFDSecond stage − 6 statesSecond stage − 12 states

(a) Close-up near the first harmonic response peak

1.8 1.85 1.90

0.02

0.04

0.06

0.08

0.1

k

FF

T a

mpl

itude


(b) Close-up near the third harmonic response peak

Figure 2.18: Moment coefficient FFT for large structural motion, two degree-of-freedom case

2.4.2 Four degree-of-freedom typical section

Even though very similar to the previous test case, this problem is presented to show the ability ofthe proposed ROM to deal with systems with a larger number of degrees of freedom, and to set thebasis for a nonlinear aeroservoelastic benchmark. In fact, the control surfaces may be used to suppressLCOs [260, 261].Assuming this case will present a nonlinear behavior similar to the previous one, the training signalshown in Figure 2.19 is given as input to the ROM training algorithm. Its frequency content is similarto the one presented in Figure 2.11a. The signals root mean square is set to 0.125 for the plunge, 0.0625radians for the pitch, 0.0125 radians for the leading edge control surface and 0.0250 radians for thetrailing edge control surface degrees of freedom. The parameter ω0 of the smoothing filter is set toobtain a reduced cut-off frequency of k = 0.4. As can be noticed from Figure 2.19b, the aerodynamicloads acting on the control surfaces are negligible compared to the lift and moment coefficients of theairfoil.As for the previous case, the ROM nonlinear performance is evaluated by considering large amplitudestructural input. The motion amplitude is set to h/c = 0.3 for the plunge, θ = 5 deg for the pitch, βLE =βT E = 8 deg for the leading and trailing edge degrees of freedom, much greater than the maximumamplitudes excited by the training signal shown in Figure 2.19. The excitation frequency is set tok = 0.7. Attention is focused again on the moment coefficient, because of its stronger nonlinearbehavior. Figure 2.20 depicts a close-up near the FFT peaks of the computed response. It can benoticed that as the number of states is increased, the ROM response gets closer to the referencesolution. Therefore, also in this case, the ROM shows good accuracy in predicting nonlinear unsteadyaerodynamic loads, even when both amplitude and frequency of the input signal are beyond the limitsconsidered during the training.

Then, coupling the aerodynamic reduced order model with the structural model, it is possibleto reproduce nonlinear aeroelastic responses. A convergence analysis is again carried out at thebifurcation point computed by the CFD code. Such a value results in Vbif = 0.635, being slightlydifferent with respect to the previous test case. In Fig. 2.21a such a convergence test is presentedgraphically. In this case, the training signal is 10 seconds long, which proved to be sufficient in theprevious section. The average training time is 6 hours, considerably longer than the 2 degree-of-freedom case, because of the increased size of the input-output data pair vectors. The smoothness

64 Chapter 2

Time [s]0 2 4 6 8

Str

uctu

ral m

otio

n

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

h/cθ [rad]β

LE [rad]

βTE

[rad]


Time [s]0 2 4 6 8

Aer

odyn

amic

load

s-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

CL

CM

CM, β

LEC

M, βTE


Figure 2.19: Training signal for the four degree-of-freedom typical section

0.69 0.7 0.71 0.72 0.73 0.740

0.1

0.2

0.3

0.4

0.5

0.6

k

FF

T a

mpl

itude


(a) Close-up near the first harmonic response peak

2.06 2.08 2.1 2.12 2.14 2.160

5

10

15

x 10−3

k

FF

T a

mpl

itude


(b) Close-up near the third harmonic response peak

Figure 2.20: Moment coefficient FFT for large structural motion, four degree-of-freedom case

2.4. Test Cases 65

4 6 8 10 120.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

System order, Na

Bifu

rcat

ion

poin

t, V

bif

ROMCFD

(a) Convergence analysis based on the bifurcation point

V0.55 0.6 0.65 0.7

g

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04 PitchPlungeTrailing edge CSLeading edge CSBifurcation point

(b) Aeroelastic V-g diagram, Na = 12

Figure 2.21: Stability properties of the linearized system

of the present convergence analysis, much more evident than for the previous test case, should benoticed. A sample of ROM-based aeroelastic V-g curves is shown in Fig. 2.21b, where it is clear thatin this particular case the two control surfaces are only lightly influenced by the aerodynamic forcescompared to the pitch and plunge degrees of freedom. A zoom near the flutter point is provided tomake clearer the instability mechanism, which resembles the classical plunge-pitch flutter [88].A comparison between the CFD and a 12 states ROM-based simulation at V = 0.74 is presented inFig. 2.22. As it can be seen, the difference between the two results is quite small, showing that thereduced order model is able to capture the basic nonlinear features of the response. It is also possibleto perform trend analyses to understand how the variations in flight speed influence the limit cycleoscillation amplitude and frequency. The results are shown in Fig. 2.23 and 2.24, where the outcomesof the ROM of [253] are also presented. It is evident from Fig. 2.23 that increasing the number of stateslead to a better correlation between CFD and ROM-based results. However, it is interesting to notethat the results with Na = 6 present the same trend of the CFD, slightly shifted to the left. Instead, theresults with Na = 12, even if showing a smaller overall error, exhibit a local trend slope quite differentfrom the reference data. From a design point of view, the results obtained with Na = 6 would bepreferred, because they are conservative (they exhibit a lower bifurcation speed) and depict the sameshape of the reference curve. Such differences may be considered as irrelevant, since all the reducedorder models capture the essential features of the system behavior. However, the ROM with Na = 12produces the best estimation of the bifurcation point, predicted to be Vbif = 0.64, and also a moreaccurate prediction of the LCO frequency. Considering only the linear part of the same model, thecomputed bifurcation speed is Vbif = 0.647, therefore proving that the assumptions of Section 2.3.1are valid in this case also. The ROM of [253] presents qualitatively the same behavior of the ROM withNa = 12, always overestimating both LCO amplitude and frequency. Such a ROM is characterized bythe following parameters: nx = 8 and nh = 10, for the details please see [253].As a final result, Table 2.3 reports a comparison of the times required to compute the previous LCOtrends. The savings are substantial, confirming again the convenience of developing ROMs from themore expensive simulations, especially in the case of optimization analyses or control law designs.Note that even if in this case the ratio between the two total times is about 3.5, such a result wouldgreatly improve as the number of analyses to be carried out is increased. As before, the ROM of [ 253]requires a longer training time, and the same conclusions drawn in the previous case are still valid.

Smooth aerodynamic nonlinearities have been considered so far, and their main effect if the

66 Chapter 2

160 170 180 190 200 210 220 230−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

τ

h/c

ROMCFD

(a) Plunge

160 170 180 190 200 210 220 230−3

−2

−1

0

1

2

3

τ

θ [d

eg]

ROMCFD

(b) Pitch

160 170 180 190 200 210 220 230−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

τ

βLE

[deg

]

ROMCFD

(c) Leading edge deflection

160 170 180 190 200 210 220 230−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

τ

βT

E [d

eg]

ROMCFD

(d) Trailing edge deflection

Figure 2.22: Comparison between CFD and ROM-based aeroelastic simulations, V = 0.74, Na = 12

CFD ROM ROM, ref. [253]

Training signal generation, hours - 12 12Training stage, hours - 6 9

Trend calculation, hours 10 × 7 points 0.0028 × 15 points 0.0023 × 15 points70 0.042 0.0345

Total, hours 70 18.042 21.0345

Table 2.3: Computational time required for computing the LCO trends, 4 degree-of-freedom typicalsection

2.4. Test Cases 67

0.6 0.65 0.7 0.75 0.80

0.05

0.1

0.15

0.2

0.25

0.3

V

h LCO

/c

CFD6 states12 statesROM, ref. 253

(a) Plunge trend

0.6 0.65 0.7 0.75 0.80

0.5

1

1.5

2

2.5

3

3.5

V

θLC

O [d

eg]


(b) Pitch trend

0.6 0.65 0.7 0.75 0.80

0.1

0.2

0.3

0.4

0.5

V

βLE

, LC

O [d

eg]


(c) Leading edge deflection trend

0.6 0.65 0.7 0.75 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

V

βT

E, L

CO

[deg

]


(d) Trailing edge deflection trend

Figure 2.23: LCO amplitude trends comparison

0.65 0.7 0.75 0.80.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

V

k

CFDROMROM, ref. 253

Figure 2.24: LCO frequency trends comparison between CFD and 12 states ROM-based simulations

68 Chapter 2

generation of limit cycle oscillations beyond the bifurcation point. The scope of the next section isto introduce also structural nonlinearities, such as free-play, in the system simulations studying theresulting response and comparing it with the LCOs obtained in this chapter.

2.4.3 Transonic aeroelasticity in presence of free-plays

Having validated the ROM versus its high-fidelity aerodynamic counterpart, structural nonlinearitiessuch as free-play are now introduced into the system. The resulting behavior is found to be morecomplex than that computed by considering only aerodynamic nonlinearities. While aerodynamicnonlinearities shape limit cycles beyond the system bifurcation point, the free-play influences theresponse at lower speeds also, making the system behavior bi-stable. Trend analyses are performed tocompare the results of the compact ROM with those coming out from CFD-based simulations.

In practice, a free-play affected actuation law is introduced in the system represented by Eq. 2.11,modifying it in the following way:

1 xθ xβLE xβTE

xθ r 2θ

j 2βLE

j 2βTE

xβLE j 2βLE

r 2βLE

0

xβTE j 2βTE

0 r 2βTE

hττ/bθττ

βLE,ττ

βTE,ττ

+

(ωh/ωθ)2 0 0 0

0 r 2θ

0 00 0 0 00 0 0 0

h/bθ

βLE

βTE

= Fext (2.13a)

with:

Fext = V 2

π

−CL(τ)2CM (τ)

2CMβLE(τ)

2CMβTE(τ)

+ (ωβ/ωθ

)2

00

f (βLE,βc ,α)f (βTE,βc ,α)

+η(ωβ/ωθ

)2 sign(β)

00

f (βLE,βc ,α)f (βTE,βc ,α)

(2.13b)

In Eq. 2.13b, the first term is the aerodynamic loads forcing contribution, the second is related to thefree-play effect, while the third is the effect of friction, which is scaled by η to be a percentage of theactual hinge moment coming from the free-play. The nonlinear function f (β,βc ,α) is so defined:

f (β,βc ,α) =

βc −β+α βc −β<−α0 |βc −β| ≤α

βc −β−α βc −β>α

(2.13c)

where α is the free-play semi-width and βc the commanded aileron position. A sketch of the presentproblem is depicted in Figure 2.25a, where the presence of a nonlinear actuation system is highlighted.In the present results, the effects of actuator dynamics are neglected, but they will be accounted forwhen a LCO suppression system will be designed.

In the following results the free-play width is first set to α = 0.1 deg for both control surfaces.Limit cycle trends with respect of variations of flight speeds are computed and sample responsescomputed by the ROM are compared by those computed through CFD simulations. The parameterη, representing the fraction of friction acting on the control surfaces hinge is set to 0.15, being this areasonable value of friction level.First of all, the effect of various forcing inputs on the response is considered. Pulses of differentamplitude, as exemplified in Figure 2.25b are given in input to the system through the trailing edgeaileron and the related response is computed, registering the resulting LCO amplitude. The computedLCO trends are displayed in Figure 2.26, where the obtained LCO amplitude is represented as a fuctionof the nondimensional flight speed and of the pulse amplitudes. As can be noticed from these results,

2.4. Test Cases 69

(a) 4-DOF typical section with free-play influenced actu-ation system

τ0 10 20 30 40 50 60

βpu

lse [d

eg]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(b) Pulse disturbances considered in the simulations

Figure 2.25: Model and disturbances used in this section

0.80.75

0.7

V0.650.61

2β [deg]

34

0.2

0.15

0.1

0.05

05

h/c LC

O

(a) Plunge trend

0.80.75

0.7

V0.650.61

2β [deg]

34

2.5

2

1.5

1

0.5

05

θLC

O [d

eg]

(b) Pitch trend

0.80.75

0.7

V0.650.61

2β [deg]

34

0.1

0.15

0.2

0.25

0.3

0.35

0.4

5

βLE

, LC

O [d

eg]


0.80.75

0.7

V0.650.61

2β [deg]

34

0.6

0.5

0.4

0.3

0.2

0.15

βT

E, L

CO

[deg

]


Figure 2.26: LCO amplitude trends with free-play and friction in the control surfaces actuation system

70 Chapter 2

depending on the flight speed and disturbance amplitude, the computed LCOs can be of differentnature. At low speeds, small amplitude LCOs are experienced because of the free-play presence. Whenthe flight speed is slightly increased, LCOs are still driven by the free-play until the disturbance reachesand amplitude greater than 3 degrees. At higher flight speeds, even a small disturbance results in a largeamplitude LCO driven by a large shock wave motion. It is therefore clear that in this case the systempresents a bi-stable behavior, where its response can be driven alternatively by the free-play, producingsmall amplitude but high frequency LCO, or by smooth aerodynamic nonlinearities, resulting in lowfrequency and large amplitude motions.

Another interesting fact is the presence of an hysteretic effect in the generation of LCO solutions. Infact, simulating a piecewise constant wind speed profile starting from speeds where LCOs are driven bythe shock wave motion and then slowing down until the free-play effect becomes dominant, it is foundout that small amplitude LCOs are obtained at a lower speed than in a case where the flight speed isincreased. These results are highlighted in Figure 2.27, where the label ’Slowdown test’ represents theoutcomes of this last numerical experiment. These results are a further proof that the combined actionof aerodynamic and structural nonlinearities leads to more complex and interesting behaviors of theaeroelastic system. Furthermore, it can be said that such an hysteretic effect is due to the free-play. Infact, the same type of simulation performed considering only aerodynamic nonlinearities would resultin the same trend computed with piecewise increments of the flight speed.

To consolidate these results, all of them provided the reduced order model presented in thischapter, a comparison with the outcomes of CFD-based simulations is provided. Both free-playand aerodynamically driven LCOs are computed, and a sample of the obtained results is shown inFigures 2.28 and 2.29 for a flight speed of V = 0.75. As can be noticed from these figures, the reducedorder model is still able to simulate correctly the dynamic response, even in presence of structuralnonlinearities. Nonetheless, it seems that aerodynamically induced LCOs are captured with a higherfidelity than those driven by the free-play, which anyway are still reproduced with a small amplitudeerror.

In conclusion, in this chapter a hybrid linear-neural ROM has been presented and tested for bothaerodynamic and aeroelastic applications. A general procedure for the design of the training signalshas been discussed and detailed.It seems that such a ROM is able to reproduce correctly nonlinear aeroelastic responses also in presenceof structural nonlinearities, such as free-play and structural friction. Thanks to these results, the ROMwill be used for the design of LCO suppression laws in the next chapters.

2.4. Test Cases 71

V0.6 0.65 0.7 0.75 0.8

hLC

O/c

0

0.05

0.1

0.15

0.2

0.25

0.3No free-playβ = 5 [deg]Slowdown test

(a) Plunge trend

V0.6 0.65 0.7 0.75 0.8

θLC

O [d

eg]

0

0.5

1

1.5

2

2.5

3

3.5

4 No free-playβ = 5 [deg]Slowdown test

(b) Pitch trend

V0.6 0.65 0.7 0.75 0.8

βLE

, LC

O [d

eg]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 No free-playβ = 5 [deg]Slowdown test


V0.6 0.65 0.7 0.75 0.8

βT

E, L

CO

[deg

]

0

0.2

0.4

0.6

0.8

1

No free-playβ = 5 [deg]Slowdown test


Figure 2.27: Comparison between numerical flight speed climbing and descent

τ

500 520 540 560 580 600

θ [d

eg]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25CFDROM

(a) Pitch response

τ500 520 540 560 580 600

βT

E [d

eg]

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2CFDROM

(b) Trailing edge response

Figure 2.28: Comparison of free-play driven LCOs

72 Chapter 2

τ

500 520 540 560 580 600

θ [d

eg]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2CFDROM

(a) Pitch response

τ500 520 540 560 580 600

βT

E [d

eg]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6CFDROM

(b) Trailing edge response

Figure 2.29: Comparison of aerodynamically driven LCOs

Structural Nonlinearities Compensation

CHAPTER 3


As mentioned in the introduction, the presence of structural nonlinearities, such as free-play, frictionand hysteresis, influences considerably the response of dynamic systems. For example, in high-precision positioning devices, the presence of a free-play between adjacent movable parts may jeop-ardize the performance of the control system [262, 263]. In more complex cases, e.g. in aeroelasticproblems, where the aerodynamic forces interact with the structural response, this kind of nonlineari-ties can lead to unstable phenomena and limit cycle oscillations [19, 26, 264].In this chapter, substantial attention will be given to the study and suppression of free-plays in actuat-ing systems, but the same approach can be extended to the compensation of friction and hysteresis, aswill be detailed later on.

A free-play can appear in mechanical systems where a driving member, i.e. the motor, is notdirectly connected with the driven member, i.e. the load. This is the case of many mechanical systems,such as printing presses, cars drive trains and industrial robots. As can be clear to understand, whenthe free-play is open, i.e. there is not a direct connection between motor and load, the motion of thislatter is autonomous, and in addition, the torque generated by the motor drives only the motor itself,and not the load.

The control of systems with free-play has been subject of studies since the 1940s. Of course linearcontrollers were first investigated, including P, PI, PID and full state feedback. In those years, the mainanalytical tool used to describe this nonlinearity has been the describing function method [265]. Inmore recent years, observer-based, fuzzy logic, neural networks and sliding mode controllers have alsobeen proposed.The preload, i.e. an approximate inverse of the free-play, has also be suggested as a remedy, both innon-adaptive and adaptive settings.Mechanical solutions have also been implemented, such as the so called anti-free-play gear thatcontains two cog wheels on the motor side connected with a stiff spring such that the free-play gap isalways closed. The price for this solution is the appearance of an additional resonance that limits theachievable closed-loop bandwidth [262].

Nonetheless, there is still a lack of meaningful results in aerospace applications regarding thesuppression of free-play effects and limit cycle oscillations of movable surfaces and flexible structures.In this effort, two control strategies are proposed. One is based on a classical approach that tackles theproblem through PID controllers combined into a proper control architecture. Because PID controllersare commonly used in practice, this methodology would permit a direct application to industrialproblems.The originality of the present control solution is its tuning. In fact, a general purpose frequency-basedtuning optimization will be presented, which, because of its data-driven nature, permits to deal withnonlinear systems also. Such a tuning algorithm requires only a one shot simulation/experiment,where the input and output variables are measured. As will be seen, starting from these data, the

73

74 Chapter 3

Figure 3.1: Open loop system with free-play between motor and load

control system is tuned to minimize the differences between a virtual closed loop response and areference one.The second approach proposed is based on the adaptive inversion of the free-play nonlinearity. Thismethod is simple to understand and allows an estimation of disturbances deriving from frictionnonlinearities.

As detailed in the following, these two controllers are not mutually exclusive, but they can becombined to improve the control system performance. In fact, the approach followed in this chapterwill be the following. At first, the PID-based controller will be designed and implemented. If this wouldprove to be robust enough, then the adaptive solution will not be considered. Instead, when the linearapproach will not be sufficient to suppress the undesired effects introduced by the free-play, then theadaptive controller will be combined into the control system to improve the closed loop behavior. Thisfact will be even more clear when aerodynamic nonlinearities will also be controlled.

3.2 Dual-Loop PID Architecture

Driven by previous experiences in the design of actuation systems for wind tunnel models, i.e. [ 266],a classical PID architecture is chosen. Such a controller relates the control action u(t) to the errorbetween the current output y(t ) and the reference signal r (t ), through the following law:

u(s) =(KP + K I

s+KD

N s

s +N

)e(s) (3.1)

where the closed loop relation is written directly in its classical frequency domain formulation. Theparameters KP , KD and K I are the proportional, derivative and integral gains respectively. To guaranteethat the realization of the controller does not deteriorate the effect of its derivative contribution, a filterwith cut-off frequency N /2π is included.A simplified model of a classical actuation system in open loop is depicted in Figure 3.1. Using thevariables defined in such a figure, a system in closed loop should be able to follow the referencecommand r (t) through the load motion θl . Unfortunately, the classical approach with the directapplication of the PID of Eq. 3.1 on the motor rotation would result in a poor tracking because of thefree-play presence, as highlighted by Figure 3.2. Here, the benchmark system that will be defined inthe following sections is considered. As clear from Figure 3.2a, when only the motor position is fedback to the PID, the system is not only incapable of following the reference signal, but it also exhibitslimit cycle oscillations. This phenomenon is due to the fact that when the free-play gap is open, theload is driven by its own inertia and continue to move until the gap is closed. However, even whenthe gap is about to close, the related process does not happen smoothly because the controller has noinformation about the load state. This results in a vibrating phenomenon that in absence of frictionpresents no damping.

In order to overcome such an inadequacy, an appropriate solution have to be devised. Thanks toits wide application in industrial high precision tracking, a dual-loop strategy based on PID controllersis here considered first [267]. This architecture employs two position sensors, one at the motor side,

3.2. Dual-Loop PID Architecture 75

Time [s]0 0.5 1 1.5 2 2.5 3 3.5

Def

lect

ion

[deg

]

-2

0

2

4

6

8

10

12

ReferenceMotorLoad

(a) Comparison between motor, load and reference posi-tions when the PID is applied on the motor side only

Time [s]0 0.5 1 1.5 2 2.5 3 3.5

Inpu

t tor

que

[Nm

]

-80

-60

-40

-20

0

20

40

60

80

100

(b) Related control input

Figure 3.2: Sample of poor tracking due to free-play

Figure 3.3: Dual loop controller.

the other at the load, as depicted in Figure 3.3. It permits to tune the motor (inner) loop as a directtorque-based PD controller, making the actuator sufficiently fast to follow abrupt speed changes, e.g.the opening of the free-play gap. The load (outer) loop is instead a PID position controller, required toassure the desired positioning precision within a reasonable bandwidth.Thanks to the feedback of θl , this architecture would permit an accurate tracking of the reference

signal also in presence of unsteady disturbances, such as acting aerodynamic loads, as will be beshown in the following applications.

Within such a framework, the contribution of this work is the optimization procedure adopted totune the above mentioned architecture, as detailed in the following section.

3.2.1 Frequency-based virtual reference tuning

The procedure used to tune the parameters of the two controllers involved in the dual loop architectureof Figure 3.3 takes inspiration from the works [268, 269, 270, 271]. Such methods are based on thegeneration of a fictitious reference signal, which is generated recursively given a set of one-shot compu-tational/experimental input-output data. This is said to be a data-driven approach. In fact, it does notattempt to identify the plant model, instead it uses the data produced by the plant to find a controller,which generally is meant to minimize some control performance criterion. Nonetheless, despitethe original idea of data-driven approaches, the closed loop transfer function can be experimentallyidentified through the collected data [271].

76 Chapter 3

Figure 3.4: Generic closed loop system.

The main difference of the present procedure with respect to the cited methods, which both workin the time domain, is the fact that the training algorithm is written directly in the frequency domainthanks to the linearity of the control system, composed by PID regulators. As will be proven in thefollowing, the mentioned approaches produce very similar results even if they are formulated indifferent frameworks.

According to [272], if a controller C(s;θ) results in a closed-loop system whose transfer function isM(s), then, when the closed-loop system is fed by any reference signal r (s), its output equals M(s)r (s)in the frequency domain. Hence, a necessary condition for the closed-loop system to have the sametransfer function as the target model is that the output of the two systems is the same for a givenreference.

Standard modern reference design methods try to impose such a necessary condition by firstselecting a reference r (t) and then by choosing C(s;θ) such that the condition is satisfied. However,for a general selection of r (t), the above task is difficult to accomplish if a model of the plant is notavailable. The basic idea of the present approaches is to perform a wise selection of the referencesignal so that the determination of the controller becomes easily achievable.

Let us first analyze the different methods and then highlight their similarities. The procedures willbe then adapted to the present control architecture.

Virtual Reference Feedback Tuning

The Virtual Reference Feedback Tuning (VRFT) was first developed in [270] and then formalizedin [272]. As previously mentioned, this approach selects a reference signal on the based of a targetsystem transfer function. This signal is then used to force the closed loop system to behave like thetarget.Using the symbols defined in Figure 3.4, suppose that the controller belongs to the class C(s;θ),where θ is the vector of parameters describing the controller. The control objective is to minimize thefollowing cost function:

JVRFT =∫ ∞

0

(P (s)C(s;θ)

1+P (s)C(s;θ)−M(s)

)2

W 2(s)ds (3.2)

where W (s) is a weight function chosen by the designer. Obviously, such a mathematical problemcannot be solved when the plant model P (s) is not known. In [272], the authors found a mathematicaltrick, used also in other contexts, to overcome such an issue.

The main idea of the method is outlined as follows. After having collected input-output timedomain data, coming from a simulation or an experiment, a reference signal rθ(t ) is considered suchthat M(s)rθ(s) = y(s). The reference rθ(t) is virtual, in fact it is not used to generate y(t). Such anoutput signal will be the desired output of the closed loop system when the reference signal is rθ(t ).Computing the tracking error e(t) = rθ(t)− y(t), even if the plant P (s) is not known, it is actuallycertain that when P (s) is fed by u(t ) it produces y(t ). Therefore, the desired controller is the one thatgenerates u(t ) when fed by e(t ). Because both signals e(t ) and u(t ) are both known, the identificationof the searched controller reduces to the identification of the dynamic relationship between these two

3.2. Dual-Loop PID Architecture 77

signals.The tuning procedure that implements this idea is summarized here below:

1. The collected input-output time domain data (u(t ) and y(t )), obtained through initial trial closedloop measurements, are transformed in the frequency domain through a Fast Fourier Transform(FFT). Such signals will be referred as U ( jω) and Y ( jω)

2. A virtual reference is computed such that Y ( jω) = M( jω)Rθ( jω), i.e. Rθ( jω) = M−1( jω)Y ( jω),where the function M(s) is a target transfer function opportunely selected

3. A tracking error is defined E( jω) = Y ( jω)−M( jω)Rθ( jω)

4. A filter operator L( jω) is applied to E( jω) and U ( jω), i.e. E( jω) = L( jω)E( jω) and U ( jω) =L( jω)U ( jω)

5. The controller parameter vector θ is then designed by minimizing the cost function:

J ′VRFT(θ) = 1

2 N f

N f∑k=1

(U ( jωk )−C( jωk ,θ)E( jωk )

)2(3.3)

where N f is the number of computed frequencies. As described in [272], the filter operator is defined

as L(s) = (1−M(s)) M(s)W (s)

φu, where φu is the spectral density of u(t). Accordingly to the authors’

comments and demonstration, such a filter function optimizes the approximation of Eq. 3.2 by meansof Eq. 3.3.

Because the input-output data is available at several frequencies, a nonlinear least-square opti-mizer, such as the Levenberg-Marquardt method [113], is chosen to solve the minimization problemdescribed by Eq. 3.3. With this method, the problem is solved compactly using matrix-vector multipli-cations.At first the residual ε( jω,θ) = U ( jω)−C( jω,θ)E( jω) is collocated at each computed frequency andstacked into a vector:

ε(θ) =

U ( jω1)−C( jω1,θ)E( jω1)U ( jω2)−C( jω2,θ)E( jω2)

...U ( jωN f )−C( jωN f ,θ)E( jωN f )

(3.4)

Then the Jacobian matrix of ε with respect to θ has to be determined. Thanks to the structure of thePID controller presented by Eq. 3.1, such a quantity can be easily computed:

C( jω,θ) = θ1 + θ2

jω+θ3

jωθ4

jω+θ4−→ ∂ε( jω,θ)

∂θ=−

[1

1

jω

jωθ4

jω+θ4

−ω2θ3(jω+θ4

)2

]E( jω) (3.5)

Again, ∂ε/∂θ is collocated at each computed frequency and stacked in a matrix:

J(θ) =−

[1

1

jω1

jω1θ4

jω1 +θ4

−ω21θ3(

jω1 +θ4)2

]E( jω1)

[1

1

jω2

jω2θ4

jω2 +θ4

−ω22θ3(

jω2 +θ4)2

]E( jω2)

...[1

1

jωN f

jωN f θ4

jωN f +θ4

−ω2N f

θ3(jωN f +θ4

)2

]E( jωN f )

(3.6)

78 Chapter 3

The update of the controller parameter vector θ is obtained through the nonlinear least square solutionof:

∆θ =−(JT J

)−1JT ε (3.7)

and the procedure is iterated until a sufficient small variation ofθ is obtained between two optimizationsteps.

Fictitious Reference Iterative Tuning

The Fictitious Reference Iterative Tuning (FRIT) [271] is an automatic tuning method that uses onlyone-shot experiment for the sake of reducing the cost and time required for arriving at the optimumparameters of the controller at hand. The similarities with the VRFT are many and will be highlightedin the following paragraphs. Differently from the VRFT, where the reference signal was specified beforestarting the optimization, in this case a virtual reference signal is computed recursively during theiterations, as its name may suggest. However, also in this case only a single experiment/simulation isrequired to tune the controller.

The tuning procedure is still proposed in the frequency domain, being this the main originalcontribution to the already known formulation. Always following the nomenclature of Figure 3.4, thenecessary steps are outlined here:

• The collected input-output time domain data (u(t ) and y(t )), obtained through first trial closedloop measurements, are transformed in the frequency domain through an FFT. Such signals willbe referred as U ( jω) and Y ( jω)

• The fictitious reference is computed as:

Rθ( jω;θ) = U ( jω)+C( jω,θ)Y ( jω)

C( jω,θ)(3.8)

• The fictitious error can be defined as: E( jω) = Y ( jω)−Rθ( jω;θ)

• The cost function to be minimized is:

JFRIT(θ) = 1

2 N f

N f∑k=1

E 2( jωk ;θ) (3.9)

Again, because of the nature of the problem, a nonlinear least-square optimizer, such as theLevenberg-Marquardt method, is chosen to solve the minimization problem described by Eq. 3.9.Therefore, the residual vector ε(θ) is defined after collocating the fictitious error at each frequency:

ε(θ) =

Y ( jω1)−Rθ( jω1;θ)Y ( jω2)−Rθ( jω2;θ)

...Y ( jωN f )−Rθ( jωN f ;θ)

(3.10)

The related Jacobian matrix can be easily determined thanks to the structure of the PID controllerpresented by Eq. 3.1. In fact ∂C( jω,θ)/∂θ was already computed in 3.5, and the Jacobian matrix can be

3.3. Adaptive Control through Nonlinearity Inversion 79

derived from Eq. 3.8 and assembled collocating the result at each frequency:

J(θ) =

[1

1

jω1

jω1θ4

jω1 +θ4

−ω21θ3(

jω1 +θ4)2

]M( jω1)U ( jω1)

C2( jω1,θ)

[1

1

jω2

jω2θ4

jω2 +θ4

−ω22θ3(

jω2 +θ4)2

]M( jω2)U ( jω2)

C2( jω2,θ)...[

11

jωN f

jωN f θ4

jωN f +θ4

−ω2N f

θ3(jωN f +θ4

)2

]M( jωN f )U ( jωN f )

C2( jωN f ,θ)

(3.11)

The update of the controller parameter vector θ is again obtained through the nonlinear least squaresolution of:

∆θ =−(JT J

)−1JT ε (3.12)

and the procedure is iterated until a sufficient small variation ofθ is obtained between two optimizationsteps.

Let us now analyze the similarities and differences between the two approaches presented. First ofall, both methods requires only a one-shot experiment to compute the controller parameters. This is agreat difference with respect to other data-based methods, like the iterative feedback tuning [273].On the other hand, they can be interpreted as different ways to formulate the optimization problempermitting to compute the controller parameters. In the case of the VRFT, it is the difference betweenthe real input and the expected one to be minimized. Because of this fact, the VRFT is said to transforma control problem into an identification one. Regarding the FRIT instead, the optimization methodminimizes the real output of the system and the one expected in closed loop. Because of this differentformulation, the VRFT requires a filter L(s) to approximate the original problem, represented by Eq. 3.2,that would be unsolvable otherwise. On the other hand the FRIT does not require any particularapproximation to solve the problem outlined by Eq. 3.9.Both methods are very flexible, in the sense that they could be employed for any controller structure,see for example [274] in the case of cascade controllers or [275, 276] for the tuning of neural con-trollers. Moreover, both method have been employed for tuning the so-called 2-degree-of-freedomcontroller [277, 278].

As will be seen in the following section, even if the two methods approach the optimization problemfrom different points of view, the resulting controller will produce similar closed loop responses. Inthis way it would be proven that such tuning algorithms produce meaningful results when a quickdetermination of non-trivial control architectures is required.

3.3 Adaptive Control through Nonlinearity Inversion

When strong nonlinear behaviors characterize the dynamic response of an aeroelastic system, classicallinear controllers may be no longer sufficient for compensating free-play effects. This could be due toarising difficulty of designing a linear controller on a nonlinear system, or to the accidental excitementof other nonlinearities, such as input saturations, which can further alter the system response.

In these cases, nonlinear controllers can be devised to improve the closed loop performance.Because of the intrinsic ’hard’ nonlinear nature of free-play and friction, discontinuous laws suchthose provided by a sliding mode controller may be employed [ 279]. Nonetheless, the application of

80 Chapter 3

Figure 3.5: Two-mass system with free-play

discontinuous controls in mechanical systems can lead to a faster degradation and to the resultingwear of the "connections" of the moving components.

Less aggressive approaches should be taken when tackling such a problem. In [262] for example, atorque observer was used in combination with a classical PID controller applied on the motor sideonly. Stable and robust results were obtained, and such outcomes will be used in the first test casestudied in this chapter for a realistic comparison. Other methodologies have been considered in recentyears, such as controller based on fuzzy logic [280].Nevertheless, the control method that seems to have introduced the real breakthrough in the com-pensation of free-play effects is the one based on the simple idea of inverting the nonlinear relationthat best approximates such system nonlinearities. The idea was first proposed by Tao and Kokotovicin [281] and then proposed again in various applications in [282, 283, 284], eventually extending it tomultiple-input, multiple-output applications [285].

Following the essential idea proposed in [281], free-play and friction are assumed to be knownin their modeling structure but unknown in the related parameters, and an on-line process updatesa set of evolving parameters in order to estimate the free-play semi-width α, the equivalent controlchain stiffness ks representing the compliance of the actuation system and a rough value of the frictionamplitude T f .The algorithm here proposed is based on some basic ideas from [211], but adopts a different parametriza-tion of the nonlinear torque resulting from free-play and friction, which permits to identify the un-known parameters separately. Furthermore, differently than [211], the problem is formulated in adiscrete time framework, thus achieving an immediate digital implementation. Following the nomen-clature introduced by Figure 3.5, the transmission torque can be written in the following form:

Ts = TF P +T f +w = ks (∆θ+α)νl +ks (∆θ−α)νr + T f sign(∆θ

)+w (3.13)

where ∆θ = θm −θl , w is the assumed measurement noise while:

νl =

1 for ∆θ <−α0 otherwise

(3.14a)

and

νr =

1 for ∆θ >α

0 otherwise(3.14b)

3.3. Adaptive Control through Nonlinearity Inversion 81

Since ks , α and T f in Eq. 3.13 are assumed to be unknown, they can be collected in the unknown vectorp =

p1 p2 p3=

ks α T f. Eq. 3.13 can then be rewritten as:

Ts = p1(∆θ+p2

)νl +p1

(∆θ−p2

)νr +p3sign

(∆θ

)+w (3.15)

where of course the variable α is substituted by p2 in the definition of νl and νr in Eqs. 3.14.The vector p is updated at each sampling instant by means of a simple gradient descent algo-

rithm [219]:

pn+1 = pn −ηGD ∂en

∂pn(3.16)

where ηGD is the classical learning rate, while en is the instantaneous output error, defined as:

en = T meass,n −Ts,n

= T meass,n −p1,n

(∆θn +p2,n

)νl ,n −p1,n

(∆θn −p2,n

)νr,n −p3,nsign

(∆θn

)+w(3.17)

T meass is the measured torque.

Because the derivative ∂en/∂pn is required by Eq. 3.16, it is provided here by its explicit definition,which can be directly obtained from Eq. 3.17:

∂en

∂p1,n=−(

∆θn +p2,n)νl ,n − (

∆θn −p2,n)νr,n =−∆θn

(νl ,n +νr,n

)+p2,n(νr,n −νl ,n

)∂en

∂p2,n=−p1,nνl ,n +p1,nνr,n = p1,n

(νr,n −νl ,n

)∂en

∂p3,n=−sign

(∆θn

)(3.18)

Following the approach proposed in [211], the desired transmission torque can be computed as:

T dess,n = p1,n∆θ

idealn (3.19)

where ∆θidealn is the ideal response of the interface between motor and load, i.e. α = 0, resulting in

∆θidealn = θm,n − (θl ,n +θC ,n), where θC ,n is the desired control input. Having a law available which

minimizes the output error of Eq. 3.17, the free-play nonlinearity can now be inverted to obtain thedesired load motion:

∆θdesn =

T dess,n −p1,n p2,n −p3,nsign

(∆θn

)p1,n

for T dess,n < 0

0 for T dess,n = 0

T dess,n +p1,n p2,n −p3,nsign

(∆θn

)p1,n

for T dess,n > 0

(3.20)

with the new control input computed as:

θC ,n =∆θn −∆θdesn (3.21)

So making explicit the control input. Such a control law should be able to "jump" across the free-play range, neutralizing its nonlinear effect. Within such a framework, the proposed compensationtechnique can be interpreted as a control augmentation algorithm which can be easily exported to anycontrol logic.

82 Chapter 3

Jm , Kg m2 cm , Nm/(rad/s) Jl , Kg m2 cl , Nm/(rad/s) ks , Nm/rad α, deg

0.4 0.1 5.6 0 3300 1

Table 3.1: Data of the two-mass system with free-play

As can be inferred from the present formulation, such an adaptive behavior comes at a price. Infact, in order to estimate the parameters describing the actual nonlinearities, the additional measureof the torque exchanged at the interface is required. Nevertheless, in realistic applications wherehigh-precision positioning is required, such a measure is often available, therefore no additionalburden is introduced to the instrumentation required to acquire the sensors data.Another fact that does not seem to be well studied in the literature is the effect of the dynamicbehavior of the actuator, which may introduce time delays that could influence decisively the trackingperformance of the closed loop system, possibly ruining also its stability properties, resulting inunstable behaviors.

3.4 Test Cases

Three different problems are tackled here to demonstrate the potentialities of the methods presentedin this chapter. The first problem is a benchmark proposed in the literature [262], fully describedin Figure 3.5. Results are compared between the methodologies proposed here and the availableresults from [262]. The second test case studies the compensation of a large free-play in the gear-boxtransmission of a high-performance motor in the actuation of a scaled aileron, to be installed on awind tunnel model. Finally, the third test case analyzes the suppression of the limit cycle oscillationsinduced by a free-play in the actuation system of a vertical T-tail, which was first studied and realizedin [19, 286].

3.4.1 Two-mass with free-play benchmark

The two-mass with free-play is a well known benchmark control problem [262]. Nevertheless, theliterature has focused mainly on the speed control of such a system.For this reason, a position control for the system of Figure 3.5 is studied, and the results are firstcompared for the different control methodologies proposed in this chapter. Then, the thus designedposition based controller is used as an equivalent speed control, eventually comparing the obtainedresults with those available in the literature.

Always referring to Figure 3.5, the data describing such a model is summarized in Table 3.1. Ascan be seen comparing the current data with that of [262], the value of cl is set to zero here, just tomake the control system design more challenging. The system dynamic behavior is described by thefollowing model:

Jm θm + cm θm + cs = Tm −Ts(θm ,θl ,α)

Jl θl + cl θl = Ts(θm ,θl ,α)−Td(3.22)

with:

Ts(θm ,θl ,α) =

ks(θm −θl +α) if θm −θl <−α0 if |θm −θl | ≤α

ks(θm −θl −α) if θm −θl >α

(3.23)

3.4. Test Cases 83

Figure 3.6: Two-mass system with free-play controlled by the dual-loop PID architecture

Such a mechanical system is extremely flexible, being characterized by a normal frequency of 3.73 Hz,making more difficult the achievement of high bandwidth performance in closed loop. In additionto this, according to the analysis of [262], the actuator dynamics include a time delay of 4 ms and alow-pass filter with time constant of 6 ms, in order to have realistic bandwidth limitations. The transferfunction of the actuator dynamics is therefore:

G(s) = e−0.004s

1+0.006s(3.24)

Notice that in such a configuration the system is non-minimum phase.The system closed in loop with the torque-based dual-loop PID architecture is depicted in Fig-

ure 3.6. At this point, the algorithms presented in Section 3.2 must be adapted to the case at hand.Following the procedure presented in [274], the controllers are tuned starting from input-outputmeasurements performed in a single experiment.Assigned two reference models Mm(s) and Ml (s) for the motor and the load respectively, and obtainedthe set of data Tm(ti ),θm(ti ),θl (ti ), i = 1, ..., Nt , the following procedure is applied in the case of theVRFT:

1. Calculate in batch:

• The collected input-output time domain data is transformed in the frequency domainthrough a FFT. Such signals will be referred as U ( jω), Ym( jω) and Yl ( jω) respectively

• Compute two virtual reference signals: Rmθ( jω) = M−1

m ( jω)Ym( jω) for the motor andRlθ ( jω) = M−1

l ( jω)Yl ( jω) for the load

• Compute two virtual errors Em( jω) = Mm( jω)Rmθ( jω)−Ym( jω) for the motor and El ( jω) =

Ml ( jω)Rlθ ( jω)−Yl ( jω) for the load

2. Apply the filter operators Lm( jω) and Lm( jω) as defined in Section 3.2.1 to Em( jω) and U ( jω)for the motor and to El ( jω) and Rmθ

( jω) for the load

3. Apply the LM-based optimization algorithm described in Section 3.2.1 to the motor and loadloops to tune the PD and PID controllers respectively

When using the FRIT instead, the procedure of Section 3.2.1 is so adapted:

84 Chapter 3

1. The collected input-output time domain data is transformed in the frequency domain through aFFT. Such signals will be referred as U ( jω), Ym( jω) and Yl ( jω) respectively

2. Then, recursively:

• Compute the fictitious input of the motor loop:

Rmθ( jω) =U ( jω)+PD( jω)Ym( jω) (3.25)

• Compute the fictitious input of the load loop:

Rlθ ( jω) = Rmθ( jω)+PID( jω)Yl ( jω)

PID( jω)(3.26)

• Compute two fictitious errors Em( jω) = Mm( jω)Rmθ( jω) − Ym( jω) for the motor and

El ( jω) = Ml ( jω)Rlθ ( jω)−Yl ( jω) for the load

• Apply the LM-based optimization algorithm described in Section 3.2.1 to the motor andload loops to tune the PD and PID controllers respectively

As mentioned in Section 3.2, the PD and PID controllers cannot be chosen arbitrarily, because theymust guarantee that the inner loop has a significantly larger bandwidth than the outer one.The two target models are chosen to be second order transfer functions of the type:

M(s) = ω20

s2 +2ω0 +ω20

(3.27)

where for Mm(s) the frequency ω0 is set to 314 rad/s while for Ml (s) is set to 62.8 rad/s.The design of the training signal plays an important part in the tuning process. For this reason a

signal able to excite a wide range of frequencies, such as a chirp signal, is chosen. Such a signal hasan amplitude of 10 deg, and it sweeps the frequency spectrum from 0.01 to 200 Hz. This is eventuallyfiltered by a second order transfer function of the type of Eq. 3.27 with a cut-off frequency of 62.8 rad/s.The signal so designed is shown in Figure 3.7a, and it should be able to excite a wide frequency spectrawhile not requiring large control inputs.Another very important aspect of the training it is its initialization. In fact, the optimization algorithmdescribed in Sections 3.2.1 and 3.2.1 requires an initial guess of the parameters, which is used togenerate the input-output set. In this work a very simple approach, based on Bessel functions, isused as in [287]. Based on the wanted closed-loop bandwidth, such a procedure permits to define thecoefficients of the PID controllers in a straightforward way. Regarding the inner PD loop, a secondorder Bessel function can be used, resulting in the following gains:

KmP = 3Jmω2

0,m KmD = 3Jmω0,m (3.28)

For the outer loop instead, a third order Bessel function has to be employed, resulting in:

KlP = 15Jlω

20,l Kl

I = 15Jlω30,l Kl

D = 6Jlω0,l (3.29)

where ω0,m and ω0,l are the requested closed-loop bandwidth, set to 62.8 and 18.85 rad/s respectively.Obviously, even when such a design could be considered reasonable and more than a second choicefor the motor side, the same cannot be said regarding the outer loop design. In fact in this case theinfluence of the motor and the free-play is completely neglected, and this would result in a possiblyworking but unsatisfactory controller design only. Nevertheless, the proposed procedure is more than

3.4. Test Cases 85

0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

Time [s]

Rot

atio

n [d

eg]

(a) Reference signal

0 1 2 3 4 5−10

−5

0

5

Time [s]

Rot

atio

n [d

eg]

ReferenceMotorLoad

(b) Output used in the training

Figure 3.7: Numerical experiment used to generate the training data

0 20 40 60 80 100−60

−50

−40

−30

−20

−10

0

10

Frequency [Hz]

Ym

otor

/Rm

otor

[dB

]

Initial designFinal design VRFTFinal design FRIT−3 dB

(a) Inner loop tuning

0 10 20 30 40 50−60

−50

−40

−30

−20

−10

0

10

Frequency [Hz]

Ylo

ad/R

load

[dB

]

Initial designFinal design VRFTFinal design FRIT−3 dB

(b) Outer loop tuning

Figure 3.8: Control system tuning

sufficient to initialize the tuning algorithm, and, as demonstrated in the following, it produces robustresults. The controller is digitally realized with a sampling frequency of 2000 Hz, and the responseof the system with such initial parameters is depicted in Figure 3.7b. As can be noticed, the initialdesign is not able to compensate the free-play presence. The output so computed is fed in input to theVRFT and FRIT algorithms described in the previous sections. The resulting closed loop response isoptimized trying to satisfy the constraint on the target reference systems, which were set previously.The outcomes of the tuning algorithm is depicted in Figure 3.8. It is evident that the two tuningmethods provide very similar results once the target systems for the inner and outer loop are set, andthe related computing time is lower of 5 seconds for both the algorithms. The obtained controllerparameters are listed in Table 3.2, while the closed loop performance are summarized in Table 3.3.The two tuning methods produce very similar outcomes in term of parameters value. Furthermore,from Figures 3.8a and 3.8b can be understood that the tuning algorithms are not able to achieve therequested bandwidth (50 and 10 Hz for motor and load respectively), but the optimize the availablecontroller structure to get as close as possible to the target data, as proven also by the results of Table 3.3.

86 Chapter 3

KlP Kl

I KlD Nl

D KmP Km

D NmD

VRFT 3.5748 24.222 0.0047 500 0.0005 40.288 500

FRIT 3.4882 27.106 0.0047 500 0.0005 44 500

Table 3.2: Parameters of the dual-loop controller computed by the VRFT and FRIT algorithms

Inner loop Outer loop Gain margin, dB Phase margin, degbandwidth, Hz bandwidth, Hz

VRFT 35.6 6.8 5.8 33

FRIT 33.8 6 5.6 30

Table 3.3: Performance and stability margins of the closed loop system

Notice also that the "noisiness" of the computed FFTs ratio is due to the free-play presence, whichinduces limit cycle oscillations when the motion frequency is high. The resulting gain margin canbe considered for both the algorithm, while the phase margin is rather limited, mainly because ofthe delays introduced by the actuator dynamics. It is interesting to see from Table 3.2 that both thetuning algorithms set the derivative pole parameter ND at a value of 500, which is the maximum valueallowed to this variable. This result is probably due to the sampling frequency with which the resultsare collected, but further analyses are needed to clarify this behavior. Nevertheless, this is clearlybeyond the scope of the present analyses.The system response to a step reference of 10 degrees is displayed in Figure 3.9. The computed resultsare compared to a LQR-based control system computed through the classical modern control theory.Such a controller presents a bandwidth of 10 Hz, with a larger gain margin and a phase margin of 36deg. Using such a controller as a reference, the controller tuned with the VRFT slightly overcomes theperformance of that tuned through the FRIT, as witnessed by Figure 3.9b. This performance comes atthe cost of a greater motion of the motor and a larger control input, as shown in Figures 3.9a and 3.10a.As shown by the last figure, the additional constraint of a maximum realizable torque of 0.1 kNm isincluded in the simulation. It is so demonstrated that this saturation does not compromise the closedloop performance. In Figure 3.10b the behavior of the free-play gap during the simulation is alsosampled. Analyzing such a response, it can be concluded that once the load is engaged by the motor,there are no significant vibrations in and out the free-play gap. If this fact happened, an annoyingacoustic noise would have been perceived. As evident, this problem is not present in the currentapplication, probably because of the high flexibility of the shaft.

In order to compare the present control architecture with the ones present in the literature, the testcase proposed in [262] is implemented. It is a classical speed control of the system shown in Figure 3.5,which is subject to a step disturbance of large amplitude applied on the load side, represented byFigure 3.12b. The reference rate is 100 rpm, and the obtained results are presented in Figure 3.11. Theresults are compared to the QFT-based design of a PI controller applied o the load side only, labeledas ’PI QFT’ in the plot. The present design presents a faster suppression of the vibrations induced bythe disturbance, reducing also the rate maximum value during the transient response, as witnessedby Figure 3.11a. Nevertheless, the greatest advantage of the present approach is without doubt theefficient suppression of the load vibrations, as demonstrated by Figure 3.11b. As could be guessed,the measure of the load position permits to greatly improve the control performance, maintaining thecontrol input limited, as shown in Figure 3.12a. In this simulation, the input saturation is neglected to

3.4. Test Cases 87

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Time [s]

Mot

or r

otat

ion

[deg

]

LQRFRITVRFT

(a) Motor rotation

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Time [s]

Load

rot

atio

n [d

eg]

ReferenceLQRFRITVRFT

(b) Load rotation

Figure 3.9: Step response of the two-mass system

Time [s]0 0.2 0.4 0.6 0.8 1

Inpu

t tor

que

[kN

m]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

LQRFRITVRFT

(a) Control Input

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

6

Time [s]

θm

− θ

l [deg

]

ResponseFree−play limit

(b) Free-play crossing

Figure 3.10: Step response of the two-mass system – Input and additional outcomes

88 Chapter 3

0 0.5 1 1.5 2 2.5 3 3.575

80

85

90

95

100

105

110

115

Time [s]

Mot

or r

ate

of r

otat

ion

[rpm

]

ReferenceLQRFRITVRFTPI QFT

(a) Motor rate

0 0.5 1 1.5 2 2.5 3 3.575

80

85

90

95

100

105

110

115

Time [s]

Load

rat

e of

rot

atio

n [r

pm]


(b) Load rate

Figure 3.11: Response of the two-mass system to the benchmark test case of [262]

Time [s]0 0.5 1 1.5 2 2.5 3 3.5

Inpu

t tor

que

[kN

m]

-0.05

0

0.05

0.1

0.15LQRFRITVRFTPI QFT

(a) Control input

0 0.5 1 1.5 2 2.5 3 3.5−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time [s]

Dis

turb

ance

torq

ue [k

Nm

]

(b) Applied disturbance

Figure 3.12: Control and disturbance applied to the two-mass system

compare the results with those of [262]. Observing in detail the various figures, it can be concludedthat the LQR-based design is the most efficient, because it is the one that exhibits the shortest transient,but also it seems to be more aggressive than the dual-loop PID controllers designed through the VRFTand FRIT algorithms, which behave in a very similar manner.

For completeness, a zoom of the response is proposed in Figures 3.13 and 3.14, where it can bebetter appreciated the improvement of the performance with respect to the results of [262] and also theeffect of the different designs. It is evident from Figure 3.13b how the present motor motion is muchmore complex than the one computed by the QFT-based PI design, but this is due to the knowledge ofthe load position, which is fed back to the controller, increasing in this way the system damping andbandwidth. Again, from both Figures 3.13 and 3.14 can be deduced that once the target system is set,the VRFT algorithm leads a controller slightly faster than that generated by the FRIT. This differencescan be considered negligible, because they do not affect in a visible way the free-play suppressionproperties of the present controller architecture.

In this section, it has been demonstrated how a dual-loop PID controller architecture can be

3.4. Test Cases 89

0 0.2 0.4 0.6 0.885

90

95

100

105

110

Time [s]

Mot

or r

ate

of r

otat

ion

[rpm

]


(a) Disturbance applied – Motor side

2 2.2 2.4 2.6 2.8 385

90

95

100

105

110

115

Time [s]M

otor

rat

e of

rot

atio

n [r

pm]


(b) Disturbance removed – Motor side

Figure 3.13: Close-up of the motor response during the two transients

0 0.2 0.4 0.6 0.885

90

95

100

105

Time [s]

Load

rat

e of

rot

atio

n [r

pm]


(a) Disturbance applied – Load side

2 2.2 2.4 2.6 2.8 395

100

105

110

115

Time [s]

Load

rat

e of

rot

atio

n [r

pm]


(b) Disturbance removed – Load side

Figure 3.14: Close-up of the load response during the two transients

90 Chapter 3

lJp , rp Jl , rl

+

k

+

2α

θpTa , θl

θm

Jm , Tm

Figure 3.15: Belt drive system with free-play in the motor gear-box

Jm , Kg m2 Kt , Nm/A Maximum torque Free-playallowed TM , Nm width α, deg

8.5·10−6 0.77 0.69 1

Table 3.4: Maxon EC22 data of interest

efficiently used in the suppression of free-play effects. However, no motion dependent disturbance,like aerodynamic loads, where applied to the present model.The scope of the second test case is to introduce linear aerodynamic forces in a more complex mecha-nism.

3.4.2 Aileron actuation system with gear-box free-play

In this section, an aileron is commanded in position through a belt drive, which is driven by a high-performance motor. The Maxon EC22, with the planetary gear-head GP 22 HP is chosen for thisapplication. Even though it can be considered a high-performance motor thanks to its good dynamiccharacteristics, it presents a gear-box free-play of α= 1 deg.The scope of the present work is to compensate the free-play effect while also minimizing the vibrationsthat occur when the free-play gap is closed during dynamic operational conditions.

The dynamic system that has to be controlled is schematized by Figure 3.15. The aim of the dual-loop control system will be to follow a certain reference r (t) through the aileron position θl , whilecommanding the motor through an input torque. The dynamic model representing the system ofFigure 3.15 is the following:

Jm θm = Kt I −Ts(θm ,θp ,α)

Jp θp +kr 2pθp −krp rlθl = Ts(θm ,θp ,α)

Jl θl +kr 2l θl −krl rpθp = Ta(θl , θl )

(3.30)

Again, the Ts function is the free-play mathematical model, already detailed by Eq. 3.23. The equivalentstiffness of the link between the motor and the gear-box is assumed to be quite high and set to a properfrequency of 100 Hz. The resulting stiffness of the free-play region is equal to ks = 1390 Nm/rad, usingthe notation of Eq. 3.23. As usual in these cases, because the motor is commanded in torque, the motorelectrical dynamics is neglected. The numerical data of the motor is extracted by the Maxon EC22data sheet and summarized in Table 3.4. Because the aileron may follow signals with high frequency,an unsteady modeling of the aerodynamics is required. To do this, the formulation proposed byTheodorsen in [288] is followed. The aerodynamic model is therefore expressed in state-space form by

3.4. Test Cases 91

b, m Aileron chord, m Aileron span, m Distance of the hingefrom the leading edge, m

0.15 0.085 0.33 0.02

Table 3.5: Aileron geometric data

Jp , Kg m2 Jl , Kg m2 rp , m rl , m k, N/m

3.77·10−8 2.502·10−4 6.33·10−3 9.16·10−3 8.0167·105

Table 3.6: Required data to define the model of Eq. 3.30

the following dynamic system:xa +aa1V∞b

xa +aa0V 2∞b2 xa = ba0θl +ba1θl +ba2θl

Ta = (ca xa +da0θl +da1θl +da2θl ) · span(3.31)

where b and V∞ are the wing semi-chord and the flight speed respectively, while the expression of theconstants aa,i , ba,i , ca and da,i , i = 1,2,3 is resumed here below:

aa0 = 0.0131, aa1 = 0.361 (3.32a)

ba0 =−0.00656V 3∞/(πb2)(T10 − lT21), ba2 =−0.114V∞/(2π)(T11 −2l T10) (3.32b)

ba1 =−0.114V 2∞/(bπ)(T10 − lT21)+0.00656V 2

∞/(2πb)(T11 −2lT10) (3.32c)

ca = ρ∞V∞b2(T12 −2l T20) (3.32d)

da0 =−ρ∞b2V 2∞/πT18 −ρ∞b2lV 2

∞/πT26 −ρ∞b2l 2V 2∞/πT28+

−0.5ρ∞b2V 2∞/π(T12 −2lT20)(T10 − lT21)

(3.32e)

da1 =−ρ∞b3V∞/πT19 −ρ∞b3lV∞/πT27 −ρ∞b3l 2V∞/πT29+−0.5ρ∞b3V∞/(2π)(T12 −2lT20)(T11 −2l T10)

(3.32f)

da2 = ρ∞b4/πT3 −2ρ∞b4l/πT2 +ρ∞b4l 2/πT5 (3.32g)

The value of the numerical constants can be directly computed from the approximation of theTheodorsen function by W. P. Jones [289], while the value of the constants Ti depends on the air-foil geometry and their expression can be found in [288]. The geometric data of the aileron is resumedin Table 3.5, while the remaining data required to completely define the model is listed in Table 3.6.The drive belt of Figure 3.15 is made of rubber reinforced by glass fibers.

To show the detrimental effect of a free-play inside the motor, the tuning of a PID on the motorside only is carried out. The same algorithms explained in the previous sections is employed, and theresults are displayed in Figure 3.16. It is clear that the free-play does not introduce any instability in thesystem, nevertheless its main effect is to reduce the precision of the closed loop system. In fact, withsuch a control architecture the motor is not able to place accurately the aileron within the free-playgap.

Having proved that a measure of the aileron position is required to achieve its high-precisionpositioning, the first step to tackle is the design of the controller architecture. As exposed by Figure 3.15,the free-play is present between the motor and the gear-box. The scope of the control system is to

92 Chapter 3

0 0.5 1 1.5 2−5

0

5

10

15

20

Time [s]

Def

lect

ion

[deg

]

ReferenceMotorAileron

(a) Comparison of the responses

0 0.5 1 1.5 2−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [s]

Inpu

t tor

que

[Nm

]

(b) Control input

Figure 3.16: Aileron control by means of a PID on the motor side only

Figure 3.17: Control of the aileron position through three loops

regulate the motion of the aileron, which is connected to the motor side by an elastic belt. Three layersof control are thus required: a dual loop for the free-play compensation of the motor-gear-box systemand a classical PID controller for an adequate positioning of the aileron. Such a concept is presentedby Figure 3.17.

Considering the tuning is this system through the algorithms explained previously, four quantitiesneed to be measured: the control current and the positions of motor, pulley and aileron. With respect tothe previous test case, an additional measure is required, which is that of the pulley position. Becausethis measure might be quite cumbersome, another solution is devised here. Starting from the measuresof motor and aileron positions, a sliding mode observer is designed to reconstruct the full state of thesystem [290, 291, 292]. Such an observer is represented by the following dynamic system:

xo = Axo +BI +Lo(y−yo

)+Hov

yo = Cxo(3.33)

where the matrices Ao , Bo and Co are constructed directly from Eq. 3.30, while the observer gain matrixis computed through classical LQR-based design and Ho is proper of this particular observer and isused to introduced the discontinuous contribution with boundary layer v, defined as:

vi =

0 yi − yo,i = 0

γoyi − yo,i

∆|yi − yo,i | ≤∆

γoyi − yo,i

|yi − yo,i ||yi − yo,i | >∆

(3.34)

3.4. Test Cases 93

Time [s]0 0.5 1 1.5 2 2.5

θp [d

eg]

-20

-15

-10

-5

0

5

10

ResponseObserver

(a) Pulley position

Time [s]0 0.5 1 1.5 2 2.5

θm[deg/s]

-150

-100

-50

0

50

100

150

200

250

ResponseObserver

(b) Motor rate

Figure 3.18: Samples of the sliding mode observer estimations

Time [s]0 1 2 3 4 5

Rot

atio

n [d

eg]

-20

-15

-10

-5

0

5

10

ReferenceMotorPulleyAileron

(a) Measured signals

Time [s]0 1 2 3 4 5

Tor

que

[Nm

]

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

(b) Control input

Figure 3.19: Tuning signal for the aileron test case

where ∆ is the boundary layer width and γo a scalar observer gain. The details related to the observerdesign will be explained in the last section of this chapter, because it will be used extensively also inthe next applications. Nevertheless, a sample of the observer response is shown in Figure 3.18, wherethe tracking of the pulley position and the motor rate is presented. As can be noticed, once tunedopportunely, the observer is able to follow fast changes of the system state.

The next step is the controller tuning. A chirp signal of amplitude 20 degrees, sweeping frequenciesfrom 0.05 to 400 Hz in 5 seconds is used. Such a references is eventually filtered by a second ordertransfer function with bandwidth of 10 Hz. A first value of the controller parameters is set using themethod based on Bessel functions described previously. The data required to tune the closed loopsystem is shown in Figure 3.19, computed at a wind speed of V∞ = 50 m/s.

Because the present controller is now a nested combination of three PID controllers, a slightmodification to the VRFT and FRIT algorithms explained previously must be implemented.

Following the same line of the previous section, after having assigned three reference mod-els Mm(s), Mp (s) and Ml (s) for motor, pulley and load respectively, and obtained the set of data

94 Chapter 3

I (ti ),θm(ti ),θp (ti ),θl (ti )

, i = 1, ..., Nt , with θp provided by the sliding mode observer, the VRFT

algorithms now reads as:

1. Calculate in batch:

• The collected input-output time domain data is transformed in the frequency domainthrough a FFT. Such signals will be referred as U ( jω), Ym( jω), Yp ( jω) and Yl ( jω) respec-tively

• Compute three virtual reference signals: Rmθ( jω) = M−1

m ( jω)Ym( jω) for the motor, Rpθ( jω) =

M−1p ( jω)Yp ( jω) for the pulley and Rlθ ( jω) = M−1

l ( jω)Yl ( jω) for the aileron

• Compute three virtual errors Em( jω) = Rmθ( jω)−Ym( jω) for the motor, Ep ( jω) = Rpθ

( jω)−Yp ( jω) for the pulley and El ( jω) = Rlθ ( jω)−Yl ( jω) for the load

2. Apply the filter operators Lm( jω), Lp ( jω), and Lm( jω) as defined in Section 3.2.1 to Em( jω) andU ( jω) for the motor, to Ep ( jω) and Rmθ

( jω) for the pulley and to El ( jω) and Rpθ( jω) for the

load

3. Apply the LM-based optimization algorithm described in Section 3.2.1 to the motor, pulley andload loops to tune the control system

When using the FRIT instead, the procedure of Section 3.2.1 is so adapted:

1. The collected input-output time domain data is transformed in the frequency domain through aFFT. Such signals will be referred as U ( jω), Ym( jω), Yp ( jω) and Yl ( jω) respectively

2. Then, recursively:

• Compute the fictitious input of the motor loop:

Rmθ( jω) =U ( jω)+PD( jω)Ym( jω) (3.35)

• Compute the fictitious input of the pulley loop:

Rpθ( jω) = Rmθ

( jω)+PIDin( jω)Yp ( jω)

PIDin( jω)(3.36)

• Compute the fictitious input of the load loop:

Rlθ ( jω) = Rpθ( jω)+PIDout( jω)Yl ( jω)

PIDout( jω)(3.37)

• Compute the fictitious errors Em( jω) = Rmθ( jω)−Ym( jω) for the motor, Ep ( jω) = Rpθ

( jω)−Yp ( jω) for the pulley and El ( jω) = Rlθ ( jω)−Yl ( jω) for the load

• Apply the LM-based optimization algorithm described in Section 3.2.1 to the motor andload loops to tune the control system

The target models are second order transfer functions with cut-off frequency of ω0,m = 502 rad/s forMm(s), ω0,p = 157 rad/s for Mp (s) and ω0,l = 94 rad/s for Ml (s).The gains computed by the two algorithms are resumed in Tables 3.7 and 3.8, while a comparisonbetween the obtained closed loop transfer functions if proposed in Figure 3.20. As can be noticedfrom Figure 3.20, both tuning algorithms are able to increase the system bandwidth up to at least 12

3.4. Test Cases 95

KpP Kp

I KpD Np

D KmP Km

D NmD

VRFT 544 48322 16.6 500 111 0.72 500

FRIT 543 48322 16.5 500 111 0.71 500

Table 3.7: Parameters of controllers of motor and pulley computed by the VRFT and FRIT algorithms

KlP Kl

I KlD

VRFT 0.281 94.35 0.002 500

FRIT 0.326 78.27 0.0043 500

Table 3.8: Parameters of the aileron (external) controller computed by the VRFT and FRIT algorithms

Frequency [Hz]0 5 10 15 20 25

Ylo

ad/R

load

[dB

]

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Initial designFinal design VRFTFinal design FRIT-3 dB

Figure 3.20: Comparison between the transfer function between load and reference computed throughthe different tuning methods

96 Chapter 3

Time [s]0 0.2 0.4 0.6

Def

lect

ion

[deg

]

-2

0

2

4

6

8

10

12

14

16ReferenceMotorPulleyAileron

(a) System response

Time [s]0 0.2 0.4 0.6

Inpu

t tor

que

[Nm

]

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

(b) Control input

Figure 3.21: Step input test

Time [s]0 0.2 0.4 0.6

θm

- θ

p [d

eg]

-1

-0.5

0

0.5

1

1.5ResponseFree-play limit

Figure 3.22: Dynamics of the free-play gap crossing for a step input

Hz, while guaranteeing ample robustness margins such as 7 dB of gain margin and more of 70 degof phase margin. As can be also noticed from Tables 3.7 and 3.8, the optimized parameters are quitesimilar in value, proving once again the similarity of the two tuning methods. Also in this case, thefilter constant of the derivative contribution is pushed to its limit of 500. Because of these similarities,in the following analyses only the results related to the VRFT-based design will be shown.

Showing sample results, the response to a step input is depicted in Figure 3.21. As evident, thecontrol system is able to place correctly the aileron without incurring in any imprecision due to thefree-play presence. However, analyzing Figure 3.22, it is evident how the crossing of the free-playregion is not monotone, but it presents three bounces in and out the free-play gap. This would result ina fastidious acoustic noise that is better to be suppressed in dynamic applications. In addition, thesefree-play bounces might also result in wear of the gear-box, which may compromise the durability andreliability of the actuation system, making the servo design unacceptable.

A further demonstration of this problem is obtained considering the response of the system toan input resembling a realistic gust suppression input, as computed in [293]. This signal presents a

3.4. Test Cases 97

Time [s]0 0.5 1 1.5 2 2.5

Def

lect

ion

[deg

]

-20

-15

-10

-5

0

5

10ReferenceMotorPulleyAileron

(a) System response

Time [s]0 0.5 1 1.5 2 2.5

Inpu

t tor

que

[Nm

]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

(b) Control input

Figure 3.23: Gust input test

Time [s]0 0.5 1 1.5 2 2.5

θm

- θ

p [d

eg]

-1.5

-1

-0.5

0

0.5

1

1.5ResponseFree-play limit

(a) Free-play gap crossing

Time [s]0.6 0.65 0.7 0.75 0.8

θm

- θ

p [d

eg]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1ResponseFree-play limit

(b) Close-up near one of the transition regions

Figure 3.24: Dynamics of the free-play gap crossing for a gust suppression input

quite limited frequency content (4 Hz), but it produces a more complex response across the free-playregion, as shown in Figure 3.24. As evident from Figure 3.23, the closed-loop system is able to followthe reference signal accurately, however, the same bouncing phenomenon in and out the free-playgap is still present, and in this case the vibrations around the closed position are more accentuated, ashighlighted by Figure 3.24b.

Obviously, in order to avoid this annoying phenomenon, proper solutions must be devised. Varioussolutions based on the finite-horizon Riccati equation were developed in [294, 295, 296], however theapplications considered there had a very limited bandwidth and requires the knowledge of the fullstate. For this reason, a relatively simple and robust approach is chosen in this case, always based on aPID architecture.The solution implemented here simply considers that, when the free-play gap is open, the motor andthe gears sides work as isolated systems, as highlighted by Figure 3.25. In this time lapse, the motortries to close the free-play gap as fast as possible in order to compensate its effect. Nevertheless, in

98 Chapter 3

+ +

2αθp

θm

Gap open

2αθp

θm

Gap closed

Figure 3.25: Closure of the free-play gap

this way the controller is not considering that when the free-play gap is open, while the motor triesto reach the gear, this latter is still moving at a different speed. Therefore, a controller that activatesonly when the free-play gap is open and tries to minimize the rate difference between motor and gear,while the motor is also approaching the free-play closure, will work fine in this case.The resulting controller in the free-play region results in:

I (t ) =KP(θp − θm

)+KI (θdes −θm) (3.38)

where θp and θm are estimated through the sliding mode observer, while θdes is defined as:

θdes =θp +α θm −θp <α

θp −α θm −θp >−α (3.39)

where again θp is estimated by the observer.Wrapping up the discussion, the three-loop controller tuned previously will be activated when thefree-play gap is closed, instead, when motor and gear are not engaged, the simpler speed controllerof Eq. 3.38 is employed, until the free-play gap is closed again. In this way, a switching controller isimplemented, and a nonlinear control system is realized for the compensation of the free-play effect.

The tuning of this secondary control strategy is carried out with the VRFT and FRIT algorithm pre-viously detailed. The resulting gains are KP = 0.0005 and KI = 31 for both algorithms. The comparisonbetween the plain three-loop controller and the switching one is proposed in Figure 3.26 for the stepinput and Figure 3.27 for the realistic gust history. As can be noticed from these figures, the use ofa switching control law permits to increase the accuracy of the closed-loop system, avoiding largeaileron vibrations once the free-play gap is closed. The transition between the two control laws is quiteevident when looking at Figures 3.26c and 3.27c, where the control activity is rather high inside thefree-play region but does not present any visible effect out of it. Such a higher control effort is relatedto the fact that when the free-play gap is closed the motor tries to reach the gear as fast as possible,trying also to minimize the forces at the eventual impact. As a final consideration, from Figures 3.26dand 3.27e the beneficial effect of the implemented switching control law is enhanced. This solutioncannot be considered as optimal, because some bounces are still present in and out the free-playregion, but it is nonetheless efficient in removing a great deal of vibrations at the free-play closure, andcan be considered a first implementation option when dealing with these kind of problematic thanksto its easy interpretation and realization.

This test case has demonstrated the great flexibility of the tuning algorithms presented whenapplied to a non-classical control problem when strong nonlinearities are present and difficult tocompensate. As an additional contribution, the problem of the vibrations resulting at the free-playclosure has been addressed, proposing a very simple solution aimed at reducing them.Nevertheless, even if motion-dependent disturbances like aerodynamic loads have been introducedhere, they have not influenced the stability properties of the system. This fact might be of essentialimportance in fluid-structure interaction problems, as will be detailed in the next section.

3.4. Test Cases 99

Time [s]0 0.2 0.4 0.6

Aile

ron

defle

ctio

n [d

eg]

0

2

4

6

8

10

12No switchSwitchReference

(a) Response comparison

Time [s]0 0.2 0.4 0.6

Rat

e [d

eg/s

]

-50

0

50

100

150

200

250No switchSwitch

(b) Motor rate response comparison

Time [s]0 0.2 0.4 0.6

Tor

que

[Nm

]

-0.1

0

0.1

0.2

0.3

0.4

0.5

No switchSwitch

(c) Comparison of the required control input

Time [s]0.2 0.22 0.24 0.26 0.28 0.3

θm

- θ

p [d

eg]

-0.5

0

0.5

1

1.5No switchSwitch

(d) Dynamics of the free-play gap – Close-up near theclosure

Figure 3.26: Response to step input – Comparison between plain controller and switching controllerwith smooth closure of the free-play gap

100 Chapter 3

Time [s]0 0.5 1 1.5 2 2.5

Aile

ron

defle

ctio

n [d

eg]

-12

-10

-8

-6

-4

-2

0

2

4No switchSwitchReference

(a) Response comparison

Time [s]0 0.5 1 1.5 2 2.5

Rat

e [d

eg/s

]

-150

-100

-50

0

50

100

150

200

250No switchSwitch

(b) Motor rate response comparison

Time [s]0 0.5 1 1.5 2 2.5

Tor

que

[Nm

]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

No switchSwitch

(c) Comparison of the required control input

Time [s]0 0.5 1 1.5 2 2.5

θm

- θ

p [d

eg]

-1.5

-1

-0.5

0

0.5

1

1.5

No switchSwitch

(d) Dynamics of the free-play gap

Time [s]0.2 0.3 0.4 0.5 0.6 0.7 0.8

θm

- θ

p [d

eg]

-1.5

-1

-0.5

0

0.5

1

1.5

No switchSwitch

(e) Dynamics of the free-play gap – Close-up

Figure 3.27: Response to gust input – Comparison between plain controller and switching controllerwith smooth closure of the free-play gap

3.4. Test Cases 101

(a) T-tail computer rendering (b) T-tail wind tunnel model

Figure 3.28: T-tail model

3.4.3 Limit cycle oscillations of a vertical T-tail

The T-tail unit considered in this study is an X-DIA aeroelastic model representative of a non-conventionalthree surfaces regional jet called Target Aircraft, intensively investigated in the last decade at the De-partment of Aerospace Science and Technology, Politecnico di Milano [215, 266].Following the work of ref. [286], The T-tail experimental model has been scaled in order to keep thesame Froude number of the Target Aircraft. The model core is composed by dynamically scaled spars,which are inserted in a series of sectors made by styrofoam covered by a carbon fiber skins. The finstructure is made by an aluminum alloy beam that is covered by five aerodynamic sectors that are usedto get the airfoil shape. The rendering of the T-tail model is proposed in Figure 3.28a, while its actualrealization is displayed in Figure 3.28b. Figure 3.29 summarizes the geometric characteristics of theT-tail. The tail structure was tested in the Small Wind Tunnel of the Department of Aerospace Scienceand Technology. It is a closed circuit low-speed wind tunnel, with a rectangular test section of 1.5 × 1 ×3 m and no heat exchangers installed. The test chamber is made of square steel tubes and replaceablewalls, which are sometimes substituted with glass windows for flow visualizations, or wooden panelsto ease the model installation.

To provide a variable amplitude freeplay, the mechanism shown in Figure 3.30 was introduced inthe control chain between the actuator and the rudder. This is composed by a rigid linkage connectedwith the rudder, which ends with a pin that is slipped into a fork connected with the gear of the electricmotor used to actuate the movable surface. The connecting fork allows to modify the behavior fromno free-play up to ± 10 deg by changing the position of the electric motor with respect to the rudderhinge axis. The angular position of the motor-fork assembly is determined by a PID controller usingthe angular encoder embedded into the motor as feedback. The rotation of the rudder is measured byan incremental encoder connected to the rudder hinge axis at the fin root. A second encoder is used tomeasure the rotation of the motor-fork assembly, as sketched in Figure 3.31.

The nonlinear system has been modeled as two connected linear sub-systems, i.e. the T-tail andthe fork-motor assembly. The T-tail sub-system state space matrices are built using an MSC.Nastranstructural finite element model and the aerodynamic doublet lattice method. The frequency domaindata are transformed into a finite state space realization by using Roger’s algorithm [297]. The structuraldynamics is governed by the classical dynamic equilibrium equation:

Ms qs +Cs qs +Ks qs = q∞fa + fNL + fc (3.40)

102 Chapter 3

Figure 3.29: T-tail geometrical data

Figure 3.30: Realization of the free-play mechanism

Figure 3.31: Measures available on the experimental model

3.4. Test Cases 103

Jm , Kgm2 cm , Nm/(rad/s) km , Nm/rad Kt , Nm/A Torque Deflectionsaturation, Nm saturation, deg

8.31·10−5 1.81·10−7 0 0.682 6.82 15

Table 3.9: Motor parameters

where Ms , Cs , Ks are the classical mass, damping and stiffness matrices, qs is the vector of generalizeddisplacements, fa are the aerodynamic forces, fNL the forces deriving from the nonlinear interactionbetween motor and rudder and fc the control forces. The latter are in this case simply represented bythe torque produced by the motor-fork system. The rotation of the rudder is part of the generalizeddisplacements, i.e. qs =

q,θl

.

On the other hand, the aerodynamic forces are modeled through the following dynamic system:c

2V∞xa = Aaxa +Baqs

fa = Caxa +Da0 qs + c

2V∞Da1 qs + c2

4V 2∞Da2 qs

(3.41)

Because the motor is commanded in current, its electrical dynamics can be neglected. Nevertheless,the dynamics of its mechanical part is governed by the usual model:

Jm θm + cm θm +kmθm = Kt I −Ts(θm ,θl ) (3.42)

A simple PID system on the motor side is first used to control the motor current in order to obtain therequired rudder position. The main motor parameters used in the following simulations are resumedin Table 3.9, while the control gains of the motor PID were set through the classical Ziegler–Nicholstuning method to KP = 200, KI = 0 and KD = 1.

Regarding the free-play nonlinearity, it can be said that when the rudder pin engages the fork, thetwo sub-systems are connected and the stiffness of the movable surface is controlled by the electricmotor. Instead, when the pin is separated by the fork, and travels freely into the gap, the two sub-systems act as independent dynamical models. The modeling of this switch system is managed byusing a penalty function approach, like the one described in [286], that computes the force exchangedbetween the two sub-systems as:

Ts =

ks (θm −θl +α)+ cs

(θm − θl

)θm −θl <−α

0 |θm −θl | ≤α

ks (θm −θl −α)+ cs(θm − θl

)θm −θl >α

(3.43)

where the same notation of Figure 3.5 is used. As can be interpreted by Eq. 3.43, this approach causesa switch that represent a discrete change in the connection stiffness between the rudder and themotor-fork systems. The parameters ks and cs are chosen according to reference [298], aiming atminimizing both the penetration as well the rebound between the two sub-systems. Their value isresumed in Table 3.10.

Preliminary open-loop simulations are carried out to understand the system behavior againstwind speed changes. Because of the interaction of the structure with the aerodynamic forces, thesystem undergoes limit cycle oscillations beyond a critical speed, which in this case is determinedboth experimentally and numerically. As shown by the LCO amplitude trend of Figure 3.32b, oncethe LCO is initiated, the related amplitudes stabilizes over a quite constant envelope, determined by

104 Chapter 3

ks , Nm/rad cs , Nm/(rad/s)

126.06 0.14

Table 3.10: Stiffness and damping of the free-play connection

Time [s]0 2 4 6 8 10

Rot

atio

n [d

eg]

-2

-1

0

1

2

3

4

(a) Rudder limit cycle at V∞ = 50 m/s. Free-play widthα= 1 deg

V∞

[m/s]30 35 40 45 50 55 60

LCO

am

plitu

de [d

eg]

0

0.5

1

1.5

2

2.5α = 1 deg, numericalα = 1 deg, experimentalα = 2 deg, numerical

(b) LCO trends for different values of α

Figure 3.32: Rudder limit cycles beyond the bifurcation point

the free-play width. Analyzing the same figure, it can be concluded that the numerical model wellreplicates the experimental behavior, presenting an error of 5.8 % in the evaluation of the bifurcationpoint and a precise simulation of the obtained LCO amplitude. Additional outputs can be analyzed,such as the motor response and the control input, as displayed by Figure 3.33. A close-up on thesetwo responses is proposed in Figure 3.34 to observe the periodic nature of the open-loop results.Comparing Figures 3.32a and 3.33a the difference in amplitude of the LCOs experiences by the twosystems can be appreciated. While in Figure 3.32a the rudder is forced by the aerodynamic loads, themotor is kept in position by the PID controller. The motor experiences an LCO anyway, because of itsinteraction with the vibrating rudder, as highlighted by Figure 3.34a

In order of suppress the LCOs induced by the free-play presence, the dual-loop architecture ofFigure 3.3 is considered. The controller is tuned on a stable response computed at V∞ = 0 m/s, thereforeavoiding the effects of the LCO induced by the interaction with the aerodynamics.The training signal is again a chirp signal of 5 degree amplitude sweeping from 0.05 to 200 Hz, filteredby a second order transfer function of bandwidth 20 Hz. The VRFT and FRIT algorithms can be directlyapplied as detailed in Section 3.4.1 and the resulting transfer function between the reference signaland the rudder output is displayed in Figure 3.35a, where a comparison between the starting andtuned transfer functions is proposed. Because the similarities between the VRFT and FRIT methodshave already been demonstrated through the previous test cases, here only the results relative to theVRFT-based tuning are shown.The target systems are second order transfer functions with a bandwidth of 50 Hz for the motor loopand 20 Hz for the rudder loop. The controller is again digitalized at a sampling frequency of 2000Hz. As can be seen, the tuning algorithm is able to increase the closed-loop bandwidth up to 15 Hz,reducing the FFT peaks associated to the free-play effect. The computed controller parameters arelisted in Table 3.11.

3.4. Test Cases 105

Time [s]0 2 4 6 8 10

Rot

atio

n [d

eg]

-1

0

1

2

3

4

5

(a) Motor limit cycle at V∞ = 50 m/s. Free-play width α= 1deg

Time [s]0 2 4 6 8 10

Tor

que

[Nm

]

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

(b) Control input at V∞ = 50 m/s. Free-play width α= 1deg

Figure 3.33: Motor limit cycles beyond the bifurcation point

Time [s]9.2 9.25 9.3 9.35 9.4

Rot

atio

n [d

eg]

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

(a) Close-up on motor limit cycle

Time [s]9.2 9.25 9.3 9.35 9.4

Tor

que

[Nm

]

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

(b) Close-up on control input

Figure 3.34: Motor limit cycles beyond the bifurcation point – Close-up on periodic behavior

KlP Kl

I KlD Nl

D KmP Km

D NmD

2.26 305 0.059 500 3.65 0.0001 500

Table 3.11: Dual-loop PID parameters for the T-tail case

106 Chapter 3

Frequency [Hz]0 5 10 15 20 25 30 35

Ylo

ad/R

load

[dB

]

-40

-30

-20

-10

0

10

20

30

40

Initial designFinal design VRFT-3 dB

(a) T-tail tuning

Time [s]0 2 4 6 8 10

Rot

atio

n [d

eg]

-1

0

1

2

3

4

5ReferenceRudder

(b) Closed-loop simulation at V∞ = 50 m/s

Figure 3.35: Response of the T-tail in closed loop with the dual-loop PID architecture

Unfortunately, as demonstrated by Figure 3.35b, relative to a closed-loop simulation at V∞ = 50 m/swith a free-play semi-width of α= 1 deg, the present controller is not able to suppress the free-playinduced LCO, which is only marginally reduced with respect to the open-loop response. This fact isbetter highlighted by the FFT comparison of Figure 3.36b, where it is evident how the LCO is onlyslightly reduced in amplitude while its frequency is increased because of the interaction with thecontrol system. For completeness, the control effort simulated in this case is shown in Figure 3.36a.

It is clear at this point that a different approach is required to tackle this problem. The adaptiveapproach of Section 3.3 based on the inversion of the free-play nonlinear model seems a naturalcandidate. For this reason, this strategy is integrated in the dual-loop control architecture as depictedin Figure 3.37. Such a solution requires the additional measure of the exchanged torque between themotor and the rudder. Therefore a torque sensor is placed on the motor side with such an aim, keepingin mind that this measure would take into account all the related contributions at the interface, suchas that coming from the free-play but also from friction and hysteresis for example. Nevertheless, thiscase emphasizes the free-play effect, therefore other contributions are not modeled but accountedthrough a large measurement noise, set as a random variation of the 25% of the current measure. Fora comparison with a plain dual-loop controller, no measurement noise is considered, but its effecton the three measures required, i.e. motor and rudder position and torque at the interface, will beevaluated through extensive analyses.

The controller of Section 3.3 is rather easy to design, because only the learning rate of the estimatorof Eq. 3.16 has to be set. Through a quick trial-and-error procedure, the learning rate is set to ηGD =0.04. As a rule of thumb, the designer should start with a fairly low value, e.g. 0.001, then increase ituntil a sufficiently fast convergence rate of the free-play parameters is obtained. From the presentexperience, low values of ηGD lead to stable but slowly converging results, while for too high values, e.g.2, the training process of Eq. 3.16 can become unstable.The free-play model is completely defined by the parameters α, ks and cs therefore Eq. 3.16 isemployed in this case to identify these three values. As will be seen in the following, the most importantidentification is that related to the free-play width, with the other parameters that play a slightly moremarginal role. Applying the same disturbing pulse signal as for Figure 3.35 at the same wind speed,the response is computed with the new architecture, and the results are shown in Figure 3.38. Insuch a simulation, a torque disturbance of 0.1 Nm is applied on the rudder for 0.1 seconds. As can

3.4. Test Cases 107

Time [s]0 2 4 6 8 10

Tor

que

[Nm

]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

(a) Control effort

Frequency [Hz]10 12 14 16 18 20

Am

plitu

de [d

eg]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Open LoopDual Loop

(b) Comparison between the response FFT in open- andclosed-loop

Figure 3.36: Further insights about the response of the T-tail in closed loop with the dual-loop PIDarchitecture

Figure 3.37: Dual-loop supported by the nonlinearity inversion control algorithm

108 Chapter 3

Time [s]0 1 2 3 4 5

Rot

atio

n [d

eg]

-1

0

1

2

3

4

5RudderReference

(a) Rudder response

Time [s]0 1 2 3 4 5

Rot

atio

n [d

eg]

-2

-1

0

1

2

3

4

5

6

(b) Motor response

Figure 3.38: Response of the T-tail in closed loop with the dual-loop PID architecture augmented withthe nonlinearity inversion-based controller

be seen from such results, the controller is able to withstand both the disturbances coming fromthe aerodynamics and external sources maintaining the rudder at the commanded position. FromFigure 3.38b is clear that the torque disturbance is strong enough to make the motor switch fromone side to the other of the free-play gap, but it is still able to stabilize the system with the controlinput shown in Figure 3.39a. The position requested by the nonlinearity inversion controller, i.e.θc ofFigure 3.37, is instead displayed in Figure 3.39b. From this last figure, it is evident that the controllerstill tries to continuously "jump" across the free-play gap to suppress its effect. Finally, in support ofthese results it is shown also the convergence of the free-play parameters α and ks in Figure 3.40. Inview of the previous results, it is evident that the correct identification of the free-play width plays amuch more important role than the other parameters.

At this point is interesting to take into account some measurement noise and run the samesimulation at different wind speeds, so reconstructing an LCO amplitude trend in open- and closed-loop for different values of the free-play width α. The obtained results are summarized in Figure 3.41for simulations with α= 1 and 2 deg respectively. It is clear from this figure that measurement noisedeteriorates the performance of the adaptive controller, which nonetheless is able to significantlyreduce the residual LCOs at high wind speeds.

Always considering the previous case with pulse perturbation at V∞ = 50 m/s, the FFT of theresponse is again compared between the different control methods in Figure 3.42. From this result canbe concluded that the current controller with nonlinearity inversion is able to suppress the free-playinduced LCO even in presence of large measurement noise. This suppression is much more efficientthan that obtained through the plain dual-loop PID controller, which reduced the LCO amplitude by asmaller factor while increasing its frequency. Instead, from the same figure, the results obtained withcontroller based on the free-play inversion seems to maintain the same frequency of the open-loopLCO, while greatly reducing the related amplitude.

In addition to the recovered stability after the bifurcation point, the rudder is now able to followgeneric signals with good accuracy. A sample of this is shown in Figure 3.43, where the gust signalsused in the previous test case is employed again, this time at a wind speed of V∞ = 60 m/s. As canbe noticed from these sample results, the simulation without measurement noise is able to followthe reference signal quite accurately without producing any LCO. On the other hand, in presence of

3.4. Test Cases 109

Time [s]0 1 2 3 4 5

Tor

que

[Nm

]

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(a) Control input (b) Desired controller position θc

Figure 3.39: Control effort computed by the dual-loop supported by nonlinearity inversion

Time [s]0 1 2 3 4 5

α [d

eg]

0.4

0.5

0.6

0.7

0.8

0.9

1

IdentificationReal value

(a) Identification of the free-play width α

Time [s]0 1 2 3 4 5

k s [Nm

/rad

]

60

70

80

90

100

110

120

130


(b) Identification of the equivalent control stiffness ks

Figure 3.40: Identification history of the main parameters describing the free-play model

110 Chapter 3

V∞

[m/s]45 50 55 60

LCO

am

plitu

de [d

eg]

0

0.5

1

1.5Open LoopClosed Loop w/o noiseClosed Loop w noise

(a) LCO trends with α= 1 deg

V∞

[m/s]45 50 55 60

LCO

am

plitu

de [d

eg]

0

0.5

1

1.5

2

2.5Open LoopClosed Loop w/o noiseClosed Loop w noise

(b) LCO trends with α= 2 deg

Figure 3.41: LCO trends in open- and closed-loop with and without measurement noise

Frequency [Hz]10 12 14 16 18 20

Am

plitu

de [d

eg]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Open LoopDual LoopNonlinearity Inversion w/o noiseNonlinearity Inversion w noise

Figure 3.42: Comparison of the response FFT between the different control methods

3.5. Sliding Mode Observer 111

Time [s]0 1 2 3 4 5

Rot

atio

n [d

eg]

-12

-10

-8

-6

-4

-2

0

2

4Referencew/o noisew noise

(a) Rudder response

Time [s]0 1 2 3 4 5

Rot

atio

n [d

eg]

-12

-10

-8

-6

-4

-2

0

2

4

6w/o noisew noise

(b) Motor response

Figure 3.43: Track of a gust suppression signal at high speed

large measurement noise, when the reference signal does not present a quick-changing profile thesystem experiences an LCO of small amplitude. For completeness, the required control effort and theidentification history of the free-play width are shown in Figure 3.44.

In conclusion, it has been shown that the dual-loop strategy used in the previous test cases is notalways able to deal efficiently with problems that involve large stability changes due to the presence offree-play nonlinearities. To overcome this problem, the support of a controller based on the nonlinearinversion of the free-play is introduced with great benefits to the closed loop response. In fact, theintegration of the two controllers results in a robustly stable closed loop system, which is also able torecover its tracking properties at wind speeds that would have produced large LCOs as soon as thesystem was perturbed.

Wrapping up the results of this chapter, it can be concluded that:

• Industrially used controllers may be combined to suppress nonlinearities in a servo system

• The frequency-based version of the VRFT and FRIT algorithms presents excellent and similarresults in the tuning of control architectures of increasing complexity

• More complex strategies must be undertaken if some practical details, i.e. the vibrations at theclosing of the free-play gap, have to be resolved

• The dual-loop strategy is not sufficient to compensate effectively the free-play induced LCOwhen large aerodynamic disturbances are present. In this case the approach based on theadaptive inversion of the estimated free-play nonlinearity produces better results, being able ofstabilizing the system over a large range of wind speeds and of recovering the tracking propertiesof the closed-loop system.

3.5 Sliding Mode Observer

Some of the previously described controller requires the availability of the system state for theirimplementation. Furthermore, the next applications, which will deal with more complex controlproblem will require the adoption of a nonlinear estimator. For this reasons, a complete description of

112 Chapter 3

Time [s]0 1 2 3 4 5

Tor

que

[Nm

]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

(a) Control input

Time [s]0 1 2 3 4 5

α [d

eg]

0.4

0.5

0.6

0.7

0.8

0.9

1


(b) Identification of the free-play width

Figure 3.44: Other details regarding the gust suppression signal at high speed – Results relative to thesimulation free of noise

the nonlinear observer implemented in this work is provided here, highlighting its properties.Thanks to the assumed type of model nonlinearities employed here, i.e. free-play and friction in thissection, while in the next chapter aerodynamic nonlinearities will be also considered, a separationprinciple can be exploited [299], so any observer-based implementation can be carried out withoutregard to the associated controller design. Moreover, since significant differences between design andverification models may cause system instabilities and performance degradation, it has then beendecided to resort to a robust sliding mode observer [290, 291, 292].

Consider a generic dynamic system, nonlinear in its states but linearly influenced by its input:x = Ax+ f (x)+Bu

y = Cx+Du(3.44)

Starting from the available measures, the full state can be reconstructed through the following slidingmode observer:

xo = Axo +Bu+Lo(y−yo

)+Hov

yo = Cxo +Du(3.45)

The matrices A, B, C, D are already available from the modeling of the dynamic system. The matrixHo is defined as Ho = T−1CT, with T that will be defined in the following. The observation gain Lo

is computed through a standard linear optimal asymptotic Kalman observer, designed through thesolution of the algebraic Riccati equation:

AΛ+ΛAT −ΛCTR−1λ

CΛ+Qλ = 0

Lo =−ΛCTR−1λ

(3.46)

with Qλ being a positive semi-definite design covariance of the system disturbances and Rλ a positivedefinite design covariance related to measurement noise.To determine v, the dynamics of the state observation error e(t ) = x−xo is considered:

e = (A−LoC)e−Hov+ f(x) (3.47)

3.5. Sliding Mode Observer 113

Let us now make the hypothesis that there exists a function ζ(x) such that f(x) = Hoζ(x). This term is adisturbance, which sums up all the disturbances and uncertainties of the system. The discontinuousswitching vector v is then determined by satisfying the stability condition associated to the Lyapunovfunction Vo = eTTe, with T a positive definite matrix, obtaining:

Vo = eTTe+eTTe = eT ((A−LoC)T T+T (A−LoC)

)e−2vTHT

o Te+2ζTHTo Te (3.48)

Therefore, after assigning an appropriate positive-definite matrix Qv and solving the following Lya-punov equation:

(A−LoC)T T+T (A−LoC)+Qv = 0 (3.49)

the sliding vector is designed to be:

vi =

0 yi − yo,i = 0

γoyi − yo,i

∆|yi − yo,i | ≤∆

γoyi − yo,i

|yi − yo,i ||yi − yo,i | >∆

(3.50)

where ∆ is the boundary layer width, taken into account to avoid possible chattering in the stateestimation, according to [61], while γo is a further scalar observer gain.At this point, proceeding with Eq. 3.48, outside of the boundary layer region we have:

Vo ≤−eTQv e−2γo ||Ce||+2||Ce||||ζ|| ≤ −eTQv e−2(γo − ζ

) ||Ce|| < 0 (3.51)

where ζ = sup(||ζ||), i.e. the estimated worst disturbance level. Therefore, to guarantee that Vo < 0,γo ≥ ζ, and this can be assured through a fast trial-and-error tuning of the observer.Because e is not available directly, it is computed by pseudo inverting the relation eo = Ce, obtaining

e = CT(CCT

)−1eo . On the other hand, when the measurement error lies within the boundary layer

region of Eq. 3.50, the time derivative of the Lyapunov function results in:

Vo ≤−eTQv e−2γo∆+2∆||ζ|| ≤ −eTQv e−2(γo − ζ

)∆< 0 (3.52)

therefore the same condition γo ≥ ζ must be met.As it is well known [160], the switching term aims at inducing a sliding motion in the state error

space So = e ∈Rn : Ce = 0

, driving it to zero in a finite time. Despite the presence of modeling

uncertainties, such a behavior can be achieved through an appropriate tuning of Qv , γo and ζ.Figure 3.45 presents a few results obtained when the proposed observer is applied to the recon-

struction of the T-tail full state, comparing the results with the outcomes of a linear observer designedthrough the classical LQR method. It evidences that an appropriate, rather easy, tuning of the observerdesign parameters results in a fast and accurate estimation of the motor rate, whereas its linear coun-terpart always overestimates such a variable. Such a design considers the following weight matricesQλ = 103I, Rλ = 10−8, Qv = 108I and γo = 0.1, where I is an identity matrix of proper dimension.

In the case of Section 3.4.2 instead the tuning parameters Qλ = 10I, Rλ = 5 ·10−11, Qv = 105I andγo = 0.05 have been set after a quick trial-and-error test campaign.

114 Chapter 3

Time [s]0 0.2 0.4 0.6 0.8 1

θm[deg]

-150

-100

-50

0

50

100

150

200

250

300RealLinear ObserverSM Observer

Figure 3.45: Estimation of the motor rate in the T-tail case

Aerodynamic Nonlinearities Compensation

CHAPTER 4


The improvement of aircraft performances through active control is a well established topic andit is likely that adaptive control systems will further enhance future airplanes stability and maneu-verability [218, 300]. Their development demands a comprehensive approach to appropriately dealwith optimized designs, covering the whole spectrum of problems integrating flight mechanics andaeroservoelasticity, e.g.: flutter, control effectiveness and divergence, maneuver and gust loads, buf-feting, flight performances [1]. Until recent decades, a somewhat inadequate computational powerhas restricted the ordinary study of aeroservoelastic systems to linear(ized) subsonic and supersonicflight regimes [5]. Nowadays, advances in computers technology and Computational Fluid Dynamics(CFD) allow to adequately evaluate nonlinear unsteady loads for inviscid and viscous flows. Therefore,the adoption of a high fidelity, CFD-based aeroservoelastic analyses is becoming more and moreviable [6], thus allowing to better deal with transonic flows and strong oscillating shocks. The fullcontrol of these, possibly dangerous, nonlinear events is of utmost importance in avoiding unac-ceptable self-induced oscillations, instabilities, limit cycles, ride-quality deterioration and fatiguefailures [1]. Different approaches to the active control of aeroelastic systems can be found in the litera-ture, to cite a few: classical LQG design [301], robust H∞ framework [302], input limiting [244, 303],immersion and invariance [239], indirect adaption [287], adaptive neural networks, both static andrecurrent [67, 215, 217]. Within such efforts, it is worth mentioning the comprehensive BenchmarkActive Control Technology (BACT) research project, conducted at NASA Langley Research Center withthe objective of measuring and archiving unsteady aerodynamics data in transonic flow. It has allowedto study, record, and experimentally validate a wide variety of active flutter suppression designs, suchas: classical and minimax [260], robust H∞ and µ−synthesis [302], robust passification [304] andneural networks [217].

Given that aeroelastic systems change their stability with flight conditions, an efficient controllermust work properly over the whole flight envelope of interest. To achieve such a result, two differentapproaches can be considered, i.e. scheduled and adaptive control. While the former requires manydesigns, covering a set of flight conditions adequate to insure a stable and smooth scheduling, thelatter, once designed and verified for a relatively few peculiar cases, should be capable to adapt even tounforeseen conditions, with the likely added advantage of a reduced design effort.

In such a view this work adopts an Immersion and Invariance (I&I) approach [73, 131, 227, 234,239, 240, 241] for stabilising an aeroelastic system beyond its flutter speed. The related theory and alarge number of applications to mechanical and aerospace systems, including airplanes trajectorytracking, can be found in [73, 227], while applications to spacecraft systems are referred in [131], wherethe concept of filter embedment is introduced for the first time. The adoption of an I&I methodology torelatively simple aeroelastic systems is considered in [239, 240, 241], where, assuming ideal actuatorsand the availability of the full state, both single and multiple input controllers are developed.

Here active flutter suppressors will be designed on realistic, linear and nonlinear, reduced order

115

116 Chapter 4

models, including sensors and actuators, the latter saturating in position, speed, and torque, verifyingthem afterward through high fidelity, CFD-based, simulations of the aerodynamic sub-system.

The whole procedure will be verified through three test cases. At first the control of a simple pitchingand plunging typical section, with a NACA 0012 airfoil, is considered. This case is characterized byhighly nonlinear unsteady aerodynamic loads, producing significant shock motions and a strong limitcycle oscillation, with a relatively high frequency. Because of the nonlinear nature of the aerodynamicresponse, the reduced order model developed in Chapter 2 is employed here. This case allows also toverify the importance of adequately modeling the dynamics of the adopted sensors and actuators.The second case considers the, already mentioned, Benchmark Active Control Technology (BACT)wing, with its fully validated models and data [260, 302], which should make it likely that a designworking on the paper will do so also in an actual implementation. Because of its mild and low frequencyflutter oscillations, it admits a linearized quasi-steady aerodynamic approximation for the designphase. Its model is also of the typical section kind but, being a true wing, the related high fidelity CFDverifications will provide a way to adequately check possible three dimensional effects.The third and final test considers the flexible Goland wing [88], with an added trailing edge controlsurface to make possible its active flutter suppression. The related aerodynamics is quite simple, i.ean experimentally validated nonlinear quasi-steady strip theory [305]. Nevertheless, by providing amore complex three dimensional case with distributed elasticity, it allows to verify the proposed I&Icontroller on a more realistic system, characterized by a significant number of degrees of freedom.

The contribution of this chapter lies in the development of a somewhat systematic approach tothe adaptive stabilisation of nonlinear aeroelastic systems, considering high fidelity aerodynamicsand saturating actuators. Moreover, the adoption of nonlinear observers, along with the simulation ofthe related digitalized implementation of the controller, addresses possible applications to even morerealistic aeroelastic problems.

4.2 Aeroservoelastic Modeling

An aeroservoelastic system is typically composed by three interconnected parts: structure, aerodynam-ics and control, and, depending on specific analysis and design needs, different model fidelities can beused in the various stages of its development. Following a standard approach, a generic linear(ized)structural model, can be discretized into the classical multi-degrees of freedom scheme:

Ms qs +Cs qs +Ks qs = q∞fa +TTβmβ (4.1)

where: Ms , Cs , Ks are the structural mass, damping and stiffness matrices, qs the generalized structuralcoordinates, whose physical meaning is determined by the assumed discretization and fa the externalgeneralized aerodynamic forces, scaled by the asymptotic dynamic pressure q∞.

To explain the term TTβ

mβ in the above formula, it is remarked that the driving degree of freedomof any control surface is typically embedded in qs , so to be easily interfaced to the aerodynamicsubsystem in the very same way as any other structural motion. Therefore, control surface rotations, β,will be defined byβ= Tβqs , Tβ being an appropriate linking kinematic matrix, so that the generalizedhinge moments, mh , associated to the external control moments, mβ, will be given by mh = TT

βmβ.

After defining with D(v) the diagonal matrix associated to a vector v, we assume that a set of

4.2. Aeroservoelastic Modeling 117

position servos, commanding β to βc , can be adequately modeled as:

xact +D(2ξactωact)xact +D(ω2act)xact =D(ω2

act)βc

mβ =D(kβ)(xact −Tβqs)

|xact| ≤ xactmax

|xact| ≤ xactmax

|mβ| ≤ mβmax

(4.2)

with ξact and ω2act defining the actuator bandwidth, kβ an assumed acceptable low frequency resid-

ualization of their dynamic compliance, the ’max’ suffixed terms indicating the related (symmetric)saturation values.

In view of the need of modeling only the transfer function of accelerometer based measures,the related acceleration output, at assigned locations, will be given by a = Ta qs , Ta being a suitabledisplacement interpolation matrix. Therefore, the related transducer dynamics (sensor, compensation,antialiasing filter) is approximated through:

xsens +D(2ξsensωsens)xsens +D(ω2sens)xsens =D(ω2

sens)a =D(ω2sens)Ta qs (4.3)

Finally, taking for granted its stability, a generic formulation of a linear-nonlinear unsteady aerody-namic subsystem is written as:

xa = fxa

(xa ,qs , qs

)fa = fa

(xa ,qs , qs

)(4.4)

where xa is the aerodynamic state, which can be either a physical entity, as in the case of a raw CFDmodel, or a generically abstract reduced order state.

Defining the extended servo-elasto-mechanical degrees of freedom q = [qs xact xsens]T and thecorresponding state x = [q q]T = [

qs xact xsens qs xact xsens]T, putting together all of what above, we are

led to the following nonlinear, strictly proper, state space formulation:x = Ax+Bcβc +q∞Ba fa

(xa ,qs , qs

)+Bs usat

xa = fxa

(xa ,qs , qs

)y = Cy x

(4.5)

where y is the measure output and the other terms are defined through the following intermediatevectors and matrices:1

sβ = linsat(kβ, linsat(1,xact )−β) sact = linsat(2ξactωact, xact)+ linsat(ωact2,xact) (4.6)

M = Ms 0 0

0 I 0−D(ω2

sens)Ta 0 I

C =Cs 0 0

0 D(2ξactωact) 00 0 D(2ξsensωsens)

usat =

sβsact

(4.7)

K =

(Ks +TTβD(kβ)Tβ) −TT

βD(kβ) 0

0 D(ω2act) 0

0 0 D(ω2sens)

Baq =I

00

Bcq =

TTβD(kβ)

00

Bsq =I 0

0 I0 0

(4.8)

1linsat(a, v) =: a v if |v | ≤ vmax; a vmax if v > vmax;−a vmax if v <−vmax; the vector case must be intended componentby component.

118 Chapter 4

so that it is possible to set the following final compacted elements of Eq. 4.5:

A =[

0 I−M−1K −M−1C

]Ba =

[0

M−1Baq

]Bc =

[0

M−1Bcq

]Bs =

[0

M−1Bsq

](4.9)

Cy =[0 0 I 0 0 0

](4.10)

It should be remarked that when high fidelity aerodynamic models are taken into account, xa canbe very large, so that the system size can be limited substantially only by using a reduced orderaerodynamics. In this case, the ROM developed in Chapter 2 is employed. Therefore, whenever a highfidelity aerodynamic implies tens to hundreds of thousands states, the controller design will be basedon a significantly more favourable ROM size.

4.3 Flutter Suppression by Immersion and Invariance Control

The basic idea of an Immersion and Invariance (I&I) controller is to achieve a stabilization by immersingthe plant dynamics into a stable target system, possibly described by a reduced number of states.Then, by introducing appropriate adaptive terms in the related controller, it is possible to achieve theinvariance of the manifold containing such a target [227]. The related theory can be found in [73],while various applications to aeroelastic systems are reported in [239, 240, 241, 261].

Regarding MIMO problems, the designs of the target dynamics, the parameters estimator and thecontrol law are intrinsically coupled, as will be shown in proving the closed-loop stability throughLyapunov function analysis. Therefore, to highlight the peculiar points of the control law development,the design of the I&I controller must be explicitly detailed.

4.3.1 Design of the target dynamics

The previously mentioned target system t is here represented by Eq. 4.11, where Λ is a strictly positivematrix, here assumed as diagonal for sake of simplicity, with positive tunable design parametersΛi ,i , while y is the controlled performance, which can be any linear combination of the system statecomponents:

t = ˙y+Λy (4.11)

Anticipating that only the displacement at key points of the structure will be taken into account, sucha performance can be written as:

y = Hx = [Hq 0

]qq

= Hq q (4.12)

with H and Hq defining the appropriate performance output matrix, specified on a case by case basis todefine the desired y. Since the performance dynamics must be asymptotically stable over the manifoldt = 0, it is sufficient to develop a control law driving t to the origin.By differentiating Eq. 4.11, the dynamics of t is driven by the following equation:

t = ¨y+Λ˙y (4.13)

where the required y derivatives can be explicitly defined exploiting Eq. 4.5:

˙y = HAx+q∞HBafa +HBs usat +HBcβc = HAx = Hq q

¨y = HA2x+q∞HABafa +HABs usat +HABcβc

(4.14)

4.3. Flutter Suppression by Immersion and Invariance Control 119

The omission of the terms containing Ba , Bs and Bc in the first equation above can be trivially inferredby looking at their definitions against that of H. Therefore Eq. 4.13 becomes:

t = HA2x+q∞HABafa +HABs usat +HABcβc +ΛHq q

=αx+q∞ωfa +γusat + Bβc +ΛHq q(4.15)

with: α= (HA2

), ω= (HABa), γ= (HABs) and B = HABc , being unknown constant matrices.

Following the approach proposed in [240, 72], the SDU decomposition of B is computed. Accordingto [72], S is a positive definite symmetric matrix, D a diagonal matrix and U an upper triangularmatrix whose diagonal elements are unitary. Once again, S, D and U are unknown matrices, whilethe elements of D are assumed to be not null, with their sign being known. In practice, D can berewritten as D = ηConD, being ηCon the control gain to be tuned during the design. Equation 4.15 isthen rewritten in the following form:

t =αx+q∞ωfa +γusat +SDUβc +ΛHq q (4.16)

Then, after defining the strictly positive matrix Ct , whose elements are design parameters, an asymp-totically stable manifold is enforced by adding and subtracting the term Ct t to Eq. 4.16, thus obtaining:

t =−Ct t+αx+q∞ωfa +γusat +SDUβc +ΛHq q+Ct t

=−Ct t+S[DUβc +S−1αx+q∞S−1ωfa +S−1γusat +S−1 (

ΛHq q+Ct t)] (4.17)

Basically Ct is introduced to guarantee the asymptotic stability of the target dynamics once the controllaw is designed to compensate all the other terms in the previous equation.Exploiting the structure of D and U, the product DUβc can be decomposed in DUβc = Dβc + Uβc ,where U is the strict upper triangular part of U, i.e. U with all the diagonal entries set to zero. Equa-tion 4.17 can be rewritten in a much more compact form:

t =−Ct t+S(Dβc +ψχ

)(4.18)

Where the matrix ψ is a block diagonal matrix of Nin rows, where each block has the following structure:

i-th block of ψ=[

xT, q∞fTa , uT

sat,(ΛHq q+Ct t

)T , βTc (i +1 : Nin)

]i = 1, ..., Nin −1 (4.19a)

Last block of ψ=[

xT, q∞fTa , uT

sat,(ΛHq q+Ct t

)T]

(4.19b)

Where the notation v(i : j

)means that all the elements of v from position i to position j , are included.

The elements of S−1α, S−1ω, S−1γ and S−1 are arranged accordingly to the structure of ψ in χ. Infact, from the stabilization point of view, the order with which the elements of χ are arranged has noimportance.In order to somewhat simplify the I&I design procedure presented in Ref. [73], t = Hq

(q+Λq

), ψ and

βc are low pass filtered and attenuated [131, 234, 241] through with the following equations:

t f =−µt f +Hq(q+Λq

)(4.20a)

ψ f =−µψ f +ψ (4.20b)

βc, f =−µβc, f +βc (4.20c)

where µ is a further positive design parameter. Given that the proposed linear filters are asymptoticallystable, it can be shown [240] that the following ordinary differential equation is satisfied asymptotically:

t f =−Ct t f +S(Dβc, f +ψ f χ

)(4.21)

120 Chapter 4

4.3.2 Design of the parameters estimator

Since the parameters vector χ is unknown, I&I approximates it through the aid of shaping terms whichwill force the stable manifold to be invariant. Within such a view, the off-the-manifold variable z [73] isdefined as:

z = (χ+δ

)− χ (4.22)

Such a variable is basically a measure of the distance between the estimated parameters(χ+δ

)and

the real, unknown ones χ, with δ(t f ,ψ f ) being a yet to be chosen shaping function, so that, defining a

control law of the form βc, f =−D−1ψ f

(z+ χ

), it is possible to cancel the unknown constant parameter

vector χ of Eq. 4.21, which becomes:t f =−Ct t f −SψT

f z (4.23)

As a side note it is remarked that for δ= 0 Eq. 4.22 recovers the classical formulation of a certainty-equivalent adaptive controller [306]. Because of Eq. 4.22 we have also βc, f = −D−1ψ f

(χ+δ

), so

that, recalling Eq. 4.20b and ˙χ= 0, βc, f =µD−1ψ f

(χ+δ

)−D−1ψ(χ+δ

)−D−1ψ f z. Therefore, using

Eq. 4.20c the control law βc =µβc, f + βc, f can be written as:

βc =−D−1ψ(χ+δ

)−D−1ψ f z (4.24a)

orβc =−D−1ψ

(χ+δ

)−D−1ψ f

(χ+ δ

)(4.24b)

thus making βc, f useless.In view of ensuring the asymptotic stability of z, let us choose the following shape function: δ =γConψ

Tf t f , with γCon being a positive design parameter. The time derivative of the off-the-manifold

variable can now be computed:

z = χ+γCon

(−µψT

f +ψT)

t f +γConψTf

(−Ct t f −Sψ f z

)(4.25)

so that, imposing the following adaptive definition of χ:

χ=−γCon

(ψT −µψT

f −ψTf Ct

)t f (4.26)

the dynamics of z is given by:z =−γConψ

Tf Sψ f z (4.27)

At this point, defining Vz = 1

2zTz we have:

Vz = zTz =−γConzTψTf Sψ f z

≤−γConλS,min||ψ f z||2(4.28)

where λS,min is the smallest eigenvalue of S.. Therefore, z = 0 is an uniformly stable equilibrium point,with z ∈L∞(0,∞). Moreover, integrating Vz the following result is obtained:

Vz (∞)−Vz (0) ≤−γConλS,min

∫ ∞

0||ψ f z||2 d t (4.29)

therefore ||ψ f z|| ∈L2(0,∞). Having obtained the proof of stability for z, we are now able to determinethe asymptotic behavior of t f defined by Eq. 4.23. Similarly to the previous calculation, defining

Vt = 1

2t f

Tt f we have:

Vt =−tTf Ct t f − tT

f Sψ f z ≤λCt ,min||t f ||2 +λS,max||t f ||||ψ f z|| (4.30)

4.3. Flutter Suppression by Immersion and Invariance Control 121

being λCt ,min the smallest element of Ct and λS,max the greatest eigenvalue of S. Exploiting Young’sinequality, we obtain:

Vt ≤−λCt ,min

2||t f ||2 +

λ2S,max

2λCt ,min||ψ f z||2 (4.31)

Therefore, since ||ψ f z|| ∈L2 (0,∞), we have also ||t f || ∈L2 (0,∞), thus proving the asymptotic stabilityof t f .

4.3.3 Final control law and proof of stability

At this point, putting together Eqs. 4.24a and 4.27, we can explicitly define the control effort βccomputed through the I&I logic:

βc =−D−1ψχ+γConD−1[−ψ f ψ

Tf t+

(µψ f ψ

Tf −ψψT

f −ψ f ψTf Ct

)t f

](4.32)

As the reader can note from Eq. 4.32, βc seems to be defined in an implicit way, because ψ and ψ fdepend on it. But thanks to the structure of ψ, shown in Eq. 4.19, it should be clear that βc is unique,since it does not depend on βc,1. In this way, starting from the last line of Eq. 4.32, the componentβc,Nin can be evaluated explicitly, so that, all the elements βc,i can be backward computed up to βc,1.

The nonlinear adaptive compensator and its control command are then resumed by Eqs. 4.20a, 4.20b, 4.26and 4.32, which are summarized here below for the reader’s convenience:

t f =−µt f +Hq(qo +Λqo

)ψ f =−µψ f +ψ

χ =−σχ−γCon

(ψT −µψT

f −ψTf Ct

)t f

βc =−D−1ψχ+γConD−1[−ψ f ψ

Tf Hq

(qo +Λqo

)+ (µψ f ψ

Tf −ψψT

f −ψ f ψTf Ct

)t f

] (4.33)

with the suffix o indicating that the related quantities are computed using the values that will beprovided by a sliding mode observer, whose implementation details were described in Section 3.5.

As can be noticed, the equation for χ has been modified, with respect to its parent Eq. 4.26, byadding an appropriate proportional feedback −σχ, with σ> 0 being a new design parameter. Sucha change is an often used fix [307] aimed at avoiding a possible long term drift associated to pureintegrations of the kind of Eq. 4.26, eventually hindering the previously demonstrated convergence andstability properties [308] because of unmodeled dynamics and disturbances. The stability improve-ment brought in by −σχ can be found in [132, 307, 308]. It is then possible to tune σ so to providean acceptable trade-off between the nominal adaption performances and a sizable robustness gainagainst the uncertainties and disturbances.

It is further remarked, as reported in [241], that the so designed controller has the interestingfeature that, defining the manifold Ω as:

Ω=(ψ f ,z

): ψT

f z = 0

(4.34)

the closed loop system is confined within Ω, with t f = −Ct t f in Ω. Such a result could also havebeen obtained through the deterministic control law βc, f =−ψT

f χ, with χ known. Therefore, thanksthe inclusion of the additional shaping term δ, it follows that an I&I based controller asymptoticallyrecovers the performances of a deterministic controller.

To demonstrate the asymptotic stability of the whole control system, a third Lyapunov function isresorted: W (t f ,z) =Vt +Vz , so that, being:

W ≤−λCt ,min

2||t f ||2 −

(γConλS,min −

λS,max

2λCt ,min

)||ψ f z||2 (4.35)

122 Chapter 4

ξ ω0, rad/s βmax, deg βmax, deg/s mβmax , Nm

Actuator 0.56 65 15 40 Steady aero-mβ at β=βmax

Sensors 1.0 190

Table 4.1: Actuator and sensors parameters

NACA 0012 BACT Goland wing

kβ, Nm/rad 4 ·104 2 ·104 105

Table 4.2: Actuator compliances adopted for the three test cases.

it can be inferred that the pair(t f ,z

) ∈L∞(0,∞), if γCon ≥ 1/(2λCt ,min

)κ (S), where κ (S) is the condi-

tioning number of S, ratio between its maximum and minimum eigenvalues. In practice, the followinginequality will be imposed:

γCon ≥ bd

2λCt ,minκ (S) (4.36)

with bd being an assigned design bound. Given that the linear filters in Eq. 4.20 are all asymptoticallystable, if t f is bounded also t and, consequently, y are bounded. Then, the whole state will be boundedand, since W is uniformly continuous, the convergence toward the origin can be proved by usingBarbalat’s lemma [306].

The meaning of the design parameters Λ, Ct , µ, γCon and ηCon should be quite clear: Λ drivesthe performance to zero when the target manifold is reached, Ct modulates the convergence towardthe stable manifold, µ determines the filtering level of the controller, γCon is the learning rate of theadaptive law and ηCon is the control gain.

4.4 Test Cases

The parameters shared by all the accelerometers and actuators of the following applications aresummarized in Table 4.1. The accelerometer parameters are mostly dictated by the assumption of asecond order anti-aliasing filter for the digital implementation, whose bandwidth is significantly belowthe one of the related sensor. Instead, plausible values for the actuators have been derived from theexperimental data of [309], albeit with a bandwidth scaled down to 65[rad/s], from 165, to take intoaccount both a more easily achievable value and a more challenging controller design. The valuesof the actuator compliances have been found to be not critical over a sensible range of values andeventually set as reported in Table 4.2.

The design of the controllers could have been carried out either interactively or through a numericaloptimization which, because of the relatively low system order and smooth dependence on a smallnumber of parameters, could have been based on an efficient gradient free optimizer, e.g. [310].Eventually, the former option has been preferred. In fact, it requires no further coding and can beeasily guided by following a simple heuristic procedure, based on the previously hinted physicalunderstanding of the design parameters. The filter parameter µ is typically chosen in relation to themaximum frequency of the open loop system response, estimated by Fourier transforming a fewopen-loop time histories. Then, after verifying that a tentative Ct = 1 can be a suitable choice, γCon iscomputed accordingly to Eq. 4.36, followed by a few analyses carried out by maintaining Λ= 1 while

4.4. Test Cases 123

0 2 4 6 8 10 12 14 16 18 20−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Time [s]

t [m

]

Ideal simulationUncertain aerodynamics

(a) t variable

0 2 4 6 8 10 12 14 16 18 20−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

t[m

/s]

Ideal simulationUncertain aerodynamics

(b) t variable

Figure 4.1: Effect of an uncertain aerodynamic model on the closed loop dynamics.

determining appropriately the values of Ct and γCon leading to a reasonable maximum control effort,eventually increasing Λ until a desired settling time is achieved. The parameter ηCon is set to 1 andthen adjusted if further improvements of the closed-loop behavior are required.

On the other hand, the sliding mode observer of Section 3.5 demonstrated to work rather well withQλ = I, Rλ = 0.1I and Qv = 10I, along with a first guess of 0.1 for γo , followed by a tuning through openloop simulations.Furthermore, a margin of bd = 2 as defined in Eq. 4.36, proved sufficient to achieve an adequate level ofrobustness and remained a common choice for all the following tests. The value of the feedback gainσ in Eq. 4.33 is set to an unit value, being this sufficient to assure a stable behavior of the parameteradaption law also in presence of disturbances.

To assure the system adaptivity and stability over a wide range of operating conditions, the con-troller parameters are tuned considering various flight speed, at least 25% greater than the openloop linearized flutter speed, combined with different type of simulations, such as the response tolarge initial conditions, to input pulses, eventually evaluating the controller adaption speed when itscompensator is switched off-on during a simulation. Because of the simplified structure assumed forthe observed ROM aerodynamic state, also an extensive set of verifications has been carried out bymarkedly biasing and randomly disturbing high fidelity generalized forces applied to the structure,so to simulate significantly different forces with respect to those used for tuning the control system.The results obtained were quite satisfactory, without any instability, contained control activity, with nosignificant saturation, except for the speed of the actuators. A sample result related to the followingtypical section, for fa = fabias + fanom (1+4r(t)), is shown in Figure 4.1. The related terms are: the biasterm, fabias , set at the steady aeroelastic solution for θ = 4 deg, fanom the nominal force provided byAeroFoam, r(t) a random normal time variation with a unit standard deviation. As can be noticed,the aerodynamic uncertainties introduce a persistent disturbance in closed loop, deteriorating theconvergence to the target dynamics. However, such a convergence is eventually recovered during thesimulations, verifying the capability of the control system to withstand large model uncertainties. Allthe tuned designs and verifications have been determined by using an explicit Runge-Kutta integratorwith adaptive step control, providing a precision adequate to allow avoiding an exact matching ofsaturation/desaturation time instants. Moreover a realistic digital implementation of the proposedcontroller has been taken into account. Through some preliminary continuous designs, it has been

124 Chapter 4

0 1 2 3 4 5 6 7 8−10

−8

−6

−4

−2

0

2

4

6

Time [s]

θ [d

eg]

Analog implementationDigital implementation, 60% delay

(a) Pitch response

0 1 2 3 4 5 6 7 8−8

−6

−4

−2

0

2

4

6

8

Time [s]

βc [d

eg]

Analog implementationDigital implementation, 60% delay

(b) Control surface deflection

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

−80

−60

−40

−20

0

20

40

60

80

Time [s]

β[deg/s]

β, Analog implementation

βc , Digital implementation, 60% delay

β, Digital implementation, 60% delay

(c) Control speed, with saturations

Figure 4.2: Comparison of a continuous and digital controller implementation

possible to verify that the sampled behavior of the continuous sliding observer and I&I compensatorcould be adequately matched at a frequency of 180 Hz, the related discretization being based on afix step Runge-Kutta-Heun integration scheme. To correctly simulate such a digitalization there isthe need to care for the processing delay (input-calculation-output), associated to the chosen dataacquisition system and control computer. Among the many carried out, Figure 4.2 shows a sample ofthe simulations comparing the controlled responses of the continuous and digitalized observer, at 180Hz with a 60% (3 ms) processing delay. However, despite the many successful verifications obtainedwith the mentioned implementation parameters, all the following simulations will be based on thesame 180 Hz sampling rate mated to a somewhat more conservative 30% (1.5 ms) processing delay.

In concluding this illustration of the features common to all the following test cases, it should beremarked that, for each of them, a vast set of simulations has been carried out against: varied ROMand finely discretized aerodynamic models, system disturbances and measurement noise, a ±20%change of most of the structural parameters. Nevertheless, for sake of brevity, only samples of therelated results will be presented, trying to blend them in a way providing as a complete as possiblepicture of some interesting findings of this work.

4.4. Test Cases 125

m Jθ JβTE Shθ ShβTE fh fθkg kg m2 kg m2 kg m kg m Hz Hz

43.41 8.14 1.628 5.43 1.0853 4.65 9.3

Table 4.3: Three degree-of-freedom typical section data

V∞

[m/s]160 170 180 190 200

h/c LC

O

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2CFDROM

(a) Plunge LCO amplitude vs. Flight speed

V∞

[m/s]160 170 180 190 200

βT

ELC

O

[deg

]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CFDROM

(b) Flap LCO amplitude vs. Flight speed

Figure 4.3: ROM vs. high fidelity CFD amplitude trends

4.4.1 Typical section with trailing edge control surface

This example has been chosen for two main reasons: it shows the ability of an I&I controller tostabilise the response of a significantly nonlinear system and demonstrates the importance of a correctmodeling in the design phase.

It is related to a plunging-pitching typical section, featuring a NACA 0012 airfoil, with a trailingedge aileron, at sea level and M∞ = 0.8, whose structural data are found in Table 4.3. The aileronextends from 75% of the chord.

This is a kind of benchmark characterized by a significantly complex unsteady nonlinear aero-dynamic behavior [102], producing an ample limit cycle having a frequency around 10 Hz, whichcannot be matched by an overly simplified aerodynamic approximation. For such a reason a referencehigh fidelity AeroFoam-Euler approximation of 12000 two dimensional cells, i.e. 48000 unknowns,will be used as the base for its validation. Two accelerometers are placed on the wing in order toestimate its full state through a sliding mode observer. One of them is set up on the wing leadingedge, while the other at 70% of the chord, i.e. just before the aileron hinge, to maximize the signal tonoise ratio, as done in [260]. After remarking that the servo-elastic subsystem will add just 12 states(6-structural, 2-actuator and 4-sensors), it should be clear that the overall system size is dominated bya huge number of aerodynamic states, which is unsuitable for the design of any active controller.

There is then the need to resort to the aerodynamic ROM of Chapter 2, so the previously presentedtraining procedure has been adopted, using a training signal with maximum frequency of kmax = 0.8.The ROM presents a converged behavior with only ten aerodynamic states. The resulting nonlinearaeroservoelastic model has twenty-two states and develops a limit cycle beyond a numerically esti-mated linearized flutter velocity VF,OL = 155 m/s. Figure 4.3 compares some trends of the ROM limitcycle parameters against their high fidelity counterparts.

126 Chapter 4

γo Λ Ct µ γCon ηCon 2CtγCon

0.5 50 30 750 0.04 1 2.4 > bd

Table 4.4: Controller parameters: typical section.

0 1 2 3 4 5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time [s]

h/c

Open loopClosed loop w/o disturbanceClosed loop w disturbance

(a) Response to input pulse of the plunge degree-of-freedom

0 1 2 3 4 5

0

1

2

3

4

5

6

7

8

Time [s]

βT

E [d

eg]


(b) Related control effort.

Figure 4.4: Input pulse response of the design model, V∞ = 190 m/s

Having available a rather robust ROM to represent nonlinear aeroservoelastic responses, it is nowtime to design a stabilization strategy able to suppress unstable behaviors. To achieve good adaptiveperformances against pulse perturbations applied through the flap, different flight speeds, up to 25%of VF,OL, are taken into account to tune the controller parameters. The target performance, Eq. 4.12,is the (linearized) vertical displacement at the leading edge of the typical section, so that, beingq = [h θ βTE xact xsens1 xsens2 ]T we have Hq = [1 lLE 0 0 0 0], where lLE is the distancebetween the elastic center of the airfoil and its leading edge.

Carrying out the design with the interactive procedure previously described the control parametersof Table 4.4 are obtained. Notice that in the inequality presented by Eq. 4.36 the term κ (S) = 1 forcontrol laws with one input only, therefore the actual inequality is the one shown in Table 4.4.

Using both the design ROM and high fidelity CFD, it is possible to show a few simulations illustratingthe effectiveness of the obtained controller, implemented through its reduced order sliding observer.At first, Figure 4.4 depicts a sample response of the controlled typical section to an input pulse appliedat the design point. A significant random disturbance, having a maximum amplitude of 1 degrees, hasbeen applied to the control surface, so showing the controller insensitivity to disturbances. Then a fewhigh fidelity responses at the off-design condition of V∞ = 215 m/s are presented in Figure 4.5, wherethe controller is switched on after t = 2.5 seconds, when the limit cycle is fully developed. Spillovereffects over the larger aerodynamic model have not been found in any of the verifications carried out.Some samples of the flow field during the limit cycle suppression are depicted in Figure 4.6, showing asignificant shock oscillation amplitudes of the order of 23% of the chord, periodically appearing anddisappearing on the upper and lower airfoil surfaces. The controller appropriately cancels the largedisturbance command applied by the control, eventually driving such a amplitude down to 2% of thechord. Nevertheless, despite the good results obtained, it can be useful to remark that the robustnessof an I&I controller can result in being inadequate against excessively simplified design models, e.g.

4.4. Test Cases 127

0 1 2 3 4 5

−5

−4

−3

−2

−1

0

1

2

3

Time [s]

θ [d

eg]


(a) Pitch response (b) Related control effort

Figure 4.5: High fidelity response to off-on control, V∞ = 215 m/s

(a) Front position of the shock during the limit cycle (b) Rear position of the shock during the limit cycle

(c) Stabilized position

Figure 4.6: Various phases during the limit cycle oscillation suppression

128 Chapter 4


0.5 50 30 125 0.1 1 6 > bd

Table 4.5: Controller parameters: ideal typical section with sensor/actuator dynamics neglected

unmodeled sensor and actuator dynamics. For example, if we totally neglect those dynamics, a designat V∞ = 190 m/s will provide good performances with the parameters of Table 4.5, both for the reducedorder and high fidelity CFD models, a sample of the surface rotation being depicted in Figure 4.7a.Instead, by verifying the very same controller after accounting for its digital implementation andthe very same actuator adopted in the previous design, we can see, Figure 4.7b, that it results in arather violent instability. As witnessed by Figure 4.7c, something similar, albeit with a somewhat softerappearance, applies also after accounting for just the previously used sensor dynamics, even if its 25Hz bandwidth is well in excess of the limit cycle frequency. Such outcomes clearly point out the needof taking into account any significant realization delay from the very inception of a design procedure.

4.4.2 BACT wing model

As is has been already hinted at in the introduction, the Benchmark Active Controls Technology(BACT) project is part of NASA Langley Research Center’s Benchmark Models Program for studyingtransonic aeroservoelastic phenomena. The BACT wind-tunnel model was developed to collect highquality unsteady aerodynamic data (pressures and loads) near transonic flutter conditions and todemonstrate the potential of designing and implementing active control systems for flutter suppressionusing flaps and spoilers [311]. Therefore, it is a well known, easy to use, detailed and fully validatedaeroservoelastic model [260, 302, 311], which has become an often referred benchmark applicationfor verifying nonlinear aerodynamic analyses and active controls design methods. It is an elasticallyconstrained rigid rectangular wing model, with NACA 0012 sections, equipped with a trailing-edgecontrol surface and upper and lower-surface spoilers, which can be controlled independently throughwell performing hydraulic actuators. Its dynamic behavior is very similar to a classical typical sectionbut, because of its low aspect ratio, it displays a not so simple three-dimensional transonic flow.However, it has been shown, e.g. [260], that the related nonlinear aerodynamic behavior is mild enoughto produce slowly growing limit cycle oscillations, which can be verified only through high fidelity CFDvalidations [121]. Because of the above remark, the literature related to the design of active controllersfor the BACT wing presents many instances of effective, experimentally validated, applications of lineardesign techniques [217, 260, 302, 304].

In such a view the fully linear model, i.e. aerodynamics included, proposed in [311] will be used todesign an I&I controller, to be verified against fully nonlinear CFD simulations afterward. Moreover, itshould be remarked that in view of a quite small flutter reduced frequency, kF ≈ 0.05, such a modeladopts also a highly simplified quasi-steady linearized unsteady aerodynamic approximation. Then,to ascertain a correct adoption of its data, the related design model, which has the same 12 state as theprevious NACA 0012 typical section, but without any added aerodynamic state, has been subjectedto a few simple flutter calculations. A sample result, at Mach M∞ = 0.77 in heavy gas R12 [311], avalue that will be used also for all the following nonlinear verifications, shows a predicted flutter speedof VF,OL = 108 m/s, only 1.3% more than the corresponding test value. Once more, targeting the tipleading edge motion, the controller is designed to stabilise the wing up to a speed 35% higher thanthe original linear flutter point, against pulse perturbations applied through the aileron. Carrying outthe usual interactive design procedure the control parameters of Table 4.6 are obtained. A sampleof the results to an input pulse with an amplitude of 5 degrees obtained during the design phase is

4.4. Test Cases 129

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

6

Time [s]

β [d

eg]

Closed Loop without disturbanceClosed Loop with disturbance

(a) Ideal control surface deflection

0 0.5 1 1.5 2 2.5 3 3.5 4−15

−10

−5

0

5

10

15

Time [s]β [d

eg]


(b) Ideal design response, after modeling the actuator

0 0.5 1 1.5 2 2.5 3 3.5 4−15

−10

−5

0

5

10

15

Time [s]

β [d

eg]


(c) Ideal design response, after modeling the accelerometers

Figure 4.7: Effects of omitting sensors and actuator dynamics in the design


0.01 50 30 30 0.4 1 24 > bd

Table 4.6: Controller parameters: BACT wing.

130 Chapter 4

0 1 2 3 4 5 6 7 8 9 10

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

θ [d

eg]

Open LoopClosed Loop without disturbanceClosed Loop with disturbance

(a) Plunge output

0 1 2 3 4 5 6 7 8 9 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

Time [s]

β [d

eg]


(b) Related control effort

Figure 4.8: Response of the design model to a sizeable input pulse

m Jθθ Jββ Shθ Shβ bending stiffness E J torsional stiffness G JKg/m Kg m Kg m Kg Kg m N m2 N m2

35.72 8.64 0.4 6.52 0.05 994506.6 100714.9

Table 4.7: Goland wing uniform inertial and elastic properties per unit length

shown in Figure 4.8. An efficient stabilisation with a limited control effort, even at a flight speed 20%greater than the open-loop flutter speed, is worth being pointed out. It should be remarked also that,differently from other references [73, 131], a value γCon ∼ 5 has been verified to be a limiting stabilitybound, whereas higher values invariably produce an unstable controller.

The verification model consists of an FV discretization, whose fineness has been determined on thebase of a steady-state convergence analysis, resulting in a mesh of 103040 cells (515200 aerodynamicstate components). As already mentioned, the control robustness has been also verified against inertiaand stiffness changes. A sample of such verifications, associated to a 20% increase of both bendingand torsional stiffnesses at 135 m/s, is shown in Figure 4.9, for an initial condition rather far awayfrom the possible uncontrolled LCO and superimposing an initial strong aileron pulse to a randomcommand. The adaption capability of the controller to appreciable changes of the nominal designshould appear clearly, even in front of a significant nonlinear aerodynamic behavior associated to alarge shock motion, spanning 20% of the wing chord during the initial part of the transient.

4.4.3 Goland wing

The Goland wing is a test case, found in the classical aeroelastic literature [88], which can exhibit bothstructural, wing and wing-store aerodynamic nonlinearities [305], of which only the one related to theaerodynamics of the clean wing will be considered here. The related geometry, aerodynamic, inertiaand structural data, taken from [88], can be found in Table 4.7. A proper modal basis, including the freerigid rotation of the here added trailing edge control surface, will be used to model the linear structure.The required 30 lower modes have been obtained through a 50 beams finite element discretization. Insuch a way the matrices Ms , Cs , and Ks will be diagonal, while, recalling that β= Tβqs , M, C and K willbe coupled through the sensors and actuator dynamics, see Eqs. 4.7 and 4.8. It might also be worth

4.4. Test Cases 131

0 2 4 6 8 10 12 14 16 18 20−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Time [s]

h [m

]

Open loopClosed loop without disturbanceClosed loop with disturbance

(a) Plunge output

0 2 4 6 8 10 12 14 16 18 20−10

−8

−6

−4

−2

0

2

4

6

8

10

Time [s]

β [d

eg]

Closed loop without disturbanceClosed loop with disturbance


Figure 4.9: High fidelity response, BACT test case

pointing out that that each modal column of Tβ is just the trivial difference between the rotation, θ, ofthe section at which the aileron is driven and the corresponding absolute modal rotation of the aileronitself.

This example considers a very simple aerodynamic model, i.e. a quasi-steady strip theory, linearin the design phase, nonlinear for the verifications. In fact, since a linear flutter analysis shows abending-torsional flutter having a reduced frequency of only 0.01, similarly to the BACT wing, theadoption of such an approximation should be justifiable. Instead, in view of its somewhat low aspectratio, a few, legitimate, doubts can be cast on the strip theory, which appears nonetheless to be used inother Goland based literature instances [305, 312]. That is likely because it is sufficiently adequate fora simpler qualitative demonstration of some nonlinear aerodynamic phenomena. In such a view, aspanwise Schrenk’s correction [313] is applied and the resulting model is exploited to verify how theproposed single input I&I controller can stabilize the nonlinear response of a test case closer to a realwing. So, calling x and y the chord and span wise running coordinates and s the wing span, we have:

CL(y) = 2π

[14s +

1

2

√1− (y/s)2

](αeff +τW (y)β

)CM (y) = 2π (xCA −xEA)

[1

4s +1

2

√1− (y/s)2

](αeff +τW (y)β

)CMh (y) =CMh ,αα+CMh ,ββ

(4.37)

with W (y) = 1 for ystarta ≤ y ≤ yend

a and W (y) = 0 elsewhere. The coefficients CL,β = 1.25 1/rad andCM ,β = -1.85 1/rad, are estimated from thin airfoil theory, as well as τ = 0.33, which is nonethelessdecreased to 0.23 to penalize the three-dimensional aileron efficiency. The angle of attack is defined

as α(y) = θ(y)+ h(y)

V∞+d3/4θ, being θ the wing torsional rotation, h the vertical displacement of the

elastic axis, positive downward and d3/4 the well known backward distance of the 3/4 chord point fromthe elastic axis.In the verification stage an aerodynamic nonlinearity is taken into account by simply replacing α withthe experimentally tuned αeff =α−10.26α3, valid for α up to ± 11 degrees [312].Using the above approximation to calculate the generalized modal aerodynamic forces, the resultinglinear terms will provide the matrices Ca and Ka , with Ma = 0, while the quadratic and cubic terms

132 Chapter 4


0.1 20 10 120 0.2 1 4 > bd

Table 4.8: Controller parameters: Goland wing

0 2 4 6 8 10 12 14 16 18 20

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time [s]

wT

ip [m

]


(a) Vertical displacement at the wing tip

0 2 4 6 8 10 12 14 16 18 20

−6

−4

−2

0

2

4

6

Time [s]

β [d

eg]


(b) Related aileron deflection

Figure 4.10: Design model response to a sizeable aileron pulse

will be gathered in fanl .Exploiting the previously presented modeling elements the design of the control parameters will

proceed with 5 normal modes, including the rigid relative aileron rotation, for a total of 16 states(10-structure, 4-sensors, 2-actuator). Then the verification phase will be carried out on a refinedmodel including many more modes and the above non linear correction of the angle of attack. Suchan approach is similar to the one taken for the BACT wing, with the exception that the verificationphase will not be based on high fidelity CFD simulations. In fact the aim of this example is directed toverifying the application of an I&I design to a servo-structural system a bit closer to a somewhat morerealistic system. Because of such an assumption a more complex and complete modelling tool willlikely not affect significantly the whole ROM based design and its high fidelity verifications.

A preliminary flutter analysis has been carried out for determining the stability of the aeroser-voelastic system. The estimated flutter speed was found to be VF,OL = 38 m/s. Once again, the targetperformance y is chosen to be the usual vertical displacement at the leading edge of the wing tip, thussynthesizing bending and torsion effects into a single variable. The needed single line target matrix Hq

can be determined after calling: qm the modes amplitudes vector, Thtip the modes displacements at tip

leading edge and defining q = [qTm xact xsens1 xsens2 ]T, so that we have Hq =

[Thtip 0 0 0

].

The controller can then be designed to stabilize the wing up to a speed 40% higher than the originallinear flutter speed, in front of pulse perturbations introduced by the aileron. So, carrying out theinteractive design procedure, the control parameters of Table 4.8 are obtained. A sample of an almostconverged design iteration is shown in Figure 4.10 for the application of a 5-degree aileron deflectionfor 0.2 seconds.

The controller robustness verification against modal spillover and model uncertainties will bebased on model with 30 vibration modes, i.e. 60 structural states, 6 sensor-actuator states and thenonlinear aerodynamic model. Figure 4.11 shows a sample of the obtained results for the response to a

4.4. Test Cases 133

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

8

Time [s]

θT

ip [d

eg]


(a) Wing tip rotation

0 2 4 6 8 10 12 14 16 18 20−10

−8

−6

−4

−2

0

2

4

6

8

10

Time [s]

β [d

eg]


(b) Related aileron deflection

Figure 4.11: Quasi-steady nonlinear aerodynamic verification to off-on control

0 5 10 15 200.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

Level of weakening in bending stiffness [%]

Max

imum

con

trol

effo

rt [d

eg]

(a) Maximum control efforts vs. bending stiffness

0 5 10 15 200.5

1

1.5

2

2.5

3

3.5

Level of weakening in torsional stiffness [%]

Max

imum

con

trol

effo

rt [d

eg]

(b) Maximum control efforts vs. torsional stiffness

Figure 4.12: Results of a robustness verification on the Goland wing

large initial condition with a random distrurbance of 2 degrees commanded by the control surface, ata flow speed 30% in excess of the open-loop flutter speed. To prove the fast adaptivity of the proposedcontroller, the control action is switched on after the limit cycle is fully developed. It should cometo little surprise that for such a speed and disturbances the initial control effort is quite significant.Nevertheless it can be verified that related response evidences again the robustness of the adaptive I&Icontroller.

As a further robustness demonstration, a series of detailed verifications have been carried out on amodified models with weakened bending and torsional stiffnesses. It is then possible to track the trendof the maximum control effort against the related stiffness changes at a fixed flight speed. The testvelocity is 35% higher than VF,OL and the applied disturbance is an aileron pulse having an amplitudeof 5 degrees and duration of 0.2 seconds. The obtained trends are shown in Figure 4.12. It should benoticed that the control effort is much more affected by the torsional stiffness changes, whereas areduction of the bending stiffness results in a decreased maximum control effort. This is likely due tothe effect that a decreasing bending stiffness has on the flutter behavior of the wing. In fact, reducing

134 Chapter 4

the frequency of the first bending mode results in a slight postponement of the bending-torsion flutteronset. Consequently, a reduction of the bending stiffness results in a slightly increased flutter speed,which should justify the trend of the control effort.

Integrated Compensation of Aerodynamicand Structural Nonlinearities

CHAPTER 5


The study of nonlinearities in control surface actuation systems is more and more relevant in activeaeroelastic control design. Therefore, an earlier introduction of nonlinear effects may lead to improvedand less expensive products [20].

Free-play nonlinearities are not only encountered in aeroservoelasticity, e.g. see the works ofKarpel [20, 31] and Vasconcellos [314] regarding the modeling of nonlinear actuators and the series oftests recently carried out by Kholodar [25]. In fact they must be studied, modeled and compensatedin most electro-fluid-mechanical applications [72, 315, 316, 317, 318, 319]. A significant improvedcomprehension of free-play phenomena in control surface mechanisms can be found in the researchescarried out at Duke University by Conner et al. [21, 22, 320]., similar studies being still underway onmore complex models [23]. Limit cycle oscillations induced by a control surface free-play have alsobeen studied on an F-16 aeroelastic model [18], simulating the effects of various free-play angles anddifferent maneuver conditions. The same type of problems has also been experienced on real aircraft,as witnessed in [321]

Nevertheless, the above mentioned works aimed mainly at the analysis of such peculiar nonlinearbehaviors, often without addressing the possibility of their compensation. It is well known thatthe presence of free-play can significatively alter the stability of an aeroservoelastic system, both inopen and closed loop [24, 31]. In particular, several studies have shown that the presence of such anonlinearity can jeopardize the properties of a control law designed without accounting for it, thusleading to closed loop responses characterized by significant residual LCOs [24, 33, 34, 35]. Therefore,the presence of free-play often worsen the performance of aeroelastic control systems, possibly leadingto multistable behaviors as in the test case considered in this work.

Following the discussion presented in Chapters 3 and 4, the problem of the compensation ofcontrol surfaces free-play is here addressed. At first, the proposed approach designs an adaptivecontrol law, based on the I&I methodology [73, 227, 239], on the ideal system, i.e. by neglecting thepresence of the free-play. Then, an additional adaptive control based on the nonlinear inversionapproach [72] is applied in order to compute the variation of the control input required to avoid beingcaught within the free-play range. To obtain such a beneficial behavior, as shown also in Chapter 3, theadditional measures required are the control surface deflection, the applied torque and the actuatorsposition, which are added to the accelerometric measures used to reconstruct the entire state by meansof the sliding mode observer of Section 3.5. Within such an approach, the free-play parameters areassumed as unknown and a real time approximator can be trained on-line to estimate their values indiscrete time, so to obtain a digital compensation scheme to be directly applied to realistic controlproblems.

The proposed method will be tested at first on a four degrees of freedom airfoil, including strongtransonic nonlinearities, taken into account through the CFD-based reduced order model of Chapter 2.This case is thus characterized by highly nonlinear unsteady aerodynamic loads, producing significant

135

136 Chapter 5

shock motions and large amplitude LCOs at a relatively high frequency, which, being coupled to anonlinear actuation servo-system, can result in a challenging test for the proposed approach. Asalready studied in Chapter 2, the presence of both aerodynamic and structural nonlinearities makesthe system bi-stable, leading to complex responses dependent on the initial conditions and the inputused to excite the system. To the author’ knowledge, this is one of the first studies of bi-stable behaviorfor this kind of aeroelastic and aeroservoelastic system, even if investigations about this phenomenonwere already carried out in the field of mechanics [ 322] and control [323]. This fact makes the designof the flutter suppression system more challenging, because of the presence of different nonlinearsources, which leads to the suppression of responses characterized by a wide range of amplitudes.The second test case will deal with the Goland wing previously studied. In this case two control surfaceswill be added to allow the design of a multi input controller, with the inclusion of free-play and frictioneffects in the related actuation system. Even if this case presents a much simpler but still nonlinearaerodynamic model, it is anyway of interest to test the performance of the present control methodwhen a larger number of states has to be controlled.

Additional original contributions presented in this chapter are the introduction and employmentof a simple estimator of the possible friction torque acting on the control surfaces hinge, as wastheoretically detailed in Section 3.3, which further helps the reduction of the residual limit cycleoscillations, and an in-depth analysis of the influence of the actuators bandwidth on the controllerperformance.

Therefore, the goal of the present part is to demonstrate a simple yet effective method for com-pensating free-play nonlinearities in control systems, while also considering other kinds of nonlineareffects, e.g. shock wave aerodynamics and structural friction. In what follows, several analyses arecarried out considering different values of free-play amplitudes and other key parameters, showingsignificant changes of the system response when they are varied. The adaptive control law is furthertested against significantly modified structural parameters, highlighting a few interesting results.

5.2 Aeroservoelastic Modeling

Again, the correct modeling of structural dynamics, aerodynamics, sensors and actuators is of utmostimportance for obtaining meaningful simulation results. Here below some technicalities about theseissues are detailed, aiming at presenting a systematic approach to the modeling of structural andaerodynamic nonlinearities.

5.2.1 Design model

Even if detailed in the previous chapter, the modeling aspects of the system are proposed again here tosimplify the presentation and to ease the introduction of additional nonlinear effects, e.g. free-playand friction.Also in this case, a generic linear structural model, can be discretized into the classical multi-degreesof freedom scheme:

Ms qs +Cs qs +Ks qs = q∞fa +TTβmh (5.1)

Through the principal of virtual work, the generalized hinge moments,mh , associated to the externalcontrol moments mβ and to its friction m f , will be given by mh = TT

β

(mβ+m f

). In general, the friction

torque will depend on the control surface control moments and rotation rates β, i.e. mf = mf

(mβ,Tββ

).

The mathematical model used to represent the free-play presence in each control surface can be

5.2. Aeroservoelastic Modeling 137

described by the usual relation:

mβ =

kβ

(∆β+α

)for ∆β<−α

0 for |∆β| <α

kβ

(∆β−α

)for ∆β>α

∆β=βext +βc −β

(5.2)

The variable ∆β is often referred as aileron dynamic response [31] and it can be decomposed into threecontributions: βext is the applied external command, βc the command computed by some control logicand β the effective aileron deflection. In addition, kβ is the equivalent stiffness connecting the pilotcommand to the control surface andα is the semi-width of the free-play, here assumed symmetric. Itshould be noticed that because of the presence of the free-play, either a local static mode or the socalled direct force [24, 31] approaches can be employed to improve the formulation of the dynamicequations of the aeroservoelastic system by accounting for highly localized deformations. Moreover,because of the large stiffness variations at the control surfaces hinge due to the free-play, a standardmodal approach might not account properly for the structural deformations at and near such movableparts. In order to improve the representation of these local deformations, the so called fictitious massapproach can be used [24, 31]. After this procedure, the stiffness matrix terms related to each controlsurface hinge are set to zero.

As clarified in the following, the friction torque will be modeled differently in the computationalmodel used by the controller and in the actual plant under control. In fact, as it will be seen later,the controller considers a simple averaged Coloumb like approximation, of the kindm f = m f sign

(β),

m f being a percentage of the maximum torque transmitted by the actuator, while a more elaborateStribeck formulation [38] will be used for more accurate system simulations.

Accelerometers, encoders and torque sensors are again modeled through second order transferfunctions, as in the previous chapters. It should be remarked that, in general, the hinge torque measurecannot separate the contributions of free-play and friction.

Finally, a generic formulation of a nonlinear unsteady aerodynamic system is written as:xa = fxa

(xa ,qs , qs

)fa = fa

(xa ,qs , qs

) (5.3)

where xa is the aerodynamic state, which can be either a physical entity, as in the case of a raw CFDmodel, or a generically abstract reduced order state, as the one detailed in Chapter 2.

Defining the extended servo-elasto-mechanical degrees of freedom q = [qs zacc zβ zm

]T and the

corresponding state x = [q q

]T = [qs zacc zβ zm qs zacc zβ zm

]T, putting together all of what above, thefollowing nonlinear, strictly proper, state space formulation is obtained:

x = Ax+q∞Ba fa(xa ,qs , qs

)+Bc m f +Bc mβ

xa = fxa

(xa ,qs , qs

)y = Cy x

(5.4)

where y is the system output and the other terms are defined through the following intermediatevectors and matrices:

M =

Ms 0 0 0

−D(ω2acc)Ta I 0 0

0 0 I 00 0 0 I

C =

Cs 0 0 00 D(2ξaccωacc) 0 00 0 D(2ξsensωsens) 00 0 0 D(2ξsensωsens)

(5.5)

138 Chapter 5

K =

Ks 0 0 00 D(ω2

acc) 0 0−D(ω2

sens)Tβ 0 D(ω2sens) 0

0 0 0 D(ω2sens)

Baq =

I000

Bcq =

TTβ

00

D(ω2sens)

(5.6)

so that it is possible to set the following final compact elements of Eq. 5.4:

A =[

0 I−M−1K −M−1C

]Ba =

[0

M−1Baq

]Bc =

[0

M−1Bcq

](5.7)

Cy =0 I 0 0 0 0 0 0

0 0 I 0 0 0 0 00 0 0 I 0 0 0 0

(5.8)

5.2.2 A comment on the design of the flutter suppression system

It should be remarked from now that the baseline I&I controller, responsible of the suppression oflarge amplitude limit cycle oscillations, is designed on the ideal system only, i.e. α= 0, and considers aslightly different model than Eq. 5.4. In fact, discarding the free-play effect, the ideal system dynamicsis governed by:

x = Ax+q∞Ba fa(xa ,qs , qs

)+Bd m f +BcI &I βc

xa = fxa

(xa ,qs , qs

)y = Cy x

(5.9)

where the matrix Bd has the same definition of Bc in Eq. 5.7, while BcI &I is defined as:

BcI &I =

0

M−1

TTβD(kβ)

00

D(ω2sens)

(5.10)

Therefore the ideal controller will be designed directly on the control surfaces deflections. The vectorkβ in Eq. 5.10 contains the nominal stiffness at each aileron hinge. With the formulation of Eq. 5.9, adesign based directly on the controlled surface deflections βc will be possible, and this result will beused as baseline by the proposed free-play compensation.

5.2.3 Verification model

It is important to remark that the actuator dynamics should be neglected in the controller computa-tional model, because, as shown in Section 3.3, its presence would not permit the explicit inversionof the free-play nonlinearity. Nevertheless, suitable actuator models are included in the controlledsystem during the simulation of the controller gains tuning, thus better matching the true closed loopoperation in flight.Therefore, the often used second order transfer function [324] of the following type will be adopted:

zact +D(2ξactωact)zact +D(ω2act)zact =D(ω2

act)βc (5.11)

The true free-play is thus experienced between the actual actuator and aileron positions, i.e. βc has tobe substituted with zact in Eq. 5.2. The real aeroservoelastic system under control is thus described

5.3. Test Cases 139

by a set of ordinary differential equations, similar to Eq. 5.4, albeit with a different internal structurebecause of the accounted the actuator dynamics. Defining the real servoelastic state xr = [

q q]T =[

qs zacc zβ zm zact qs zacc zβ zm zact]T, the final set of equations describing the system dynamics is so

modified:

Mr =[

M 00 I

]Cr =

[C 00 D(2ξactωact)

]Kr =

[K 00 D(ω2

act)

](5.12)

Baq,r =

I0000

Bmq,r =

TTβ

00

D(ω2sens)

0

Bcq,r =

0000

D(ω2act)

(5.13)

Ar =[

0 I−M−1

r Kr −M−1r Cr

]Ba,r =

[0

M−1r Baq,r

]Bm,r =

[0

M−1r Bmq,r

]Bc,r =

[0

M−1r Bcq,r

](5.14)

Leading to the following dynamic system:xr = Ar xr +q∞Ba,r fa

(xa ,qs , qs

)+Bm,r m f +Bm,r mβ+Bc,rβc

xa = fxa

(xa ,qs , qs

)yr = Cy,r xr

(5.15)

with the output equation now including also the measure of the actuator position:

Cy,r =

0 I 0 0 0 0 0 0 0 00 0 I 0 0 0 0 0 0 00 0 0 I 0 0 0 0 0 00 0 0 0 I 0 0 0 0 0

(5.16)

5.3 Test Cases

In all the following simulations, the sensors parameters are set to ξ = 1 and ωacc,ωsens = 220 rad/s.Such values are mostly dictated by the assumption of a second order anti-aliasing filter for the digitalimplementation, whose bandwidth is significantly below the one of the related sensor.

It is recalled that the presented results are related to the real system, i.e. the one including theservo dynamics, while the control law is designed without considering it. It must however be stressedthat, even if the actuator dynamics does not appear in the I&I control law formulation, its effect is fullytaken into account, i.e. see Eq. 5.15, during the controller tuning, because, as proved in Chapter 4, abasic modeling of the actuation system is required to achieve a good level of robustness of the adaptivelaw against variations of the actuator bandwidth. Thus, the controller will be tuned on the systemincluding such additional elements, verifying in this way also the strength of the proposed methodagainst unmodeled dynamics. In what follows, the nominal value of the actuators parameters is set toξact = 0.65 and ωact = 110 rad/s. The actuators saturate in position at 15 deg and in rate at 45 deg/s. As itwill be seen in the following results, such a bandwidth is not sufficient to guarantee a complete removalof the residual LCOs from the closed loop response. Even if the assumed bandwidth is a somewhatconservative specification, being clearly more demanding than what is strictly needed to suppress asimple flutter, a higher performance is required when dealing with free-plays, because they requirea sort of fast ’jump’ between the limits of the deadzone band, thus requiring a very quick actuatorresponse to follow the computed control effort. In fact, it is anticipated that sensitivity analyses will

140 Chapter 5

assess the influence of the actuator bandwidth on the compensation performance of the controller,showing, as expected, that the faster an actuator is, the better results can be achieved.

It is also remarked that, since aerodynamic loads are directly acting on the control surfaces,the adoption of a dual loop PID controller for the suppression of free-play driven LCOs would beunsuccessful, as highlighted in Section 3.4.3. In fact, when aerodynamic loads disturb the servo systemof a control surface already influenced by the presence of free-play, this is no longer able to compensatesuch a nonlinear effect. As shown in Section 3.4.3, the dual loop PID controller was only able to slightlyreduce the free-play induced LCO, and the cases studied in this chapter are no exception. Therefore, amore complex approach, such as the nonlinearity-inversion-based compensator needs to be takeninto account, as done in the cited section.

Regarding the I&I controller of Chapter 4, the filter parameter µ is typically chosen in relation tothe maximum frequency of the open loop response, estimated by Fourier transforming a few timehistories. Then, after verifying that a tentative Ct = diag(1) can be a suitable choice, γCon is computedaccordingly to Eq. 4.36, followed by a few analyses carried out by maintaining Λ = diag(1), whiledetermining appropriately the values of Ct , γCon and ηCon leading to a reasonable maximum controleffort, eventually increasing the value of diagonal elements of Λ until a desired settling time is achieved.Values of bd and σ equal to two proved sufficient to achieve an adequate level of robustness.

To assure the system adaptivity and stability over a wide range of operating conditions, the con-troller parameters are again tuned over various flight speeds, combined with different type of simula-tions, such as the response to large initial conditions, to input pulses, and evaluating the controlleradaption speed when its compensator is switched off-on during a simulation, eventually consideringvarious free-play widths and friction models.

Through some preliminary continuous designs, it has been possible to verify that the sampledbehavior of the continuous compensator can be adequately matched at a frequency of 180 Hz, withthe related discretization based on a fix step Heun integration scheme. Moreover, it has been pre-ferred to envisage a somewhat more realistic implementation, where the free-play cancellation isembedded within the control surface servos, which are mostly self contained black boxes with theirown independently digitalized realization. Thus, in order to provide a negligible phase shift againstdeflection commands, provided at a different (in general lower) rate, the control servos are mostlyrun at a higher sampling rate, which will be here assumed to be 500 Hz. To simulate correctly such adigitalization there is the need to care for the processing delay (input-calculation-output), associatedwith the chosen data acquisition system and control computer, in this case set to 1.5 ms.

As described previously, an equivalent average Coloumb like approximation of structural frictionis identified by the control law. Instead, the actual plant is characterized by a more realistic Stribeckmodel [38], whose parameters are tuned to give a maximum friction torque comparable to the onecomputed by the simpler model.A substantial set of simulations has been carried out for various ROM and finely discretized aerody-namic models, measurement noise, disturbances and large changes of the free-play parameters.

5.3.1 Four degree-of-freedom typical section with free-play

The Immersion and Invariance-based technique of Section 4.3 and the compensator by nonlinearityinversion of Section 3.3 are applied to a plunging and pitching NACA 0012 airfoil, flying at M∞ = 0.8 inair. The airfoil is equipped with fully unbalanced leading and trailing edge controls, whose hinges areplaced at 15% and 80% of the chord respectively. A schematic representation of the system is given inFigure 2.25a, while the related structural data can be found in Table 5.1.

For this case, a reduced order model was already developed in Sections 2.4.2 and 2.4.3, showingthat in absence of free-play the linearized flutter point is encountered at V∞,bif = 160.68 m/s, and an

5.3. Test Cases 141

m Jθ JβLE JβTE Shθ ShβLE ShβTE fh fθ kβLE kβTE

kg kgm2 kgm2 kgm2 kgm kgm kgm Hz Hz Nm Nm

43.41 8.14 1.221 1.628 5.43 0.814 1.0853 4.65 9.3 23330 31107

Table 5.1: Dimensional structural model data of the 4 degree-of-freedom typical section

γo Λ Ct µ γCon ηCon

0.5 diag(150) diag(50) 750 0.1 100

Table 5.2: Controller parameters: typical section with 4 degree-of-freedom and free-play

aerodynamic reduced order model with twelve states was precise enough to describe the nonlineardynamic behavior of the system. Moreover, in the same sections was shown that the inclusion offree-play and friction leads to a complex bi-stable behavior.

After being validated in the previous section, the reduced order aeroservoelastic model is hereemployed in the design of the control law. The total number of states of the compacted model is 32:8 structural, 12 aerodynamic, 8 for the sensors and 4 for the actuators. The total number of statesrequired to describe the high fidelity, CFD-based system is equal to 128020, showing once again that adesign carried out on such a model would have been impractical.

The following results are so organized: first it will be shown that even an I&I controller designed onthe ideal system is able to suppress the main limit cycle behavior of the system, yet does not removethe residual LCO due to the control surface free-play. Then the results related to the design modelwith active free-play compensation system will be presented. Finally, a few verification tests where theaerodynamic sub-system is simulated by a full CFD solver will be discussed.

To achieve good adaptive performances against pulse perturbations applied through the trailingedge control surface, different flight speeds, up to 25% higher than V∞,bif, are taken into account totune the controller parameters. The target performances are the plunge and pitch of the typical section.Two accelerometers are placed at the 20% and at the 70% of the chord respectively, maximizing in thisway the signal to noise ratio and thus making the reconstruction of the structural state easier.Carrying out the design on the ideal system, i.e. α = 0, with the interactive procedure previouslydescribed, the control parameters of Table 5.2 are obtained. Being κ (S) = 1, the inequality of Eq. 4.36is satisfied, guaranteeing the stability and robustness of the controller. A comparison of open andclosed loop responses for the ideal system with the parameters of Table 5.2 is shown in Figure 5.1. Allthe coefficients are tuned considering responses to input pulses of the trailing edge control surface,over flight speeds ranging fromV∞,bif to 1.15 V∞,bif. As can be seen, the system is efficiently stabilizedby the designed controller. At this point free-play and friction are introduced into the system model.Computing a few open loop responses due to input pulses, the amplitude of friction torque on bothhinges has been set to 20% of the actual torque transmitted by the actuator. Introducing the free-playnonlinearity in the control surface actuation, the closed loop response is partially deteriorated, aswitnessed by Figure 5.2. The designed control law is still able to reduce the amplitude of the limitcycle oscillations, but residual vibrations due to the presence of the control surface free-play show up,making the closed loop response unacceptable.

The free-play compensation is then introduced, with the aim of further reducing the residual LCOin the closed loop response. The only parameter that should be set in this case is the learning rateηGD of Eq. 3.16. As a rule of thumb, the designer should start with a fairly low value, e.g. 0.01, thenincrease it until a sufficiently fast convergence rate of the free-play parameters is obtained. Following

142 Chapter 5

0 1 2 3 4 5 6 7

−0.2

−0.1

0

0.1

0.2

Time [s]

h [m

]

Open loopClosed loop

(a) Plunge response

0 1 2 3 4 5 6 7

−3

−2

−1

0

1

2

3

Time [s]

θ [d

eg]


(b) Pitch response

0 1 2 3 4 5 6 7−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time [s]

βLE

[deg

]

Open LoopClosed loop


0 1 2 3 4 5 6 7

−1

0

1

2

3

4

5

6

Time [s]

βT

E [d

eg]



Figure 5.1: Ideal system response to an input pulse, V∞ = 188 m/s

0 1 2 3 4 5 6 7

−3

−2

−1

0

1

2

3

Time [s]

θ [d

eg]


(a) Pitch response

0 1 2 3 4 5 6 7

0

1

2

3

4

5

Time [s]

βT

E [d

eg]


(b) Trailing edge deflection

Figure 5.2: System response to an input pulse, free-play present but not compensated, V∞ = 188 m/s

5.3. Test Cases 143

0 2 4 6 8

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

h/c

[−]


(a) Plunge response

0 2 4 6 8

−3

−2

−1

0

1

2

Time [s]

θ [d

eg]


(b) Pitch response

0 2 4 6 8−0.4

−0.2

0

0.2

0.4

Time [s]

βLE

[deg

]



0 2 4 6 8

0

2

4

6

8

Time [s]

βT

E [d

eg]



Figure 5.3: System response to an input pulse with compensated free-play, V∞ = 188 m/s

this simple procedure, in the present case the best results are obtained with ηGD = 1.25. From authors’experience, low values of ηGD lead to stable but slowly converging results, while for values of ηGD toohigh, e.g. 2, the training process of Eq. 3.16 can become unstable.The initial value for the unknown free-play parameters has been chosen through some guided trials:their value is not known, but their rough estimate can be easily assumed. With this idea in mind, weset randomly the initial value for the free-play parameters, oscillating from −50% to +50% of theirtrue value. The closed loop response with the free-play compensator maintained active during thesimulation is shown in Figure 5.3: the free-play effect is reduced to a remarkably low level, with almostno residual oscillations. The convergence history of the unknown free-play parameters can be tracked,their history being shown in Figure 5.4. A fairly small final error has been obtained, and this canbe considered a very good result for two main reasons. First: for each control surface, we are tryingto estimate three different parameters starting from only a single error measure, which combinesthem. Second: the system is not "persistently excited" [72, 306] and this may have led to inaccurateidentification results. For better appreciating the compensation effect of the proposed control law aphase space plot of the control surfaces deflection is shown in Figure 5.5.

144 Chapter 5

0 2 4 6 8 102

2.2

2.4

2.6

2.8

3x 10

4

Time [s]

kβ[N

m]

True ValueEstimated Value

(a) Convergence of the leading edge command stiffness

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

Time [s]βFP[deg]

True ValueEstimated Value

(b) Convergence of the trailing edge free-play width

Figure 5.4: Convergence history for the unknown parameters of the free-play model

−0.4 −0.2 0 0.2 0.4−30

−20

−10

0

10

20

30

β [deg]

β[deg/s]


(a) Comparison of leading edge LCOs

−1 −0.5 0 0.5 1−60

−40

−20

0

20

40

60

β [deg]

β[deg/s]


(b) Comparison of trailing edge LCOs

Figure 5.5: Phase space plots of the control surfaces deflection.

5.3. Test Cases 145

0 2 4 6 8

−4

−3

−2

−1

0

1

2

Time [s]

θ [d

eg]


(a) Pitch response

0 2 4 6 8

−2

0

2

4

6

8

Time [s]

βT

E [d

eg]



Figure 5.6: Verification response to off/on control, V∞ = 205 m/s

As a verification of the results obtained with the design model, the compensator is tested on asystem excited by aerodynamic loads computed directly by using a fine Euler-based CFD solution. Inthis test the system is left free to reach an LCO condition. Then the control law is activated, eventuallydriving the response to zero. This kind of test can be useful for assessing possible dynamics that havenot been captured by the reduced order model. Furthermore, switching on the controller only after theLCO has been reached permits one to evaluate its robustness in a case where the system nonlinearitiesare completely developed. To make the test even more challenging, a random disturbance with a rootmean square of 0.1 deg is directly applied to the trailing edge control surface and the free-play of bothcontrol surfaces is set to α= 0.2 deg. The response obtained is shown in Figure 5.6, where the controllaw is activated after four seconds of simulation. It should be noted that the control law is still able tostabilize the response, even if a small residual LCO is still present, as witnessed by Figure 5.7.

A further verification is carried out considering larger values of the free-play width, now set toα= 0.25 deg for each control surface and a 30% reduction of their stiffness. As evidenced by Figure 5.8,now the open loop LCO is driven by the free-play nonlinearity, leading to much smaller motionamplitudes. It is therefore interesting to observe how the designed controller is also able to adaptand stabilize the system in such very different circumstances. Furthermore, as shown in Figures 5.8cand 5.8d, a sizeable free-play reduction can be otained by discarding the friction torque from theidentification procedure, even if the estimation of its average effect helps to further reduce the residualLCOs in closed loop. The free-play induced LCO is reduced to a fairly low amplitude even if thefree-play width is relatively large in this case, proving once again the effectiveness of the proposedcompensator. In addition, the proposed controller is quite insensitive even with a not so precisemodeling of the friction torque, nevertheless identifying a comparable averaged value of its amplitudein any performed simulation, thus helping in the compensation of the nonlinearities present in theactuation system.

The robustness of the compensation system is now verified in face of variations of the actuatorsbandwidth, while the flight speed is maintained constant at V∞ = 195 m/s. The results are shown inFigure 5.9, where the bandwidth is varied from 32 to 8 Hz. As expected, a reduction of the bandwidthresults in a reduction of the control system effectiveness. For frequencies lower than 15 Hz, usuallysufficient in classical flutter stabilizations [302], the residual LCOs in closed loop are comparable to theone experienced in open loop. It is therefore clear that the suppression of control surfaces free-play

146 Chapter 5

−0.4 −0.2 0 0.2 0.4−30

−20

−10

0

10

20

30

β [deg]

β[deg/s]


(a) Comparison of leading edge LCOs

−1 −0.5 0 0.5 1−100

−50

0

50

100

β [deg]

β[deg/s]


(b) Comparison of trailing edge LCOs

Figure 5.7: Phase space plots of the control surfaces deflection, V∞ = 205 m/s

0 2 4 6 8

−0.05

0

0.05

Time [s]

h/c

[−]


(a) Plunge response

0 2 4 6 8−2

−1

0

1

2

Time [s]

θ [d

eg]


(b) Pitch response

0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time [s]

βLE

[deg

]

Open LoopClosed Loop without Friction CompensationClosed Loop with Friction Compensation


0 2 4 6 8 10−1

0

1

2

3

4

Time [s]

βT

E [d

eg]

Open LoopClosed Loop without Friction CompensationClosed Loop with Friction Compensation


Figure 5.8: Verification response with large free-play width, V∞ = 205 m/s

5.3. Test Cases 147

10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

fAct

[Hz]

βLE

[deg

]

Free−play LCOFree−play widthResidual LCO

(a) Sensitivity analysis on the leading edge control surfaceactuator

10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

fAct

[Hz]

βT

E [d

eg]

Free−play LCOFree−play widthResidual LCO

(b) Sensitivity analysis on the trailing edge control surfaceactuator

Figure 5.9: Sensitivity of the LCO compensation effectiveness versus the actuators bandwidth

requires higher performance servos.

Finally, the ability of the controller to widen the flutter-free flight range is investigated. Severalsimulations with increasing velocities are carried out. As seen in Figure 5.10, the flutter point ismoved forward to 57% of its initial value. This result is obtained by considering off/on simulations,which have proven to be the most challenging tests for the presented controller. Now, consideringpulse perturbations with different amplitudes, indicated byβ in the figure, the flutter-free envelope iswidened even more, moving back to the off/on results as the pulse amplitude is increased. It is alsointeresting to see how the flutter trends change with the presence of the controller: while the openloop system shows a range where the LCO amplitudes gradually increase with the flight speed, theclosed loop envelope presents a sort of ’linear’ instability point, beyond which the system producesvery large oscillations, as for the unstable response of a linear system. Analyzing again Figure 5.10, it isclear that going towards the bifurcation point the basin of attraction of the stable solution is shrinking,so LCOs are obtained for smaller and smaller perturbations.

5.3.2 Goland wing with free-play

The Goland wing of the previous test case is here equipped with two trailing edge control surfaces totest the capability of the coupled I&I – nonlinear inversion controller on a problem with a large numberof degrees of freedom. Again, the present aerodynamic model is rather simple, based on strip theory,but this simplicity is compensated by the presence of many elastic modes that makes the dynamicresponse more complex. The inner control surface extends from the 30% to the 50% of the span, whilethe outer one goes from 70% to 95%. Both surfaces have a chord of 25% the wing chord.The actuation system of the control surfaces is still subjected to friction and free-play, which is set to 1deg for the outboard control surface and 0.5 deg for the inboard one. A sample of the normal modes ofthe present model is depicted in Figure 5.11, where the mode shapes are computed using the fictitiousmass method [31]. The main structural data that defines the model is the same listed in the previouschapter. Because of the presence of two control surfaces, the additional data required are the relatedinertia Jβ and control compliance kβ. Their value is resumed in Table 5.3.

Four accelerometers are placed at the leading edge and at 70% of the chord, at the wing tip and at

148 Chapter 5

150 200 250 300 350 4000

0.05

0.1

0.15

0.2

V∞

[m/s]

h LCO

/c [−

]

Open Loopβ = 1 [deg]β = 5 [deg]β = 10 [deg]Off/On

(a) New plunge flutter envelope

150 200 250 300 350 4000

0.5

1

1.5

2

2.5

V∞

[m/s]

θLC

O [d

eg]

Open Loopβ = 1 [deg]β = 5 [deg]β = 10 [deg]Off/On

(b) New pitch flutter envelope

Figure 5.10: Closed loop flutter envelopes, each one obtained considering different perturbations

−1−0.5

00.5

11.5

0

2

4

6

8−0.05

0

0.05

0.1

0.15

0.2

X [m]

Mode number 1, Frequency = 2.4445 [Hz]

Y [m]

Z [m]

(a) Bending mode

−1−0.5

00.5

11.5

0

2

4

6

8−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

X [m]


Y [m]

Z [m]

(b) Torsion mode

−1−0.5

00.5

11.5

0

2

4

6

8−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

X [m]


Y [m]

Z [m]

(c) Outer control surface mode

−1−0.5

00.5

11.5

0

2

4

6

8−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

X [m]


Y [m]

Z [m]

(d) Inner control surface mode

Figure 5.11: Principal normal modes of the Goland wing

5.3. Test Cases 149

Jβ,In, Kg m2 Jβ,Out, Kg m2 kβ,In, Nm/rad kβ,Out, Nm/rad

0.6096 0.8534 2.1659 ·104 3.4501·104

Table 5.3: Ailerons structural data

γo Λ Ct µ γCon ηCon

0.1 diag(100) diag(70) 2000 diag(0.01, 0.005) 100

Table 5.4: Controller parameters: Goland wing with two control surfaces and free-play

50% of the span. These measures are used by a sliding mode observer to reconstruct the full state. Thedesign model includes 8 normal modes, including the rigid relative ailerons rotation, for a total of 28states (16-structure, 8-sensors, 4-actuator). The verification phase will be then carried out on a refinedmodel including many more modes, the previously mentioned non linear correction of the angle ofattack and a larger value of the free-play width.The controlled performance y is composed by the usual vertical displacement at the leading edgeof the wing tip and by the vertical displacement of the leading edge at 50% of the span. The neededtarget matrix Hq can be determined after calling: qm the modes amplitudes vector, Thtip the modesdisplacements at tip leading edge, Th50 the modes displacements of the leading edge at 50% span, and

defining q = [qTm xact xsens]T, so that we have Hq =

[Thtip 0 0Th50 0 0

].

The controller can then be designed to stabilize the wing up to a speed 45% higher than the originallinear flutter speed in absence of free-play and friction, in front of pulse perturbations introduced bythe ailerons. So, carrying out the usual trail and error-based design procedure, the control parametersof Table 5.4 are obtained. As can be realized from the actual value of the gain γCon, a higher controlauthority is assigned to the outboard control surface, while the inboard aileron has a more limitedassigned control effort.

As in the previous example, the closed loop system is first tested without the compensation offree-play and friction. The response to an input pulse of 7.5 degrees at V∞ = 45 m/s is shown inFigure 5.12. As can be seen from these results, the control system is able to stabilize the response ofthe aeroelastic system, which otherwise would be linearly unstable. Nevertheless, the presence of thefree-play limits the performance of the control system to a residual LCO induced by it. Therefore, acompensation strategy aimed at removing this unwanted behavior is required. From these figurescan be noticed that the linearity of the aerodynamics makes the whole system unstable even if thefree-play is present. In fact, as soon as the wing motion undergoes very large amplitudes, also thecontrol surfaces are driven by these instability, without being affected by the free-play limitation.

The compensator of Section 3.3, based on the nonlinearity inversion of free-play and friction istherefore integrated in the control loop. The learning rate ηGD is set to 0.2, with this value representinga good compromise between a fast adaptivity of the controller and a robustly stable behavior also whenthe first guess of the free-play and friction parameters are not very precise. The closed loop responseobtained considering the same pulse input is shown in Figure 5.13. As can be seen, the free-play effectis well compensated by this strategy, confirming the good results obtained in the previous applications.In Figure 5.14 additional data relative to the free-play nonlinear compensator is displayed. It is clearfrom it that a correct identification of the free-play width is of great importance for its compensation.On the other hand, the command compliance kβ does not need to be identified with great precision,since its contribution to the results seems to be rather limited, as also represented by Figure 5.14c.

150 Chapter 5

0 5 10 15 20−2

−1

0

1

2

3

Time [s]

wT

ip [c

m]



0 5 10 15 20−5

0

5

Time [s]

θT

ip [d

eg]


(b) Rotation at the wing tip

0 5 10 15 20

−2

−1

0

1

2

Time [s]

βIn

[deg

]


(c) Inner control surface deflection

0 5 10 15 20−8

−6

−4

−2

0

2

4

6

8

Time [s]

βO

ut [d

eg]


(d) Outer control surface deflection

Figure 5.12: Response to an input pulse without compensating the free-play effect

5.3. Test Cases 151

0 5 10 15 20

−1

0

1

2

3

Time [s]

wT

ip [c

m]



0 5 10 15 20

−3

−2

−1

0

1

2

3

Time [s]

θT

ip [d

eg]



0 5 10 15 20

−1

−0.5

0

0.5

1

Time [s]

βIn

[deg

]



0 5 10 15 20

−2

0

2

4

6

8

Time [s]

βO

ut [d

eg]



Figure 5.13: Response to an input pulse with compensated free-play effect

Moreover, compared with the control effort computed without the free-play compensation, the presentone is remarkably lower probably thanks to the contribution of the nonlinear inversion of the structuralnonlinearities, as highlighted by Figure 5.14d.

Having tuned the controller on a linear, low order model, it is now tested on a system with anincreased number of elastic modes where nonlinear aerodynamic effects are introduced empirically, asdone in the previous chapter. To make such a test more challenging, the free-play widths are doubledup, obtaining a free-play of 2 degrees for the outboard aileron and 1 degree for the inboard aileron. AStribeck friction model is employed in this case for a higher fidelity representation of the related effect.Having said this, the verification model uses the first forty elastic modes, including the rigid relativeailerons rotation, for a total of 92 states (80-structure, 8-sensors, 4-actuator).As first test, the system is left free after imposing a large initial condition at a flight speed higher thanthe previous cases, V∞ = 55 m/s. The controller is active since the beginning of the simulation, and theresponse quickly converges to the origin, as depicted by Figure 5.15. The effect of the free-play andfriction on the open loop response is quite clear from these figures, if compared with the responsescomputed in the previous chapter, where structural nonlinearities were not modeled. The chaoticmotion of the control surfaces makes the response quite irregular when looking at both displacement

152 Chapter 5

0 5 10 15 200.35

0.4

0.45

0.5

0.55

Time [s]

αIn

[deg

]

True valueEstimated value

(a) Identification of the free-play width of the inboardaileron

0 5 10 15 200.95

1

1.05

1.1

1.15

1.2

1.25

Time [s]

αO

ut [d

eg]


(b) Identification of the free-play width of the outboardaileron

0 5 10 15 203.4

3.5

3.6

3.7

3.8

3.9

4x 10

4

Time [s]

k β, I

n [Nm

]


(c) Identification of the compliance of the inboard aileron

0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4

Time [s]

βc co

rr

[deg

]

External CSInternal CS

(d) Control effort correction because of the free-play pres-ence

Figure 5.14: Additional data available from the free-play compensator

5.3. Test Cases 153

0 5 10 15 20

−20

−15

−10

−5

0

5

10

15

20

Time [s]

wT

ip [c

m]



0 5 10 15 20

−10

−5

0

5

10

Time [s]

θT

ip [d

eg]



0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4

Time [s]

βIn

[deg

]



0 5 10 15 20

−10

−5

0

5

10

Time [s]

βO

ut [d

eg]



Figure 5.15: Response to large initial conditions with compensated free-play effect

and rotation of the wing tip.

The second and final test considers the response of the system when the controller is switchedon after five seconds of simulation. This test is still performed at V∞ = 55 m/s. No modifications areintroduced in the control law and all the gains are set at the values computed through the design model.As can be seen from Figure 5.16, the controller is sufficiently robust to take the response, already closeto the steady-state limit cycle behavior, and drive it to a stable condition.Additional results, such as the convergence of the identification process of the free-play width for boththe control surfaces is available in Figure 5.17. From these results, it is clear that the convergence ofthese variables to their real value is rather quick, permitting a ready and robust compensation of thefree-play effect.

Concluding, in this chapter it has been presented a control method for the simultaneous com-pensation of aerodynamic and structural nonlinearities. Such a methodology is based on the serialapplication of the I&I technique used in Chapter 4 for suppressing aerodynamically driven nonlin-earities and the nonlinear inversion-based compensator used in Chapter 3 to compensate nonlinearbehaviors induced by the presence of free-play and friction. With the help of a sliding mode observerand the additional measure of the hinge torque, this approach has proven to be rather efficient and

154 Chapter 5

0 5 10 15 20

−20

−15

−10

−5

0

5

10

15

20

Time [s]

wT

ip [c

m]



0 5 10 15 20

−10

−5

0

5

10

Time [s]

θT

ip [d

eg]



0 5 10 15 20

−6

−4

−2

0

2

4

6

Time [s]

βIn

[deg

]



0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

Time [s]

βO

ut [d

eg]



Figure 5.16: Response to off/on control switching with compensated free-play effect

0 5 10 15 200.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

Time [s]

αIn

[deg

]


(a) Inboard aileron free-play

0 5 10 15 201.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

Time [s]

αO

ut [d

eg]


(b) Outboard aileron free-play

Figure 5.17: Identification of the free-play width when the controller is activated after 5 seconds ofsimulation

5.3. Test Cases 155

robust when tested against different problems. In fact, in the four degree-of-freedom typical sectioncase, the structural model was quite simple, while the modeling of the aerodynamic system was ratherinvolved. On the other hand, the modeling of the Goland wing with two control surfaces requireda more complex definition of the structural model, eventually refined through the method of thefictitious masses to take into account the influence of the free-play, while the aerodynamic systemwas based on a much simpler, experimentally tuned model. Nevertheless, the controller was ablein both cases to stabilize the aeroservoelastic system of interest without presenting any particularproblem regarding its stability and robustness. The additional contribution of a friction estimator wasimplemented, showing the improvements that such a solution may bring to the compensation of thesystem nonlinearities.Therefore, a rather systematic approach to the suppression of nonlinear instabilities in aeroservoelasticsystems was presented, setting the basis for a generalized method for studying nonlinear aeroelasticsystems in closed loop.

Conclusions and Final Recommendations

CHAPTER 6

Throughout this work a systematic procedure for modeling and control nonlinear aeroservoelasticsystems has been presented. With the detailed methodologies, arbitrary aerodynamic and structuralnonlinearities can be taken into account during the design and analysis of a large number of aeroelasticsystems of interest.Problems of increasing complexity have been considered, showing that the studied approaches candeal with typical aeroelastic problems where the system stability is of main interest. The implementedmethods proved to be robust enough in the face of unmodeled dynamics, external disturbancesand variations of the system key parameters. This can be considered a good starting point for thedevelopment of a general purpose tool for the analysis and control of nonlinear dynamic systems,where initial information is extracted from full-order models to generate reduced order design models,then used to design and tune opportune control systems. The resulting controller may be then exportedback to the initial full order model to validate the results.

Several interesting results have been obtained, and it is worth to highlight them in a structuredway:

• Aerodynamic modeling – A reduced order model based on neural networks has been developedand validated versus full order simulations. The method has been designed considering severaldetails, from the design and selection of the training signal, based on physical considerations, tothe performance assessment in the face of large variations of the boundary conditions

• Modeling of structural nonlinearities – These kind of effects have been modeled through muchsimpler static relations. Nonetheless, they have proven to be adequate enough to representresponses that were also validated through experiments. In fact, one of the most interestingfindings of this thesis has been the discovery of a bi-stable behavior of aeroelastic systems whereaerodynamic and structural nonlinearities are both present

• Compensation of structural nonlinearities – Industrial approaches based on PID controllershave been used to compensate the presence of friction and free-play and avoid limit cycleoscillations in the closed loop response. A data-driven optimization strategy is developed inthe frequency domain in order to tune all the necessary gains. Such a technique has provento be applicable to any linear controller. However, when aerodynamic forces are present andaffect the system stability even without the presence of structural nonlinearities, more complexapproaches are required. In this work a controller based on the dynamic inversion of the systemnonlinearity is developed and applied, showing the necessity of additional measures if limitcycle oscillations want to be effectively suppressed

• Suppression of aerodynamic nonlinearities – This kind of nonlinearities have shown to bemore subtle to suppress because of their intrinsic dynamic nature in typical aeroelastic appli-cations. An approach based on a recent technique called immersion and invariance has beendeveloped, additionally highlighting practical aspects of the related implementation, such as

157

158 Chapter 6

the adoption of nonlinear observers for estimating the required system state and discussingabout the importance of modeling sensors and actuators. The robustness of the method hasbeen proven through several off-design analyses

• Integrated nonlinear aeroservoelastic design – A controller that can be systematically designedhas been proposed for stabilizing nonlinear aeroelastic systems when both structural and aerody-namic nonlinearities are present in the model. The developed approach combines the previouslystudied methods to show how they can be interfaced in such a problem, resulting in an efficientcontroller for the suppression of limit cycle oscillations of different nature. The method hasproven to be robust in the face of model uncertainty, always providing an effective suppressionof all the nonlinear effects present in the system

Nonetheless, even if a lot of work has been done to develop methods as systematic as possible formodeling and control nonlinear dynamic systems, a great effort is still required to fill the gaps withrespect to already well known methods and to tackle problems that are solved on a daily basis by theindustry.Some possible extensions to the work presented in this thesis are listed below trying at least to identifysome paths that can be followed to improve the state of the art in nonlinear aeroservoelastic modeling:

• Aerodynamics – A good understanding of limit cycle oscillations with Euler-based flow modelshave been achieved. Nevertheless, additional analyses using more realistic models such asReynolds averaged Navier Stokes should be carried out to understand the influence of viscosityon limit cycle oscillations when large angle of attack are reached. The present idea is that onceflow separation is present in the system, limit cycle oscillations would be much more limited inamplitude, therefore easier to control through the classical use of ailerons.

Even if problems up to four degrees of freedom have been analyzed and successfully modeled,it would be interesting to see if the designed reduced order model will work as well with morecomplex problems with a larger number of inputs and outputs, such as flexible wings. Of coursethe training time will experience a large increase mainly because of the longer time required togenerate the training data. Nonetheless, the use of parallel simulations can solve this problem,allowing to test the proposed identification algorithm for much more complex configurations,such as free flying flexible aircraft.

Another point to improve is the robustness of the reduced order model in the face of Machnumber variations. This parameter has presented the greatest influence on the performance ofthe designed reduced order model. This means that a model trained for subsonic Mach numbermay not work well at transonic Mach numbers. It is thought that this fact should be addressedthrough the use of some parameters interpolation or by tackling the problem from a differentpoint of view, developing robust models using the techniques used in the field of uncertaintyquantification. Nevertheless, up to the present date, an adequate formulation have not beenpresented yet in the literature.

In addition, the reduced order modeling technique presented here could be employed in experi-mental applications to fit the experimental data is some simulation environment. Its potential isquite appealing and nonlinear behaviors can be captured from real phenomena and importedin design tools used to run parametric analyses

• Controls – This thesis has mainly focused on the stabilization of unstable aeroelastic systems.One first natural extension would be the design of controllers aimed at alleviating unsteadyloads coming from gusts. In those cases the aircraft would be already stabilized and therefore

159

the control goal would be to modify the system dynamics in order to behave like a model withincreased generalized stiffness and damping. Immersion and invariance has never been testedin such a case and therefore this would be a good case to test the convergence properties of thispromising technique.

Another point to improve is the tuning of the control laws, which could be made automaticthrough the use of ad-hoc optimization techniques. Data-driven approaches have shown goodpotential in this but their application to nonlinear control system seems somewhat limited andproblem-dependent. Other approaches based on gradient-free algorithms, e.g. Nelder-Meadalgorithms, may be used, but their effectiveness still needs to be proven in nonlinear problems.

As a final stage in a simulation environment, all the presented controllers should be integratedin a realistic flight control system as control augmentation algorithms, providing their help onlywhen requested by the master system. True design specifications related to stability and responseperformances should be considered in the control law tuning, balancing the performance of theflutter suppression and gust alleviation systems with that of the whole control structure

Finally, the designed adaptive control laws should be implemented on hardware, testing therobustness of their discretization on real-world problems, possibly considering real-time con-straints in their implementation, keeping the sampling frequency as low as possible.

Concluding, this work has tried to set the basis for a generic tool to model, analyze, simulate andcontrol large order nonlinear dynamic systems through the use of reduced order models. The resultsobtained in this thesis can be considered as a starting point for some more advanced design tools basedon the concepts presented here. In particular, when considering problems proper of the aeronauticalindustry, the early consideration of nonlinearities in the design of aerospace systems may lead in thefuture to lighter, cheaper and more robust products to be used in commercial and military aviation,both manned and unmanned.

Virtual Realization of Actuation SystemNonlinearities

APPENDIX A

A.1 Problem Description

As discussed in the previous chapters, even if during the design of control loops the actuators areusually considered linear, their behavior is intrinsically nonlinear. In fact, during normal operationalconditions, they may saturate in position, e.g. the maximum displacement or rotation reachable, inrate, e.g. the maximum speed that an actuator is able to achieve, and force, e.g. the maximum loadthat the actuation system is able to produce.

Usually these kind of saturations are intrinsically embedded in the physics of the actuator. Some-times though, when some sort of scaled testing is required, i.e. a wind tunnel experiment, suchsaturations have to be reconstructed through ad-hoc control laws.This is the case of the problem analyzed in the present appendix.

The aim of Gust Load Alleviation techniques assessment on wind tUnnel MOdel of advanced Re-gional aircraft (GLAMOUR) proposal [325] is a technological optimization and experimental validationthrough an innovative aeroservoelastic wind tunnel model of gust load alleviation control systems forthe advanced Green Regional Aircraft [326]. The expected benefits of such technologies are mainly themitigation of gust load responses, the reduction of peak stresses so to potentially decrease sizing loadsand consequently increase the weight saving.In this case, the requirements that affect the design of the GLAMOUR servo–control systems descenddirectly from the reference aircraft, that is a full–scale aircraft, and include the related maximumbandwidth, that must be robust in the face of possible uncertainties and disturbances, and the deflec-tion and rate saturation of each control surface. In order to reproduce these effects, the wind tunnelactuation system is composed by two independent motors which drive the ailerons by reinforcedrubber belts, while the elevator is directly actuated by one motor placed outside the wind tunnelmodel.

Considering the reference aircraft, its actuation system is realized by means of electro-hydraulicactuators that present various saturations due to physical limitations. Because the wind tunnel modelis constrained in size, small electrical motors are chosen to actuate the control surfaces. Such motorsdo not present any position saturation, while they may saturate in velocity but at values that arenever reached in aeronautical applications. They also present a torque saturation that is related to themaximum current that can be applied to the motor. In order to adapt the present performance to theone of the real actuator, scaled down to be applied in the wind tunnel model, a control law is requiredto realize the saturations experienced by the real system and to allow an adequate bandwidth.

This appendix is devoted to the design and implementation of the control law of the mentionedcontrol surfaces. Particular focus will be given to the design related to one of the ailerons, with theothers that can be considered very similar. The frequency-based optimization described in Section 3.2will be employed again here, proving that such an algorithm is able to tune robust control laws thatcan be also tested experimentally.

161

162 Chapter A

Figure A.1: Wind Tunnel Model and actuation system layout

Figure A.2: Belt drive system

A.2 Control Architecture and Tuning

Because of the lack of room inside the wing, the actuation system of the two ailerons is made up by a setof two independent drive belts shown in Figure A.1, which are nevertheless driven by the same inputsignal. Regarding the elevator instead, because of its massive inertia properties, a direct connectionwith the motor is chosen, in order to avoid possible positioning inaccuracies due to the flexibilityintroduced by the drive belt. A control approach with PID regulators is taken into consideration, thanksto their simplicity and intrinsic robustness. In order to guarantee good tracking accuracy and therealization of the required deflection rate saturation, a dual loop architecture, of the type described inChapter 3, is chosen. This solution permits to tune the motor (inner) loop as a speed controller, makingit sufficiently fast to follow abrupt speed changes, e.g. rate saturations, while the load (outer) loop isa position loop, required to assure the desirable tracking precision. The deflection rate saturation isintroduced in between the two control loops.

A simplified model of the belt drive system used for the actuation of the ailerons is depicted inFigure A.2. Using the variables defined in such a figure, the closed loop system should be able tofollow the reference command r (t ) through the aileron rotation θ2. Unfortunately, as also shown in theprevious chapters, the classical approach with the direct application of a PID on the motor rotation θ1

would results in a poor tracking because of the combined action of the belt flexibility, indicated as k inFigure A.2, and the aerodynamic load acting on the aileron, Ca , which is in fact a disturbance in thepresent design.

A.2. Control Architecture and Tuning 163

Figure A.3: Dual loop controller

To overcome such an inadequacy, an appropriate solution have to be devised. Thanks to its wideapplication in industrial high precision tracking, a dual-loop strategy based on PID controllers isimplemented [267], as done in Chapter 3. Thanks to the feedback of θ2, this architecture would permitan accurate tracking of the reference signal also in presence of unsteady disturbances, such as theaerodynamic torque. As can be seen from Figure A.3, this cascade dual-loop architecture allows theintroduction of a rate saturation in between the two loops, so emulating the behavior of the realelectro-hydraulic actuator installed on the reference aircraft. This would have resulted impossible ifonly a single control on the motor side was implemented.

The tuning of the control system is carried out using the frequency-based optimization proceduredetailed in Section 3.2, nevertheless, a first estimation of the control gains will be provided through ananalytical approach based on pole-placement theory. Therefore, an initial estimation of the controlgains will be provided by the following pole-placement-based design, while the optimization will becarried out to further improve the performance of the closed loop system.

Before beginning the analysis, a first comment is required. In the following passages, the motor rateis supposed to be measured. However, as clarified by Figure A.3, only the motor position is measured,and its speed is estimated through the filterNs/(s +N) that has to be designed. Nevertheless, the ratespeed will be supposed to be known, and the value ofN will be set equal to the same value computedfor its outer loop parent.Now, consider the motor loop at first. This will be described by the following model in the frequencydomain:

J sω1 +Cω1 = Tm =(Km

P + KmIs

)(ωr −ω1) (A.1)

where J is the motor inertia, which considers also the one of the controlled surface, while C is themotor damping. The various Km are the motor gains and ωr is the reference motor rate as can beobserved from Figure A.3. Equation A.1 can be rewritten as:(

s2 + C +KmP

Js + Km

IJ

)ω1 =

(KmPJ

s + KmIJ

)ωr (A.2)

To assign the system eigenvalues, a pole-placement approach based on Bessel functions is adopted.Considering the second order Bessel filter:

s2

ω20,i n

+3s

ω0,i n+3 = 0 (A.3)

and imposing the equality of the two transfer functions coefficients, the following gains can be com-puted:

KmP = 3Jω0,i n −C

KmI = 3Jω2

0,i n

(A.4)

164 Chapter A

where ω0,i n is the inner loop desired bandwidth, which has to be divided by the coefficient Cin =1.36, for converting the Bessel parameters that have been normalized to unit delay at ω = 0 to 3 dBattenuation [327].

At this point, the inner loop gains are designed and held constant. The outer PID loop is designedto perform a tracking of a reference deflection, coming from the gust suppression control law. Thesystem in closed loop now reads:

(s2 +a1s +a0

)sτθ2 = (b1s +b0)ωr −→

(s2 +a1s +a0

)sθ2 =

(b1

τs + b0

τ

)ωr (A.5)

with obvious meaning of the constants ai and bi , i = 0, 1. The constant τ is the ratio between r2 andr1 of Figure A.2 required to transport the aileron rotation on the motor side. This transformation isrequired because the signal tracking is performed through the aileron position, which differs from themotor one by the factor τ due to the different diameters of the two pulleys employed. The motor rate isrepresented as the time derivative of its position to allow the design of the outer position loop. Thecontrol law is designed such that:(

b1s + b0)

sωr =

(Kl

P+ KlI

s+Kl

DNs

s +N

)(r −θ2) (A.6)

with bi = bi /τ. In this way, a pole-zero cancellation is allowed to obtain the desired closed loopbehavior. Thanks to this assumption, the system in closed loop with the outer PID results in thefollowing model: (

s2 +a1s +a0)θ2 =

(Kl

P+ KlI

s+Kl

PNs

s +N

)(r −θ2) (A.7)

or equivalently:[s4 + (a1 +N) s3 +

(a0 +a1N+Kl

P+KlDN

)s2 +

(a0N+Kl

PN+KlI

)s +Kl

IN]θ2 =[

Ns3 +(Kl

P+KlDN

)s2 +

(Kl

PN+KlI

)s +Kl

IN]

r(A.8)

Once again, to assign the closed loop eigenvalues, a fourth order Bessel filter is considered:

s4

ω40,out

+10s3

ω30,out

+45s2

ω20,out

+105s

ω0,out+105 = 0 (A.9)

where ω0,out is the outer loop desired bandwidth, which has to be divided by the coefficient Cout =2.11 for converting the Bessel parameters that have been normalized to unit delay at ω = 0 to 3 dBattenuation [327]. Imposing the equality between the coefficients of Eq. A.9 and those of the left handside of Eq. A.8, the following gains can be computed:

N = 10ω0,out −a1

KlI =

105ω40,out

NKl

P = 1

N

(105ω3

0,out −a0N−KlI

)Kl

D = 1

N

(45ω2

0,out −a0 −a1N−KlP

)(A.10)

A.3. Numerical Simulations 165

Jm , kg m 2 Ja , kg m 2 C, Nm/rad/s r1, m r2, m k, N/m b, m

1.8308·10−4 2.5048·10−4 0.03 7.285·10−3 1.1105·10−2 5.8378·105 0.15

Table A.1: Belt drive main data

Once the gains are computed, the outer loop controller is so realized:

ωr =

1

b1s(

s + b0

b1

) (Kl

P+ KlI

s+Kl

DNs

s +N

)(r −θ2) (A.11)

thus performing the pole-zero cancellation previously mentioned. This method gives a first estimationof the controller gains that can be refined through the frequency-based optimization detailed inSection 3.2. Nevertheless, numerical experiments proven that such an estimate is already a goodstarting point for the closed loop design.In the following simulations a comparison between the results obtained with a first estimate of thegains and those obtained with the refined controller will be provided.

A.3 Numerical Simulations

The mathematical model related to the system of Figure A.2 can be written following basic physicalprinciples:

Jm θ1 +C θ1 +αkr 21θ1 −αkr1r2θ2 = Tm

Ja θ2 −αkr1r2θ1 +αkr 22θ2 = Ta

xa +aa1V∞b

xa +aa2V 2∞b2 xa = ba0θ2 +ba1θ2 +ba2θ2

Ta = ca xa +da0θ2 +da1θ2 +da2θ2

(A.12)

As can be noticed from the last two equations of Eq. A.12, an unsteady aerodynamic model is usedto compute the hinge moment acting on the control surface. The model is developed starting fromthe work of Theodorsen [288], transforming the frequency-based model presented in such a workinto a state-space one. Therefore, the constants ca , da,i , aa,i and ba,i can be directly computedfollowing [288] as done in Section 3.4.2, while all the other required variables are defined in Table A.1.The constant α is introduced to take into account the fact that, in the rubber belt of Figure A.2, itscompressed side might not work at all as the other one is under tension (α = 1) or it could presentsome kind of residual stiffness (1 <α≤ 2) thanks to the presence of an internal reinforcement. Severalanalyses have been carried out to study the influence of such a parameter, while the results presentedhere are all related to α= 1, to show that the control system is able to perform its functions properlyalso in this pessimistic situation.

Starting from the initial guess estimated through the previously detailed pole-placement procedure,the frequency-based optimization of Section 3.2 is applied to refine all the controller parameters, i.e.Km

P , KmI , Kl

P, KlI, Kl

D, N, b0 and b1.The controller so designed is then applied through its discrete implementation, performed at a controlfrequency of 2500 Hz. Furthermore, because the response of the aeroelastic system described byEq. A.12 depends on the wind speed, a rough gain scheduling of the controller has been required.

166 Chapter A

KmP Km

I KlP Kl

I KlD N b1 b0/b1

Pole-placement 0.17 24 9772.4 4.6522·106 61.301 89 200 140

Optimization 0.06 7.1 20130 8.7314·106 40.36 295 124 300

Table A.2: Controller gains

Frequency [Hz]0 50 100 150 200

Am

plitu

de

-60

-50

-40

-30

-20

-10

0

10ReferenceClosed loop-3 dB

(a) Inner loop design

Frequency [Hz]0 50 100 150 200

Am

plitu

de

-40

-35

-30

-25

-20

-15

-10

-5


(b) Outer loop design

Figure A.4: Final closed loop response of the optimized system

Three speeds have been considered: 0, 25 and 50 m/s. The tuning simulations shown here are relativeto 0 m/s, while a verification test at the maximum wind speed will be presented later.

As was done in Section 3.4.2, the employed training signal resembles a chirp excitation of 1 degreeup to 150 Hz, filtered by a second order transfer function of bandwidth 20 Hz. The closed loop systemis first designed by the pole-placement method to have an inner bandwidth of 50 Hz and an outerbandwidth of 20 Hz. Starting from the collected input-output data, the frequency-based optimizationrefines the controller gains obtaining the frequency responses of Figure A.4. Gain values are reportedin Table A.2, comparing them before and after the refinement through the optimization algorithm. Ascan be noticed from the latter, the optimization algorithm mainly modifies the inner loop gains andthe value of the filters constants, i.e. N and b0/b1, taking them to the maximum value allowed by theoptimization. From Figure A.4 can be seen that the optimized control loop is able to guarantee thedesired closed loop transfer functions for both motor and aileron loops.

The closed loop system is now tested in two cases. The first considers the tracking of a simplesine of amplitude 10 degrees and frequency 3 Hz. Such a case presents large and continuous ratesaturations and therefore it is a good benchmark for the present control system to reproduce thewanted saturations without introducing any kind of instability. The results are reported in Figure A.5,where a comparison between the controller designed through the pole-placement technique andthe optimized one is considered. In the present appendix, all the results are normalized with respectto their related saturation values. As can be seen, the rate saturation does not allow an adequatetracking of the command signal, and both designs lead to very similar outputs. However, the designedinner loop is sufficiently fast to permit an efficient rate saturation, as shown in Figure A.5c. Alwaysconsidering the same figure, it is clear that the optimized controller is able to suppress the oscillationsaround the saturation value much more efficiently than the pole-placement design method. This is


Time [s]0.5 0.6 0.7 0.8 0.9 1

Non

dim

ensi

onal

aile

ron

rota

tion

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4ReferencePole-placementOptimized

(a) Command tracking

Time [s]0.5 0.6 0.7 0.8 0.9 1

Non

dim

ensi

onal

torq

ue

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Pole-placementOptimized

(b) Control input

Time [s]0.5 0.6 0.7 0.8 0.9 1

Non

dim

ensi

onal

aile

ron

rate

-1.5

-1

-0.5

0

0.5

1

1.5Pole-placementOptimizedSaturation limit

(c) Aileron rotation rate

Time [s]0.25 0.3 0.35 0.4 0.45 0.5

Non

dim

ensi

onal

aile

ron

rate

0.8

0.9

1

1.1

1.2


(d) Close-up near the saturation

Figure A.5: Tracking of a sine of amplitude 10 degrees and frequency 3 Hz

due to the fact that optimized controller is tuned on the real input-output data of the system, whilethe pole-placement design considers a very approximate model of the system dynamics that does noteven take into account the presence of the elastic belt.

Once proven that the control system is able to simulate the presence of rate saturations, thetracking properties are tested on a more realistic signal coming from the gust suppression law designedin [293]. Sample results are shown in Figure A.6. Again, the two designs lead to very similar trackingresponses, nevertheless the optimized controller improves the performance near the rate saturation,eliminating the oscillations introduced by the pole-placement design. The resulting gain margin is6.22 dB at 34 Hz, while the phase margin is 45 deg at 11 Hz. These values guarantee a good level ofrobustness in the face of possible system uncertainties.

Because the aerodynamic loads act as a disturbance in the present control design, and becausethey depend on the wind speed, the controller gains are scheduled at three different speeds, e.g. 0, 25and 50 m/s using the same procedure detailed previously. A test performed at V∞ = 30 m/s is shown inFigure A.7a. As can be seen from it, the controller presents good tracking properties, even if it is notable to follow fast deflections changes such as the one between the two largest peaks. This is mainlydue to the limited capability of reproducing rate saturations at large wind speeds, as highlighted

168 Chapter A

Time [s]0 0.5 1 1.5 2 2.5 3

Non

dim

ensi

onal

aile

ron

rota

tion

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2ReferencePole-placementOptimized


Time [s]0 0.5 1 1.5 2 2.5 3

Non

dim

ensi

onal

torq

ue

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Pole-placementOptimized

(b) Control input

Time [s]0 0.5 1 1.5 2 2.5 3

Non

dim

ensi

onal

aile

ron

rate

-1.5

-1

-0.5

0

0.5

1



Time [s]0.4 0.45 0.5 0.55 0.6

Non

dim

ensi

onal

aile

ron

rate

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08


(d) Close-up near the saturation

Figure A.6: Tracking of a gust suppression signal


Time [s]0 0.5 1 1.5 2 2.5 3

Nor

mal

ized

rot

atio

n

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1 ReferenceAileron


Time [s]0 0.5 1 1.5 2 2.5 3

Nor

mal

ized

inpu

t tor

que

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

(b) Control input

Time [s]0 0.5 1 1.5 2 2.5 3

Nor

mal

ized

aile

ron

rota

tion

rate

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1AileronSaturation limit


Figure A.7: Tracking of a gust suppression signal at V∞ = 30 m/s

by Figure A.7c. In this conditions, the controller is not able to follow precisely the requested ratesaturation, but it stabilizes at a lower value as soon it reaches such a limitation. This is due to thepresence of the aerodynamic disturbance. In fact the outer PID loop is able to reject efficiently constantor slowly varying disturbances only, thanks to its integral term.The problem is that, while the controller tries to maintain the deflection rate constant during thesaturation, the aerodynamic hinge moment changes because it depends also on the control surfacedeflection, as well represented by Eq. A.12. Because of this, the controller is not able to performefficiently the required task.

A possible solution to this problem would by the introduction of an additional control term able tocompensate time-varying disturbances. In this work a solution based on the use of a reduced orderobserver is devised, as in [328]. Starting from the measure of the motor position θ1,meas the observerintegrates the following dynamic model:

I1θ1 = Tm +d

d = w(A.13)

Where d is the disturbance perturbing the motor actuation, whilew is a white noise signal. Therefored includes the effect of the aerodynamic disturbance and other possible effects, e.g. dry friction, that

170 Chapter A

Figure A.8: Dual loop controller with integrated observer for disturbance rejection

may deteriorate the controller action. An estimation of such a disturbance and its following integrationinto the control law would result into a beneficial effect, especially near saturating conditions. Theobserver has the following structure:

θ1

θ1

d

=0 1 0

0 0 I−11

0 0 0

θ1

θ1

d

+ 0

I−110

Tm +0

01

w +L1

L2

L3

(θ1,meas −θ1

)(A.14)

where Li , are the elements of the observer gain matrix. This system is tuned through a classical LQGdesign [329], where the weight matrices used in this case are Q = diag

(0,0,7 ·105

)and R = 1 ·10−6. As

can be noticed from Eq. A.14, the observer not only provides an estimate of the disturbance but alsoa direct calculation of the motor rate θ1, which has proven to be more accurate and robust than theestimation computed by the linear filter of Figure A.3. In view of these results, the final architecture ofthe controller is depicted in Figure A.8.Considering again the reference signal of Figure A.7, the results relative to the new architecture arepresented in Figure A.9. As can be noted from Figure A.9a, the tracking properties of the controllerare improved using this new architecture. Comparing also Figure A.9b with A.7b, it is clear that thecontrol effort required now is also slightly lower than in the previous case. Nevertheless, the maindifference is spotted comparing Figure A.9c with A.7c. With the integration of an observer that rejectstime-varying disturbances, the controller is now able to follow the saturation in rate with a sufficientlygood accuracy. The robustness margins of the present controller are even better than the previouscase, with a gain margin of 7.04 dB at 35 Hz, while the phase margin is 50 deg at 11 Hz.Because of all of these improvements, the present controller architecture will be the one used in therealization of the actuation system of the wind tunnel aircraft that will be employed for the validationof different gust suppression control laws within the GLAMOUR project.

A.4 Experimental Validation

Simulation results were used to define the design requirements in terms of bandwidth and maximumtorque acting on the motor. In the case considered, an Harmonic Drive RSF-5B with a gear ratio of 50and an additional gear ratio due to the pulleys contribution allows to produce the required torque.The complete hardware for the test rig servo–controller of Figure A.10 is composed by one PC, anelectric motor connected to a drive and two power supplies. The drive needs two power lines, one forthe electronics and the other one for the motor power. The additional power supply is connected tothe digital PCI board and to the digital channels of the drive. The PC is equipped with four PCI boards:the digital board, the analog input board, that allows to receive the command input from the gustalleviation control system, the analog output board and the encoder board.The motor position is measured through the encoder already embedded in the motor, while the aileronposition is measured by a potentiometer with a resolution of 0.2 degrees. This last signal is conditionedthrough an anti-aliasing filter, whose cut-off frequency is set at 200 Hz.

A.4. Experimental Validation 171

Time [s]0 0.5 1 1.5 2 2.5 3

Nor

mal

ized

rot

atio

n

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1 ReferenceAileron


Time [s]0 0.5 1 1.5 2 2.5 3

Nor

mal

ized

inpu

t tor

que

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

(b) Control input

Time [s]0 0.5 1 1.5 2 2.5 3

Nor

mal

ized

aile

ron

rota

tion

rate

-1

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Figure A.9: Tracking of a gust suppression signal at V∞ = 30 m/s with torque compensation

Figure A.10: Aileron test rig

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Figure A.12: Optimization of the control loops with experimental data

The control law is implemented through the real time environment RTAI [330] and the gainsdesigned through the simulator are directly imported in the test rig apparatus. Nevertheless, it isinteresting to test the optimized tuning procedure presented in Section 3.2 on experimental data.Therefore, the same training chirp signal used by the simulator is tracked by the test rig, and the outputsignal is recorded, as shown in Figure A.11. This data is used by the optimization algorithm presentedpreviously, always setting the target inner loop bandwidth at 50 Hz and the outer loop bandwidth at20 Hz. The resulting optimized closed loop transfer functions are depicted in Figure A.12, while theobtained gains value are listed in Table A.3 and compared with their initial guess coming from thesimulator. As can be noticed from Figure A.12, even if the data is noisy, the optimization is able to bringthe closed loop response close enough to the target. To do this, the inner loop gains are maintainedalmost unchanged from the parent values coming from the simulator-based design, while the outerloop gains are slightly modified, in particular the value of Kl

P, as witnessed by Table A.3. This changecould be due to the fact that the estimated belt stiffness used in the simulator may be smaller of thereal one, therefore a smaller proportional gain is required to maintain the same performance on the


KmP Km

I KlP Kl

I KlD N b1 b0/b1

Simulator optimization 0.06 7.1 20130 8.7314·106 40.36 295 124 300

Test rig optimization 0.052 4.97 4651 3.6017·106 37 300 60.3 300

Table A.3: Experimental controller gains

experimental apparatus.Various reference signals are now tracked by the test rig to verify if it presents a similar behavior to

the one predicted by the simulator. At first, a sine signal of frequency 2 Hz and an amplitude sufficientlysmall to avoid rate saturation is used as reference. This test is used to verify if the apparatus presentsreasonable performance and does not exhibits unusual behaviors. The results are displayed in FigureA.13, where the experimental output signal is compared to the tracked reference and the outcome ofthe simulator due to the same input signal. As can be seen, the servo system presents good trackingproperties, with a response very similar to that predicted by the simulator. Nevertheless, even if ananti-aliasing filter is present, the potentiometer signal appears to be rather noisy, and this influencesthe estimates of the aileron rate of rotation, which also appears noisy. This signal noise may be due toa not perfect set up of the electronics, but at the moment no modifications to the acquisition systemhave been implemented yet. Nonetheless, the estimated aileron rate follows with good correlation thesimulator outcome, and this fact confirms that the simulator is capturing the main dynamic behaviorof the implemented servo.With respect to rotation and rate of rotation, the torque signal presents a higher level of noise. This canbe due to two main reasons. First, the sampling frequency is rather high (2500 Hz), and this allowsthe servo to act very quickly to assure good tracking performance, even in presence of unmodeledphenomena. Second, a quite high level of dry friction has been experienced on the test rig, i.e. about10% of the saturation torque. This fact influences the control effort, which is slightly larger than theone predicted by the simulator, presenting also a less regular shape.

The second test considers as reference a sine signal of frequency 4 Hz that will induce rate sat-uration in the system. This test is used to verify if the experimental apparatus presents a good ratesaturation performance. The results are displayed in Figure A.14, where the experimental output signalis compared to the tracked reference and the outcome of the simulator due to the same input signal.This verification provides satisfactory results, reproducing a rate saturated signal, which anyway doesnot lose the tracking of the reference signal. Differences with respect to the simulator outcomes can bespotted. For example, it seems that the simulator allows the servo to reach larger amplitudes than whatis actually achieved by the test rig. This can be still due to the noisy aileron rotation measure. In fact,this noise influences the rotation rate estimate, which appears rather irregular around the saturationlimit, nonetheless respecting this value in average.

Finally, the last test considers the gust suppression input previously used in the simulator design.This verification is used to compare the test rig output to that of the simulator in a realistic condition.The outcomes are shown in Figure A.15, where the experimental output signal is compared to thetracked reference and the response of the simulator due to the same input signal. Also in this case thetracking of the reference signal is satisfactory, even in presence of a visible measurement noise. Theaileron rate estimation is still not very accurate but the saturation limits are satisfied in average. Thegreatest difference between test rig and simulator is once again represented by the control effort. Inthis case, the torque signal is noisier than the simulator one, like in all the previous cases, but here itpresents also a very strange behavior close to the signal tail, where spurious oscillations are recorded.Such oscillations are completely absent in the simulator calculations. This result can be interpreted

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Figure A.13: Experimental tracking of 2 Hz sine


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Figure A.15: Experimental tracking of a gust suppression signal


considering the signal rate of change during the final 0.4 seconds. In this period, the rate of rotationof the aileron is very small, approximately zero, and in this case dry friction effects can be dominant,inducing to an unforeseen control effort.

Concluding, the test rig setting has to be optimized to reduce the signal acquisition noise andto allow a probably better estimation of the aileron rate of rotation. Nevertheless, these preliminaryresults show that a simulator-based design of the servo system leads to a rather robust experimentalimplementation, avoiding unstable behaviors and respecting the design constraints of the wind tunnelmodel actuation system.

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