Lqr Control With Matlab

7/23/2019 Lqr Control With Matlab

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Control of a Tailless Fighter usingGain- Scheduling

N.G.M. Rademakers

Traineeship report

Coaches: Associate prof. C. Bil, RMITDr. R. Hill, RMITSupervisor: Prof. dr. H. Nijmeijer

Eindhoven, January, 2004


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Abstract

In this report the nonlinear model and flight control'system (FCS) for the longitudinaldynamics of a tailless fighter aircraft are presented. The nonlinear model and flightcontrol system are developed as an extension of earlier work on a conceptual designproject at W I T Aerospace Engineering, referred to as the AFX TAIPAN. The aircraftis based on a hypothetical specification aimed at replacing the rapidly ageing RAAFfleet of F-111 and F-18 aircraft with a single airframe.The presented six degree of freedom model consists of six second order differentialequations. The lateral and the longitudinal dynamics are highly coupled and nonlinear.Based on this full order model, the longitudinal dynamics of the aircraft are formulated.A gain scheduled controller is developed for the longitudinal dynamics of the aircraft.The gain scheduled controller is based on 25 operating points. An approach is providedto select the operating points systematically, such that the stability robustness of theoverall Gain-Scheduled (GS) control is guaranteed a priori. The longitudinal modelis linearized around these operating points and a linear quadratic regulator (LQR)controller is designed for each operating point to stabilize the aircraft. These controllersare interpolated using spline interpolation. Nonlinear simulation of the aircraft andflight control system is executed to analyze the performance of the flight control system.


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Contents1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . ..1 Motivations and Objective 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..2 Research Approach 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3 Outline 32 Model of the AFX-TAIPAN 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1 AFX.Taipan 42.2 Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..2.2 Axes Systems 6

. . . . . . . . . . . . . . . . . ..2.3 State, input and output variables 8. . . . . . . . . . . . . . . . ..2.4 Equations of Motion in Body Axes 10

. . . . . . . . . . . . . . . . . . . . . . . . ..2.5 Forces and Moments 123 LQR Control 15

3.1 Linear Quadratic Regulator (LQR) State Feedback Design . . . . . . . . 153.2 Longitudinal LQR-Control application . . . . . . . . . . . . . . . . . . . 17

3.2.1 Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Operating point . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.3 Stabiiity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


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Contents iii

4 Gain Scheduling 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1 Introduction 22. . . . . . . . . . . . . . . . ..2 Stability of the Gdr, Scheddec! Cmtrdler 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3 Stability Radius 23

4.4 Se!edion of the operating points . . . . . . . . . . . . . . . . . . . . . . 245 Longitudinal Gain Scheduling Control 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1 Scheduling Variables 27. . . . . . . . . . . . . . . . . . . . . ..2 Selection of the Operating Points 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3 Scheduling 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..4 Simulation 32

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..4.1 Flight path 32. . . . . . . . . . . . . . . . . . . . . . . . . ..4.2 Simulation Results 32

6 Conclusions and Recommendations 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1 Conclusions 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..2 Recommendations 37A Parameters of the AFX-Taipan 38B Specification of the operating points 41@ Matlab Files 44Bibliography 47


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Chapter 1

Introduction1.1 Motivations and ObjectiveIn 1999, a conceptual design project of a new fighter aircraft, referred as the AFXTAIPAN, was started at RMIT Aerospace Engineering, based on a hypothetical spec-ification aimed at replacing the rapidly ageing RAAF fleet of F-111 and F-18 aircraftwith a single airframe. This has resulted in a design without a traditional verticaltailplane. The main advantage of a tailless design is the reduction of the radar crosssection. In addition, the weight of the aircraft is reduced significantly. However, amajor disadvantage of a tailless design is the tendency of autorotation and the spinningdue to insufficient damping. The aircraft features two engines with thrus t vectoringcapability to enhance its manoeuvrability. It was proposed to investigate if the thrus tvectoring could be used t o artificially augment the stability of the aircraft without avertical tailplane. Moreover, the vectored thrust property enables additional controlauthority.Subsequent to the design of the aircraft in 1999, a new research project was star ted in2001 to derive the nonlinear mathematical model of the aircraft to s tudy the stability.From this study it appeared that the AFX-TAIPAN can be stabilized with a LQR sta tefeedback for a certain flight condition. However, the simple LQR technique to designa controller for a certain operating point, is not sufficient to ensure the controllabilityof the aircraft over its entire flight envelope. Due to the large changes in the aircraftdynamics, caused by the significant changes in lift, altitude and speed of the aircraft, aswell as aerodynamic angles or the change in the aircraft mass, a dynamic mode that isstable and adequately damped in one flight condition may become unstable, or at leastinadequately damped in the other flight condition.


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Chapter 1. Introduction 2

The objective of this study is to develop a Gain Scheduled controller for the longitudinaldynamics of the AFX-TAIPAN. A gain scheduled controller is a linear feedback con-troller for which the feedback parameters are variable. The feedback gains are scheduledar,r,nr&ng the instachneol~c : ahes srhechllng v~ ia Mes j hich are signals or variableswith which an operating condition of the plant is specified.

1.2 Research ApproachTo achieve the objective, the following research approach is followed.

First, the Tailless aircraft is considered and a dynamic model for the six degreeof freedom is derived. ivioreover, the iongitudinai dynamics are formulated.Subsequently, an equilibrium point is determined using nonlinear optimizationtechniques. This equilibrium point is used as the nominal operating point, whichis considered to be the middle of the region in which the controller is designed.The longitudinal model of the AFX-TAIPAN is linearized around th e nominaloperating point and a LQR controller is designed to stabilize the aircraft aroundthis operating point.A method is developed to automatically generate a grid of operating points aroundthe nominal operating point, such that the stability in the entire control area isguaranteed. This method is based on the stability radius concept. The controlarea is determined by the grid of operating points.For all operating points LQR controllers are designed to stabilize the aircraftwithin a certain area around the operating points. Using gain scheduling tech-niques, a global controller is designed, for which the sta te feedback gains of theindividual LQR controllers are scaled according to the operating point in thecontrol area.

0 Simulation of the flight control system and the nonlinear model of the longitudinaldynamics of the AFX-TAIPAN is executed to analyze the performance of the flightcontrol system.


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Chapter 1. Introduction

1.3 OutlineIn Chapter 2 the aircraft which is considered in this report, the AFX-TAIPAN, isdiscussed. First the description of the taiiiess aircrah is provided, after which themodelling is treated. The equations of motion and the force equations of the full sixdegree of freedom model are presented. The subject of Chapter 3 is Linear QuadraticReguiator (LQR) control. LQR control in general is discussed and is applied for thelinearized longitudinal dynamics of the AFX-TAIPAN. In Chapter 4 Gain Schedulingis studied. A method to select the operating points is presented, such that the stabilityof the entire control area is guaranteed. The gain scheduled controller is applied for thelongitudinal dynamics of the AFX-TAIPAN in chapter 5. Finally, some conclusions aredrawn and recommendations are given in Chapter 6.


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Chapter 2

Model of the AFX-TAIPANIn this chapter the AFX-TAIPAN is introduced. First , a general description the aircraftis provided. Subsequently, the six degree of freedom dynamic model of the aircraft isderived. The assumptions which are needed for this derivation are presented, just asthe axes systems used in this report. The equations of motion and the force model aredescribed in body axes.

The AFX-Taipan was designed in 1999 to be a multirole aircraft. The aircraft is basedon a hypothetical specification aimed a t replacing the rapidly ageing RAAF fleet ofF-111 and F-18 aircraft with a single airframe. The airframe is a fighter-bomber, whichis capable to carry a large and diverse payload for the strike role. To attain super-manoeuvrablility in air, the thrust-to-weight ratio is larger than one. This resulted ina tailless design. The aircraft is depicted in Figure 2.1. The full specification of theaircraft is given in [GPT99].The aircraft is equipped with two JSF119-611 Afterburning Turbofans. These enginesenable the use of a 3D thrust vectoring system in order to improve the manoeuvrabilityof the aircraft. The maximum pitch and yaw angle of the nozzles are fixed to 20.Moreover, the control system of the aircraft contains outboard split ailerons. Theapplication of thrust vectoring allowed the designers to consider a tailless design. Themain adva ~tag esf a tailless configuration are:

o By removing the tail and shorteniag of the fnselage, the drag of the aircraft c mbe reduced by 20% - 35%.The structural weight of the aircraft can be reduced with 10%

0 The range for a given fuel load can be increase because of the reduced drag.


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Chapter 2. Model of the AFX-TAIPAN 5

IFigure 2.1: Multi-view diagram of the AFX-Taipan

0 The stealth characteristics can be improved because of the reduced surface area.However, a major disadvantage of a tailless design is the tendency of autorotation andthe spinning due to insufficient damping. Moreover, the tail provides a downward forcethat counteracts the pitching tendency of an aircraft. Therefore a flight control systemhas to be designed to stabilize the aircraft.

2 .2 Aircraft ModelFor the design of a flight control system for the AFX-TAIPAN, a mathematical descrip-tion of the dynamical behaviour is needed. In this section the equations of motion arederived and the force model is provided. The model is an improvement of the modelderived [Pan02].

2.2.1 AssumptionsFor the modelling of the aircraft dynamics, several assumptions are used. These as-sumptions follow from [Pan02].

0 The aircraft is considered as a rigid body with six degrees of freedom; threetranslational and three rotational degrees.

0 The mass of the aircraft is constant.


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The Earth is flat and therefore the global coordinate system is fixed to it. Con-sequently, the Earth is considered as an inertia system and Newton's second lawcan be applied.

0 O X b and OZb are planes of symmetry for the aircraft, therefore: Ix y= J yz dmand Ixy= xy dm are equal to zero and I,, = I,, in the inertia tensor matrix.

0 The atmosphere is iixed to the Earth.0 The airflow around the aircraft is assumed quasi-steady. This means that the

aerodynamics forces and moments dependent only on the velocities of the vehiclerelative to the air mass.

0 As the aerodynamics is quasi-steady, the forces and moments of the aircraft areconsidered to act with respect to the aircraft centre of gravity (CG).

0 The vector thrust, aerodynamic forces and moments can be resolved into body-axis components a t any instant of time.

0 The atmosphere is still (no wind).0 The ailerons are symmetrical with respect to the plane that is spanned by the

x-axis and z-axis of the body fixed axes system of the aircraft.The contribution to the inertial coupling of all portions of the control system otherthan the aerodynamic surfaces is neglected.

0 For the purpose of calculating the inertia coupling, the control surfaces are ap-proximated as laminae lying in the coordinate planes.

0 The control systems are frictionless.0 Each control system has one degree of freedom relative to the body axes.0 Each control system consists of a linkage of rigid elements, attached to a rigid

airplane.The acceleration due to gravity is considered as constant with a value equal to9.81 m/s2.

0 There are no gyroscopic effects acting on the aircraft.

2.2.2 Axes SystemsThere are three primary axes systems considered, which are depicted in figure 2.2.


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Chapter 2. Model of the AFX-TAIPAN

Figure 2.2: Aircraft axes systems

1. The first axes system is the Earth initial axes system(F).ecause2;s pointedtowards the North, 2;owards the East and 2; ownward, this reference frameis also known as the NED axes system. The inerial frame is required for theapplication of Newton's laws.

2. The second axes system is the aircraft-carried inertial axes system (z).hisaxis system is obtained if the Earth inertial axes frame is translated to the centerof gravity of the aircraft with a vector

3. The body axes system (zb)s also an aircraft-carried axes system, with 2;pointed towards the nose of the aircraft, 2; owards the right wing and 2;to the bottom of the aircraft. The axis system is obtained through successiverotations of the aircraft-carried inertial frame with Tait-Bryant angles $ 8 and6. The velocity vectors along these axes are u, v and w and the angular velocityvectors are respectively: roll rate q, pitch rate q and yaw rate r. These velocityvectors are sho-WIIin Egiire 2 .3


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Figure 2.3: Body axes frame and velocities

2.2.3 State , input and output variablesThe motion of an aircraft considered as a rigid body can be described by a set of sixcoupled nonlinear second order differential equations. In general the model can bedescribed as

where g E Rn is the state vector, 2 E Rm is the input vector and - E R' s the outputvector.

The elements of the state and output vector are described in table 2.1, while the elementsof the input vector are shown in table 2.2. The translational velocities, u, v and w areselected instead of the to tal velocity, &, the angle of attack, a, and the sideslip angle,p, which are chosen in [Pan02]. The relation between the velocities of the aircraft withrespect to the inertia axis frame can be expressed in terms of th e velocities with respect


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Table 2.1: State definitions

I w I z component of velocity in body axes I m / s IP I Roll rate in body axes I r a d l sa I Pitch rate in bodv axes I r a d l s

Unitmlsm / s

St&,esuv

Ix I x ~ositionf CG in inertial frame m

Defi~iticr,x component of velocity in body axesv component of velocity in body axes

Yz

1 I Yaw ande

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Table 2.2: Input Definitions

y position of CG in inertial framez ~ositionf CG in inertial frame

UnitThrust force engine 1

mm

Roll anglePitch angle

y,~ I Yaw angle of nozzle 1 I r a dT? I Thrust force enaine 2 1 N

r a dr a d

- - Ipn2 Pitch angle of nozzle 2 r a dvnz Yaw of nozzle 2 , r a d ,


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to the body fixed axes frame using the direction cosine matrix.

where:cos$ cos 8 sin q5sin 8 cos$ - cos4 sin$ cos4 sin 8 cos$ + sin q5 sin .JIsin$cosO sinq5sin8sin$+cosq5cos$ cos~sin8sin$-sinq5cos?l,- ( - in B sin 4 cos 13 cos q5cos 8

(2.5)The angular velocity vector gives the relation between the Tait-Bryant angles and theirderi-mtiws 2nd the re!! Y nit&- ( n )Y / a d aw (r) rates of the aircraft.

2.2.4 Equations of Motion in Body AxesThe non-linear model of the tailless aircraft can be derived using Newton-Euler equa-tions. Newton's second law is applied for the translational dynamics and Euler equationsare used to describe the rotational dynamics of the aircraft.

Newton's second lawNewton's second law for a rigid body is:

where C2 s the resultant of all external forces applied to the body with mass rn.is the linear velocity vector of the center of gravity (CG) of the body relative to

the Earth inertial frame. Because the mass of the aircraft is considered to be constant,(2.7) can be written as

where ?cG is the linear acceleration vector of the body relative to the Ear th inertialfxm e. Based on (2.8)' the eq ~at fon sf motion derived in the body mes system are:


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with

where (2.6)is the angular velocity vector of the aircraft with respect to the bodyfixed frame. This results in the following force equationsCF!= m(iL+qw-rli)

CF; m(ir+ru-pw)nL@' m(w+pv-qu)

Euler EquationsEuler equations are applied for the rotational dynamics of the aircraft. The rotationaldynamics can be described as

g c o = 2co (2.12)where C g C ~s the resultant of the external moments applied to a body relative toits center of gravity. is the angular momentum vector relative to the center ofgravity of th e aircraft, which can be described by

~ C GJ C G ~ (2.13)where JCMs the inertia tensor of the rigid body. From (2.12) and (2.13),the rotationalequations of motion derived in the body axes system are:

Tnis results in the following equations.


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2.2.5 Forces and MomentsThe forces and moments a t the center of gravity of the aircraft have components dueto the ga-$Ita~ iona l ects, ~erodjrna+~csffect m d the f~rces ~ dcmezts due t c the3d thrust vectoring system. The force model is based on the force model derived in[Pan02].

Gravitational forcesThe components of the gravitational forces in the aircraft center of gravity with respectto the body axes are:

= -mg sin 6jF : ~ = mg sin(+) cos(0)F ; ~ = mg cos(+) cos(6)

Aerodynamic forces and momentsThe aerodynamic forces acting on the aircraft are basically the drag, lift, side-force andthe aerodynamic rolling, pitching and yawing moment.

where 7j= & v , ~ s the dynamic pressure of th e free-stream, S the wing reference area, bthe wing span and mac the mean aerodynamic chord. The values of these parameters aredetermined in [GPT99] and given in table A.l of Appendix A. CD, CL,Cy Cl,CmandCn are the dimensionless coefficients of drag, lift, side-force, rolling-moment, pitching-moine~t d yzwing-moment respectively. The aerodynamic coeEcier;ts tire mainly afunction of the angle of attack and the side-slip angle. Although the dependency ofthe aerodynamic coefficients to these parameters can be very nonlinear in high Machnumber region, a linear approximation is used. The aerodynamic coefficients are derived


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in [GPT99] and shown below.

The values for the aerodynamic parameters ci are given in table A.2. In (2.19) theparameters Peg,aeq,,, and ueq are obtained from [Pan021 and are given in table A.3.Since the drag force is defined in negative flight direction and the lift force perpendic-ular to the flight direction pointed to the top of the aircraft, the components of theseaerodynamic forces in the body fixed frame are:

Thrust forces

The thrust forces are generated by a 3D thrust vectoring system. The thrust forces inbody axes frame are derived using goniometric relations

where TI and T2denote the magnitude of the thrust force, pnl and p,2 the pitch angle ofthe rmzz!es a d nl &ridyn2 the yaw ar,gle of the nozzles. The moments on the aircraftgenerated by the thrust forces are


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where d is the distance between the nozzle exit center and the aircraft vertical planeand XCg he distance from the nozzle exit to the aircraft center of gravity.With this, the dynamical model of the aircraft is derived. The combination of theequations of motion and the force model yields in six coupled nonlinear second orderdifferential equations. The model can be used for the determination of the response ofthe aircraft.


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Chapter 3

LQR ControlIn this chapter the design of a Linear Quadratic Regulator (LQR) controller is discussed.First, LQR control in general is studied and then applied for the longitudinal controlof the aircraft. A nominal operating point is selected and the input is adjusted suchthat remaining accelerations in the operating point are minimized. Subsequently, thenonlinear longitudinal model is linearized around this nominal operating point and thestability of the open-loop and closed-loop system is discussed.

3.1 Linear Quadratic Regulator (LQR) State Feedback De-sign

A most effective and widely used technique of linear control systems design is the optimalLinear Quadratic Regulator (LQR).A brief description of LQR state feedback designis given below. For more details, see [Lew98].Consider the linear time invariant system

with state vector, x(t) E Rn, input vector, u(t) E IRm and output vector y(t) E Rz. Ifall the states are measurable, the state feedback

with sta te feedback gain matr ix, K E BmZn, an be applied to obtain desirable closedloop dynamics


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Chapter 3. LQR Control

x = (A - BK)x- A&For LQR control the following cost function is defined:

Substitution of (3.2) into (3.4) yields:

The objective of LQR control, is to find a state feedback gain matrix, K , such thatthe cost function (3.5) is minimized. In (3.5), the matrices Q E RnXnand R E Rmxmare weighting matrices, which determine the closed-loop response of the system. Thematrix Q is a weighting matrix for the states and matrix R is a weighting matrix for theinput signals. By the choice of Q and R a consideration between response time of thesystem and control effort can be made. Q should be selected to be positive semi-definiteand R to be positive definite.To minimize the cost function, (3.5) should be finite. Since (3.5) is an infinite integral,convergence implies x(t) + 0 and u(t)+ 0 as t -+w. This in turn guarantees stabilityof the closed-loop system (3.3). To find the optimal feedback, K , i t is assumed tha tthere exists a constant matrix P such that:

Substituting (3.6) into (3.5) results in

If the closed-loop system is stable, x( t) -+ 0 for t + w. Now, J is a constant thatdepends only on the matrix P and the initiai conditions. ~t this point, it is possibieto find a state feedback, K , such that the assumption of a constant matrix P holds.Substituting the differentiated form of (3.6) into (3.3) yields:


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and thereforeA ; P + P A ~ + Q + K ~ R K = O

Substitution of (3.3) into (3.9) yields

the following result can be obtained.

This result is the Algebraic Riccati Equation (ARE). It is a matrix quadratic equation,which can be solved for P given A,B,Q and R, provided that (A,B) is controllable and(Q4 ,A ) is observable. In th at case (3.12) has two solutions. There is one positive definiteand one negative definite solution. The positive definite solution has to be selected.Summarizing, the procedure to find the LQR state feedback gain matrix K is:

Select the weighting matrices Q and R

Solve (3.12) to find P.

Compute K using (3.11).

With this, the optimal feedback u = -R-lBTPx is obtained.

3.2 Longitudinal LQR-Control applicationThe above LQR-theory is applied for the longitudinal flight control of the tailless air-craft. First the longitudinal dynamics are formulated and a nominal operating point isselected. The nonlinear model is linearized around this operating point and an LQRcontroller is designed.


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3.2.1 Longitudinal dynamicsThe full model of the tailless aircraft is described in Chapter 2. In this chapter only theiongitudinai dynamics are v~ua I--~ ~ ~ ~ b ~ d i ~ r d:A-- ' mode! of t h e AFX-TAIPAX ir,body axes are defined as & = fion k&-m ~zopz) (3.13)

Y = glon(Z~m, (3.14)where gz, E IR4 = [8 u w q]T is the state vector, uz, E R2 = [T, pniIT is theinput vector and y E R4 = [8 u w q]T is the output vector. It should be notedthat for the longitudinal model, there are three degree of freedom and two independentinputs. The longitudinal dynamics are considered for a side velocity v = 0, roll angle4 = 8 a ym - sngk $ = O. The mgz!ar dcxit ies p ESK!r ere dsc! eqm! t o zero.The thrust force and pitch angle of both engines are assumed to be equal and the yawangle for the nozzle is considered to be zero. The dynamic equations, flon (a,,,,)are defined as follows:

m - q w

1+- (2Ti cos (pni)-mg sin (8))m - q w1- C L ~- LI (arctan (E) a,,) - pc~zqZfHSti) =-+ q u (1+ $)+

~ f g ~CDO+cm (arctan ( f )- aeq)+ :-m +qu (1+$)1+-m + qu (2Ti sin (pni)+mg cos (8))

1+- (2% sin (pni)Xq)I5rY


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Table 3.1: Equilibrium states and remaining accelerations

The vdues of the parameters can be found in table A .2 and table A.3 of Appendix A.

3.2.2 Operating pointIn order to design a LQR controller, the nonlinear model (3 .15) has to be linearizedaround a certain equilibrium point. :p:n is an equilibrium point of (3 .13) if there existsueq such tha t fi,(gpQ,, &) = 0 .-1 onBecause it is not possible to find an equilibrium analytically, nonlinear optimizationtechniques are used to find values for the elements of the inputs Ti and pni such thatthe sum of the squared remaining acceleration is minimized for a given operating point.The objective function 0 is:

The upper and lower bound for the optimization parameters are:

20-- 20180T ad < pni 5 -T ra d180The next thing to do is to determine an initial operating point. For the initial operatingpoint it is desired that this is an equilibrium point. From (3 .13) it can only be concludedthat the pitch velocity, q, has to be equal to zero. Further, the inputs Ti nd p,iare kept at a fixed value of respectively 50.000 N and 0 rad. The remaining statesu, w and 0 are used as optimization parameters to minimize the sum of the scparedaccelerations on the aircraft (3 .16) . The results are listed in table 3.1. This operatingpoint is considered as the nominal operating point. It can be observed that the maximalremaining acceleration is -3.3358 rad / s2 . This means that no equilibrium pointis found.


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Table 3.2: EigenvaluesI Open-loop eigenvalues I Closed-loop eigenvalues1 0.5607 1 -9.5580 1

3.2.3 StabilityNGWhe n~mina ! perzting poht is determined, the !ongitudind mnde! I-?----5) is 1 i ~ -earized around this point. Subsequently an LQR controller is designed using the function'lqr' in the MatLab control toolbox. The weighting matrices Q and R that are used are:

It should be noted tha t the values for the weighting matrices are chosen arbitrary. In thisreport performance requirements for the aircraft are not specified. The only requirementis to stabilize the aircraft. However, if performance requirements are taken into account,they can be satisfied by adjusting Q and R. The selected weighting matrices result inthe following feedback gain for the nominal operating point.

The eigenvalues of the open-loop and the closed-loop systems are given in table 3.2.From this it can be concluded that the unstable modes are now fully stabilized using aLQR controller. In figure 3.1 the closed loop response of the four states on a step in thepitch angle of the nozzle is depicted. From figure 3.1 it can be observed that a steadystate value is reached after 50 seconds. Also from this figure it can be concluded tha tthe s tate feedback matrix K , can stabilize the the system.Summarizing, it can be observed that the longitudinal dynamics of the AFX-Taipanare derived in this chapter and linearized around an operating point. The nominaloperating point is found using optimization techniques, but it is not an equilibriumpifit. In addition it, can be concluded that a LQR controller can be used to stabilizethe system around the nominal operating point.


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lo-5 Step Response

time time

time time

Figure 3.1: Response to a step change in command pitch angle of the nozzle


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Chapter 4. Gain Scheduling 23

The last step is the performance assessment. This can be done analytically orusing extensive simulation.

4.2 Stability of the Gain Scheduled ControllerConsider the nonlinear system

ii= f (:,IdLet I' c Rq denote the set of GS-parameters, such that for every y E I? there is anequilibrium (x7, u7) of 4.1. Consider the control law for system (4.1)

where Ky E Rmxn is a feedback gain that strictly stabilizes the linearized plant atequilibrium (g7, 7).The stability and the robustness of the nonlinear plant (4.1) with controller (4.2) asthe system moves from one equilibrium to another along an arbitrary path is studiedin [AkmOl] Theorem 4.2 in [AkmOl] proves th at the control law (4.2) is robust withrespect to perturbations if the scheduling variables vary slowly.

4.3 Stability RadiusIn most gain scheduling applications, the operating points are selected heuristically. In[AkmOl] a systematic way to select the operating points, such th at the stability of theentire control area is guaranteed, is explored, based on the idea of the complex stabilityradius. The automatic construction of a regular grid of operating points for the controlarea of the AFX-TAIPAN, is based on this technique.Consider the linear system of the form:

where (A,B) is assumed to be stabilisable. Applying the state feedback stabilizingcontroller

u(t) = -Kg(t) (4.4)(4.3) can be written as

~ ( t )Ad:(t) (4.5)where ACl= A - BK. Since Ad is stable, the eigenvalues of Ad are in the left halfplane.


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Chapter 4. Gain Scheduling 24

Consider a perturbation of the closed-loop system with a structured perturbation E A H ,where E E WnxP,H E Rqxn are scale matrices tha t define the structure of the pertur-bation and A is an unknown linear perturbation matrix. The closed-loop perturbedsystem then becomes

k ( t )= (Ad +EAH) :(t) (4.6)According to [EK89], the compkx structnred stability r a d i~ sf (4.6) is defi~e d y

rc(A,E, H ) = inf 1 1 All; A E Wpxq, a(Ad f EAH) nC+# 0)where C+ denotes the closed right half plane and IlAll the spectral norm (i.e. largestsingular value) of A. Further, in [HK89] it is proved that the determination of thecomplex stability radius (4.7) is equivalent to the computation of the Hco-norm

of the associated transfer function G(s) = H (sI-A&)-' E.4.4 Selection of the operating pointsUsing the concept of the complex stability radius, new operating points can be selectedin such a way that the stability and performance robustness of the overall design isguaranteed.Given the complex stability radius, r,(yi), of operating point yi, sphere, Byi (a) , canbe constructed in which the state feedback gain matrix Kyi can be used to stabilizethe aircraft. This methode is described in [AkmOl] and is depicted in figure 4.1 for the2D case with 2 scheduling variables. In this figure the construction of a circle aroundyi, contained in the stability radius, is shown. Mathematically the construction of thesphere can be formulated as follows:

The next thing t o do is to find a systematic way to fill the operating space with operatingpoints, such that the stability of the whole control area is guaranteed. The idea is tofill the control area with a grid of adjoining squares of the same size. The size of thesquares is determined by the point in the control area for which the radius of the sphere


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Chapter 4. Gain Scheduling

Figure 4.1: Construction of By,i(a)

in which the system can be stabilized is the smallest. The centers of the squares are thenew operating points. This approach is depicted in figure4.2. With this, the procedureto construct a regular grid of operating points is completed.


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Chapter 4. Gain Scheduling

u [ m/sl

Figure 4.2: Neighbourhood


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Chapter 5

Longitudinal Gain Scheduling

In this chapter the gain scheduling technique will be applied for the longitudinal controlof the AFX-TAIPAN. The longitudinal model is described in Chapter 3. A regular gridof operating points is constructed according the technique of Chapter 4. For theseoperating points LQR controllers are designed. The resulting global gain scheduledcontroller is applied to the longitudinal flight control system of the AFX-TAIPAN. Infigure 5.1 a schematic representation of the design approach is depicted.

5.1 Scheduling VariablesThe first step in the design approach is the selection of the scheduling variables. In[AkmOl] it was proved that the scheduling variables should vary slowly to maintainstability when the aircraft moves from one equilibrium point to another. From @ug91]and [RSOO] it also follows that the scheduling variables should be slowly varying andcapture the nonlinearities of the system. However, this is only a qualitative result.In this report the forward velocity, u, and the downward velocity, w, are chosen asscheduling variables. Intuitively, these variables are slowly time varying and capturethe dynamic nonlinearities. However, not all nonlinearities are captured. To captureall nonlinearities, the density of the air and the pitch angle should also be considered.In this report the pitch angle and the density of the air are kept on a wed value, suchthat the nonlinearities caused by these parameters not occur.

5.2 Selection of the Operating PointsThe next step is the selection of the operating points as described in Section 4.4. Thenominal operating point was earlier determined in Section 3.2.2. The nominal values of


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Chapter 5. Longitudinal Gain Scheduling Control

Selection of !scheduling variables 1

i', II Selection of I' operating points I' 4-I I1I I

I/ LinearizationI

I LQR ControllerDesign

+ I/ Scheduling 1

electionOperatingPointsiInitial operatingpoint i

Selection of thecontrol area I/ Find Control Neighbourho odI * Input Opitrnization* Linearization* LQRControl Design* Stability Radius C omputation* Construction of Stability Ball

Construction grid ofoperating points-Figure 5.1: S t e ~ ~ ~ r o a c hain Scheduling


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the scheduling variables are

which corresponds to a total velocity of 166.59 m/s and an 4.4' angle of attack. Thesevalues are chosen because the remaining accelerations are minimized for this combina-tion of scheduling variables. The nominal operating point is considered to be in themiddle of the area for which the controller is developed. The ranges of the schedulingvariables for which the controller is developed are:

It should be noted t ha t this is only a very small area of the entire flight envelope ofthe AFX-Taipan. However, the principle of the selection of the operating points can beapplied in the same way for the entire envelope.For a regular grid of 25 points within this control area, the radius of the sphere isdetermined, in which the state feedback of this point is able to stabilize the system. Foreach of these 25 points the input is optimized to minimize the remaining accelerationsand the model is linearized around these equilibrium points. Subsequently the stabilityradius is computed. For the computation of the stability radius the structure of theperturbations matrices E and H has to be defined. Because there is no uncertaintyin the first equation of (3.15), 8 = q, and unstructured disturbance is assumed, theperturbation matrices are:

0 0 0 0

The singular values of G(s) = H (sI- ~d)-' E are computed between radlsand i~h.racl/sor all the poil?ts. The stability radius Is the Inverse of the lzrgestsingular value. Now the stability radius is known, the radius of the sphere B%(Q)can be constructed as was s h c t ~ I~ figwe 4.1. In eight directicns the value fe r a isdetermined such that Byi(a) ri. his is illustrated for the nominal operating pointin figure 5.2.The maximal values for a for each of the 25 points are depicted in figure 5.3. From thisfigure it appears that a,i, = 0.002 for all the points in the entire control area. This


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Figure 5.2: Determination of E

Figure 5.3: Stability Radius for the control area


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from the current point y o operating point i is calculated as follows:

The equilibrium input matrix and feedback gain matrix are computed using a percentageof the U and K matrices according to the distance.

where

with q the number of operating points. Using (5.4) the resulting controller becomes:

5.4 Simulation5.4.1 Flight pathIn order to analyse the tracking capacities of the gain scheduled controller, a flightpath is defined. The flight path has to lie in the control area of figure 5.4. The initialoperating point is:

The first 100 seconds, it is desired to stay in this initial operating point. During thenext 100 seconds a step of 5 . m/s2 acceleration is applied in xb direction and astep of -5. m/s2 n yb direction. After that the desired accelerations in xb and wbdirection are put to zero again. This means th at after 200 seconds the desired valuesfor u and w are constant again. It is desired to stay in tha t operating point for 100seconds. This makes it possible to analyse the steady state response.

5.4.2 Simulation ResultsUsing the nonlinear longitudinal model of the AFX-TAIPAN and the gain scheduledflight control system, simulations are executed. In figure 5.5 the desired and the actual


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Actual and desired states

. . . .

- 3278 . . . . . . . . . . . . . . . . . . . . .U .mh'0@ 0.3278 . . . . . . . . . . . .:.

. . . . . . . . . ..3278 . . . . . . .

time [s] time [s]

time [s] time [s]

Figure 5.5: Desired response and actual response


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time [s] time [s]

L 0 100 200 300time [s]

-21 I0 100 200 300time [s]

Figure 5.6: Error as function of time


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Chapter 5 . Longitudinal Gain Scheduling Control 35

response are depicted during a 300 seconds simulation. From this figure it appearsthat the desired trajectory and the actual trajectory do not agree. The error, which isdefined as the difference between the actual value of the states and the desired value ofthe states, is depicted in figare 5.6. lt caE bbe observed that zi steady s tz te ermr rem&i.lns.The reason for the steady state error is that the operating points which are used arenot equilibrium points. From table 3.1 it is shown that for the initial operating pointsthe acceleration of the aircraft is not equal to zero. Remaining accelerations for theoperating points are equivalent to a constant disturbance which acts on equilibriumpoints. That is why a steady state error can be observed in figure 5.6. The error issmall enough such that the aircraft stays in the control area. This means tha t the gainscheduled controller is able to stabilize the aircraft. However, if the desired trajectoryis chosen at the boundary of the control area, it could be possible that the remainingaccelerations take the aircraft out of the area for which the controller is deveioped. Thiscan lead to instability.Summarizing, it can be concluded that the gain scheduled controller has been success-fully applied to th e stabilization of the aircraft. However, a steady state error remains.


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Chapter 6

6.1 Conclusions0 In this report the six degree of freedom dynamical model of the AFX TAIPAN

is derived. Based on this six degree of freedom model, the longitudinal dynamicsare formulated.In order to find an equilibrium point for the longitudinal dynamics of the aircraft,optimization techniques are use to minimize the remaining accelerations. However,it appeared that the remaining accelerations are non-zero. This means th at theoperating point found, is not a real equilibrium point.

0 LQR Control can be used to stabilize the AFX TAIPAN around an operatingpoint. The concept of the stability radius is used to determine the neighbourhoodof a n operating point in which the LQR controller is able to stabilize the nonlinearsystem.

0 An approach to construct a regular grid of operating points for the gain scheduledcontroller is provided to assure global stability of the entire control area. This ap-proach is based on the smallest control neighbourhood in a predetermined controlarea.

0 The gain scheduled controller, based on 25 LQR controllers, is applied for thelnngitudina! dyrmmic of the AFX-TAImN. The gain scheduled controller is ableto stabilize the aircraft. However, because the control area does not consist of alink up of equilibrium points, a steady state error can be observed.


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Chapter 6. Conclusions and Recommendations

6.2 Recommendations0 In this report gain scheduling is applied for the longitudinal dynamics of the

AFX-TAIPAN. The scheduiing variables used are the forward and the downwardvelocity of the aircraft. However, to capture all nonlinearities the pitch angle, 8 ,and the density of the air, p , should also be considered. Moreover, gain schedulingcan be applied for the fuii order model.The idea of gain scheduling is to move through the operating space along a p athof equilibrium points. However, the operating points which are found are notequilibrium points. Additional control surfaces are needed to ensure that theoperating points become equilibrium points and that the gain scheduling theoryc= be applied.

0 The stability radius of the closed-loop system is very small. Since the stabilityradius determines the number of operating points in the control area, the relationbetween controller design and stability radius can be explored. After all, theselection of the weighting matrices Q and R determines the state feedback gainmatrix, K, and the placement of the poles. By exploring the relation between thecontroller design and the stability radius, the controller can be designed in sucha way that the stability radius is maximized.The way to construct a regular grid as shown in Chapter 4, is based on the smalleststability radius in the control area. If the stability radius changes a lot within thecontrol area this results in a large overlap of the stabilizing areas of the operatingpoint. More research can be done on minimize the overlap area.

0 In this report the performance requirements for the aircraft are not taken intoaccount. However, in general military standard requirements are specified. Theonly requirement was to stabilize the system. Further research is required to finda control design and an approach to select the operating points, such that theperformance requirements are satisfied.


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Appendix A

Parameters of the AFX-TaipanThe physical parameters of the AFX-Taipan as derived in are listed in Table A.1. A fulldescription of th e aircraft is given in [GPT99]. In table A.2 the aerodynamic parametersare listed as derived in [GPT99]. The equilibrium parameters for the aerodynamiccoefficients are depicted in A.3. These parameters follow from [Pan02].


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Appendix A. Parameters of the AFX-Taipan

Table A.3: Equilibrium parametersSymbolQeqQen- zUeqU n

NameEquilibrium angle of attackEquilibrium side slip angle

Value00

m l sm/s

- -UnitsradradNominal forward velocity

Equilibrium forward velocity205205


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Appendix B

Specification of the operating

The operating points which are selected for the gain scheduled controller are listed intable B.1. For these operating points the thrust force and the pitch angle of the nozzleare optimized to minimize the largest remaining acceleration. The acceleration for eachoperating point are given in table B.2.


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Appendix B. Specification of the operating points

Table B.l: State values of the operating pointsq [ rad l s ]

0000

w [ m l s ]12.753012.755512.758012.7605

Operating PointOP 1OP 2=I? 3OP 4

0 [rad]0.32780.32780.32780.3278

u [ m l s ]166.1001166.1001166.1001166.1001


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Appendix B. Specification of the operating points

Table B.2: Derivatives of th e states in the operating pointsq rad/s2]

-0.0223-0.0144

w m/s2]0.00050.0002

u m/s2]-0.3438-0.1617

1 07 3

- Operating PointOP 1OP 2

0 I 0.0198 1 0.0001 1 n nncF- V . V V V V I

8 [radls]00


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Appendix C

Matlab FilesThis appendix contains a short description of the MATLAB iles which are used.

Model Derivationparameters.mThe file parameters.m contains the parameters of th e aircraft as specified in[PanO2].Kinematics.mThe program Kinematics.m is used to formulate the kinematics of the 6th ordermodel of the AFX-Taipan. The file dir-cos-matrix is used for the computation ofthe direction cosine matrix. The kinematics are saved in th e file Kinematics.mat.F0rces.mIn th e file F0rces.m is used to formulate the force model of the AFX-Taipan. Theforce model is saved in the file Forces.mat.nonlinear6dof.mIn the file nonlinear6dof.m, the force model is substituted in the kinematics. Thisresults in the 6 dof dynamic model of the AFX-Taipan.Aircraft1on.mThe program Aircraft-lon is used to compute the derivatives of the longitudinalstates as a function of the values of the states and inputs.Linearization1on.mThe file Lictearizati:~ct~Im.ms used to compute the !inear !ongitudinal dynamicsof the AFX-Taipan.matrixAlon.m and matrix-B-1on.mThe files matrix-A-1on.m and matrix-B-1on.m are used for the evaluation of theA-matrix and B-matrix as function of the values of the states and inputs.


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Appendix C. MATLAB iles

Gain-Scheduling0 gs-cont rol1er.m

The file 9s-cantroller.%. is wed to compute the control input for a, raandom pointin the control area of the aircraft according to the distance to the individualoperating points. The LQR Controllers are saved in the file gs,con.mat.

0 gssim.md1This program is a SIMULINKrogram which is used for the simulation of thelongitudinal model of the aircraft and the gain scheduled controller.

0 run-GSsim.mThe program run-GS-sim.m is used to set all parameters for the simulation ofthe b~gitiididmde! of the &reraft m d the gain schedu!ed contro!!er. The filep1otaircrajLlon.m is used for the visualization of the desired and actual responsof the AFT-Taipan.


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Bibliography[AkmOl] Akmeliawati, R., Nonlinear Control for Automatic Flight Control Systems,University of Melbourne, PhD Thesis, 2001[APHB02a] Akmeliawati,. R., Panella, I., Hill, R., Bil., C., Nonlinear Modeling and

Stabilization of a Tailless Fighter, RMIT University, 2002[APHB02b] Akmeliawati,. R., Panella, I., Hill, R., Bil., C., Stabilization of a Tailless

[RSOO]

Fighter using Thrust Vectoring, RMIT University, 2002Garth, J., Phillips, M., Turner J., AFX- Taipan, RMIT University, Under-graduate Thesis in Bachelor of Engineering (Aerospace), 1999Hinrichsen, D., Kelb, B., An Algoritm for the computation of the StructuredComplex Stability Radius, Automatica, Vol. 25, No 5 p. 771-775, 1989Lewis, F.L., Linear Quadratic Regulator (LQR) State Feedback Design,http://arri.uta.edu/acs/Lectures/lqr.pdfNelson, R.C., Flight Stability and Automatic Control, McGraw-Hill, 1989Panella, I. Application of modern control theory to the stability and ma-noeuvrability of a tailless fighter, RMIT University, Undergraduate Thesis inBachelor of Engineering (Aerospace), 2002Rugh, W.J., Analytical Framework for Gain Scheduling, IEEE Control Sys-tems Magazine, p. 79-84, January 1991Rugh, W.J. , Shamma, J.S., Research on gain scheduling, Automatica, Vol.36 , p. 1401-1425, 2000

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Lqr Control With Matlab