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AN ADAPTIVE ROBUST APPROACH TO ACTUATOR FAULT-TOLERANT
CONTROL IN PRESENCE OF UNCERTAINTIES AND INPUT CONSTRAINTS
A Dissertation
Submitted to the Faculty
of
Purdue University
by
Shreekant Gayaka
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
December 2008
Purdue University
West Lafayette, Indiana
ii
TABLE OF CONTENTS
Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Need for actuator fault-tolerant control . . . . . . . . . . . . . . . . 11.2 Existing techniques: Scope and Limitations . . . . . . . . . . . . . . 2
1.2.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Desired features of a New Fault-Tolerant Controller . . . . . 4
1.3 An Adaptive Robust Approach to Actuator Fault-Tolerant Control 41.4 Assumption on Input Saturation . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Input saturation and its effect of actuator fault-tolerant control 51.4.2 Robust global stabilization of an integrator chain . . . . . . 61.4.3 Saturated Adaptive Robust Actuator Fault-Tolerant Control 6
1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Accommodation of Unknown Actuator Faults Using Output Feedback basedAdaptive Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Output Feedback based ARFTC . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Observer Canonical Form . . . . . . . . . . . . . . . . . . . 142.3.2 State Estimation . . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 Parameter Projection . . . . . . . . . . . . . . . . . . . . . . 222.3.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Simulation Example: Linearized Boeing 747 Model . . . . . . . . . 332.4.1 Detailed ARFTC Controller design . . . . . . . . . . . . . . 36
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Output Feedback based Adaptive Robust Fault-Tolerant Control for a Classof Uncertain Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Output Feedback based ARFTC . . . . . . . . . . . . . . . . . . . . 46
3.3.1 State estimation . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.2 Parameter Projection . . . . . . . . . . . . . . . . . . . . . . 503.3.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 50
iii
Page3.4 Simulation Example: A Nonlinear Hypersonic Aircraft Model . . . . 593.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 A Backstepping based Approach to Robust Global Stabilization of a Chainof Integrators with Input Saturation . . . . . . . . . . . . . . . . . . . . . 684.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Motivation and Problem Formulation . . . . . . . . . . . . . . . . . 714.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Convergence to Unsaturated Region . . . . . . . . . . . . . . 754.3.2 Controller Parameter Selection . . . . . . . . . . . . . . . . 80
4.4 Simulation Example: Third Order Integrator Chain . . . . . . . . . 874.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Saturated Adaptive Robust Actuator Fault-Tolerant Control for FeedbackLinearizable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 975.3 Adaptive Robust Actuator Fault-Tolerant Control . . . . . . . . . . 99
5.3.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . 1035.3.2 Controller design . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.3 Controller Parameter Selection . . . . . . . . . . . . . . . . 1115.3.4 Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . 118
5.4 Simulation Example: Nonlinear Hypersonic Aircraft Model . . . . . 1205.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 1266.0.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.0.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
iv
LIST OF FIGURES
Figure Page
2.1 Reference tracking, control signals and tracking error for MRAC versusARC based fault-tolerant schemes in absence of disturbances . . . . . . 40
2.2 Reference tracking, control signals and tracking error for MRAC versusARC based fault-tolerant schemes in presence of disturbances . . . . . 41
3.1 Structure of ARC and RAC based fault-tolerant controllers . . . . . . . 53
3.2 Reference tracking, control signals and tracking error for RAC versus ARCbased fault-tolerant schemes in absence of disturbances . . . . . . . . . 65
3.3 Reference tracking, control signals and tracking error for RAC versus ARCbased fault-tolerant schemes in presence of disturbances . . . . . . . . . 66
4.1 Saturation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Comparative results for stabilization in absence of disturbances . . . . 91
4.3 Comparative results for stabilization in presence of disturbance . . . . 92
4.4 Tracking in presence of large disturbance . . . . . . . . . . . . . . . . . 93
5.1 Saturation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Comparative results for stabilization in absence of disturbances . . . . 123
5.3 Comparative results for stabilization in presence of disturbance . . . . 124
v
ABSTRACT
Gayaka, Shreekant Ph.D., Purdue University, December 2008. An Adaptive RobustApproach to Actuator Fault-Tolerant Control in Presence of Uncertainties and InputConstraints . Major Professor: Bin Yao, School of Mechanical Engineering.
In this work, we develop adaptive robust schemes for actuator fault-tolerant con-
trol in presence of uncertainties and input saturation. The type of faults considered
in the present work encompass hardover-failure, loss in efficiency and stuck actua-
tors. The two chief ways in which the system performance can degrade following an
actuator-fault are undesirable transients and unacceptably large steady-state tracking
errors. Adaptive control based schemes are ideal for handling the jump in parameter
values following an actuator fault, and can guarantee good final tracking accuracy.
However, such schemes may not be able to suppress the transients due to sudden
change in system parameters. Furthermore, the performance of adaptive control
based schemes deteriorate significantly in presence of unknown modeling errors and
disturbances. Robust control based schemes, on the other hand, can guarantee desired
transient response due to the sudden jump in system parameters and attenuate the
effect of modeling uncertainties on the tracking error. But, in face of large paramet-
ric uncertainties due to actuator faults, the final tracking accuracy of robust control
based schemes may degrade as they cannot reduce the extent of parametric uncer-
tainties. In the present work, we claim that an adaptive robust fault-tolerant control
scheme can solve both the problems, as it seamlessly integrates adaptive and robust
control design techniques. Comparative simulation studies are performed using linear
and nonlinear aircraft models to illustrate the superior performance of the proposed
scheme over robust MRAC and robust backstepping based adaptive control designs
respectively.
vi
One of the standard assumptions made in the design of adaptive fault-tolerant
control is that the healthy actuators have sufficient control authority despite faults
to recover desired closed-loop performance. In reality, however, the controller could
generate large control commands to suppress the undesired transients, leading to ac-
tuator saturation. Furthermore, in direct adaptive schemes, the estimator may fail
to generate reliable parameter estimates due to saturation. This could further de-
grade the performance of a actuator fault-tolerant control. As a first step towards
developing an approach which can deal with input constraints, we propose a concep-
tually different approach for global stabilization of a chain of integrators. A novel and
elegant approach to solve this problem was proposed by Teel [1] using saturation func-
tions and coordinate transformation. With Teel’s work as foundation, many results
have been proposed to improve the performance of tracking/stabilizing controllers for
chain of integrators. Naturally, all such approaches also inherited the limitations of
Teel’s approach. Most importantly, in presence of uncertainties, such a transforma-
tion would considerably shrink the region where the controller is unsaturated, and in
some cases, may even render the task of designing a stabilizing controller impossible.
We combine the backstepping based design with saturation functions to develop a
simple controller which does not rely on coordinate transformation and meets all the
desired objectives. Furthermore, necessary and sufficient conditions for the existence
of the proposed control law, as well as a systematic way of choosing the controller
parameters is also presented. Comparative simulation studies are performed on a
third order integrator chain which shows the effectiveness of the proposed scheme.
Finally, an actuator fault-tolerant controller is designed which combines the pro-
posed backstepping based saturation functions approach with a least-square estima-
tor. The indirect scheme ensures that the adaptation mechanism is not affected
adversely due to actuator saturation. Simulation studies performed on a hypersonic
aircraft model demonstrate the effectiveness of the proposed scheme in addressing
actuator faults in presence of input constraints.
1
1. INTRODUCTION
1.1 Need for actuator fault-tolerant control
On 4 October 1992, El Al Flight 1862, a Boeing 747 cargo plane of the Israeli airline
El Al, crashed into the Groeneveen and Klein-Kruitberg flats in the Bijlmermeer
neighbourhood of Amsterdam, Netherlands. This unfortunate incident resulted in
43 casualties, consisting of the plane’s crew of three and a non-revenue passenger in
a jump seat, plus 39 persons on the ground. Among many factors that caused this
disaster, one of the chief reasons was the damaged control surfaces on the right wing.
On 22 September 1981, in a similar incident, Easter Airlines L-1011 flying from
Newark, New Jersey to San Juan, Puerto Rico, suffered a massive failure of its number
two (tail) engine. A shrapnel from the engine damaged 3 out of 4 hydraulic systems
in the tail structure. But, the fluid which remained pressurized in that 4th system
enabled the captain to land the plane safely at John F. Kennedy International Airport,
with remaining healthy components of the control system. There were no injuries.
Thus, the additional 4th hydraulic control system saved the plane and all on board.
Unfortunately, there are more examples like the El Al Flight where final outcome
was a disaster, and few success stories like that of Eastern Airlines. This naturally
raises the question - after a failure, is it at all possible to use the remaining healthy
components of the control system in a fashion to avoid complete breakdown? Many
researchers have tried to simulate the conditions which lead to such incidents, and
tried to come up with schemes which can avoid such disasters. In [2], it was concluded
that had a fault-tolerant control scheme been in place, the ill fate of the El Al Flight
could have been avoided. One of the reasons Easter Airlines L-1011 could be saved is
2
because the captain and the crew had sufficient reaction time, (i.e., time required to
detect, isolate and identify the fault) to reconfigure the remaining actuators to land
the aircraft safely. From the two incidents cited above, it can be concluded that the
key to avoid a catastrophic outcome is to use the working actuators effectively. Ac-
tuator fault-tolerant controller precisely accomplishes this by exploiting the available
redundancy in modern machines e.g., multiple control surfaces on an aircraft.
1.2 Existing techniques: Scope and Limitations
1.2.1 Literature review
From the preceding section, it should be evident that as complexity of modern
day machines increase, it may not be safe to rely on human interference to detect,
isolate and compensate faults. Ensuring safety and reliability in machines with large
number of subsystems and components call for advanced algorithms. Thus, in the
past few decades, many researchers have focused their attention to this important
problem, leading to a steady increase in literature devoted to actuator fault-tolerant
control. The effect of such actuator faults on the system dynamics can be captured
as unknown, sudden change in system parameters, and it can degrade the system
system performance in two chief ways: (a) it can cause large transients, which may
eventually cause instability and, (b) it may result in unacceptably large steady-state
tracking errors.
Most of the available literature on this topic can broadly be categorized in two
groups - adaptive control and robust control based designs. Adaptive schemes are a
promising approach to deal with such failures as it can learn the change in system
parameters by virtue of their on-line learning capability. Not surprisingly, many
adaptive schemes have been developed to solve this problem. Tao et al. proposed
a model reference adaptive control (MRAC) based scheme for unknown actuator
failure compensation for linear systems in [3]. They further extended their direct
fault-compensation scheme to various classes of nonlinear systems in [4], [5] using
3
backstepping based adaptive control. Another popular adaptive approach to solve
this problem is multiple model adaptive control (MMAC), switching and tuning [6].
But, none of these papers considered unstructured or non-parametric uncertainties
e.g., unknown nonlinearities and disturbances, which can be a limiting-factor of the
achievable system performance. Robust adaptive control (RAC) based schemes, that
can guarantee boundedness of closed loop signals in presence of unknown modeling
uncertainties and disturbances, were investigated in [5,7,8]. But, RAC is a variant of
adaptive control and lacks two desirable properties inherent to robust control based
techniques. First, there is no convenient and transparent way to attenuate the effect
of non-parametric uncertainties, like external disturbances, on system response and
steady-state tracking error. Second, such techniques are not well suited to suppress
the undesirable transients following a sudden change in system parameters due to
unknown actuator faults. As poor transients in adaptive control based schemes can
be attributed to the learning phase of the controller, it may appear that increasing
the adaptation gain can improve the transient response as it speeds up the learning
process. In fact, this result has been claimed in many articles (see [9], [10]). However,
as projection type of robustness modifications are present in RAC based techniques
to avoid parameter drift, the use of high adaptation gains may cause the estimated
parameters to bounce back and forth between the present upper and lower limits.
This could introduce a high frequency component in the control signal, which may
ultimately excite the high-frequency ignored dynamics. Thus, even though all signals
can be shown to be bounded, obtaining guaranteed transient response in a RAC
framework still remains a challenging problem. Robust control based schemes [11–
14], on the other hand, have guaranteed transient response in presence of various
uncertainties. Furthermore, the effect of such disturbances can be attenuated to any
desired extent on the steady-state tracking error. However, in face of large parametric
uncertainties due to actuator faults, such schemes will either lead to undesirable
control input chatter or poor steady-state tracking error due to smoothing techniques.
4
1.2.2 Desired features of a New Fault-Tolerant Controller
A critical review of the existing literature reveals that adaptive and robust control
based fault-tolerant schemes can each address a part of the problem, but not all the
issues associated with actuator fault-tolerant control when used individually. Thus,
a fault-tolerant control should possess the desirable properties of both the schemes
in order to satisfactorily solve the problem. More specifically, the control objectives
a new fault-tolerant control scheme are
1. ability to handle large parametric uncertainties with desired transients
2. ensure small steady-state tracking error, despite actuator faults and non-parametric
uncertainties like external disturbances and unknown nonlinearities
3. stable controller with desired closed-loop properties, in case the uncertainties
disrupt the functioning of estimation module
1.3 An Adaptive Robust Approach to Actuator Fault-Tolerant Control
Given the need for stability in safety critical missions, the large parametric un-
certainties introduced due to unknown actuator failures and the inherent limitations
of adaptive control, the idea of safe adaptive control is coming to forefront. Safe
adaptive control ensures certain stability properties even without adaptation [15,16].
ARC based schemes have already resolved this issue [17,18] and may be classified as
the so-called safe adaptive control. In the proposed ARC based fault-tolerant scheme,
adaptive and robust control designs are integrated seamless fashion. This allows all
control objectives, enlisted in the preceding section, to be achieved. In fact, switching
the adaptation off at any instant converts the adaptive robust controller into a de-
terministic robust controller with guaranteed transient performance. Moreover, the
design procedure allows us to calculate explicit upper bound for tracking errors over
the entire time history in terms of certain controller parameters and achieve prespec-
5
ified final tracking accuracy. Thus, ARC based fault-tolerant schemes are natural
choices for safety sensitive systems over conventional adaptive and robust schemes.
1.4 Assumption on Input Saturation
An important assumption in the design of fault-tolerant control in the literature
cited above is that the working actuators have sufficient authority to accommodate the
fault and fulfill the desired control objective. In reality, however, as actuators fail, it
may not be possible to achieve all performance criteria, as there is only limited control
authority. Especially, in safety critical missions like flight control systems, saturation
can not only deteriorate the closed loop performance significantly [19], but also lead
to catastrophic outcomes in the worst case scenario. A good example of harmful
effects of saturation is the crash of JAS 39 Gripen in 1993. Post analysis revealed
that there were no component malfunction which could have caused the accident. It
was concluded after further analysis that Pilot Induced Oscillation (PIO) caused the
aircraft to crash. The PIO was a direct consequence of overlooking the saturation
limits in the design of the control systems. The situation becomes much more grave
when there are actuator faults.
1.4.1 Input saturation and its effect of actuator fault-tolerant control
In order to understand how actuator faults and actuator saturation negatively
reinforce their concomitant destructive effects, we must first understand the inter-
play of the actuator fault-tolerant control and the actuator faults. Following actuator
faults, the tracking errors tend to stray away from zero due to undesired transients.
If the controller is designed without any regard to saturation, it may generate large
control signals to attenuate the effect of undesired transients, thereby saturating the
actuators. As the actuators saturate, the error further increases, which in turn in-
creases the commanded control input, so on and so forth. This may ultimately lead
the closed-loop system to become unstable. Second, in direct adaptive schemes, the
6
state estimation error – the difference in actual state and the reconstructed state
which uses estimated parameters, correlates directly to the mismatch in estimated
parameters. This allows the actuator-fault tolerant controller to learn the change in
system parameters due to faults, and adjust the model compensation accordingly,
thereby resulting in small steady-state tracking error. However, in presence of satu-
ration, in addition to the parameter mismatch, the difference between the actual and
commanded control input is also responsible for the state-estimation error. Hence, the
adaptation can no longer guarantee reliable estimates for the unknown parameters.
As a first step towards designing a saturated adaptive robust actuator fault-tolerant
control, we investigate the design of saturated controller for an integrator chain with
input disturbances.
1.4.2 Robust global stabilization of an integrator chain
Controller design in presence of input saturation is a theoretically challenging
problem with deep practical implications. The significance of this problem cannot be
overemphasized, as proper functioning of any control system depends on actuators,
all of which have physical limitations e.g., limited valve opening, available voltage
for servo-motors etc. Especially, in safety critical missions like flight control systems,
input saturation can not only deteriorate the closed loop performance significantly
[19], but also lead to catastrophic outcomes in the worst case scenario [20]. Not
surprisingly, many design techniques which account for this input nonlinearity have
been proposed (see [21], [22], [23] and the references there in).
1.4.3 Saturated Adaptive Robust Actuator Fault-Tolerant Control
From the preceding discussion, it should be clear that if a fault-tolerant scheme
can be designed which (a) relaxes the performance criteria when the states deviate
far away from zero following an actuator fault, and (b) ensures reliable adaptation
in spite of actuator faults, then the actuator fault-tolerant scheme will become more
7
practical. In the proposed approach, bounded feedback control laws are designed
such that performance is compromised when the error-variables are far away from
zero and the actuator is saturated, and more emphasis is put on bringing the error-
variables in a region where the controller is unsaturated. This implies unrealistically
high control inputs are not demanded of the saturating actuators after a fault. Once
within the unsaturated region, the desired performance can be recovered. To tackle
the second problem, we use an indirect adaptive scheme with controller and estimator
modularity. Many schemes have been proposed in the literature to avoid the harmful
effects of saturation on adaptation. For example in [24,25], a modified tracking error
was used to avoid the harmful effect of saturation on adaptation. Pseudo Control
Hedging (PCH) based technique was used in [26], which involves altering the com-
manded reference to a level that allows controller operation without saturation. Such
techniques unnecessarily increase the complexity, without significantly improving the
performance. An indirect scheme, in contrast, does not require such modifications,
as the model structure of the system does not change despite saturation. This facil-
itates an uninterrupted and reliable estimation of parameters. Surprisingly, indirect
adaptive schemes, which are the simplest and surest way of avoiding the effect of
saturation on adaptation, have largely been overlooked.
1.5 Contributions
In this dissertation, standard assumptions and design approaches for actuator
fault-tolerant control and saturated control problem were carefully examined to ex-
plore their drawbacks in terms of achievable performance. A quantitative analysis of
the shortcomings naturally lead us to consider these problems from a fundamentally
different viewpoint, which resulted in novel control design techniques. Following prob-
lems were recognized as fundamental issues which have not been effectively addressed
in the existing literature
8
1. accommodation of unknown actuator faults and the limitations of existing tech-
niques in effectively addressing these issues,
2. degradation in performance of an actuator fault-tolerant control in presence of
saturation,
3. conservativeness of the conventional coordinate transformation based approach
for controller design in presence of saturation and input disturbance.
Following schemes have been developed in this dissertation to solve the aforemen-
tioned problems,
1. proposed a performance oriented adaptive robust control (ARC) based fault-
tolerant design which can accommodate actuator faults with guaranteed tran-
sient response and small steady-state tracking error,
2. developed a conceptually different approach to design a globally stable controller
for integrator chain with input disturbance,
3. designed an indirect adaptive robust actuator fault-tolerant control scheme
which explicitly considers the effect of actuator saturation.
1.6 Organization
This dissertation is divided in six chapters. The contents of chapters are summa-
rized as follows:
1. Chapter 1, Introduction: motivates the study of actuator fault-tolerant control.
This chapter gives an overview of the existing techniques, which leads to the
research objectives and problem formulation. Research approach and some
features of the techniques proposed in this dissertation are also discussed.
2. Chapter 2, Accommodation of Unknown Actuator Faults Using Output Feedback
based Adaptive Robust Control: presents an output feedback based adaptive ro-
9
bust fault-tolerant scheme to accommodate unknown actuator faults. Compar-
ative simulation studies are also presented to show the superior performance of
the proposed design.
3. Chapter 3, Output Feedback based Adaptive Robust Fault-Tolerant Control for
a Class of Uncertain Nonlinear Systems: addresses the problem of unknown
actuator fault-accommodation for a class of nonlinear systems. Comparative
simulation studies demonstrate that the shortcomings of a robust adaptive back-
stepping based design can be overcome by using the proposed scheme.
4. Chapter 4, A Backstepping based Approach to Robust Global Stabilization of a
Chain of Integrators with Input Saturation: proposes a novel way of stabilizing
an integrator chain in presence of input disturbances. It is shown through sim-
ulations that improved disturbance rejection properties and faster convergence
rate can be achieved by using the proposed controller.
5. Chapter 5, Saturated Adaptive Robust Actuator Fault-Tolerant Control for Feed-
back Linearizable Systems: addresses the problem of actuator fault-tolerant
control in presence of uncertainties and input saturation. The results obtained
indicate the effectiveness of the proposed scheme in accommodating actuator
faults, despite input saturation.
6. Chapter 6, Conclusions and Future Work: discusses the results and summarizes
the thesis work. Some ideas for future research are also presented.
10
2. ACCOMMODATION OF UNKNOWN ACTUATOR FAULTS USING
OUTPUT FEEDBACK BASED ADAPTIVE ROBUST CONTROL
2.1 Introduction
In this chapter, we solve the problem of fault accommodation for unknown actua-
tor failures in an uncertain linear system with unknown parameters and subjected to
bounded disturbances. This problem is of prime importance as conventional feedback
control design for complex systems may result in unacceptable degradation of perfor-
mance or even instability in the event of actuator failures. The faults are modeled as
actuators experiencing bounded disturbance about an unknown constant value, loss
in actuator efficiency or a combination of the two. We do not assume the knowledge
of failed actuators or the instant of failure. Fortunately, adaptive schemes, by virtue
of their on-line learning capability can bypass this problem. Consequently, many
adaptive schemes have been developed to solve this problem.
A novel approach for solving the problem of unknown actuator failure compen-
sation was proposed in [3] for linear systems. But, the schemes use conventional
MRAC [3] which suffer from poor transients during the learning phase and have diffi-
culty in ensuring tracking error bounds in absolute sense in the presence of exogenous
disturbances. Robust schemes for actuator fault accommodation, which can handle
such disturbances and unstructured uncertainties with guaranteed transient perfor-
mance, include LMI based techniques [27] and sliding mode control (SMC) based
approaches [13]. But, robust control based direct fault-tolerant schemes lack the abil-
ity to learn the large extent of parametric uncertainty introduced by failed actuators.
Thus, such schemes rely on high gain to attenuate the effect of uncertainties on the
steady-state tracking error. But, as all actuators have limited bandwidth, this may
lead to degraded tracking performance. One approach which potentially alleviates
11
these problems is multiple model adaptive control (MMAC), switching and tuning.
In [28], the authors propose a MMAC based actuator fault accommodation scheme.
The proposed scheme, however, did not take into account modeling uncertainties and
used full state-feedback.
In the present work, we develop an output feedback based Adaptive Robust Fault-
Tolerant Control (ARFTC) scheme for accommodation of unknown actuator faults.
Actuator faults manifest themselves as jump in parameter values in the system model
and pose a twofold problem. First, it can introduce undesirable transients due to the
jump in parameter values and second, it can lead to unacceptable steady-state track-
ing error. Robust schemes can address the first issue and adaptive control can deal
with the second one, but none of them can resolve both issues simultaneously. In
the present work, we claim that ARFTC can tackle both the problems at the same
time. This is to be expected as an Adaptive Robust Control (ARC) based fault-
tolerant scheme combines robust and adaptive control design philosophies seamlessly
(see [17], [18]). In fact, parameter projection is used not only to guarantee the bound-
edness of the estimated parameters but, also in the design of the robust component
of the control law. Thus, the known extent of the jumps in parameter values is
incorporated in the controller design to guarantee desired transient response. Fur-
thermore, as ARFTC design puts more emphasis on the underlying robust control law
design, switching the adaptation off at any instant converts the ARFTC controller
into a robust controller which guarantees desired transient performance and certain
steady-state accuracy. This feature is extremely desirable for safety critical systems.
The adaptation mechanism of the proposed controller, on the other hand, tries to
learn the change in system parameters due to actuator faults and adjusts the model-
compensation of the control law accordingly. By doing so, an improved steady-state
tracking performance – asymptotic tracking in the presence of parametric uncertain-
ties only – is achieved. Also, it is seldom the case that a parametric model can
completely capture the effect of fault. In the proposed approach, an unknown time
varying disturbance is added to the parametric fault model, and the underlying ro-
12
bust controller is designed in a way to attenuate its effect. This considerably enlarges
the class of faults that can be addressed using the proposed scheme. Furthermore, in
addition to robust feedback for dealing with unstructured uncertainties, we also use
a priori information about the exogenous disturbances to achieve better disturbance
rejection properties.
In this chapter, a linearized Boeing 747 model is used to show the effectiveness of
the proposed scheme in dealing with unknown actuator faults. Additionally, compar-
ative studies performed with respect to a robust MRAC based fault-tolerant design
clearly demonstrate the superior performance in terms of guaranteed transient re-
sponse and steady-state tracking error.
2.2 Problem Statement
In the present work, we consider systems which can be represented in the input-
output form as follows,
y(t) =
q∑
j=1
Bj(s)
A(s)uj(t) +
D(s)
A(s)∆(y, t) + dy(t) (2.1)
where, A(s) = sn + an−1sn−1 + . . . + a1s + a0, Bj(s) = bjms
m + . . . + bj1s + bj0 and
D(s) = dlsl + . . . + d1s + d0, l ≥ m. Note that l ≥ m implies we do not restrict
ourselves to matched uncertainties only. The plant parameters ai and bi are unknown
constants. The coefficients di corresponding to the disturbance distribution are as-
sumed to be known but, the results can be readily extended to the case where they are
unknown constants. uj(t) represents the actual output from the jth actuator. dy(t)
represents the output disturbance, and ∆(y, t) represents any disturbance coming
from the intermediate channels of the plant. An implicit assumption in the system
representation (2.1) is,
A1: The relative degree ρ = n−m is known and same for any input uj.
13
In this work, we will consider actuator failures which can be modeled as,
uj(t) =
uj + uj(t), ∀t ≥ Tf ,
if jth actuator gets stuck at Tf
ηjju∗j(t), ∀t ≥ Tf ,
if jth actuator loses efficiency at Tf
(2.2)
where u∗j(t) represents the control command to the jth actuator, uj is an unknown
constant value at which the actuator gets stuck, uj(t) is a bounded unknown signal
about uj, Tf is the unknown instant of failure and ηjj represents actuator loss in
efficiency with ηjj ∈ [(ηjj)min, 1], (ηjj)min ≥ 0. Note that conventional MRAC based
schemes are not well suited to handle time-varying disturbances like uj(t). Hence,
the class of actuator faults addressed here is larger and more practical than what is
typically considered. It is noted that the fault model presented here encompass both
lock-in-place and hard-over failure actuator faults. Without actuator redundancy,
actuator faults cannot be accommodated. This is formally stated in the following
assumption,
A2: System (2.1) is such that the desired control objective can be fulfilled with
up to q − 1 stuck actuators and any number of actuators with loss in efficiency.
Now, the problem we attempt to solve in this work can be stated precisely as
follows. For the system described by (2.1), subjected to unknown actuator failures
(2.2) and bounded disturbances, the goal is to design an output feedback control
law such that the steady-state output tracking error converges exponentially to an
acceptable bound and has a guaranteed transient performance.
In addition to actuator fault compensation, it is also desirable that the closed-
loop system posses good disturbance rejection properties. In the present approach,
such properties are achieved by using robust feedback as well as explicitly tak-
ing into account ∆(y, t) (see [29]). We use prior information about the nature of
disturbance to construct a nominal disturbance model ∆n(y, t) = q(y, t)T c, where
q(y, t) = [qp(y, t), . . . , q1(y, t)]T ∈ R
p represents the vector of known basis shape func-
tions and c = [cp, . . . , c1]T represents the vector of unknown magnitudes. Thus, the
14
disturbance can be represented as, ∆ = ∆n + ∆, where ∆ is the modeling error.
Adaptation will be used to compensate for the effect of ∆n on the output tracking
performance and ∆ will be dealt with via certain robust feedback for robust perfor-
mance.
2.3 Output Feedback based ARFTC
2.3.1 Observer Canonical Form
In the present work, we will assume that control commands to all actuators are
the same i.e., u∗1 = . . . = u∗q = u∗. With this choice of control inputs, and the fault
model described by (2.2), the healthy and faulty actuators can be parameterized in
the following way,
uj(t) = ηjj(1 − σjj)u∗(t) + σjj(uj + uj(t))
⇒ uj(t) = κju∗(t) + σjj uj + σjjuj(t)) (2.3)
where
σjj =
0 before jth actuator gets stuck
1 after jth actuator gets stuck
ηjj =
1 before jth actuator loses efficiency
[(ηjj)min, 1] after jth actuator loses efficiency
κj = ηjj(1 − σjj)
15
Now, system (2.1) can be rewritten as,
y(t) =
q∑
j=1
κj
Bj(s)
A(s)u∗j(t) +
q∑
j=1
σjj
Bj(s)
A(s)uj
+
q∑
j=1
σjj
Bj(s)
A(s)uj(t) +
D(s)
A(s)∆(y, t) + dy(t) (2.4)
=bfms
m + . . .+ bf1s+ bf0sn + an−1sn−1 + . . .+ a1s+ a0
u∗(t)
+bfms
m + . . .+ bf1s+ bf0sn + an−1sn−1 + . . .+ a1s+ a0
1
+
q∑
j=1
σjj
bjmsm + . . .+ bj1s+ bj0
sn + an−1sn−1 + . . .+ a1s+ a0uj(t)
+dls
l + . . .+ d1s+ d0
sn + an−1sn−1 + . . .+ a1s+ a0∆ + dy(t) (2.5)
where,
bfi = κ1b1i + · · ·+ κqbqi, i = 0, · · · , m
bfj = σ11u1b1j + · · · + σqquqbqj j = 0, · · · , m (2.6)
Note that as a consequence of assumption A1 and A2, bfm 6= 0 for any fault.
Before we proceed to present the observer canonical form of the system (2.5), we
recall the standard result that for a transfer function
G(s) =βms
m + ... + β0
sn + αn−1sn−1 + ... + α0
(2.7)
the observer canonical form is given by
x =
−αn−1 1 0 . . . 0
−αn−2 0 1 . . . 0...
. . ....
.... . .
...
−α0 0 . . . 0 1
x+
0...
βm
...
β0
u (2.8)
16
This is a standard result and can be found in any book on linear systems like [30].
Now we present an observer canonical realization of the above input-output model
which is more suitable for the controller design technique presented here,
y(t) =
q∑
j=1
κjBj(s)
A(s)u∗(t) +
q∑
j=1
σjjBj(s)
A(s)uj +
D(s)
A(s)∆(y, t)
+
q∑
j=1
σjjBj(s)
A(s)uj(t) + dy(t)
Now, expanding each term in the transfer function, we have
q∑
j=1
κjBj(s) =
q∑
j=1
κj(bjmsm + . . .+ bj1s+ bj0)
=
(
q∑
j=1
κjbjm
)
sm + . . .+
(
q∑
j=1
κjbj1
)
s+
(
q∑
j=1
κjbj0
)
= bfmsm + . . .+ bf1s+ bf0 , using the definition of bfj
q∑
j=1
σjjBj(s)uj =
q∑
j=1
σjj(bjmsm + . . .+ bj1s+ bj0)uj
=
(
q∑
j=1
σjjbjmuj
)
sm + . . .+
(
q∑
j=1
σjjbj1uj
)
s+
(
q∑
j=1
σjjbj0uj
)
= bfmsm + . . .+ bf1s+ bf0 , using the definition of bfj
D(s) = dlsl + . . .+ d1s+ d0
q∑
j=1
σjjBj(s) =
q∑
j=1
σjj(bjmsm + . . .+ bj1s+ bj0)
A(s) = sn + an−1sn−1 + . . .+ a1s+ a0
which leads to
y(t) =bfms
m + . . .+ bf1s+ bf0sn + an−1sn−1 + . . .+ a1s+ a0
u∗(t) +bfms
m + . . .+ bf1s+ bf0sn + an−1sn−1 + . . .+ a1s+ a0
1
+dls
l + . . .+ d1s+ d0
sn + an−1sn−1 + . . .+ a1s+ a0
∆(y, t)
+
q∑
j=1
σjj
bjmsm + . . .+ bj1s+ bj0
sn + an−1sn−1 + . . .+ a1s+ a0uj(t) + dy(t)
= Y1 + Y2 + Y3 +
q∑
j=1
Y j4 + dy(t)
17
The corresponding observer canonical form for TF =∑q
j=1κjBj(s)
A(s)is given by
Y11
Y12
...
Y1n
=
−an−1 1 0 · · · 0
−an−2 0 1 · · · 0...
.... . .
. . ....
−a0 0 0 0 0
Y11
Y12
...
Y1n
+
0...
bfm...
bf0
u∗
Y1 = Y11
For TF =∑q
j=1σjjBj(s)uj
A(s)
Y21
Y22
...
Y2n
=
−an−1 1 0 · · · 0
−an−2 0 1 · · · 0...
.... . .
. . ....
−a0 0 0 0 0
Y21
Y22
...
Y2n
+
0...
bfm...
bf0
u0
Y2 = Y21
where u0 = 1.
For TF = D(s)A(s)
,
Y31
Y32
...
Y3n
=
−an−1 1 0 · · · 0
−an−2 0 1 · · · 0...
.... . .
. . ....
−a0 0 0 0 0
Y31
Y32
...
Y3n
+
0...
dl
...
d0
∆
Y3 = Y31
18
And similarly for TF =σjjBj(s)
A(s), for j = 1, 2, ..., q
Y j41
Y j42
...
Y j4n
=
−an−1 1 0 · · · 0
−an−2 0 1 · · · 0...
.... . .
. . ....
−a0 0 0 0 0
Y j41
Y j42
...
Y j4n
+
0...
bjm...
bj0
σjjuj(t)
Y j4 = Y j
41
Next, we define a coordinate transformation and rewrite the dynamics in terms
of the transformed coordinates as follows
xi = Y1i + Y2i + Y3i +
q∑
j=1
Y j4i
Let us pick i < min(l,m) for clarity of presentation. Then, taking the derivative of
xi, we obtain
xi = Y1i + Y2i + Y3i +
q∑
j=1
Y j4i
= Y1(i+1) − an−iY11 + Y2(i+1) − an−iY21 + Y3(i+1) − an−iY31 +
q∑
j=1
(Y j
4(i+1) − an−iYj41)
=
(
Y1(i+1) + Y2(i+1) + Y3(i+1) +
q∑
j=1
Y j
4(i+1)
)
− an−i
(
Y11 + Y21 + Y31 +
q∑
j=1
Y j41
)
= xi+1 − an−ix1
19
Similar expressions can be derived for all i = 1, . . . , n by substitution, which gives us
the following state-space realization
x1 = x2 − an−1x1
...
xn−l−1 = xn−l − al+1x1
xn−l = xn−l+1 − alx1 + dlqT (y, t)c+ dl∆
...
xρ−1 = xρ − am+1x1 + dm+1qT (y, t)c+ dm+1∆
xρ = xρ+1 − amx1 + bfmu∗(t) + bfm
+
q∑
j=1
σjjbjmuj(t) + dmqT (y, t)c+ dm∆
...
xn = −a0x1 + bf0u∗(t) + bf0 +
q∑
j=1
σjjbj0uj(t)
+d0qT (y, t)c+ d0∆
y = Y11 + Y21 + Y31 +
q∑
j=1
Y j41 + dy(t) = x1 + dy(t) (2.9)
In addition to the assumptions made previously, we will make the following real-
istic assumptions. The first assumption is standard in adaptive control designs.
A3: The polynomial bfmsm + · · ·+bf0 is Hurwitz and the sign of the high frequency
gain (sign(bfm)) is known, irrespective of the failed actuators.
A4: The extent of parametric uncertainties, modeling error ∆(y, t), output dis-
turbance dy(t) as well as derivative dy(t) satisfy,
θ ∈ Ωθ , θ : θmin < θ < θmax
∆ ∈ Ω∆ , ∆ : |∆(y, t)| ≤ δ(t)
dy ∈ Ωd , dy : |dy(t)| ≤ δd(t)
dy ∈ Ωf , dy : |dy(t)| ≤ δf (t) (2.10)
20
where
θ = [−an−1, ...,−a0, bfm, .., b
f0 , cp, .., c1, b
fm, . . . , b
f0 ]
T ∈ R2m+n+p+2, θmin and θmax
are assumed to be known, and δ(t), δd(t), and δf(t) are unknown but bounded func-
tions. Note that ∆(y, t) is assumed to be globally bounded w.r.t y. Thus, in the
present work we consider only bounded uncertainties.
2.3.2 State Estimation
In this section, we will describe the design of K-filters [31] for state estimation.
The state-space equations (3.3) can be rewritten as,
x = A0x+ (k − a)x1 + dqT (y, t)c+ bfu∗ + bf + ∆
y = x1 + dy(t) (2.11)
where,
A0 =
−k1
... In−1
−kn 0 . . . 0
k =
k1
...
kn
d =
0(n−l−1)×1
dl
...
d0
bf =
0(ρ−1)×1
bfm...
bf0
bf =
0(ρ−1)×1
bfm...
bf0
a =
an−1
...
a0
(2.12)
and
∆ =
0(l−1)×1
dl∆...
dm∆ +∑q
j=1 σjjbjmuj(t)...
d0∆ +∑q
j=1 σjjbj0uj(t)
(2.13)
21
represents the bounded disturbances.
The observer matrix A0 can be made stable by a suitable choice of k. Thus, there
exists a symmetric positive definite matrix P such that,
PA0 + AT0 P = −I, P = P T > 0 (2.14)
For the purpose of state-estimation, the following set of K-filters is defined,
ξn = A0ξn + ky
ξi = A0ξi + en−iy, 0 ≤ i ≤ n− 1
υi = A0υi + en−iu∗, 0 ≤ i ≤ m
ψi = A0ψi + dqi(y, t), 1 ≤ i ≤ p
ζi = A0ζi + en−i, 0 ≤ i ≤ m (2.15)
where ei denotes the ith standard basis vector in Rn and ξn, ξi, υi, ψi and ζi ∈ R
n×1.
Note that due to the special structure of A0, the order of the K-filters described above
can be reduced by using the following two filters and certain algebraic expressions,
η = A0η + eny
λ = A0λ+ enu (2.16)
Now, the ξi and υi filter states can be obtained using the following expression,
ξn = −An0η
ξi = Ai0η 0 ≤ i ≤ n− 1
υi = Ai0λ 0 ≤ i ≤ m (2.17)
Using the above filters, the state estimate x ∈ Rn×1 is given by,
x = ξn −n−1∑
i=0
aiξi +m∑
i=0
bfi υi +m∑
i=0
bfi ζi +
p∑
i=1
ciψi (2.18)
Let εx = x − x be the estimation error. Then, using (4.46), (3.10) and the filters
described above, the estimation error dynamics is given by,
εx = A0εx + (a− k)dy + ∆ (2.19)
22
Using assumption A4 and taking derivative of the positive semi-definite (psd)
function Vεx = εTxPεx,
Vεx = εTx (AT
0 P + PA0)εx + 2εTx ((a− k)dy + ∆)
≤ −|εx|2 + 2|εx|δ(t) = −|εx|22
− 1
2(|εx| − 2δ)2 + 2δ2
≤ −|εx|22
+ 2δ2 ≤ − Vε
2pmax
+ 2δ2 (2.20)
where δ = ||(a − k)|δd + |∆||, pmin = mineig(P ), pmax = maxeig(P ) and using
pmin|εx(t)|2 ≤ Vεx(t) ≤ pmax|εx(t)|2. Using comparison lemma, from (2.20) we obtain
Vεx(t) ≤ exp
(
− t
2pmax
)
Vεx(0) + 4pmax‖δ‖2∞
(
1 − exp
(
− t
2pmax
))
⇒ |εx(t)|2 ≤ pmax
pminexp
(
− t
2pmax
)
|εx(0)|2 + 4pmax
pmin‖δ‖2
∞
(
1 − exp
(
− t
2pmax
))
⇒ |εx(t)| ≤√
pmax
pmin
exp
(
− t
4pmax
)
|εx(0)| + 2
√
pmax
pmin
‖δ‖∞
√
(
1 − exp
(
− t
2pmax
))
(2.21)
In (3.15), the first term is exponentially vanishing and the second term is a bounded
function of time. Thus, we have
εx ∈ Ωε , εx : |εx| ≤ δε(t) (2.22)
i.e., the estimation error remains bounded within a ball of unknown radius δε.
2.3.3 Parameter Projection
Let θ denote the estimate of θ and θ = θ−θ denote the estimation error.Parameter
projection is used to ensure that the parameter estimates remain within a known
bounded region. The update law and the projection mapping used here have the
following form,
˙θ = Projθ(Γτ) (2.23)
Projθi(•i) =
0 if θi = θi,max and •i > 0
0 if θi = θi,min and •i < 0
•i otherwise
(2.24)
23
where Γ > 0 is a diagonal matrix, and τ is any adaptation function. The projection
mapping guarantees that the following two properties are always satisfied,
P1 θ ∈ Ωθ = θ : θmin ≤ θ ≤ θmax (2.25)
P2 θT (Γ−1Projθ(Γτ) − τ) ≤ 0, ∀τ (2.26)
2.3.4 Controller Design
In this section, we present the design of output feedback based Adaptive Robust
Fault-Tolerant Control (ARFTC) scheme. The main idea is to synthesize a virtual
control law at each step which will replace the estimated state and, the estimation
error will be dealt with at each step using robust feedback.
Step 1: The derivative of the output tracking error z1 = y − yr is given by,
z1 = x2 − an−1y + an−1dy + dy − yr (2.27)
But, x2 is not measured and is replaced by its estimate using (3.10),
x2 = ξn,2 −n−1∑
i=0
aiξi,2 +m∑
i=0
bfi υi,2
+m∑
i=0
bfi ζi,2 +
p∑
i=1
ciψi,2 + εx2 (2.28)
where εx2 is the estimation error of x2, and denote
ξ(2) = [ξn−1,2, . . . , ξ0,2], υ(2) = [υm,2, . . . , υ0,2],
ζ(2) = [ζm,2, . . . , ζ0,2], ψ(2) = [ψp,2, . . . , ψ1,2]. (2.29)
Substituting (3.21) back in (3.20), we obtain
z1 = bfmυm,2 + ω0 + θT ω − yr + ∆1 (2.30)
where ωT = [ξ(2), υ(2), ψ(2), ζ(2)] + e∗T1 y, ω = ω − e∗n+1υm,2, ω0 = ξn,2, ∆1 = an−1dy +
dy + εx2 and e∗Ti is the ith standard basis vector in Rn+2m+p+2. (3.23) suggests a
24
natural choice for the virtual input is υm,2, which will be used for synthesizing the
virtual control law α1,
α1(y, η, λm+1, ψ, θ, t) = α1a + α1s,
α1a = − 1
bfmω0 + θT ω − yr (2.31)
where λi = [λ1, . . . , λi]T is obtained from (2.16). In (3.24), α1a is the model compen-
sation component of the control law used to achieve an improved model compensation
through on-line parameter adaptation. Thus, the fault is partly accommodated using
model compensation, as [θn+m+p+2, ..., θn+2m+p+2]T = [bfm, ...,
ˆbf0 ]T . Let z2 = υm,2 − α1
denote the input discrepancy. Substituting (3.24) into (3.23), we get
z1 = bfm(z2 + α1s) − θTφ1 + ∆1, φ1 , ω + e∗n+1α1a (2.32)
We now present the design of the robust component of the control law α1s, which
suppresses the potential destabilizing effect of parameter estimation transients.
α1s = α1s1 + α1s2 + α1s3, α1s1 = − 1
(bfm)min
k1sz1 (2.33)
where (bfm)min = minj(ηjjbjm) = (θn+1)min and k1s is a nonlinear gain, such that
k1s ≥ g1+ ‖ Cφ1Γφ1 ‖2, g1 ≥ 0 (2.34)
in which Cφ1is a positive definite constant diagonal matrix to be specified later
and g1 is a constant. Consider the positive semi-definite (p.s.d) function V1 = 12z21 .
Substituting (3.26) and (3.27) in (3.25), V1 is given by
V1 ≤ bfmz1z2 − k1sz21 + z1(b
fmα1s2 − θTφ1)
+z1(bfmα1s3 + ∆1) (2.35)
As a result of assumption A4, we have
‖ θTφ1 ‖≤‖ θM ‖‖ φ1 ‖ (2.36)
25
where θM = θmax − θmin. Thus, ‖ θTφ1 ‖ is bounded by a known function, which
ensures that there exists a robust control function satisfying the following conditions
[17]:
(a) z1bfmα1s2 − θTφ1 ≤ ǫ11
(b) z1α1s2 ≤ 0 (2.37)
where ǫ11 is a positive design parameter.
Remark 1: Condition (a) of (3.31) shows that α1s2 is synthesized to attenuate the
effect of parametric uncertainties θ with the level of control accuracy being measured
by ǫ11. Condition (b) is to make sure that α1s2 is dissipative in nature so that it does
not interfere with the functionality of adaptive control law α1a.
Similarly, from assumption A4 and (3.14), we obtain
|∆1| ≤ δ1(t) , |an−1|δd(t) + δf (t) + δε2(t) (2.38)
Note that δ1 is a bounded function, and the same strategy as in (3.31) can be used
to design a robust control law. However, since the bound of ∆1 is not known, it is
impossible to prespecify the level of control accuracy exactly. So, a more relaxed
requirement compared to the condition (a) of (3.31) is sought to be satisfied,
z1bfmα1s3 + ∆1 ≤ ǫ12δ21 (2.39)
where ǫ12 is a controller parameter which can be freely tuned. This implies that by
choosing a sufficiently small ǫ12, right hand side of equation (3.37) can be made smaller
than any desired value, even though it cannot be prespecified exactly. Examples of
smooth α1s2 and α1s3 satisfying (3.31) and (3.37) respectively are given by [32],
α1s2 = − h1
4(bfm)minǫ11z1, α1s3 = − 1
4(bfm)minǫ12z1 (2.40)
where h1 ≥‖ θM ‖2‖ φ1 ‖2. Also, it should be noted that the effect of unstructured
uncertainties like uj(t), dy(t), and ∆(y, t), which manifests itself in δε2, is compensated
by α1s3 i.e., robust feedback .
26
Step 2: From (3.24), (2.16), (2.17) and rearrangement of (3.20-3.23), the derivative
of α1 can be written as,
α1 = α1c + α1u
α1c =∂α1
∂y(ω0 + θTω) +
∂α1
∂ηη +
m+1∑
j=1
∂α1
∂λj
λj
+
p∑
j=1
∂α1
∂ψj
ψj +∂α1
∂t
α1u =∂α1
∂y(−θTω + ∆1) +
∂α1
∂θθ (2.41)
Using (2.15) and (2.16), α1c is calculable and can be used in the design of control
functions. However, α1u is not calculable due to various uncertainties and hence, will
be dealt with via robust feedback in this step. From (2.15) and (3.39), the derivative
of the z2 = υm,2 − α1 is
z2 = υm,3 − k2υm,1 − α1c − α1u (2.42)
Now, consider the augmented p.s.d function V2 = V1 + 12z22 . From (3.29) and (3.40),
the derivative of V2 is given by
V2 ≤ V1|α1+ z2bfmz1 + υm,3 − k2υm,1 − α1c − α1u (2.43)
where V1|α1= −k1sz
21 + z1(b
fmα1s2 − θ1φ1) + z1(b
fmα1s3 + ∆1). As in (3.24), the ARC
control function α2 for the virtual control input υm,3 in (3.40) consists of
α2(y, η, λm+2, ψ, θ, t) = α2a + α2s
α2a = −bfmz1 + k2υm,1 + α1c
α2s = α2s1 + α2s2 + α2s3, α2s1 = −k2sz2
k2s ≥ g2 +
∥
∥
∥
∥
∂α1
∂θCθ2
∥
∥
∥
∥
+ ‖Cφ2Γφ2‖2 (2.44)
where g2 > 0 is a constant and Cθ2 and Cφ2 are positive definite constant diagonal
matrices, α2s2 and α2s3 are robust control functions to be chosen later. As mentioned
previously, due to use of discontinuous projection, we cannot use tuning functions
27
which anticipates and compensates for the effect of˙θ. The last two terms in the
inequality for k2s compensates for this loss of information. Substituting (3.42) and
(3.39) in (3.41), and using similar techniques as in (3.25), we have
V2 ≤ V1|α1+ z2z3 − k2sz
22 + z2(α2s2 − θTφ2)
+z2(α2s3 + ∆2) − z2∂α1
∂θ
˙θ (2.45)
where z3 = υm,3 − α2 represents the input discrepancy and
φ2 = e∗n+1z1 −∂α1
∂yω , ∆2 = −∂α1
∂y∆1 (2.46)
From (3.36), it follows that ∆2 ≤ |∂α1/∂y|δ1. Similar to (3.31) and (3.37), the robust
control functions α2s2 and α2s3 are chosen to satisfy
(a) z2(α2s2 − θTφ2) ≤ ǫ21
(b) z2(α2s3 + ∆2) ≤ ǫ22δ21
(c) z2α2s2 ≤ 0 , z2α2s3 ≤ 0 (2.47)
where ǫ21 and ǫ22 are positive design parameters. As in step 1, α2s2 and α2s3 can be
chosen as,
α2s2 = − h2
4ǫ21z2 , α2s3 = − 1
4ǫ21
(
∂α1
∂y
)2
z2 (2.48)
where h2 is any smooth function satisfying h2 ≥‖ θM ‖2‖ φ2 ‖2. From (3.29) and h2
defined above, the derivative of V2 satisfies
V2 ≤ z2z3 −2∑
j=1
kjsz2j + z1(b
fmα1s2 − θ1φ1)
+z1(bfmα1s3 + ∆1) + z2(α2s2 − θTφ2)
+z2(α2s3 + ∆2) −∂α1
∂θ
˙θz2 (2.49)
Step i (3 ≤ i < ρ): For any j ∈ [3, i− 1], let zj = υm,j − αj−1 and recursively define
φj = −∂αj−1
∂yω , ∆j = −∂αj−1
∂y∆1 (2.50)
28
At step i, choose the desired ARC control function αi as
αi(y, η, λm+i, ψ, θ, t) = αia + αis
αia = −zi−1 + kiυm,1 + α(i−1)c
αis = αis1 + αis2 + αis3, αis1 = −kiszi
kis ≥ gi +
∥
∥
∥
∥
∂αi−1
∂θCθi
∥
∥
∥
∥
+ ‖CφiΓφi‖2 (2.51)
where gi > 0 is a constant, and Cθi and Cφi are positive definite constant diagonal
matrices, αis2 and αis3 are robust control functions satisfying,
(a) zi(αis2 − θTφi) ≤ ǫi1
(b) zi(αis3 + ∆1) ≤ ǫi2δ21
(c) ziαis2 ≤ 0 , ziαis3 ≤ 0 (2.52)
and
α(i−1)c =∂αi−1
∂y(ω0 + θTω) +
m+i+1∑
j=1
∂αi−1
∂λj
λj
+
p∑
j=1
∂αi−1
∂ψj
ψj +∂αi−1
∂ηη +
∂αi−1
∂t(2.53)
Then, the ith error subsystem is
zi = zi+1 − zi−1 − kiszi + (αis2 − θTφi)
+(αis3 + ∆i) −∂αi−1
∂θ
˙θ (2.54)
and the derivative of the augmented p.s.d function Vi = Vi−1 + 1/2z2i satisfies,
Vi ≤ zizi+1 −i∑
j=1
kjsz2j + z1(b
fmα1s2 − θTφ1)
+i∑
j=2
zj(αjs2 − θTφj) + z1(bfmα1s3 + ∆1)
+
i∑
j=2
zj(αjs3 + ∆j) −i∑
j=2
∂αj−1
∂θ
˙θzj (2.55)
29
Step ρ: In this final step, the actual control law u∗ will be synthesized such
that υm,ρ tracks the desired ARC control function αρ−1. The derivative of zρ can be
obtained as
zρ = υm,ρ+1 + u∗ − kρυm,1 − α(ρ−1)c
−∂αρ−1
∂y(−θTω + ∆1) −
∂αρ−1
∂θ
˙θ (2.56)
If υm,ρ+1 + u∗ were the virtual input, (2.56) would have the same form as the inter-
mediate step i. Therefore, the general form, (3.48-2.56) applies to step ρ. Since u is
the actual control input, it can be chosen as,
u∗ = αρ − υm,ρ+1 (2.57)
where αρ is given by (3.49). Then, zρ+1 = u∗ + υm,ρ+1 − αρ = 0.
Theorem 1. Let the parameter estimates be updated using adaptation law (5.20)
in which τ is chosen as
τ =
ρ∑
j=1
φjzj (2.58)
If diagonal controller gain matrices Cθj and Cφk are chosen such that c2φkr ≥ ρ
4
∑ρ
j=1 1/c2θjr,
where cθjr and cφkr are the rth diagonal element of Cθj and Cφk respectively. Then,
the control law (3.51) guarantees that,
1. In general the control input and all internal signals are bounded. Furthermore,
Vρ is bounded above by,
Vρ(t) ≤ exp(−λρt)Vρ(0)
+ǫρ1 + ǫρ2 ‖ δ1 ‖2
∞
λρ
[1 − exp(−λρt)] (2.59)
where λρ = 2ming1, . . . , gρ, ǫρ1 =∑ρ
j=1 ǫj1, ǫρ2 =∑ρ
j=1 ǫj2 and ‖ δ1 ‖2∞
stands for the infinity norm of δ1.
2. If after a finite time t0, ∆ = 0, uj = 0 and dy = 0 (i.e., in the presence of
parametric uncertainties and modeled disturbances only) then, in addition to
results in (1), asymptotic output tracking control is also achieved.
30
Proof. Using (3.51), we known zρ+1 = 0. From (3.49), we have
αρ(y, η, λn, ψ, θ, t) = αρa + αρs
αρa = −zρ−1 + kρυm,1 + α(ρ−1)c
αρs = αρs1 + αρs2 + αρs3, αρs1 = −kρszρ
kρs ≥ gρ+ ‖ ∂αρ−1
∂θCθρ ‖ + ‖ CφρΓφρ ‖2 (2.60)
Substituting in (2.56), we obtain
zρ = −zρ−1 − kρszρ + (αρs2 − θTφρ)
+(αρs3 + ∆ρ) −∂αρ−1
∂θ
˙θ (2.61)
Also, substituting i = ρ− 1 in (2.55) gives
Vρ−1 ≤ zρ−1zρ −ρ−1∑
j=1
kjsz2j + z1(b
fmα1s2 − θTφ1)
+
ρ−1∑
j=2
zj(αjs2 − θTφj) + z1(bfmα1s3 + ∆1)
+
ρ−1∑
j=2
zj(αjs3 + ∆j) −ρ−1∑
j=2
∂αj−1
∂θ
˙θzj (2.62)
Taking derivative of the Lyapunov function Vρ = Vρ−1|αρ−1+ 1
2z2
ρ and using (2.61)
and (2.62) we obtain
Vρ = Vρ−1|αρ−1+ zρzρ
≤ −ρ∑
j=1
kjsz2j + z1(b
fmα1s2 − θTφ1)
+
ρ∑
j=2
zj(αjs2 − θTφj) + z1(bfmα1s3 + ∆1)
+
ρ∑
j=2
zj(αjs3 + ∆j) −ρ∑
j=2
zj
∂αj−1
∂θ
˙θ (2.63)
31
Now, using the fact that robust component of the virtual control law is designed to
satisfy inequalities (3.31),(3.45) and (3.50), we have
Vρ ≤ −ρ∑
j=1
(gj +
∥
∥
∥
∥
∂αj−1
∂θCθj
∥
∥
∥
∥
+ ‖ CφjΓφj ‖2)z2j
+
ρ∑
j=1
(ǫj1 + ǫj2δ21) −
ρ∑
j=2
zj
∂αj−1
∂θ
˙θ (2.64)
By AM-GM inequality y1+...+yp
p≥ p
√y1.y2...yp, the last term of the above inequality
satisfies
−ρ∑
j=2
zj
∂αj−1
∂θ
˙θ ≤
ρ∑
j=2
|zj |∣
∣
∣
∣
∂αj−1
∂θCθjC
−1θj
˙θ
∣
∣
∣
∣
≤ρ∑
j=2
(
∥
∥
∥
∥
∂αj−1
∂θCθj
∥
∥
∥
∥
2
z2j +
1
4‖C−1
θj
˙θ‖2
)
(2.65)
Once again, using AM-GM inequality we have
‖C−1θj
˙θ‖2 = ‖C−1
θj Proj(Γτ)‖2 ≤ ‖C−1θj Γτ‖2
≤ (
ρ∑
k=1
‖C−1θj Γφkzk‖)2 ≤ ρ(
ρ∑
k=1
‖C−1θj Γφk‖)2z2
k (2.66)
Putting (2.65) and (2.66) together, we get
−ρ∑
j=2
zj
∂αj−1
∂θ
˙θ ≤
ρ∑
j=2
∥
∥
∥
∥
∂αj−1
∂θCθj
∥
∥
∥
∥
2
z2j
+ρ
4
ρ∑
k=1
ρ∑
j=2
‖C−1θj Γφk‖2z2
k
≤ρ∑
j=2
∥
∥
∥
∥
∂αj−1
∂θCθj
∥
∥
∥
∥
2
z2j
+
ρ∑
k=1
‖CφkΓφk‖2z2k (2.67)
Finally combining (2.64) and (2.67), we have
Vρ ≤ −ρ∑
j=1
gjz2j +
ρ∑
j=1
(ǫj1 + ǫj2δ21)
≤ −λρVρ + ǫρ1 + ǫρ2δ21 (2.68)
32
which proves first part of the theorem.
To prove the second part, we note that in presence of parametric uncertainties
only i.e., ∆ = 0, dy = 0 and uj = 0, ∆1 = ǫ2. From (3.37) and condition b of (3.50),
we have |z1(bfmα1s3 +∆1) ≤ ǫ12ε22| and |zj(αjs3 +∆j) ≤ ǫj2ε
22|, j = 2, ..., ρ. Thus, from
(3.54), (2.55), condition b of (3.31), and (3.45), Vρ satisfies
Vρ ≤ −ρ∑
j=1
(θTφ1zj + gjz2j − ǫj2ε
22)
= −ρ∑
j=1
gjz2j − θT τ + ǫρ2ε
22 (2.69)
Now, we use the augmented positive definite function
Vθ = Vρ +1
2θT Γ−1θ + γεTPε (2.70)
where γ ≥ ǫρ2. Now using (5.24) and (2.14), the derivative of Vθ is given by
Vθ ≤ −ρ∑
j=1
gjz2j − θT τ + ǫρ2ε
22 + θT Γ−1θ − γ‖ε‖2
≤ −ρ∑
j=1
gjz2j (2.71)
It may appear that we have neglected the ρ+1 to n states in the present analysis.
But, due to the assumption of stable zero dynamics and bounded disturbances as
well as actuator faults, it can be easily proved using standard adaptive control argu-
ments that all internal signals remain bounded and do not interfere with the tracking
performance.
Remark 2: In context of actuator fault compensation, first part of theorem 1
guarantees that the jump in parameter values due to failed actuator does not interfere
with the desired transient performance. By using trajectory initialization techniques
as described in [17], we can set Vρ(0) = 0 and then, from (3.52) we have
|z1(t)| ≤√
2(ǫρ1 + ǫρ2 ‖ δ1 ‖2∞)
λρ
[
1 − exp(−λρt)
2
]
(2.72)
This equation provides an upperbound for the output tracking error z1 and charac-
terizes the transient response. ǫρ1, ǫρ2 and λρ are controller parameters which we
33
can choose. Note that from remark 1, we know by properly tuning ǫρ2 we can make
ǫρ2 ‖ δ1 ‖2∞ as small as desired. In this sense, we have guaranteed transient response.
This result on transient response of the system is a direct consequence of underlying
robust filter structure of the ARC controller.
2.4 Simulation Example: Linearized Boeing 747 Model
For simulation purposes, we will use the linearized model of Boeing 747. It should
be noted that the same example was used in [3, 33], and thus, provides a platform
to compare the MRAC in [3] and its robust adaptive control version in [33] with the
backstepping based ARC algorithm proposed here for actuator fault accommodation.
Plant Model: The linearized model for the lateral motion of Boeing 747 without
disturbances can be represented as,
x(t) = Ax(t) +Bu(t), y(t) = x2(t) = yr(t)
x(t) = [β, yr, p, φ]T , B = [b1, b2, b3] (2.73)
where, β is the side-slip angle, yr is the yaw-rate, p is the roll rate, φ is the roll
angle, y is the output which needs to follow the reference trajectory r(t) and u is
the control input vector consisting of three control signals representing three rudder
servos δr1, δr2, δr3. Note that the B matrix has been augmented by b2 and b3 for
studying actuator failure compensation properties of the proposed algorithm. From
34
the data provided in [34] for Boeing 747 in horizontal flight at 40,000 ft and nominal
forward speed 774 ft/s, the perturbation dynamics matrices are
A =
−0.0558 −0.9968 0.0802 0.0415
0.598 −0.115 −0.0318 0
−3.05 0.388 −0.465 0
0 0.0805 1 0
(2.74)
b1 =
0.00729
−0.475
0.153
0
b2 =
0.01
−0.5
0.2
0
b3 =
0.005
−0.3
0.1
0
(2.75)
It can be easily verified that this plant satisfies the assumptions A1-A4.
Simulation results: Simulations are done using r(t) = 0.02sin(0.2t) as the refer-
ence signals for MRAC and ARC based fault compensation techniques. Three faults
are introduced in the system during the simulations: the third actuator gets stuck
at u3 = 0.5 rads at 40 seconds and the second actuator loses 90% efficiency at 60
seconds. These faults can be represented as follows,
u2(t) =
u∗(t) for t ≤ 60 sec
0.1u∗(t) for t > 60 sec
u3(t) =
u∗(t) for t ≤ 40 sec
0.5 rads for t > 40 sec(2.76)
and u1 is assumed to be healthy for all times i.e., u1(t) = u∗(t), ∀t. For the state
estimators, the gain was chosen to be k = [4, 6, 4, 1]T , which places all the poles of
A0 at -1. The initial conditions for the plant was x = [0, 0.02, 00]′
. The controller
parameters for the MRAC based scheme can be obtained from [3]. The details of
ARC controller is given at the end of this section. Two scenarios are studied through
simulations: with and without disturbance. In the first case, as all disturbances are
assumed to be zero, we have ∆(y, t) = dy(t) = 0. As can be seen in Fig.1, both the
systems perform well initially, and have similar commanded control input profiles.
35
Note that the magnitude of the commanded control input increases with each fault.
This is to be expected as initially three actuators were performing the task, whereas
with each failure, the same task needs to be done with fewer remaining healthy
actuators. As the faults chosen are fairly severe, they cause large jump in the plant
parameters. Consequently, the transient tracking error deviates significantly for the
MRAC based scheme, but stays close to zero for the proposed adaptive robust fault-
tolerant control (ARFTC). This can be explained as follows. The design of robust
component of the ARFTC law has already incorporated the extent of such jumps
in parameter values, and hence, is better suited to handle the transients introduced
by actuator failures. This shows the guaranteed transient performance of ARFTC
which cannot be achieved using any adaptive control based scheme. On the other
hand, it should also be noted that due to the learning capability of the ARFTC,
the steady-state error tends to zero. This cannot be achieved using a fault-tolerant
scheme which relies on robust feedback only.
In the next set of simulations, intermediate disturbance ∆(y, t) = 0.004sin(2.1t)
with d = [1, 1.50, 0.5625, 0.0625]′
and completely unknown output disturbance dy(t) =
0.002sin(4t) are added to the system. Note that in context of aircraft control, many
disturbances can be modeled using harmonic functions, e.g., wind-shear [35] which
corresponds to the ∆y term. Also, rate-gyros which are used for measuring yaw-rates
suffer from harmonic disturbances (see [36]) as well. Hence, sinusoidal disturbances
were used for simulation studies. Fig.2 shows result with the same set of faults as
described earlier for r(t) = 0.02sin(0.2t). To make the simulation studies more mean-
ingful, robustness modifications and disturbance estimation was added to the MRAC
based design, as outlined in [33], with bounds given by [θi,min, θi,max] , [0.80θ∗i , 1.20θ∗i ].
These simulations demonstrate the strength of ARFTC in attenuating the effect of
disturbance and modeling error in addition to large parametric uncertainties. In fact,
even though the magnitude of the control signals remain comparable for MRAC and
ARFTC based designs, there is an order of magnitude difference in the tracking error.
As seen in Fig. 2, even with robust adaptation mechanisms and disturbance estima-
36
tion, MRAC based schemes can neither attenuate the effect of unknown modeling
errors like dy(t) on the steady-state error nor guarantee desired transient response.
On the other hand, due to the underlying robust controller structure of ARFTC,
it can attenuate the effect of such uncertainties on the steady-state error as well as
guarantee desired transient response.
2.4.1 Detailed ARFTC Controller design
The plant model in the transfer function (TF) form is given by
[y(t)] =−0.4750s3 − 0.2479s2 − 0.1187s− 0.0563
s4 + 0.6358s3 + 0.9389s2 + 0.5116s+ 0.0037[u1(t)]
+−0.5s3 − 0.2608s2 − 0.1223s− 0.0583
s4 + 0.6358s3 + 0.9389s2 + 0.5116s+ 0.0037[u2(t)]
+−0.3s3 − 0.1564s2 − 0.0747s− 0.0355
s4 + 0.6358s3 + 0.9389s2 + 0.5116s+ 0.0037[u3(t)]
+s3 + 1.50s2 + 0.5625s+ 0.0625
s4 + 0.6358s3 + 0.9389s2 + 0.5116s+ 0.0037[∆(y, t)] + dy(t)(2.77)
where ∆(y, t) = 0.004 sin(2.1t), dy(t) = 0.002 sin(4t). The corresponding observer
canonical form is given by
x1 = x2 − 0.6358x1 − 0.4750u1 − 0.5u2 − 0.3u3 + 0.004sin(2.1t)
x2 = x3 − 0.9389x1 − 0.2479u1 − 0.2608u2 − 0.1564u3 + 0.0.006sin(2.1t)
x3 = x4 − 0.5116x1 − 0.1187u1 − 0.1223u2 − 0.0747u3 + 0.0023sin(2.1t)
x4 = −0.0037x1 − 0.0563u1 − 0.0583u2 − 0.0355u3 + 0.0003sin(2.1t)
y = x1 + dy(t)
For the filters, we use k = [4, 6, 4, 1], which places all the poles of A0 at −1. The
filters are given by
η1
η2
η3
η4
=
−4 1 0 0
−6 0 1 0
−4 0 0 1
−1 0 0 0
η1
η2
η3
η4
+
0
0
0
1
y
37
from which we can obtain the required ξi, i = 0, 1, 2, 3, 4 filter-states as follows
ξ0 = η, ξ1 = A0η, ξ2 = A20η, ξ3 = A3
0η, ξ4 = −A40η
The next set of filter is given by
λ1
λ2
λ3
λ4
=
−4 1 0 0
−6 0 1 0
−4 0 0 1
−1 0 0 0
λ1
λ2
λ3
λ4
+
0
0
0
1
u
from which we can obtain υi, i = 0, 1, 2, 3
υ0 = λ, υ1 = A0λ, υ2 = A20λ, υ3 = A3
0λ
ψ1
ψ2
ψ3
ψ4
=
−4 1 0 0
−6 0 1 0
−4 0 0 1
−1 0 0 0
ψ1
ψ2
ψ3
ψ4
+
1
1.5
0.5625
0.0625
sin(2.1t)
Note that in this case it assumed that D(s) = s3 +1.50s2 +0.5625s+0.0625 is known.
But, the controller can be designed when D(s) has unknown coefficients, with suitable
modifications, the details of which were presented in the previous response.
The last set of filters is given by
ζi = A0ζi + e4−i, i = 1, 2, 3, 4 (2.78)
where e4−i is the standard basis in R4
Now, we are ready to present the controller design. Using the controller parametriza-
tion proposed (see equation (3)), the derivative of the error is given be
z1 = y − yr = (x1 + dy) − yr
z1 = x2 − 0.6358x1 − 0.4750u1 − 0.5u2 − 0.3u3 + 0.004sin(2.1t) + dy − yr
= x2 − 0.6358x1 + [0.4750η11(1 − σ11) + 0.5η22(1 − σ22) + 0.3η33(1 − σ33)]u∗(t)
+0.4750σ11(u11 + u11(t)) + 0.5σ22(u22 + u22(t)) + 0.3σ33(u33 + u33(t))
+0.004sin(2.1t) + 0.6358dy + dy − yr
38
The unmeasured state x2 is replaced by the estimated state x2 = ξ4(2)−ξ(2)a+υ(2)bf +
ζ(2)bf +ψ2c+ εx2, where the unknown parameters a, bf , bf and c are as defined in the
manuscript. With this, the dynamics can be rewritten as
z1 = ξ4(2) − ξ(2)a+ υ(2)bf + ζ(2)b
f + ψ2c+ εx2 − 0.6358y + 0.004 sin(2.1t) + 0.6358dy + dy − yr
= b3υ3,2 + ω0 + θT ω − yr − ∆1
where ωT = [ξ(2), υ(2), ψ(2), ζ(2)] + e∗T1 y ∈ R13, ω = ω− e∗5υ3,2, (e∗j is jth standard basis
in R13×1) ω0 = ξ4,2, ∆1 = 0.004 sin(2.1t) + 0.6358dy + dy + εx2.
The control law is given by
α1 = α1a + α1s
α1a =1
bf3ω0 + θT ω − yr
The robust component of the control law is designed as
α1s1 = − 1(
bf3
)
min
k1sz1
α1s2 = − h1
4(bfm)minǫ11z1
α1s3 = − 1
4(bfm)minǫ12z1
(2.79)
where k1s ≥ g1 + ‖Cφ1Γφ1‖, g1 ≥ 0. However, as the relative degree of the system is
1, there is no restriction on the choice of g1 and Cφ1, as long as both are greater than
0. Thus, they can always be selected such that g1 + ‖Cφ1Γφ1‖ ≤ 60 = k1s. The other
parameter h1 is chosen as
h1 = ‖θM‖‖φ1‖
where φ1 = ω + e∗5α1a and θM = ‖θmax − θmin‖2. The other parameters are given by
ǫ11 = ǫ22 = 0.2.
˙θ = ProjθΓτ1, where τ1 = φ1z1, Γ = 50I13×13
39
θ = [−a3,−a2,−a1,−a0,−bf3 ,−bf2 ,−bf1 ,−bf0 , c, bf3 , bf2 , bf1 , bf0 ]
θmin = [0.5086, 0.7511, 0.4093, 0.0030, 0.0240, 0.0125, 0.0060, 0.0028, 0.0032,
−1.506,−0.7856,−0.3728,−0.1773]
θmax = [0.7630, 1.1267, 0.6139, 0.0044, 1.5300, 0.7981, 0.3788, 0.1801, 0.0048,
1.506, 0.7856, 0.3728, 0.1773]
This completes the design of the ARC based fault-tolerant control for the linearized
Boeing 747 model.
40
0 20 40 60 80−0.01
0
0.01
0.02
0.03
0.04
time (sec)y
and
ym (r
ad/s
ec)
Plant output and reference signal
MRACARCRef
0 20 40 60 80−0.01
0
0.01
0.02
0.03
time (sec)
e(t)
(rad
/sec
)Tracking error
MRACARC
0 20 40 60 80−0.6
−0.4
−0.2
0
0.2
0.4
time (sec)
u(t)
(rad
)
Control signals: MRAC
u1(t)
u2(t)
u3(t)
0 20 40 60 80−0.6
−0.4
−0.2
0
0.2
0.4
time (sec)
u(t)
(rad
)
Control signals: ARC
u1(t)
u2(t)
u3(t)
Figure 2.1. Reference tracking, control signals and tracking error forMRAC versus ARC based fault-tolerant schemes in absence of distur-bances
41
0 20 40 60 80−0.01
0
0.01
0.02
0.03
0.04
time (sec)y
and
ym (r
ad/s
ec)
Plant output and reference signal
Robust MRACARCRef
0 20 40 60 80−0.01
0
0.01
0.02
0.03
time (sec)
e(t)
(rad
/sec
)Tracking error
Robust MRAC
ARC
0 20 40 60 80−0.6
−0.4
−0.2
0
0.2
0.4
time (sec)
u(t)
(rad
)
Control signals: Robust MRAC
u1(t)
u2(t)
u3(t)
0 20 40 60 80−0.6
−0.4
−0.2
0
0.2
0.4
time (sec)
u(t)
(rad
)
Control signals: ARC
u1(t)
u2(t)
u3(t)
Figure 2.2. Reference tracking, control signals and tracking error forMRAC versus ARC based fault-tolerant schemes in presence of distur-bances
2.5 Conclusion
In this work, an output feedback based Adaptive Robust Fault-Tolerant Control
(ARFTC) scheme was presented for the fault accommodation of uncertain linear sys-
42
tems with a larger class of unknown actuator faults. Adaptation and robust feedback
are used simultaneously to maintain tracking performance in face of large parametric
uncertainties introduced due to failing actuators, exogenous disturbances and other
modeling uncertainties. Comparative simulation studies were done using a linearized
model for lateral motion of Boeing 747 which confirmed the superior performance of
the proposed ARFTC strategy over conventional robust MRAC based schemes. In
summary, some of the salient features of the ARFTC scheme presented in the chapter
are, (1) capability to handle large parametric uncertainties due to unknown actuator
failures with guaranteed transient performance with desired steady-state tracking er-
ror, (2) better disturbance rejection properties and (3) guaranteed robust performance
when adaptation is switched off.
43
3. OUTPUT FEEDBACK BASED ADAPTIVE ROBUST FAULT-TOLERANT
CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
3.1 Introduction
In this chapter, we extend the output feedback based adaptive robust fault-tolerant
control (ARFTC) developed in the previous chapter to a class uncertain nonlinear
systems. The faults considered here comprise of stuck actuators, loss in actuator
efficiency or a combination of the two faults. Tao et al. proposed direct fault-
compensation schemes to deal with such faults in [4], [5] using backstepping based
adaptive control. Another popular adaptive approach to solve this problem is mul-
tiple model adaptive control (MMAC), switching and tuning [6]. Robust adaptive
control (RAC) based schemes, that can guarantee boundedness of closed loop signals
in presence of unknown modeling uncertainties and disturbances, were investigated
in [5,7,8]. As, backstepping based RAC schemes accommodate the effect of parame-
ter estimation transients ( ˙θ) through tuning functions, they have better performance
than MRAC based techniques, which rely on certainty-equivalence principle. How-
ever, they still lack the ability to attenuate the effect of non-parametric uncertainties
like external disturbances and unknown nonlinearities on the tracking error. Fur-
thermore, as discussed in the first chapter, using high adaptation gain in RAC based
design in presence of disturbances to improve transient performance can results in
high frequency oscillations in the control input, which is extremely undesirable.
A nonlinear Hypersonic aircraft model is used to demonstrate the effectiveness
of the proposed scheme. Note that in this chapter, the comparative studies are car-
ried out with respect to a robust adaptive backstepping based design, whereas in
our previous chapter, the comparative studies were conducted with a model reference
adaptive control (MRAC) based scheme. As MRAC design is fundamentally different
44
from backstepping based designs, it was not possible to compare and contrast the
two designs at each step. In this chapter, however, the comparison is done between
two backstepping based approaches, which makes it possible to emphasize the under-
lying subtle but important differences in the design of the two fault-tolerant control
strategies. This makes the comparative studies much more compelling.
3.2 Problem Statement
In the present work, we will consider systems which can be represented in the
following output feedback form subjected to output dependent uncertainties
x1 = x2 + ϕ0,1(y) +
p∑
j=1
ajϕ1,j(y) + ∆1(y, t)
...
xρ−1 = xρ + ϕ0,ρ−1(y) +
p∑
j=1
ajϕρ−1,j(y) + ∆ρ−1(y, t)
xρ = xρ+1 + ϕ0,ρ(y) +
p∑
j=1
ajϕρ,j(y) +
q∑
j=1
bm,jβj(y)uj(t) + ∆ρ(y, t)
...
xn = ϕ0,n(y) +
p∑
j=1
ajϕn,j(y) +
q∑
j=1
b0,jβj(y)uj(t) + ∆n(y, t) (3.1)
where ρ = n − m is the relative degree, uj is the control input, y = x1 is the
measured output, ϕ0,i(y) and βj(y) are known smooth functions of y. Furthermore,
it will be assumed that |βj(y)| > 0 and is uniformly bounded above by a constant
i.e., |βj(y)|≤βj for any y. ∆i(y, t) represents uncertainties e.g., modeling error and
disturbances. ai, bi,j are unknown constants such that sign of the high frequency gain
(sgn(bm,j)) is known.
45
System (3.1) we will be subjected to actuator faults which can be represented as
uj(t) =
uj + uj(t), ∀t ≥ Tf ,
if jth actuator gets stuck at Tf
ηjju∗j(t), ∀t ≥ Tf ,
if jth actuator loses efficiency at Tf
(3.2)
where u∗j represents the control command to the jth actuator, Tf is the unknown
instant of failure, uj is an unknown constant value at which the actuator gets stuck,
uj(t) represents an unknown but bounded disturbance about uj, and ηjj ∈ [(ηjj)min, 1]
represents actuator loss in efficiency. Note that the class of faults considered here is
more practical and significantly larger than what has been considered in the literature
due to the presence of uj(t).
Without actuator redundancy and sufficient control authority, actuator faults can-
not be accommodated and the same is stated formally in the following assumption
A1. System (3.1) is such that the desired control objective can be fulfilled with
up to q − 1 stuck actuators and any number of actuators with loss in efficiency.
The problem we attempt to solve in this work can now be stated as follows. For
the uncertain nonlinear system (3.1), subjected to faults (3.5) the goal is to design an
output feedback based control strategy such that the output tracking error converges
exponentially to a prespecified bound and has a guaranteed transient performance.
Remark 1. In the present work, a simple fault model (3.2) was chosen in order
to clearly illustrate the advantages of using ARFTC over robust adaptive control
(RAC) based scheme, without unnecessarily complicating the presentation. Any fault
which can be linearly parameterized in terms on known basis functions (i.e., uFj (t) =
∑rk=1 bkuj,k(t)+ uj(t), where uF
j (t) is the signal from faulty actuator, bk are unknown
constants, uj,k(t) are known basis functions) can be addressed using the ARFTC
framework.
46
3.3 Output Feedback based ARFTC
Control signals uj(t) are designed such that βj(y)uj(t) = u∗(t). With fault model
(3.2) and the chosen actuation scheme, we can rewrite the control inputs as follows,
uj(t) =ηjj
βj(y)(1 − σjj)u
∗(t) + σjj(uj + uj(t)) (3.3)
for j = 1, . . . , q where σjj = 1 corresponds to stuck actuators, σjj = 0, (ηjj)min ≤ηjj ≤ 1 represents actuator loss of efficiency and σjj = 0, ηjj = 1 corresponds to
healthy actuators.
With this, we can rewrite the system as follows
xi = x2 + ϕ0,i(y) +
p∑
j=1
ajϕi,j(y) + ∆i(y, t)
for i = 1, 2, .., ρ− 1, and
xρ = xρ+1 + ϕ0,ρ(y) +
p∑
j=1
ajϕρ,j(y) +
q∑
j=1
µm,jβj(y)
+κmu∗(t) +
q∑
j=1
bm,jσjjβj(y)uj(t) + ∆ρ(y, t)
...
xn = ϕ0,n(y) +
p∑
j=1
ajϕn,j(y) +
q∑
j=1
µ0,jβj(y)
+κ0u∗(t) +
q∑
j=1
b0,jσjjβj(y)uj(t) + ∆n(y, t) (3.4)
where
κi =
q∑
j=1
ηjj(1 − σjj)bi,j,
µi,j = σjj ujbi,j, i = 0, 1, .., m j = 1, 2, .., q
Note that κi is the unknown measure of actuator effectiveness after faults and µi,j is
the unknown measure of the fault magnitude which needs to be compensated. Thus,
the system experiences jump in parameter values and bounded disturbances with the
47
occurrence of each new fault. The proposed scheme accommodates such faults by
estimating µi,j and κi, and relies on robust feedback scheme to deal with the bounded
disturbances uj(t) and the jump in parameter values.
We will make the following realistic assumptions regarding the uncertainties present
in the system
A2. The extent of parametric uncertainties and uncertain nonlinearities satisfy
ai ∈ Ωa , ai : (ai)min < ai < (ai)max
κi ∈ Ωκ , κi : (κi)min < κi < (κi)max
µi,j ∈ Ωµ , µi,j : (µi,j)min < µi,j < (µi,j)max
∆i ∈ Ω∆ , ∆i : |∆i(y, t)| ≤ δi(t) (3.5)
where (ai)min, (ai)max, (κi)min, (κi)max, (µi,j)min, (µi,j)max are known and δi(t) is an
unknown but bounded function.
We will make another standard assumption which guarantees stability of the zero
dynamics
A3. The polynomial κmsm + κm−1s
m−1 + . . . + κ0 is a stable polynomial and
sign(κm) is known, irrespective of the failed actuators.
3.3.1 State estimation
We need to construct state-estimator for
x = A0x+ ky + ϕ0(y) + Φ(y)a+m∑
i=0
en−iκiu∗ +
q∑
j=1
m∑
i=0
en−iµi,jβj(y) + ∆(y, t)
y = Cx, C = [1 0 0 . . . 0] (3.6)
where
A0 =
−k1
... In−1
−kn 0 . . . 0
k =
k1
...
kn
ϕ0(y) =
ϕ0,1
...
ϕ0,n
(3.7)
48
and
Φ(y) =
ϕ1,1 · · · ϕ1,p
.... . .
...
ϕn,1 · · · ϕn,p
∆(y, t) =
∆1
...
∆ρ−1
∆ρ +∑q
j=1 bm,jσjjβj(y)uj(t)...
∆n +∑q
j=1 b0,jσjjβj(y)uj(t)
(3.8)
Note that A = A0 +kCT and the observer matrix A0 can be made stable by a suitable
choice of observer gain k such that there exists a symmetric positive definite matrix
P satisfying
PA0 + AT0 P = −I, P = P T > 0 (3.9)
We will define the following set of filters for the purpose of state-estimation,
ξ0 = A0ξ0 + ky + ϕ0(y), ξ0 ∈ Rn×1
ξ = A0ξ + Φ(y), ξ ∈ Rn×p
ϑi = A0ϑi + en−iu∗, ϑi ∈ R
n×1
ψi,j = A0ψi,j + en−iβj(y), ψi,j ∈ Rn×1 (3.10)
where i = 0, 1, .., m and j = 1, 2, .., q.
Due to the special structure of A0, the order of K-filters can be reduced by using
the following two filters
λ = A0λ + enu∗
ζj = A0ζj + enβj , j = 1, 2, .., q (3.11)
and the following algebraic equations
ϑi = Ai0λ
ψi,j = Ai0ζj, i = 0, 1, .., m (3.12)
49
The estimated state can be written as
x = ξ0 + ξa+
m∑
i=0
κiϑi +
q∑
j=1
m∑
i=0
µi,jψi,j (3.13)
Let ε = x− x be the estimation error. Then, the state-estimation error dynamics
is given by
ε = A0ε+ ∆(y, t) (3.14)
From (4.46) and using βj(y) ≤ βj, we know there exists an unknown but bounded
function which upperbounds ∆(y, t) i.e.,
|∆(y, t)| ≤ δ(t) (3.15)
where | • | denotes the L2 norm.
Vεx = εTx (AT
0 P + PA0)εx + 2εTx ∆(y, t)
≤ −|εx|2 + 2|εx|δ(t)
= −|εx|22
− 1
2(|εx| − 2δ(t))2 + 2δ2(t)
≤ −|εx|22
+ 2δ2(t) (3.16)
using (3.15), pmin = mineig(P ), pmax = maxeig(P ) and pmin|εx(t)|2 ≤ Vεx(t) ≤pmax|εx(t)|2. Using comparison lemma, from (3.16) we obtain
Vεx(t) ≤ exp
(
− t
2pmax
)
Vεx(0) + 4
∫ t
0
exp
(
− t− τ
2pmax
)
δ(τ)2dτ (3.17)
⇒ |εx(t)|2 ≤ pmax
pmin
exp
(
− t
2pmax
)
|εx(0)|2 +2
pmin
∫ t
0
exp
(
− t− τ
2pmax
)
δ(τ)2dτ
⇒ |εx(t)| ≤√
pmax
pminexp
(
− t
4pmax
)
|εx(0)| +√
2
pmin
[∫ t
0
exp
(
− t− τ
2pmax
)
δ(τ)2dτ
]
(3.18)
In (3.18), the first term is exponentially vanishing, and the second term is unknown
but bounded. Hence, the state-estimation error remains bounded and converges ex-
ponentially to a residual-ball whose size depends on the extent of unknown modeling
uncertainties i.e.,
|εi| = |xi − xi| ≤ δεi(t) for i = 1, ..., n (3.19)
50
3.3.2 Parameter Projection
Discontinuous projection, as defined in the previous chapter, will be used in this
work.
3.3.3 Controller Design
The output feedback based controller design presented here combines the output
feedback based adaptive backstepping [31] and discontinuous projection based ARC
[18], which uses state-feedback. The main idea is to synthesize a virtual control law
which will drive the error to a small residual ball. But, as in this case only a single
state is available for measurement, the synthesized virtual control law will replace
the reconstructed state at each step, and the state estimation error will be dealt
with via robust feedback. Also, it should be noted that the use of discontinuous
projection implies a tuning function based backstepping cannot be used, and hence
a stronger robust control law is needed to negate the effects of parameter estimation
transients. For advantages of discontinuous projection based technique over smooth
modifications of adaptive law like smooth projection and, the full-state feedback based
ARC controller design, the reader is referred to [18].
Step 1: The derivative of the output tracking error z1 = y − yd is given by,
z1 = y − yd
= x2 + ϕ0,1(y) +
p∑
j=1
ajϕ1,j(y) − yd + ∆1(y, t)
= x2 + ϕ0,1(y) +
p∑
j=1
ajϕ1,j − yd + ∆1(y, t)
= ω0 + ωTθ + ∆1(y, t)
= κmϑm,2 + ω0 + ωTθ + ∆1(y, t) (3.20)
51
where
ω0 = [ξ0,2 + ϕ0,1],
ω = [ξ(2) + Φ(1), ϑm,2, ϑm−1,2, .., ϑ0,2,
ψm,1(2), .., ψm,q(2), .., ψ0,1(2), .., ψ0,q(2)]T
ω = ω − e∗p+1ϑm,2
θ = [a1, a2, .., ap, κm, .., κ0,
µm,1, .., µm,q, .., µ0,1, .., µ0,q]T
∆1(y, t) = ∆1(y, t) + ε2 (3.21)
and e∗k is the kth basis vector in Rp+m+qm+1.
If ϑm,2 were the input, we would synthesize a virtual control law α1 to make z1 as
small as possible
α1(y, ξ0, ξ, λm+1, ψi,j,2, θ, t) = α1a + α1s
α1a = − 1
κm
ω0 + ωTθ − yd (3.22)
In (3.22), α1a is the model compensation component of the control law used to achieve
an improved model compensation through on-line parameter adaptation. Thus, the
fault is partly accommodated using model compensation, as
[θp+m+2, ..., θp+m+qm+1]T = [µm,1, .., µm,q, .., µ0,1, .., µ0,q]
T
Now, as we assumed the sign of κm is known, without loss of generality, one can as-
sume κm > 0 and it is lower bounded by a non-zero positive constant i.e., (κm)min =
(θp+1)min > 0 where (θp+1)min is independent of the failure pattern. Also, that
(θp+1)min is known from assumption A4. Then, the projection mapping (5.22) guaran-
tees that κm ≥ (κm)min > 0, which implies that the control law (3.22) is well defined.
Substituting (3.22) into (3.20), we get
z1 = κm(z1 + α1s) − θφ1 + ∆1 (3.23)
52
The robust component is designed to compensate for the potential destabilizing effect
of the uncertainties on the right hand side of (3.23) as follows
α1s = α1s1 + α1s2 + α1s3, α1s1 = − 1
κm,min
k1sz1 (3.24)
where k1s is a nonlinear gain, such that
k1s = g1 + ||Cφ1Γφ1||2, g1 ≥ 0 (3.25)
in which Cφ1is a positive definite constant diagonal matrix to be specified later. As
discontinuous projection is used, tuning functions cannot be used to compensate for
parameter estimation-error transients.
Substituting (3.25) in (3.23), we obtain
z1 = κmz2 −κm
κm,min
k1sz1 + κm(α1s2 + α1s3) − θTφ1 + ∆1 (3.26)
Define a positive semi-definite (p.s.d) function V1 = 12z21 . Its derivative is given by
V1 ≤ κmz1z2 − k1sz21 + z1(κmα1s2 − θTφ1) + z1(κmα1s3 + ∆1) (3.27)
From assumption A1,
‖ θTφ1 ‖≤‖ θM ‖‖ φ1 ‖, θM = θmax − θmin (3.28)
As ‖ θTφ1 ‖ is bounded by a known function, there exists a robust control function
satisfying the following conditions
(a) z1κmα1s2 − θTφ1 ≤ ǫ11
(b) z1α1s2 ≤ 0 (3.29)
Similarly, from assumption A2 and (3.19), we have
|∆1| ≤ |ε2| + |∆1| = δε2(t) + δ1(t) , δ1(t) (3.30)
Now, we can follow the same strategy as in (3.29) to design a robust control law. But,
as δ1(y, t) is unknown, we cannot prespecify the level of control accuracy. Hence, we
seek to achieve the following relaxed conditions
(a) z1κmα1s3 + ∆1(y, t) ≤ ǫ12δ21
(b) z1α1s2 ≤ 0 (3.31)
53
Remark 2. Condition (a) of (3.29) shows that α1s2 is synthesized to attenuate the
effect of parametric uncertainties θ with the level of control accuracy being measured
by ǫ11. Condition (b) ensures that α1s2 is dissipative in nature so that it does not
interfere with the functionality of adaptive control law α1a. One smooth example of
α1s2 satisfying (3.29) is
α1s2 = − h1
4κm,minǫ11z1, h1 ≥ ‖θM‖2‖φ1‖2 (3.32)
Similarly, an example of α1s3 satisfying (3.31), which is synthesized to attenuate the
effect of unstructured uncertainties ∆1(y, t), is given by given by
α1s3 = − 1
4κm,minǫ12z1 (3.33)
Remark 3. There are subtle but important design differences between an ARC
Figure 3.1. Structure of ARC and RAC based fault-tolerant controllers
and RAC based fault-tolerant control scheme. Figure (3.1) shows the underlying
structure of ARC and RAC based designs. Note that the ARC based fault-tolerant
controller, the emphasis is on the inner loop robust controller, and the adaptation
mechanism in the outer loop is used to reduce the extent of modeling uncertainties.
As unknown actuator faults introduce severe estimation error (θ) and estimation
error transients (˙θ), it is necessary to suppress their undesirable effect on the system
dynamics. The coordination mechanism ensures that the potential destabilizing effect
54
of θ and ˙θ are effectively suppressed by the robust controller. Furthermore, the
bounded uncertainties are also attenuated to desired extent by the robust controller.
Thus, desired transient response is guaranteed. On the other hand, the RAC based
designs use an adaptive controller in conjunction with robustness modifications to the
adaptation scheme to guarantee the boundedness of all the signals. They lack the
extra design freedom present in ARC due to the underlying robust controller. Thus,
they cannot guarantee desired transient response.
Step 2: From (3.11-3.13) and (3.20-3.22), we can obtain the derivative of α1 as
follows
α1 = α1c + α1u
α1c =∂α1
∂yω0 + ωT θ +
∂α1
∂ξ0A0ξ0 + ky + ϕ0(y)
+∂α1
∂ξA0ξ + Φ(y) +
m+1∑
i=1
∂α1
∂λi
λi +
q∑
j=1
m+1∑
i=1
∂α1
∂ζi,jζi,j +
∂α1
∂t(3.34)
α1u =∂α1
∂y(−θTω + ∆1) +
∂α1
∂θ
˙θ (3.35)
α1c is calculable and will be used for control function design. α1u, however, is not
calculable and will be dealt with via certain robust terms. From (3.10), the derivative
of z2 = ϑm,2 − α1 is given by
z2 = ϑm,3 − k2ϑm,1 − α1c − α1u (3.36)
Define a p.s.d function V2 = V1 + 12z22 . Then, derivative of V2 using (3.27) and (3.36)
is given by
V2 ≤ V1|α1+ z2κmz1 + ϑm,3 − k2ϑm,1 − α1c − α1u (3.37)
55
where V1|α1= −k1sz
21 + z1(κmα1s2 − θTφ1) + z1(κmα1s3 + ∆1). Similar to (3.22), we
can now define α2 for ϑm,3 as follows
α2(y, ξ0, ξ, λm+2, ψi,j,3, θ, t) = α2a + α2s
α2a = −κmz1 + k2ϑm,1 + α1c
α2s = α2s1 + α2s2 + α2s3, α2s1 = −k2sz2
k2s ≥ g2 + ‖∂α1
∂θCθ2‖ + ‖Cφ2Γφ2 ‖2 (3.38)
where g2 ≥ 0 is a constant, Cθ2 and Cφ2 are positive definite constant diagonal
matrices, α2s2 and α2s3 are robust control functions to be synthesized later. Due
to use of discontinuous projection, we cannot use tuning functions which anticipates
and compensates for the effect of parameter estimation transients. α2s1 is the robust
control term which compensates for this loss of information. The reason for choosing
this form for α2s1 will become apparent in the proof of theorem 1. Substituting (3.38)
in (3.37), we obtain
V2 ≤ V1|α1+ z2z3 − k2sz
22 + z2(α2s2 − θTφ2) + z2(α2s3 + ∆2) − z2
∂α1
∂θ
˙θ (3.39)
where z3 = ϑm,3 − α2 represents the input discrepancy and
φ2 = e∗n+1z1 −∂α1
∂yω, ∆2 = −∂α1
∂y∆1 (3.40)
From (3.30), it follows that ∆2 ≤ |∂α1/∂y|δ1. Similar to (3.31) and (3.37), the robust
control functions α2s2 and α2s3 are chosen to satisfy
(a) z2(α2s2 − θTφ2) ≤ ǫ21
(b) z2(α2s3 + ∆2) ≤ ǫ22δ21
(c) z2α2s2 ≤ 0 , z2α2s3 ≤ 0 (3.41)
where ǫ21 and ǫ22 are positive design parameters. As in (3.32) and (3.33), α2s2 and
α2s3 can be chosen as,
α2s2 = − h2
4ǫ21z2 , α2s3 = − 1
4ǫ21
(
∂α1
∂y
)2
z2 (3.42)
56
where h2 is any smooth function satisfying h2 ≥‖ θM ‖2‖ φ2 ‖2. From (3.27) and h2
defined above, the derivative of V2 satisfies
V2 ≤ z2z3 −2∑
j=1
kjsz2j + z1(κmα1s2 − θ1φ1)
+ z1(κmα1s3 + ∆1) + z2(α2s2 − θTφ2) + z2(α2s3 + ∆2) −∂α1
∂θ
˙θz2 (3.43)
Step i (3 ≤ i < ρ): Mathematical induction will be used to prove the general result
for all the intermediate steps. At each step i, the ARC control function αi will be
constructed for virtual control input ϑm,i+1. For any j ∈ [3, i−1], let zj = ϑm,j −αj−1
and recursively design
φj = −∂αj−1
∂yω , ∆j = −∂αj−1
∂y∆1 (3.44)
Lemma 1: At step i, choose the desired ARC control function αi as
αi(y, ξ0, ξ, λm+i, ψk,j,i+1, θ, t) = αia + αis
αia = −zi + kiϑm,i + α(i−1)c
αis = αis1 + αis2 + αis3 αis1 = −kiszi
kis ≥ gi+ ‖ ∂αi−1
∂θCθi ‖ + ‖ CφiΓφi ‖2 (3.45)
where gi > 0 is a constant, and Cθi and Cφi are positive definite constant diagonal
matrices, αis2 and αis3 are robust control functions satisfying,
(a) zi(αis2 − θTφi) ≤ ǫi1
(b) zi(αis3 + ∆i) ≤ ǫi2δ21
(c) ziαis2 ≤ 0 , ziαis3 ≤ 0 (3.46)
and
α(i−1)c =∂α1
∂yω0 + ωT θ +
∂α1
∂ξ0A0ξ0 + ky + ϕ0(y)
+∂α1
∂ξA0ξ + Φ(y) +
m+1∑
i=1
∂α1
∂λi
λi +
q∑
j=1
m+1∑
i=1
∂α1
∂ζi,jζi,j +
∂α1
∂t(3.47)
57
Then, the ith error subsystem is
zi = zi+1 − zi−1 − kiszi + (αis2 − θTφi) + (αis3 + ∆i) −∂αi−1
∂θ
˙θ (3.48)
and the derivative of the augmented p.s.d function Vi = Vi−1 + 1/2z2i satisfies,
Vi ≤ zizi+1 −i∑
j=1
kjsz2j + z1(κmα1s2 − θTφ1) +
i∑
j=2
zj(κmαjs2
−θTφj) + z1(κmα1s3 + ∆1) +
i∑
j=2
zj(αjs3 + ∆j) −i∑
j=2
∂αj−1
∂θ
˙θzj (3.49)
The lemma can be easily verified by recursively writing the various expressions and
substituting the expressions obtained in step 1 and 2.
Step ρ: In this final step, the actual control law u∗ will be synthesized such that
ϑm,ρ tracks the desired ARC control function αρ−1. The derivative of zρ can be
obtained as
zρ = ϑm,ρ+1 + u∗ − kρϑm,1 − α(ρ−1)c −∂αρ−1
∂y(−θTω + ∆1) −
∂αρ−1
∂θ
˙θ (3.50)
If ϑm,ρ+1 + u∗ were the virtual input, (3.50) would have the same form as the inter-
mediate step i. Therefore, the general form, (3.44-3.50) applies to step ρ. Since u∗ is
the actual control input, it can be chosen as,
u∗ = αρ − ϑm,ρ+1 (3.51)
where αρ is given by (3.49). Then, zρ+1 = u∗ + ϑm,ρ+1 − αρ = 0.
Theorem 1. Let the parameter estimates be updated using adaptation law (5.20)
in which τ is chosen as
τ =
ρ∑
j=1
φjzj (3.52)
If diagonal controller gain matrices Cθj and Cφk are chosen such that c2φkr ≥ ρ
4
∑ρ
j=1 1/c2θjr,
where cθjr and cφkr are the rth diagonal element of Cθj and Cφk respectively. Then,
the control law (3.51) guarantees that,
58
1. In general the control input and all internal signals are bounded. Furthermore,
Vρ is bounded above by,
Vρ(t) ≤ ǫρ1 + ǫρ2 ‖ δ1 ‖2∞
λρ
[1 − exp(−λρt)] (3.53)
where λρ = 2ming1, . . . , gρ, ǫρ1 =∑ρ
j=1 ǫj1, ǫρ2 =∑ρ
j=1 ǫj2 and ‖ δ1 ‖2∞
stands for the infinity norm of δ1.
2. If after a finite time t0, ∆(y, t) = 0 (i.e., in the presence of parametric uncer-
tainties only) then, in addition to results in (3.53), asymptotic output tracking
control is also achieved.
Proof. The proof is similar to that of theorem 1 of chapter 1.
Remark 4. In context of actuator fault compensation, first part of theorem 1
guarantees that the jump in parameter values due to failed actuator does not interfere
with the desired transient performance. By using trajectory initialization techniques
[17], we can set Vρ(0) = 0 and then, from (3.52) we have
|z1(t)| ≤√
2(ǫρ1 + ǫρ2 ‖ δ1 ‖2∞)
λρ
[
1 − exp(−λρt)
2
]
(3.54)
This equation provides an upperbound for the output tracking error z1 and charac-
terizes the transient response. ǫρ1, ǫρ2 and λρ are controller parameters which we
can choose. Note that from remark 1, we know by properly tuning ǫρ2 we can make
ǫρ2 ‖ δ1 ‖2∞ as small as desired. Thus, we can tune the parameter ǫρ2 to make |z1(t)|
smaller than any predetermined bound. In this sense, we have guaranteed transient
response. This result on transient response of the system is a direct consequence
of underlying robust filter structure of the ARC controller. The second part of the
theorem guarantees asymptotic output tracking in presence of actuators failures and
parametric uncertainties only i.e., ∆i(y, t) = 0. As desirable properties of robust and
adaptive control designs are preserved, we get desired transient response with small
steady-state tracking error in spite of actuator faults.
59
3.4 Simulation Example: A Nonlinear Hypersonic Aircraft Model
The proposed scheme is implemented on a nonlinear longitudinal model of hy-
personic aircraft cruising at a velocity of 15 mach, at an altitude of 110, 000 feet.
The control objective is to track a desired trajectory in presence of various modeling
uncertainties and elevator segment failures. Nominal model of the system is
α = q − γ
q =Myy
Iyy
γ =L+ T sinα
mV− (µ− V 2r) cos(γ)
V r2(3.55)
where
α = angle of attack, rad
γ = flight-path angle, rad
V = velocity, ft/sec
q = pitch rate, rad/sec
T = thrust, lbf
L = lift, lbf
Myy = pitching moment, lbf.ft
Iyy = moment of inertia, slug.ft2
Details of the full order model can be found in [37]. Note that the same reduced order
model was used by Tang et al. [5] and thus will provide a platform to compare the
robust adaptive control (RAC) based fault-tolerant control (FTC) and the adaptive
robust fault-tolerant control (ARFTC) schemes. The nominal model of the system
does not take into account any unstructured modeling uncertainties and external
disturbances. As modeling uncertainties are inherent to any realistic system model,
60
unmatched uncertainties will be introduced in order to make the simulation studies
more meaningful. The state-space representation is given by
x1 = x2 + a1y + a2 sin(y) + a3y2 sin(y) + a4 cos(x3) + ∆1(y, t)
x2 = a5y2 + a6y + (a7 + a8y + a9y
2)x2 + b1u1 + b2u2 + ∆2(y, t)
x3 = a10 cos(x3) − a1y − a2 sin(y)
y = x1 (3.56)
where [x1, x2, x3] = [α, q, γ] and ∆i represents uncertain nonlinearities and distur-
bances. The numerical values of the nominal plant parameters are
a1 = −0.0427, a2 = −3.4496×10−4, a3 = 5×10−5, a4 = 0.0014, a5 = −4.2006,
a6 = 1.0821, a7 = −3.6896, a8 = 0.1637, a9 = −0.1242, a10 = 0.0014, b1 = 0.8, b2 = 0.8
The initial conditions are set to x(0) = [0, 0.01, 0]T . Due to the presence of the
term a4 cos(x3) in x1, (5.64) is not in the output-feedback form. However, since
|a4 cos(x3)| ≤ a4, it can be considered a bounded uncertainty and can be dealt with
using robust feedback.
The reference command chosen, yd(t) = 0.01 sin(0.1t) is in accordance with [5].
Details of the ARFTC controller is given at the end of this section. Details of RAC
based FTC can be obtained from [5]. Additionally, the controller parameters for
ARFTC scheme were chosen such that the control input profiles would be comparable
for both schemes.
Using the parametrization (3.3), before actuator faults the actual control signal is
given by
u1(t) = 1b1u∗(t) rads
u2(t) = 1b2u∗(t) rads
for t < 50 secs
61
Two faults are introduced: at t = 50 seconds the first actuator loses 40% efficiency
and at t = 75 seconds, the second actuator gets stuck at u2 = 0.1 radians. These can
be represented in terms commanded control signals as follows
u1(t) = 0.6b1u∗(t) rads
u2(t) = 1b2u∗(t) rads
for t ≥ 50 secs
and
u1(t) = 0.6b1u∗(t) rads
u2(t) = 0.1 rads
for t ≥ 75 secs
Two cases are considered in order to illustrate the effectiveness of the proposed
scheme. In the first set of simulations, all unstructured modeling errors and distur-
bances are assumed to be zero i.e., ∆i = 0 for i = 1, 2. This provides a level ground to
compare the transient performance of the two schemes after an actuator fails. After
the first fault, the tracking error remains close to zero for both schemes. However,
after the second fault, which is more severe as compared to the first one, tracking
error remains close to zero for ARFTC, but deviates significantly for RAC based FTC
and can be explained as follows. The second fault causes a large jump in parameter
value which can lead to significant transient tracking error. In the ARFTC scheme,
however, the robust component of the control law is designed to suppress the effect
of such parameter jumps (αis2 in equations (3.29), (3.31), (3.41) and (3.46)). An
explicit upper bound for the transient tracking error can also be obtained in terms
of controller parameters when an upper bound for the unstructured uncertainties is
known a priori. RAC based FTC, on the other hand, can only guarantee the bound-
edness of all signals at best. Also, as a4 cos(x3) is present even when when other
uncertainties are set to zero, asymptotic tracking cannot be achieved using either
of the schemes. But, in ARFTC, its effect on the final tracking accuracy can be
attenuated by adjusting the gain of the robust component of the control law in a
transparent manner. Another interesting observation is that the steady-state track-
ing error is small in ARFTC without control chattering. Any scheme which relies on
62
robust control only, would either lead to control input chattering in presence of such
large parametric uncertainties, or result in poor steady-state tracking accuracy due
to smoothing techniques. In contrast, the learning mechanism of ARFTC leads to
improved model compensation and guarantees acceptable steady-state tracking error,
as seen from Fig.1. Thus, desired transient response is guaranteed due to robust
filter structure of the ARFTC. Smaller steady-state tracking error is also achieved
due to the combined effect of reduction in modeling uncertainties through parameter
adaptation, as well as the use of gain to attenuate the effect of bounded uncertainties
on steady-state error.
A second set of simulations were performed where disturbances were introduced
to test the performance of the two schemes in presence of unstructured bounded
uncertainties. We set ∆1(y, t) = 0.01 sin(2t) and ∆2(y, t) = 0.01 sin(3t). Although,
these disturbances are chosen for illustrative purposes only, it is worth mentioning that
many disturbances which affect the aircraft dynamics can indeed be modeled using
harmonic basis functions e.g., wind-shear [35]. Faults and controller parameters are
same as used in the previous case. As evident from Fig.2, the performance of the
RAC based scheme deteriorates significantly in presence of unstructured modeling
uncertainties, which are inherent to any realistic system model. From the tracking
error plots, we see that not only the transient error, but the steady-state error is also
unacceptably large. The limitation of adaptive schemes in presence of disturbances,
even with robustness modifications, becomes obvious by comparing Fig.1 and Fig.2.
It can be explained as follows. Although, parameter projection is used in the RAC
based scheme, it can neither improve the transient performance, nor can it attenuate
the effect of disturbances on the steady-state tracking error. On the other hand, in
ARFTC, the parameter bounds are used not only for projection, but also to design
the robust component of the control law. The parametric uncertainty (which also
accounts for the jump in parameter values) and unstructured uncertainty bounds
are incorporated in the design of the baseline robust controller in ARFTC (αis3 in
equations (3.29), (3.31), (3.41) and (3.46)), guaranteeing desired transient response
63
and acceptable steady-state tracking error. Therefore, the achievable performance
using the proposed scheme is superior to that of RAC based schemes.
ARC based fault-tolerant controller
The unknown parameter vector θ is
θ = [a1, a2, a3, a5, a6, κ0, µ0]T
and the initial values and bounds for the parameter estimates are
θ(0) = [−0.05,−4 × 10−4, 0,−4.0, 0.9, 1.5, 0]
θmin = [−0.06,−5 × 10−4, 4 × 10−4,−5, 0.8, 0.2,−1]
θmax = [−0.03,−2 × 10−4, 7 × 10−4,−3.5, 1.2, 2, 1]
The gain matrix for parameter estimation is given by Γ = diag1, 1, 1, 1, 1, 5, 0.1.The observer gain matrix is chosen to be k = [2, 1]T and the filters used for state-
reconstruction are
ξ0 = A0ξ0 + ky + ϕ2(y) − yϕ2(y)e1
ξ = A0ξ + Φ(y) + ϕ2(y)ξ
ϑ0 = A0ϑ0 + e2u∗ + ϕ2(y)ϑ0
ψ = A0ψ + ϕ2(y)ψ + e2 (3.57)
where
Φ(y) =
y sin(y) y2 sin(y) 0 0
0 0 0 y2 y
ϕ2(y) = a7y2 + a8y + a9
64
Initial condition for all filter states are set to 0. The control law is given by
α1 = α1a + α1s, z1 = y − yd
α1a = − 1
κ0(ω0 + ωT θ − yd), α1s = −k1sz1
α1c =∂α1
∂y(ω0 + ωT θ) +
∂α1
∂ξ0,2
˙ξ0,2 +∂α1
∂ξ(2)ξ(2)
+∂α1
∂ϑ0,2
ϑ0,2 +∂α1
∂ψ2
ψ2 +∂α1
∂t
u∗ = u∗a + u∗s, z2 = ϑ0,2 − α1
u∗a = k2ϑ0,1 + α1c − κ0z1, u∗s = −k2sz2 (3.58)
where k1s = −50, k2 = 1 and k2s = −90. The parameter update law is given by
(5.20) where
τ = z1φ1 + z2φ2
φ1 = ωT + eT6 α1a, φ2 = eT
6 z1 −∂α1
∂yω (3.59)
65
0 20 40 60 80 100 120 140−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
time (sec)
y an
d y
m (rad
/sec
)
Plant output and reference signal
RefRACARC
0 20 40 60 80 100 120 140−0.01
−0.005
0
0.005
0.01
time (sec)
e(t)
(rad/
sec)
Tracking error
RACARC
0 20 40 60 80 100 120 140−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
time (sec)
u(t)
(rad)
Control signals: RAC
u1(t)
u2(t)
0 20 40 60 80 100 120 140−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
time (sec)
u(t)
(rad)
Control signals: ARC
u1(t)
u2(t)
Figure 3.2. Reference tracking, control signals and tracking error for RACversus ARC based fault-tolerant schemes in absence of disturbances
66
0 20 40 60 80 100 120 140−0.03
−0.02
−0.01
0
0.01
0.02
0.03
time (sec)
y an
d y
m (rad
/sec
)
Plant output and reference signal
RefRACARC
0 20 40 60 80 100 120 140−0.02
−0.01
0
0.01
0.02
0.03
time (sec)
e(t)
(rad/
sec)
Tracking error
RACARC
0 20 40 60 80 100 120 140−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
time (sec)
u(t)
(rad)
Control signals: RAC
u1(t)
u2(t)
0 20 40 60 80 100 120 140−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
time (sec)
u(t)
(rad)
Control signals: ARC
u1(t)
u2(t)
Figure 3.3. Reference tracking, control signals and tracking error for RACversus ARC based fault-tolerant schemes in presence of disturbances
67
3.5 Conclusion
In this work, an adaptive robust output feedback based scheme is presented for
unknown actuator fault accommodation for a class of uncertain nonlinear system.
Adaptation and robust feedback are used simultaneously to maintain tracking perfor-
mance in face of large parametric uncertainties introduced due to failing actuators,
exogenous disturbances and other modeling uncertainties.
A nonlinear model of hypersonic aircraft is used for simulation studies, which
clearly demonstrates the effectiveness of the proposed scheme in accommodating ac-
tuator faults. In summary, some of the salient features of the fault accommodation
scheme presented in this chapter are,
1. capability to handle large parametric uncertainties due to unknown actuator
failures like stuck actuators and actuator loss in efficiency with guaranteed tran-
sient performance
2. guaranteed robust performance when adaptation is switched off
3. calculable upper bound for tracking error based on controller parameters and
ability to achieve prespecified final tracking accuracy
68
4. A BACKSTEPPING BASED APPROACH TO ROBUST GLOBAL
STABILIZATION OF A CHAIN OF INTEGRATORS WITH INPUT
SATURATION
4.1 Introduction
In the previous two chapters, an adaptive robust fault-tolerant scheme was devel-
oped to accommodate unknown actuator faults. An important underlying assumption
was that the healthy actuators had sufficient actuator authority to compensate for
the unknown actuator faults. In reality, however this may not be the case. In fact,
achieving small transient tracking error in presence of actuator faults may require
large control signals, which can easily saturate the healthy actuators. Thus, it is ex-
tremely important that the actuator limits be taken into consideration while designing
an actuator fault-tolerant controller.
As a first step towards developing a saturated fault-tolerant controller, we develop
a backstepping based controller for a celebrated problem - stabilization of a chain of
integrators in presence of input saturation. There are two important reasons for
the popularity of this problem within the input saturation community. First, an
integrator chain is not a stable system. For a long time, most researchers focused
on global stabilization of stable systems only. In fact, it was shown by Sontag in [],
that a third or higher order integrator chain cannot be stabilized by linear control
laws. Thus, integrator chain is complex enough that simple linear controllers cannot
work, but at the same time, owing to its simple form, the analysis remains clear and
tractable. Second, many schemes which were proposed for stabilizing the integrator
chain have been successfully generalized to classes of nonlinear systems.
Among various approaches for dealing with input saturation, anti-windup schemes,
nested saturation functions and model predictive control are most popular. In anti-
69
windup based schemes ( [38], [39], [40], [41]), first a controller is designed without
any regard to the actuator limits, and then a modification is introduced to minimize
the adverse effects of saturation. But, such designs tend to be more conservative in
terms of achievable performance. Model predictive control (MPC), which involves
solving an open-loop optimization problem at each step, is adept at dealing with
hard constraints and is fast becoming a useful tool in dealing with saturation prob-
lems [42], [43]. The main drawback and challenge of MPC based schemes is to incor-
porate the modeling uncertainties, which are inherent to any realistic system model.
Lyapunov function based techniques for stabilizing a system in feedforward ( [44])
and feedback form ( [45], [46]) with bounded control have also been proposed. Note
that backstepping based approaches suffer from the so called “explosion” of terms,
and showing boundedness in such a setting is very challenging. This problem was
circumvented by imposing some restrictions on the growth rate of various nonlinear-
ities and assuming that bounded control laws and control Lyapunov functions are
known a priori for the reduced order system. However, they do not provide very
clear and practical guidelines for designing a controller in presence of uncertainties.
Nested saturation functions and small-gain theorem are amongst the most widely
used tools for dealing with input saturation. This methodology was first proposed by
Teel in [1] for a chain of integrators. The first step of the design involved a coordinate
transformation, which transformed the system to a feedforward form. In the second
step, saturation functions were used to construct a nested control law in terms of the
transformed coordinates. Subsequently, this approach has been extended to various
classes of nonlinear systems in feedforward form under various assumptions [47], [48].
For chain of integrators, many modifications to Teel’s original design have been
proposed to improve the transient response and robustness of the controller [49], [50],
[51]. But, as all of these designs were based on [1], they also inherited the limitations
of that approach. Particularly, in presence of bounded input disturbance, the region
where the controller is unsaturated shrinks drastically, and in the worst case, renders
the controller design impossible even when magnitude of the disturbance is less than
70
the available control input. This is to be expected as all the transformed coordinates
depend on the last state of the integrator chain, the dynamics of which includes the
input disturbance. Thus, the input disturbance affects the dynamics of every state
in the transformed coordinates. This implies that the parameters for all saturation
functions need to be chosen conservatively to accommodate the effect of disturbance.
Furthermore, the coordinate transformation makes it more difficult to tune the con-
troller parameters to achieve a desired objective. For example, say the objective is for
the output x1 to track yd with desired transient response in the unsaturated region.
In Teel’s approach, as x1 depends on all the transformed coordinates yi, i = 1, ..., n,
the tracking problem for x1 translates to designing a controller with desired transient
response for all the states - a more challenging problem which can further add to
the conservativeness of the overall design procedure. Moreover, it is expected that
in presence of input saturation, there will be trade-off between high controller gains
and the region where it operates unsaturated. The coordinate transformation makes
it difficult to understand these trade-offs and thus, reduces the design flexibility.
From the preceding discussion, it should be clear that the design conservativeness
in presence of disturbances can be reduced and it can be made more transparent if
a controller can be designed without the coordinate transformation. In the present
work, we take a conceptually different approach and solve the problem without using
coordinate transformation. We combine a backstepping based approach with satura-
tion functions to design a globally stabilizing controller for a chain of integrators. As
no transformation is required, the input disturbance appears only in the last channel
and thus, affects the choice of only the last saturation function parameters. Also,
in the proposed technique, bounded virtual control laws are designed at each step
to drive the intermediate error variable to zero. This makes the design more trans-
parent, as we need not deal with a combination of states but only one state at each
step of the design to achieve the desired transient response. In fact, the proposed
scheme not only preserves the clarity present in backstepping based designs, but also
71
allows design of stabilizing controllers in presence of bounded disturbances when a
coordinate transformation based approach fails.
In addition to being less conservative, the proposed design can also guarantee
desired closed-loop performance within the unsaturated region of controller operation.
However, this implies that the controller gains be selected in a fashion that allows the
closed-loop poles to be placed at desired locations. Thus, the procedure for choosing
the controller parameters become a non-trivial and fairly challenging problem, as they
need to satisfy two sets of constraints - first, a set of inequalities which guarantee that
all the states can be steered to the unsaturated region of controller operation, and
second, a set of constraints which ensure that desired closed-loop performance can be
achieved. To this end, we provide necessary and sufficient conditions for the existence
of the proposed control law, and propose a systematic way of choosing the controller
parameters based on rigorous analysis.
Comparative simulation studies are presented in this chapter show the effective-
ness of the proposed scheme. Note that for comparative studies, we use the technique
proposed in [50], which is based on coordinate transformation. The results demon-
strate that better disturbance rejection capabilities and faster convergence rate can
be achieved using the proposed design.
4.2 Motivation and Problem Formulation
Consider a chain of integrator with input disturbance
x1 = x2
x2 = x3
...
xn = u+ d(t)
y = x1 (4.1)
72
where |u| ≤ uM , |d(t)| ≤ dM and dM is a known constant. The objective of the
present work is to design a controller such that
1. x1 tracks yd with steady-state error |ess| ≤ δ
2. ess(t) should have desired transient performance in the unsaturated region.
We will make the following assumption regarding the extent of input disturbance
and the reference trajectory
A1: The extent of disturbances and the desired trajectory is such that
∣
∣
∣
∣
dnyd
dtn
∣
∣
∣
∣
≤ λM
uM > dM + λM (4.2)
Let us first investigate the effect of this disturbance on Teel’s approach. In all such
approaches, the coordinate transformation used takes the following generic form [49]
yi =
n∑
j=i
tij xj , j = 1, . . . , n
(4.3)
where xj = xj − y(j−1)d and T = [tij] is the transformation matrix given by
tij =
1, 1 ≤ i ≤ n, j = n
0, i = n, 1 ≤ j ≤ n− 1∑n
m=i+1 kmtm,j+1, i ≥ 1, j ≤ n− 1
With this, the dynamics of the transformed states can be expressed as
y1
...
yn−1
yn
=
0 k1 · · · 1...
. . .. . .
...
0 · · · 0 1
0 · · · · · · 0
y1
...
yn−1
yn
+
1...
1
1
u(t) +
1...
1
1
d(t) (4.4)
The proposed control law takes the form [1]
u = −σn(knyn + σn−1(kn−1yn−1 + ...+ σ1(k1x1))...) (4.5)
73
where the saturation functions σi are given by
(1) sσi(s) ≥ 0, ∀s 6= 0
(2) σi(s) = s, ∀|s| < Li
(3) |σi(s)| ≤ Mi, ∀s ∈ R, Li < Mi (4.6)
To account for the disturbances, the inequalities proposed in [1] can be modified as
follows to ensure that there is available control input for steering the tracking error
to a neighborhood of zero
|y(n)d | ≤ Ln+1 −Mn
Li+1 ≥ 2Mi + dM , i = 0, ..., n− 1 (4.7)
with M0 = 0. From (4.4), we see that due to the coordinate transformation, the
disturbance affects all the states, although it was present only in the input channel in
the original coordinates. Consequently, the linear unsaturated region (i.e., the region
where the linear controller is unsaturated) of all the transformed states shrink by
dM , as seen from (4.7). This makes the design procedure very conservative. In fact,
even when the extent of uncertainties is less than the available control input, the
coordinate transformation may render the design of a stable controller impossible, as
shown below
uM > Mn+1 > Ln+1
> Mn > Ln
> dM + 2Mn−1 > dM + 2Ln−1
> dM + 2(dM + 2Mn−2) > dM + 2(d+ 2Ln−2)...
...
> dM + 2dM + · · ·+ 2n−1dM = (2n − 1)dM
(4.8)
From the above inequalities, it should be clear that if d(t) is such that dM ≥ uM/(2n − 1),
then this approach would not work. For example, consider the stabilization problem
for a third order chain of integrator with uM = 10, and dM = 4. Then, dM = 4 > 10/7,
74
which implies (4.8) cannot be satisfied. This shows the conservativeness of any ap-
proach which relies on coordinate transformation. In this work, we will show that such
conservativeness can be removed by using a backstepping based technique, which does
not require coordinate transformation. In contrast to the condition uM > (2n − 1)dM
given by (4.8), we need a much less restrictive condition uM > dM for the stabilization
problem, as given by assumption A1.
4.3 Main Result
In this section, we present the backstepping based controller design for robust
global tracking for an integrator chain and present the main results. In the first
step, a set of inequalities is proposed such that when satisfied, it would guarantee
that for any given set of initial conditions, the error can be driven to an invariant
set where the proposed controller is not saturated. Once within the unsaturated
region, the desired properties of a linear controller, e.g. exponential convergence,
desired transient response and arbitrary disturbance attenuation, can be guaranteed.
However, this implies that the chosen controller gains should not only satisfy the
inequalities proposed in the first step, but should also be able to place the closed-loop
poles at the desired location. In the second step, we provide necessary and sufficient
conditions for the existence of such a control law, and also present a systematic way
of choosing the controller gains to achieve the desired closed-loop performance.
75
4.3.1 Convergence to Unsaturated Region
We will use the coordinate transformation xi = xi−y(i−1)d to simplify the analysis.
The designed virtual control law αi, error variables zi and the xi-dynamics in terms
of αi and zi are given by
zi = xi − αi−1(zi−1)
αi = −σi(zi)
˙xi = zi+1 + αi (4.9)
where the σi(zi), i = 1, ..., n (see figure (5.1)) are saturation functions.
Remark 1. Note that the form of the virtual control laws designed is signifi-
cantly different, and at the same time much simpler than the typical “cancellation”
backstepping design. In cancellation backstepping design, the detrimental effect of
the virtual control law of step i− 1 i.e., αi−1 is completely canceled at step i, by in-
corporating the appropriate terms in the design of αi. This results in a simple linear
dynamics for the error variables, such that the closed-loop poles can be placed at any
desired location. But, this simplicity comes at the expense of a complicated control
law, with exponentially growing number of terms as the order of the system increases.
In the context of bounded control, it is extremely difficult to show the boundedness
of such a control law. For chain of integrators, as we shall see, the error dynamics can
be stabilized with a much simple control law σi(zi), and without canceling the terms
resulting from the derivative of αi−1. Although linear, the resulting error-dynamics
without cancellation leads to a set of constraints that the controller parameters must
satisfy for global stabilization. This makes the parameter selection process slightly
more complicated than the cancellation backstepping design.
The saturation functions used in this work play a significant role in guaranteeing
the overall system stability. In fact, it is only through a careful and rigorous analysis
of the effects of saturation function parameters on system dynamics that a set of
76
conditions will be derived, which will ensure all desired objectives can be achieved.
The saturation functions are defined as
(a) ziσi(zi) ≥ 0, ∀zi (4.10)
(b) σi(zi) = kizi, ∀|zi|≤li,
σi(zi) = Mi, ∀|zi|≥Li (4.11)
(c) |σi(zi)| ≤Mi, ∀zi (4.12)
(d)∂σi
∂zi
≤ ki, ∀zi (4.13)
Also, li = βiLi with βi ≤ 1, and Mi = kili(1 + γi) with γi > 0. The interval for
zi is divided into three different regions - Ωi1 = zi : |zi| ≤ li, Ωi2 = zi : |zi| ≤Li and Ωi3 = zi : |zi| > Li. Note that the nonlinear transition region of the
saturation function (Ωi2\Ωi1) needs to be at least second order differentiable, as the
backstepping design involves taking derivatives of σi. σn, however, need not have a
smooth transition region, as this appears in the last step. Hence, we choose γn = 0,
βn = 1 for σn, which implies Ωn1 = Ωn2. Substituting (4.9) in (4.1), the error dynamics
Figure 4.1. Saturation function
77
can be written as
z1 = z2 − σ1(z1)
· · ·
zi = zi+1 − σi(zi) +i−1∑
j=1
[
j∏
r=1
∂σi−r
∂zi−r
]
(zi−j+1 − σi−j(zi−j))
· · ·
zn = u+ d(t) − y(n)d +
n−1∑
j=1
[
j∏
r=1
∂σn−r
∂zn−r
]
(zn−j+1 − σn−j(zn−j))
(4.14)
Now, we are ready to state the main result of the present work.
Theorem 1. Consider system (4.14). Let the control input be
u = y(n)d − σn(zn) (4.15)
If the controller parameters can be chosen such that
kili > li+1 + ki−1Ni, i = 1, 2, ..., n− 1 (4.16)
knln > kn−1Nn + dM (4.17)
knln ≤ uM − λM , u′M (4.18)
where
Ni , Li +Mi−1 +i−2∑
j=1
[
(
j∏
r=1
ki−1−r)(Li−j +Mi−1−j)
]
and k0 , 0, ln+1 , 0. Also, Mn used in the design of σn(zn) is chosen such that
λM + Mn = uM . Then, for any set of initial conditions, all states reach the linear
unsaturated region i.e., Ωi1 in a finite time.
Proof. Consider the following claims.
Claim1. For any initial condition zn(0)
(a) if |zn(0)| > Ln, then it reaches Ωn1 in a finite time
(b) if zn(0) ∈ Ωn1, then zn(t) ∈ Ωn1, ∀ t > 0
Claim 2. Assume |zi+1| ≤ li+1, then
(a) if |zi(t0)| > Li, then it reaches Ωi2 in a finite time
(b) if zi(t0) ∈ Ωi2\Ωi1, then it reaches Ωi1 in a finite time
(c) if zi(t0) ∈ Ωi1, then zi(t) ∈ Ωi1 ∀ time t > t0
78
Assume both claims are true. Then, from claim 1, we have |zn| < Ln = ln after
a finite time. Theorem 1 then follows from a recursive application of claim 2 to the
intermediate states zj , j = n − 1, · · · , 1. Thus, we will be done if we can show that
both the claims are true.
Proof of Claim 1, Part (a). Without loss of generality (w.l.o.g), assume
zn(0) ≥ Ln. Then, u = −Mn + y(n)d and from (4.9) we get
˙xn = −Mn + d(t)
⇒ xn(t) ≤ xn(0) − (Mn − dM)t
⇒ zn(t) + αn−1(t) ≤ zn(0) + αn−1(0) − (Mn − dM)t
⇒ (Mn − dM)t ≤ zn(0) − zn(t) + αn−1(0) − αn−1(t)
Using |αn−1| ≤Mn−1, and substituting zn(t) = Ln an upperbound for the time taken
to reach Ωn1 is given by
tn32 ≤zn(0) − Ln + 2Mn−1
Mn − dM
(4.19)
From assumption A1, it is easy to see that as Mn = uM − λM > dM , zn(t) reaches
Ωn1 in a finite time.
Remark 2. Before we proceed further, we note the following facts about zn−dynamics
1. zi, for i < n, affects the zn−dynamics only through bounded terms like σi,∂σi
∂zi
and multiplication of ∂σi
∂ziwith other bounded terms. Note that ∂σi
∂zizi is also
bounded, as it vanishes for |zi| > Li.
2. zn is also upper bounded by Ln in Ωn1.
As zn-dynamics is affected by only bounded terms once |zn| ≤ Ln, it is possible to
use a bounded feedback which can make the tangent vector point inwards at Ωn1
boundary, which is essential in proving the next part of the claim.
Part (b). From (4.17), we know there exists δn > 0 such that
kn(ln − δn) =n−1∑
j=1
[
(
j∏
r=1
kn−r)(Ln−j+1 +Mn−j)
]
+ dM (4.20)
79
In order to show that any trajectory starting in Ωn1 will never leave this set, we need
to show that the tangent vector points inward at the boundaries i.e., at zn = ±Ln.
Assume w.l.o.g zn > 0 Then, at zn = Ln, znzn is given by
znzn = zn
(
−σn(zn) +n−1∑
j=1
[
j∏
r=1
∂σn−r
∂zn−r
]
(zn−j+1 − σn−j(zn−j))
+ d(t)
)
≤ |zn|(
−knln +n−1∑
j=1
[
(
j∏
r=1
kn−r)(Ln−j+1 +Mn−j)
]
+ dM
)
(4.21)
where we have used ziσi(zi) > 0, ∀ zi. Thus, combining (4.20) and (4.21) we get
znzn ≤ −|zn|knδn < 0 (4.22)
This shows that the tangent-vector points inward at the boundaries and completes
the proof of the claim.
Proof of Claim 2, Part (a). When zi ∈ Ωi3 i.e., |zi| > Li, we have αi = −Mi
(w.l.o.g assume zi > 0) from the definition of σi. Using the assumption of claim 2
i.e., |zi+1|≤li+1 and substituting αi = −Mi in (4.9), we obtain
˙xi = zi+1 + αi
≤ li+1 −Mi = −(Mi − li+1)
⇒ xi(t) ≤ xi(t0) − (Mi − li+1)(t− t0)
⇒ (Mi − li+1)(t− t0) ≤ xi(t0) − xi(t) + αi−1(t0) − αi−1(t)
⇒ ti32 ≤ xi(t0) − Li + 2Mi−1
Mi − li+1(4.23)
where we substituted zi(t) = Li and ti32 , t − t0, the time taken to reach Ωi2 from
Ωi3. It follows from (4.16) that Mi > kili > li+1. This proves the first part of the
claim.
Part (b). To prove the next two parts, we proceed as done in the proof of first
claim. From (4.16), there exists a δi such that
ki(li − δi) = li+1 + ki−1Ni (4.24)
80
Without loss of generality, assume zi(t0) > 0. Then, from the zi-dynamics (4.14), we
have
zi = zi+1 − σi(zi) +i−1∑
j=1
[
j∏
r=1
∂σi−r
∂zi−r
]
(zi−j+1 − σi−j(zi−j))
≤ li+1 − kili +i−1∑
j=1
[
(
j∏
r=1
ki−r)(Li−j+1 +Mi−j)
]
≤ li+1 − kili + ki−1Ni (4.25)
= −kiδi
⇒ ti21 ≤ Li − likiδi
(4.26)
where we have used the fact that Li − li≥|zi(t) − li|, ∀ zi(t) ∈ Ωi2\Ωi1. Thus, there
exists a finite time-interval ti21, within which any trajectory starting in Ωi2\Ωi1 reaches
Ωi1. This completes the proof of second part of the claim.
Part (c). In order to show that any trajectory starting in Ωi1(t0) stays there ∀t > t0, we need to show that zizi < 0 at zi = ±li. For zi = li, we know σi(zi) = kili
and using the assumption |zi+1|≤li+1, from (4.14) the zi-dynamics can be written as
zizi = zi
(
zi+1 − σi(zi) +i−1∑
j=1
[
j∏
r=1
∂σi−r
∂zi−r
]
(zi−j+1 − σi−j(zi−j))
)
≤ |zi|(
li+1 − kili +i−1∑
j=1
[
(
j∏
r=1
ki−r)(Li−j+1 +Mi−j)
])
≤ −ki|zi|δi (4.27)
This shows that Ωi1 is positively invariant. This completes the proof.
4.3.2 Controller Parameter Selection
There are two important questions which need to be answered next. First, when
can the existence of a solution to inequalities (4.16)-(4.18) be guaranteed. Note that
even when such a solution exists, it only guarantees the convergence of all intermediate
error variables to the linear unsaturated region (an invariant set in the neighborhood
of zero). The system response within the linear unsaturated region is governed by the
81
gains ki, and leads to the second important question - how to select the gains such
that the desired closed-loop performance can be achieved, without violating (4.16)-
(4.18). It is natural to first select a set of gains ki such that desired performance
can be achieved, and then seek a set of lis such that inequalities (4.16)-(4.18) can
be satisfied. However, such an approach may not work as after fixing the gains ki,
solving the inequalities (4.16)-(4.18) may result in negative lis, which will make the
control law unrealizable. Hence, in the following section, we first state and prove
the main result regarding the necessary and sufficient condition for the existence of a
solution to the inequalities. Then, a systematic way of choosing the controller gains in
accordance with this theorem is proposed, such that desired closed-loop performance
can be achieved.
Necessary and Sufficient Conditions for the Existence of Controller Pa-
rameters
After a series of derivations, (4.16)-(4.17) can be rewritten in a matrix form
AL < D, (4.28)
where L = [l1, l2, · · · , ln−1, ln]T , D = [0, 0, · · · , 0,−dM ]T . And A is a function of ki
given by
A =
−k1 1 0 · · · 0
k21(1 + γ1)
(
k1
β2− k2
)
1 · · · 0
· · · · · · · · · · · · · · ·an1 an2 an3 · · ·
(
kn−1
βn− kn
)
, (4.29)
where
aij = ki−1kj
i−j−1∏
r=1
ki−r−1(1 + γj) + ki−1
i−j∏
r=1
ki−r−11βj, ∀i > j.
aij = −ki + ki−1
βi, ∀i = j
aij = 1, ∀j = i+ 1
aij = 0, ∀j > i+ 1
(4.30)
82
For any γi > 0 and 0 < βi ≤ 1, if we fix a set of positive kis, then the control law
is feasible if and only if the there exist l1, l2,..., ln > 0 such that (4.18) and (5.44)
are satisfied. In other words, at least one solution to (4.18) and (5.44) should lie in
the region (l1, · · · , ln) : li > 0. The following theorem gives the necessary and
sufficient condition for kis such that the control law is feasible.
Theorem 2. For any γi > 0 and 0 < βi ≤ 1, with a set of positive kis, at least
one solution to (4.18) and (5.44) lie in the region L ∈ (l1, · · · , ln) : li > 0 iff the
kis satisfy the following set of inequalities:
k1 > 0,
k2 >a21p1+
k1β2
p2
p2= (1 + γ1 + 1
β2)k1,
· · ·
ki >∑i−1
j=1aijpj+
ki−1
βipi
pi, ∀i < n
· · ·
kn > uM−λM
uM−(λM +dM )·∑n−1
j=1anjpj+
kn−1
βnpn
pn,
(4.31)
where, the coefficients pis are computed recursively using the formula
p1 = 1
pi = −∑i−1
j=1 ai−1jpj
(4.32)
Proof 2. =⇒ (Necessary condition):
Assume there exists a solution satisfying (4.18) and (5.44), which lies in the region
L ∈ (l1, · · · , ln) : li > 0. Denote it by
L′ ∈ (l′1, · · · , l′n) : l′i > 0 (4.33)
Let us first prove the following claim using mathematical induction.
Claim 3.
pil′i−1 > pi−1l
′i
pi > 0,(4.34)
for all i ≤ n.
Proof of Claim 3.
(i) Looking at the first row of AL′ < D, we have −k1l′1 + l′2 < 0. From the definition
83
of p1 and p2 given by (5.48), and noting that a11 = −k1, we have p2l′1 > p1l
′2. Since
l′1, l′2 > 0 and p1 > 0, we obtain p2 > 0. Thus, the claim is true for j = 2.
(ii) Suppose the claim is true for all 2 ≤ j < i. Then, we have
l′j >pjl
′j+1
pj+1>pjpj+1l
′j+2
pj+1pj+2> · · · > pjl
′i−1
pi−1(4.35)
Now, looking at the i− 1 row of AL′ < D, and using aij > 0, ∀i > j, we get
i−1∑
j=1
ai−1jl′j + l′i < 0
⇒i−1∑
j=1
ai−1jpj l′i−1
pi−1+ l′i < 0
⇒ −pil′
i−1
pi−1+ l′i < 0, (using (5.48))
⇒ pil′i−1 > pi−1l
′i.
(4.36)
Since l′i−1, pi−1, l′i > 0, we get pi > 0. Thus, the claim is also true for j = i.
This procedure can be continued until i = n, as the R.H.S of the first n− 1 rows
of AL′ < D are all zeros. This completes the proof of the claim.
Next, it will be shows that the condition on gains, given by (5.47), follows from
(5.44) through the use of claim 3.
For each i < n, from the i-th row of AL′ < D we have
i−1∑
j=1
aijl′j +(
ki−1
βi− ki
)
l′i + l′i+1 < 0
⇒i−1∑
j=1
aijpj l′ipi
+(
ki−1
βi− ki
)
l′i + l′i+1 < 0, (l′j >pj l′ipi, ∀j < i using claim 3)
⇒i−1∑
j=1
aijpj l′ipi
+(
ki−1
βi− ki
)
l′i < 0
⇒[
i−1∑
j=1
aijpj
pi+(
ki−1
βi− ki
)
]
l′i < 0
⇒ ki >∑i−1
j=1aijpj+
ki−1
βipi
pi
(4.37)
From the last row of AL′ < D, we get
n−1∑
j=1
anjl′j +(
kn−1
βn− kn
)
l′n < −dM)
⇒n−1∑
j=1
anjpj l′npn
+(
kn−1
βn− ki
)
l′n < −dM
⇒[
kn −n−1∑
j=1
anjpj
pn− kn−1
βn
]
l′n > dM
(4.38)
84
Applying (4.18), which is uM − λM ≥ knl′n, to the above inequality and eliminating
l′n:(
kn −n−1∑
j=1
anjpj
pn− kn−1
βn
)
[uM − λM ] ≥ kndM
⇒ kn >uM−λM
(uM−(λM +dM ))·∑n−1
j=1anjpj+
kn−1
βnpn
pn,
(4.39)
which completes the proof of necessary condition.
⇐= (Sufficient condition):
Suppose that kis satisfy inequalities (5.47). Then, for i ≤ n− 1,
ki >∑i−1
j=1aijpj+
ki−1
βipi
pi
⇒∑i−1
j=1aijpj+
ki−1
βipi−kipi
pi< 0
⇒ −pi+1
pi< 0, ( using definition of pi and aii = −ki + ki−1
βifrom(5.46))
⇒ pi+1
pi> 0.
(4.40)
Since p1 = 1 > 0, we have pi > 0, ∀i ≤ n.
Next, we are interested in finding the solution of the linear matrix equation AL′ =
D. Using the first n− 2 algebraic equations of AL′ = D, we can represent all the l′is
with i > 1 in terms of l′1:
l′1 = p1l′1
l′2 = p2l′1
· · ·l′n = pnl
′1.
(4.41)
Substituting them into the last equation of AL′ = D, we have:
n−1∑
j=1
anjpjl′1 + (kn−1
βn− kn)pnl
′1 = −dM
⇒ l′1 = −dMn−1∑
j=1
anjpj+(kn−1
βn−kn)pn
.(4.42)
From the last equation of (5.47), since 0 < (uM − (λM + dM)) < uM − λM , we have
kn >∑n−1
j=1anjpj+
kn−1
βnpn
pn
⇒n−1∑
j=1
anjpj + (kn−1
βn− kn)pn < 0.
(4.43)
85
Thus, l′1 > 0. As a result, all the l′is are greater than zero. Multiplying both sides of
(5.47) by (uM − (λM +dM))pn, and rearranging the terms to factor out uM , we obtain
knl′n = knpnl
′1 =
−dMknpn
n−1∑
j=1
anjpj + (kn−1
βn− kn)pn
< uM − λM (4.44)
Thus, we have shown that the only solution to the linear matrix equation AL′ = D
lie in the region (l′1, · · · , l′n) : l′i > 0, knl′n < uM which is an open set. Since A is a
n by n matrix, the solution to inequality AL < D must exist and is a a sector region
with the acme being the point L′ which solves equation AL′ = D. Then, we can
always pick a solution of AL < D to be arbitrarily close to L′ such that L also lies in
the open region (l′1, · · · , l′n) : l′i > 0, knl′n < uM − λM. After choosing that L, we
have
(i) AL < D;
(ii) knln < uM − λM ;
(iii) all the lis are greater than zero.
Thus, there exists at least one solution satisfying (4.16)-(4.17), which lies in the
region (l′1, · · · , l′n) : l′i > 0. This proves the sufficiency condition of the theorem.
Remark 3. Condition (5.47) plays an important role in the selection of controller
parameters kis, and can be explained as follows. It can be easily shown that the
closed-loop transfer function from d(t) to z1 (which is also the tracking error y − yd)
inside the linear unsaturated region Ω11 is
[
z1(t)
d(t)
]
=1
sn + knsn−1 + knkn−1sn−2 + · · ·+n∏
j=2
kj · s +n∏
j=1
kj
. (4.45)
Due to the conditions imposed by (5.47), the closed-loop poles of (5.61) cannot be
assigned arbitrarily. However, as pi depends only on kjs for j < i, from (5.47) we can
easily work out a recursive way of choosing the controller parameters. Specifically, we
first choose k1, and then choose k2 large enough such that the first inequality of (5.47)
is satisfied. Continuing in this fashion, we can get a set of kis that are permissible.
In the next subsection, we clearly outline such a technique for choosing kis such that
86
all closed-loop requirements are met, without violating the constraints imposed by
(5.47).
Controller Gain Selection: A Recursive Root-Locus Design
As mentioned above, the closed-loop poles of (5.61) cannot be arbitrarily assigned
because of the constraints (5.47) on the controller parameters ki. However, (5.47)
implies that ki can be chosen arbitrarily large, which makes it possible to satisfy the
closed-loop performance requirement. In the linear unsaturated region, the desired
closed-loop performance can be achieved by placing the poles sufficiently far to the
left of imaginary axis e.g., if the required steady-state error is δ, and the transient
response criteria dictates that the slowest closed-loop pole be to the left of p0, then
it is sufficient to place all the poles to the left of pcl = min−√
dM
δ,−p0. In the
following, we propose a recursive root locus design to meet the above requirement.
• Step 1: Select k1 > −pcl, then the root of the equation s+ k1 = 0 lie to the left
of s = pcl.
• Step 2: Let the virtual open-loop system be k2s+k1
s2 , then the virtual closed-loop
characteristic equation is s2 + k2s + k1k2 = 0. To determine k2, draw the root
locus of k2s+k1
s2 . This open-loop system has two poles at origin and one zero at
−k1, left to s = pcl. From the general guidelines for drawing a root-locus, there
exists a k2 large enough such that: (a) the first inequality of (5.47) is satisfied
and, (b) all the roots of s2 + k2s+ k1k2 = 0 lie to the left of s = pcl, on the real
axis.
.............
.............
.............
• Step n: Let the virtual open-loop system be kn
sn−1+kn−1sn−2+···+n−1∏
j=1
kj
sn , then the
virtual closed-loop characteristic equation is exactly the same as that of the
87
actual system, i.e., knsn−1 + knkn−1s
n−2 + · · · +n∏
j=1
kj. To determine kn, draw
the root locus of kn
sn−1+kn−1sn−2+···+n−1∏
j=1
kj
sn . This open-loop system has n poles
at origin and n− 1 zeros to the left of s = pcl. As the difference in the number
of poles and zeros is one, there always exists an asymptote along the negative
real axis. Thus, for sufficiently large gain kn, there is a branch of the root-locus
on the negative real axis. This implies (a) the last two inequalities of (5.47) are
satisfied and (b) all the closed-loop poles lie to the left of s = pcl on the real
axis.
Thus, we can choose the controller gains such that the desired closed-loop perfor-
mance is achieved, as well as the conditions imposed for the existence of a feasible
control law given by (5.47) are also satisfied simultaneously.
4.4 Simulation Example: Third Order Integrator Chain
A 3rd order chain of integrators is used to demonstrate the effectiveness of the
proposed scheme. The proposed scheme is compared with a transformation based
controller design technique [50] to underline the superior achievable performance. In
the first set of simulations, it is assumed that there is no disturbance present in the
system. The results show that the achievable convergence rate is better than that
obtained in [50]. Simulation studies conducted in presence of disturbances - second
and third set of simulations, demonstrate the robustness of the proposed scheme
against input disturbance.
Simulation 1: The goal of this simulation study was to investigate the convergence
rate of the proposed scheme against a transformation based approach for large initial
88
conditions in absence of input disturbance. The third order chain of integrator given
by
x1
x2
x3
=
0 1 0
0 0 1
0 0 0
x1
x2
x3
+
0
0
1
u (4.46)
was used in [50] and thus, provides a level platform to compare the two approaches.
The initial conditions are same as used in the cited paper x(0) = [2,−2, 3]. We use
the control law proposed in Theorem 3 of [50] for comparison. The controller param-
eters for the proposed scheme are: [k1, k2, k3] = [0.2, 0.9, 20], [l1, l2, l3] = [4, 0.8, 0.05],
[β1, β2, β3] = [0.909, 0.909, 1], [γ1, γ2, γ3] = [0.01, 0.01, 0]. The parameter selection
scheme proposed in the previous section was used as a starting point, and then it
was tuned to improve the performance. As seen from fig. (5.2), we achieve slightly
faster convergence with the proposed scheme. This shows that the performance of
the proposed technique in terms of achievable convergence rate is at least as good as
that of a transformation based approach. However, the true strength of the proposed
controller and its robustness against disturbances is demonstrated in the next set of
simulations when large disturbances are considered.
Simulation 2: The purpose of this simulation study was to compare the conver-
gence rate and steady-state error for a third order integrator chain in presence of
input disturbance. The input disturbance and initial conditions were chosen to be
d(t) = 0.1 sin(πt2) and x(0) = [0.2,−0.2, 0.3] respectively. The initial conditions were
chosen to be smaller than the previous case in order to highlight the effect of distur-
bance on steady-state error. Controller parameters are same as used in the previous
simulation. The disturbance attenuation capability of the proposed scheme over a
coordinate transformation based approach is evident from fig. (5.3) and can be ex-
plained as follows. First, note that there is no transparent and easy way of tuning the
controller proposed in [50] to attenuate the effect of disturbances on the states. In
fact, as the stability analysis in the aforementioned paper is performed in the trans-
formed coordinates, there is no convenient way of studying the effect of disturbance
89
on the original states. Second, in our design it is easy to understand and account
for the effect of disturbances. The closed loop characteristic equation of the system
in the linear unsaturated region is given by s3 + 20s2 + 18s + 3.6 = 0. Thus, using
the final value theorem, we know the disturbance should be attenuated by a factor
of 3.6, which matches the simulation result. In fact, the effect of disturbance can
be attenuated to any desired extent by choosing high controller gains, as outlined in
the previous section. Also, note that the convergence of states depend on two fac-
tors: first, how fast the states reach the unsaturated region, and second, how fast the
states go to zero once within the unsaturated region (or a residual ball in presence
of disturbances). Thus, if we choose large li, we can make the convergence to the
unsaturated region faster, but then we cannot choose very high controller gains once
within the unsaturated region, as inequalities (4.16-4.18) must be satisfied. In fact, in
the context of bounded control, such a trade off is expected. Due to the transparency
of these trade-offs in the proposed scheme and the available freedom in choosing the
controller parameters, a better controller can be designed when the initial conditions
are known e.g., when the initial conditions are not too large, we can select a relatively
small li and choose high gains to attenuate the effect of disturbances and vice-versa.
Simulation 3: In this case, we present an example where the conventional ap-
proach fails to yield a stable controller. In the problem formulation, we had al-
ready shown that when uM = 10, the transformation makes it impossible to de-
sign a stable controller with dM = 4. In contrast, using the proposed technique,
we can not only design a stable controller, but also achieve desired disturbance
attenuation. For this set of simulations, the desired trajectory is a filtered a 0.5-
step command and the objective is to track it with a steady-state error less than
0.01 in presence of large input disturbance d(t) = 4 sin(πt2). Initial conditions are
set at x(0) = [0, 0, 0.2]. Controller parameters used for the tracking problem are:
[k1, k2, k3] = [1, 8, 50], [l1, l2, l3] = [0.12, 0.1, 0.2], [β1, β2, β3] = [0.909, 0.909, 1], and
[γ1, γ2, γ3] = [0.01, 0.01, 0]. Note that the closed loop characteristic equation of the
system in the linear unsaturated region is give by s3 + 50s2 + 400s+ 400 = 0, which
90
ensures that ess ≤ 0.01. As can be seen from fig. (4.4), the steady-state error is less
than 0.01 i.e., the desired performance is achieved. This example clearly illustrates
the effectiveness of the proposed scheme in reducing the design conservativeness over
a transformation based approach.
91
0 10 20 30 40 500
2
4
6
8
10
12
time (sec)
norm
(x(t)
)
Proposed designMarchand and Hably
0 10 20 30 40 50−1
−0.5
0
0.5
time (sec)
u
Proposed designMarchand and Hably
0 10 20 30 40 50−2
0
2
4
6
8
10
12
time (sec)
x 1
Proposed designMarchand and Hably
0 10 20 30 40 50−2
−1
0
1
2
3
time (sec)
x 2
Proposed designMarchand and Hably
0 10 20 30 40 50−1
−0.5
0
0.5
1
1.5
2
2.5
3
time (sec)
x 3
Proposed designMarchand and Hably
Figure 4.2. Comparative results for stabilization in absence of distur-
92
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
time (sec)
norm
(x(t)
)
Proposed designMarchand and Hably
0 10 20 30 40 50−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
time (sec)
u
Proposed designMarchand and Hably
0 10 20 30 40 50−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
time (sec)
x 1
Proposed designMarchand and Hably
0 10 20 30 40 50−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
time (sec)
x 2
Proposed designMarchand and Hably
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
time (sec)
x 3
Proposed designMarchand and Hably
Figure 4.3. Comparative results for stabilization in presence of distur-
93
0 10 20 30 40 50 60 70 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (sec)
x 1 and
x1d
x1
x1d
0 10 20 30 40 50 60 70 80−0.05
0
0.05
0.1
0.15
time (sec)
e(t)
0 10 20 30 40 50 60 70 80−12.0
−8.0
−4.0
0.0
4.0
8.0
time (sec)
u
0 10 20 30 40 50 60 70 80−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time (sec)
x 2
0 10 20 30 40 50 60 70 80−1
−0.5
0
0.5
1
time (sec)
x 3
Figure 4.4. Tracking in presence of large disturbance
94
4.5 Conclusion
The main contribution of the present work lies in proposing a conceptually dif-
ferent approach to solve the problem of global stabilization for a chain of integrators
in presence of input disturbance with desired performance in the unsaturated region.
Based on Teel’s work, many modifications have been proposed in the literature to
improve the performance of the controller. In our analysis, it was clearly shown that
all such schemes exhibit poor robustness properties with respect to input disturbance
and leads to conservative design. In fact, a quantitative analysis revealed that even
when the magnitude of the disturbance is less than that of the available control input,
the coordinate transformation can render the design of a stabilizing controller impos-
sible. These limitations cannot be overcome by any modification based on Teel’s
work, as coordinate transformation is an essential step in all such designs. In order
to remove these limitations, we take a fundamentally different viewpoint and propose
a scheme which does not rely on coordinate transformation, and is based on back-
stepping design. The resulting controller is easy to implement and tune, as we only
deal with the original coordinates. Furthermore, the proposed design can also cap-
ture the trade-offs which are inherent to any input-saturated control design problem
- choosing high controller gains versus enlarging the unsaturated region of linear con-
troller operation. Comparative studies have been performed on a third order chain
of integrator to show the superior performance of the proposed technique. The first
set of studies, performed in absence of disturbances, revealed that the convergence
rate of the proposed scheme is at least as good as that proposed by Marchand and
Hably, which is based on Teel’s work. In presence of disturbances, however, signif-
icant differences could be seen in terms of disturbance attenuation and convergence
of the states. A tracking problem with large disturbance, where a stable controller
could not be designed due to coordinate transformation, was also solved to show the
effectiveness of the proposed scheme.
95
5. SATURATED ADAPTIVE ROBUST ACTUATOR FAULT-TOLERANT
CONTROL FOR FEEDBACK LINEARIZABLE SYSTEMS
5.1 Introduction
In this chapter, an adaptive robust fault-tolerant scheme is proposed to deal with
unknown actuator faults in presence of disturbances and input magnitude constraints.
Among various approaches, adaptive control based fault-tolerant schemes have found
popularity among researchers, as they have the ability to learn the change in system
parameters due to actuator faults. However, input magnitude constraint - one of the
most important factors which can limit the performance of any control system, has
largely been overlooked in the literature.
The harmful effect of actuator faults on the system response increases manifold in
presence of actuator saturation. A control system which does not explicitly take into
account input saturation could generate large control signals to suppress the tran-
sients. However, as all actuators have limited authority, this could lead to actuator
saturation. Furthermore, direct adaptive scheme, which are preferred over indirect
scheme due to their lower dynamic order, may generate unreliable parameter esti-
mate once the actuators saturate. This could further degrade the performance of an
actuator fault-tolerant scheme.
In this chapter, the backstepping and saturation function based approach devel-
oped in the last chapter is combined with a least-squares estimator to develop an
indirect saturated adaptive robust fault-tolerant scheme. x-swapping lemma is used
to design the least-squares estimator. The bounded feedback control generated using
saturation functions compromise the performance when error-variables are far away
from zero, and lays more emphasis on bringing the error-variables in a region where
the controller is unsaturated. A carefully chosen set of controller gain ensures that
96
desired closed-loop performance is achieved, once within the unsaturated region. Ad-
ditionally, the indirect scheme ensures that parameter estimation is unharmed despite
actuator saturation and the proposed controller is shown to ISS with respect to para-
metric uncertainties and disturbances in the linear region. This results in improved
tracking performance, and guarantees asymptotic stability in presence of parametric
uncertainties only.
Controller design with actuator constraints is a challenging problem in itself. It
is worth mentioning that in the present work, we focus our attention to only those
class of nonlinear systems which are feedback linearizable with matched uncertain-
ties. Feedback linearization [52] based control techniques have become a useful tool
in the repertoire for flight control systems design e.g., [53], [54] and [55]. One of the
main reasons for the popularity of this technique is that it allows linear control design
techniques to be applied to nonlinear systems, and relieves the burden of nonlinear
systems stability analysis. In the present work, we add another powerful tool to this
collection in the form of adaptive robust actuator fault-tolerant controller, which ex-
plicitly considers the input magnitude constraint. Note that we do not claim that the
proposed scheme can accommodate faults of any magnitude. Furthermore, for general
nonlinear systems, it is not possible to design a globally stable bounded controller.
Thus, the initial conditions for the system trajectories and the extent of parametric
uncertainties and disturbance will be assumed to be bounded by known constants.
To summarize, three main contributions of the proposed work are - (i) design of a
control law that accommodates unknown actuator faults with desired closed-loop per-
formance in the unsaturated region, (ii) explicit consideration of actuator limits in
the controller design, which allows the controller to pull out of saturation phase after
faults and, (iii) uninterrupted adaptation regardless of saturation.
The performance of the fault-tolerant controller proposed in this chapter is com-
pared with the one developed in the second chapter, using the nonlinear hypersonic
aircraft model. The results clearly indicate the harmful effect of saturation on the
closed-loop stability. The controller which does not consider input constraints in
97
the design leads to unbounded signals, where as the proposed scheme goes into the
saturation mode temporarily, but returns to the unsaturated region quickly.
5.2 Problem Formulation
In this work, we consider systems which can be written as
x = f(x) + g(x)[b1u1 + · · ·+ bquq] + w(x)d(t)
y = h(x), |uj(t)| ≤ uM , j = 1, ..., q (5.1)
where w(x) is the distribution matrix for the disturbance. It will be assumed that
f(x), g(x), w(x) and h(x) are smooth functions. The plant parameters bi, which
are assumed to be unknown, belong to a known region i.e., bi ∈ [bi,min, bi,max]. In the
present analysis, we focus our attention to the class of systems for which the following
assumption holds.
A1: System (5.1) has a well-defined relative degree m with respect to the output
y = h(x) such that there exists a diffeomorphism T (x) which transforms the x to
[ζ, η]′ coordinates as follows
ζ
η
=
ζ1
ζ2...
ζm
η1
...
ηn−m
= T (x) =
h(x)
Lfh(x)...
Lm−1f h(x)
T1(x)...
Tn−m(x)
(5.2)
98
Thus, the dynamics can be rewritten as
ζ1 = ζ2
ζ2 = ζ3...
ζm = Lmf h(x) + LgL
m−1f h(x)[b1u1 + · · ·+ bquq] + LwL
m−1f h(x)d(t)
η = fη(ζ, η, d(t)) (5.3)
where it is assumed that the disturbance distribution matrix is such that LwLjf = 0
for all j = 1, ..., m− 2 and LwLm−1f 6= 0, and Lm
f h(x) 6= 0, for all x ∈ Rn.
In the present work, we will also make the following assumption about the zero-
dynamics.
A2: The η-dynamics is input to state stable (ISS) with respect to ζ, η, d(t).
Assumption A2 guarantees the boundedness of all closed-loop signals when a stable
controller can be designed for the ζ-dynamics.
In this work, we will consider actuator faults which can be modeled as
uj(t) =
uj, ∀t ≥ Tf ,
if jth actuator gets stuck at Tf
ηjju∗j(t), ∀t ≥ Tf ,
if jth actuator loses efficiency at Tf
(5.4)
where u∗j(t) represents the control command to the jth actuator, uj is an unknown
constant value at which the actuator gets stuck, Tf is the unknown instant of failure
and ηjj represents actuator loss in efficiency with ηjj ∈ [(ηjj)min, 1], (ηjj)min ≥ 0. It
will be assumed that uj belongs to a known interval i.e., uj ∈ [uj,min, uj,max].
For the system described by (5.1), subjected to unknown actuator faults (5.4)
and bounded disturbances, the objective is to design a control law such that the
output tracking error converges to a sufficiently small neighborhood of zero where the
controller is unsaturated, and has desired closed-loop performance and disturbance
attenuation properties within the unsaturated region. It is also desired that the
tracking error asymptotically converges to zero in absence of disturbances.
99
5.3 Adaptive Robust Actuator Fault-Tolerant Control
Denote fζ(x) = Lmf h(x), gζ(x) = LgL
m−1f h(x) and wζ = LwL
m−1f h(x). In the
present work, we will assume that control commands to all the actuators are the
same i.e., u∗1 = · · · = u∗q = u0. With this control input, and fault model described by
(5.4), the healthy and faulty actuators can be parameterized in the following way
uj(t) = ηjj(1 − σjj)u0(t) + σjj uj
⇒q∑
j=1
bjuj(t) =
(
q∑
j=1
bjηjj(1 − σjj)
)
u0(t) +
(
q∑
j=1
bjσjjuj
)
⇒q∑
j=1
bjuj(t) = κu0(t) + µ (5.5)
where
σjj =
0 before jth actuator gets stuck
1 after jth actuator gets stuck
ηjj =
1 before jth actuator loses efficiency
[(ηjj)min, 1] after jth actuator loses efficiency
κ =
q∑
j=1
bjηjj(1 − σjj), µ =
q∑
j=1
bjσjj uj
We will make the following practical assumption regarding the extent of uncertainties
present in the system.
A3: The unknown parameters θ , [κ, µ]T and disturbance satisfy,
θ ∈ Ωθ , θ : θmin ≤ θ ≤ θmax (5.6)
wζ(x)d(t) ∈ Ωd , wζ(x)d(t) : |wζ(x)d(t)| ≤ dM (5.7)
Now, that we have established a parametric fault-model, we are ready to present
the bounded control laws to be used in the backstepping based design. We will use
100
the coordinate transformation xi = ζi − y(i−1)d to simplify the analysis. The designed
virtual control law αi and the corresponding error zi are given by
zi = xi − αi−1(zi−1),
αi = −σi(zi),
˙xi = zi+1 + αi, i = 1, ..., n− 1 (5.8)
where σi(zi) are saturation functions (see figure (5.1)). The saturation functions used
in this work play a significant role in guaranteeing the overall system stability. In fact,
it is only through a careful and rigorous analysis of the effects of saturation function
parameters on system dynamics that a set of conditions will be derived, which will
ensure all desired objectives can be achieved. The saturation functions are defined as
(a) ziσi(zi) ≥ 0, ∀zi (5.9)
(b) σi(zi) = kizi, ∀|zi|≤li,
σi(zi) = Mi, ∀|zi|≥Li (5.10)
(c) |σi(zi)| ≤Mi, ∀zi (5.11)
(d)∂σi
∂zi
≤ ki, ∀zi (5.12)
Also, li = βiLi with βi ≤ 1, and Mi = kili(1 + γi) with γi > 0. The interval for
zi is divided into three different regions - Ωi1 = zi : |zi| ≤ li, Ωi
2 = zi : |zi| ≤Li and Ωi
3 = zi : |zi| > Li. Note that the nonlinear transition region of the
saturation function (Ωi2\Ωi
1) needs to be at least second order differentiable, as the
backstepping design involves taking derivatives of σi. σm, however, need not have a
smooth transition region, as this appears in the last step. The lm and Lm parameters
used in the definition of σm depend on the extent of uncertainties, and will be defined
later.
101
Figure 5.1. Saturation function
102
Now, substituting (5.8) in (5.3), the error dynamics can be written as
z1 = z2 − σ1(z1)
...
zi = zi+1 − σi(zi) +
i−1∑
j=1
[
j∏
r=1
∂σi−r
∂zi−r
]
(zi−j+1 − σi−j(zi−j))
(5.13)
...
zm = fζ(x) + κgζ(x)u0 + µgζ(x) + wζ(x)d(t) − y(m)d
+m−1∑
j=1
[
j∏
r=1
∂σm−r
∂zm−r
]
(zm−j+1 − σm−j(zm−j))
(5.14)
Remark 1. Note that the form of the virtual control laws designed is signifi-
cantly different, and at the same time much simpler than the typical “cancellation”
backstepping design. In cancellation backstepping design, the detrimental effect of
the virtual control law of step i − 1 i.e., αi−1 is completely canceled at step i, by
incorporating the appropriate terms in the design of αi. This results in a simple
linear dynamics for the error variables, such that the closed-loop poles can be placed
at any desired location. But, this simplicity comes at the expense of a complicated
control law, with exponentially growing terms as the order of the system increases.
In the context of bounded control, it is extremely difficult to show the boundedness
of such a control law. For chain of integrators, as we shall see, the error dynamics can
be stabilized with a much simple control law σi(zi), and without canceling the terms
resulting from the derivative of αi−1. Although linear, the resulting error-dynamics
without cancellation leads to a set of constraints that the controller parameters must
satisfy for global stabilization. This makes the parameter selection process slightly
more complicated than the cancellation backstepping design. Next, we present the
adaptation mechanism used in the present work.
103
5.3.1 Parameter Estimation
We will use x-swapping lemma (Ch.6, [31]) to implement least-squares estimation
scheme, and then use discontinuous projection to ensure that the parameters stay
within a known region in presence of disturbances. Note that as all unknown param-
eters appear in the mth channel in the ζ coordinates, we need only ζm dynamics for
estimation purposes.
The ζm-dynamics can rewritten as
ζm = fζ(x) + κgζ(x)u0 + µgζ(x) + wζ(x)d(t)
= fζ(x) + φ(x, u)Tθ + wζ(x)d(t) (5.15)
where θ , [κ, µ]T and φ(x, u)T = [gζ(x)u0, gζ(x)]. Following the standard steps of
x-swapping, we define the following filters
Ω0 = A(Ω0 + ζm) − fζ(x) (5.16)
ΩT = AΩT + φ(x, u)T (5.17)
Define the prediction error as ǫ = ζm + Ω0 − ΩT θ, which is calculable. It is shown
in [31] that ǫ can be rewritten as
ǫ = ΩT θ + ǫ (5.18)
where ǫ is governed by ˙ǫ = Aǫ, which exponentially converges to zero. Thus, we have
a static model (5.18), that is linearly parameterized in terms of θ with an additional
term ǫ which exponentially decays to zero. With this static model, various estimation
algorithms can be used to estimate the unknown parameters. In the following, we
present the least-squares estimation scheme which will be used in the present work.
˙θ = Γτ
τ =Ωǫ
1 + νtrΩT ΓΩ
Γ = −ΓΩΩT
1 + νtrΩT ΓΩΓ, Γ(0) = Γ(0)T > 0, ν ≥ 0 (5.19)
104
In order to guarantee certain desired properties, we will use the following discon-
tinuous projection algorithm.
˙θ = Projθ(Γτ) (5.20)
Projθi=
0 if θi = θi,max and •i > 0
0 if θi = θi,min and •i < 0
•i otherwise
(5.21)
This guarantees that the parameters do not drift away and stay within known bounded
region even in presence of disturbances. Additionally, we will use the following rate-
limiting projection scheme
satθM(ξ) = s0ξ, s0 =
1, ‖ξ‖ ≤ θM
θM
‖ξ‖‖ξ‖ > θM
(5.22)
where θM is a pre-set rate limit. Thus, the final parameter update law used takes the
following form
˙θ = satθM
(Proj(Γτ)) , θ(0) ∈ Ωθ (5.23)
Using the properties of the projection operator in Lemma E.1 in [31], and noting
that s0 is a positive scalar, it is to verify that the following desirable properties hold
P1 θ ∈ Ωθ = θ : θmin ≤ θ ≤ θmax (5.24)
P2 θT (Γ−1Projθ(Γτ) − τ) ≤ 0, ∀τ (5.25)
P3 ‖ ˙θ‖ ≤ θM , ∀t (5.26)
Property P1 guarantees that the estimates stay within a known bounded region, and
P3 guarantees that the parameter update rate is uniformly bounded.
5.3.2 Controller design
In this section, we present the proposed adaptive robust fault-tolerant controller
and prove the overall stability of the system using the following steps:
105
1. In the first step, it is shown that for any set of initial conditions zi(0), all
error-variables can be driven to an invariant region where the controller is un-
saturated, as long as a set of controller parameters exist which satisfy certain
inequalities.
2. Next, sufficient and necessary conditions for the existence of the controller pa-
rameters is proposed and proved.
3. In this step, a recursive root-locus design is proposed such that once the con-
troller is unsaturated, desired closed-loop performance e.g., disturbance atten-
uation and desired transients can be achieved.
4. In the last step, asymptotic convergence of the adaptive system is proved in
absence of disturbances within the unsaturated region.
Define ua = −fζ(x) − µgζ(x). Control law to be used
u0 = σm
[
1
gζ(x)κ(σa(ua) + y
(m)d − kmzm)
]
(5.27)
where
σa(ua) =
ua, for |ua|≤Ma
sign(ua)Ma, for |ua| > Ma
and
σm(u0) =
u0, for |u0|≤uM
sign(u0)uM , for |u0| > uM
Note that as system (5.3) may not necessarily be stable, the model-compensation
component ua can easily become unbounded. In the context of bounded control,
some assumptions must be made on the growth rate of the nonlinearities to make the
stabilization/tracking problem feasible.
A4: In the present work, it will be assumed that the nonlinearities and the esti-
mated parameters are such that
(i) when the model-compensation is saturated i.e., σa(ua) = Ma, the difference can
106
be bounded above by a known constant i.e., |σa(ua) − ua| ≤ uaM
(ii) the discrepancy due to parameter estimation mismatch is bounded by a known
constant i.e., |φT (x, u)θ| ≤ hM .
(iii) gζ(x) is such that gζ,min ≤ |gζ(x)|≤gζ,max.
Consider the ζ−dynamics of system (5.3) along with assumptions (A1-A4). The
following theorem states that in spite of unknown actuator faults (5.5), the error
dynamics can be driven to small neighborhood around zero, where the controller is
unsaturated and desired closed-loop performance can be recovered.
Theorem 1. Consider the error-dynamics represented by (5.14). With the control
law given by (5.27), and the chosen parameter update law (5.23), if a set of controller
parameters can be chosen such that
kili > li+1 + ki−1Ni, i = 1, ..., m− 1 (5.28)
kmlm > km−1Nm + hM + dM + uaM , (5.29)
kmlm ≤ κmingζ,minuM − (Ma + λM) (5.30)
where
Ni = Li +Mi +
i−2∑
j=1
[
(
j∏
r=1
ki−1−r)(Li−j −Mi−1−j)
]
and k0 = 0 then, the error variables zi reach a region where the controller is unsat-
urated in a finite time (i.e., z ∈⋂m
j=1 Ωj1), for any set of initial conditions and any
fault pattern.
Proof. Consider the following claims
Claim 1. u0 is unsaturated for
|zm| ≤ lm ,gζ,minκminuM − (Ma + λM)
km
(5.31)
and saturated for
|zm| ≥ Lm ,gζ,maxκmaxuM + (Ma + λM)
km
(5.32)
107
Claim 2. For any initial condition zm(0),
(a) if zm(0) ∈ Ωm3 i.e., |zm(0)| > Lm, then it reaches Ωm
2 in finite time.
(b) if zm(0) ∈ Ωm2 \Ωm
1 i.e., lm < |zm(0)|≤Lm, then it reaches Ωm1 in finite time.
(c) if zm(0) ∈ Ωm1 , then zm(t) ∈ Ωm
1 , ∀t > 0.
Claim 3. Assume zi+1≤li+1, then
(a) if |zi(t0)| > Li, then it reaches Ωi2 in a finite time
(b) if zi(t0) ∈ Ωi2\Ωi
1, then it reaches Ωi1 in a finite time
(c) if zi(t0) ∈ Ωi1, then zi(t) ∈ Ωi
1 ∀ time t > t0
Assume all the claims are true. The first claim establishes the bounds for zm such
that the control law given by (5.27) is unsaturated/saturated. Then, from the second
claim, we have zm≤lm after a finite time, and it can never leave this set. Theorem 1
then follows from the recursive application of claim 2 to the intermediate states zj ,
j = m− 1, ..., 1.
Proof of Claim 1. Assume w.l.o.g that zm > 0. Then, for zm≤lm, from (5.31)
we have the following inequalities
kmzm ≤ kmlm ≤ gζ,minκminuM − (Ma + λM) ≤ gζ(x)κuM − (Ma + λM)
⇒ kmzm +Ma + λM
gζ(x)κ≤ uM
⇒ kmzm − σa(ua) + y(m)d
gζ(x)κ≤ uM
⇒ u0 ≤ uM
Thus, the controller is unsaturated. In order to show that the controller is saturated
for zm > Lm, consider the following inequalities
kmzm ≥ kmLm ≥ gζ,maxκmaxuM + (Ma + λM)
⇒ kmzm − (Ma + λM) ≥ uMgζ(x)κ
⇒ kmzm − σa(ua) + y(m)d
gζ(x)κ≥ uM
Thus, the control law u = u0 is saturated.
108
Proof of Claim 2, part (a). Assume w.l.o.g zm(0) > 0. Then, we have the
following inequalities for zm ∈ Ωm3
˙xm = fζ(x) + κgζ(x)u0 + µgζ(x)
−y(m)d + φT (x)θ + wζ(x)d(t)
= −ua − y(m)d + φT (x)θ + wζ(x)d(t) + κgζ(x)u0
≤ Ma + λM + hM + dM − κmingζ,min(x)uM
⇒ xm(t) ≤ xm(0) − (κmingζ,min(x)uM
−(Ma + λM + hM + dM))t
⇒ zm(t) + αm−1(t) ≤ zm(0) + αm−1(0) − (κmingζ,min(x)uM
−(Ma + λM + hM + dM))t
Using |αm−1(t)| ≤ Mm−1 and substituting zm(t) = Lm, an upperbound for the time
taken to reach Ωm1 can be found as
tm32 ≤zm(0) − Lm + 2Mm−1
(κmingη,min(x)uM − (Ma + λM + hM + dM))(5.33)
From (5.29) and (5.30) we also have
κmingζ,minuM − (Ma + λM) ≥ km−1Nm + hM + dM + uaM
⇒ κmingζ,minuM ≥ hM + dM +Ma + λM .
Thus, zm reaches Ωm2 in a finite time.
Proof of Claim 2, part (b). Note that when zm ∈ Ωi2\Ωi
1, the control law
can be saturated or unsaturated depending on the relative magnitudes of the various
uncertainties (e.g., φT θ, wζ(x)d(t)), the model-compensation (ua) and the robust
component (−kmzm). However, the error-variables can be made to converge in this
region, as long as conditions (5.29) and (5.30) are satisfied. Both the cases i.e.,
109
|u0| > uM and |u0|≤uM , will be considered separately. Assume w.l.o.g zm > 0. The
derivative of zm can be written as
zm = fζ(x) + µgζ(x) − ymd + κgζ(x)u0
+ φT (x)θ + wζ(x)d(t) +
m−1∑
j=1
[
j∏
r=1
∂σm−r
∂zm−r
]
(zm−j+1 − σm−j(zm−j))
≤ −ua − y(m)d + κgζ(x)u0 + km−1Nm + hM + dM (5.34)
Case A. u0 is not saturated.
In this case, substituting (5.27) in (5.34) and using condition (5.29), we obtain
zm ≤ σa(ua) − ua − kmzm + hM + dM + km−1Nm
≤ −kmlm + km−1Nm + hM + dM + uaM
< 0
Thus, zm is negative in this case.
Case B. u0 is saturated.
Substituting u0 = uM , we get the following inequalities from (5.34)
zm ≤ −ua + κgζ(x)u0 + km−1Nm + hM + dM + λM
≤ Ma − κmingζ,minuM + km−1Nm + hM + dM + λM
For zm to be negative, we must have
uM >km−1Nm +Ma + hM + dM + λM
κmingζ,min(5.35)
Note that we have following inequalities from (5.29) and (5.30)
κmingζ,minuM − (Ma + λM) > kmlm > km−1Nm + hM + dM + uaM
⇒ uM >km−1Nm + hM + dM + λM +Ma
κmingζ,min(5.36)
Thus, condition (5.35) is automatically satisfied. Thus, zm is negative for zm ∈ Ωi2\Ωi
1,
irrespective of whether the control input is saturated or not.
110
Proof of Claim 2, part (c). For Ωm1 to be invariant, zmzm should be negative
at zm = lm. Note that from (5.29), there exists a δm such that
km(lm − δm) = km−1Nm + hM + dM + uaM (5.37)
Now, substituting zm = lm in zm, and using (5.37), we get
zm = −kmzm + σa(ua) − ua + φT (x)θ + wζ(x)d(t)
+m−1∑
j=1
[
j∏
r=1
∂σm−r
∂zm−r
]
(zm−j+1 − σm−j(zm−j))
⇒ zmzm ≤ |zm|(−kmlm + km−1Nm + uaM + hM + dM)
≤ −km|zm|δm < 0 (5.38)
Thus, any trajectory starting in Ωm1 will remain in this set.
Proof of Claim 3, part (a). When zi ∈ Ωi3 i.e., |zi| > Li, we have αi = −Mi
(w.l.o.g assume zi > 0) from the definition of σi. Using the assumption of claim 2
i.e., |zi+1|≤li+1 and substituting αi = −Mi in (5.8), we obtain
˙xi = zi+1 −Mi
≤ li+1 −Mi = −(Mi − li+1)
⇒ xi(t) ≤ xi(t0) − (Mi − li+1)(t− t0)
⇒ (Mi − li+1)(t− t0) ≤ xi(t0) − xi(t) + αi−1(t0) − αi−1(t)
⇒ ti32 ≤ xi(t0) − Li + 2Mi−1
Mi − li+1(5.39)
where in the last step we substituted zi(t) = Li and ti32 , t − t0, the time taken to
reach Ωi2 from Ωi
3. This proves the first part of the claim.
Proof of Claim 3, part (b). To prove the next two parts, we proceed as done
in the proof of first claim. From (5.28), there exists a δi such that
ki(li − δi) = li+1 + ki−1Ni (5.40)
111
Without loss of generality, assume zi(t0) > 0. Then, from the zi-dynamics (5.13), we
have
zi = zi+1 − σi(zi) +i−1∑
j=1
[
j∏
r=1
∂σi−r
∂zi−r
]
(zi−j+1 − σi−j(zi−j))
≤ li+1 − kili +i−1∑
j=1
[
(
j∏
r=1
ki−r)(Li−j+1 +Mi−j)
]
≤ li+1 − kili + ki−1Ni (5.41)
= −kiδi
⇒ ti21 ≤ Li − likiδi
(5.42)
where we have used the fact that Li − li≥|zi(t) − li|, ∀ zi(t) ∈ Ωi2\Ωi
1. Thus, there
exists a finite time-interval ti21, within which any trajectory starting in Ωi2\Ωi
1 reaches
Ωi1. This completes the proof of second part of the claim.
Proof of Claim 3, part (c) In order to show that any trajectory starting in
Ωi1(t0) stays there ∀ t > t0, we need to show that zizi < 0 at zi = ±li. For zi = li, we
know σi(zi) = kili and using the assumption |zi+1|≤li+1, from (5.13) the zi-dynamics
can be written as
zizi = zi
(
zi+1 − σi(zi) +
i−1∑
j=1
[
j∏
r=1
∂σi−r
∂zi−r
]
(zi−j+1 − σi−j(zi−j))
)
≤ |zi|(
li+1 − kili +
i−1∑
j=1
[
(
j∏
r=1
ki−r)(Li−j+1 +Mi−j)
])
≤ −ki|zi|δi (5.43)
This shows that Ωi1 is positively invariant, and also completes the proof of the theorem
1.
5.3.3 Controller Parameter Selection
There are two important questions which need to be answered next. First, when
can the existence of a solution to inequalities (5.28)-(5.30) be guaranteed. Note that
even when such a solution exists, it only guarantees the convergence of all intermediate
112
error variables to the linear unsaturated region (an invariant set in the neighborhood
of zero). The system response within the linear unsaturated region is governed by the
gains ki, and leads to the second important question - how to select the gains such
that the desired closed-loop performance can be achieved, without violating (5.28)-
(5.30). It is natural to first select a set of gains ki such that desired performance
can be achieved, and then seek a set of lis such that inequalities (5.28)-(5.30) can
be satisfied. However, such an approach may not work as after fixing the gains ki,
solving the inequalities (5.28)-(5.30) may result in negative lis, which will make the
control law unrealizable. Hence, in the following section, we first state and prove
the main result regarding the necessary and sufficient condition for the existence of a
solution to the inequalities. Then, a systematic way of choosing the controller gains in
accordance with this theorem is proposed, such that desired closed-loop performance
can be achieved.
Necessary and Sufficient Conditions for the Existence of Controller Pa-
rameters
After a series of derivations, (5.28)-(5.29) can be rewritten in a matrix form
AL < D, (5.44)
where L = [l1, l2, · · · , lm−1, lm]T , D = [0, 0, · · · , 0,−(hM + dM + uaM)]T . And A is a
function of ki given by
A =
−k1 1 0 · · · 0
k21(1 + γ1)
(
k1
β2− k2
)
1 · · · 0
· · · · · · · · · · · · · · ·am1 am2 am3 · · ·
(
km−1
βn− km
)
, (5.45)
113
where
aij = ki−1kj
i−j−1∏
r=1
ki−r−1(1 + γj) + ki−1
i−j∏
r=1
ki−r−11βj, ∀i > j.
aij = −ki + ki−1
βi, ∀i = j
aij = 1, ∀j = i+ 1
aij = 0, ∀j > i+ 1
(5.46)
For any γi > 0 and 0 < βi ≤ 1, if we fix a set of positive kis, then the control
law is feasible if and only if the there exist l1, l2,..., lm > 0 such that (inequality3)
and (5.44) are satisfied. In other words, at least one solution to (inequality3) and
(5.44) should lie in the region (l1, · · · , lm) : li > 0. The following theorem gives
the necessary and sufficient condition for kis such that the control law is feasible.
Theorem 2. For any γi > 0 and 0 < βi ≤ 1, with a set of positive kis, at least
one solution to (inequality3) and (5.44) lie in the region L ∈ (l1, · · · , lm) : li > 0iff the kis satisfy the following set of inequalities:
k1 > 0,
k2 >a21p1+
k1β2
p2
p2= (1 + γ1 + 1
β2)k1,
· · ·
ki >∑i−1
j=1aijpj+
ki−1
βipi
pi, ∀i < m
· · ·
kn > uM−(Ma+λM )uM−(Ma+hM+uaM+λM+dM )
·∑n−1
j=1anjpj+
kn−1
βnpn
pn,
(5.47)
where, the coefficients pis are computed recursively using the formula
p1 = 1
pi = −∑i−1j=1 ai−1jpj
(5.48)
Proof. =⇒ (Necessary condition):
Assume there exists a solution satisfying (5.44) and (5.30), which lies in the region
L ∈ (l1, · · · , lm) : li > 0. Denote it by
L′ ∈ (l′1, · · · , l′m) : l′i > 0 (5.49)
Let us first prove the following claim using mathematical induction.
114
Claim 3.
pil′i−1 > pi−1l
′i
pi > 0,(5.50)
for all i ≤ n.
Proof of Claim 3.
(i) Looking at the first row of AL′ < D, we have −k1l′1 + l′2 < 0. From the definition
of p1 and p2 given by (5.48), and noting that a11 = −k1, we have p2l′1 > p1l
′2. Since
l′1, l′2 > 0 and p1 > 0, we obtain p2 > 0. Thus, the claim is true for j = 2.
(ii) Suppose the claim is true for all 2 ≤ j < i. Then, we have
l′j >pjl
′j+1
pj+1>pjpj+1l
′j+2
pj+1pj+2> · · · > pjl
′i−1
pi−1(5.51)
Now, looking at the i− 1 row of AL′ < D, and using aij > 0, ∀i > j, we get
i−1∑
j=1
ai−1jl′j + l′i < 0
⇒i−1∑
j=1
ai−1jpj l′i−1
pi−1+ l′i < 0
⇒ −pil′
i−1
pi−1+ l′i < 0, (using (5.48))
⇒ pil′i−1 > pi−1l
′i.
(5.52)
Since l′i−1, pi−1, l′i > 0, we get pi > 0. Thus, the claim is also true for j = i.
This procedure can be continued until i = m, as the R.H.S of the first m− 1 rows
of AL′ < D are all zeros. This completes the proof of the claim.
Next, it will be shows that the condition on gains, given by (5.47), follows from
(5.44) through the use of claim 3.
For each i < n, from the i-th row of AL′ < D we have
i−1∑
j=1
aijl′j +(
ki−1
βi− ki
)
l′i + l′i+1 < 0
⇒i−1∑
j=1
aijpj l′ipi
+(
ki−1
βi− ki
)
l′i + l′i+1 < 0, (l′j >pj l′ipi, ∀j < i using claim 3)
⇒i−1∑
j=1
aijpj l′ipi
+(
ki−1
βi− ki
)
l′i < 0
⇒[
i−1∑
j=1
aijpj
pi+(
ki−1
βi− ki
)
]
l′i < 0
⇒ ki >∑i−1
j=1aijpj+
ki−1
βipi
pi
(5.53)
115
From the last row of AL′ < D, we get
m−1∑
j=1
amjl′j +(
km−1
βm− km
)
l′m < −(uaM + hM + dM)
⇒m−1∑
j=1
amjpj l′mpm
+(
km−1
βm− ki
)
l′m < −(uaM + hM + dM)
⇒[
km −m−1∑
j=1
amjpj
pm− km−1
βm
]
l′m > (uaM + hM + dM)
(5.54)
Applying (5.30), which is uM − (Ma + λM) ≥ kml′m, to the above inequality and
eliminating l′m:
(
km −m−1∑
j=1
amjpj
pm− km−1
βn
)
[uM − (Ma + λM)] ≥ km(uaM + hM + dM)
⇒ km > uM−(Ma+λM )(uM−(Ma+hM+uaM+λM +dM ))
·∑m−1
j=1amjpj+
km−1
βmpm
pm,
(5.55)
which completes the proof of necessary condition.
⇐= (Sufficient condition):
Suppose that kis satisfy inequalities (5.47). Then, for i ≤ m− 1,
ki >∑i−1
j=1aijpj+
ki−1
βipi
pi
⇒∑i−1
j=1aijpj+
ki−1
βipi−kipi
pi< 0
⇒ −pi+1
pi< 0, ( using definition of pi and aii = −ki + ki−1
βifrom(5.46))
⇒ pi+1
pi> 0.
(5.56)
Since p1 = 1 > 0, we have pi > 0, ∀i ≤ m.
Next, we are interested in finding the solution of the linear matrix equation AL′ =
D. Using the first m− 2 algebraic equations of AL′ = D, we can represent all the l′is
with i > 1 in terms of l′1:
l′1 = p1l′1
l′2 = p2l′1
· · ·l′m = pml
′1.
(5.57)
116
Substituting them into the last equation of AL′ = D, we have:
m−1∑
j=1
amjpjl′1 + (km−1
βm− km)pml
′1 = −(uaM + hM + dM)
⇒ l′1 = −(uaM+hM+dM )m−1∑
j=1
amjpj+(km−1
βm−km)pm
.(5.58)
From the last equation of (5.47), since 0 < (uM − (Ma + uaM + hM + λM + dM)) <
uM − (Ma + λM), we have
km >∑m−1
j=1amjpj+
km−1
βmpn
pm
⇒m−1∑
j=1
amjpj + (km−1
βm− km)pm < 0.
(5.59)
Thus, l′1 > 0. As a result, all the l′is are greater than zero. Multiplying both sides of
(5.47) by (uM − (Ma + uaM +hM +λM +dM))pm, and rearranging the terms to factor
out uM , we obtain
kml′m = kmpml
′1 =
−(uaM + hM + dM)knpn
m−1∑
j=1
amjpj + (km−1
βm− km)pm
< uM − (Ma + λM) (5.60)
Thus, we have shown that the only solution to the linear matrix equation AL′ = D
lie in the region (l′1, · · · , l′m) : l′i > 0, kml′m < uM which is an open set. Since A
is a m by m matrix, the solution to inequality AL < D must exist and is a a sector
region with the acme being the point L′ which solves equation AL′ = D. Then, we
can always pick a solution of AL < D to be arbitrarily close to L′ such that L also lies
in the open region (l′1, · · · , l′m) : l′i > 0, kml′m < uM − (Ma + λM). After choosing
that L, we have
(i) AL < D;
(ii) kmlm < uM − (Ma + λM);
(iii) all the lis are greater than zero.
Thus, there exists at least one solution satisfying (5.28)-(5.28), which lies in the
region (l′1, · · · , l′m) : l′i > 0. This proves the sufficiency condition of the theorem.
117
Remark 3. Condition (5.47) plays an important role in the selection of controller
parameters kis, and can be explained as follows. It can be easily shown that the
closed-loop transfer function from d(t) to z1 (which is also the tracking error y − yd)
inside the linear unsaturated region Ω11 is
[
z1(t)
d(t)
]
=1
sm + knsm−1 + kmkm−1sm−2 + · · ·+m∏
j=2
kj · s +m∏
j=1
kj
. (5.61)
Due to the conditions imposed by (5.47), the closed-loop poles of (5.61) cannot be
assigned arbitrarily. However, as pi depends only on kjs for j < i, from (5.47) we can
easily work out a recursive way of choosing the controller parameters. Specifically, we
first choose k1, and then choose k2 large enough such that the first inequality of (5.47)
is satisfied. Continuing in this fashion, we can get a set of kis that are permissible.
In the next subsection, we clearly outline such a technique for choosing kis such that
all closed-loop requirements are met, without violating the constraints imposed by
(5.47).
Controller Gain Selection: A Recursive Root-Locus Design
As mentioned above, the closed-loop poles of (5.61) cannot be arbitrarily assigned
because of the constraints (5.47) on the controller parameters ki. However, (5.47)
implies that ki can be chosen arbitrarily large, which makes it possible to satisfy the
closed-loop performance requirement. In the linear unsaturated region, the desired
closed-loop performance can be achieved by placing the poles sufficiently far to the
left of imaginary axis e.g., if the required steady-state error is δ, and the transient
response criteria dictates that the slowest closed-loop pole be to the left of p0, then
it is sufficient to place all the poles to the left of pcl = min−√
dM
δ,−p0. In the
following, we propose a recursive root locus design to meet the above requirement.
• Step 1: Select k1 > −pcl, then the root of the equation s+ k1 = 0 lie to the left
of s = pcl.
118
• Step 2: Let the virtual open-loop system be k2s+k1
s2 , then the virtual closed-loop
characteristic equation is s2 + k2s + k1k2 = 0. To determine k2, draw the root
locus of k2s+k1
s2 . This open-loop system has two poles at origin and one zero at
−k1, left to s = pcl. From the general guidelines for drawing a root-locus, there
exists a k2 large enough such that: 1) the first inequality of (5.47) is satisfied
and 2) all the roots of s2 + k2s + k1k2 = 0 lie to the left of s = pcl, on the real
axis.
.............
.............
.............
• Step m: Let the virtual open-loop system be km
sm−1+km−1sm−2+···+m−1∏
j=1
kj
sm , then
the virtual closed-loop characteristic equation is exactly the same as that of the
actual system, i.e., kmsm−1 + kmkm−1s
m−2 + · · ·+m∏
j=1
kj . To determine kn, draw
the root locus of km
sm−1+km−1sm−2+···+m−1∏
j=1
kj
sm . This open-loop system has m poles
at origin and m− 1 zeros to the left of s = pcl. As the difference in the number
of poles and zeros is one, there always exists an asymptote along the negative
real axis. Thus, for sufficiently large gain km, there is a branch of the root-locus
on the negative real axis. This implies (i) the last two inequalities of (5.47) are
satisfied and (ii) all the closed-loop poles lie to the left of s = pcl on the real
axis.
Thus, we can choose the controller gains such that the desired closed-loop perfor-
mance is achieved, as well as the conditions imposed for the existence of a feasible
control law given by (5.47) are also satisfied simultaneously.
5.3.4 Asymptotic Stability
In the linear unsaturated region, the dynamics can be represented as
119
z1...
zi
...
zm
=
−k1 1 · · · 0...
... · · · ...
−ki−1...k21 −ki−1...k2(k2 − k1) · · · 0
...... · · · ...
−ki−1...k21 −km−1...k2(k2 − k1) · · · −(km − km−1)
z1...
zi
...
zm
+
0...
0
1
(φ(x, u)T θ + wζ(x)d(t))
⇒ z = Aclz +Bθθ +Bdwζ(x)d(t) (5.62)
where
Bθ =
0 0...
...
0 0
φ1(x, u) φ1(x, u)
, Bd =
0...
0
1
From the preceding discussion, we know Acl is stable and ‖φ(x, u)T‖ is bounded, as
|gζ(x)| is bounded below and above by known positive constants, |u| ≤ uM . Thus, it
can be easily verified using a Lyapunov function Vz = zT z that z(t) is Input to State
Stable (ISS) with respect to the inputs θ and wζ(x)d(t). Now we are ready to state
an important result which shows that the proposed controller preserves the desired
property of an adaptive controller - asymptotic tracking in presence of parametric
uncertainties only.
Theorem 3. In presence of parametric uncertainties only i.e., when d(t) = 0, by
using the control law given by (5.27), and the parameter update law (5.23) along with
filters (5.16-5.17), asymptotic output tracking is also achieved i.e., z → 0 as t→ ∞.
Proof. The proof follows from the fact that the controller is ISS w.r.t θ and θ ∈ L∞
for least-square estimation using x-swapping lemma (see (Ch.6, [31])). This, in turn
implies that z ∈ L∞, and from Barbalat’s lemma, we obtain z → 0 as t→ ∞.
120
5.4 Simulation Example: Nonlinear Hypersonic Aircraft Model
An adaptive robust control (ARC) based approach was recently proposed to solve
the unknown actuator fault accommodation for linear [56] and nonlinear [57] systems.
The superior performance of an ARC based approach in achieving desired transient
response, as well as small steady-state tracking error over a robust adaptive control
based design was demonstrated through comprehensive simulation studies. In the
present work, we compare the performances of the proposed scheme and the recently
developed ARC based fault-tolerant controller in presence of saturation. A nonlinear
longitudinal model of hypersonic aircraft cruising at a velocity of 15 mach, at an
altitude of 110, 000 feet is used to test the effectiveness of the proposed scheme.
Nominal model of the system is
α = q − γ
q =Myy
Iyy
γ =L+ T sinα
mV− (µ− V 2r) cos(γ)
V r2(5.63)
where
α = angle of attack, rad
γ = flight-path angle, rad
V = velocity, ft/sec
q = pitch rate, rad/sec
T = thrust, lbf
L = lift, lbf
Myy = pitching moment, lbf.ft
Iyy = moment of inertia, slug.ft2
Further, details of the model can be found in [37]. Note that the nominal model of
the system does not take into account any unstructured modeling uncertainties and
121
external disturbances. As modeling uncertainties are inherent to any realistic system
model, unmatched uncertainties will be introduced in order to make the simulation
studies more meaningful. The state-space representation is given by
x1 = x2 + a1y + a2 sin(y) + a3y2 sin(y) + a4 cos(x3)
x2 = a5y2 + a6y + (a7 + a8y + a9y
2)x2 + b1u1 + b2u2 + ∆(t)
x3 = a10 cos(x3) − a1y − a2 sin(y)
y = x1 (5.64)
where [x1, x2, x3] = [α, q, γ] and ∆(t) = 0.02 sin(3t) represents the input disturbance.
The nominal plant parameters are
a1 = −0.0427, a2 = −3.4496×10−4, a3 = 5×10−5, a4 = 0.0014,
a5 = −4.2006, a6 = 1.0821, a7 = −3.6896, a8 = 0.1637,
a9 = −0.1242, a10 = 0.0014, b1 = 0.8, b2 = 0.8
The initial conditions are set to x(0) = [0, 0.01, 0]T .
In ARC based technique, once the controller saturates, it cannot return to the un-
saturated region of controller operation. On the other hand, in the proposed design,
the controller saturates temporarily, but returns to the unsaturated region. This can
be explained as follows. Following an actuator fault, the performance requirements
necessitate that the transients be suppressed using large feedback action. Note that
undesirable transients can increase the demanded control input on two accounts: first,
it induces a large feedback action due to the proportional type controller i.e., −kizi,
and second, since the model-compensation depends on z as well i.e., ϕ(x) = ϕ(xd+z),
as z increase, ϕ(x) may increase correspondingly. However, as the control input is
limited, the transients cannot be suppressed effectively after model-compensation,
which further increases the transient error. This, in turn, demands larger control
input, leading to a controller saturation scenario, which is unsalvageable. In the pro-
posed approach, on the other hand, when the error is large, we sacrifice the model
compensation to certain extent, and use the available control input to supply maxi-
122
mum possible feedback action −kizi, such that zi can be made negative, and the error
can be reduced to an extent which allows the controller to be unsaturated. This can
also reduce the required model-compensation.
As can be seen from fig. 1, in the ARC based approach, once the controller gets
saturated, the adaptation mechanism breaks down, and the estimated parameter
assumes the minimum or maximum value used in the projection algorithm. This is
not surprising, as the adaptation mechanism is driven by the error, which has now
two sources - mismatch in estimated parameters, and error due to the saturation
effect. But, as in the proposed approach an indirect scheme is used, the model
structure does not change whether the controller is saturated or not, and this results
in accurate estimation.
123
0 20 40 60 80 100 120−20
−15
−10
−5
0
5
time (sec)
y an
d y
d (rad
)
Commanded versus actual angle of attack
RefARC
20 40 60 80 100−0.5
0
0.5
0 20 40 60 80 100 120
−0.2
−0.1
0
0.1
0.2
0.3
time (sec)
u(t)
(rad
)
Control signal (elevator angle): ARC
u1(t)
u2(t)
max(u)
min(u)
Fault
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
time (sec)
κ est
Estimated controller gain ( κest
) vs. time
κestmax(κ)
min(κ)
0 20 40 60 80 100 120−1.5
−1
−0.5
0
0.5
1
1.5
time (sec)
µ est
Estimated fault ( µest
) vs. time
µest
min(µ)
max(µ)
Figure 5.2. Comparative results for stabilization in absence of distur-bances
124
0 20 40 60 80 100 120−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
time (sec)
y an
d y
d (rad
)
Commanded versus actual angle of attack
ReferenceOutput
0 20 40 60 80 100 120
−0.2
−0.1
0
0.1
0.2
0.3
time (sec)
u(t)
(rad
)
Control signal (elevator angle): Saturated ARC
u1(t)
u2(t)
min(u)
max(u)
Saturation
Fault
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
time (sec)
κ est
Estimated controller gain ( κest
) vs. time
κest
min(κ)
max(κ)
0 20 40 60 80 100 120−0.1
−0.05
0
0.05
0.1
time (sec)
µ est
Estimated fault ( µest
) vs. time
µest
Figure 5.3. Comparative results for stabilization in presence of distur-bance
125
5.5 Conclusion
In this chapter, an indirect adaptive robust scheme was proposed to accommodate
unknown actuator faults in presence of actuator magnitude constraints. Feedback
linearizable systems with matched uncertainties were considered in the present work.
Actuator saturation disrupts the functioning of an actuator fault-tolerant controller
in two ways - first, the working actuators can saturate due to the undesired transients
following an actuator fault and second, once the actuators saturate, the adaptation
may become unreliable. A backstepping based approach was proposed to deal with
the first issue, which explicitly takes into account the actuator saturation limits. In
the proposed approach, performance requirements were relaxed once the actuator
saturates, and more emphasis was put on returning the actuator to the unsaturated
mode. Second, it was shown that an indirect adaptive scheme, with controller and
estimator separation, would yield reliable parameter estimates despite actuator satu-
ration. Another important feature of the proposed scheme is that once all the states
are within a pre-determined region where the controller is unsaturated, certain desired
closed-loop properties, like disturbance attenuation to any extent could be recovered.
Furthermore, it was shown that the error would asymptotically converge to zero in
in spite of actuator faults, in presence parametric uncertainties only. Comparative
simulation studies showed that in presence of actuator saturation limits, which would
make an otherwise stable fault-tolerant control scheme unstable, the proposed design
could achieve desired control objectives.
126
6. CONCLUSIONS AND FUTURE WORK
6.0.1 Conclusions
In this dissertation, an output feedback based Adaptive Robust Fault-Tolerant
Control (ARFTC) scheme was presented for unknown actuator fault accommoda-
tion. The proposed scheme is applicable to a class of uncertain linear and nonlinear
systems and addresses a large class of actuator faults. A critical review of the existing
literature revealed two problems associated with actuator fault-tolerant control - (i)
undesired transients and (ii) unacceptably large steady-state tracking errors. Both
the problems cannot be addressed by any adaptive control or robust control based
fault-tolerant scheme when used individually. In the proposed approach, adaptation
and robust feedback were used simultaneously to achieve desired transients and main-
tain tracking performance in face of large parametric uncertainties introduced due to
failing actuators. For the linear case, comparative simulation studies were done us-
ing a linearized model for lateral motion of Boeing 747. For nonlinear systems, a
Hypersonic aircraft model was used to evaluate and compare the performance of the
proposed design with robust backstepping based design. It could be concluded based
on the comprehensive simulation studies that the proposed scheme can achieve su-
perior performance over a conventional robust MRAC or robust backstepping based
based technique.
One of the assumption made in the first part of work was that the healthy ac-
tuators always had sufficient control authority to accommodate the actuator faults.
In fact, this has become a standard assumption in the adaptive control based fault-
tolerant control designs. However, ignoring actuator saturation can lead to a disas-
trous outcome, as it can easily disrupt the functioning of an adaptive control based
fault-tolerant scheme in the following ways. First, the working actuators can satu-
127
rate due to the undesired transients following an actuator fault and second, once the
actuators saturate, the adaptation may become unreliable. As the first step towards
solving this problem, we solved the problem of global stabilization of an integrator
chain using a conceptually different approach. This problem was first solved by Teel in
1992. Based on Teel’s work, many modifications had been proposed in the literature
to improve the performance of the controller. In our analysis, it was clearly shown
that all such schemes exhibit poor robustness properties with respect to input distur-
bance and leads to conservative design. In fact, a quantitative analysis revealed that
even when the magnitude of the disturbance is less than that of the available control
input, the coordinate transformation can render the design of a stabilizing controller
impossible. These limitations could not be overcome by any modification based on
Teel’s work, as coordinate transformation is an essential step in all such designs. In
order to remove these limitations, we took a fundamentally different viewpoint and
proposed a scheme which did not rely on coordinate transformation, and was based
on backstepping design. Comparative studies were performed on a third order chain
of integrator to show the superior performance of the proposed technique. The first
set of studies, performed in absence of disturbances, revealed that the convergence
rate of the proposed scheme is at least as good as that proposed by Marchand and
Hably, which is based on Teel’s work. In presence of disturbances, however, signif-
icant differences could be seen in terms of disturbance attenuation and convergence
of the states. A tracking problem with large disturbance, where a stable controller
could not be designed due to coordinate transformation, was also solved to show the
effectiveness of the proposed scheme.
Finally, we combined the adaptive robust and input saturated control designs to
propose a saturated adaptive robust actuator-fault tolerant controller. Feedback lin-
earizable systems with matched uncertainties were considered in the present work.
Controller design using feedback linearization is a popular technique for flight control
applications, as it makes it possible to apply linear control theory to a nonlinear sys-
tem. As mentioned earlier, there were two chief problems associated with actuator
128
saturation - saturation of healthy actuators due to undesired transients and unreliable
adaptation. A backstepping based approach was proposed to deal with the first issue,
which explicitly takes into account the actuator saturation limits. In the proposed
approach, performance requirements were relaxed once the actuator saturates, and
more emphasis was put on returning the actuator to the unsaturated mode. Sec-
ond, it was shown that an indirect adaptive scheme, with controller and estimator
separation, would yield reliable parameter estimates despite actuator saturation. An-
other important feature of the proposed scheme is that once all the states are within a
pre-determined region where the controller is unsaturated, certain desired closed-loop
properties, like disturbance attenuation to any extent could be recovered. Further-
more, it was shown that the error would asymptotically converge to zero in in spite
of actuator faults, in presence parametric uncertainties only. Comparative simulation
studies performed on a Hypersonic aircraft model showed that in presence of actuator
saturation limits, which would make an otherwise stable fault-tolerant control scheme
unstable, the proposed design could achieve desired control objectives.
6.0.2 Future Work
1. In the present work, an adaptive robust approach was proposed to accommodate
unknown actuator faults. A similar approach can also be developed to address
a large of sensor faults. Some common type of sensor faults include constant
bias failures, drift or additive-type sensor failures and multiplicative-type sensor
failures [58]. As the effect of these faults can be captured by a parametric
model, a similar framework as developed in this thesis can be used to address
these faults.
2. The backstepping based approach was proposed in this work to deal with in-
put saturation, which differs significantly from the conventional “cancellation”
backstepping design. This resulted in a simpler control law, which was shown to
be bounded for an integrator chain, and for a class of feedback linearizable non-
129
linear system with some assumptions on the nonlinearities and uncertainties. A
natural extension of the proposed technique would be the design of a bounded
controller for systems in parametric feedback form with bounded uncertainties
and nonlinearities. Furthermore, based on the bounds of the nonlinearities, ap-
proximation schemes can be developed for the region of attraction, which would
be a valuable addition to existing literature on saturated control.
3. Based on the work proposed in this dissertation and the previous work on
adaptive robust fault-detection [59], an active fault-tolerant design seems like a
natural extension. The proposed fault-tolerant control accommodates the fault
without an active FDI module. As in many practical applications it may be
necessary to identify and isolate the fault, an adaptive robust fault-tolerant
design with room for exchanging information with an FDI module should be
investigated.
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130
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