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OUTPUT REGULATION OF NONLINEAR SYSTEMS USING
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Attaullah Y. Memon

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OUTPUT REGULATION OF NONLINEAR SYSTEMS USING CONDITIONAL
SERVOCOMPENSATORS

By
Attaullah Y. Memon

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Electrical Engineering

2009

ABSTRACT

OUTPUT REGULATION OF NONLINEAR SYSTEMS USING
CONDITIONAL SERVOCOMPENSATORS

By
Attaullah Y. Memon

The design of output feedback controllers that solve the output regulation problem
for nonlinear systems is considered, with emphasis on improving their transient per-
formance. We concentrate on the output regulation of nonlinear systems using con-
ditional servocompensators, an idea that was introduced in the earlier work of Khalil
and co—researchers. The conditional servocompensator acts as a traditional servo-
compensator in a neighborhood of the zero-error manifold while acting as a stable
system otherwise, leading to improvement in the transient response while achieving
zero steady-state regulation error. The idea was introduced in a sliding mode control
framework. We extend the technique of conditional servocompensators to more gen-
eral feedback controllers by using Lyapuno'u redesign and saturated high-gain feedback.

The striking feature of our approach is the flexibility of starting with any stabi-
lizing state feedback controller and then including a conditional servocompensator to
achieve zero steady-state regulation error without degrading the transient perform-
nace. We give regional as well as semi-global analytical results for error convergence.
The flexibility offered by our design approach allows us to consider as a special case,
the output regulation problem for control-constrained linear systems. Such a design
approach, however, assumes the availability of all state variables to meet the control
objectives, and the output feedback control can only be implemented using a full-
order observer. An extension of methodology to nonlinear systems would necessitate
the development of a full-order nonlinear observer. We develop a full-order high-gain
observer for nonlinear systems, extending earlier work on reduced-order high-gain

observers for nonlinear systems and full-order observers for linear systems.

To my father... and his fond memories!

iii

ACKNOWLEDGMENTS

I am highly indebted to my advisor Dr. Hassan K. Khalil, whose keen insight
and great vision in the subject provided me with invaluable advice and guidance to
conduct the research work presented in this dissertation. His infinite patience as a
teacher, endless encouragement as a mentor and continuous support has made the
time span of last four years truly memorable. I can not thank him enough for all he
has done for me.

I would like to thank National University of Sciences & Technology (NUST),
Pakistan, for the partial financial support of my PhD. studies.

I am thankful to many friends and colleagues for their help through the ups and
downs during the past several years. Especially to my group members J eff Ahrens and
Shahid Nazrulla, and to my friends Sajjad Zaidi and Abrar Siddiqui, to whom I owe
the pleasure of many things, that. contributed to my Ph.D., often in very unpredictable
ways.

And, I have no words to thank the very special person in my life, my mother,
for her love and unconditional support. I owe my gratitude to my best friends, my
brother and my sister, for their love and encouragement, and a profound appreciation
to my wife, Rashda, for years of boundless love, unflinching support and inestimable

patience and understanding that have made all this possible.

iv

TABLE OF CONTENTS

List of Figures ................................ vii
Introduction 1
1.1 Preliminaries ............................... 3
1.1.1 The Nonlinear Output Regulation Problem ........... 3
1.1.2 Lyapunov Redesign [29] ..................... 6
1.1.3 High-Gain Observers ....................... 7
1.2 Background and Motivation ....................... 9
1.3 From the Conventional to the Conditional Servocompensators . . . . 11
1.4 An overview of the Dissertation ..................... 13

Regulation of Minimum-Phase Nonlinear Systems Using Conditional

Servocompensators 15
2.1 Introduction ................................ 15
2.2 System Description and Assumptions .................. 16
2.3 Control Design .............................. 22
2.4 Performance ................................ 32
2.5 Example .................................. 38
2.6 Conclusions ................................ 43

Output Feedback Regulation of Input-Output Linearizable Nonlin-

ear Systems 44
3. 1 Introduction ................................ 44
3.2 System Description and Assumptions .................. 45
3.3 Control Design and Analysis ....................... 47
3.4 Examples ................................. 51
3.5 Conclusions ................................ 61
Regulation of Linear Systems Subject to Input Constraints 62
4.1 Introduction ................................ 62
4.2 Low-gain design for linear systems .................... 63
4.3 Problem Statement and Control Design ................. 66
4.4 Closed-Loop Analysis ........................... 68
4.5 Output Feedback Design ......................... 73
4.6 Example .................................. 84
4.7 Conclusions ................................ 88

5 Full-Order High-Gain Observers for Minimum Phase Nonlinear Sys-

tems 89
5.1 Introduction ................................ 89
5.2 System Description and Problem Formulation ............. 94
5.3 Performance Recovery .......................... 98
5.3.1 Recovery of the Boundedness and Convergence of Trajectories 98
5.3.2 Recovery of the Asymptotic Stability of the Origin ...... 100

5.4 Discussion on the main results ...................... 100
5.5 Example .................................. 102
5.6 Conclusions ................................ 108
6 Conclusions 109
6.1 Synopsis of Results ............................ 110
6.2 thure Work ................................ 112
Bibliography ..................................................... 113

vi

1.1

2.1

2.2

3.1

3.2

3.3

4.1

4.2

5.1

5.2

5.3

5.4

LIST OF FIGURES

A general setup for the solution of output regulation problem ..... 10
Trackng error during the transient period ............... 41
Steady-state Tracking error ....................... 42
Regulation error during the transient period .............. 55
Steady—state regulation error ....................... 56

Performance of the two control designs (a) Regulation error ’e’ during
the transient period (b) Control input ’u’ ................ 60

Performance comparison of the two control designs (a) Regulation error
’e’ during the transient period (b) Corresponding control input, ’u’ . 86

Transient performance of the two control designs when control coeffi-
cient is perturbed by 40 percent (a) Design II - Nominal vs Perturbed
(b) Design I - Nominal vs Perturbed .................. 87

Performance under state feedback, partial-state feedback, and output
feedback, with c = 0.1 .......................... 104

Performance under state feedback, partial-state feedback, and output
feedback, with e = 0.01 .......................... 105

Estimation errors 7);;2 = 2:2 — i2, and 772 = z — 2, with e = 0.01 . . . . 106

Closed-loop Performance under model uncertainty, with e = 0.01, 61 =
1.25, 03 = 0.75 ............................... 107

vii

Chapter 1

Introduction

This dissertation concentrates on the design of feedback controllers that solve the
output regulation problem for a class of minimum-phase nonlinear systems, with
emphasis on improving the transient performance. For this class of systems, robust
continuous feedback control techniques like Lyapunov redesign (min-max control) or
sliding mode control can be used to ensure convergence of the tracking error to a small
ball around the origin of the closed-loop system, while rejecting bounded disturbances.
However, rendering the tracking error arbitrarily small requires the use of high-gain
feedback near the origin, which is traditionally achieved by using the classical idea of

including a servocompensator with the stabilizing compensator.

We concentrate on the output regulation of nonlinear systems using conditional
servocompensators, an idea that was introduced in the earlier work [52, 53, 54] of
Khalil and co—researchers. The key feature of this idea is that the conditional servo-
compensator acts as a traditional servocompensator only in a neighborhood of the
zero-error manifold, while it is a bounded-input-bounded-state stable system whose
state is guaranteed to be of the order of a small design parameter. The use of condi-
tional servocompensators enables us to achieve zero steady-state tracking error with-

out degrading the transient response of the system. The idea was introduced in [52]

1

and [53] in a sliding mode control framework, where [52] dealt with the special case
of conditional integrator for constant exogenous signals, while the more general case
of time-varying signals was treated in [53]. We focus on extending the technique
of conditional servocompensators [53] to more general feedback controllers by using

Lyapunov redesign and saturated high-gain feedback.

The flexibility offered by the Lyapunov redesign framework of starting with any
stabilizing state feedback controller and then including a conditional servocompen-
sator, allows us to consider as a special case, the output regulation problem for linear
systems subject to control constraints. The presence of saturation in the input chan-
nel imposes strong limitations to the achievable control objectives such as transient
performance. Because of the constraint on the control, the mechanism of solving
the stabilization problem through Algebraic Riccati Equation (ARE) necessitates the
use of full-state feedback. The sliding mode control design of [53] uses partial state
feedback, and therefore, can not be used to address this problem. We exploit the
two-time-scale approach to the observer design described in [13] to design an output
feedback control to achieve regulation in the presence of input constraints. Naturally,
the first thing that comes to mind is to extending the methodology to nonlinear sys-
tems. However, such a design approach assumes the availability of all state variables
in order to meet the control objectives. Consequently, the output feedback control
will be implemented using a full-order nonlinear high-gain observer. This motivates

us to develop a full-order high-gain observer for minimum-phase nonlinear systems.

In Section 1.1, we briefly review some of the main background elements of this
dissertation. These include the problem of nonlinear output regulation, Lyapunov
redesign, and output feedback using high-gain observers. In Sections 1.2 and 1.3,
we provide the background and evolution of the problem of output regulation for
nonlinear systemsland the motivation of this work. Finally, Section 1.4 gives an

overview of this dissertation.

1 . 1 Preliminaries

1.1.1 The Nonlinear Output Regulation Problem

One of the most fundamental problems in control theory is the output regulation
problem, alternatively known as the servomechanism problem, which deals with the
design of a feedback controller for a fixed plant that yields a prescribed steady-state
response to every external command in a given family of functions. This includes the
problem of rendering the output y of a fixed plant to asymptotically track a reference
signal r in a certain class of functions, as well as the problem of asymptotically
rejecting a disturbance signal d in a certain class of disturbances. The objective is to
force the difference between the reference input r and the actual output y decay to
zero as time tends to infinity, for every reference input r and every disturbance signal
d over prespecified families of functions.

Consider a nonlinear system modeled by equations

:i: = f(:c,u),u) (1.1)

e z (1(33, w) (1.2)

where a: E R" is the state of the plant, u E Rm is the control input, to 6 RT
represents the exogenous input variables which includes refrences to be tracked and / or
disturbances to be rejected, and e E Rm denotes an error variable expressed as
a function of the state a: and of the exogenous input to. It is assumed that the
functions f (x, w, u) and h(:c, w) are sufficiently smooth, f (0, 0, 0) = 0, and h(0, 0) = 0.
The exogenous input signals w(t) are assumed to be the solutions of a homogenous
differential equation

u) = S w (1.3)

where S has distinct eigenvalues on the imaginary axis. This mathematical model

3

that generates all possible exogenous input functions is called the exosystem. Suppose
there exists a feedback controller which can process the information available from

the plant, and whose output it is a function of a: and to, given by

u = 70?. W) (1-4)

The interconnection of (1.1)-(1.2) and (1.4) yields a closed-loop system characterized

by the equations

ti) = 5w (1.5)

11': == f(a:,w,'y(:r,w)) (1.6)

The closed-loop system (1.5)-(1.6) is said to have the property of output regulation if
the control u can be designed such that for every exogenous input 11) (in a prescribed
family of signals) and for every initial state in some neighborhood of the origin, the

output 6 decays to zero as time tends to infinity.

The structure of the feedback controller usually depends on the amount of in—
formation available for feedback. If all the components of the state a: of the plant
and the state to of the exosystem are available, then the control law takes the form
of equation (1.4), which is a memoryless system. A more realistic situation is when
only the error e is available for measurement. In this case, the control signal u is

synthesized by means of a dynamical nonlinear system

S = nae) (1.7)
u = 9(€) (1-8)

with internal state 5. The interconnection of (1.1)-(1.2) and (1.7)—(1.8) yields a closed-

4

loop system characterized by the equations

u'; = Sw (1.9)
5 = U(€,h($,w» (1-11)

The above formulation is known as error feedback output regulation problem. Together
with the notion of immersion, it is shown in [24] that the error feedback output
regulation problem is solvable if there exists a continuously differentiable mapping
a: = 7r(w), with 7r(0) = 0, and a continuous mapping x(w), with x(0) = 0, that solve

the nonlinear regulator equations

Egg—(25w = f(7r(w),w, x(w)) (1.12)

0 = h(7r(w),w) (1.13)

In other words, the above condition means that the error feedback output regulation
problem is solvable if it is possible to find a mapping x(u)) that renders the identities
of equations (1.12)-(1.13) satisfied for some 7r(w), and is such that control input

generated by the autonomous system

u) = Sw (1.14)

u 2 Will (1.15)

is precisely the control it required to achieve zero steady-state error, in the presence

of any exogenous input 211.

1.1.2 Lyapunov Redesign [29]

The term Lyapunov redesign refers to a nonlinear control technique where a stabilizing
state feedback control can be constructed with knowledge of the Lyapunov function
of a nominal system, resulting in a control design which is robust to large matched

uncertainties. Consider the system

:i: = f(t,:r) + G(t,a:)[u + 6(t,:2:, u)] (1.16)

where a: E R" is the state and u 6 RP is the control input. Assume that the
functions f, G, and 6 are piecewise continuous in t and locally Lipschitz in a: and
u. The functions f and G are known precisely, while the function 6 is an unknown
function which satisfies the matching condition and represents various uncertain terms
pertaining to model simplification, parameter uncertainty etc. A nominal model of

the system can be taken as

it = f(t,a:) + G(t,:1:)u (1.17)

Suppose we can design a stabilizing state feedback control law u = ib(t, 9:) such that

the origin of the nominal closed-loop system

1: = f(t,:1:) + C(t, warm) (1.18)

is uniformly asymptotically stable. Suppose further that we know a continuously

differentiable Lyapunov function V(t, :13) for (1.18) that satisfies the inequalities

a1(ll:rll)s var) 3 again) (1.19)
85?- + gin...) + Gamma] s a3(|lxll) (1.20)

where (11, a2, and 013 are class [C functions. Now assume that, with u = $03, 1:) + u,

the uncertain term 6 satisfies the inequality

“Mani/20.56)) + U” 5 90,95) + fiollvll, 0 S no <1 (121)

where p is a nonnegative continuous function, which represents a measure of the size of
the uncertainty. It is shown in [29] that with the knowledge of the Lyapunov function
V(t, :c), the function p, and the constant no in (1.21), the additional feedback control
component i) can be designed such that the overall control u = 211(t, :13) + u stabilizes
the actual system (1.16) in the presence of the uncertainty 6. The design of u is called

Lyapunov redesign.

1.1.3 High-Gain Observers

High-Gain observers provide an important technique for the design of output feedback
controllers for minimum-phase nonlinear systems. A high-gain observer is essentially
an approximate differentiator that robustly estimates the derivatives of the output.
The gain of a high-gain observer depends on a small parameter c, which can be
adjusted to guarantee that the estimation error decays to an 0(6) value arbitrarily
fast. High-gain observers are applicable to a class of nonlinear systems that can be

transformed into the normal form

2 = tb(z, :17) (1.22)
:i: = Ax + B¢(z, as, u) (1.23)
y = Ca: (1.24)

where z E R1 and a: E R7" are the system states, u E R is the control input and y E R

is the measured output. The r x r matrix A, the r x 1 matrix B and the 1 x r matrix

7

C, are given by

 

 

 

 

‘0 1 0] ”01
0 0 l O 0
A: , B: ,
0 0 1 0
.0 0.1 -1-
C: 1 0 0

Over the past several years, many researchers have contributed toward the inves-
tigation of output feedback control for the class of systems of the form (5.1)—(5.3). Of
significant relevance are the works [2, 3, 13, 29], which solve the problem of robust
output feedback stabilization, in the large, of the input-output linearizable nonlinear
dynamic systems by means of bounded partial-state feedback control (e.g. u = 7(a))
and high-gain observer, with subsequent substitution of the estimate of 2:, provided
by the high-gain observer, in the feedback. The boundedness of the control protects
the state of the plant from peaking when the high-gain observer estimates are used
instead of the true states. As a result, the output feedback control recovers the per-
formance under the state feedback control. One of the important consequences of
this technique is the ability to separate the design of output feedback control for
nonlinear systems into a state feedback design followed by the design of the high-gain
observer. Teel and Praly [57] developed a generic separation principle, which showed
that (semi)global stabilizability via state feedback plus uniform observability imply
semiglobal stabilizability via output feedback. Atassi and Khalil [2] provided a more
comprehensive separation principle and showed that the output feedback controller
recovers the performance of the state feedback controller in the sense of recovering
asymptotic stability of an equilibrium point, its region of attraction, and its trajec-

tories. An exhaustive survey of the use of high-gain observers in nonlinear control

8

can be found in [31, 34]. A review of the many approaches to design observers for

stabilization of nonlinear dynamical systems appears in [16].

1.2 Background and Motivation

The output regulation problem was first studied for multivariable linear systems un-
der various names, such as the robust servomechanism problem by Davison [11] and
the structurally stable output regulation problem by Francis and Wonham [14]. The
solvability conditions for the output regulation problem were worked out either in
terms of the solvability of a system of two linear matrix equations called the regu-
lator equations or in terms of the location of the transmission zeros of a composite
system that incorporates the plant and the exosystem. A salient result of the theory
is the observation that any structurally stable (robust) controller which solves the
regulation problem can always be viewed as the interconnection of two components;
a servocompensator and a stabilizing compensator. The servocompensator is a device
that incorporates an internal model of the exosystem, a model capable of generating
the reference and disturbance signals produced by the exosystem. The role of the sta-
bilizing compensator is to stabilize the augmented system comprising of the plant and
the servocompensator. A general setup is shown in Figure 1.1. The above mentioned
property is known as the internal model principle, which reduces in the special case of
constant references and disturbances to the classical integral control. The significance
of the internal model principle is that it enables formulation of the output regulation
problem into the well-known stabilization problem for an augmented linear system
formed of the plant and the servocompensator.

Over the past two decades many researchers have contributed toward the inves-
tigation of the nonlinear output regulation problem, among whom we specifically

mention Isidori and co-researchers [4, 5, 6, 7, 8, 25, 26, 45, 46, 49, 50, 51], Huang and

9

‘—

 

 

 

T _ S eruo— Stabilizing u y

 

 

 

 

 

+ Compensator Controller Plant t

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured Signals

 

 

Figure 1.1: A general setup for the solution of output regulation problem

co—researchers [9, 10, 20, 21, 22, 23, 36] and Khalil and co—researchers [30, 32, 40, 53].
The pioneering work of Isidori and Byrnes [26] showed how the results of Francis and
Wonham [14] could be extended to nonlinear plants and nonlinear neutrally stable
exosystems with their formulation of the nonlinear regulator equations. The re-
sults in [26] were local and required smallness of both the exogenous signals and the
initial states. Huang and Rugh [22] pursued a computational approach involving a
power series expansion of the solution of the nonlinear regulator equations. Their
method allowed for large exogenous signals, but was still local in terms of the initial
states. Regional and semi—global results first appeared, for the case of fully lineariz-
able systems, in the work of Khalil [30]. An important contribution of [30] was the
observation that in the nonlinear case the internal model must be able to generate
not only the trajectories of the exosystem, but also a number of their higher-order
harmonics. This idea was also elaborated independantly by Huang and Lin [23] and
Priscoli [44]. Together with the idea of immersion, Isidori [24] provided a complete
set of necessary and sufficient conditions for the existence of a solution to the 10-
cal output regulation problem. An interesting generalization of the results of [30],
in terms of the structurally stable output regulation approach of [24] and the robust
control approach of [30], was presented by Isidori [25]. A good account of the available
results for the nonlinear output regulation problem can be found in [7, 20, 24, 27].

A simplification of the robust servomechanism design of [40] was presented in [32],

10

where the only precise information that is required in the design of the controller is
the relative degree of the plant, the sign of its high-frequency gain and the linear
internal model. Some results that relax the assumption of input-to—state stability of
the zero dynamics and allow for the frequencies of the exosystem to be unknown,
thereby making use of an adaptive internal model, can be found in [50, 51]. A re-
sult that relaxes the assumption that the solution of the regulator equations be a
polynomial in the exogenous signals can be found in [10]. Recent works have focused
on the identification of design procedures yielding nonlinear internal models. In this
respect, [9] focuses on an internal model that is constituted by a linear system having
a nonlinear output map, whereas [5, 45, 46] focus on the design of nonlinear internal
models having nonlinear observability forms. A more recent result [41] considers the
practical nonlinear output regulation with respect to the dimension of the internal

model and to the gain of the stabilizing compensator near the zero-error manifold.

1.3 From the Conventional to the Conditional Ser-
vocompensators

The classical idea of the internal model principle design has been used by Khalil and
co—researchers in [30, 32, 33, 39, 40] to achieve asymptotic output regulation for a. class '
of nonlinear systems. For the case of constant references and disturbances [33, 39], the
servocompensator is simply an integrator driven by the tracking error and its inclusion
creates an equilibrium point at which the tracking error is zero. For the more general
case [30, 32, 40], a linear internal model is identified which generates the trajectories
of the exosystem and, along with them, a number of higher-order harmonics generated
by the system nonlinearities. This is then used to synthesize a servocompensator, the
inclusion of which creates an invariant manifold on which the regulation error is zero.

In order to achieve nonlocal stabilization of the disturbance-dependant equilibrium

11

point or zero-error manifold, the stabilizing compensator is designed using robust
continuous feedback control techniques like min-max control or sliding mode control
(SMC). The error feedback controller is designed using the separation approach of
Esfandiari and Khalil [13], where a state feedback control is designed first and then a
saturated high-gain observer is used to recover the performance of the state feedback

design.

While the above described designs achieve robust output regulation, the steady-
state performance achieved often happens at the expense of degradation of the tran-
sient performance. This is due in part to the increase in system order as a result of
servocompensator, and in part to the interaction of the servocompensator with the
control saturation. To address the issue of transient performance degradation in the
conventional integrator and servocompensator designs of [30, 32, 33, 39, 40], the idea
of conditional integrators and servocompensators was introduced by Seshagiri and
Khalil [52, 53], in a sliding mode control framework. The key feature of this idea is
that the conditional servocompensator acts as a traditional servocompensator only in
a neighborhood of the zero-error manifold, while it is a bounded-input-bounded-state
system whose state is guaranteed to be of the order of a small design parameter. The
use of conditional servocompensators makes it possible to achieve zero steady-state
tracking error without degrading the transient response of the system. The special
case of conditional integrator for constant exogenous signals was addressed in [52],
while the more general case of time-varying signals was treated in [53]. To extend
the design beyond the sliding mode control, [54] developed the conditional integrator
using Lyapunov redesign and saturated high-gain feedback. Starting with any stabi-
lizing state feedback controller, [54] shows how to include a conditional integrator by

modifying the original controller.

12

1.4 An overview of the Dissertation

In this dissertation, we focus on the design of controllers that solve the output reg-
ulation problem for a class of minimum-phase nonlinear systems with emphasis on
improving the transient performance. We extend the technique of conditional servo-
compensators [53] to more general feedback controllers by using Lyapunov redesign

and saturated high-gain feedback.

In Chapter 2, we consider the problem of state feedback regulation of nonlinear
systems using conditional servocompensators. We use the Lyapunov redesign and
saturated high-gain feedback approach to design the stabilizing compensator, and in-
clude a conditional servocompensator by modifying the original controller that yields
asymptotic error regulation without degrading the transient performance. We prove
that the trajectories of the closed-loop system under saturated high-gain feedback
control with a conditional servocompensator approach those of a closed-loop system
under saturated high-gain feedback control without a servocompensator. We pro-
vide analytical results for a compact set of initial conditions, which can be chosen

arbitrarily large if all the conditions hold globally.

In Chapter 3, we consider the output regulation problem for a class of minimum-
phase input-output linearizable nonlinear systems. The state feedback control design
of Chapter 2 is specialized to partial state feedback control design. This partial
state feedback controller can be viewed as an intermediate step towards the output
feedback controller of Chapter 3, which is implemented using a reduced-order high-
gain observer. We also prove that the output feedback controller with conditional
servocompensator recovers the performance of a state feedback controller that does

not include any servocompensator.

In Chapter 4, we consider the output regulation problem of linear systems subject

to control constraints. We apply the Lyapunov-redesign-servocompensator approach

13

of Chapter 2 to the linear output regulation problem under input constraints. The
output feedback control is implemented using a two-time—scale full-order observer
design of [13] and the performance recovery is shown using the separation principle
of [1, 3].

In Chapter 5, we study the problem of state estimation of a minimum-phase
nonlinear system using a full-order high—gain observer. The observer comprises two
components, a slow open-loop observer that estimates the state of the internal dy-
namics, and a fast observer that estimates the state of the external dynamics. The
observer design approach is based on the two-time scale observer design of Esafandiari

and Khalil [13], and the performance recovery is shown using the separation principle

of Atassi and Khalil [2].

14

Chapter 2

Regulation of Minimum-Phase
Nonlinear Systems Using

Conditional Servocompensators

2.1 Introduction

We consider the problem of state feedback regulation of nonlinear systems using
conditional servocompensators, an idea that was introduced in an earlier work [52, 53,
54] of Khalil and co-researchers. The key feature of this idea is that the conditional
servocompensator acts as a traditional servocompensator only in a neighborhood
of the zero-error manifold, while it is a bounded~input-bounded-state stable system
whose state is guaranteed to be of the order of a small design parameter. The use of
conditional servocompensators enables us to achieve zero steady-state tracking error
without degrading the transient response of the system. The idea was introduced in
[52] and [53] in a sliding mode control framework, where [52] dealt with the special case
of conditional integrator for constant exogenous signals, while the more general case

of time-varying signals was treated in [53]. To extend the design beyond the sliding

15

mode control, [54] developed the conditional integrator using Lyapunov redesign and
saturated high-gain feedback. Starting with any stabilizing state feedback controller,
[54] shows how to include a conditional integrator by modifying the original controller.
We aim to extend the development of [54] by addressing the servomechanism problem

for time-varying exogenous signals.

2.2 System Description and Assumptions

Consider the nonlinear system

C' = f(c.w)+5¥(c.w>u
e = h((,w) (2.1)

where C E R" is the state and u E Rm is the control input. The plant is subjected
to a set of exogenous input variables to that belong to a compact set W E 12‘”,
which include unknown disturbances to be rejected and references to be tracked. The
variable e E R” denotes the regulation error. The functions f, C and h are sufficiently
smooth in C on a domain E C R" and are continuous in w for w E W. Our goal is to

design a controller to asymptotically regulate e to zero.

Assumption 2.1. w(t) is generated by the known exosystem
ti) = 3011) (2.2)

where SO has distinct eigenvalues on the imaginary axis.

The requirement that So has distinct eigenvalues on the imaginary axis implies
that w(t) is bounded and persistent in time, i.e. w(t) 4+ 0 as t —+ 00. In the context

of output regulation problem it is particularly meaningful when the exogenous signals

16

are persistent in time, as it is in the case of any periodic (and bounded) function. In
these cases, the system is expected to exhibit a steady-state response that is itself
a persistent function of time, and whose characteristics depend on the specific input
imposed on the system rather than the state in which the system was at the initial
time. For example, the following choice of SO can generate bounded sinusoidal signals

wl and w2 of frequency w and a constant signal w3

 

 

 

 

 

 

ill} 0 w 0 w 1
tbz = -—w 0 0 102
’LL‘3 0 0 0 w3

We only require SO to be known which is equivalent to precisely knowing the frequen-
cies of the sinusoidal signals generated by the exosystem (2.2). The amplitude and
phase of the sinusoidal signals are allowed to be unknown since we do not require the
initial conditions w(0) to be known. For the case where the frequecies of the exosys-
tem are unknown, an alternate design, that makes use of an adaptive internal model
whose natural frequencies are automatically tuned to match those of the unknown

exosystem, can be found in [51].

Assumption 2.2. There exist a continuously differentiable mapping C = 7r(w), with

7r(0) = 0, and a continuous mapping x(w) that solve the equations

87r(w)
8w

 

Sow f(7r, w) + C(n, w)X(w)

~

0 = h(7r, w) (2.3)

for all w E W.

The above assumption states a necessary and sufficient condition for the solution

of the output regulation problem. It essentially means that C = 7r(w) is a zero-error

17

f

invariant manifold and x(w) is the steady-state control that maintains the motion on

this manifold, in the presence of any exogenous input w.

Assumption 2.3. There exists a set of real numbers c0, ..., cq_1 such that x(w) sat-
isfies the identity

—1
LEI-x = cox + ClLsX + ' - - + cq—1L3 X (2-4)

for all w E W, where Lsx = (Bx/8w) Sow and the characteristic polynomial

pq _ cq_lpq_1_ . . . _ CO
has distinct roots on the imaginary axis.

Motivation for the above assumption comes from the nonlinear version of the
internal model principle, which recognizes that in the nonlinear case, the controller
must be able to reproduce not only the trajectories generated by the exosystem,
but also a number of higher order nonlinear harmonics, an idea that was elaborated
independantly by Khalil [30], Huang and Lin [23] and Priscoli [44]. Assumption 2.3,
along with the notion of immersion [24], allows the construction of a finite-dimentional

linear internal model as follows. Defining

 

 

 

 

- 0 1 0 - - x -

0 0 1 0 LsX
S: i T:

0 0 1 [lg-2X

.60 Cq-I. _Lg—1x4

and F = [1 0 0]1xq, it can be shown that the steady-state control x(w) is gener-

ated by the internal model

8r(w)
0w

 

Sow = Sr(w), x(w) = I‘r(w) (2.5)

18

The internal model (2.5) is valid only when x(w) contains a finite number of har-
monics, which will always be the case if x(w) is a polynomial function of w.1 The
coefficients c0,c1, - -- ,cq_1 in (2.4) are required to be known, even in cases when
x(w) is uncertain. To further elaborate on this, let x(w) = awl + burg, where a and
b are unknown constants, and w1 = sin(wt + (t), in which to is the frequency and d)
is the phase. It can be shown that in this case Equation (2.4) can be satisfied with
Lgx(w) = —9w4x(w) — 10w2Lgx(w). It is important to note that the coefficinets of
the polynomial (in this case, c0 = —9w4, and c2 = —10w2) only depend on frequency
w, and not on the unknown constants a and b. Thus, by construction, the internal

model (2.5) includes harmonics of the sinusoids generated by the exosystem (2.2).

With the change of variables x = C — 7r, the system (2.1) can be represented by
5i: = f (:6, w) + C(x, w)lu - X(W)l (2-6)

where f(x, w) = f(x + 7r,w) — f(7r,w) + [C(x + 7r, w) — C(n,w)]x(w) and G(x,w) =
C(x + 7r,w). The system (2.6) is in the form where the state feedback regulation
problem can be formulated as a state feedback stabilization problem by treating

x(w) as matched uncertainty. We assume that a stabilizing state feedback controller

is available for the system
:i: = f(x, w) + G(x,w)u

Assumption 2.4. There exists a locally Lipschitz function 7,0(x, w), with ib(0, w) = 0,
and a continuously difierentiable Lyapunov function V(x, w), possibly unknown, such
that

ai(||$||) S V(WU) S a2(||$||) (2-7)

 

1An example where this assumption is not satisfied is when x(w) = sin(w), where w = sin(at),
in which a is a constant.

19

a—wsow + 989m. w) + G<x,w>w<x,w)1 s —W(a:> (28>

Vx E X C R” and Vw E W, where (11 and 012 are class [C functions, W(x) is
a continuous positive definite function, and X is a given domain that contains the

origin.

The system (2.6) can be written as

x = f(x,w)+G(x,w)t/)(x,w)

+ C(x, w)u — C(x, w)]x(w) + w(x, w)] (2.9)

In what follows, we use Lyapunov redesign [29] to construct the robust stabilizing
feedback control to deal with the uncertain term x(w). Towards that end, let 9 =
{supwEwV(x,w) 3 Cl} C X, for some c1 > 0, and 6(x) be a continuous function
such that

[lx(w) + 2b(x, w)“ s 5(x) Vx e a, Vw e w (2.10)

The derivative of V(x, w) along the trajectories of (2.9) is

V = a—dSow + %[f(x,w) + C(x, w)i/)(x,w)]
+ g—ZGCC, w)u - g—:G(x, w)[x(w) + t/1(£II, w)]

—W<x> + 33,—Eco, w)u — get, w)[x(w) + m, w)] (2.11)

l/\

The first term on the right-hand side of (2.11) is due to the nominal closed-loop
system. The second and third terms represent, respectively, the effect of the control
u and the uncertain term [x(w) +ib(x, w)] on V(x, w). Due to the matching condition,
this uncertain term appears on the right-hand side exactly at the same point where
u appears. Consequently, it is possible to design u to cancel the (destabilizing) effect

of [x(w) +tb(x, w)] on V(x, w). Before proceeding further, we introduce the following

20

assumption.
Assumption 2.5. (BV/Bx)G(x, w) can be expressed as
(8V/Bx)G(x, w) = vT(x)H(x,w) (2.12)

where v(x) is a known, locally Lipschitz function, with v(O) = 0, and H (x,w) is a,

possibly unknown, function that satisfies
HT(x, w) + H(x,w) 2 2A1m, ]|H(x,w)|| g k; k 2 A > 0 (2.13)

Vx E Q and Vw E W, where [m is m x m identity matrix.

The inequality (2.13) means that the uncertainity in the term (8V/3x)G(x,w)
can be bounded by a known positive constant A. The matrix H will simply be an

identity matrix if there is no uncertainty in the term (8V/ 8x)G (x, w). Taking
v
u = ——a(x)gb (—) (2.14)
p
where the continuous function a(x) is chosen such that

a(x) Z 6(x) + a0, do > 0 (2.15)

VIP?

the saturation function d (fi) is defined as

v/llvll if |lv||2 ii
<25 (—) = (2.16)

v/iu if ”U HS it

and ,u > O is a small design parameter. With this control, we can see that when

21

Hull 2 p, the equation (2.11) yields

V

l/\

-W(Ir) - a($)/\ llvll + 19503) III)”

< —W(x) + pAozo (2.17)

The same result can be obtained when “v” S )1. Hence, with control (2.14), the
derivative of V(x, w) along the trajectories of the closed-loop system (2.9) is negative
over the domain of interest. In what follows, we will refer to this control law as

saturated high-gain feedback control.

2.3 Control Design

The robust state feedback control designed in the previous section will achieve prac-
tical regulation in the sense that the regulation error will be 0(p), so that it can
be made arbitrarily small by choosing p. small enough. However, a very small u will
induce chattering. Therefore, we can not rely on reducing u as a mechanism to render
the error arbitrarily small. We refer to the eralier work of Khalil and co—researchers
[30, 40, 52, 53] where a servocompensator is used to ensure that the regulation error
converges to zero, without requiring a to be arbitrarily small so as to render the reg-
ulation error arbitrarily small. Motivated by the latest results of Seshagiri and Khalil
[53], we introduce the conditional servocompensator via the saturated high-gain feed-

back controller

u = —a(x)q§ G) (2.18)

where s = v(x) + K10, and o is output of the conditional servocompensator

.=(s_....).....(;) (2...)

22

in which J = [0, - -- ,0,1]T, and K1 is chosen such that S — J K 1 is Hurwitz, which
is always possible since the pair (S, J) is controllable.

By appropriately choosing the initial condition 0(0), the solution of (2.19) will be
0(a) for all t Z 0. In particular, consider the Lyapunov function V0(o) = oTPoo,
where P0 is the symmetric positive definite solution of the Lyapunov equation POAa +

AZPO = —I , in which A0 =: S — J K 1. The derivative of V0 satisfies the inequality

- 2
V0 5 - IIUII + 2i! “all IIPOJII

Therefore, V0 5 0 on the surface V0(o) = p2c2 for the choice c2 = 4 ||P0J||2 AW(P0).
Hence, the set {V0(o) S p2c2} is positively invariant, and we require 0(0) to belong

to it.

The closed-loop system is given by

if} = 8011)
x = f(x, w) + C(x, w)w(x, w) — a(x)G(x, w)¢ (i)
- G($,W)IX(1U) + Wit, 10)] (220)

a = Ago + pqu (2)

We will now show that, for sufficiently small p, the set \II = 52 x {V0(o) S p2c2} is a
subset of the region of attraction, and for all initial conditions in \II, every trajectory
of the closed-loop system (2.20) asymptotically approaches an invariant manifold on
which the error is zero. The forthcoming analysis shares many points in common
with the ones in [54], apart from various technical differences due to the presence
of the time-varying signal w(t). We start by showing that the set \II is positively
invariant and there is a class [C function p such that every trajectory in \II enters the

set ‘11” = {|[x|| S p(,u)} x {V0(o) S u2c2} in finite time and stays thereafter. The

23

derivative of V along the trajectories of the closed-loop system (2.20) satisfies

- 6V V
V = EU—Sow + %;[f(x, w) + C(x, w)i,/)(x, w)]

—- -f;—:G(x,w)a(x)¢ (E) — %G(x,w)[x(w) + W33 w)]

2 5172.301); + %E[f(x, w) + C(51), w)u/(33,110]

T s T i ’ x w
—v <x>H<x.w>a<x>¢ (12) —v (x)H(x,w)[x(u)+v( . )1

I/\

—W(z> — a(x)[s - KlalTHa. wn (g)

— [s — KialTHu. w)[x(w) + Maui]

= —W(x> — a<x>sTH(x. w)¢ (g) + athKwiTHa. wk (5)

— sTH<x,w>1x<w> + use. w)] + (KtaTHe. w)[x(w) + «is, w)]

Inside ‘1’, [[0]] S p cz/Amin(P0). Using this along with (2.13), (2.16) and (2.15), it

can be shown that when [Isl] 2 p we have

V S -W(Iv) — A0433) H-S‘ll + l€501?) IISII
+ llKill llUll k[61(33) + 5(3)]

S —W(11?) + #71 (2-21)

where '71 = maxxEQ kk0[a(x) +6(x)] and k0 = [|K1[[ \/c2//\m,-n(P0). Similarly, when
[[3]] g p we have

- Ms”2 A

V < —W(f€) - (W(x) # + k(5(117) ”SH

8
+ as) ”Kin lloll k”—,,—”- + an) 1K1“ Hall k

S —W(r) + #72 (2.22)

24

where 72 = maxxeg kko [a(x) + 6(x)(1 + 1/k0)] Z 71. From (4.18) and (4.19),
V S -W($) +u72. V030) E ‘1’

Hence, from [29, Theorem 4.18], for sufficiently small p, \I! is positively invariant and

all trajectories starting in ‘11 enter \II,‘ in finite time and stay thereafter.

Next, we use V1 = %sTs and Assumption 2.6, below, to show that the trajectories

reach the boundary layer {[[s[[ S p} in finite time.

Assumption 2.6. N(x,w) =: (av/0x)G(x,w) satisfies
N(x,w) + NT(x, w) 2 2ip1m, “N(x,w)[l g k, (2.23)

where kp 2 AP > 0, Va: E {[[xl] S p(p)} and Vw E W. Moreover, 0(0) 2 7136(0) +

010, (10 > 0.

Remark 2.1. The above assumption is automatically satisfied in the special case
when V = xTPx and G = B, where P and B are known constant matrices. In this
case, from Assumption 2.5, we get v = (8V/8x)B = 2BTPx, and H (x,w) = I .
Applying Assumption 2.6 yields us N = (3v/8x)B = 2BTPB, which is a positive

definite matrix, independant of (x, w).

For (x,o) E ‘11), ,

sTs = —a(x)sTN(x,w)¢ (E) + sT-g—:[f(x, w) + C(x, w)w(x, w)]

- sTN(x, w)lX(w) + 111(13, w)] + STKIAOU + ”STKIMS (i)

25

When [[3]] 2 u we have

sTe s -a($)/\pI|S|l+|IN($.w)|lIIX(W)+¢($.w)IIIISII
[331mm + G<ew>e<ew>1 Ilsll

+ (IIUII ||K1|| ”Ag” + it llK1ll llJll) IISII (2-24)

..[....

 

 

 

 

Inside \Ilp, [[0]] S p\/c2//\mm(P0). Also, the function g—Z—[f(x,w) + G(x,w)t/J(x,w)]
is continuous such that 3% f(O, w) + G(0,w)w(0,w)] = 0. Therefore, the norm

 

 

gave. w) + G<ew>ei<ew>1|| together with the norms Ilall ”Kin 1.4.11, u 1K1” IlJll.
[[a(x) — 0(0)“, and [[6 (x) — 6(0)” can be bounded by a class [C function p1(p). Hence,

sTs' S -a(0)r\p||8|| +kp5(0)||8|| +P1(#l[[5ll

=> V1 < —/\p [cm — $32] [[5]] (2.25)

Thus, for sufliciently small u, all trajectories inside \IIfl would reach the boundary
layer {[[s [[3 p} in finite time.

Finally, we show that inside the boundary layer the trajectories of the closed-loop
system asymptotically approach an invariant manifold on which the error is zero.

In order to prove local convergence to this manifold, we shall make the following

assumption.
Assumption 2.7. There exist non-negative constants k1 to k6 such that

llw(=r,w)|| S k1 ”v(x)“ + k2 W(x)

 

 

anew) + G<e,w)w(e.w)1|[ 3 k3 ”v(x)” + k4 WW

— 0
[W] s is “v(x)” + k6\/—W($)
\7’w E W, in some neighborhood of x = 0.

26

Remark 2.2. If W(x) 2 k,- [[x[[2, k, > 0, near the origin x = 0, then any locally
Lipschitz function can be bounded by mm for some kj > 0. Furthermore, if the
equilibrium point of the closed-loop system is exponentially stable, such that Equations
(2. 7) and (2. 8) are satisfied with al, a2 and W being quadratic in [[x[[, and all other

functions are locally Lipschitz, then the above assumption is automatically satisfied.

Inside the boundary layer, the closed-loop system (2.20) is given by

a = Sow
:12 = new) +G<e.w>e(e.w> —e<e>G<e,w)<s/e>

- C(x,u’)[x(w) +w<ew>1 (2.26)
a = Sa+Jv(x)

From [53], there exists a unique matrix A such that
SA=AS and —K1A=I‘
We define
M), = {x = 0,0 :2 0(w)}

where 0(w) = (p/a(0))A r(w). It is easy to verify by direct substitution that M j) is

an invariant manifold of (2.26) for all w E W.
Defining 0 = 0 — 0(w) and s = v + K10, the closed-loop system inside the

boundary layer can be written as

tb=SOw

x = f(x,w)+

~

010 W) (33 (W) — 04100013 W);
+ C(x, w) (a___(x)—;);1(0)) x(w) — G(x,w)w(x, w) (2.27)
0 = A00+Js = S0+)Jv

27

Consider the Lyapunov function candidate

b
V2 = V(x, w) + zaTPOa + $ng (2.28)

where b and c are positive constants to be chosen, and P0 is the symmetric positive

definite solution of the Lyapunov equation POAU +A£P0 = —I. Calculating V2 along
the trajectories of the system (2.27), we obtain

' ' b ... I. .1 .. ~ .1.
V2 = V + E [0TP00 + 0TP00] + csTs

The first term of V2 can be written as

V = EESOw + %%[f(x,w) + C(x,w)i,l2(x, w)]

8_V x w —a x E _____a(x) __ 01(0) w
+axG(. )[ ”Hf 0(0) )x< )]
— 5500’" w)w(x,w)

g —W(x) — vTH(x, w)9-Li)(v + K10)

+ vTH(x, w) (W) x(w) — vTH(x, w)w(x, w)

Using Assumptions 2.4 - 2.7 yields

v s —W<e>—eo(A/e)ne12+Ilvnue/mlmnuen
+ueuk W9]

 

 

 

le(w)|| + Ilvll k ||¢(w)l|
s —W(e) — [eon/u) — k7] W + k8 llvll «W(x)

+ (kg/u) HUN ”5“ (2-29)

where (it is an upper bound on a(x) and k7 to k9 are some positive constants. Similarly,

28

the second term of V2 can be written as

 

b . . b b
— [0TP00 + 0TP00] = —— “all2 + —[0TP0J§ + sTJTPg‘a]
it it u
b ~ 2 20km ., ~
S -;||0|| + # IIUIIHSII (2-30)

where (910 = Ama$(P0). Next, we have

%lf(x, w) + C(x, WWW: w)]

+ 250a, w) [-a(x)-:; + (W) x(w)]
(9v

— 550(23’ w)w(x, w) + K1(50 + Jv)

Cm-
ll

c§T§ = cs'T-g—Z[f(x, w) + C(x, ’lU)‘t/J($, w)]
+ csTN(x) [—a(x)i + (W) x(w)[
— chN(x)v(x, w) + c's'TN(x)K1(S0 + Jv)

..T; .. 2 .. ..
cs 8 S ‘CO‘OO‘p/fllllsll +ck11|lsllllall

+ ck12 ll§l| llvll + ck13|I§|| \/ W(x) (2-31)

29

for some positive constants km to M3. From (2.29), (2.30) and (2.31), we have

1'3 _<. -W(93)-la0(/\/#)-k7lllv||2--le5||2
— mow/e) new2 + k8 llvll «W(x) + 3.13 Hen x/W(93)

+(k9/1u)||vl| ||5|| +(2bk10/u + ck11)||5ll “5H + den ”3'“ llvll (2-32)

The right-hand side of (2.32) can be arranged in the following quadratic form of

H=[\/W llvll ”5H ll§lllT1
V2 3 —HTAH (2.33)

where the symmetric matrix A is given by

\
t—|
“LI
Phi
O
kg
0
kl
___/

\1
V
I
{315’

l
“if

:8 (es-e

_k e -125ka )
0 if ,. ( u

 

 

use an (Lam—eke) 392 i

If the principal leading minors of A can be made positive by choosing the constants
b and c appropriately, and by choosing p sufficiently small, then V2 will be negative
definite. This would imply that, inside the boundary layer, the trajectories of the

closed-loop system will asymptotically approach M p as t —> 00. Towards that end,

30

T

we partition the matrix A as

 

 

1 _qfll“2
A =
-<112 fiQ22 + A22
where
T
(112 = ( 528 0 95213 )
{ 00A —%9 0 \
Q22 = J? b —bk10
k 0 —bk10 caOAp )
and
f ‘k7 0 if” )
A22 = 0 0 —Ck

 

 

Heal oi

(2.34)

(2.35)

(2.36)

(2.37)

From (2.36), it is easy to see that by choosing b and c, we can successively make the

principal leading minors of Q22 positive. First, b is chosen large enough to make the

2 x 2 minor positive, and then, c is chosen large enough to make the 3 x 3 minor

31

i,

positive. Finally, choosing ,u small enough will render

1 —qfi’;
det > 0 (2.38)

-412 fiQ22 + A22

Consequently, V2 will be negative definite. Therefore, the trajectories of the closed-
loop system will asymptotically approach M ,1 as t —+ 00. Our conclusions are sum-

marized in the following theorem.

Theorem 2.1. Suppose Assumptions 2.1 - 2.7 are satisfied and consider the closed-
loop system (2.20). Suppose w(0) E W. Then, there exists p* > 0 such that ‘v’p E
(0, M], the set \IJ = Q x {V0(0) S u2c2} is a subset of the region of attraction, and

for all initial conditions in \II, the state variables are bounded and limtAoo e(t) = 0.

Remark 2.3. The analytical results are provided for a compact set of initial condi-
tions, which can be chosen arbitrarily large if all the conditions hold globally. For
example, if Assumptions 2.4 - 2.6 hold globally, then the compact set of the initial

conditions it can be chosen arbitrarily large, and will yield semi-global results.

2.4 Performance

In section 2.3, we introduced a saturated high-gain feedback controller that incorpo-
rated a conditional servocompensator, and subsequently showed that trajectories of
the closed-loop system under this scheme are bounded and yield zero steady-state
regulation error. In this section we show that the conditional servocompensator does
not degrade the transient response of the system. The trajectories of the closed-loop
system under saturated high-gain feedback control with servocompensator approach

those of a closed-loop system under saturated high-gain feedback control without

32

servocompensator. The latter control is given by

u = —a(x)¢ (10%)) (2.40)

which is similar to (2.18) except that v(x) + K10 is replaced by v(x), and yields the

closed-loop system

ID = 3011)
x = f(x, w) + C(x, w)r,b(x, w) — a(x)G(x, w)¢ (Bit—x2)
- 000110)]me 1141310)] (2-41)

In both cases, the trajectories eventually enter a boundary layer, which is [[v(x) [[ S p
for the closed-loop system (2.41). We will denote the solution of (2.41) by x* to
distinguish it from the solution of (2.20). There could be a period of time when
trajectories enter and leave their boundary layers before eventually settling inside.
The challenging part in showing the closeness of the trajectories of the two systems
is to keep track of when trajectories enter and leave the boundary layers since the
entry and exit times will be different for the two systems. For convenience, we impose
conditions which ensure that once the trajectories enter the boundary layer they do
not leave. Recall that in the proof of Theorem 1 we showed that trajectories in ‘11,,
enter the boundary layer in finite time. The new feature here is that we need to show
this property for all trajectories in Q and not just ‘11,). Towards that end, the choice
of a(x) in the control laws (2.18) and (2.40) is modified to

a(x) 2 max {§6(x), £36m) + Aipfi(x)} + 00 (2.42)

33

‘ 1'

where B (x) is a known continuous function such that

8v

%[f(x, w) + G(:r. which w)][[ E W)

 

 

for all x E 0. Suppose that the inequality (2.23) holds for all x E fl. Then, outside

the boundary layer {[[s[[ 2 p}, the trajectories of (2.20) satisfy

eTs' s we». Ilsll + are» Ilsll + we) Ilsll
+<nenuK11 “A.“ + ullK1ll an) llsll
-—/\p [am —- £36m — fine) — rue. Ilsll

—A. [00 — urea] llsll (2.43)

|/\

|/\

for some positive constant ka. Thus, for sufficiently small [1, the trajectories starting
inside 9 would reach the boundary layer in finite time and stay there for all future
time. A similar result can be obtained for the system (2.41) using Vl’“ = %v(x*)Tv(x*).
Inside the boundary layer the systems (2.20) and (2.41) exhibit slow and fast dynamics
for small p. To show closeness of their trajectories, we will compare the equations of

the two systems represented in the singularly perturbed form.

Assumption 2.8. There exists a mapping Tw (x), continuously difierentiable in (x, w),

such that the Jacobian QIEéI—Q is nonsingular, uniformly in w, and

0
m0(x,w) = ; Vfol;‘v’wEW
8x I
m
The change of variables (p = 901 = Tw (x) transforms the system (2.6) into
‘P2

34

the form

9’51 = 7r1(<.01,<P2,IU)

r52 = 7r2(<.01.r2,w) + [u-X(w)]

In the new variables, v(x) 3:111:10” == v1(<,ol, 902, w).

Assumption 2.9. The equation v1(<p1,<,o2,w) + K10 = 0 has a unique solution

902 = w(cpl, 0, w) for all x E fl and w E W, and the system

a = Ago (2.44)

901 = 7T1(<P1,w(901, 0, w),w) (2.45)

has an exponentially stable equilibrium point at cpl = 0.

Remark 2.4. Exponential stability of (2.44) - {2.45) implies exponential stability of

901 = W1(<P1,W(<P1, 0,110,110 (2-45)

0n the other hand, exponential stability of (2.46) together with continuous differen-
tiability of 7rl(<p1,w(cp1, 0, w),w) implies exponential stability of (2.44) - (2.45).

Outside their respective boundary layers, the two systems can be represented by

 

 

x = f(x, w) — C(x,w)x(w) — a(x)G(x, w) [”1583 :filzlll (2.47)
_ _ 0 v(x) + K10
" ‘ A” + “J luvs) + Kleul (2'48)

and

 

15* = f(x*,w) - G(x*,w)x(w) — a(x*)G(x*,w) ll]:i::llll (2.49)

35

Suppose to is the time instant when one of the two systems reaches its boundary
layer. Since [[0 [[= 0(p) , the difference in the right-hand side of state equations
for x and 2* is 0(p) Vt E [0,t0]. From [29, Theorem 3.4], [[x(t) — x*(t) [[= 0(p)
Vt E [0, to]. Therefore, when the trajectory of one system reaches the boundary layer,
then for sufliciently small p, the trajectory of the other system will be in some 0(a)
neighborhood of its boundary layer. Since V1 and V; are strictly negative outside
their boundary layers, uniformly in p, it must be true that the trajectory of the other
system also reaches its boundary layer in time t1 = to + 0(a). For all t 2 t1, when
the trajectories of the two systems are inside their boundary layers, the two systems

can be represented by

v(x)+K10) (2.50)

x = f(x, w) — G(x,w)x(w) — a(x)G(x, w) ( I1
0 = A00 + J [v(x) + K10] (2.51)

and

2* = f(x*,w) — C(x*,w)x(w) — a(x*)C(x*,w) (Elli—*2) (2.52)

With the change of variables cp = Tw(x) and cp’“ = Tw(x*), the two systems can be

written in the singularly perturbed forms

(7 = A00+lei(901,902,w)+K10l (2-53)
901 = 7T1(<P1,902,w) (2-54)

(“p2 = #lfi2((p17 992:1”) _ X(w)l _ 010101, Lp2iw) lvl (901) @2210) + K10] (2‘55)

36

I

for (2.50)-(2.51) and

231‘ = W102i. 902, w) (2-56)

use; — #[r2(<pi.<p§.w)-X(w)l-adrimiwlwlwiviwl) (2-57)

for (2.52), where 01(901, (p2,w) = e(x) x—T_1(cp)' For the singularly perturbed sys-
_ w
tem (2.53)-(2.55), the slow model is
‘pl 2 7].10101713010120.) ’UJ),’LU) (2'59)
and the fast model is
(1892 _
71;- — —01(<P1,902.w) [UI(<P1,<P2,w) + K10] (2-60)

where T = t / p. Similarly, for the singularly perturbed system (2.56)-(2.57), the slow

model is

 

«21‘ = r1lri.tU(s0i.0.w).w) (2-61)
and the fast model is
dcp“
6,72 = -a1(90’{.<p§.W)v1(90’f.</9§,w) (2-62)

The definitions of the slow and fast models follow the procedures in [35]. We require

the two boundary-layer systems to be exponentially stable.

Assumption 2.10. The two boundary-layer systems given by (2.60) and ( 2. 62) have

37

exponentially stable equilibrium points at (<p1,<p2) = (0,0), and (90;,tp3) = (0,0),

respectively.

Because 0 is 002), [29, Theorem 9.1] confirms that the solutions of (2.58)-(2.59)
and (2.61) are 0(p) close to each other. Furthermore, [29, Theorem 11.2] shows the
closeness of the solutions of the systems (2.53)-(2.55) and (2.56)-(2.57) to those of
their respective slow and fast models given by (2.58)-(2.60) and (2.61)-(2.62). There-
fore, [|x(t) — x*(t) [[= 0(a) Vt Z 0. The foregoing conclusions can be summarized
in the following theorem, which confirms that the controller (2.18) recovers the per—

formance of the controller (2.40).

Theorem 2.2. Let (x(t),0(t)) be the state of the closed-loop system (2.20), and
x*(t) be the state of the closed-loop system without servocompensator (2.41). Let
x(0) = x*(0) E Q. Suppose Assumptions 2.8 - 2.10 are satisfied and a(x) satisfies
(242). Then, under the hypotheses of Theorem 2.1, there exists if > 0 such that
V# E (Owl‘l; ”113(0- 58*(75) ||= 001) W Z 0-

Remark 2.5. The performance recovery provided by Theorem 2.2 is with respect to
a system controlled by the saturated high-gain feedback controller (2.40), rather than

the original stabilizing controller w(x, w).

2.5 Example
Consider a second-order system modeled by the equations

$1 = 1132
5&2 = —61(21—e§/6)+62u (2.63)
y = 531

38

where 61 and 62 are (possibly uncertain) constants. with the reference signal r(t) =

rosin(wt), which is generated by the exosystem

0 w T 0
w = w, w (0) = , r(t) = w1
-w 0 r0

With the change of variables x1 = 5:1 — r, x2 = 5132 -— 7", we have

:1: = Ax + B[f(:r, w) + C(x. w)(u — WW]

6 = Cx (2.64)
where
0 1 0
A = , B = , C = (1 0) ,
0 0 1

f(x,w) = —01x1 + 96]- [(x1 + w1)3 — wfl, C(x, w) = 62, and

x(w) = 91-2- [91101 - $61013 - w2w1]. It can be verified that x(w) satisfies the identity
Lg‘x = —9w4x — 10w2Lgx.

We compare the performance of saturated high—gain feedback stabilizing controller
without a servocompensator (Design 1), with two control designs that use servocom-
pensators (Design 2 and Design 3). Design 2 uses the conditional servocompensator
(3.13), while Design 3 uses a conventional servocompensator [40]. In Design 3, we
augment a fourth-order conventional servo-compensator 0 = S 0+J e, with the system

(2.64) to obtain an augmented system of the form

C = A1§+Bl[f(x,w)+G(x,w)w(x,w)]
e = Cl; (2.65)

39

f. .

where

SJC
0A

, B1=(0 B)T, Cl =(0 C) (2.66)

33
u

T
and C = ( 0 x ) is the state. A stabilizing feedback controller is designed via

Lyapunov redesign [40], which yields
1 6 1
w(mv) = — 012:1 — 4 ((el + as - w?) — —¢(w/#) (2.67)
62 k
where w = 2BTPC, and P = PT > 0 is the solution of the Lyapunov equation
P(A1 + 311(2) + (A1 + BIK2)TP = —I

in which K 2 is chosen such that Al +Bl K 2 is Hurwitz. For the conditional servocom-
pensator design (Design 2), Assumption 2.4 is satisfied with V(x) = %(3x¥ + 2x1x2 + 2x%)
and w(x, w) = 915 [01x1 — Qg-[(x1+w1)3 — wil] —- klxl — k2x2] where k1 = k2 = 1. As-
sumption 2.5 is satisfied with v(x) = x1 + 2x2 and H (x, w) = 1. We use the following
numerical values in the simulation: 61 = 1, 62 = 3, w = 0.5 rad/s, r0 = 1, k = 1/5,
and p = 0.1. For the conventional servocompensator design (Design 3), K2 is chosen
so as to assign the eigenvalues of A1 + 31K2 at —0.5, —1, —1.5, —2, —2.5 and —3.
For the conditional servocompensator (Design 2), K1 is chosen so as to assign the

eigenvalues of .S' — JKl at —0.5,—1,—1.5 and —2. The control u is given by

 

u: _10 sat (5B1 +2132+K10>

u

Figure 2.1 shows the tracking error during the transient period and Figure 2.2

shows the steady state tracking error for the three designs. The transient response of

40

 

Tracking Error ('I’renelent)

 

 

 

 

 

 

 

 

 

0.8 r
if“!
0.6 P _.‘ ‘3
.' a
' a
0.4 '— 1. 3|
I I
0.2 '- ; 3.. .5" ‘2‘
5' 3. 5' 3.
o _ I . 1 2 ‘ T
.5. l, ' ‘0 ; -“‘uo'i
-0.2 r i ‘| .‘i
\‘ _'
-o.4 e ‘3 f
3""!
-0.0 ~
- - - Deeign1: Sat. High-Gain Control without Servo
-0 8 _ —Deslgn 2: Sat. High-Gain Control with Cond. Servo
- 1-‘-' Design 3: Sat. High-Gain Control with Conv. Servo
-1 1 l I ' I
0 0.5 1 1.5 2 2.5

Time (sec)

Figure 2.1: Tracking error during the transient period

41

Tracking Error (steady-state)

 

 

 

4 X 10 r i i i i I T i i
3 _ _
2 '- ‘ a ‘ ------- Q ~ A
1 " I’ "
’l
I, - _ .- _ _
o--------—-‘---'--T-"Vi 'l ___-_---'-- ""~-
’I
’I
-1 _ ’ , ’ a
-2 - ~ " """"" ‘ ‘
- - - Design 1: Sat. High-Gain Control without Servo
"3 ‘ —- Design 2: Sat. High-Gain Control with Cond. Servo ‘
----- Design 3: Sat. High-Gain Control with Conv. Servo

 

 

 

 

 

 

'32 53 54 55 56 57 58 59 so 61
Time (sec)

Figure 2.2: Steady—state Tracking error

42

62

the controller design without a servocompensator (Design 1) is close to the one with
conditional servocompesator (Design 2), but it does not result in asymptotic error
convergence. Asymptotic error convergence to zero is achieved with the conventional
servocompensator design (Design 3), however, at the expense of a degraded transient

performance.

2.6 Conclusions

We have extended the state feedback regulator of [54], which uses conditional in-
tegrators, Lyapunov redesign, and saturated high—gain feedback, to a more general
case of time-varying signals by using conditional servocompensators. We showed that
the use of conditional servocompensators enables us to achieve zero steady-state reg-
ulation error, in the presence of time-varying exogenous signals that are generated
by a known exosystem. Analytical results are provided for a compact set of initial

conditions, which can be chosen arbitrarily large if all the conditions hold globally.

43

 

Chapter 3

Output Feedback Regulation of
Input-Output Linearizable

Nonlinear Systems

3.1 Introduction

We consider the output feedback regulation problem for minimum-phase nonlinear
systems that can be transformed into the normal form. We first design a partial-
state feedback control that regulates the error to zero in the presence of time-varying
refrence and disturbance signals. The output feedback control is then implemented
using a high-gain observer. We show that the output feedback controller with condi-
tional servocompensator recovers the performance of a state feedback controller that

does not include any servocompensator.

44

3.2 System Description and Assumptions

Consider the nonlinear system

2‘: = f($,w)+G(w,w)[u-X(W)l

e = h(x, w) (3.1)

where e is the measured output that we want to regulate to zero. This problem
relates to a more realistic and practical situation when only the error variable e is
available for measurement, rather than the complete state x as in Chapter 2. It is
well known [24] that if the above system has a well-defined vector relative degree and
the distribution span {g1, . . . ,gm} is involutive, where 91, . . . ,gm are the columns of

G, then it can be transformed into the normal form

5 = A€+B{f1(§,z,w)+GI(C,z,w)[u—X(w)[}
2" = f2(C,z,w) (32}
e = CC

where C and z belong to the sets X E C Rn“? and X ,3 C RT, respectively. The r x r

matrix A, the r x m matrix B and the m x r matrix C, given by

A = block diag [A1, . . . ,Am],

 

 

0 1 0
0 0 1 0
21,:
0 0 1
L0 - 0 Jrixri

45

B = block diag [B1,...,Bm], B,- =

 

 

['1 ‘1‘in

C = block diag [01,. - -,Cm[,

02-:[1 0 ... ... 0]
eri

where 1 S i S m and r = r1 + . .. + rm, represent m chains of integrators. In the

new coordinates, the zero-error manifold can be written as
M” = {C = 0,2 = 0,0 = 0(w)}

In what follow, we first state the assumptions required to address the output feedback

regulation problem, and then we elaborate on how they are related to those in Chapter

2.
Assumption 3.1. The following inequalities hold for all (C, z,w) E X g x X z x W:

GI(€1 Z7 1U) + 0%..(63‘23211) Z 2AlIm: ’\1 2 A > 0
BTPIBG'1(0,0, w) + G'{(o, o, w)BTPlB 2 2mm, A2 2 A > o

where P1 = PIT is the solution of the Lyapunov equation P1 (A—BK )+(A—BK )TP1 =
-In and K is chosen such that A — BK is Hurwitz.

Assumption 3.2. There exists a Lyapunov function Vz(z, w) such that for all (C, 2, w)

EXgXXzXW

 

 

k12||2|l2 s vz<z.w) s 1:13 Men2 (33)
av. av. ‘2
< — z .
aw Sow+ agrees)- 1614 Hz“ +1.1, n Illléll (34)

for some positive constants k12, k13, km and k15.

46

Let V(C,z,w) = Vz(z,w) + k16[CTP1C] for some positive constant k15. In this
case, Assumption 2.4 is satisfied with w(C,z,w) = Gl-1(C,z,w)[-f1(C,z,w) — KC]
and inequalities (2.7) and (2.8) are satisfied with al, a2 and W which are quadratic
[E z [T -
Assumption 2.5 is satisfied with vT(C) = 2k16CTP1B and H(C, z, w) = C1 (C, 2, w).

in Consequently, the set M p is exponentially stable, uniformly in w.

 

 

 

 

Remark 3.1. For a $150 nonlinear system in the normal form, Assumptions 3.1 -
3.2 imply Assumptions 2.4 - 2. 7, and boil down to the requirements that the system
is minimum-phase and the control coefficient a(.) is positive and bounded away from

2870.

3.3 Control Design and Analysis

We first derive a partial state-feedback control which will then be implemented as an
output feedback control using a high-gain observer. Towards that end, we re-write

the inequality (2.10) as
llx(w) + ¢(€.0,w)ll s 5(6) V(t, 2. w) e Xg >< X2 x W (36)
Consequently, (2.15) can be modified as
e(t) 2 £86 + a0. a0 > o (3.6)
and from (2.18), a partial state feedback control can be taken as

u = -a(€)¢ (ii) (3.7)

47

where 31 = v(C) + K10. Under the partial state feedback control, the closed-loop

system is represented by

i = A8 + B {f1(€.z.w) + 01a. 2. w)l-a(€)¢ (3‘) — x(w)l] (3.8)
23 = f2(C,z,w) (3-9)
0 = (S — JK1)0 + qub (87]) (3.10)

The control (3.7) can be implemented as an output feedback controller that uses the

high-gain observer

6: AC + L(e — or?) (3.11)

to estimate C, where the observer gain L is chosen as

L: blockdiag[L1,...,Lm], Li:

 

 

l
m
s

- r-xl

in which 6 is a positive constant and the positive constants a; are chosen such that

the roots of
Ari + afiiri-1+---+ afrlk + 0%. = 0

are in the open left-half plane, for all i = 1, . . . ,m. The output feedback control is

given by

u ——- —e<é>e (ii) (3.12)

48

where §1 = v(C) + K10 and 0 is the output of the conditional servocompensator

a = (s — JK1)0 + 2J2 (ii) (3.13)

Next, we show that the output feedback control (3.12) recovers the performance of
the state feedback control for sufficiently small 6. For the purpose of analysis, we
replace the observer dynamics by the equivalent dynamics of the scaled estimation

error D17 = C —— C, where

D = block diag [D1, - - - , Dmli

Di = [eri_1,eri_2,--- ,1]

Ti X’I‘i

The closed-loop system can be represented by

6' = A6 + B{f1(€. z,w) +
G1(€.z. eat—ea - Dme (WE ‘ DZ) + K”) — X(w)l} (3.14)

2 = f2(€_Dniziw) (315)
(v(C - D71) + K10)
p

 

 

(3.16)
0'7 = Aon+€Bff1(€.2.'w)+

G1(€,Z,’lU)[—a(€ _ Dn)¢ (v(g — D77) + K10"

p

 

) — x(w)l} (3.17)

where A0 = e D—1(A — LC)D is an r x r Hurwitz matrix. The system (3.14) - (3.17)
is a standard singularly perturbed system, and r) = 0 is the unique solution of (3.17)
when 6 = 0. The reduced system is the closed-loop system under state feedback
control. For convenience, we write CT = [CT zT]. Let the initial states be 9(0) =
90 E 8, 0(0) = 00 E {Vb(0) S pzcz} and C(0) = Co E Q, where B is a compact set
that contains M”, and Q is any compact subset of RT. Let (9(t,e),0(t,e),n(t,c))

denote the trajectory of the system (3.14) - (3.17) starting from (80,00,C0). The

49

 

system (3.14) - (3.17) fits into the framework of Atassi & Khalil [3] so that the
results provided therein can be conveniently applied. In particular, the system (3.14)
- (3.17) satisfies Assumptions 1 and 2 of [3], which require the functions f1 (C ,z, w),
f2(C, z, w) and Cl (C, 2, w) to be locally Lipschitz in their arguments uniformly in w

over the domain of interest.

Let (8r(t), 0,-(t)) be the solution of the closed—loop system under state feedback,
starting from (90,00). Using [3, Theorems 1 &. 3], it can be shown that, the trajec-
tories of (3.14) - (3.17) are bounded, and e(t, 6) — 9r(t) and 0(t, 6) — 0,»(t) converge
to zero, as 6 —> 0, uniformly in t, for all t 2 0. Using [3, Theorems 2 86 5] , we
can show that there is a neighborhood N of M l, x {n = 0}, independent of 6, and
62 > 0 such that for every 0 < 6 S 62, the set M), x {r} = 0} is exponentially
stable and every trajectory in N converges to this set as t ——> 00. Furthermore, from
[3, Theorems 1, 2 & 5], there is 63 > 0 such that for every 0 < 6 S 63, the solu-
tions starting in B x {V0(0) S p262} x Q enter N in finite time. Hence, for every
0 < 6 S 64 = min {62,63}, the set M,“ x{n = 0} is exponentially stable and B
x {V0(0) S p262} x Q is a subset of the region of attraction. Thus, for sufficiently
small 6, the closed-loop system (3.14) - (3.17), under the output feedback controller
(3.12), is uniformly exponentially stable with respect to the set M j, x {C - C = 0},
and limt_.oo e(t) = 0. The foregoing conclusions are summarized in the following

theorem.

Theorem 3.1. Suppose Assumptions 2.1 - 2.3 and 3.1 - 3.2 are satisfied. Then,
given any 7 > 0, there exists [2* > 0 such that Vp E (0, [1*], there exists 6*(p) > 0
such that V6 E (0, 6*], the closed-loop system under the output feedback {3.14) - (3.17)

50

is uniformly exponentially stable with respect to the set M I, x {C — C = 0}, and

505:5) {r(t) 5)
z(t, 6) i “ Zr(t,€) S 7
0(t,6) 07-(t, 6)

 

 

 

 

foralltZOandwEW.

3.4 Examples

In this section, we present simulations for two examples. In the first example we
consider the magnetic suspension system [29] to illustrate the performance of the sat-
urated high gain feedback control using conditional servocompensator and compare
it with the earlier results of Mahmoud & Khalil [40] , that involved the traditional
approach of augmenting a servocompensator with the plant. In the second example,
we revisit the simulation example treated in Serrani, Isidori & Marconi [50] , to com-

pare the performance of our control design with that of [50].

Example 3.4.1. Magnetic suspension system

Consider the magnetic suspension system [29] given by

L0 22
2 +fd

"=-—k' ____________
my y +mg 2a(1+y/a)

where m is the mass of the ball, y 2 0 is the downward vertical position of the ball
measured from a reference point y = 0, k is a viscous friction coefficient, 9 is the
acceleration due to gravity, L0 is the inductance of the electromagnet, a is a positive
constant and fd is an external disturbance force. Let y* be a nominal equilibrium

point and i* be the corresponding nominal value of the current i that maintains the

51

equilibrium condition

7ft _ aLO 2*
g— 2(a+y"‘)2

where m is the nominal value of the mass of the ball. Defining C1 = 153.5, C2 = 13L—

y wn ’

u = 7—d— and r = wnt, yields the normalized model

C1 (2 (3-18)

(2 = -bC2 + N41) + 9(6)“ + d(t) (3-19)

where An) = 1— 2%)2. 9(C1) = 1 — Act). b = 7.5;. d = 4.1,,

* 2
my run

(11,2, = g/y* and (.) denotes the derivative with respect to 7'. It is desired to balance
the ball at a certain constant position r0 in the presence of a sinusoidal disturbance

signal d(t) = dosin(wt). These signals are generated by the exosystem

0 w 0
w = -40 0 0 w, wT(0) = (d0,0,r0), d(t) = wl, r0 = w3
0 0 0

With the change of variables x1 = C1 -— w3, x2 = C2, we have

3'31 = 332
i‘2 = -b:v2 + N131 + 7113) + 9(531 + wslu + W1 (3-20)
6 = 2:1

52

The system (3.20) can also be written as

x = Ax + B[f2(x, w) + 9(5131 + w3)(u — X(w))l

e = Cx (3.21)

where

fete. w) = —be2 + Am + ...,) + w1+ g(:v1+ wane). and X(w) = — (W).
It can be verified that x(w) satisfies the identity Lgx = —w2Lsx. We compare
the performance of a saturated high-gain feedback stabilizing controller without a
servocompensator (Design 1), with two control designs that use servocompensators
(Designs 2 and 3). Design 2 uses the conditional servocompensator (3.13). For this
design, Assumption 2.4 is satisfied with V(x) = %(3x¥ + 2x1x2 + 2xg) and w(x, w) =
Wng — A(x1+ w3) — g(x1+w3)x(w)— w] - k1x1 — k2$2] where k1 = k2 = 1.
Assumption 2.5 is satisfied with v(x) = x1 + 2x2 and H (x, w) = 1. Assumption 2.7 is
also satisfied since the system (3.21) is locally exponentially stable (see Remark 2.2).

The control u for Design 2 is given by

 

a = (s — JK1)0 + ,quat (21+ 2‘”? + K1”)

p

u = —12 sat (m1+2x2+K10) (3.22)
p

 

The state estimate 22 is provided by the high-gain observer

551 = 532 + 91(331 - I371)/€, 3:2 = 92(331 - 530/62 (3-23)

53

where 91 and g2 are chosen such that the polynomial A2 + 91/\ + 92 is Hurwitz. The

control u for Design 1 is given by
2.
u z _12 ,at (2:32)
p

which is similar to (3.22) except that it does not have a servocompensator included.
In Design 3, we augment a third-order conventional servo—compensator 0 = S0 + J 6,

with the system (3.21) to obtain an augmented system of the form

C = A15 + Bl[—b22 + A(x1 + w3) + wl + 9(21 + w3)u]

e = 015 (3.24)

where

T 5 JC T
€=(0T xT), A1: A ,Bl=(0 B), Cl=(0 C)
0

A stabilizing feedback controller, designed via Lyapunov redesign [40], is given by

1

u = mlbxz - N331 + w3) — gfxl + w3)X(w) - wl - “kiflw/iill

where km is a positive constant, 23 = 2BTPC, the state estimate :32 is provided by the
high-gain observer (3.33) and P = PT > 0 is the solution of the Lyapunov equation

P(A1 + 311(2) + (A1 + BlK2)TP = -I

in which K 2 is chosen such that (A1 + Bl K2) is Hurwitz.

We use the following numerical values in the simulation: in = m = 0.2 kg,

k = 0.001, a = 0.01, g = 9.81 m/s, L0 = 1, w = 2 rad/s, d0 = 1, y* = 0.1m,

54

(a)
I I I I I 1 I I
------ Design 1: Sat. High-Gain Control without Servo
’ \ — Design 2: Sat. High-Gain Control with Cond. Servo
- - - Design 3: Sat. High-Gain Control with Conv. Servo“

 

 

 

 

 

 

Regulation Error (Transient)

 

I l l l

 

 

 

 

 

 

 

20 25 30 35 40
Time, s
(b)
[ [ I 1 J 1 J
10 r ------- Design 1: Sat. High-Gain Control without Servo
' -— Design 2: Sat. High-Gain Control with Cond. Servo
5 If - - - Design 3: Sat. High-Gain Control with Conv. Servo.
I

Control Input, u

 

 

 

I I l l l l I

O 5 10 15 20 25 30 35 40
Time, s

 

Figure 3.1: Regulation error during the transient period

55

L-

 

0.8“

0.6-

0.2-

(u,

 

Regulation Error (Steady-State)

 

 

 

~ Design 1: Sat. High-Gain Control without Servo
—Design 2: Sat. High-Gain Control with Cond. Servq

 

 

---Design 3: Sat. High-Gain Control with Conv. Servo

 

 

94 95 96 97
Time, s

Figure 3.2: Steady-state regulation error

56

98

km = 1/10, u = 0.1, 91 = 2, 92 = 1 and 6 = 0.01. K1 is chosen so as to assign the
eigenvalues of (S —— JKI) at -0.5,—1,—1.5 and —2. K2 is chosen so as to assign the
eigenvalues of (A1 + BlKg) at —0.5, —1, —1.5, —2 and —2.5.

Figure 3.1(a) shows the regulation error during the transient period and Figure 3.2
shows the steady-state regulation error for the three designs. The transient response
of the controller design without a servocompensator (Design 1) is close to the one with
conditional servocompesator (Design 2), but it does not result in asymptotic error
convergence. Asymptotic error convergence to zero is achieved with the conventional
servocompensator design (Design 3); however, at the expense of degraded transient

performance.

Example 3.4.2.

Consider the single-input single-output nonlinear system [50]

z = —z+x1+zx1—wlz

331 = $2

- _ 2 2

x2 — z + x2 — x1w2 + u (3.25)
y = $1

It is desired that the output y of the system (3.25) asymptotically tracks a desired
constant set point wl, while rejecting the sinusoidal disturbance 102. These signals

are generated by the exosystem

000
w: 00ww
0—w0

With the change of variables C1 = x1 — wl, C2 = x2, we have

2‘ = —-z+(w1+l+z)C1
51 = 52

e = e+s—tn6+iu—xum 626
6 =51

where x(w) = —wl + wlwg is the steady-state control input. It can be verified that
x(w) satisfies the identity Lgx = —w2Lsx. It can be seen that the zero dynamics are

globally exponentially stable. The system (3.26) can be written as

2.: = fl (ztgiw)
t = A»: + Bif2(z.t. w) + (u - X(w))l (3.27)
e = CC
where
0 1 0
A: ,3: ,C=(1 0) (3.28)
0 0 1

f1(zi€)w) : —Z +(1U1 + 1 + Z)€17and f2(z1€)w) : Z + 5% - £1103
We compare the performance of two output feedback control designs. Design 1
implements the dynamic output feedback control law of [50] with all design parameters

chosen as presented therein, and is given by

7') = P72 + C26 (3.29)
C = <I>C + Nsat(l, d) (3.30)
u = —Ksat(l, 6“) + 6KMC (3.31)

58

“I

where d = n,- + kT-laon + - - ~ + ka,_2n,._1, in which r denotes the relative degree of

the system, the saturation function sat(l, d) is defined as

. 6 if [é[Sl
satl,0= . .
( ) ‘3 if [0[>l
[ii
0 1 0 1
—400 1 —400
1’: .Q= ,<I>= 0 01 ,N= 0
—(200)2 0 —(200)2
0 —4 0 0

The following design parameters (as given in [50]) are used in the simulation: k = 10,
K = 25, e = 2, and l = 50. Design 2 uses the conditional servocompensator
(3.13). Assumption 3.1 is satisfied since the system is minimum-phase and the zero
dynamics are globally exponentially stable. Assumption 2.4 is satisfied with V(C) =
tax? + 2€1€2 + 5%) and w(ew) = -53 + as - his -— as where 1w = 162 = 1.
Assumption 2.5 is satisfied with v(C) = C1 + 2C2 and H (x, w) = 1. Since the closed-
loop system is locally exponentially stable, Assumption 2.7 is also satisfied. The

control it for Design 2 is given by

 

u = -—12 sat (£1 + 2:2:- K10) (3.32)

The state estimate C2 is provided by the high-gain observer

fl = 32 + 91(61 -€1)/€, £2 = 92(51 —€A,1)/€2 (3-33) -

where 91 and 92 are chosen such that the polynomial A2 + 91A + 92 is Hurwitz. We

use the following numerical values in the simulation: u = 0.2, 91 = 2, g2 = 1 and

59

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I I I I
0.2
0
e
E -o.2
LU
S
E -0.4
3
8
r: -0.6
'0'8 - - - Design 1 - High-Gain Approach (Serrani, Isidori. & Marconi, 2000) 3
-— Design 2 - Conditional Servo Approach
-1 —-1
l l l l I l l
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (sec)
(b)
I I I I I I I
25 r' - - - Design 1 - High-Gain Approach (Serrani, Isidori. 81 Marconi, 2000) ‘
l. —- Design 2 - Conditional Servo Approach
20 —', —
|
3 t
‘5' 15_. e
g I
Z |
2 10 ~ 1 e
S t
o 1
\
5 - a
\
\
o — X
l l 1 l l l l
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (sec)

Figure 3.3: Performance of the two control designs (a) Regulation error ’e’ during the

transient period (b) Control input ’u’

60

e = 0.01. The gain vector K1 is chosen so as to assign the eigenvalues of (S — J K 1)
at —0.5, —1,-1.5 and -2.

Figure 3.3(a) shows the regulation error during the transient period and Figure
3.3(b) shows the control input for the two designs. The transient response of the con-
troller design using high-gain approach (Design 1) has a better rise-time but results
in overshoot, which is not present in the case of the controller design using a condi-
tional servocompensator (Design 2). Notice that, in case of Design 1, a large control
effort is needed to regulate the error to zero, which in case of Design 2, is saturated
to a pre-determined threshold. The use of high—gain feedback, by its nature, leads
to a large spike in the control during initial period, and this can be seen in Figure
3(b). It turns out that the trajectories of the closed-loop systems of Design 1 and
Design 2 are close to each other under high-gain feedback, so that for the same set
of initial conditions or for the same given region of attraction (achieved by adjusting

the saturation level), the performance of both designs are close to each other.

3.5 Conclusions

We considered output feedback regulation problem for a class of minimum-phase,
input-output linearizable, nonlinear systems, where the states of the system are regu-
lated to a disturbance-dependent invariant manifold on which the regulation error is
zero. We showed that the use of conditional servocompensators enables us to achieve
zero steady-state regulation error, in the presence of time-varying exogenous signals
that are generated by a known exosystem. The output feedback control is imple-
mented using a high-gain observer. The state feedback controller of Chapter 2 can
be viewed as an intermediate step towards the output feedback controller of Chapter
3. Analytical results are provided for a compact set of initial conditions, which can

be chosen arbitrarily large if all the conditions hold globally.

61

Chapter 4

Regulation of Linear Systems

Subject to Input Constraints

4. 1 Introduction

We consider the problem of output regulation for linear systems subject to input
saturation. Stabilization of linear systems under input constraints has been exten-
sively studied over the past decade, c.f., [15, 17, 38, 47, 55, 56]. When the open-loop
eigenvalues are in the closed left-half plane, global or semi-global stabilization can be
achieved by low-gain feedback or by a combination of low-gain and high-gain feedback.
The techniques are extended to the servomechanism problem in [37]. Although more
recent results [12, 19] have also dealt with cases with right-half plane eigenvalues; we
consider here the case of left-half-plane eigenvalues. The presence of saturation in
the input channel imposes strong limitations to the achievable control objectives such
as transient performance. In order to achieve desired control objectives we cast the
output regulation problem for linear systems subject to input constraints in the Lya-
punov redesign framework as presented in Chapter 2. The key feature of this idea is

that the conditional servocompensator acts as a traditional servocompensator only in

62

a neighborhood of the zero—error manifold, while it is a bounded—input-bounded-state
stable system whose state is guaranteed to be of the order of a small design parame-
ter. The use of conditional servocompensators enables us to achieve zero steady-state
tracking error without degrading the transient response of the system. The goal of
this work is to apply the Lyapunov-redesign—servocompensator approach of Chapter 2
to the linear regulation problem under input constraints and compare its performance

with the approach presented in [37].

We extend the state feedback design to output feedback by using a full-order
high—gain observer. This is different from Chapter 3 where a reduced-order high-gain
observer was used because the state feedback control was a partial one. In the current
problem, because of the control constraint, the mechanism of solving the stabilization
problem through ARE necessitates the use of full-state feedback. We use the singular

perturbation approach to the observer design described in [13].

4.2 Low-gain design for linear systems

In this section, we briefly review the approach presented in [37] for the semiglobal
output regulation problem of linear systems subject to input saturation. Consider a

single-input single-output linear system

C = AC+Bg(u) +Ew

e = CC+Fw (4.1)

where C E R" is the state, 11. is the control input, 6 is the regulation error and w(t) is
an exogenous input that belongs to a compact set W E Rw, and is generated by the
internal model

21') = Sw (4.2)

where S has distinct eigenvalues on the imaginary axis. The function g is defined as

y if lylsl
e(y)= —1 if y<—1 (4.3)

1 if y>1

Assumption 4.1. The matrix A has all eigenvalues in the closed left-half plane, the
pair (A, B) is stabilizable, and the pair (A, C) is detectable.

Assumption 4.2. There exist matrices II, I‘ such that
IIS=AII+BF+E, 0=CH+F (4.4)
where [I‘m] 3 1 — 6 for all 111 E W, for some 0 < 6 <1.

It is shown in [37] that if C and 11) were available for feedback, a stabilizing state

feedback control law can be taken as
u = —K(A)C + [K(/\)H + F]w (4.5)

where K (A) = BTPO‘), and P(A) is the positive definite solution of the Riccati
equation

P(>\)A + ATPo) — P(A)BBTP()\) + Q(A) : 0 (4.6)

in which Q()\) is a positive definite matrix that satisfies lim,\_,0 Q(/\) = 0. The
positive parameter A is chosen small enough that the control does not saturate over

the domain of interest. Assuming that

AE
[CF],
0 S

64

is detectable, the state feedback design is extended to an error feedback design by

using the observer

6 A E 6 B
= + 2(a)
a 0 s a 0
L . .
+ A (e—[C F] C ) (47)
L3 If)

where the matrices L A and L S are chosen such that the matrix

_ A—L C E—L F
A—é A A (4.8)
-LSC S—LSF

is Hurwitz. The error feedback control law is given by
u = —K(A)C + [K(A)H + I‘]zf) (4.9)

With the change of variables, C = C — Hw, C = C — C and if) = w — if), the closed-loop

system can be written as

6 = Ag + Bel—mm + W — w) + K<A><€ — 11229)]
+ (AU — US + E)w

(A — L Ami + (E — L Ana (4.10)

A).
II

{a

= —LSCC + (S — LSF)zD

The key feature of the approach [37] is designing an observer with much faster dy-
namics than those of the original system. Since the last two equations of (4.10)

are homogeneous, designing the observer dynamics arbitrarily fast yields rapid error

65

convergence to zero. Hence, lirnt_.00 e(t) = O.

4.3 Problem Statement and Control Design

We now cast the output regulation problem for linear systems subject to input con-
straints in the Lyapunov redesign framework as presented in Chapters 2 and 3. Our
goal here is to design an output feedback controller for the system (4.1) to stabilize
the system when w = 0 and to asymptotically regulate e to zero when w # 0. With

the change of variables x = C — 7r, the system (4.1) can be written as

:i: = Ar + B[g(u) — Fw]

e = Cr (4.11)

The system (4.11) is in a form where the state feedback regulation problem can be
formulated as a state feedback stabilization problem by treating Fw as a matched
uncertainty. We design a low-gain feedback control law to achieve stabilization and
then introduce a conditional servocompensator through a saturated high-gain feed-
back. Towards that end, let K (A) = BTPO‘) be the state feedback gain matrix. The
derivative of the Lyapunov function V(x) = xTP()\):I: with respect to the nominal

system at = [A — BK()\)]:1: is
V(x) = —:rT[Q(/\) + P()\)BBTP(/\)]a: (4.12)

For convenience we write (0V/8x)B = v(x), where v(x) = 2BTP(A)$. The system

(4.11) can be written as

:i: = (A — BK)1L‘ + Bp(u) + B[K:z: — I‘w] (4.13)

66

We design a saturated high-gain feedback controller for this system to deal with the

uncertain term I‘w. Let Q = {V(x) S Cl} C X be a compact set for some c1 > O and
|K(/\)a:—I‘w|31—60, (50>O VxEQ,VwEW (4.14)

We introduce the conditional servocompensator via the saturated high gain feedback

it = — (E) (4.15)

where s = v(x) + K10 and a is the output of the conditional servocompensator

controller

a = (S — JK1)0 + [1.79 (E) (4.16)

where a > 0 is the width of the boundary layer, (S, J) is controllable and K 1 is chosen
such that S — J K 1 is Hurwitz. Equation (4.16) is a perturbation of the exponentially
stable system ('7 = (S — J K 1)0, with the norm of the perturbation bounded by ,u. In

order to show that a is always 0(a), we define the Lyapunov function
V0(0) = UTP00’

where the symmetric positive definite matrix P0 is the solution of PoAa +AZP0 = —I
and Ag Q S - JK 1. Consider the compact set {a : V0(a) S p2c2}, where c2 is a

positive constant. Let 0(0) belong to this set. Using the inequality
' 2
V0(U) S - ”all + 2M Hall HPOJH

it is easy to show that 170(0) 3 0 on the boundary V0(0) = a2c2 for the choice

c2 = 4 ”POJI]2 Amax(P0). Hence, the set {a : V0(0) _<_ u2c2} is positively invariant.

67

4.4 Closed-Loop Analysis

In this section, we will show that, for sufficiently small it, every trajectory of the
closed-loop system asymptotically approaches an invariant manifold on which the
error is zero. The forthcoming analysis follows the procedure of Section 2.3 and is
presented mainly to delineate the more sharper results that can be obtained in case

of the linear systems. Towards that end, the closed-loop system is given by

ib=Sw

is = [A—BK]a:—Bg(£)+B[Ka:—Fw] (4.17)

(7 = Add +ng (i)
y

We start by showing that the set \II = Q x {V0(0) S a2c2} is positively invariant
and every trajectory in \II reaches the positively invariant set \IJp = {V(x) S p(/.i)} x

{V0(a) S a2c2} in finite time, where p is a class [C function.1

- (9V 8V 3 (9V
V — E[A — BK]17 — -a—x-BQ (p) + 51:3[K13 — Fw]

- —xT T :r— s— a i
— iQ<A>+P<A>BB PW] < K1 h(u)
+ (s — K10)[K:r - Fw]

= —:L‘T[Q(A) + P()\)BBTP(A)]17 — 39 (E)

+ K109 (E) + s[K:r — Fw] - K10[K:z: — I‘w]

 

1The set \II“ is defined as a positively invariant set, unlike the definition used in Chapter 2 where
‘11,, = {leH S p(p)} x {Vo(0) S p2C2}. The sharper definition used here is due to the fact that V(x)
is independent of w.

68

 

Inside ‘11, ”all S pfig/Amin(P0). When Is] 2 a, from (4.3) and (4.14), we have

V

|/\

-$T[Q(/\) + PWBBTPO‘W - |8|+ ||K1ll IIUII +|8| +||K1||l|0||

|/\

-2:T[Q(A) + P(A)BBTP(A)]x + 1171 (4.18)

 

where 71 = 2||K1|| \/CQ/)\min(P0). Similarly, when [3] S ,u

2
v s -leQ(/\) + P<A>BBTP<A>14 — '7' + “Kin Hall % + Isl + 1K1” “an
s 4042(4) + P<A>BBTP<A>14 + m (4.19)
where 72 = 71 + (1/4). From (4.18) and (4.19), we have
v g —:1:T [42(1) + P(/\)BBTP(A)]1: + m, V(x,a) e 11 (4.20)

Hence, from [29, Theorem 4.18], for sufficiently small a, ‘1! is positively invariant and

all trajectories starting in ‘11 enter the positively invariant set ‘11” in finite time.

Next, we use V1 2 $32 to show that the trajectories reach the boundary layer
{Isl S p} in finite time. Since P(A) is positive definite, BTP(A)B > 0. Let kp =
2BTP()\)B. For (33,0) 6 \Ilfl, we have

ss 2 —2sBTPBg (E) + 2sBTP[A — BK]:z:

+ ZSBTPB[K11: — I‘w] + sKle + asKng (2)
Outside the boundary layer, i.e. when [3] 2 ,u, we have

55‘ g —k,,|s| + |]2BTP[A — BK]1:|||s| + klem — Fst]
+ (lloll IIK1II IIAall +ullK1ll IIJll) Isl (4-21)

69

 

Inside II”, [[0]] S H\/62//\min(P0)- Also, the function [A — BK (A)]:z: is continuous
and vanishes at a: = 0. Therefore, the norm [IZBTP()\)[A - BK (A)]:c I] together with
the norms ||0|| ”Kl” [[A0]] and a ||K1|] [[1]] can be bounded by a class [C function

(9104)- Hence,

s -kp|s| + kp(1- 60)Isl + plansl

_k, [60 - (1,55) Isl

|/\

 

=> V1

Thus, for sufficiently small a, all trajectories inside ‘11), would reach the boundary

layer {Isl S a} in finite time.

Finally, we show that inside the boundary layer the trajectories of the closed-loop
system asymptotically approach an invariant manifold on which the error is zero.

Inside the boundary layer, the closed-loop system (4.17) is given by

u') = Sw
:i‘ = [A—BK]:c—B(;)+B[K:c—Fw] (4.22)

('7 = S0 + Jv(:z:)
From [53], there exists a unique matrix A such that
SA = AS, —K1A = I‘ (4.23)

We define N,“ = {:c = 0,0 = 6}, where 6 = aAw. It is easy to verify by direct
substitution that N” is an invariant manifold of (4.22) for all 21) E W. Defining

6 = a — 5 and s' = v + K16, the closed-loop system inside the boundary layer can be

70

written as

if) = Sow
i‘ = [A-BK]$—B(E)+BK:U (4.24)

a = A06+J§ = 85+JU
Define the Lyapunov function

1)
V2 = V(x) + Z635005 + E? (4.25)

where b and c are positive constants to be chosen. Calculating V2 along the trajectories

of the system (4.24), we obtain
. . b . . .
V2 = V + — [5TP05 + anoa] + c§§ (4.26)
p

Calculating V along the trajectories of (4.24), we have

- 6V 8V (9' 8V
2 [U]2 ka ~
S-Amin(Q)llxll - 7 + Ilvl “all + kolvlllrll (427)

where kc and k0 are the upper bounds on ”K 1” and “K”, respectively. The second

term of V2 satisfies the inequality

b - - b
— [6TP06+6TP06] g ——||&||2
,u H
2bk 1 -
+ 7110“ Ishmael) (4.28)

where k1 is the upper bound on “.1“. Next, we have

71

; av 01) s 8v
3 — EM — BK()\)]:1: — DEB (Z) + 553K003:
+ K1(S5 + .111)
c§§ S -C(kp/#)|§|2 + ck2|§|||517|| + ck3|§| ”5”

-l- ck4|§||v| (4.29)

where k2, k3 and k4 are some positive constants. From (4.27), (4.28) and (4.29), we

have

' 2 1 2 b ~2 ~2
V2 S -)\min(Q)|l$l| —;|U| —;H0|| “C(kp/flllsl
k ~ - ~
+ kol’UHlIH + gal”! ”0“ + Ck4|$|lvl + ckzlsllll‘ll

+ [(Zbkl/iulAmax(P0l + ck3l ||5|| |§| (4-30)

The right-hand side of (4.30) can be arranged in the following quadratic form of
H = I lell lvl “5'“ [El 1T2
V2 3 —HTAH (4.31)

where the symmetric matrix A is given by

( )‘min(Q) "£31 0 #2 \

 

 

K 1‘2 )

Similar to Chapter 2, the leading principal minors of A can be made positive by

first choosing b large enough, and then, choosing c large. Finally, by choosing a

72

 

sufficiently small, V2 will be negative definite. Therefore, inside the boundary layer,
the trajectories of the closed-loop system will exponentially approach the zero-error

manifold N“ as t —+ 00. Our conclusions can be summarized in the following theorem.

Theorem 4.1. Suppose Assumptions 4.1 - 4.2 are satisfied and consider the closed-
loop system comprising of the system (4.11), the servocompensator (4.16) and the
state feedback control (4.15). Then, there exists if“ > 0 such that Va 6 (0, ,u*], the set
the set ‘11 = Q x {V0(o) S u2c2} is a subset of the region of attraction, and for all

initial conditions in \II, the state variables are bounded and limtmoo e(t) = 0.

4.5 Output Feedback Design

In this section, we extend the state feedback controller of the previous section to out-
put feedback by using a full-order high-gain observer. This is different from Chapter
3 Where a reduced-order high-gain observer was used because the state feedback con-
trol was a partial one. In the current problem, because of the control constraint, the
mechanism of solving the stabilization problem through ARE necessitates the use of
full-state feedback. We use the singular perturbation approach to the observer design
described in [13].

The state feedback control (4.15) is implemented as an observer-based controller

:f: = Ai- + Bg(u) + L(e — 02)

u = 4 (S) (4.32)

where L is the vector of observer gains to be designed and s = 2BTP()\)§: + K10.

The estimation error, 77 = x - 5:, satisfies the equation

7') = (A — LC)77 — BFw (4.33)

73

The observer design starts by transforming the system into the normal form. There

(Ea
is a nonsingular matrix T such that :c = T transforms the system (4.11) into

33f
the form
ia = Aaa-Ta + Aafy (4°34)
if = Afrcf + Bf[Ea:ca + Efxf + 9(a) — I‘ur] (4.35)
y = CfIL‘f (4.36)

where (A f, B f, Cf) represents a chain of integrators. The eigenvalues of A00 are the
zeros of the triplet (A, B, C). The observer gain L is designed as [13]

A0,
M(€)Lf

L(e) = T (4-37)

where L f assigns the eigenvalues of (A f — L fC' f) in the open left-half plane, M (e) =
blkdiagE, 212, - - - , glq], q = dim(:cf), and E is a small positive constant. The observer
gain L(e) assigns the observer eigenvalues into two groups: (n — q) eigenvalues are
assigned at the open-loop invariant zeros and q eigenvalues are assigned at 0(1/5)
locations, approaching the eigenvalues of (A f — L fC f) /5 as e —> O. The state com-

ponent :ra is estimated by the observer
fa = Aaai‘a + Aafy (4.38)
and the state scf is estimated using the high-gain observer

if =Afif+M(8)Lf(y-Cf§3f)+Bfg(u) (4.39)

74

 

The change of variables 77 = ( T1 T2 ) 773 transforms the error equation (4.33)

17f
into

Tls = Aaans (4.40)

77f = [Af—M(E)Lfo]77f+Bf[Ea773+Ef7—)f-Fw] (4.41)

where 773 2 (ca — 5:0 and i7 = :1: — :7: . To bring the system 4.40 - 4.41 into the
f f f

standard singularly perturbed form we need to scale 17f as

77,. = N‘1(s)r7f (4.42)

where N (5) = blkdiag[£q_l, ~ -- ,5, 1]. With the special structure of the matrices
N(e), Af, Bf, Cf, M(5) and Lf, it is shown in [13] that

N—1(e)Bf = Bf

N_1(€)[Af — M(€)Lfo]N(€) = :‘lAf — Lfo]

where [A f — L fC' f] is Hurwitz. The scaling (4.42) transforms (4.40)-(4.41) into the

standard singularly perturbed system

773 = Aaans (4.43)

Eflf=[Af - Lfo]77f + EBf[Ea7]3 + EfN(€)77f — I‘w] (4.44)

We notice that the perturbation term I‘w is multiplied by 5 so that its effect diminishes

asymptotically as 5 —+ 0.

We now analyze the closed-loop system composed of (4.11), (4.16) and (4.32).

75

 

Using (11:,0, 173,1”) as the state vector, the closed-loop system is given by

it:[A—BK]$+Bg[—-/1:(s—2BTP(A)[T1 T2N(€)] "3 )] +B[K:c—I‘w]
77f

o 1 77
U=AUU+MJQ[—(S_ZBTP()‘) [ T1 T2N(5):| 8 )]
u n
f
m : Aaans (4.45)

enf = [Af — LfC'f]77f +EBf[Ea773 + EfN(c)77f — Fw]

The system (4.45) is a standard singularly perturbed system with (1:,0,773) as the
slow variable and 17f as the fast variable. The slow model of (4.45) is obtained by
setting E = 0 in the last equation of (4.45). Since [A f — L fC’f] is Hurwitz, hence
non-singular, we obtain the unique root 77f = 0. Substitution of 77f = 0 in (4.45)

results in the slow model

:i: = [A - BK]:r + Bg[ — i(s — 2BTP(A)T1773)] + B[K:L‘ — Fw] (4.46)

q.
H

Ago + ,ng [% (s — 2BTP(/\)T1ns)] I (4.47)

7:]3 : Aaans (4.48)

which appears as the cascade connection of (4.48) and the closed-loop system under
the state feedback (4.17). We will now show that, for sufficiently small it, every
trajectory of the system (4.46)-(4.48) exponentially approaches an invariant manifold

on which the error is zero. Towards that end, we define the Lyapunov function

where as > 0 and V3 = nzPsns, in which R; is the positive definite solution of the

76

 

Lyapunov equation
PsAaa + AZGPS : _I (4.50)

The function V3 2 nzPsns satisfies Vs S -—[]275||2. Let $20 2 {V3 S C11} C X be
a compact set for some C11 > 0. We start by showing that the set E = 90 x
{V0 (0) S p2c2} is positively invariant and every trajectory in 2 reaches the positively
invariant set 2], = {V3 S p0(u)} x {V0(o) S u2c2} in finite time, where p0 is a class

IC function.

- 8V , 8V 3 8V
V3 — -8—r[A — BA]:1: - 539 (F) + 5:3[K1‘ — I‘m]

9 (5) _ Q (s — 2BT:(A)T1n.)

= —:cT[Q()\) + P()\)BBTP(A)]:r — (s — K10)g (E) + (s — K10)[K;r — I‘w]

, (i) _ Q ( — 245mm...)

= —:cT[Q + PBBTP]IL‘ — 39 (i) + K109 (E) + s[K:c — Put]
,a

S s— T s
_ K10[K:c— I‘w] +3 [Q (if) _ 9( 28 13007177)

- 1...].(5) _ . (. was...) “

Inside 2, ”a“ S ,u c2/Amm(P0). When Is] 2 u, from (4.3) and (4.14), we have

8V
+ EB

 

+ 013173073)

 

 

+ (3 " K10) + (Isl/3073)

 

 

 

 

 

 

+ cal/3(7).) (4.51)

 

V3 S -/\mm(Q)l|xll2-lS|+||K1|l||0||+ISI+||K1||l|0|l
+{||v(x)|l+||K1IIIIUII}L1llnsll+||K1IIIIUIIL1||773|| - as llnsll2

S -/\mz°n(Q) HIE”2 + 2Amax(P)L1 Ill?” llflsll - as ”Us“2 + #73 (4-52)

77

 

where 73 = 2]]K1|| Cg/Amin(P0)(1 + L1 HUSH), in which L1 is a positive Lipschitz

constant that satisfies the inequality

s—QBTPAT s
4( if ) 1’73) —g(;) _<_ L1 llnsll

Similarly, when ls] S u, we have

 

2
l

l_8 l

|___s

V3 S ‘Amin(Q )le “2-
+{llv(=r)ll+||K1||||0lllL1||nsll+||K1||||0llL1Ilnsll - as llnsll2

s -Amin(Q)|l$l|2+2Amax(P)L1H1?”“77s“*asllnsllgiL/i’m (4.53)

+||K1||||0||—+|S|+I|K1|l||0||

where 74 = '73 + (1/4). From (4.52) and (4.53), we have

V3 S Amm(Q) HHIHZ +2>xmasic(1’)L1||13|l||77sH-0¢s||77s||2 +1474. V(I.0,ns) E 3 (454)

The right-hand side of (4.54) can be arranged in the following quadratic form of

H1 = [”33” llnslllT=

V3 S -HfAIH1+#74 (4-55)

where the symmetric matrix A1 is given by

)‘min(Q) “Ama$(P)Ll
—/\maz(P)L1 as

A1:

By choosing as large enough the matrix A1 can be made positive definite. Then,
from [29, Theorem 4.18], for sufficiently small )4, 2 is positively invariant and all
trajectories starting in 2 enter the positively invariant set 2” in finite time. In the

next step, we use V1: 232 to show that the trajectories reach the boundary layer

78

 

{|s| S p} in finite time. For (23,0,723) E 2,), we have

33' = 2sBTP[A — BK]$ — 2sBTPBg (i) + 23BTPB[Kx — Fw]

g (Ii) _ Q (s — 287mm.)

3 —- zBTP(A)T1ns)
u

+ 2sBTPB

 

 

 

 

+ sKlAa + usKng(
Outside the boundary layer, i.e. when |s| 2 ,a, we have

33 s II2BTP[A — BK]:c Ills] — kplsl + kleiz: — I‘st]

+ kal ”as” [SI + (Hall ||K1|| llAall + H ”Kl” NJ” ”77le ISI (4-56)

 

Inside 2,), [[0]] S p\/c2/Am,-n(P0). Also, the function [A — BK(A)]:1: is continuous
and vanishes at :1: = 0. Therefore, the norm ||2BTP(/\)[A — BK (A)]a: I] together with
the norms kal “173“ , [[0]] ||K1|| “Ag” and u||K1H ||J|| can be bounded by a class [C

function pg ()4). Hence,

I/\

V1: 33 — plsl + kp(1— 60)Isl + p2(u)lsl

-k, [60 — 32%] ISI

|/\

Thus, for sufficiently small ,u, all trajectories inside 211 would reach the boundary

layer {[3] S u} in finite time.

Finally, we Show that inside the boundary layer the trajectories of the closed-loop

system (4.46)-(4.48) exponentially approach an invariant manifold on which the error

79

is zero. Inside the boundary layer, the closed-loop system (4.46)-(4.48) is given by

j; = [A — BK]x — B (E) + B (zBTPffles) + B[Ka: - I‘w] (4.57)

 

 

T
('7 = So + Js — J (28 PLA)T1778) (4.58)
773 = Aaans (4.59)
Next, we define
Z,,={:c=0,0=6,173=0}, 6=qu (4.60)

which is an invariant manifold of (4.57)-(4.59) for all 21) E W. Defining 6 = o — 6 and

s = v + K16, the closed-loop system inside the boundary layer can be written as

i: = [A — BK]:I: — B (E) + B (zBTPLAWWS) + BKx (4.61)

A05 + J5 (4.62)

 

Q»
II

773 = Aaans (4.63)

Define the Lyapunov function

As

V4 = V(x) + Flam.) + 2571155 + 52 (4.64)

MIG

where 63, p and q are positive constants to be chosen. Calculating V4 along the

trajectories of the system (4.61)-(4.63), we obtain

fis

V4 = V + 7V8 + 5- [an05 + 3TP06] + q§§ (4.65)

80

f

Calculating V + Eff/S along the trajectories of (4.61)-(4.63), we have

 

V+%‘—Vs = —IA— BK(A)]x—%Y-B(§ )+ +6VB(2 BTP<A>T1772>
8:6 ,1
+%EBK(A)3:+ %+V.
s —m.-..<Q )le II2— '—”'2+ €va IIIv II+2—-——*m22(P )Iv Illnsll
+ koIvIIIvII - 9; “72.12 (4.66)

where ka and [Co are the upper bounds on ”KI” and IIK II, respectively. The third

term of V4 satisfies the inequality

_2_Pk1

BIarpoa+3TPOaI s —§IIvII2+ II5|IISI/\max(Po) (4.67)

where k1 is the upper bound on IIJ II Next, we have

,-, _ 8'0 81) " BUB 2BTP(/\)T17)3 ~
3 — a—xArr x—8_B(§ )+ 571: B( [J +K1(SO‘+JU)
qksI'S'IIIl‘II - 4(kp/#)I§I2 + 2Q(kp/N))‘max(P) |§I ”Us“

+ qk6I§I ”5” + qk7I§IIvI (4-68)

 

Q
Cm
Cm-

IA

where k5, [CG and 167 are some positive constants. From (4.66), (4.67) and (4.68), we

have

I__UI2+ 2Am__a__:c(P )

. k -
v. s —m.-..<o >IIv II2— ++fIvIIIvII Iv Wall

5 2 P ~
+ ICOIUIIISCII — :3 IIflsII " ; IIUII2+ ——II: II IngmaxIPOI

+ qkslélllxll — (1(kp/rt)|§|2 + 2q<kp/v)/\m..(P )lsl Ilnsll

+ qk6I§I ”5” + qk7I§IIv| (4.69)

2P__’_€_1

81

The right-hand side of (4.69) can be arranged in the following quadratic form of

H2 = I ”1‘” IIflsII IUI II5II I§I IT:
V4 S —H§A2H2 (4.70)

where the symmetric matrix A2 is given by

{ Amin(Q) 0 _EQO' #5" \

Q2 __ Amara?)
p

0

p 0

= 1 _k
A2 4 It

2

u

l 212.2 I

As before, if the principal leading minors of A2 can be made positive by choosing

 

 

the constants 63, p and q appropriately, and by choosing p sufficiently small, then
V4 will be negative definite. This would imply that, inside the boundary layer, the
trajectories of the closed-loop system will exponentially approach Z), as t —+ 00.

Towards that end, we partition the matrix A2 as

Amman —d¥‘2
A2 = (4.71)
-d12 £022 + A22

where

d12=(0 530 0 i’g‘i)T (4.72)

82

 

l (is —Ama.(P) 0 —qkp4m..(P))

k
I — -f‘ 0
022 = (4.73)

P ‘Pkl Amax (P0)

) 2’27 /

 

 

and

I 0 0 0 0 )
0 0 0 ‘2’“
A22: 71 (4.74)

ooo—i'gfi
(000 0)

From (4.73), it is easy to see that by choosing 63, p and q, we can successively make

 

 

the principal leading minors of D22 positive. First, 63 is chosen large enough to make
the 2 x 2 minor positive, then, p is chosen large enough to make the 3 x 3 minor
positive, and then, q is chosen large enough to make the 4 x 4 minor positive. Finally,

choosing In small enough will render

Ammo) ~42; ,
det > 0 (4.75)
-d12 $1722 + A22

Consequently, V4 will be negative definite. Therefore, inside the boundary layer, the
trajectories of the closed-loop system (4.61)-(4.63) will exponentially approach the

zero-error manifold Z”.
Let the initial states (3(0), 0(0), 713(0)) 6 g and 77f(0) E H, where g is a compact
set which contains Z”. Using Theorems 2 and 5 of Atassi & Khalil [3], we can show

that there is a neighborhood N of Z ,, x {77f = 0}, independent of e, and 81 > 0 such

that for every 0 < e S 51, the set Z) x {W = 0} is exponentially stable and every

83

trajectory in N converges to this set as t —-> 00. Furthermore, from [3, Theorems
1, 2 8c 5], there is 52 > 0 such that for every 0 < e S 52, the solutions starting in
Q x H enter N in finite time. Hence, for every 0 < e S 83 = min {51,82}, the
set Z” x {77f = 0} is exponentially stable and g x H is a subset of the region of
attraction. Thus, for sufficiently small 5, the closed-loop system (4.45), under the
output feedback controller (4.32), is uniformly exponentially stable with respect to
the set Z” x {71f = 0}. Hence, limtmoo e(t) = 0. The foregoing conclusions are

summarized in the following theorem:

Theorem 4.2. Suppose Assumptions 4.1 - 4.2 are satisfied and consider the closed-
loop system comprising of the system (4.11), the servocompensator (4.16) and the
observer-based feedback control (4.32). Then, there exists 6* > 0 such that V5 6

(0, 5*], the closed-loop system, is uniformly exponentially stable with respect to the set

Z), X {77f = 0}.

4.6 Example

Consider a minimum-phase linear system, with the transfer function from u to e

s + 1
T = —
(8) 32(3 + 5)
that corresponds to (4.1) with
0 1 0 O 0 0

A: 00 1 ,3: 0 ,E= 00
00—5 1 10

 

with the signal to generated by the exosystem

0 w T
ii}: 71), w (O)=(0,w0)

—w 0
We show the performance of two designs: Design I incorporates the saturated high-
gain feedback using a conditional servocompensator and the full—order observer (5.12)-
(5.13). For this design, K1 is chosen so as to assign the eigenvalues of S — JK1 at
—0.5 and —1, and the observer gain L(e) is designed such that the eigenvalue of (5.12)
is assigned at the location of the invariant zero of the triplet (A, B, C), i.e. at —1,
1
and L f = g is chosen such that the polynomial 1272 + 91:27 + 92 is Hurwitz.

92
Design 11 is based on the linear observer-based error-feedback control approach [37],

reviewed in Section III. For this design, a fifth-order linear observer of the form
(4.7) is constructed where the matrices L A and L S are chosen such as to assign the
eigenvalues of the matrix A at [—22, —23, —24, -25, -26]. We use the following
numerical values in the simulation: (.2 = 1 rad/s, wo = 0.5, ,u = 0.1, 91 = 2, g2 = 1,

A = 0.05 and e = 0.05.

Figure 4.1(a) shows the regulation error during the transient period for the two
designs and Figure 4.1(b) shows the corresponding control input. The regulation
error goes to zero sharply in the case of Design I, where as in Design II, the same
oscillates before eventually converging to zero. Note that due to the fact that a higher
dimensional (fifth-order) observer is used in Design II, in order to achieve reasonable
performance, the observer gains were required to be pushed very high e.g. 0(107),
in contrast to 0(103) for those in Design I. Figure 4.2 shows the performance of the

two control designs when the control coefficient is perturbed by 40% (from 1 to 1.4).

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 I I r I I l
0 ’\ —-Design-I
E 1-" \‘ ---Deslgn-II1
m ’ \
c ’ ‘
o 0.5 \ ‘
3 \
g \
O \ A' ..... ~
«I 0* “ / -—-.. ----- __
m ‘ ’,
\.'o
_05 l l l l l l
0 10 20 30 40 50 60 70
TIme(sec)

(b)

1_ I I I I I I _
j. 05— . . ~
3 l ‘ f
2’ r ‘ r
_ g g .-
§ 07‘, 1‘ '
‘E 1 ,'
8 -0.5rl v

'1E'J 1 a 1 1 I l ‘
0 10 20 30 40 50 60 70
TIme(sec)

Figure 4.1: Performance comparison of the two control designs (a) Regulation error
’e’ during the transient period (b) Corresponding control input, ’u’

86

 

 

 

 

 

 

 

 

 

 

 

 

 

2 I I I I I I T I I
S. — Nominal
g - - -Perturbed _
o
C
2
E _ _ _ _
3 - _, .. - _ _. _.
U)
o
I!
m l l
70 80 90 100
Time (sec)
(b)
0.6 I F I I I I r I I
— Nominal
0,4] - - - Perturbed ‘

 

 

 

 

Regulation error, e
o
N
I
l

O
I

 

 

l l l I l l l l I

60 70 80 90 100

 

O
A
O
N
o
8
A
O
8

Time (sec)

Figure 4.2: Transient performance of the two control designs when control coefficient
is perturbed by 40 percent (a) Design II - Nominal vs Perturbed (b) Design I -
Nominal vs Perturbed

87

4.7 Conclusions

In this chapter, we considered the output regulation problem of linear systems subject
to input constraints. We presented a novel control design that includes a conditional
servocompensator, introduced via Lyapunov redesign and saturated high-gain feed-
back. The use of a conditional servocompensator enables us to achieve zero steady-
state regulation error, without degrading the transient response. The output feedback
control is implemented using a two-time-scale observer design of [13] and the perfor-
mance recovery is shown using the separation principle of [3, 1]. The performance of

the control design is demonstrated by simulation.

88

 

Chapter 5

Full-Order High-Gain Observers
for Minimum Phase Nonlinear

Systems

5.1 Introduction

Most nonlinear control techniques assume availability of all state variables to achieve
control objectives like stability or asymptotic tracking. Since, in many practical
problems we cannot measure all state variables due to technical or economic reasons,
a state observer is used to estimate the system states from the output measurements.
An exhaustive review of the many approachas to design observers for stabilization of
nonlinear dynamical systems appears in [16]. Among these, one popular approach
is the high-gain observer which is attractive because of its ability to estimate the
unmeasured states while rejecting the effect of disturbances. High-gain observers are

applicable to a class of nonlinear systems that can be transformed into the normal

89

2 = 7,!)(z, x) (5.1)
x = Ax + B¢(z, x, u) (5.2)
y = Cx (5.3)

where z E R1 and x 6 RT are the system states, it E R is the control input and y E R
is the measured output. The r x r matrix A, the r x 1 matrix B and the 1 x r matrix

C, are given by

 

 

 

 

0 1 0 0
0 0 1 O 0
A: , B: ,
O 0 1 0
L0 O_ _1_
C: 10 0

Over the past several years, many researchers have contributed toward the investiga-
tion of output feedback control for the class of systems of the form (5.1)-(5.3). Of
significant relevance are the works [2, 3, 13, 29], which solve the problem of robust
output feedback stabilization, in the large, of the input-output linearizable nonlinear
dynamic systems by means of bounded partial-state feedback control (e.g. u = v(x))
and high-gain observer, with subsequent substitution of the estimate of as, provided
by the high-gain observer, in the feedback. The boundedness of the control protects
the state of the plant from peaking when the high-gain observer estimates are used
instead of the true states.

In certain applications, feedback control synthesis requires availability of the full

state vector (i.e. x and 2:), rather than just the partial state x. One such example

90

 

is the output regulation of linear dynamic systems under input contraints treated
in Chapter 4. Because of the control constraint, the mechanism of solving the sta-
bilization problem through Algebraic Riccati Equation (ARE) necessitates the use
of full-state feedback. In order to design an output feedback control to achieve the
desired control objective in the presence of input constraints, the full-order observer
design described in [13] is exploited. Extensions to nonlinear systems of the form (5.1)-
(5.3) would necessitate the development of a full-order nonlinear high-gain observer.
A rather similar situation arises in applications that incorporate optimal stabilizing
controllers. The optimality of the stabilizing controller yields strong robustness prop-
erties by virtue of its design through the existence of a Control Lyapunov Function
(CLF) [48]. However, the optimal control design approach assumes the availability of
all state variables in order to meet the control objectives. Consequently, the output

feedback control can only be implemented using a full-order observer.

A few researchers have considered the estimation of the full state of nonlinear sys-
tems using high-gain observers. Esfandiari and Khalil, in [13], use a pole placement /
singular perturbation approach to design a one-parameter observer gain, in order to
recover the robustness properties of a state feedback controller designed to stabilize

a fully linearizable system of the form

x = Ax+B¢(x)u (5.4)

y = C(33) (5.5)

The state feedback control u = F (x) is implemented as an observer-based controller

:2 = .42 + ng(x)u + L(y — cs) (5.6)
u = 177(2) (517)

91

where L is the observer gain. The observer design starts by transforming the system

x
into the normal form. There is a nonsingular matrix T such that x = T a
17f
transforms the system (5.4)-(5.5) into the form
i‘a = Aaa-Ta'I‘Aafy (5-8)
if = AfIL‘f-I-BfIEaIL‘a-I-Efl‘f-I-UI (5.9)
y = Cfx f (5.10)

where (A f, B f, C f) represents a chain of integrators. The eigenvalues of Am are the

zeros of the triplet (A, B, C). The observer gain L is designed as

Aaf
M(€)Lf

L(e) = T (5.11)

where L f assigns the eigenvalues of (A f — L fC f) in the open left-half plane, M (e) =
block-diag [$212, - -- ,E—lq], q = dim(xf), and e is a small positive constant. The
observer gain L(e) assigns the observer eigenvalues into two groups: (n—q) eigenvalues
are assigned at the open-loop zeros and q eigenvalues are assigned at 0(1/8) locations,
approaching the eigenvalues of (A f — L fC f) / e as e —> 0. The state component xa is

estimated by the open-loop observer
(La = Aaaia + Aafy (5.12)
and the state x f is estimated using the high-gain observer

23f=Afif+M(€)Lf(y—Cf:8f)+Bfg(u) (5.13)

92

This design results in a standard two-time scale singularly perturbed system. Subse-
quent analysis in [13] reveals the potential of using this approach towards development
of full-order high-gain observers for the class of nonlinear systems under investigation
here.

A recent result [28] designs a full-order observer for the class of nonlinear systems

of the form (5.1)-(5.3). The observer is designed as

2 = 2AM) (5.14)

a»
II

Ax + 86(2, 5:, u) + H(y — 02) (5.15)

where the observer gain H is chosen as

 

 

21
C
2%
C
H = '
O
_ Bf .
in which 6 > 0 is a design parameter and the positive constants 07,-,2' = 1,. . . ,r are

chosen such that the roots of
Ar+alAT_1+---+ar_1)\+ar =0

are in the open left-half plane. Assuming that the state component x can be rapidly
estimated by the observer (5.14)-(5.15), reduces the problem to only that of ensuring
that the observation error of the state component 2 (the internal dynamics) goes to
zero. The analysis in [28] focuses on the observer design, and the performance of
the closed-loop system under output feedback is not considered. The drawback of

the above approach is that the observer design relies on the exact knowledge of the

93

 

nonlinear functions w(z,x) and d(z,x, u), and does not take into consideration the
effects of model uncertainties. Moreover, the effect of peaking is not investigated,
since the functions w(z,x) and d(z,x,u) are required to be globally Lipschitz in x
and z. This yields global results and, therefore, the problem of destabilization due
to peaking does not arise, though peaking is present, and can lead to a degraded
transient behavior of the closed-loop system under output feedback.

The goal of this chapter is to design a full-order observer for a class of nonlinear
systems of the form (5.1)-(5.3). Our design approach is based on the earlier work of
Esafandiari and Khalil [13]. In contrast to [28], we allow for model uncertainties of
the functions w(z, x) and d(z, x, u), and do not require them to be globally Lipschitz
in x and 2. We establish the performance recovery properties of the output feedback
design using the separation principle of [2].

The rest of the chapter is organized as follows. Section 5.2 states the problem
formulation and motivation for the observer design which comes from the earlier
work [13, 2] and the more recent result [28]. Section 5.3 recalls the separation results
of [2] in order to establish the performance recovery properties of the output feedback

design. Section 5.4 discusses the main results.

5.2 System Description and Problem Formulation

Consider the nonlinear system given by equations (5.1)-(5.3). The synthesis process
of an output feedback controller for this system involves two steps. First, a state
feedback controller that uses measurements of the states (z, x) is designed to asymp-
totically stabilize the origin. Then, a state observer is designed to estimate (z,x).

The system (5.1)-(5.3) is required to satisfy the following assumption:

Assumption 5.1. o The functions i/J and (b are locally Lipschitz in their argu-

ments for (z, x, u) E DZ x D1, xR, where Dz C RIC" and Du,- C R" are domains

94

that contain their respective origins.
o w(z,x) = (0,0), and ¢(2,x, u) = (0,0,0).

The stabilizing state feedback controller takes the form

79a 2 Fa(19a,z,m) (5.16)

u = 7(I9a,z,x) (5.17)

The state feedback design is required to satisfy the following properties:

Assumption 5.2. o The functions Pa and '7 are locally Lipschitz functions in

their arguments over the domain of interest.
0 Pa and 7 are globally bounded functions of x.
o I‘a(0,0,0) = 0, and ’y(0,0,0) = 0.

The closed-loop system under the state feedback controller (5.16)-(5.17) is given

by
i = w(z,x) (5.18)
x = Ax+qu(z,x,'y(i9a,z,x)) (5.19)
19}, = r(ea,z,m) (5.20)

Consider now the full-order observer, given by the equations

211003155) (5.21)

i: = 42 + B¢0(2,:’i~, u) + H(y — .21) (5.22)

N)-
II

where w0(z, x) and ¢0(z, x, u) are, respectively, nominal models of the nonlinear func-

tions w(z,x) and d(z,x,u). The functions ’l/J0(Z,:L‘) and ¢0(z,x,u) are required to

95

 

satisfy the following assumption.

Assumption 5.3. o The functions 7120(25, x) and (b0(z, x, 7(79, z,x)) are locally Lip-

schitz in their arguments over the domain of interest.
0 The functions w0(z,x) and ¢0(z,x,7(i9, z,x)) are globally bounded in x.
o w0(z,x) = (0,0), and ¢0(z,x,u) = (0,0,0).

Remark 5.1. We require the functions w0(z, x) and ¢0(z, x, u) to be globally bounded
in x to avoid the efi'ect of peaking, which occurs in the observed states 5: and propagates

to the state variables (z,x, 2) through the control law.

The observer (5.21)-(5.22) comprises two subsystems, each with different dynam-
ics. Equation (5.21) is the observer for the internal dynamics and, by design, is an
open-loop observer, yielding in its slow dynamics. Equation (5.22) is a high-gain ob—
server, with fast dynamics. We proceed with the analysis of the closed-loop system in
two steps. In the first step, we consider a situation when x is available for feedback,

and all we need is the estimate of 2, which is provided by the observer

.2 = 770(23, 2:) (5.23)

The state feedback control (5.16)-(5.17) is modified to include the observer (5.23) as

via = I‘a(790,2,x) (5.24)
2 = 7002.23) (525)
u = 7(i9a,2,x) (5.26)

96

We define

19 F
19:: a , F=: a (5.27)

V0

N)

Using (5.27), the partial-state feedback control (5.24)-(5.26) becomes

19 = I‘(i9, x) (528)

u 2 7(19, x) (5.29)

Then, the closed-loop system under the partial-state feedback control (5.28)-(5.29) is

given by

2 = w(z,x) (5.30)
x = Ax+B¢(z,x,'y(79,x)) (5.31)
19 = P(19,IL‘) (5.32)

In this step of the analysis we combined the slow dynamics of (5.21) with those of
(5.18)-(5.20). The closed-loop system (5.30)—(5.32) is required to satisfy the following

assumption.

Assumption 5.4. The origin (x = 0, r9 = O) is an asymptotically stable equilibrium

point of the closed-loop system {5.30)-(5.32).

We will elaborate on the implication of this assumption in Section 5.4. In the
second step of the analysis, we bring-in the high-gain observer (5.22) to estimate x.
The dynamic output feedback controller is given by equations (5.28)-(5.29), with x

replaced with :2.

97

5.3 Performance Recovery

The purpose of the following analysis is to show that under the stated assumptions,
the origin (2 = 0,11: = 0,190 = 0,2 = 0,57: = 0) of the closed-loop system under
output feedback is asymptotically stable for sufficiently small 6. For the closed-loop
analysis, the observer dynamics are replaced by the equivalent dynamics of the scaled

estimation error [2]
D(e)17 = x - :3: (533)

where
D(e) = [Er—1,€r_2, - ~ ,1]

The closed-loop system under the output feedback controller can be written as

2 = 442,2) (5.34)
:5 = Ax+B¢(z,x,7(i9,x—D(e)n)) (5.35)
6 = I‘(i9,x—D(e)r}) (5.36)
a) = Aon+eB6(z,x,i9,D(e)77) (5.37)

where the matrix %A0 2: (A-HC) is Hurwitz and 6(2, x, i9, D(e)r)) = d(z, x, 7(79, 53))-
¢0(57,7(19. 53))-

5.3.1 Recovery of the Boundedness and Convergence of Tra-
jectOries
Let R be the region of attraction of the closed-loop system (5.30)-(5.32). Let the

initial states be (z(0),x(0),r9(0)) = (x0,zo,i90) E S, and 513(0) = x0 6 Q, where S

is any compact set in the interior of R, and Q is any compact subset of RT. With

98

this formulation, the system (5.34)-(5.37) fits in the framework of [2]. In particular,
the system (5.34)-(5.37) is in standadrd singularly perturbed form, and n = 0 is the
unique solution of (5.37) when 6 = 0. Substituting 77 = 0 in (5.34)-(5.36) yields the
reduced system exactly as the closed-loop system (5.30)-(5.32). For convenience, we

write the system (5.34)-(5.36) as

X = fr(Xa D(€)77) (538)

where x = [zT, xT, 19TIT, and x(0) = [33, x31, 193;]? The reduced system can now be

written as

X = fr(X10) (5.39)

The boundary-layer system, obtained by applying to (5.37) the change of time variable

7 = t/e then setting 6 = 0, iS given by

d0
d7 0?? (5 0)

Let (x(t,e),77(t,e)) denote the trajectory of the system (5.34)-(5.37) starting from
(x(O),r)(0)). The following result is due to [2, Theorems 1, 2 & 3].

Theorem 5.1. Consider the closed loop system (5.34)-(5.37). Let Assmuptions 5.1
- 5.4 hold. Then, the following results hold.

0 There exists e’i‘ > 0 such that, V6 6 (0,51‘], the trajectories (x,r)) starting in

S x Q, are bounded for allt > 0.

0 Given anyC > 0, there exists 6; = 65(C) > 0 and T1 = T1(C) such that,

99

V6 6 (0,65], we have

||X(t. €)I| + ”00.6)“ S 6, Vt 2 T1 (541)

0 Let Xr(t) be the solution of (5.3.9) starting from x(0). Then, given anyC > 0,

there exists 6}; > 0 such that, V6 E (0, 65], we have

IIX(tv€) - 260*)” S E. W > 0 (5.42)

5.3.2 Recovery of the Asymptotic Stability of the Origin

Theorem 5.1 guarantees that the trajectories of the system (5.34)-(5.37), starting in
S x Q, enter a small ball of radius C > 0 around the origin (x, 77) = (0, 0) after a finite

time and stay thereafter. We now show asymptotic stability.

Assumption 5.5. The origin (x = 0,19 = O) is an exponentially stable equilibrium

point of the closed-loop system ( 5. 30)-( 5.32).
The following result is due to [2, Theorem 5].

Theorem 5.2. Suppose the function fr(x, 0) is continuously difierentiable around the
origin. Let Assumptions 5.] - 5.5 hold. Then, under the hypotheses of Theorem 1,
there exists 6:; > 0 such that, V6 6 (0,62], the origin of the system (5.34)-(5.37} is

exponentially stable and S x Q is a subset of its region of attraction.

5.4 Discussion on the main results

The results in Theorem 5.1 guarantee that the trajectories of the closed-loop system

under the output feedback (5.34)-(5.37), starting in S x Q, are bounded and asymp—

100

totically converge to the trajectories of the closed-loop system (5.30)-(5.32). Notice,
however, that the system (5.30)-(5.32) is different from the closed-loop system under
state feedback (5.18)-(5.20), in that the former consists of not only the state feedback
control (5.16)-(5.17) but, in addition, contains the slow part (i.e. Equation (5.21)) of
the full-order observer (5.21)-(5.22). Thus, the trajectories of the closed-loop system
(5.34)-(5.37) do not necessarily converge to the trajectories of the closed-loop sys-
tem under state feedback (5.18)-(5.20).. The result of Theorem 5.2 establishes that,
under prescribed conditions, the exponential stability properties of the origin of the
closed-loop system (5.30)-(5.32) can be recovered under the output feedback, in the

presence of modeling error in the function d(z, x, u).

The uncertainity in the function w(z,x) is implicitly taken care of by assuming
that the origin of the closed-loop system (5.30)-(5.32) is asymptotically stable. In
other words, any uncertainity in the function w(z,x), which does not destroy the

asymptotic stability of the closed-loop system (5.30)-(5.32), is allowed.

In contrast to [28], we do not require the functions ¢(2,x,u) and w(z,x) to be
globally Lipschitz in x and 2. Our work showed the need for global boundedness of
these functions in x, in order to avoid the effect of peaking, an issue that was not
addressed in [28]. Our results hold on any compact sets and, therefore, are not just
limited to global results presented in [28]. Furthermore, we allow for uncertainty in
the nonlinear functions 06(2, x, u) and w(z, x), whereas the state observer synthesis in

[28] assumes perfect knowledge of these functions.

101

5.5 Example

Consider the nonlinear system

:5: = —z + 01x1 (5.43)
x1 = x2 (5.44)
x2 = z + 02x3 + 03a (5.45)

y = x1 (5.46)

The system (5.43)-(5.46) is in the normal form (5.1)-(5.3), with

7162,13): —2 + 91231.

0 1
A: ,B=[01]T,C=[10],
0 0

¢(2,x, u) = z + 02x3 + 03u

in which 61, 62 and 63 are some positive constants. This system can be globally

stabilized by the state feedback controller
1 3
u = -6—(—z — 62x2 — x1 — .732) (5.47)
3

We compare the performance of three control designs. Design 1 uses the state feed-

back control (5.47). In Design 2, we modify (5.47) as a prtial-state feedback control

1
u = 6—(— ‘ -— 02x3 -— x1 — x2) (5.48)
3

where 2 is provided by the observer

2 = —2+01y (5.49)

102

 

Design 3 incorporates the output feedback controller given by

1
u = 6—(— “ — sat (02:23 — 21 — 22)) (5.50)
3

where 2, 21, and 22 are provided by the full-order high-gain observer that comprises

of (5.49) together with

1‘1 = i2 + 91(331 - 531)/6 (5-51)

920:1 — vI)/e2 (5.52)

H
to
II

in which gl and 92 are chosen such that the polynomial A2 + 91A + 92 is Hurwitz. In
the simulation, we use the numerical values: 61 = 62 = 63 = 1, 91 = 1, 92 = 2 and

6 = 0.1.

Figures 5.1 and 5.2 show the performance of the closed-loop system under the
three control designs, with 6 = 0.1, and 6 = 0.01, repectively. The states 2, x1,
and x2, exhibit the expected transient behavior; namely, the response under output
feedback (Design 3) approaches the response under the partial-state feedback (Design
2) as 6 decreases from 0.1 (in Figure 5.1) to 0.01 (in Figure 5.2). In fact, in the latter
case, the response under output feedback is indistinguishable from the same under
partial-state feedback. As expected, the trajectories under output feedback (Design
3) do not approach to those under state feedback (Design 1). Figure 5.3 shows the
estimation errors 77:52 = x2 — 22, and 772 = z — 2, under output feedback (Design 3),
with 6 = 0.01. The estimation errors 7);, = x1 — 21 (not shown in Figure 5.3), and
77332 converge rapidly to zero, in contrast to the estimation error 777., which converges
to zero, slowly. As the value of 6 decreases, the speed of convergence of the estimation

errors 71331, and 77222 increases.

103

 

 

 

 

 

 

 

 

 

 

 

 

 

1 I I I 1 r

wwDeelgn 1: StateFeedback

— Design 2: Partial-state Feedback

- - -Deelgn 3: Output feedback

. - -.- :1: 1 l
8 10 12
8 '.. I T I I '
0. .‘ u‘ '-*--Deelgn1:StateFeedback

— Design 2: Partial-state Feedback)
- - -Deelgn 3: Output feedback ]

 

 

 

—l

 

 

l l

 

 

 

 

 

8 1o 12
'---'Deelgn1:8tateFeedback
—Deslgn 2: Partial-state Feedbmk'
- "Design 3: Output feedback

 

 

 

 

8 10 12

 

Figure 5.1: Performance under state feedback, partial-state feedback, and output
feedback, with 6 = 0.1

104

 

 

 

1 I I 4 .
mm-DeelgntSteteFeedbed
0.8 —-Deslg12:PatIeI-mteF
---Deelgi 3:0utpumerlieek

I

 

 

P
-'-'—I-x—._._

I
10 12

 

 

 

“bananas-room
-Deelg12:Pertlel-steteF
---Deeim3:0utput1eedbeck

 

 

 

 

I

I
10 12

 

 

 

‘-'-'Deeim1:SteteFeedbeck
—Deelgn2:Pertlel-eteteF
---ongI3:0quumom

 

 

 

 

 

 

Figure 5.2: Performance under state feedback, partial—state feedback, and output

feedback, with 6 = 0.01

105

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 I I T I I I I I I
1 - _
v I 4
F’ 0
-1 .— ..
_2 I I I I I I I I
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (sec)
I I I I T I I I I
0.5 -
0.4- _
0.3- _
N
f: 02» I
0.1+ ~
.I
”0.1 l I 1 I L L l I J
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (sec)

Figure 5.3: Estimation errors 1732 = x2 — 22, and r); = z — 2, with 6 = 0.01

106

 

o.—
C

 

  
  

 

'-'-‘Deelgn1:StateFeedbeck
—DeelgI2:Pertlel-eteteI-'

 

 

 

 

 

 

 

   

 

 

 

 

 

 

M "-0.091: 3:01:1me
"‘ 0.4 .
0.2,- . . . ' Ir. a —
o— L I J I I l...,-l-I'-l-'- - .1- -.- . -
0 6 3 10 12 14 16 18 20
'I'ImeIsec)
I I I I f j l
0.8}: 1-I-2Deeim 1:8tete Feedback
-—Deeim 2: PerdeI-eteteF
“-5 "-11%“:me
x“ 0.4 _
0.2 Dr.,.’- 'l
0" i-'-‘-.-'.'-""l-III-v-I-x-nunollIII-I-III-i-v-I-you-III-mI
I I l I I I_ 4L
0 0 3 10 12 14 10 10 20
TimeIsec)

 

 

 

 

'---»Deeim1:8tateFeedback
—Deelm2: Partial-MP
- "Deelm 3:0utputfeedbeck

 

 

 

 

'0-4 I I L I I I I
0 6 3 10 12 14 10 18 20
Time (see)

Figure 5.4: Closed-loop Performance under model uncertainty, with 6 = 0.01, 61 =
1.25, 03 = 0.75

107

Figure 5.4 shows the closed-loop performance under the three control designs,
when the coefficients 01, and 03 are perturbed from 1 to 1.25 and 0.75, respectively.
It can be seen that the output feedback (Design 3) recovers the performance of the

partial-state feedback (Design 2), in the presence of the model perturbation.

5.6 Conclusions

In this chapter, we considered the problem of state estimation of a minimum-phase
nonlinear system using a full-order high-gain observer. The observer comprises of
two components, a slow open-loop observer that estimates the state 2 of the internal
dynamics, and a fast observer that estimates the state x. From a dynamic state
feedback controller, we define a dynamic partial—state feedback controller that assumes
availability of x but uses the estimate 2. It is shown that the performance of such
a dynamic partial-state feedback control can be recovered by the dynamic output
feedback control using a sufficiently fast high-gain observer, in the presence of model
uncertainty. The observer design approach is based on the two-time scale observer
design of Esafandiari and Khalil [13], and the performance recovery is shown using
the separation principle of Atassi and Khalil [2]. The performance of the observer

design is demonstrated by simulation.

108

Chapter 6

Conclusions

This dissertation concentrates on the problem of output regulation for a class of
minimum-phase nonlinear systems, with emphasis on improving the transient per-
formance. We have extended the technique of conditional servocompensators in a
sliding mode control framework [53] to more general feedback controllers by using
Lyapunov redesign and saturated high-gain feedback. The issue of transient perfor-
mance is significant in output regulation problem because conventional approaches
to designing servocompensators often result in poor transient performance. A con-
ditional servocompensator, in contrast, provides servo action only in a neighborhood
of the zero-error manifold while acting as a stable system otherwise, thus, leading to
improvement in the transient response while achieving zero steady-state regulation

€I'I'OI‘ .

The striking feature of our approach is the flexibility of starting with any stabi-
lizing state feedback controller and then including a conditional servocompensator to
achieve zero steady-state regulation error without degrading the transient perform-
nace. We have proved that the trajectories of the closed-loop system under saturated
high-gain feedback control with a conditional servocompensator approach those of

a closed-loop system under saturated high-gain feedback control without a servo—

109

 

compensator. Analytical results are provided for a compact set of initial conditions,
which can be chosen arbitrarily large if all the conditions hold globally. A précis of

the results presented in this dissertation is given in the following section.

6.1 Synopsis of Results

In Chapter 2, we considered the problem of state feedback regulation of nonlinear
systems using conditional servocompensators. We used the Lyapunov redesign and
saturated high-gain feedback approach to design the stabilizing compensator. We
showed that the inclusion of conditional servocompensators in the Lyapunov redesign
framework enables us to achieve zero steady-state regulation error, in the presence of
time-varying exogenous signals that are generated by a known exosystem. Analyti-
cal results are provided for regional and semi-global output regulation, and for the
performance recovery of a saturated high—gain feedback controller without a servo-
compensator. Advantages of the proposed framework over the conventional approach
were shown by simulation.

In Chapter 3, we considered the output regulation problem for a class of minimum—
phase input-output linearizable nonlinear systems. The state feedback control design
of Chapter 2 is specialized to partial state feedback control design for this class of
systems. This partial state feedback controller can be viewed as an intermediate step
towards the output feedback controller of Chapter 3, which is implemented using a
reduced-order high-gain observer. We also proved that the output feedback controller
with conditional servocompensator recovers the performance of a state feedback con-
troller that does not include any servOcompensator. We have included simulation
results for two examples to demonstrate the advantages of the proposed framework
over the conventional servocompensator design approaches.

In Chapter 4, we considered the output regulation problem of linear systems sub-

110

 

ject to control constraints. The presence of saturation in the input channel imposes
strong limitations to the achievable control objectives such as transient performance.
We applied the Lyapunov—redesign-conditional-servocompensator approach of Chap-
ter 2 to the linear output regulation problem under input constraints, to achieve
desirable transient performance. Because of the control constraint, the mechanism
of solving the stabilization problem through Algebraic Riccati Equation (ARE) ne-
cessitates the use of hill-state feedback. Therefore, the output feedback control is
implemented using a two-time-scale full-order observer design of Esfandiari & Khalil
[13] and the performance recovery is shown using the separation principle of Atassi &
Khalil [1, 3]. Advantages of the proposed approach over the conventional approach

presented in Lin et al. [37] were shown by simulation.

In Chapter 5, we considered the problem of state estimation of a minimum-phase
nonlinear system using a full-order high-gain observer. The motivation comes from the
desire of extending the methodology of Chapter 4 to nonlinear systems, which neces-
sitates the development of a full-order nonlinear high-gain observer. A rather similar
situation arises in applications that incorporate optimal stabilizing controllers, where

the output feedback control can only be implemented using a full-order observer.

The observer designed in Chapter 5 comprises two components, a slow open-loop
observer that estimates the state of the internal dynamics, and a fast observer that es-
timates the state of the external dynamics, in the presence of model uncertainty. The
observer design approach is based on the two—time scale observer design of Esafandiari
and Khalil [13], and the performance recovery is shown using the separation principle
of Atassi and Khalil [2]. The performance of the observer design is demonstrated by

simulation.

111

 

6.2 Future Work

One special feature of the proposed Lyapunov redesign + saturated high-gain feed-
back framework is that it allows us to start with any stabilizing controller and then
include a conditional servocompensator by modifying the original controller to achieve
the desired control objectives. Identifying and implementing stabilizing controllers
with some built-in special features such as robustness properties, optimality etc. in
this framework, and investigating to see if the overall performance of such nonlinear
control designs can be improved when applied to the nonlinear servomechanisms, is
an interesting line of future work.

A promising direction of future research work would be understanding whether
the proposed framework allows the flexibility of incorporating controller designs that
extend the class of systems considered in this dissertation, or relax the assumptions
we have made. Of particular significance, in this regard, are the controller designs for
the output regulation of nonminimum phase nonlinear systems [50] and the controller
designs that incorporate adaptive or nonlinear internal models [51, 9, 5, 45, 46].

The saturated high-gain feedback designs naturally accomodate the applications
with control constraints, however, with a significant trade-off between the region of
attraction and the speed of convergence. An interesting, yet chellenging, direction of
future research would be investigating the output regulation of constrained nonlinear
systems, with the goal of semi-global regulation, when the control level is fixed apriori.

Last but not the least, further research work needs to be done on understanding
how to tune the controller parameters in order to achieve specific control objectives,

and to identifying possible limitations on the achievable performance.

112

 

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