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ROBUST TRACKING CONTROL FOR
NONLINEAR SYSTEMS USING OUTPUT
FEEDBACK

By

N azmz’ A. Mahmoud

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1994

ABSTRACT

ROBUST TRACKING CONTROL FOR
NONLINEAR SYSTEMS USING OUTPUT
FEEDBACK

By

N azmz' A. Mahmoud

In this work we use output feedback to study the regional as well as the semi-global
tracking and disturbance rejection problems. The class of systems that we are dealing
with is that of uncertain minimum phase single-input, single-output nonlinear sys-
tems that are transformable into the normal form, uniformly in a set of disturbances
and uncertain parameters that belong to a known compact set. The uncertainties
encompass both parameter uncertainty as well as modeling errors.

We first address the regulation problem where parameters and/or disturbances are
constant. We show that with the addition of an integrator driven by the tracking
error we create an equilibrium point at which the tracking error vanishes for all ad-
missible uncertainties. We then design a partial state feedback controller that relies
on feedback of some of the states to stabilize this unknown equilibrium point.

Next, we tackle the more general tracking problem where disturbances and references

are, in general, time-varying and generated by a linear exosystem. In this approach

we surpass the issue of partial state feedback by extending the system with the in-
troduction of m integrators at the input. With this we change an n-th order system
to an (n + m)-th order system with m states available for measurement. Instead of
giving a specific controller we present conditions that will characterize a class of state
feedback controllers.

To recover the asymptotic properties achieved under state feedback in both cases, we
saturate the state feedback control over a compact set of interest then implement it
as an observer-based control using a linear high gain observer. We provide estimates
of the region of attraction that are not shrinking. On the contrary, they are limited
only by the region of validity of our model. If this region encompasses the whole state
space, then the estimates can be chosen arbitrarily large and our semi-global result
follows directly. Finally we test the two design methodologies through simulations
on some examples, both physical and contrived. The results obtained from those

simulations are in good agreement with the predicted behavior of the system.

To The Memory of My Father: Abdelfattah H. M. Sabi

iv

ACKNOWLEDGEMENTS

I would like to express my gratitude with appreciation to my mentor professor
Hassan Khalil whom without his help, guidance and encouragement this project would
not have been possible. I gratefully acknowledge his substantial support to me during
the years that I worked with him. Thanks are also due to my committee members;
professors: F. Salam, R. Schlueter, C. MacCluer and P. Fitzsimons. In particular, I
would like to acknowledge with appreciation the support that I received from Professor
Schlueter during my years at Michigan State. Special thanks go to professor C.
MacCluer for helping in the proof of Inequality 2.47 given in the appendix. Last, but
not least I wish to thank my wife Salam for her complete support, understanding and
patience. I am also thankful to my daughters Simren, Yasmin and Sarah for their

moral support.

TABLE OF CONTENTS

LIST OF TABLES viii
LIST OF FIGURES ix
1 Introduction 1
2 Asymptotic Regulation of Minimum Phase Nonlinear Systems 5
2.1 Introduction ............................... 5
2.2 System Description .......................... 8
2.3 Integral Control ............................ 12
2.4 Output Feedback Controller ..................... 26
2.5 Semi-global Regulation ........................ 35
2.6 Stabilization Result .......................... 36
2.7 Time-Varying External Signals ................... 37
2.8 Conclusions ............................... 41
3 Robust Tracking for Nonlinear Systems Represented By Input-
Output Models 44
3.1 Introduction ............................... 44
3.2 Preliminaries .............................. 46
3.3 State Feedback Design ........................ 53
3.4 Recovering State Feedback Performance ............. 58
3.5 Examples of Controllers ....................... 65
3.5.1 Example 1 ............................ 66
3.5.2 Example 2 ............................ 68
3.6 The Class of Nonlinear Systems .................. 68
3.7 A Special Case: Nonlinear Systems with Polynomial Nonlin-
earity ................................... 71
3.8 A Design Example ........................... 73
3.9 Conclusions ............................... 80

vi

4 Output Regulation of a Field-Controlled DC Motor
4.1 Introduction and Problem Definition ...............
4.2 Method 1 ................................
4.3 Method 2 ................................
4.4 Conclusions ...............................

5 Conclusions and Future Work
5.1 Conclusions ...............................
5.2 fixture Work ..............................
5.2.1 Multi-Input Multi-Output Systems ............
5.2.2 Unmatched Uncertainties ..................
5.2.3 Internal Model .........................

A Proof of Inequality 2.47

BIBLIOGRAPHY

vii

100
100
101
101
102
102

103

104

LIST OF FIGURES

3.1 Tracking error 61 and the control 45‘ ...................
3.2 Observer performance; top: 61 (solid) and its estimate; bottom: 82
(solid) and its estimate ..........................
3.3 Saturation of the control 45’ .......................
3.4 Control ¢’ and é; for c = 0.01 (solid) and c = 0.001 ..........
3.5 Tracking error when a = 0.1 and b = 12 .................
3.6 Tracking error when a = —0.1 and b = 8 ................
4.1 The domain Nd in the (e, z)-coordinates ................
4.2 -¢o(0,z); (nominal) ...........................
4.3 Tracking error and control (0 — 1 s) ...................
4.4 The states a and x1 (0 — 1 s) ......................
4.5 Tracking error and control (0 — 20 s) ..................
4.6 The states a and 1:1 (0 — 20 s) ......................
4.7 The tracking error when the parameters take their highest values
4.8 The tracking error when the parameters take their lowest values . . .
4.9 Tracking error and control of the nominal system ...........
4.10 The states $2 and 1:3 ...........................
4.11 6; (solid) and its estimate él .......................
4.12 6; (solid) and its estimate ég .......................
4.13 Saturation of the control ¢‘ .......................
4.14 The control «V and é; for e = .01 (solid) and c = .005 .........

4.15 The tracking error when the parameters take their upper limits
4.16 The tracking error when the parameters take their lower limits . . . .

viii

77

77
78
79
79
80

84
85
87
88
89
90
91
91
95
95
96
96
96
97
98
98

CHAPTER 1

Introduction

Most engineering systems encountered in practice exhibit significant nonlinear be-
havior. For systems exhibiting nonlinearities, the normal design procedure in the
past has employed a linearized approximation of the model followed by the applica-
tion of linear control methodology. However, this procedure can yield unsatisfactory
performance, especially when the system is highly nonlinear and operates over wide
nonlinear regimes as it is the case in aircraft control and many chemical processes.
During the past fifteen years, motivated by progress in the nonlinear differential ge-
ometry, techniques have been developed to solve the feedback linearization problem.
Feedback linearization utilizes state feedback, after a possible change of coordinates,
to transform a given nonlinear system into a linear and controllable one. Then already
developed linear control tools are available for design. This application is character-
ized in the nonlinear literature as exact linearization.

This idea is probably an old one. An illustration is the so called computed torque

method in robotics. Consider a single-link manipulator model [1]

(81:13:,

5:; = —asin(:rl)+u

L

 

 

The control u can be chosen to cancel the nonlinear function, i.e., u = a sin(a:1) + 2).
Then we can use the linear control v 2 k1 3:1 + k2$2 where 1:], 1:; are chosen to stabilize
the resulting matrix.

Exact linearization techniques can be classified as follows: input-output linearization
where the objective is to get a linear response between the input and the output;
input-state linearization where the objective is to get a linear response between the
input and the state of the system; and finally full linearization, where the desire is to
achieve an input-state linearization as well as a linear mapping between the states and
the output. The class of input-output feedback linearizable systems is more general
and our emphasis here is on this class of nonlinear systems.

Exact linearization techniques have been applied to solve many practical engineering
problems. Without quoting numerous references, we only list a representative sample
of the work that has been done using these techniques. In [2] the tools were used to
design an automatic flight controller for an aircraft; in [3] they controlled a brush-
less DC motor for direct-drive robotic application; in [4] they controlled the position
of PM stepper motor, and in power systems, [5] employed feedback linearization to
enhance transient stability and achieve voltage regulation. Adaptive control schemes
were also developed for this class of systems; see for example [6, 7, 8, 9].

An attractive structural property of linearizable systems is the fact that it can be
transformed into a form (normal form) where the the nonlinearity satisfies the match-
ing condition, and this permits cancellation of the nonlinearity using state feedback.
As in any cancellation scheme, exact mathematical cancellation can not be achieved
due to parameter variations as well as hardware limitations. Hence the issue of robust-
ness, i.e., insensitivity to parameter perturbations, must be addressed; also modeling
errors will have to be taken into account.

In this work we focus on the robust tracking problem for minimum phase input-output

linearizable uncertain systems. We allow parameter uncertainty as well as modeling

errors, hence the robustness issue is one of our primary concerns. Since measurement
of the states is not always feasible, we use dynamic output feedback in our design ap-
proach through building an observer. The choice of dynamic instead of static output
feedback is motivated by the fact that most systems can not be stabilized through
the use of static output feedback (e.g., 5:1 = 2:2, i2 = u, y = $1).

The details of the design procedure is carried out in two steps. first, a robust state
feedback control is designed to achieve the control task, and in the second step we
implement it as a globally bounded control through designing an observer to recover
the robustness and asymptotic properties achieved under state feedback.

In Chapter 2 we study in detail the regulation problem. More precisely, we present a
solution to the problem of asymptotically regulating the output of a nonlinear system
to a constant reference with zero error, in the presence of uncertain constant param-
eters/disturbances. We do this through the introduction of an integrator driven by
the tracking error. This will create an equilibrium point at which the tracking error
is zero for all admissible perturbations. We then stabilize the system via the use of a
min-max state feedback controller although, other methods such as variable structure
control or high-gain control could have been used as well.

In Chapter 3 we address the more general tracking problem for systems that are repre—
sented by input-output models where disturbances and / or references are time-varying
and generated by a linear exosystem. In this chapter we utilize the idea of [10] in
extending the system with the advantage of being able to implement less restrictive
control schemes. The nonlinearities of the system are restricted to introduce only a
finite number of harmonics of the original modes. This will enable us to identify the
internal model as a linear servo-compensator. In this chapter we present conditions
that will characterize a class of state feedback stabilizing controllers as opposed to
presenting only one specific design of controllers.

In Chapter 4 we illustrate the controllers of Chapters 2 and 3 by solving a speed

regulation problem for a field-controlled DC motor in the presence of uncertain pa—
rameters. The results of the simulations are in good agreement of what we expected
and give a good example of how the theory presented can be utilized in solving engi-
neering problems.

There is a fundamental difference between the two design methodologies implemented
in this work. In Chapter 2 no feedback from the zero dynamic states was utilized,
while in Chapter 3 we were able to use feedback from the extended states. This is done
with a price, since the strategy adopted in Chapter 3 will, in general, lead to a more
complex controller with higher dimension. This is evident in Chapter 4 where both
strategies are tested. Using the approach of Chapter 2 we did not need an observer
while it was needed using the other approach. Common to both methodologies, how-
ever, is the choice of the linear high gain observer, which is basically an approximate
differentiator of the output [11], and the idea of globally bounded control introduced
first in [12]. Finally, in Chapter 5 we give some concluding thoughts and prospects

for future work.

CHAPTER 2

Asymptotic Regulation of
Minimum Phase Nonlinear

Systems

2.1 Introduction

One of the important problems in control systems is the servomechanism problem;
that is, to get the plant output to asymptotically track a reference while asymptoti-
cally rejecting disturbances, when both the reference and disturbance signals satisfy
a given differential equation model. For linear systems this problem was extensively
studied by many researchers; see for example Davison [13] and Davison and Ferguson
[14]; a self-contained exposition is found in Desoer and Wang [15]; a different approach
to the solution of this problem was presented by Francis [16]. Since our focus in this
work is on nonlinear systems, we highlight some of the previous contributions to the
solution of this problem. In [17] Desoer and Wang studied this problem for a class of
nonlinear distributed systems where the nonlinearity appears as causal operators at
the input and output channels of a linear system. Desoer and Lin [18] studied this

problem using PI controllers for exponentially stable plants having a strictly increas-

ing dc steady—state input-output map with references and disturbances tending to
constant vectors. Isidori and Byrnes [19] provided necessary and sufficient conditions
for the local solution of the problem for a general case where the disturbance and
reference signals can be time-varying but small, and initial states were required to
be small. Huang and Rugh [20], [21], using the method of extended linearization,
designed dynamic output feedback controllers for the solution of this problem. In
[20] they considered the case of sufficiently small constant or slowly-varying external
signals. In [21], they allowed large slowly—varying external signals but still required
proximity of the initial states to the zero-error manifold. In [22] Priscolli provided
a local solution, with robustness, to the general problem addressed in [19]. Except
for [17], [22] and the integral control of [20], the other papers [18, 19, 21] did not
explicitly address robustness of asymptotic tracking to modeling errors. Aside from
these servomechanism papers, tracking and disturbance rejection problems for feed-
back linearizable systems have been tackled by many authors. Related to this work
are the results of [9, 23]. In [9] Marino and Tomei used adaptive control techniques
to solve this problem globally for a class of systems that is characterized by geo-
metric conditions where the zero dynamics are restricted, in suitable coordinates, to
be linear. In [23] Khalil studied a special case of this problem for a class of single-
input, single-output (SISO) nonlinear systems with relative degree 7' = n that admit
a disturbance-stm’ct-feedback form, but he allowed time-varying disturbances and ref-
erence signals. For feedback linearizable systems, robust continuous feedback control
laws can be designed to ensure convergence of the tracking error to a small ball while
rejecting bounded disturbances. However, making the error arbitrarily small requires
the use of high gain feedback near the origin; see for example [24, 25, 26, 27].

In this work we use integral control to ensure asymptotic regulation in the case of con-
stant references, for a $180 minimum phase nonlinear system that is transformable

into the normal form, uniformly in a set of constant disturbances and uncertain pa-

rameters. The introduction of the integrator creates an equilibrium point at which
the tracking error is zero for all possible parameters and /or disturbances that belong
to a known compact set. We provide estimates of the region of attraction that are
limited only by the region of validity of our model. If the domain becomes global,
those estimates can be made arbitrarily large. As a consequence of this, we have the
semi-global result which follows directly. In this case we do not impose global linear
growth conditions on the nonlinearities nor do we require global exponential stability
of the zero dynamics.

In order to recover the asymptotic properties achieved under state feedback we sat-
urate the state feedback control over a compact set of interest then implement this
globally bounded control as an observer-based control using a linear high gain ob-
server.

This chapter is organized as follows: The class of systems that will be considered
is presented in Section 2.2 Asymptotic regulation is achieved under state feedback;
this is shown in Section 2.3. In Section 2.4, output feedback is used to recover the
robustness and asymptotic regulation properties of the state feedback controller. The
main result of this section is Theorem 2.1. In Section 2.5 we present the semi-global
result. A stabilization result is given in Section 2.6 for a special case of the class of
systems. In Section 2.7 we investigate the performance of the integral control in the
presence of certain time-varying signals. Finally, some concluding remarks are given

in Section 2.8.

2.2 System Description

Consider a 8130 nonlinear system, modeled by

6' = 115.6) + g(:.0)u
hw)

(2.1)

Q
ll

whereC 6 R" is the state, 11 E R is the control input, 31 E R is the measured output, 0
is a vector of unknown but constant parameters and disturbance inputs which belongs
to a compact set 9 C R'. We consider a case where the output y(t) is to track a
constant reference V 6 I‘ where I‘ C R is compact. For all 0 E 9, we assume the
following: f, g are smooth vector fields on U9, an open subset of R" that might depend
on 9, h is a smooth function in 5 from U9 -> R.

In this work we are interested in input-output linearizable minimum phase nonlinear
systems where f, h do not necessarily vanish at the origin, i.e., f (0, 0) 525 0, h(0, 0) ¢ 0.
We consider the case where the system has a well defined normal form and possibly
nontrivial zero dynamics. With this in mind, we assume the following about the

system (2.1).

Assumption 2.1 V V E F and V 0 E 9, there exist an equilibrium point {oh/,0) and

a control u(u,9) such that

0 = “50049),”+9(€o(V,9),9)u(Va9)
V = h(€0(Vao)v0)

Moreover, {0(11, 9) is the only such equilibrium point in U9.

The existence of £001, 9) and u(u, 0) satisfying (2.2) is a necessary requirement for the

system to maintain equilibrium at y = V.

Assumption 2.2 V 0 E 6 there exists a mapping

= use) (2.3)

which is a difieomomhism of U9 onto its image, that transforms {2.1) into the normal

form

 

 

 

 

it; = 225+],ISiST—l
it, = -x,z,9 +‘:c,z,0u
f( ) g( ) (2.4)
2 = ¢(:c,z,0)
y = $1
or, more compactly,
.i: = A3 + B[f(:r, 2,0) + §(:1:, z, 9)u]
2 = ¢(z.z.0) (2.5)
y = C2:
where _ _
0 1 - 0 0
0 0 1 0 0
A = ,B = i
0 0 1 0
0 . 0 1
H rXr L . rxl
1Xr

Conditions under which Assumption 2.2 holds locally or globally, when 0 = 00
(known), are given in [28, Proposition 3.2b, Corollary 5.6]. Global conditions are

also given in [28, Corollary 5.7] for the case when 43 = ¢(a;l, 2). We point out here

~1

\\

 

10

that, in contrast to a local or a global diffeomorphism, the mapping (2.3) is required
to be a diffeomorphism of a domain U9 onto its image. A result of [29] gives necessary
and sufficient conditions for a smooth mapping that maps U into V to be a diffeo-
morphism of U onto V. A smooth mapping F : U —+ F (U ) is a diffeomorphism of U
onto its image if and only if (a) deth 74 0 through out U and (b) F is a proper map
of U into F (U ); where J9 denotes the Jacobian matrix of F at a general point 5 E U.
At this juncture, we recognize that there are no results available in the literature that
will guarantee the existence of the mapping T for a given domain U9.

Usually one starts from local conditions and in the process of transforming the system
into the normal form (2.5) a region over which the normal form holds is identified.
Requiring the normal form (2.5) to hold uniformly in 9 is clearly more restrictive than
requiring it to hold for a given value of 9. For many examples of physical systems
which are transformable into the normal form, it is indeed true that the normal form
holds uniformly in the system parameters, at least over a compact set of these param-
eters; see for example the field-controlled DC motor and the robot arm examples of
[30, section 4.10]. There is also the result of [6] which characterizes a class of systems
for which such representation is valid. That paper considers a special case of (2.1)

where 9 appears linearly in the model, i.e.,

t = f(£)+g(€)u+2l=1pa(€)9e
y = h(C)

(2.6)

For this class of systems it was shown in [6] that if there exists a parameter-
independent global diffeomorphism that, when 9,- = 0, transforms (2.6) into the nor-
mal form (2.5) with Q} = 45(11, - - - , $91.1, z) for some integer 1 S q _<_ r — 1 then, under

certain geometric conditions on the vector fields p,(£), the same diffeomorphism will

 

 

 

 

 

 

tl

dc

 

 

11

transform (2.6) when 9.- ;£ 0 into the parametric-strict—feedback form:

ii = xi+1+0T7i($la°”vxi)a 13139—1
it = $i+1+9T7i($1,"',$i,Z), (13137—1
if = f(:r,z)+§(:c,z)u+9T7,(a:,z) i (27)

é = ¢(xla° ° ° axq-Haz) "l" Ej=10i7iz($1a' ' ' a$q+laz)

 

9:171

where 7,-(.) 6 R',V 1 S i S r, 7f(.) 6 R"",V 1 S i S I. It can be easily shown
that, using a parameter-dependent transformation, system (2.7) is transformable into
a global normal form uniformly in 9.

Now, let us introduce a new set of coordinates in terms of the tracking error and its

derivatives. Let

el = 2:1 — u
€£+1 = é; = n+1, IS i S 7‘ — 1 (2'8)
2 = 2

Let us denote the transformation (2.8) by

e \Ilo(:r, V)

= = W(:c,z,u)
z z

Id

\\

 

 

12

and rewrite (2.4), in the newly defined error coordinates, as

 

éi = eg+1,ISiSr—l l
(E, = f(el + V,€2," - ,e,,z,9) +§(el + V,€2,°' - ,e,,z,9)u
def
= f1(e z,V,9) +gl(e,z,V,9)u
’ > (2.9)
2': = ¢(e1+V,e2,---,e,,z,9)
dé! ¢o(€, 2, V9 0)
ym = 81 J
where ym denotes the measured output, and e = [e1, - - - , e,]T. We remark that the

transformation (2.8) is a diffeomorphism for all V E F. To simplify the notation, we

setd=(V,9)andD=I‘x(-).

2.3 Integral Control

We augment system (2.9) with an integrator driven by the tracking error, i.e.,

a = ‘/:(y(r) — V) dr

The augmented system is given by

a = e1

e,- = €g+1,IStST—l

e, = f1(e,z,d) + g1(e,z,d)u
z' = ¢o(e, 2, d)

V

(2.10)

 

ym=€1

13

Rewrite (2.10) in the compact form

4' = AC+B[fl(e,z,d)+g1(e,z,d)u] (2.11)
2" = ¢o(e,z,d) (2.12)
gm = Ce
where
0 C 0 a
A = a B 2 ’ C:
0 A B e

We remark that in (2.11) f1 and g1 satisfy the matching condition, and the pair (.4, B)

is controllable. We assume the following:

Assumption 2.3 There exists a domain N9 C R", that contains the origin, such

that (\II o T)'1(N9) C U9 and contains the point {0, for all d E D.

Let M9 = R x N9. We restrict our analysis to the domain of interest M9.

We note that as a consequence of Assumptions 2.1 and 2.3 together with the fact that
the composite transformation \II o T is a diffeomorphism, equation (2.12), with e = 0,
has a unique equilibrium point in the domain of interest, M9, that will be denoted by
20 = 1(a), i.e., ¢o(0, A(d),d) a o, for all d e D.

Before stating the next assumption, let S C R"H and U C R"‘r be open sets such
that S x U C M9. Also define the balls, 80 = {C E S : IICII < r1} and 110 = {z E
u : ||z — 20” < r2} where H." denotes the Euclidean norm and r,- > 0, (i = 1,2) are

chosen to give the maximum balls in S and LI, uniformly in d.

Assumption 2.4 With 6 as a driving input to (2.12), there exist a C1 properfunction
W : U —+ 72+ and class [C functions, a,- : [0,r2) —-+ R+, (i = 1,2,3) and 71 : [0,r1) —+
R+ such that

C!1(||~? - Z0”) S W(Z) S a(2(IIZ - 20”) (2-13)

14

9W
Enema) 3 -a.(nz — 2°“), v llz — 2°” 2 Mllell) (2.14)

V (C,Z,d) 6 So Xuo X D.

0 is asymptotically stable.

This assumption implies that when e = 0, the equilibrium 2
Moreover, when e 75 0 but bounded it can be shown [31, Theorem 4.10] that the

solution z(t) satisfies the estimate
”z(t) — 20” S 91(||Z(0) - ZOIIJ) + 7080, V t Z 0 (2.15)

where [cl = sup{||e(t)|| : t Z 0}, 91 is a class ICL function and '7 is a class [C function;
see [31] for the definition of these classes of functions.

We point out that for semi-global results, to be considered in section 2.5, Assumption
2.4 must hold globally. In this case the estimate (2.15) implies that the system (2.12)
is input to state stable, (ISS ) for short, as it is defined in [32]. In light of a recent
result of [33], this assumption is also necessary for the system to be ISS. However,
for regional results it becomes less restrictive. To see this, we consider the following

example where the z-dynamics, with :r as input, are given by
é=—z+(22+1)z

This system, as it was indicated in [32], is not bounded-input bounded-state (BIBS)
stable, hence it is not ISS. However, if we restrict our domain of interest to the region
defined by {(z,:r) : [2] S 1,|:c| S .25} we can see that W(z) = (1/2)22 satisfies
(2.13)-(2.14) since

W S -(1/2)22. V lzl Z 4|$|

There are also some global results available in the literature that guarantee the ex-

istence of W(z) satisfying (2.13)-(2.14) when the zero dynamics 2': = ¢o(0,z) are

15

globally exponentially stable and 450 is globally Lipschitz in 2:; see for example [34],
[35] and [7]. However, Assumption 2.4 is less restrictive. To illustrate, consider the

example

2 = —z - 223 + (z2 +1).7:2 = ¢0(x,z) (2.16)

When a: = 0, the system has a globally exponentially stable equilibrium point at
z = 0, but 450 is not globally Lipschitz in 2:. It can be shown that W(z) = %22

satisfies Assumption 2.4 with
W s —(1/2)22. v Izl 2 2le”

We proceed now to design a robust state feedback controller assuming that the state
e is available for measurement. This is not a reasonable assumption because of the
dependence of the transformation (2.3) on the unknown parameter 9. However, the
final controller will be an observer-based controller and no measurement of the state
will be used. For feedback linearizable systems where the uncertain terms f1 , 91 satisfy
the matching condition, several methods are available to design robust state feedback
controls such as min-max, high gain and variable structure control. In this work we
chose to use the min-max controller of [26], but other methods could have been used
as well.

We start the design procedure by choosing K such that (.A + BK) is Hurwitz. Let

P=PT > 0 be the solution of the Lyapunov equation
P(.A+BK)+ (A+BK)TP = —I (2.17)
Take V(()=(TPC, and for c,- > 0, (i = 1,2) define

0.. .__.. {c e R'“ = WC) 3 c.} (2.18)

16
DC, déf {z E R"" : lV(z) S c2} (2.19)

Since our assumptions are required to hold in a given region, we require both QC, and
QC, to belong to the domain of validity of our assumptions, i.e., 961 x QC, C 80 x U0.
Moreover, since e acts as a driving input to (2.12), we need to choose c1 and C; such
that if C is contained in QC, for all t, then 2 will be contained in {26, for all t. Inside

QC, we have
Cl

2
ncn s 1....(P)

 

For 961 to be in the interior of So, we require
c1 < /\mm(P)7Ԥ (2.20)

when C is contained in QC, for all t, inequality (2.14) holds outside the ball {“2 —z°|| S

71 (‘/c1/Am,-,,(P) )}. To contain this ball inside it”, we require

 

0. (den/Amid”) S 62
where 04 = a; o 71. Again for DC, to be in the interior of U0 we must have
02 < 01(1‘2)

Therefore, c2 must satisfy

 

a4 (Vol/AWJPO S C; < 01(r2) (2.21)

Hence, to satisfy both (2.20) and (2.21) we choose c1 small enough to satisfy

c. < min{ Am,,,(P)rf,/\m,-,,(P)(a;1 o a.(.~.))2 } (2.22)

17

From now on, we fix c1, c2 as chosen above. Thus we are guaranteed, under Assump-
tion 2.4, that as long as C(t) 6 51¢; , QC, will be a positively invariant set.

Let fo(e, V) and go(e, V) be known nominal models of f1(e, 2, d) and g1(e, 2, d), respec-
tively, that are not allowed to depend on 2. If no such nominal functions are known,

we can take fo=0 and go=sgn(g1).

Assumption 2.5 There exist a scalar nonnegative locally Lipschitz function p(() and

a positive constant lc, both known, such that

[f1(€,Zad) ‘91(€,Z,d)90-1(€aV)f0(€,V)- KCl S F(C) (2-23)

91(6a2ad)931(6,l/) Z k (2.24)
V (Cw’ad) E 9c, X 9c, X D.

Since Inequality (2.23) is required to hold on a compact set, it is always possible to
find p(() that satisfies (2.23). In fact we can always take p to be constant. However,
allowing p(() to depend on C will lead, in general, to less conservative bounds. We
note also that p is not allowed to depend on 2 even though the left-hand side of
(2.23) is a function of z. This is a built-in conservative measure of our design which is
adopted to allow the use of a partial state feedback independent of z. Inequality (2.24)
is a sign definiteness requirement on g1 that is sometimes referred to in the literature
as the high frequency gain assumption [36]. Continuing the design procedure, we

consider the following partial state feedback control [26]

u = 90(Ci V) = -gO-l(69 V)fo(€, 1’) _ £90403, V)”(C)N#(Sa ”(CD (225)

18

where n(C) 2 9(4) and

~37, ifn|s|>p>0
M4392): " (.

nfi. ifUISISV

[\D
(O
C)

with s=2BTPC and p is a design parameter whose role will be discussed soon.

As it was emphasized in [36], handling local convergence to the equilibrium point
separate from showing boundedness of solutions has some advantages, as certain
assumptions on the nonlinearities of the system are required to hold only locally on
a small set around that equilibrium point. Then, if we force the trajectories, using
nonlinear or high gain control, to enter in finite time a positively invariant set that
is in the interior of the set of validity of our local assumptions, one can utilize these
assumptions in the local analysis to show convergence to the equilibrium point. Our
control (2.25) achieves this task by using the nonlinear term —(1/k)g3117N,,, which
is preferable to using a high gain term that results in a larger control effort far from
the origin. In light of this discussion, our immediate task is to perform regional
analysis to show boundedness of solutions. To do this, we first show using Lyapunov
techniques that the state (C ,2) will enter and thereafter remain inside a positively
invariant set around (0, 2°) which can be made arbitrarily small. Thus, if the system
has an equilibrium point it has to be inside this residual set. To that end, use V(C)

as a Lyapunov function candidate for the system

. , 1 _

C: (v4 + [3ch + 3(f1 — gigdlfo — 1‘ C) - £39190 177(09/14 (2-27)
then for all (C,z,d) E {C : {17(C) Isl > p}flflc,} x no, x D, we have

V s —IICII2

19

while for all (C,z,d) E {C : {7)(C) |s| S u}flflc,} x QC, x D, we have

. 2 32
v s —ncn" + no Isl - 1‘7?—
< — 2 5‘-
- ncn + 4
Hence, we conclude that
V _<_ —||(||2 + g, v (C,z,d) e 52., x 11., x D (2.28)
For )4 S 4c1/aAm99(P), with a > 1,
V < 0, V (C,z,d) 6 DC, x (L, x D and V(C) = Cl (2.29)
From (2.14)
W < 0, v (C,z,d) e 9., x 12., x D and W(z) = c; (2.30)

From (2.29) and (2.30) we conclude that 09, x 06., is a positively invariant set; i.e.,
all trajectories (C(t), z(t)) starting at (C(0),z(0)) 6 52¢, x 09, will remain in QC, x QC,
for all t Z 0. Note also that for any positive constants a1 and a2 such that al < c1

and

a
C!4( m _<_02<CQ

we can repeat the foregoing argument to show that for p S 4a1/aAm99(P), the set
99, x ('29, is a positively invariant set.

Now, choose a1 = 51=(Am9,(P)ap)/4 for some a > 1 and a2 = fl; = 014 (J%),
and take )1 S 4b1/a/\m9,(P), where 0 < b1 < c1. This particular choice of a1, a2 and

it implies that 91 S b] < c1 and 92 < c2. Then the previous argument shows that

R. “é’ “a x 09. c m. x n... c n. x 0.. (2.31)

 

€11

v~
ll‘

 

 

20
is a positively invariant set.

Lemma 2.1 All trajectories (C(t),2(t)) that start in (C(0),z(0)) 6 (L, x 962 will

enter the set 72,, in finite time.

Proof: From (2.28) we have
Vg—————+$vgpununxngxo

When V Z 91, we have

Vg—gm—i) (2a)

which implies that V will be bounded by a bound that tends to zero in finite time.
Hence, it will clearly tend to 91 in finite time. This proves that the trajectories
(C (t), z(t)) will enter the set 99, x DC, in finite time and remain thereafter. Note that
(2.32) explains why we included the factor a > 1 in the choice of 91. Now, for all

trajectories inside 09, x QC, but outside R9 = 99, x (29,, we have from Assumption

2.4

WS-03071( 73-12-17)

which shows that W will reach ,82 in finite time. Thus we conclude that (C (t), z(t))
will enter the set 72,, in finite time. 0

As a consequence of this lemma, we see that (C, 2 — 2°) are ultimately bounded with
ultimate bounds which are class IC functions of p; hence the trajectory can be made
arbitrarily close to (0, 2°) by choosing )1 small enough.

We turn now to the behavior of the trajectory inside R”. We want to establish that
there is an equilibrium point inside R9 at which the tracking error is zero, and that
every trajectory in ’R“ converges to this point as t —» 00. Our approach to that
end is to force the state C to enter and thereafter remain inside the boundary layer

{1)(C) Isl S p} and to have the nonlinear term 17(C)N,,(s,77(C)) reduce to a linear

 

 

w]

 

for

1a}

He

i0 :-

aLC

21

function of 3 inside this boundary layer. We consider the function V = %s° whose

derivative along the trajectories of (2.27) satisfies

i/ = s[2BTP(A + 3104 + 2BTPB(f1 — glgglfo — KC)

-(2/ k)3TPBglgo' 112(C )Nul

Using (2.23), (2.24) and BTPB = p, where p is the (r + 1)-th diagonal element of P,

 

we get

2 s 2|s|[IIBTP(A+BK)CII+pp(C)l-2sm(C)N.. (2.33)
T
s 2P|8|maxcen.,{P(C)+ ”3 1094:3109}
—23W(C)Nu (234)
Let

17(6) = max{no.p(C)} (235)

where

+ ||BTP(A + 3104"} + ”I (2.36)

 

710 Z Ina-R609, iP(C)

for some [41 > 0, so that inside 99,, 17(C) = ’70- This step simplifies the boundary

layer inside flg, to {770 [s] S p}. Also, it follows from (2.36) that
V S 217 ISI (no - #1) - 282017046. (237)

Hence

9 s -2|3|PV1 (2.38)

for all 770 [s] 2 p. This implies that the set 5,, (lg {’70 [s] S p} is positively invariant

and all trajectories inside 0,3, will enter it in finite time. Thus, C will approach the

 

 

 

 

(m

"v-o

l‘.‘ I.

 

22

a!
set A” é

99, fl 5,, in finite time. Inside A“, 17(C)N,,(s,n(C)) will be linear in s.
i.e., 171V 9:353. Therefore, the trajectory (C (t), z(t)) enters the positively invariant set

A“ x (2,9, in finite time. From that time on the closed-loop system is given by

C' = (A + BKX + BU. - 9.05% - KC]
- ”—g’éngalsTPc (239)

2' = ¢o(e, 2, d)

The next step is to establish the convergence of the trajectories of (2.39) to an equi—
librium point at which e = 0. Before doing that, we will impose the following local

growth assumption that is required to hold in the neighborhood of 2°.

Assumption 2.6 There exists a C1 function V and a continuous function 19, both

positive definite in 2 = z - 2° and vanish at the origin, such that V d E D

%§¢O(Oa Z) d) S —00¢(§)3 00 > 0

|f.(0.z.d)—9.01.2,d)gr‘(0.z°.d)f.(0.z°.d)l s klwz)

0 < a 51/2, k1> 0 (2.40)

817 - 1— b
El¢0(eiz,d) — ¢0(09z)d)l S k2¢b(2)”e“c, 0 < b <11 C = 7) k2 > 0

This local growth assumption is adopted from [37]. It is less restrictive than local
exponential stability of the zero dynamics. In particular, if the zero dynamics are
locally exponentially stable, then by the converse Lyapunov Theorem [31, Theorem
4.5] there is a Lyapunov function V which satisfies Assumption 2.6 with a = 1/ 2, b =
1 / 2, c = 1 and 19(2) = ||§||2. Assumption 2.6 is also less restrictive than the quadratic-

type Lyapunov function assumption used in [38]. In [38] the emphasis is on completing

 

 

 

 

 

 

23

squares, while here we follow [37] in repeatedly using the following fact which is a

special case of Young’s Inequality [39, Theorem 156].

Fact 2.1 Vx,y€R+,Vp>1andVeo>0
1
.y S g; lep+ (cow 013’: (2.41)

where p0 = :37.

Lemma 2.2 There exists p‘ > 0 such that V p S ,u" and V d 6 D, the closed-loop
system (2.11), (2.12) and (2.25) has a unique equilibrium point (0' = &,e = 0,2 =
2°) and all closed-loop trajectories starting in QC, x QC, at t = 0 will remain in

QC, x QWV t Z 0 and will converge to this equilibrium point as t —-> 00.

Proof: We have already established that trajectories starting in QC, x QC, enter
A,, x (1,3, in finite time. We choose p‘ small enough to ensure that for p S u" the
set A,, x (29, is in the interior of the set of validity of Assumption 2.6. We start by
showing the existence of an equilibrium point in A” x 99,. At equilibrium, we have

from (2.39) and Assumptions 2.1 and 2.3 that e = 0,2 = 2° and

BTPSa = §,;—‘3-[go(o,u)gr‘(o.z°.d)f.(o.2°.d)—fo(o,u)1 (2.42)

‘12! it:
2173 “((1)

where P is partitioned as

P11 P12
P17; P22

P:

with dim(Pu) = 1. Using (2.23), (2.24) and (2.36) it can be verified that k |a(d)| < '70
for all d 6 D, and this ensures that 6 is confined to the boundary layer 5,, .

Equation (2.42) has a unique solution if the scalar constant (BTPS) is nonzero. To

24

show this, observe that from equation (2.17) we have 2KIBTP1T2 = —1 where K is
partitioned as K = [ K1 K2 ]. The fact that (A + BK) is Hurwitz implies that
K1 74 0. Hence, BTPS 7t 0. Combining this with Assumptions 2.1 and 2.3, we
conclude that (0' = 6,e = 0,2 = 2°) is the unique equilibrium point of the closed-
loop system (2.39). To study convergence to this equilibrium point, we shift it to the

origin by the change of variables

Then the closed-loop system becomes

L ~ ~ 2
c = (.4 + 8104+ 8160:, 2. d) — K41 - [#:3919314 (2.43)

— ¢o(e, 2 + 20,41) (2.44)

N1-
|

where 3 = 2BTPC and

6(69 Z, d) = fl(ei 2’ d) — 91(61 Z, d)go-l(ea V)fo(€, V)
-91(e. 2. 0096103. V)go(0. V)gf1(0. 2°. d)f1(0. 2°. 60

+9103, 2’ (090-103, V)f0(0, V)

Observe that 6(0,2,d) = f1(0,2,d) - g1(0,2,d)gf'1(0,2°,d)f1(0,z°,d) which is re-
quired to satisfy inequality (2.40). After adding and subtracting the term 6(0,2,d)
to the bracketed term in (2.43), it follows from Assumption 2.6 and smoothness of

the nonlinearities that

|6(e.z.d)—K&| s koueu+k.¢°(a+uKunén

s k1¢°(5) + aIIIEII

25

s 4.012%) + Hill)

where 61 Z max{k1,a1}. Let V(C,2) = 17(2) + /\(CTPC)7,7 = 1/2a 2 1, be a

Lyapunov function candidate for the system (2.43)-(2.44). It can be shown that

. ~ ~ ~C ~ 2x\a ~ _ -
v s —ao¢(z)+kzwb(z>n<u -/\a2IICII”‘yQ,.—2|ICII”

+)\7lll’ll"'1”5|er2 l6 - K5

 

|§| (2.45)

where a2 = 7(A,,,,-,,(P))7"1. Using the inequalities -—[|C||2”'2 S —a3 |5|2"‘2 (for some

as > 0) and Hill S (¢“(5) + llfll), we set

. ~ ~ - - A ~
12 s —aov(z)+k2¢"(z)l|C|l°—Aa2IICI|2"--:—°Isl”

+MIIPII""&1 |§| (W(E) + Hill)?"1

where p0 := qgagag. We apply (2.41) to wb(2)||C||° and ls] (w“(2) + ||C||)2'Y'l, with

p = 1 / b and p = 27, respectively, to obtain

. ~ k , ~ ~ A -
v s -a.4(2)+Iii-4(2)+would“44.11012"—721914”
mums-1

 

I‘s'l”

+(éo)"51/\7IIPII""(119(5) + Hill)”1 +

(2.46)

where p0 = b/(l - b), p1 = 1/(27 — 1) and pmeo are arbitrary positive constants.
Choose no large enough so that do — if 2 9'21. Then choose A large enough so that

Aag - £72040)” 2 A33-; we get

V S -51(¢(5) + ”5“”) + (60)”‘6147||P|l”"(¢“(5) + Hill)”

MIIPIP-l 141'“ _ fl I94
60

+

 

26
(2.47)

where a = min(ao / 2, ha; / 2). Using the inequality "
(W(E) + Hill)” S 60(11)(5)+ Hill”), Co > 0

we obtain
gums) + Hill)” +(60)”‘5147IIPII”“(¢“(5)+llfll)”

1'2 3
&,A7||P||"" ~2 4P0 ~2
lsl l — 7 [3| 7

+
60
Choose e0 small enough such that
6
— (Calmaft/VIP“?1 2 2:2;

QIDI

Finally, choose p“ small enough so that
)‘Po _ {ll/\Vllplll”1 > E
Co - 21!.

I‘.

Then, for all p S u‘, we have
v < ——5 (1/2°(5)+ "(Eu)“ - LP” Isl“ .
2co 2;:

2.4 Output Feedback Controller

To implement the controller of the previous section as an observer-based controller,

we need to estimate the state C. However, since a is available as the output of the

‘see Appendix A for the proof of (2.47).

27

integrator, we will only estimate the state e using the following observer [40]

8" = ég+1+%(81—é1),i=1,'°°,T—1

(2.48)
e, = %,£(e1 — 61)
where e > 0 is a design parameter to be specified, and the positive constants o,- are

to be chosen such that the roots of
s' + 0113"1 + - - - + a,_ls + a, = 0 (2.49)

are in the open left half plane. This is a high-gain observer motivated by our desire
to recover the robustness properties of the state feedback control. To eliminate the
peaking phenomenon associated with this high-gain observer, we implement the idea
of [12] in saturating the control u = <p(C,V) defined by (2.25) outside the region of

interest (L, X I‘. Let S = maxKecher) [99(C, V)| and define

. _ 90(6, V)
.(.,.)_s...(_, )

where sat(.) is the saturation function defined by

si 11 x , if x > 1
sat(x)= g ( ) I I (2.50)

2:, otherwise
then (p'(C, V) = <,o(C,V), V C E Re, and the conclusions obtained earlier for the
state feedback control u = cp(C, V) hold for the state feedback control u = (,0‘(C, V).
The output feedback controller will be taken as u = cp’(o, «E, V). Define the scaled

estimation error

x. = ——.(e.- — é.). ls 2' s r (2.51)

 

from
as thr‘

singu.

 

Where

 

 

lt car.-
by th.
model
1002 5:
Condh
behari
lunctt
like b.

Scaled

[0. T
COuter
allram
undEr

fifille l

 

28

From (2.51) it follows that é = e — D(e)x, where D(e) is a diagonal matrix with er"
as the ith diagonal element. Then, the closed-loop system will have the following

singularly perturbed form

t = AC + B[f1(e, 2, d) + g1(e, 2, (1)9303, a, V)] (2.52)
2. = ¢o(€,2, d) (2.53)
e); = (A — HC)X + eB[f1(e, 2, d) + g1(e, 2, d)<,9"(o, é, V)] (2.54)
where F q -
X1 01
X = E ,H =
-Xr.rx1 larnxl

 

 

 

 

It can be shown that the characteristic polynomial of the matrix (A — H C ) is given
by the left hand side of (2.49). Hence, (A — H C) is Hurwitz. The boundary-layer
model of (2.52)-(2.54) is exponentially stable, and the reduced model is the closed-
loop system under state feedback. Moreover, in view of the scaling (2.51), the initial
conditions of the fast variables are of order 005"“). This causes an impulsive-like
behavior in x [12]. Since x enters the slow equation (2.52) through the bounded
function <p‘(a, e — D(e, V)X), the slow variable C does not exhibit a similar impulsive-
like behavior. However, we will show that after some arbitrarily small time T1, the
scaled estimation error will decay to a level where “x“ is of the order 0(6) (Lemma
2.3) and then approaches zero asymptotically (Theorem 2.1). During the time period
[0, T 1] the estimate of C could be very far from the actual state; this is not of any
concern since the control saturates and the state (C, 2) will remain in the region of
attraction. In the following lemma and the subsequent analysis we will establish that
under output feedback and by choosing 6 small enough, the state C will, after some

finite time, enter a positively invariant set that is in the interior of the set A ,,. Again,

 

Len

We k:-

JA‘)
1

deri Va

Where

 

29

Assumption 2.4 will imply the boundedness of 2.

Lemma 2.3 Let I), be a positive constant such that 0 < bl < b1 < c1. Suppose that
(C(0),z(0)) E 951 x He, and let T; be the first time (C(t),2(t)) exits from the set
99, X (26,. Then T; > 0 and for sufi‘iciently small e, there exists a finite time T1

(0 < T1 < T2) such that Vt 6 [T1,T2), ”x” is of order 0(6).

Proof: We know that (C(0), 2(0)) 6 {251 x QC, C 99, x 519,. Since

at ., M . )..
Amm(P) Amin(P)

we know from Assumption 2.4 that

 

 

W< 0, V(C,2,d) 6 99, x (2,, x D and W(z) = c2

Hence, the trajectory (C (t), z(t)) can leave the set 99, x QC, only through the boundary
V(C) = b1. Since <p‘(a, 6, V) is globally bounded, we have

||f1 + 9199’ - KCII S ks (2.55)

V (C,2,d) e 09, x 09, x D, for some k3 > 0. Taking V(C) = CTPC and evaluating its

derivative along the trajectories of (2.52), we obtain

V S -||C|l2+2kallP3llll(|l (2.56)

g —272v + 253W, v (C,z,d) e 0., x (2., x D (2.57)
where 72 = 1/2Amu(P) and 93 = k3||PB||/‘/).m.-,,(P). From (2.57) we have

W 3 WW + $0 — .421)

 

T2

in

30

Since V(O) S 51 < b1, there exists a finite time T2, independent of 6, such that
T2 2 T2. Now, let us turn to the fast equation (2.54) and study its solution over the
interval [0, T2). Let W = xTPx, where P = PT > 0 is the solution of the Lyapunov
equation

F(A — HC) + (A — HC)T13 = —I

From (2.55) we have

“f1 + 9199’“ .<_ (“4

V (C,z,d) E 05, x QC, x D, for some 1:, > 0. Then, similar to (2.5G)-(2.57), it can be
shown that V t E [0,T2),

W S _ 1 _ 2“138”,? M
eAma1(P) Amin(P)

_<_ —-:3W, for W 2 62,34

where m = 16||PB||2kZ/\?MI(P)/Amgn(15) and 73 = 1/2Ama,(P). Thus, as long as
(C,Z,d) 6 05, X 9.3, x D and W 2 62,64, we have

ks

e-‘vat/f
627—2

W) s wane-73‘“ s

for some k5 > 0. Let 61 be small enough such that

 

6 165 1 —
= — < —
T1(C) 73 In (,6462') _ 2T2

V 6 6 (0, 61]. This is possible since the left-hand side of the foregoing inequality tends
to zero as 6 —r 0. Hence, for all 6 e (0, 61], there exist T1 S §T2 such that Vt 6 [T1, T2),
W(t) _<_ 6234 and this implies that le|| is of order 0(6). In case T2 = 00, the foregoing

analysis implies that "x” is of order 0(6) for all t 2 T1. 0

 

and
thei
Equa
state
of Hi

Heno

fOI SO

 

We 0b

This 9
(Cl/1 l. :

Repea

 

31

In the forthcoming analysis, our purpose is to establish that under output feedback
C will enter the set A1 dag (23, F) {C : 770 Isl S m} in finite time, where p; < p is a.
positive constant independent of 6. We note that the set A1 is chosen with #2 < a to
ensure that when Isl < £3“: we will have I.§I < #0- for sufficiently small 6; consequently,
the nonlinearity 17(5 )/\/,,(§,1)((,2 )) will reduce to 173-3. To that end, let us study the

slow equation (2.52) over the time interval [T1, T2). Rewrite (2.52) as

C = AC + Blf1(e,z,d) + 91(e,z,d)s9’(0»6w)l

+Bgl(e,z,d)[9o’(o, é, V) — cp‘(0',e,u)] (2.58)

and observe that “(E — e|| = IID(€)XII S k66, (I|D(6)II = 1,V 6 S 1). Since (2),, is in
the interior of QC, , choosing 6 small enough ensures that whenever C E 91,, , C 6 Gel.
Equation (2.58) can be viewed as a perturbation of the closed-loop equation under
state feedback over the time period [T1, T2), with the perturbation term of the order
of ”x“. Define fl, = {W S 6234}, and recall that V t E [T1,T2), leII is of order 0(6).

Hence we can revise the estimate of V given in (2.28) to
V S _IICII2 + % + [€769 V (CszaXad) E Qbi X Q<32 X QC X D

for some to; > 0. For

c s (a —1)-8% (2.59)

we obtain

VS-%(a-1),VVZfll

This shows that 93, is a positively invariant set. Since B1 S b1, the trajectory
(C(t), z(t), x(t)) can not leave the set 01,, x QC, x (I, through the boundary V(C) = 5,,

Repeating previous arguments, it can be shown that the set S15, x QC, x Q, is a

tt

The

When

RX

[4

We ha

Where
Will e
Small
and .J
by

 

 

32

positively invariant set, and trajectories inside 9),, x QC, x Q, reach the positively
invariant set Ru x Q, C (25, x QC, x (L in finite time. Therefore, the set (lb, x QC, x Q.
has no finite exit time, i.e., To = 00; hence, ”x” is 0(6) for all t 2 T1.

The next step is to establish the attractivity of 51 43‘ {C : no IsI S #2}. By repeating
the analysis carried out for the state feedback case, taking into consideration the effect

of 0(6) perturbation due to output feedback, one can show that, inside 72,, x 0.,

I7 = zsIBTPM + 8104+ BTPBU. + 91¢ — K0]

+2sBTP391 [90%, «E, u) - Mam/)1 (2-60)

The first bracketed term of (2.60) is the expression we had under state feedback, while
the second term represents the perturbation due to output feedback. Therefore, using

(2.37) we obtain
)7 s 2 Islp<no — m) — 25P770Nu + 2:2 Isl ksksIaé (2.61)

where L; is a Lipschitz constant for cp’((,u) and k8 = max{|g1(e,z,d)I : (C,z,d) 6
72,, x D}. Notice that L1 depends on 1/ )1. It can be shown that for all 6 S p1/2k6k8L1
we have

; 1
V 5 -§ ISIPM, when 170 ISI 2 #2

where p; = p(l — fut) < p. Thus, we obtain the desired result, namely the trajectory
will enter Al x QB: x Q( in finite time and remain inside thereafter. By choosing 6
small enough we can ensure that for all (C, z,x) 6 Al x 9,3, x 0,, we have nu") = no
and No(.§,no) = no-E. Therefore inside A; x (to, x Q. the closed-loop system is given

by

E = (A + 8106 + 8m + 9199’(C,V) — K61+ 39. [9916, u) — m. u)] (2.62)

 

be a

chos

kn'a

and

9&6,

 

5(0, 3

 

 

33

2:? = ¢o(€,2,d) (2.63)

65c = (A - HC)X + eB[f1 + 9190‘“, 11)] + 6391 [W(CW) - 99’“, 11)] (2-64)

where in cp‘((, u) and cp‘((j, 11) above, No(n,s) = no-f; and No(n,.§) = no%. The system
(2.62)-(2.64) is a singular perturbation of (2.39) and has an equilibrium point at
(E = 0,2 = 2°,x = 0).

The last step in our analysis is to show asymptotic regulation, i.e., all trajectories

inside Al x no, X (lo approach the equilibrium point as t —§ 00. Let

~ ~

W(c, 2.x) = 17(2) + METPE)" + (£er

be a Lyapunov function candidate for the system (2.62)-(2.64) where A and 7 are

chosen as in the state feedback case. It can be shown that

i 5 - ~ APO ~ 2
- 2%(4’ (z) + Inc”) 2” Isl

~ A06 a3
Macho)" ||C||27 + —||X||2“' - —||X||2”
’70 C
+09”le + amllxllh‘l |f1 + also’l (2-65)

for all p S p“, where

06 = 27IIPII7k8L1a08 = 7(Amin(P))7-19

09 = 27k8L1IIPII7fllo = Z'YIIplPa

and ‘70 is some arbitrary positive constant. Define F (o, e, 2, d) = f1(e, 2, d) +
gl(e, z,d)cp’(o,e, 11). Then, it can be shown that inside Al x 0,3, x {26, F(6,0, z,d) =
6(0,z,d). Hence by adding and subtracting the term F (6,0, z,d) to F (o, e, z,d) we

 

ob

wh‘

Whe:

CHOU

Fina}

Then

 

 

34

obtain from (2.40) and smoothness of the nonlinearities

IF(a,e.z.d)I s klw°(é)+a7II6II
_ a7(¢°(s)+IIEII)

(2.66)

/\

where ('17 > 0. Using (2.66),(2.41) in (2.65) we get

; _A
W _ —<—-(¢(2)+IICII) —2’;°Is sI“

01007

+/\ae(7o)”‘(¢°(2) + IICI|)2"+ —3-Z(¢“( )+ IICII)”

ha
-I— " a9 — 0106—17013)”1 — -—6III: ”27

where #3 is some arbitrary positive constant. Choose ‘70 small enough and no large

enough so that

 

 

6 01057 a
_ _ A m _ > _
260 Gem) #3 — 460
Finally, choose 6‘ S 1 small enough so that
A
'a—f — 09 — 6110617013)”l — —a—6 2a
6 ’70 6

Then for all 6 S 6" and for all u S u‘, we have
‘i' a ~ ~ APO ~ 2 (18
< __ a 2‘1 _ __ ‘7 _ _ 21.
w _ 4600/) (2) + IICII) 2y IsI 2€lell

Our conclusions are summarized in the following theorem.

Theorem 2.1 Suppose that Assumptions 2.1 through 2.6 are satisfied and consider
the closed-loop system formed of the system (2.10), the observer (2.48) and the output

feedback control u = $16, u). Suppose (C(0), 2(0)) 6 951 x 062 and 6(0) = 0(0) is
é(0)

bounded. Then, there exist p" such that V p E (0,;1‘] there is 6" = 6"(p) S 1 such that

 

We
indé
natt

reqL

"I
“’11;

lntl
ofi

outp

COP!
and'

and

 

 

 

35

V 6 6 (0, 6"] all the state variables of the closed-loop system are bounded and e(t) —+ 0

ast—ioo.

We remark that although the region of attraction in the transformed coordinates is
independent of the parameters, it will be parameter-dependent in the original coordi-
nates due to the dependence of the transformation (\11 o T)”1 on the parameters. To
require this to hold uniformly in the parameters in the original coordinates as well

will further restrict the allowable size of uncertainty.

2.5 Semi-global Regulation

In this section we will consider the semi-global case. Precisely, given any compact set
of initial conditions of f, we can design an output feedback controller that ensures

output regulation for all initial states in that set.

Corollary 2.1 Suppose that U9 = R" for all 0 6 O, the functions a,-(.), (i = 1,2,3)
and 71(.) are class Koo functions, Assumptions 2.1, 2.2, 2.3, 2.4 and 2.6 are satisfied,

and that Assumption 2.5 is satisfied for all c1, c2 > 0 such that

 

a4 (Vol/AWAPO S 62,

then for any given compact set N and for all initial states {(0) E N there exists
p‘ > 0 such that V p 6 (0,p‘] there is 6‘ = 6"(u) S 1 such that V 6 6 (0,6‘], the
states of the closed-loop system, consisting of the system (2.10), the observer (2.48),

and the output feedback control u = 99’“, V), are bounded and e(t) -+ 0 as t —+ 00.

Proof: In the previous section we have already proved this result for initial states
in 951 X It”, 0 < b1 < b1 < C]. Thus, It is enough to show that b1,c2 can be chosen
arbitrarily large to include any compact set N in 951 X 0”. We have U9 = R" and

T(U9,0) = R", for all 0 6 9; clearly, Assumption 2.3 will be automatically satisfied.

IleI
rtfl
be

larg

2.1

trol
that
out;
(Lip
the:
dyna
lZ€()
\Ie c
the 1
back
usut
Safisf
stabh

Smne

éo.b

 

LipSC
fora
butx'

Putt

 

36

Hence, M4 = R"+1 for all d 6 D. Moreover, S = R”1 and U = 12"". Therefore,
r,-, (i = 1, 2) can be chosen arbitrarily large. It follows from (2.22),(2.21) that c1, c2 can
be chosen arbitrarily large, respectively. Consequently, b; can be chosen arbitrarily

large. 0

2.6 Stabilization Result

In solving the regulation problem we exercised special care in the choice of the con-
trol strategy and the analysis tools so as to arrive at a semi-global regulation result
that avoids four restrictive features that appear in some earlier global or semi-global
output feedback control of input-output linearizable systems; namely, global growth
(Lipschitz) conditions, global exponential stability of the zero dynamics, linearity of
the zero dynamics and restrictions on the way y and its derivatives appear in the zero
dynamic equation (2.12). To put our contribution in a better perspective, we special-
ize our design to the thoroughly investigated output feedback stabilization problem.
We consider the system (2.1) with 9 = 0o (known) and f(0,0o) = h(0,0o) = 0, i.e.,
the origin is an open-loop equilibrium point. The goal is to design an output feed-
back control that achieves global or semi-global stabilization of the origin. Earlier
results are available in [34, 35, 41, 42, 12, 40, 37, 36]. In [34], 45o, f1 were required to
satisfy global Lipschitz conditions and the zero dynamics were globally exponentially
stable. Also, only the output y was allowed to appear in (to. In [35] they used the
same controller of [34] and allowed the output and its time derivatives to appear in
¢o, but required global exponential stability of the zero dynamics and imposed global
Lipschitz conditions on (to, f1, as in [34]. In [41] and [42] a solution was presented
for a certain class of nonlinear systems that is characterized by geometric conditions
but the zero dynamics, in suitable coordinates, were linear. In [12], a dynamic out-

put feedback controller was designed to stabilize a fully linearizable nonlinear system

37

where the zero dynamics, in suitable coordinates, were also linear. The results of
[34], [35], [41] and [42] were global in nature. In a recent work, Teel and Praly [37],
[36] presented a semi-global solution without imposing global growth conditions, but
they allowed only the output y in (to. Also in [43] Praly and Jiang considered systems
that admit a global normal form and, as [37], [36], only the output and not its time
derivatives were allowed to appear in the nonlinearities. The controllers of [34], [35],
and [37] were linear.

To develop our control strategy for this stabilization problem, we start from the nor-
mal form (2.5). The main difference from our regulation control is that we do not
need to use integral control; recall that in the regulation problem we try to stabilize
an uncertain equilibrium point where the tracking error is zero; a task that requires
integral action. Without integral control, and with the reference V = 0, equations
(2.11)-(2.12) coincide with (2.5). From this point on we proceed to design the output
feedback controller as in the regulation problem case, leading to a semi-global result
that does not suffer from any of the four restrictions stated at the beginning of this

section. A concise description of this stabilization result can be found in [44].

2.7 Time-Varying External Signals

Although the integral control is used to ensure asymptotic tracking when the distur-
bances and references are constant, one would intuitively expect it to be effective for
some cases of time-varying signals. Two such cases are time-varying signals which
tend to constant limits as t —» co, and slowly-varying signals. In this section we study
the tracking problem for these two cases. We start with time-varying signals which
tend to constant limits. We assume that d(t) E D, Vt Z 0 and d(t) —> d as t —+ 00.
Since D is compact, Je D. Let 5: déf I o _ 5(3), (3; _ 5(3))T, (z _ 3(3))1', XT IT and

d 4;; d - d, where 6(.), 5:(.) and A(.) denote the equilibrium point that corresponds to

SI.

frc

wh

equ

Hen
Lya]

that

 

 

38

d = d. Define Dd = R X \II‘1(Nd) X R'; then, in the .i' coordinates, the closed-loop

system can be represented by

~

=f(x, 3+ d) (2.67)

v 5: e D,l "'3

{5: E R"+'+1: ”5:” S r1} where r1 is chosen small enough to ensure
that V 5: 6 D,1 , (xT, 2T)T E \Il‘1(Nd). The expression for f can be easily determined

from the previous section. The right hand side of (2.67) can be rewritten as

food) + g(x, d, J) (2.68)

H!-
II

where g(x, d ,d)= f(x, d)— f(:i:,d). Due to the smoothness off we have I|g(.)II S
mlIIdII, m1 > 0. Since at 6 D, we know that 5: = 0 is an asymptotically stable

equilibrium point of the system

if: = f(5:,d’) (2.69)

Hence, using the Converse Lyapunov Theorems (see [31, Theorem 4.7]) there exists a
Lyapunov function Vo : D,o d-i-f {5: E R"+'+1 : ”x” S ro} —» [2+ for the system (2.69)

that satisfies the inequalities:

a«(II-"13“) S Vo(i') S 05(IIiII)

a—.-V°i(i d)- < -ae(ll:r||)

6V
ll— JaillSadlli II)

where ro is chosen small enough to ensure that the solution is always in D,.1 and 0,,
(i = 4,5,6, 7) are class [C functions defined on [0,ro]. Now, let {Tn} be a sequence

such that Tn -» 00 as n —> co and T1 = to = 0. With Vo as a Lyapunov function

 

lion

whet

that

 

 

\NheI

39

candidate for the system (2.68), it can be shown that

- ~ ~ _ m a r ~ _
V0 < —(1 - 90)ae(llxll). v llrrll 2 a.‘(—‘—£—Qlldll). o < a. <1 (2.6)
From (2.70) we conclude that
- , ~ _ m a r ~
V0 < —(1 - oo)a.(IIxII), v Her” 2 oat—“5:13 I‘ll.» W 2 T. (2.71)

where IdIT = sup{|Id(t)|I : t 2 Tn}. Applying [31, Theorem 4.10] it can be shown

that the solution 5(t) satisfies
”z(t)” s B(lli(T.)|l.t - T.) + R(IJITn). v t 2 T.
where )6, E: are class KL and class IC functions, respectively, and

Ilfé(T.)|| s B(Ila”:(0)ll. T.) + atlciln)

V 53(0) 6 Dro. Since S(II:E(Tn)|I,t) —* 0 as t -+ 00, given any 6 > 0, there exist no such
that Vt Z Tn0
5(I|5=(71"n)||.t - T.) < 6/2

Also, since d —i 0 as t -—r 00, there exist n1 such that V t Z Tn1
“(in)” < k"(c/2) => IJITM < rc"(e/2)

Therefore, there exists T > 0 such that IIi:(t)II < 6, Vt 2 T. This implies that
x(t) —» 0 as t -> oo. Observe that 552(t) = 61(t), therefore e1(t) —+ 0 as t —+ 00 or
y(t) -> :7 as t —> 00.

Next, we consider slowly-varying external signals where we assume that d is bounded,

Len

repre

wher
that
D is

exist

and

Here

estah

Leta

SlflCe

EXISI.

 

40

i.e., ”d” S c3. Recall from (2.67) that our system, in the 5: coordinates, can be

represented as

é=f@u) (2n)

where :E, f are as defined earlier and d 6 D is a parameter. We have shown previously
that V d 6 D, the equilibrium point 5: = 0 of (2.72) is asymptotically stable. Since
D is closed and bounded, it follows from [45, Auxiliary Lemma] that V 6 > 0 there

exists 6(6)(independent of d) such that V d E D
"530)” < 6. V t .>_ to. if ||5?(to)I| < 5(6)

and

pflfln=atumum<6m

Here, we need to establish that the passage to the limit is uniform in d which was not

established in the auxiliary lemma of [45]. Given n > 0 we have V d E D
lli(t,d)ll < 1). Vt 2 to, if II55(to,d)II < 5(7)) (2-73)
Let do 6 D. Since 5:(t,do) —) 0 as t —> 00, there exists T(n,do) Z to such that
I|x(t,do)I| < 6/2, Vt Z T

Since the solution 5:(t, d) depends continuously on the parameter d, given a > 0 there

exists 7 > 0 such that V d 6 {d: ”d — doII < 7},

”53(t, d) — 53(t,do)II < O, V t E Ito,T]

 

lhk
Bl.l
£IUI

dsuc

 

41

This implies that, at t = T, IIx(T,d)II < 6 (by choosing a S 6/2).
Hence, from (2.73) and since the system (2.72) is autonomous, we conclude that
I|5:(t,d)|| < n, Vt Z T, for all 61 E {d : I|d — doII < 7}. Since D is compact, it is
covered by a finite number of these neighborhoods. Hence, there exists T (n) such
that

||5:(t,d)|| < n, Vt 2 T, v d e D

This shows that a = 0 is asymptotically stable uniformly in d. Following the proof of
[31, Lemma 4.2], it can be shown that there exist a class IC function, ’61, and a class
.C function, 01 (see [46, pp 7] for the definition of class .6 functions) , independent of
d such that

“50,00” S ~1(||53(to)||)01(t - to), V d 6 D

V 53(to) 6 D,.2 déf {it 6 R"+’+1 : IIxII S r2} C D,,. Thus, assumptions 1,2 and 3 of
Hoppenstead’s lemma [47, Lemma 1] are satisfied. By an application of this lemma,

together with an argument similar to the one used in [47], we conclude that
||5:(t,d)|| S 31(II5(10)I|.1)+ 710630. V d E D. W 2 to

where 31, 71 are class [CL and class 16 functions, respectively. Hence, the tracking
error is ultimately bounded by a class [C function of c3. Therefore, for slowly—varying
signals where IdI is small, the tracking error will be small. If, in addition, d —> 0 as
t —+ 00, then an argument similar to the one used earlier in this section will show

that 5:(t) —+ 0 as t -—> co and this implies that the tracking error -+ 0 as t —+ 00.

2.8 Conclusions

The basic idea of this work is the use of integral control to achieve asymptotic reg-

ulation and disturbance rejection for constant references and disturbances. By in-

42

troducing the integrator we create an equilibrium point at which the tracking error
is zero, since the integrator is driven by the tracking error. The main task of the
controller after that is to stabilize the equilibrium point, which is a challenging task
because the equilibrium point depends on the unknown parameter 0. This is not
an issue in linear servomechanism where stabilizing the equilibrium point does not
require knowledge of that point. There it is merely stabilization of a matrix which
represents the homogeneous part of the closed-loop system. If we were interested
in local results that would hold only for sufficiently small initial states and external
signals, the stabilization problem for the nonlinear servomechanism would reduce to
stabilization of a matrix. It is the desire to obtain regional and semi-global results
that makes this stabilization problem a challenging one. Our approach to stabilize
this unknown equilibrium point is to use a nonlinear robust controller to drive the
trajectories toward the point ((,z) = (0, 2°). As far as that nonlinear controller is
concerned, it is not trying to stabilize the closed-loop equilibrium point. It is only
pushing trajectories to a small residual set around the point (0,20), and by doing
so it automatically pushes the closed-loop equilibrium point inside that set. Since
the size of the residual set can be made arbitrarily small by the choice of a certain
design parameter, we choose it small enough to ensure that inside the residual set the
controller will act as a high-gain controller that stabilizes the closed-loop equilibrium
point. At no point in our design were we required to know the exact location of the
unknown closed-loop equilibrium point. This whole idea would not have been possible
if we were to use state feedback, since the states in the error coordinates are again
dependent on 0, and here comes the role of the high-gain observer and the globally
bounded control idea of [12] to implement the controller using only measurement of
the tracking error.

An important reason for using integral control is its robustness to model uncertainties.

For linear servomechanism, it was shown that for any plant perturbation that does

tirr

as E

 

 

 

43

not destroy the asymptotic stability of the closed-loop system, asymptotic tracking
and disturbance rejection will be achieved. In this work, as long as the uncertain
parameter 0 does not change the basic structure of the plant; namely, it preserves
the relative degree and the minimum-phase property, asymptotic regulation will be
achieved.

The control strategy used to solve the regulation problem was shown to solve the cor-
responding stabilization problem when the origin is an open-loop equilibrium point.
Also, the semi-global result comes as a natural extension of the regional one. Our
global assumption (for the semi-global case) relaxes some of the global growth condi-
tions available in the literature, as we did not require exponential stability of the zero
dynamics nor did we require the nonlinearities to satisfy global Lipschitz conditions.
Asymptotic tracking was also achieved, using integral control, for a certain type of
time-varying signals. these signals may represent references and disturbances as well

as some time-varying system parameters.

rn

In

acl.

HOI
stu<
the
In t]
tion
time
inpu

by .

d

 

0t (I.

 

CHAPTER 3

Robust 'IYacking for Nonlinear
Systems Represented By
Input-Output Models

3.1 Introduction

In linear servomechanism theory, conditions under which asymptotic tracking is
achieved using state or output feedback control have been completely established; see,
e.g., [13, 14, 15, 16]. The nonlinear servomechanism problem, on the other hand, is
not completely understood and continues to be an active area of research. It has been
studied before under various assumptions on the class of systems being studied and / or
the exogenous signals being tracked and /or rejected; see, e.g., [17, 18, 19, 20, 21, 23].
In this work we extend our previous work of Chapter 2. where we studied the regula-
tion problem, to the more general tracking problem where the exogenous signals are
time-varying. We consider a class of $130 nonlinear systems that are represented by
input-output models. Using the idea of [10], we extend the dynamics of the system
by augmenting a series of integrators at the input side. This makes the derivatives

of the input available for feedback. With this approach we overcome one of the lim-

44

it;

de

45

itations of Chapter 2, namely being able to use only partial state feedback. In our
design methodology we start by identify the internal model; a task that hinges on two
basic assumptions. The first is the solvability of a partial differential equation, and
the second is an equality that will be automatically satisfied if the nonlinearity is of
polynomial type. After augmenting the original system with the servo-compensator,
we design a state feedback control that stabilizes the augmented system. However.
instead of using a specific stabilizing control, which is another limitation of Chapter
2, our objective here is to allow flexibility in the design of state feedback control.
Toward that end, we present a set of conditions that a state feedback control should
satisfy in order to be stabilizing. Then we proceed to design an observer to recover
the asymptotic tracking properties achieved under state feedback. For that we used
the technique introduced in [12] that comprises a high-gain observer with a globally
bounded implementation of the control.

This chapter is organized as follows: In Section 3.2, we present some preliminaries
and some basic assumptions. In Section 3.3, we give general characteristics of a state
feedback stabilizing control. In Section 3.4, we present the observer structure and
give our main result. In Section 3.5 we give examples of controllers. Since state-space
models are prevalent in control theory, in Section 3.6, we give conditions under which
a given state space model will have the input-output model assumed in our work.
In Section 3.7, we illustrate our results by giving a special input-output model for
which most of our assumptions are automatically satisfied. In Section 3.8 we present
a design example with simulation results. Finally, we make some concluding remarks

in Section 3.9.

46
3.2 Preliminaries

We consider a 8180 nonlinear system which has an input-output model given by the

n-th order differential equation:

g(n) = f(y,...,y(n-l),u,_..,u(m—l),w(t))

.I. g(y, . . . , 3101-1), u, . . . , uIm-ll, w(t))u(m) (3.1)

where y“) and u“) are the i-th derivatives, with respect to time, of y and u, respec-
tively, w(.) is a continuous time-varying disturbance signal, assumed to be contained
in a compact set D C R”, and g(.) at 0, for all possible values of its arguments in a
domain U1 X U2 X D where U1 C R“ and U2 C R”; let U = U1 X U2.

Let r(t) be a time-varying reference signal that together with w(t) are generated by

a pI—dimensional exosystem

V(t) = 3.1/(t) (3.2)

where all the eigenvalues of So are on the imaginary axis and distinct. Clearly, V(t)
belongs to a compact set 01 C R" .

We note that for feedback linearizable systems with relative degree less than n the
state feedback control, in some of the existing literature, uses only a part of the state
vector, i.e., the states that describe the observable subsystem; see for example [37,
36, 44, 48] and the work presented earlier in Chapter 2. To avoid this in the present
work, we utilize the idea advanced in [10] of augmenting a series of m integrators at

the input side of the system. Define:

$i+l = y“), 2:0,m,n—1

(1+1 = “(1'), j=09"°9m—1

 

Augt‘.

dime

wher
feedl

estin

Whei

 

Obs.
torn
ll] U

Ass

47

Augmenting a series of m integrators at the input side, and using (2:, C) as an (n+m)-

dimensional state vector, a state model of the system is given by

6,- = x,-+1,i=1,-~,n—1 ‘

in = f($,C,V(t))+§($,C.V(t))v

C} = Cj+1,j=l,'--,m—l ) (3.3)
Cm = v

y = $1 I

 

where x = (x1, - - - ,xn)T, C = (C1, - - - ,Cm)T. We note that the state C is available for
feedback. Hence in output feedback control, as it is the case here, we need only to

estimate the state x. We rewrite the last m-equations of (3.3) as

C°=A24+Bzv (3.4)
where _ . . q
0 1 - 0 0
0 0 l 0 0
A2: : i 9 82:
0 0 1 0
L0 O‘mxm lemXI

 

 

 

 

Observe that system (3.3) is input-output linearizable, but it is not in the normal
form [30] due to the appearance of the control v in the last state equation. With this

in mind, we state the following assumption

Assumption 3.1 There exists a diffeomorphism

I (I:

= - gnaw) ' (3.5)
2 T1($,C,V)

W ' v'.‘m-L'm-.u.'vdr—-i

that

0H3

u'hic

“‘60

The

 

 

 

 

48

that maps (x,C) into (x,z) for all V 6 D1, and transforms the last m state equations
of (3.3) into
é : ¢0($,Z,V(t)) (36)

which, together with the first n state equations of (3.3), define a normal form which

we assume to hold in the domain D déf T(U X D1) C RM”.

Remark 3.1 The above assumption requires the normal form to be valid on a given
region rather than being valid only locally. A similar assumption was used in Chapter
2; see Assumption 2.2 and the subsequent discussion for more details. If g(.) is
independent of xn and Cm, i.e., g(.) = g(x1,-~,x,,_1,C1,-~,Cm_1,w(t)), then the

transformation

2; = (is i=1)'°°1m—1

‘Q

will satisfy Assumption 3.1.

Define the tracking error 61 = x1 — r(t). Let

Then, in the new coordinates, the system (3.3) becomes

6, = e,+1,i=1,--~,n—1 (3.8)
6,, = for:1 +r(t),-~,en+r("’1)(t),C.V(t))

+ r(e1+ r(t). - - - . e. + 1‘"""(t). c. v(t))v — r‘"’(t)

‘é’ fte.<.u(t))+g(e,<.u(t))v (3.9)

C = A2C+Bzv (3.10)

wh-

eqt
the

for

Ass

Par

 

 

49

gm = 61 (311)

where e = (e1, - - - , en)T and ym is the measured output.
Our objective is to identify the internal model. Of particular interest to us are the
equations that govern the system (3.8)-(3.11) on the zero-error manifold. These are

the equations that will result when we restrict e] E 0, => 6 E 0. Hence, we get

 

f(0.C.V(t)) +g(0.C.V(t))v = 0 (312)
° _ -f(09CaV(t))
C— A2C+ BzI g(O.C.v(t)) I (3.13)

Equation (3.13) is referred to in the literature as the tracking dynamics [8]. It gives the
zero dynamics of the system (3.8)-(3.11) when the output is the tracking error. In gen—
eral, the state feedback controller will take the form v = g;1(e, C, V(t))I—fo(e, C, V(t))+
17], where go(.) and fo(.) are nominal functions of g(.) and f (), respectively, that are
allowed to depend on e,C and V(t) (possibly the reference signal and its derivatives).
We observe that although the state C is measurable, the state 2 is not so, due to the
dependence of the transformation T on the disturbance. Substituting the expression

for v in (3.12), we obtain

0 = go(O,C.V(t))9’l(0,C,V(t))f(0.C.V(t))

-fo(0.C. V(t)) + 6 (3.14)

Assumption 3.2 (a) There exists a unique mapping C = Ao(V) which solves the

partial difi'erential equation

0A0 _ _f(09A0(V)aV)
'575011 — A2A0(V) + 82 g(o, A0(V), V) 1 (3.15)

 

(“Oh

 

 

50

(b) There exist a q X q matrix S, q 2 p1, with distinct eigenvalues on the imaginary

axis, and a 1 X q constant matrix 1" such that

90(0, /\0(V(t)). V(t))g'lma I00/00), V(t))f(0. AMI/(0). 1/(t))

—fo(0,Ao(V(t)),V(t)) = FV(t) (3.16)
V t E R, where V(t) is a solution of the linear difierential equation
V(t) = SV(t) (3.17)

Remark 3.2 Note that ifC = Ao(V), on the zero-error manifold, then it follows that
z is a well defined function of V(t) through T1(.), e.g, z = T1(el + r(t),---,en +
r(“’1)(t),)\o(V),V)Ie=o dg A(V). In the special case of constant exogenous signals that
we studied in Chapter 2, Assumption 3.2(b) is automatically satisfied while 3.2(a)

reduces to an assumption on the existence of an equilibrium point, see Assumptions

2. 1 and 2. 3.

Assumption 3.2(b) was first used by Khalil [23] and it is fundamental to this work.
Essentially this assumption requires the nonlinearities in the system to generate only
a finite number of harmonics of the original modes as modeled by equation (3.2).
This being the case, then the q-dimensional exosystem can be modeled by a linear
differential equation. Clearly, V(t) belongs to a compact set; D2 C R". To simplify
the notation we will drop, from now on, the time variable t from V(t) and V(t),

although the fact remains that they are time dependent.

Remark 3.3 Related to this assumption is the one used by [22]. The paper [22]

considers a nonlinear system

5: = f(x) + g(x)u + 19(le

61

an
coc
obt

p01.

Rev

 

51

with an error equation

6 = h(It) + 9(2r)

where the signal V satisfies a linear equation V = SoV and the matrix So has all
its eigenvalues on the imaginary axis. The paper assumes that for all admissible
perturbations of the system, there exist functions 7r(V) and p(V) which satisfy the
equations

375°” = My» +9(T(v))p(u) +p(«(u))u (3.18)

o = h(7r(V))+q(V) (3.19)

and p is a polynomial in V. Applying these equations to our system in the error

coordinates with W(V) = and v = ggl(—fo + p), it can be verified that we
1‘00’)
obtain (3.15) and the function p is the left-hand side of {3.16). The fact that p is a

polynomial in V guarantees that (3.16) is satisfied.

Rewrite the transformation (3.7) as

e III—LIV def \Il1(.’II,l/)

Z Z 2'

where L1 is an n X p1 matrix. We assume the following

Assumption 3.3 There exists a domain N = N1 x N2, that contains the origin, such

that \I’f1(N1) C U1 and T1’1(N2) C U2 for all V 6 D1.

Basically, this assumption will restrict the size of the signal V. To see this, consider
the transformations 6 = x — r(t), z = C — xr(t), and suppose the sets U1, U2 and D1
are given by U. = {. = IzI s 1}, U. = {c = ICI s 1) and D. = {r(t) = W)! s ..}.
Clearly V ro > 1 both N1 and N2 will be empty.

lbrc

Let.
whet
flat}

mph

 

 

52

Let X1 = R9 X N1. Also, define a new variable 3: :2 : - A(V). For the purpose of
stating the next assumption, we need to express the system in the (e, E)-coordinates.

First, observe that from (3.6) we have

2: = ¢(6,Z,V) (3.20)

where ¢(.) = wo(e + L111, z, V). Then the system in the (e, §)-coordinates becomes

éi : €5+1,i=1,°",n—1
én = f(ea (9 V) ‘I' g(e, (a VleC=T,"(e+L1 “5449),” (3°21)
i: = ¢o(e,§, V)

where ¢o(.) = 65(6, 2 + A(V), V) — %Sou. We state the following assumption.

Assumption 3.4 Assume for the system 2 = ¢o(e,§, V) there exists a 01 function

W : Rm —» R... which satisfies

01(||5||) S W(15)S az(l|5||)

6W

a7,6,2...) 5 -¢1(||5|I). v 121 2 7(Ilell)

for all (e,z,V) 6 N1 X N2 x D1, where 61,-, (i = 1,2), 451 and 7 are class [C functions.

Let Q“, = {z : W(E) S a2} where a2 is chosen such that no, C N2. This assumption,
when e = 0, implies that the origin of the system 2 = ¢o(0,§,V) is asymptotically
stable, i.e., the zero dynamics have the minimum phase property. It also implies

input-to—state stability from the input e to the state 5 [32].

 

We

 

53

3.3 State Feedback Design

We augment the linear servo-compensator
6=So+Je1 (3.22)

where (S, J) is a controllable pair, with (3.8)-(3.11) to obtain the augmented system:

 

 

 

 

67 = 50' + J61 (3.23)
e = Ac + BIfte, c. u) + g(e. c, u)v] (3.24)
C. = AzC + 320 (3.25)
gm = Ce (3.26)
where q q
0 l - 0 0
0 0 1 0 0
A = 18 = ,
0 0 1 0
O . 0 - an L 1 d nXl
1Xn

We consider a state feedback control of the form

v = 951(e.(. u)I—fo(e, c, u) + «as. t. V)] dé’ 9(6 4, u)

tvhe

and
To I

insx

cont
ifth

COD)

Ass

Unfi

and

 

 

54

where ¢(.) is locally Lipschitz in C uniformly in C, V. For compactness of the presen-

tation we rewrite the closed-loop system as

t = A5+BIf(e.c,u)+g(e.<,u).o(5.<,u)1

dé’ h1(£.C.V) (3.27)
c' = A2C+Bzr(€.C.I/) (3.28)
gm = Ce
where
S .10 0 o
A: a B: ., 6:
0 A B e

and h is defined in an obvious way.

To motivate the upcoming assumption, we recall the design strategy as documented
in some of the literature; see for example [23, 44, 37] and our earlier work in Chapter
2. The essence of this strategy is to achieve the control task in two steps. First the
controller will ensure that the trajectories of the system are ultimately bounded, then
if the system satisfies certain local properties, the controller will achieve asymptotic

convergence to an equilibrium point.

Assumption 3.5 Assume for the system C = h1(C,C,V) there exists a 01 function

V : R”? —» R... which for all C 6 X1 satisfies

31(IICII) S V(t) S 52(Iléll)

gig/11366:!) s —¢2(I|€||). v V(t) 2 9(9) (3.29)

uniformly for all (C, V) 6 U2 X D1, where 3, 3,, (i = 1, 2) and (to are class IC functions

and p > 0 is a design parameter.

55

Define {26, = {C : V(C) S a1} where a1 > 0 is chosen such that 00, C X1.

Assumption 3.5 implies that C(t) will be eventually confined to the set A“ 3;; {C 6
X1 : V(C) S fl(p)}, a neighborhood ofC = 0, where the size of this set can be made
arbitrarily small by choosing p small enough. In the upcoming analysis we will use
Assumptions 3.4 and 3.5 to establish that z(t) will also be confined to a small set in
the neighborhood of A(V). To do this we first express the closed-loop system in the

(C, E) coordinates, i.e.,

6. = A6 + B[f(8, Ca V) + 9(6) (7 1099(6) C3 V)] IC=Tl—l(e+L1V,i+A(V),V)
dé’ 215+ B[f1(e,§.V) + 91(6.5.V)991(£. 2. V)] (3.30)

g = ¢0(6,§, V) (331)

then show that the product set Qa, X (2..., is positively invariant. Recall that we have
a similar situation in Chapter 2 except their it is done for a specific controller. In the
forthcoming analysis we will show that the set 52,, X (to, is positively invariant for a
more general setup. However in our argument we will highlight the main points of
the proof and refer the reader to Chapter 2 for more details.

To that end, we note that Choosing a; 2 02 o 7 o flf 1(611) will guarantee that as long

as C(t) 6 9a,, z(t) 6 52“,. From Assumption 3.5 we have V V Z ,B(p)
V 3 —¢. 0 fl;‘(V)
Hence, V 6(a) < a1 we have

V < o, v (5,2,u) e 9,, x 9,, x D, and v = a1 (3.32)

56

From Assumption 3.4 and the choice of no we have
W < 0, V (C,z,V) 6 (2,, X (In, X D, and W = a», (3.33)

From (3.32) and 3.33) we conclude that the set (to, X 0,, is positively invariant. Now,
we choose p such that fl(p) S b1 < al. This is needed for output feedback in order
to guarantee that whenever C E 0),, , the estimate of C, i.e., C belongs to (20,. Again,

we can show that the set

n, “‘2 A,, x r,

is positively invariant, where F,, = {W S a; o 7 o flf1(fl(p))}; (see Chapter 2, pp 19
for more details). To show that the trajectories (C (t), z(t)) will enter the set 72,, in

finite time, observe that V V Z 6(a) we have

VS —¢20fl2—1°5(fl)

andVWZaoo7oflflofl(p) wehave
WS—¢l°7oflf10fl(l‘l

From now on, an argument identical to the one used in the proof of Lemma 2.1 will

complete the proof.

Assumption 3.6 There exists a compact positively invariant set 5,, C A,, such that
C(t) will enter 8,, in finite time and inside 5,, the control component ¢(C,C,V) takes

the form

97> = K06 + MC V)

where f; is, in general, a nonlinear function ofC and V that satisfies f2()\o(V),l/) =

L21).

57

Lemma 3.1 Suppose that Assumptions 3.2 and 3.6 hold, then there exists a q X q
matrix L such that the set {0' = LV, 8 = 0, C = Ao(V)} is an integral manifold of the

closed-loop system (3.23)-{3.26).

Proof: It was shown in [23], using results from [49] and [13], that there exists a

unique matrix quq that solves the following two matrix equations:
I‘ = —(I\'o1L + L2), LS = SL

The rest of the proof follows by direct substitution. 0

Our purpose from now on is to design controllers, using only measurement of the
tracking error, to establish regional as well as semi-global asymptotic convergence
of the trajectories to this set. An obvious consequence of this is that 61(t) —+ 0 as
t -+ 00.

To study attractivity of the zero—error manifold, let 6 = a — LV. Writing the closed-

loop system in terms of the shifted variables 5’ and :2, we get

5 = AE+BIfI(€,5,V)
+ 91(6959V)¢1(5+LV,6,5,V)I (3.34)

¢o(e, '2', V) (3.35)

M:-
II

where C: (6T, eT)T. For convenience, we rewrite (3.34)—(3.35) as

7i = 72107.10 (336)

where n = (CT, ET)T and h1(.) can be easily defined.
We point out that the origin is an equilibrium point of (3.36). In the following

assumption we impose another requirement on the state feedback controller; namely,

58

we require it to be locally asymptotically stabilizing.

Assumption 3.7 The origin of the system 1'] = h1(1],V), is locally asymptotically

stable.

Implicit in this assumption is the requirement that the system be locally minimum
phase which is guaranteed under Assumption 3.4. This point will be made clear in
Example 1. The following lemma is an obvious consequence of the requirements we

imposed on the state feedback control.

Lemma 3.2 Suppose Assumptions 3.] through 3.4 are satisfied, then under any stabi-
lizing state feedback control that satisfies Assumptions 3.5, 3.6 and 3. 7, and V V 6 D1.

all states will remain bounded and el(t) —+ O as t -—+ 00.

3.4 Recovering State Feedback Performance

In this section we want to study output feedback implementation of the class of state
feedback controllers described in the previous section. In other words, suppose we
were successful in designing a state feedback control that satisfies Assumptions 3.5,
3.6 and 3.7, the question is: can we recover the asymptotic properties of the state
feedback control using an observer. With this understanding and since we have only
measurements of the tracking error, we will consider an estimator with the objective
to recover the properties of the state feedback control, vis a vis convergence of the
trajectories to a residual set first, then attractivity of the manifold inside this set.
Consider the following estimator [40]

_ a a’ * '_
ei — €g+1+;‘(81—€1),3—1,"',n—1

(3.37)

8n = %£(€1—é1)

59

where e is a design parameter to be specified, and the positive constants a.- are to be

chosen such that the roots of

s"+als"'1+---+an_1s+an=0

are in the open left half plane.

This is a high-gain observer that is known to exhibit peaking in the estimates é. To
eliminate this peaking phenomenon, we saturate the control outside a compact set of
interest that will be made more precise in the sequel; see [12, 44, 36, 50].

Define the scaled estimation error

 

.(e; — é.), 1 g i g n (3.38)

then the closed—loop system, under output feedback, will take the following singularly

perturbed form

5 = A6 + Blf(e,C,V) +9(6,C,V)<p(aaé,C,t/)l

“é’ Ms, c, x, u) (3.39)
c‘ = Ag( + 32940, é, (, u) (3.40)
)2 = (1/6)A1X + B[f(e, c, v) + g(e, c, uma, 6:, C, u)]

“é’ (1/€)A1x + 122(5, 9 x, v) (3.41)

where (E = e — D(e)x, 0(a) is a diagonal matrix with 6"“ as the i-th diagonal element,
A1 is an n x n Hurwitz matrix, h1(£,(,0, V) = h1(£.C,z/) and h2(.) is defined in
an obvious way. To answer the question posed at the beginning of this section we
first demonstrate that there exists a finite time To, the value of which can be made

arbitrarily small by choosing 6 small enough, such that V t 2 To the estimation error

60

x will be of 0(e), i.e., ||x|| S koe.

Before delving into this analysis, we need to saturate our control 99(5, C, 11) over the
compact set (In, x Q“, x D1. Let 5'1 = max lap“, C, u)|<=Tl.1(c+Lsz/), for all (5, 2:, V) E
Q“, x 902 x D1. Then the globally bounded control will be taken as

was) = 51 sat (___"°(5’Sf"’l)

where sat(.) is the saturation function defined in (2.50). Replace so in the closed-loop
system (3.39)-(3.41) by (p’.
Let V(€o) < b2 < b] < a1, where b; is chosen such that 95, C 95,. Then there exists

a finite time T1, independent of e, such that
V(§(t,e)) 3 b1, V t e [0,T1)

Now, we look at the solution of (3.41) over this time period. By repeating previous
analysis which was carried out in Chapter 2 (page 30) using the same Lyapunov

function candidate for the fast dynamics, we conclude that V To g t < T1,
||X(t)l| 5 leaf, ko > 0

Next, we look at the effect of output feedback on Inequality (3.29). Calculating the

derivative of V with respect to the system (3.39), it is easily shown that

V S -¢2(ll£ll) + [C55, V V(() _>_ [BULL V t E [T09Tl)$ k5 > 0

|/\

Since 432 vanishes only at the origin, it is strictly positive in the set {6 : 5(a)
V({) S a1}. Hence, (1/2)¢2(||{||) 2 c3 > 0 and it follows that V c < 7:}, V S
—(1/2)¢2(§), V V({) _>_ fly). This shows that the set A” is positively invariant.

61

Notice that A” C 95,. Again, repeating previous arguments, we can show that the
set 0),, x Ila, x Q, is a positively invariant set, and all trajectories inside 0),, x Q0, x Q,
will reach the positively invariant set 72,, x Q. in finite time. From this we conclude
that ({(t),z(t),x(t)) E 95, x Q“, x 9,, Vt 2 To and “x“ S has, V t 2 To. This
implies that the set {25, x (In, x Qc has no finite exit time, i.e., T1 = 00.

In terms of the shifted variables, the closed-loop under output feedback will take the

form

E = A£+B[f1(e,z,v)+gl(e,2,u)w1(é,2,u)1 (3.42)
.2; = ¢o(e,§,u) (3.43)
)2 = (1/€)A1x+B[f1(e,§,V)

+gx(e, 5', 1030105, 5, l/)] (3.44)

where f = [a é]T. For convenience, we rewrite (3.42)-(3.44) as

X = (1/€)A1X+712(77aX,V) (3-46)

Notice that for x E 0, we obtain the state feedback system and h3|X=o = in. To study
the system (3.45)—(3.46), we need some interconnection conditions between the slow

(7]) and the fast (x) variables to hold.
Assumption 3.8 There exists a C 1 function V : Rn+°+m —+ R). that satisfies

8V~
377,130),an S _90¢3(7l)a (IO > 0 (3°47)

62

where ¢3(17) is continuous and positive definite in 77. Moreover the following intercon-

nection conditions are satisfied

||tg(n,o,V)|| s (1033(7)), (1., > 0, 0 < a g 1/2 (3.48)
6‘7 "' ~ - b c
Emmy) - h3(’7»0» V)] S a¢3(n)llxll (3.49)

0<b<1,c=lT'bandd>0.V(£,z,x,V)€S,,xI‘,,xfl¢xD2

We remark that the existence of a C 1 function satisfying (3.47) follows, in light of
Assumption 3.7, from The Converse Lyapunov Theorem; see [31, Theorem 4.7]. The
additional element here is that we require this function together with (#3 to satisfy
(3.47)-(3.49) simultaneously.

To motivate Assumption 3.8 it is important to note that for a singularly perturbed
system with asymptotically stable reduced system and exponentially stable boundary
layer model, it is not true, in general, that the composite system will be asymptotically
stable for sufficiently small 6. To guarantee asymptotic stability of the full system,
in this case, some interconnection conditions similar to (3.48)-(3.49) will be needed.
These conditions are adopted from [36]. They are less restrictive than the quadratic-
type Lyapunov conditions used in [38]. To illustrate the need for such conditions,
consider the singularly perturbed system

' _ 5 2

ex = -x+<€n2

Observe that the origin is an equilibrium point of (3.50). Let V(n) = 172”. It can

be verified that with 433(7)) = 112"“ and go = 2n, Inequalities (3.48) and (3.49) are

satisfied with a = 51:17, b = ifi-‘i and c = 1. However, 1 — b yé a. Hence (3.50)

violates Assumption 3.8. The point that we want to make here is that the origin of

63

(3.50) is not locally asymptotically stable. To assert this we use the Center Manifold

Theorem to show that the reduced system is given by

1i = -n5 + 677‘ + 0(lnl6) (3-51)

Clearly, the origin of (3.51) is not locally asymptotically stable for all e > 0 since the
second term dominates near the origin. Hence, the origin of the full system (3.50) is
not locally asymptotically stable.

To show that we can recover asymptotic stability under output feedback, we first
let W0 = V(n) + (XTPlx)‘7 be a Lyapunov function candidate for the system (3.45)-
(3.46), where 7 = (1/2a) Z 1. Then it can be shown, using (3.47) and Assumption

3.8, that
Aw—l
° — c Amin P
Wu 5 -qo¢3(n)+a¢§(n)llxll- ( --—-1-)llx||"""
+2~r»\;.;:,(a)nanaoqsgmnlxns-l (3.52)

V (£,z,x,V) e S" x I‘,‘ x (I, x D2.
We apply (2.41) to the second and last term of (3.52) with p = l/b and p = 27

respectively, to obtain

W. s -qo¢3(n)+£¢3(n)+5t(€o)”°||X||2”

+2700Amax(Pl)llP1ll
70
,M‘Y-1(Pl

mm

(#3 (n) + 2700A321(P1)IIP1||(70)" IIXII”

 

)le n”

where 60 > 0, 70 > 0, p0 = b/(l -— b) and p1 = 1/(27 — 1). Choose 60, 70 large enough

such that
E + 2700Amair(Pl)llP1ll<

50 ‘70

 

NIH

64
then choose 6 small enough such that

7Ajn:n(Pl)

a(éo)Po + 27’0‘0Amax(f)1)llI31”(’70)Pl < 26

xy—l
=wo_<__—(¢3(n+ )+ m‘;(P———'——‘— ————IIx II”)

This implies, that n(t),x(t) —> 0 as t —+ 00. Hence e1(t) -+ 0 as t —) 00. The main

result of this section is summarized in the following theorem

Theorem 3.1 Suppose that Assumptions 3.1 through 3.8 are satisfied. Consider the
closed-loop system consisting of the extended system (3.3), with v = (p’, together with
the observer (3.37). Suppose (60,20) 6 (lb, x (la, and {(0) is bounded, then there
exists 44" such that V p 6 (0,;4'] there is e“ = 6‘01) 3 1 such that V e 6 (0, 6“] all state

variables are bounded and el (t) —> 0 as t —» 00.

At this point we recall the remark, following Theorem 2.1, concerning the dependence
of the region of attraction in the original coordinates on the exogenous signals. Again,
to require uniformity of the region of attraction will further restrict the size of ad-
missible exogenous signals. As a consequence of Theorem 3.1, we have the following

semi-global result:

Corollary 3.1 (Semi-global Tracking) Suppose U = Rm”, also suppose Assump-
tions 3.1, 3.2, 3. 6, 3.7 and 3.8 are satisfied and that Assumptions 3.4, 3.5 are satisfied
with W,V which are radially unbounded, then for any compact set N C 1‘3”” and for
all initial states (x(0),C(0)) E N there exists p" > 0 such that V p E (0,u‘] there is
6" _<_ 1 such that V e 6 (0,6‘] the states of the closed-loop system consisting of (3.3),

(3.37) , and the control cp‘, are bounded and e1(t) —> 0 as t —> 00.

Proof: For the class of state feedback controllers satisfying Assumptions 3.5, 3.6,

3.7 and 3.8, we showed earlier in this section that we achieved tracking under output

65

feedback for all initial states in (lb, x Ga, where b; < b1 < a1. Clearly, Assumption
3.3 will be satisfied with N1 = R" and N2 = Rm. Therefore, X1 = 12“". Hence.
aha; can be chosen arbitrarily large . It follows that b2,a2 can be made arbitrarily
large and the inverse image of the region of attraction can also be made arbitrarily

large. a

3.5 Examples of Controllers

In this section we give two examples of stabilizing state feedback controllers that
satisfy Assumptions 3.5, 3.6, 3.7 and 3.8. We first require our system to satisfy a

local assumption. Let

6003,01!) = f(e,C,V)-g(e,C,V)96‘(6,C,V)fo(e,C,l/)
- g(e,C,V)gE’(e,C,V)go(0,Ao(l/)x”)9"(0,Ao(l/)xV)f(0.Ao(V),V)

+ 9(61C3V)g0—1(eaCa l/)fo(0, ’\0(V)a V) (3-53)

then substitute for C = T1'1(e + L111, 2, V) and A0(V) = T1’1(L1V, A(V), V) in the right-
hand side of (3.53). Denote the resulting expression by 6(e, 2, V).
Local Assumption: Suppose there exists a C1 function W which is decrescent and

positive definite in E and a continuous function it» that is also positive definite in 5

such that V V E D; , they satisfy
3W ~ - -
$450“), 29”) S _00¢(z)a 00 > 0 (354)

|5(0,2,V)I S k1¢°(5) (3-55)

66

whereO<aSl/2, lcl>0
—I$o(e,2, u) — 430(0,s,u)1 s kzi""(5)llellc (3.56)

where0<b< 1, cle’b, k2>0.
Inequality (3.54) by itself follows from assumption 3.4. However, what we require in
this assumption is the existence of a Cl function such that (3.54)-(3.56) are simulta-

neously satisfied.

3.5.1 Example 1

Consider the nonlinear system (3.27)-(3.28). Let V(C) = CTPC, be a Lyapunov func-
tion candidate for the system (3.27 ), where P = PT > 0 is the solution of Lyapunov

equation

P(.A + BK) + (A + BIOTP = —1

Notice that we added and subtracted the term KC to (3.27) where K is chosen such
that .A + BK is Hurwitz.

Suppose the following two inequalities hold
|f(e, c, u) — ate. 4, male, c, woe, c, u) - ml 3 pa, C)

g(exC,V)gEI(6,C,1/) Z k > 0

V (C, C, V) 6 X1 x U2 x D1. Observe the dependence of the function p on the state
C. Also, we emphasize that what is required here is the knowledge of p and lc since

their existence is always guaranteed. Using the min-max control

_ 1
so = 90 l(-fo - Erma)

67

where

—’-, if s> >0
m: "' "H it (3.57)

8

with s=23TPC, 17(C,C) Z p(C, C) and p a design parameter, it can be shown that

We) 5. ins”? 42’ —¢2(II€II), v we 2 A-..(P)

MIX:

Thus Assumption 3.5 is satisfied with A“ = {C : V(C) S gkm,n(P)}. Let S“ = {C :
17 [3| 5 u} 0 A”. Then we can exploit the technique used in Chapter 2 (page 21) to
show that C (t) will converge to S“ in finite time. Inside S“ the control will assume the
structure so = go'1(-fo — (173/1023). Notice that (,3 is linear in C and independent of C
inside S“, hence Assumption 3.6 is satisfied with Ko = —(2173 / ku)BTP and f2 = 0.

Furthermore, using V(n) = W(E) + A(C~TPC)", 7 = 1/2a, /\ > 0 as a Lyapunov
function candidate for the system (3.34)-(3.35), it can be shown that (see Lemma 2.2

for details)

9(a) s -%<w°(z) + Hill)” dé’ -qo¢3(n), etc. > 0

To verify Inequality (3.48), note that

712lx=0 : f(ea Ca V) + g(ea Ci ”)998(€3 Ca V)

Define F(o,e,C,V) = f(e,C,V) + g(e,C,V)I,o"(o,e,C,V). Then it can be shown that

inside the set A“ x P" we have
F(LV, 0, C, V)|C=Tl-1(L1u,z,y) = 6(0) Z, V)
Hence by adding and subtracting this term to F (a, e, C, V)l(=Tl-1(c+L1V,z.V)’ we obtain

IF(a,e,c, Alex-1th....) s k1¢°(2)+ aoIIEII _<_ anew) with a = 1/27

68

To verify Inequality (3.49), we know that V(n) = W(E) + MCTPC)”. With a straight-
forward calculations we can show that it will be satisfied with 6 = 2x\7||P||X"1 (P)b0

max

where ho is a Lipschitz constant, and b = (27 — l) / 27. Hence with 7 = l, Inequalities
(3.47), (3.48) and (3.49) will be satisfied.

3.5.2 Example 2

Instead of using a min-max controller, we can use the nonlinear high-gain controller

2 S
,. =g.;‘(—fo— ” (if) )

 

Again, using V(C) = CTPC as a Lyapunov function candidate for the system (3.27),

it is easily shown that

17(4) s guru? = —¢2(4), v we 2 gnu-4P)

Again, using the same technique of Chapter 2 (page 21) we can ensure that 17( .)
becomes constant inside A“. We note that in this case S“ = A". Using V(n) of
Example 1, we can establish the same result obtained in Example 1 with the same

(:53 function.

3.6 The Class of Nonlinear Systems

Since the models of most nonlinear systems are specified in terms of state space
models, in this section we give conditions under which they can be transformed to

the input-output model (3.1). Consider the system

f(1'0)+ 9010)" + (1(1‘0x woltll
h(.’L’o)

to

(3.58)

c
H

69

where wo(t) 6 (20 C R”; 00 is compact, y,u E R and x0 belongs to a domain

Uo C R". Notice that in (3.58) q(.) is possibly a nonlinear function of we.

We assume the following

(1) the system has a uniform relative degree p0.

(2a) L,Lf,h(xo) = 0, 0 S i S V0 — 2, V x0 6 U0

(2b) LqL?-lh(xo) 76 0, for some wo(t) 6 00, some x0 6 U0

where V0 S p0. Suppose we can choose the functions 451‘ such that

<d¢j,g>=0, V$0€U0,j=l,"',n—po

and define a change of coordinates:

-.' Li'lh
Z - 1 ($0) déf To($o) (3-59)

NI
II
I

Epo+j ¢j(30)

- - - ,po, j = 1, - - - ,n — p0. Then, in the new coordinates, we have
5,- = 25+], i=1,---,Vo—1

2,0,,- = 2.0,,“ + LqLSW-l+i)h(xo)IzozTo-n(§), i = 0, - - - ,po — V0 — 1

dg §w+i+1 + 7m+i(5a t00(1))
2,, = 14° h(foo) + law-“(1‘0)” + LqLi°-lh($0)|xo=T,-‘I2)

“'é’ 5(2) + 91(2):: + ms, wom)

Emu = L1¢j($o)+ Lq¢j(30)lxo=To“(§)i j = 1,...” — P0
“‘é’ <I>.-(z.wo(t))

9:21

70

Note that 71, - - - ,7,,0_1 E 0 (follows from (2a)). Assume that
d(LqL‘fh) e {dh,---,d(L‘,h)}, i = V0 —1,-..,p0 — 1
then we have

g(.) = 5H1, i=0,"°,V0—1
_ . , _ - _ (i)
y — zVo-i-i-H ‘l’ ¢z(zla ° ' ' a zuo+nw0v ' ° ' ’wO )

i=0,~--,po—V0—1

g(Po) = hbpo—Vo(zrsila'°' 95003w09°"9w(()p0_uo)) +gl(2)u
3,000+!) = twang“...,gm,wo,...,w(po-Vo+i),u,...,u(£—1))
+91(5)"(i)

2:19H'an—p0-11 .7 :Po-Vo+'i, Er:(§Po+la"')zn

(3.60)

(3.61)

—)T

Observability Assumption: Suppose that equations (3.63) are explicitly solvable

for 2,. in terms of y,u,wo and their derivatives.

This assumption together with the fact that the states 21, - - . ,2” can be explicitly

solved for in terms of y, wo and their derivatives, implies that the system is observable

in the sense of [52]. From this we obtain

g(n) = f—(ya ° ° ' ) g(n-1)1 us ’ ° ' a u(n—po-l), wOa ' ' ' ,wgn-Vo))

(n-Po-I),

+§(y? " ay(n-1)aua' ° ° ’21

wo, . . . , wan-V0))u(n_90)

Let m = n — p0 and w(t) = (wg, - - ~ , (wan—W))T)T, to obtain the system (3.1).

71
3.7 A Special Case: Nonlinear Systems with
Polynomial Nonlinearity

In this section we give an example of a class of systems for which the transformation
(3.5) is explicitly known and also Assumption 3.2(b), which is fundamental to this

work, is satisfied. Consider the following system

y(n) = f(y,---,y("'1),w(t)) +2110“) (364)
.=0

where
o f is a polynomial nonlinearity in its arguments.

The polynomial lms’" + + lo with Im 775 0 is Hurwitz, i.e., all its zeros have

negative real parts.

0 0 = 00(y)u and oo(y) # 0 for all y E U1 C R

w(t) E Q C R”; Q is compact

It can be shown that the class of systems characterized by geometric conditions in
[41] is transformable into the input-output model (3.64). This is a minimum phase

system where the zero dynamics can be expressed, in suitable coordinates, as a linear

Bounded-Input-Bounded-State (BIBS) system. Define

$i+1 = y“)ai=07'”9n—l
C,“ = g(i),;=0,...,m_1 (3.65)

v = 90")

72

then the extended system will be given by

 

ii = $i+1,i=1,"',n—1
in = f($,w(t)) + 31C + lmv
C} = (.41, i=1,-~,m—1 > (3.66)
ém = v
y = 371 l
where Bl = (lo, l1, - , lm_1). The change of coordinates
2": {_anm-H, —l, .m

takes the system (3.66) into the normal form

x,- = x,-+1,i=1,~-,n—1 ‘
6:, = f(x,w(t)) + 31C + lmv
5'. = z,-+1,i=1,---,m—1 I (3.67)
2m = —(1/lm)Blz - f1(x,w(t))
y = 171 J

 

 

where f1(x,w(t)) = (l/lm)f(x,w(t))+(1/l?n)le and :7: = (xn-m+1, - - - , xn)T. Clearly,
the zero dynamics are BIBS stable. Let U = 1;},(‘f0 — 30C + 6) where (mo, f0 and Bo
are the nominal models of 1",, f and B, respectively. Then on the zero-error manifold
we get

[mllmofioa V) + (ImllmoBl "' BOX. — f0“), V) + 6 = 0

2i = ZH-li z=l,---,m—1

2.3m -‘-— ~(1/lm)BIZ—fl(0,I/)

73

and C = z +(1/lm)17 where V = (r(n'm),~ - ,r("‘1))T.

Since 2 is a solution of a linear system that is driven by an input which has a finite
number of modes, it will be a linear combination of those modes. Hence, C will also be
a linear combination of the same modes. Thus, we conclude that Assumption 3.2(b)
is satisfied. Assumption 3.2(a), on the other hand, will not hold automatically for

this class of systems and its satisfaction will be problem dependent.

3.8 A Design Example
Consider the second order system
yz—y+ay3—y+u—w+bu=f+bu (3.68)

where a 6 [—0.1,0.1], b 6 [8,12] and w is a constant disturbance input such that
w 6 [—1,1]. The nominal values of a0, b0 are 0,10 respectively. Notice that in (3.68)

f is a polynomial function of y, y, u and w. The extended system is given by

i‘l = 1'2
532 = -$1 + 03:13- x2 + x3 - w + bv (3.69)
(33 = ‘0

Let 6; = x1 -- r0 sin(t), e; = x2 — r0 cos(t) and z = x3 — 1’51 where r0 sin(t) is the
reference signal. Then, in the new coordinates, the system will be in the normal

form:

él = 62
('32 = —el + a(el + r0 sin(t))3 —(1—1/b)(€2 + r0 cos(t)) + z — w + bv

2 = —(1/b)z +(1/b)[e1 + To sin(t) — “(61 + 7‘0 sin(t))3

74

+(1—1/b)(e2 + r0 cos(t)) + w]

Let v = «p = g;l(—fo + (6). With yo = 10, f0 = 0 we have (,9 = 01¢. On the

zero-error manifold we have
argsin3(t)—(1—1/b)rocos(t)+ A(V) — w + 0.1b55 = 0 (3.70)

2" = —(1/b)z + (1/b)[ro sin(t) - a(ro sin(t))3 +(1—1/b)r0 cos(t) + w]

With the given disturbance and reference signals we have

 

 

 

 

0- 1 0 r0 sin(t)
So = —1 0 0 and V = r0 cos(t)
0 0 0 J L w
Let
/\(V) = alVll + 021/21 + 031/121 + 041/31 + (Isl/111121 + V31 + 061’131
+0711; + (181/121 V21 + 091/3111“ (3.71)

then we use Assumption 3.2 and equate coefficients of equal powers of the components
of V to obtain a3 = a4 = a5 = 0. Since the exact knowledge of the matrix I‘, which
will be a function of the parameters a and b, is not needed in the controller design we

now substitute for /\(V) in (3.70) its expression in (3.71) to obtain the vector V, i.e.,

r .
sin(t)

3 . 3 cos(t)
ar0 sm (t) — (l — 1/b)ro cos(t) + A(V) — w = F
sin(3t)

 

 

cos(3t)

75

where
F“ = 0.75ar3 + alro + 0.75a6r3 + 0.25a9rg,
F12 = 0.9r0 + (121‘0 + 0.75a7r3 + 0.25a3r3,
F13 = —0.25ar3 -— 0.25a6r3 + 0.25a9r3,
and

F14 = 0.25071‘3 — 0.256187%

where b = b0 = 10 is used. From this the matrix S will be given by

 

 

T
we take J = [ 1 0 0 1 ] . Notice that (S, J) is controllable.

Choosing the gain matrix K as

K=]—3.4959 —2.789 4.4719 0.0462 —8.8743 —5.2677

we found that s is given by

s=[0.4117 0.201 —0.4498 0.2443 0.8104 0.3437]£

and the function p is given by

News) = 0-1II£II3 + 0.3IIEII2 + (109644 + 0.3r3)II£II + Ixal + 1 + ro + 0.11%

76

Notice the dependence of p on the extended state x3. Setting the initial states of the

servo—compensator to zero and selecting compact sets of initial conditions for x1, x2

and x3 as |x1(0)| S 0.5, |x2(0)| S 0.5 and |x3(0)| S 0.5, we found that C(O) belongs

to the set C(0)TPC(0) S c with c = 1.5529. Now, using the state feedback control

—0.1p-:—, ifpls] > p
90(67 33) = I I

-0-1p2i, if plSl S u
we found that it will saturate at :l:15. Hence the output feedback control will be

given by
. _ M
(b — 15 sat ( 15 )

where

€=[01 02 03 04 é1 éz]

and§=[0.4117 0.201 —0.4498 0.2443 0.8104 0.3437]&

With a] = 3, a; = 2 and after scaling the observer states such that q] = él and

qz = eég, we obtain the observer model:

all = qi+3(€1-€h)

642 = 2(61—91)

With p = 0.1 and e = 0.01, the simulations were performed for a = 0.0, b = 10 and
r0 = 0.1. The tracking error and the control are shown in Figure 3.1 which clearly
shows asymptotic tracking. The performance of the observer is shown in Figure 3.2.
Notice the peaking in 62. In Figure 3.3 the control is shown to saturate at —15 during
the same period in which 62 peaks. To study the effect of pushing 6 small, simulations
were carried out for two values of e and the behavior of both the control d)’ and E2

was recorded. This is shown in Figure 3.4. Observe that for e = 0.001 the period

77

 

 

 

 

 

 

 

 

 

 

 

40 . ................................................................................................ 1
.1 L 1 1 1
fl) 5 10 15 20 25
Hmeu)

 

 

 

 

 

 

 

 

 

 

 

0 OJ 02 03 0A 05 05 O] 08 Q9 1
fimoh)

Figure 3.2. Observer performance; top: e1 (solid) and its estimate; bottom: e2 (solid)
and its estimate

78

 

 

 

 

 

 

 

E
‘E
O
o . - ............................ ..
.10.. ................................................................... ............................ ..
.15 i 111' 111,1

0 0.1 02 03 0.4 05 0.5 0.7 0.3 0.9 1

Tlme(8)

Figure 3.3. Saturation of the control 03‘

over which the control saturates decreases while the peaking in 8:, over the same
period, increases. To investigate the robustness of the control, simulations-were also
performed for two sets of parameters, i.e., a = 0.1, b = 12 and a = —0.1, b = 8. The
tracking error for these two sets is shown in Figures 3.5 and 3.6 respectively, which

clearly indicates that asymptotic tracking is achieved.

79

 

 

 

 

 

 

 

Estimate 0!. 2

 

 

 

 

"""0 0.02 0.04 0.06 0.00 0.1
Time (s)

Figure 3.4. Control ¢' and 62 for e = 0.01 (solid) and e = 0.001

fl
.0 l j T I

 

p
.5

Tracking Error
0
to

 

 

 

 

 

0
‘on 1 l l l
‘0 5 10 15 20 25

“(00(5)

Figure 3.5. Tracking error when a = 0.1 and b = 12

80

 

 

 

 

 

 

0.6 l T I I
§ 0A.. ................................................................................................ .,
Ill
3 02 ................................................................................................ ,
t
E
l- 0 v_' -

.on 1 L l i

‘0 5 10 15 20 25

“me (5)

Figure 3.6. Tracking error when a = -0.1 and b = 8

3.9 Conclusions

In this paper we have extended the nonlinear servomechanism theory to include sys-
tems that are represented by input-output models. We allowed a time-varying exoge-
nous signals that are not necessarily small. This aspect of the exogenous signals is
considered an extension over our work in Chapter 2.

In this work instead of designing a specific state feedback control, we presented a
characterization of a class of stabilizing controllers. The introduction of m integra-
tors at the input, made the state C available for feedback. With this we avoided the
more restrictive partial state feedback approach. Implementing the controller, using
a linear high-gain observer, we showed that for a class of locally Lipschitz, globally
bounded state feedback control, we can recover the asymptotic properties achieved
under state feedback. With globally bounded control, only semi—global results are
possible which, from a practical point of view, is not a serious limitation. The design
procedure was illustrated on an example that we worked out all through including
simulations. the results of those simulations are in good agreement with the predicted

behavior of the system.

CHAPTER 4

Output Regulation of a
Field-Controlled DC Motor

4.1 Introduction and Problem Definition

As an illustration, we will study the nonlinear model of a field-controlled DC motor
using two design methods. In the first method we will follow the procedure of Chap-
ter 2 where we augment an integrator driven by the tracking error, then we design
a controller to stabilize the augmented system. In the second method we follow the
procedure of chapter 3 where we first extend the original system by adding one in-
tegrator at the input, then we identify the internal model for the type of reference
and disturbance signals present in the system. For this case study we will consider
uncertain constant parameters. Hence the internal model is simply an integrator.
However, we will show that the presence of an integrator at the input will exclude
the necessity of adding an extra integrator. Then we design a stabilizing controller.

Consider the nonlinear model

i] —0111 — 62113211 + 03

i2 = —04x2+05x1u

81

82

y = 932
where 91 = %, 92 = 51:1, 03 = 113, 0.. = g, 95 = 5}, x1 is the armature current (A),

$2 is the angular velocity (rad/s), V is a fixed voltage (V) applied to the armature
circuit and u is the field winding voltage (V) and represents the control variable.
The constants R, L and K1 are, respectively, the resistance, the inductance and the
torque constant of the armature circuit, while the parameters J and B are the load’s
moment of inertia and the viscous damping coefficient. This system was studied
in [53] using measurement of the state x, while here we assume that we have only
measurement of the speed x2. Our desire is to get the output of the motor, in this
case the speed, to follow a given constant reference in the presence of uncertain
parameters. The following nominal values of the parameters, taken from [53], will be
used in the simulations: R = 79, L = 0.12H, V = 5(V), B = 6.04 x 10’6N — m —
s/rad, J = 1.06 x 10‘6N — m — s2/rad and K1 = .0141 N — m/A. To investigate
the robustness of the controllers, the following changes in the parameters are allowed:
001i0.15901, 002:1:0.9002, 0032!:0.2003, 904:1:0.15004, 905i0.2005, where 00;, i = 1, - - - , 5,

denote the nominal values.

4.2 Method 1

Let V be the desired constant Operating velocity. Let (Ia, VF) be the corresponding

equilibrium values for ($1, u). Then we can find Vp from the equation
xz—bx+1=0 (4.1)

where x = Kle/VBR, b = V/(VVBR) and 1,, is given by

V/R
L, =
1+3:2

 

83

Thus Assumption 2.1 is satisfied. Also, the above system has a relative degree one
provided x1 94 0. Thus, we will impose the constraint: x1 2 1m,“ and define the set.
U9 = {x : x1 _>_ Inn-n} to be our domain of interest. For (4.1) to have real roots, we
must restrict V to the admissible range V E [0, 57%]122 1‘. Choosing x2 < 1 implies
that In > V/2R. Hence we choose Inn-n = V/2R such that only one solution of (4.1)
will be associated with the domain of validity U9. With this we ensure the uniqueness

of the equilibrium point in U9. The change of variables \Il(x, V) is given by

e = :13; —V
K1 K1 K1 K1
2 = 7x? +— Lx x2— J —I2— L —V2

This change of variables will shift the desired equilibrium point to the origin. Its
choice is motivated by our desire to transform the system into the normal form.
Before proceeding with the design, we augment an integrator driven by the tracking
error with the plant to obtain the matrices A and B and then investigate the validity

of our assumptions.

 

 

 

 

          

('7 = e
B K1

. = __ _ _ __ _ 2
e J(€+V)+— :fll/KSILI ”)1’10-1-{El-+JI+ V211]
z — 2 Kz+2(JL —KL(JB)(:)V-+) L( éIa+LV)

K1 2 V2

+ J I +—— ]

“é’ ¢o(e.z,d)K
The constraint x1 2 1mm implies that
> — .
z__ L(e+V)— JI— LV + JImm

84

 

 

 

 

 

Figure 4.1. The domain Nd in the (e, z)-coordinates

From this we can define our domain of interest N; that will be valid for all admissible
values of the parameters. This is shown in Figure 4.1. It can be verified that choosing
In > Imgn guarantees that N; will contain the origin for all values of the parameters
and the reference. From Figure 4.1 we identify the sets: 8 = R x (—442.8,42.8),
Ll = (—4237, 00), 80 = R x (—42.8, 42.8) and 110 = (—4237, 4237).

The function —¢o(0, 2, d), for the nominal system, is shown in Figure 4.2. It is a first-
quadrant third-quadrant nonlinearity over a certain domain and it will retain this
property for all possible values of the parameters. Using W(z) = - f; ¢o(0,y,d)dy
as a Lyapunov function candidate for the system 2" = ¢o(0, 2, d) it can be shown that
the origin is asymptotically stable with a region of attraction, that is valid for all
allowable values of the parameters, given by {z : z _>_ —4237}. Also when 6 74 0, it

can be shown, after some algebraic manipulations, that

. 1 2
W s —§¢3(o,z,d), v Izl 2 —k°70(le|)
m0

85

V 2 6 M» where 7006!) = .2 + quIIeI, k0 = 21B.“ — —2—| + x-af—I and m. =

I"

V—
2—&2’iL—. Hence Assumption 2.4 is satisfied. Furthermore, a linearization
I.+ I3-O.26544

around the origin will reveal that this equilibrium point is exponentially stable, this

implies that Assumption 2.6 is also satisfied. We proceed now with the design. The

 

 

 

 

 

Figure 4.2. —¢o(0, z); (nominal)

matrix K is chosen as K = [ -31623 -004 1, we then obtain the matrix P

1.2662 0.1581
0.1581 0.1155

and s = [ 03162 0,231 ] C. We take f0 = 0, yo = 2 x 104 and the function p is found
to be

p = (3.1623 + 0.15004)||€|| +1.1500,|u|

With C = (a, e). Setting the initial conditions of (a,x1,x2) = (0,0.5805, 300), we find
that C(O) = [ 0 100 ], and it belongs to DC, with c1 = 1.155 X 103, the set RC, is

86

given by {z : |z| S 4237}. With 11 = 0.5 , the control is given by

u: -96‘p(€).-:., ifp(€)|6|>0-5 (4.3)

-2961P2(€)8x if F(C) ISI 5 0.5
The simulations were performed for the nominal system with a desired reference
V = 200 rad/s starting from a steady state speed of 300 rad/s. Figure 4.3 shows the
tracking error and the control, while Figure 4.4 shows the states a and .731 over a one
second period. Figures 4.5 and 4.6 show the same quantities over an extended period
of time which clearly indicates that asymptotic regulation is achieved and all other

states converge to their respective calculated equilibria.

0 (MI!)

8

87

 

as

o8688

vac-0.... .............................................................................................

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

 

 

 

11m} (s)

 

0.1

 

 

 

Jm

 

0.5
Tlmo (s)

Figure 4.3. Tracking error and control (0 -— 1 s)

 

 

88

 

 

 

 

 

 

 

 

 

Figure 4.4. The states a and x1 (0 — 1 s)

89

 

 

 

0 (MB)

 

 

 

 

 

 

 

V I
3
......................................................................................... J
.................... (, ..(
...................................................................................................... d
3
DUI-ooloauoncv-auwnOoon...In.DIG-II-c-IoCI-anolouo-o-Ono-III... oooooooooooooooooooooooooooooooooooooooo fl
2 3
1 1
15 20 25

 

Tlmo (s)

Figure 4.5. Tracking error and control (0 — 20 s)

90

 

 

 

 

 

 

 

 

 

 

 

Figure 4.6. The states a and x1 (0 — 20 s)

91

To show the robustness of the controller, we recorded the tracking error for two
different sets of parameters, namely when the parameters take their highest and lowest

values; this is shown in Figures 4.7 and 4.8 respectively.

 

‘00 I 1 T T

8

 

 

e (rad/s)
o

 

 

 

T1010 (5)

Figure 4.7. The tracking error when the parameters take their highest values

 

 

 

 

 

Tune (5)

Figure 4.8. The tracking error when the parameters take their lowest values

92

4.3 Method 2

In this section we illustrate the design procedure of Chapter 3. The input-output

model, with the speed as an output, can be obtained from the given state model as

y = —(91 + 94)!) — 919431 + 9395‘“

' + 0 .
420531.12 + ”—2191.

Let x1 = y, x2 = 3), x3 = u and v = u, then the extended system is given by

231 = $2
552 = —(91 + 94012 — 910411 + 939513 (4 4)
—9205$1$§ + WI)

233:?)

 

The system (4.4) has a uniform relative degree of 2 on the domain

U={x:x37$0, x2+94x1¢0}

This will be taken to be our region of interest. The following change of variables will
transform (4.4) into the normal form and shift the equilibrium point of interest to

the origin.

81 = 1'1 — V
62 2 1'2 (4.5)
2 = _23_ __ _fa.

93

where V is a constant reference speed and 5133 is the value of x3 at equilibrium which

can be found from (4.4)

4RJV2
V2

V

1:1: 1—
2K1V

 

 

(4.6)

T3:

 

 

It can be verified that the root with the negative sign is the one associated with the
minimum phase zero dynamics [53]; hence, we use it in (4.5). In the new coordinates

the system (4.4) will be given by

él = 82
éz = —(91 + 94062 — 9194(61 + V) + 9305[(2 + 5.23;)(62 + 94(81.+ V))]
4
_ _ 2 2 0w
0295(61 + V)(Z + 94”) (82 + 94(61 '1' V)) + 941,2 + £31)
. 573 5'3 3
z = 01(2 + —) + 0205(e1 + V)(z + —) (e2 + 04(e1 + V))
04V 04V
_ E 2
9395(2 + 04”)
0 1 0
We identify the matrices .A = and B = . We choose K =
0 0 1

[ _2.2361 _3,0777 ]. The function 3 is found to be 3 = 0,4472 0.4702 ] e, where
L

T

With a; = 3, 02 = 2 and after scaling the observer states such that q1 = e] and

 

q; = Eég, we obtain the observer model

all = (12+3(€1—<11)

642 = 2(61-Q1)

94

Using )1 = 0.5, go = 10“, p = 10, and the control

—123
v=-g p-
0 ll

then assuming |x1(0)| S 300, [1172(0)] S 0.5 and 0.024244 S x3(0) S 1.29721, we find
that the initial state e(0) belongs to DC with c = 1.2162 x 104 and that the control

will saturate at :l:10. Hence we end up with the controller

3 A _ —‘U(S)
¢(3)—10sat( 10)

 

where 5 = [ 0.4472 0.4702 ] «‘2 and é = [ a, a. ]T.
Setting 6 = 0.01, simulations were performed for the nominal system, using the same
data we used earlier, with a desired speed of 200 rad/s and starting at the initial
condition (x1, x2, x3, él, £2) = (300, 0.5, 1.2, 0, 0) which corresponds to a steady
state speed of 300 rad/s. The tracking error and the control are shown in Figure 4.9
where, clearly, asymptotic regulation is achieved and the control converges to zero
which is the equilibrium value of v in (4.4). Also, the states x2 and x3 are shown, in
Figure 4.10, to converge to their respective equilibria. Recall that x3 corresponds to
the original control variable, i.e., the field voltage and it is shown here to converge to
the same value that the control 11 converged to in Method 1.

To study the performance of the observer, we simulated the nominal system over
a tenth second period. The results are shown in Figures 4.11 and 4.12. Also the
control 05’ is shown in Figure 4.13 over the same period of time. Notice that there is a
one to one correspondence with'respect to the period of time over which the peaking
in ég occurs and the period during which the control saturates.

To investigate the effect of making 6 small on is; and the control 45’, we simulated

the nominal system for two different values of 6; namely, 6 = 0.01 and e = 0.005.

95

CAR

 

1”

 

 

u.
988888

 

 

 

 

 

   

Figure 4.9. Tracking error and control of the nominal system

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M T’ V V M V
8 I ' l
. ......... , .......... . .......... 'I ......... . .......... ', ......... , ...............
. .
. ;
= x
......... ......._.
: :
$
1'
t 8
Q .
......... .L........... ........._.,... ...........'.. ...,. .. ........ .........q
.......... x..........q....................4..........’..........i.......... ..-.o-.-~
. .
1 i
.......... ‘.,,, ...........
. .
2 l
A J A A A A
3 O I 7 C II ‘80
70000.1.)
V V V f T_ ‘r T
. ; E
. .
................... 8..........o.......o..$.........J~........-|.........(a.-......-|-.-....o—r
. .
s ’- 3 X 8
. 2
.......... r.........’..........:................ ..., .. .. ..é...................‘
I i
3
............. .... ... ......... ........ 1 ....... d
.
I t l
.
............................................... 1 .... ...f..........5........_.(
8 8 8
8
......................................................................... §----.--.q
1
.
' t
l A A A A I
a II I 7 I 0 1o
Tim-(o)

Figure 4.10. The states x2 and x3

96

 

 

 

i a

' I
60-) ...,

i s

l E
wpi....................................................-.........................................: ........ .q

" i

l :
m7" ............................................................................................. S ........ .4

. 5

j_

 

 

00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.“ 0.1
Time (s)

Figure 4.11. 61 (solid) and its estimate él

 

 

 

 

 

 

_‘m A 1 l I A 1
0 0.01 0.02 0.!!! 0.04 0.05 0.06 0.07 0.00 0.09 0.1
Time (0)

Figure 4.12. e2 (solid) and its estimate é;

 

0 1 v 1 T 1 1 r T 1

-2. ................................................................................................... .,

_4, ................................................................................................... _,
—

 

 

 

 

-8 . ............................................................
_a p .................................................................................................... ..
-10 I i J i L i l l i

0 0.01 0.02 0.03 0.04 0.05 013 0.07 0.00 0.00 0.1

Time (s)

Figure 4.13. Saturation of the control 43‘

97

The results are shown in Figure 4.14 where the solid line represents the case when

6 = 0.01. Observe that the period over which saturation (peaking) occurs is smaller

for smaller 6.

 

 

 

 

 

 

 

 

-1O

 

 

 

‘ MMh 1
.15 0.02 0.025 0.00 0.035 0.04 0.045 0.35
11mm

Eulmdo d o_2

 

Figure 4.14. The control 03’ and é; for e = .01 (solid) and e = .005

To investigate the robustness of the controller, we simulated the system for two
different sets of parameters; namely, when the parameters assume , during the course
of operation, their highest and lowest values. The tracking error for both cases is

shown in Figures 4.15 and 4.16 respectively which clearly indicates that asymptotic

regulation is achieved.

98

 

Time (s)

Figure 4.15. The tracking error when the parameters take their upper limits

    

5
Time (s)

Figure 4.16. The tracking error when the parameters take their lower limits

99

4.4 Conclusions

In this case study, we demonstrated the application of the tools presented in designing
robust nonlinear controllers to solve a physical problem. The control strategies used
proved to be effective in regulating the speed of the field-controlled DC motor in the
presence of uncertain parameters. All parameters of the motor were allowed to vary

and the simulation results are quite encouraging.

CHAPTER 5

Conclusions and Future Work

5.1 Conclusions

In the work presented here we provided a solution to one of the fundamental prob-
lems in control theory for a class of nonlinear systems. The introduction of integral
control in Chapter 2 to achieve robust asymptotic regulation for a class of feedback
linearizable systems regionally as well as semi-globally is the major contribution of
this chapter. The tools used to establish the results are standard Lyapunov techniques
familiar to most control engineers. Our solution of the corresponding stabilization
problem is shown to relax many of the restrictions imposed on the system under study
as it is documented in some of the available literature. The integral control is shown
to work, although locally, for a class of time varying signals.

In Chapter 3 we laid the foundation for the development of internal models for nonlin-
ear systems. In this work we identified a subclass of input-output linearizable systems
for which the internal model is known and linear. In conjunction with that we pre-
sented a class of state feedback stabilizing controllers that can be used to stabilize
the zero-error manifold.

The case study treated in Chapter 4 indicates that the control schemes devised in

this work can be successfully implemented to solve practical engineering problems.

100

101

5.2 Future Work

In this section we go over some of the topics that need further investigation and could

be pursued in the future.

5.2.1 Multi-Input Multi-Output Systems

In our work here we dealt with 8180 nonlinear systems only. Extending the results of
Chapters 2 and 3 to the more general MIMO nonlinear systems could be done after
carefully analyzing the relevant assumptions to determine the necessary modifications
needed in order to handle the MIMO case. We conjecture that if Assumptions 2.1-2.3
hold for a MIMO system, Inequality (2.23) holds as a norm inequality and Inequality

(2.24) is modified to account for the MIMO nature of the system, i.e.,
[IGl(e,z,d)G;1(e,V) — I” S 101 <1

where G1 is an m X m input matrix, then the analysis leading to the boundedness of
the state C can be repeated with little modifications. We may assume that the MIMO
system is square, i.e., the number of inputs equal to the number of outputs=m. Again,
using V = (l/2)sTs, we can show attractivity of the boundary layer. Note that As-
sumption 2.4 is independent of the nature of the input or output. To prove asymptotic
stability we need an assumption similar to Assumption 2.6 with the necessary modi-
fication on Inequality (2.40), i.e., replacing f1, gl by F1, G1 respectively, where F1 is
an m x 1 vector. The system (2.6) was given as a special case of the class of systems
(2.1) that can be transformed uniformly in the parameters into a normal form via
transforming the system into a strict feedback form first. Developing a MIMO version
of (2.7) is yet to be done. Although partial results are available, the issue deserves

further investigation to establish necessary and sufficient geometric conditions that

102

will characterize a class of systems. We think that recovering the asymptotic proper-
ties using a high-gain observer can be achieved with little or no difficulty.

In Chapter 3, we conjecture that all relevant assumptions can be modified after a.
careful study. It is our intention to pursue this issue along with the derivation of ge-

ometric conditions that will characterize the class of systems that are transformable

into a MIMO strict feedback form.

5.2.2 Unmatched Uncertainties

Due to the feedback linearizability property of the system, the uncertain terms will
end up satisfying the matching condition. The tracking and disturbance rejection
problems for systems with unmatched uncertainties need further study. We expect
our results to hold for sufficiently small uncertainties. The issue then is to find bounds

for which the results hold.

5.2.3 Internal Model

In linear servomechanism theory, the internal model is given and this provides a com-
plete solution to the asymptotic tracking and disturbance rejection problem. Because
this will specify the servo-compensator, then stabilization follows using any of the
techniques available for linear systems design. This is not the case for nonlinear sys-
tems due to the harmonics that might be generated by the nonlinearities present in
the system. Aside from the work presented in Chapter 3 which provides the inter-
nal model for a subclass of systems, identifying the internal model for any nonlinear

system remains a challenging and open problem.

 

APPENDICES

APPENDIX A

Proof of Inequality 2.47

Need to show that there exists a constant c > 0 such that
(a + b)’1 S 00!“1 + b"), 7 21
where a, b are positive constants. Hélder’s Inequality states that

IlfgIIISIIfllpllgllq, + =1

‘UIH
ale-0

Set f = (a,b) and g = (1,1)T, then

a-1+b-15(a”+bP)I/P-(1"+1")‘/°

i.e.,

a + b s 2”“(a’ + (1’)"?

(a + b)" S 2""1(a'y + b”)

103

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