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OUTPUT FEEDBACK STABILIZATION OF
FULLY-LINEARIZABLE SYSTEMS

By

Farzad Esfandiari

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

Summer 1990

(0453- ”Ibex

ABSTRACT

OUTPUT FEEDBACK STABILIZATION OF
FULLY-LINEARIZABLE SYSTEMS

By

Farzad Esfandiari

In this work, we study the problem of output feedback control of nonlinear
systems which are fully-linearizable via static state feedback, left-invertible, and
minimum-phase. The output feedback controller proposed is an observer-based
control, whose state feedback component consists of two parts: An inner loop to
cancel the nonlinearities (either exactly or approximately), and an outer loop which is

a robust stabilizing control law such as variable structure control, or min-max control.

To implement such state feedback controllers using an observer-based control,
the observer should be designed to reject disturbances caused by model uncertainties,
as well as by estimation error. Observer designs with such a disturbance rejection
property are high-gain observers, where certain observer gains are pushed
asymptotically towards infinity to locate some observer poles far to the left in the
complex plane. When observer poles are assigned in this way, the trajectory of the
closed-loop system exhibits an impulsive-like behavior, which is known as the
peaking phenomenon. The peaking phenomenon which is generally present in

systems of relative degree higher than one, has a destabilizing effect on the behavior

of the closed-loop system.

In this work, we design such high-gain observers using a singular perturbation
approach. In this approach peaking exhibits itself through certain scalings which are
dependent on the singular perturbation parameter. We prove a new singular
perturbation result on the behavior of the closed-loop system in the presence of such
scalings. Then, as a corollary of this reult, we show that presence of saturating
nonlinearities at the plant input eliminates the destabilizing effect of peaking, since it
provides a buffer that prevents the impulsive-like behavior of the observer from

passing to the plant.

To my mother, Parvin Rastegar

and to the memory of my father, Mohammad Mehdi Esfandiari

iv

 

Acknowledgements

I am indebted to Professor Hassan K. Khalil for his guidance, support, and

patience during the course of this work.

Table of Contents

List of Figures ........................................................................................................... vii
Nomenclature ............................................................................................................ viii
1 Introduction ........................................................................................................... 1
2 Full Linearization .................................................................................................. 4
3 State Feedback Control ........................................................................................ 15
3.1 Introduction .................................................................................................. 15
3.2 Continuous Implementation of Variable Structure Control ....................... 22
3.3 Variable Structure Control for Fully-Linearizable Systems ...................... 34
4 Observer-Based Control ....................................................................................... 40
4.1 Introduction .................................................................................................. 40
4.2 A Special Coordinate Basis ........................................................................ 42
4.3 Observer Design .......................................................................................... 45
4.4 Closed-loop Stability Analysis .................................................................... 47
5 Peaking Phenomenon ............................................................................................ 55
6 Globally Bounded Control ................................................................................... 65
6.1 Introduction .................................................................................................. 65
6.2 A Singular Perturbation Result ................................................................... 65
6.3 Stability Result ............................................................................................ 77
7 Future Work .......................................................................................................... 86
Appendix A- Proof of Proposition 3.1 ..................................................................... 89
Appendix B- More on Special Coordinate Basis of Section 4.2 ............................ 91
List of References ..................................................................................................... 93

vi

List of Figures

Figure 5.1- Pendulum of Example 5.1

Figure 5.2- Simulation Results of Example 5.1

Figure 5.3- Phase portrait of state feedback control of Example 5.2

Figure 5.4- Time profiles of state variables for output feedback
control of Example 5.2, with e = 0.014

Figure 5.5- Time profiles of state variables for output feedback
control of Example 5.2, with e = 0.013

Figure 6.1- Example 5.1 with bounded control

Figure 6.2- Time profiles of the state varaibles for Example 5.2,

with bounded control in the boundary layer
Figure 6.3- Time profiles of the state varaibles for Example 5.2

with bounded control

vii

56
59
62

'63

83

84

85

[V]i

lell

llxlll

“15"..

)‘maxug )

)‘min(P )

Nomenclature

the i-th component of vector v

the 2-norm of x

the 1-norm of x

the infinite norm of x

the maximum eigenvalue of matrix P

the minimum eigenvalue of matrix P

u1(t)=u2(t) forall 1

viii

1 Introduction

In the last two decades exact linearization of nonlinear systems via feedback have
received considerable attention in nonlinear control community [Isidori (1989)].
Roughly speaking, feedback-linearizable systems are those classes of nonlinear sys-
tems which can be made to behave linearly under the effect of an appropriate state
feedback and a possible change of coordinates. The problem of state feedback con-
trol of feedback-linearizable systems have been extensively studied in the literature
(refer to Isidori’s book (1989) for a survey. For more recent results, refer to the
work of Sussmann-Kokotovic (1989) ). However, few results are available on the
problem of output feedback control of linearizable systems. we briefly go over the

available results:

In the work of Marino (1985) a high-gain static output feedback control is proposed
for stabilization of single-input single-output linearizable systems which have a rela-
tive degree of one. Assuming that the relative degree of the system is one, excludes
most of the physical problems of interest. For instance, the equation of a robotic arm
has a relative degree of two (Refer to Example 2.1 in Chapter 2). The case of sys-
tems whose relative degree is higher than one has been studied by Khalil-Saberi
(1987), Isidori-Byrnes (1990) and Isidori (1989), where lead-lag compensators are
proposed for local stabilization of linearizable systems. In this work, we address the

problem of nonlocal output feedback stabilization of a class of linearizable systems,
namely systems which are fully-linearizable via static state feedback, left-invertible,
and minimum-phase. In chapter 2, we study this class of nonlinear systems in detail.
Since most of the physical problems to which techniques of exact linearization have
been applied are fully-linearizable systems for a meaning choice of output variables
(Refer to Chapter 2), we focus on the class of fully-linearizable systems, rather than

the more general class of input-output linearizable systems.

The proposed output feedback controller is an observer-based control, whose
state feedback component consists of two parts: An inner loop to cancel the non-
linearities (either exactly or approximately), and an outer loop which is a robust sta-
bilizing control law such as variable structure control, or min-max control. In
Chapter 3, we study the problem of designing such a state feedback control. Then, in
chapter 4, we study the problem of observer design. To ensure that the observer-
based control preserves the stability properties of the state feedback control, the
observer should be designed to reject the effects of model uncertainties and estima-
tion errors. Observer designs with such a disturbance rejection property are high-gain
observers, where certain observer gains are pushed asymptotically towards infinity to
locate some observer poles far to the left in the complex plane. In section 4.3, we
will design such an observer by transforming the system into a canonical form that
exhibits the finite and infinite zero structure of the linearized system. Then, in sec-
tion 4.4, we perform the closed-loop stability analysis using singular perturbation
theory.

The closed-loop stability results of section 4.4 are local results in most cases.
In chapter 5, we will argue that the locality of the stability results is due to what is
known in linear system dreary as peaking phenomenon. When some of the observer
poles are located far to the left in the complex plane, the trajectory of the closed-

loop system exhibits an impulsive-like behavior, which is known as the peaking
phenomenon. Singular perturbation theory provides an elegant framework for the
analysis of peaking phenomenon, since peaking exhibits itself through certain scal-
ings which are dependent on the infinite zero structure of the system and the singu-
lar perturbation parameter. In Chapter 6, we prove a singular perturbation result on
the behavior of the closed-loop system in the presence of such scalings. As a corol-
lary of this result, we show that the presence of saturating nonlinearities at the plant
input eliminates the peaking phenomenon, and hence the local nature of the stability
results which is caused by peaking.

2 Full Linearization

Consider the nonlinear syStem

é=f<§> +g<§> u +g<§> [13f(§.t) + liner) u + Amen]
y=mo 0“

where fie RP , u e R", and y e R’ are state, input and output vectors, respec-
tively. 2t,(.,.), and 25“...) represent parametric uncertainties in f(.) and g(.),
respectively, while 3,, (.,.) represents exogenous disturbances. Note that all the
uncertainties and disturbances satisfy the matching condition, i.e., they enter the
right-hand side of the state equation at the same point as the input. In this chapter,
we define the class of nonlinear systems which is under study in this work. Con-
sider the following nominal model for (2.1), obtained by setting 3,, Ag, and 8,,
to zero:
{EZQEEW‘W
Definition 2.1: [Cheng, et. al., (1988)] System (2.2) is said to be fidly-
linearizable, if there exist an open connected set ‘1‘ c R" containing the origin, a
diffeomorphic transformation T : ‘I’ —9 RP , smooth mappings a : ‘P -> R4 ,
4

B : ‘I’ -+ R4 xR‘l, with B(§) invertible for all fie ‘P, such that the state feedback
control u = mg) + B(§)v and the change of coordinates z = T (é) transform sys-

tem (2.2) into a controllable linear system:

i = A2 + Bv
{

y=Cz

Local necessary and sufficient conditions for full linearization are given in the work

of Cheng-et. al. (1988).

Definition 2.2: [Hirschom (1979)] Let y(t,§o,u (t )) be the output of system
(2.2) for the initial condition 2,0 and the input u (t ). System (2.2) is said to be
left-invertible on ‘P c RP , if for all £0 e ‘I’

y(t,§o,ul(t) ) = y(t,§o,u2(t)) for all :20 : u1(t) = u2(t) for all :20

Definition 2.3: [Isidori-Moog (1986)] Suppose there exists a set ‘I’ c RP con-
taining the origin and a smooth submanifold N o of ‘1' containing the origin with
the following properties:

i) No C Ker h(E,)

ii) There exists a state feedback control u = Ki), defined on ‘I’, such that
f' (t) := f (a) + s (but) is tangent to No

iii) N 0 is maximal, i.e., any submanifold of ‘I’ which contains the origin and
satisfies conditions (i) and (ii) is contained in N 0.

Then the vector field of N0 defined as the restriction of f‘ to N0 is said to be a

local zero dynamics of system (2.2).
Definition 2.4: [Isidori-Moog (1986)] System (2.2) is said to be minimum-

phase, if the vector field of N 0 of Definition 2.3 is asymptotically stable.

Proposition 2.1: Suppose that system (2.2) is fully-linearizable. Then system
(2.2) is left-invertible and minimum-phase over the domain ‘I’ if and only if the

linear system (2.3) is left-invertible, mrmmum-phase, and detectable.
Proof of Proposition 2.1:

Sufficiency: The coordinate transformation 2 = T(§) transforms system (2.2) into

{2' =Az +B B'1(z) [u -a(z)] (24)

y = C2
where

a2=ao T-1 , 52:80 T-1
So, without loss of generality, we prove that system (2.4) is left-invertible and
minimum-phase, if system (2.3) is left-invertible, minimum-phase, and detectable.
Let 2 (t , 20, u (t) ) denote the solution of the state equation of (2.4) for the initial
condition vector 20, and the input function u(t). Similarly, let x(t, zo, v(t))
denote the solution of the state equation of (2.3) for the initial condition vector 2 0,
and the input vector v(t ).
i) System (2.4) is left-invertible on RP .
Proof of ( i ): Suppose that (2.4) is not left-invertible on RP . Then there exist an ini-
tial condition vector 20 e R" , input functions u 1(t) and u2(t ), such that

u, éuz, but
C z(t,zo,u1)EC z(t,zo, 14;) (2.5)
Define

21(t) := z(t, 20, ul)

22(1) i=2“, 20, “2)

vim := B‘ltzr) [arm — can]
v2(t) := [3‘1(22) [u2(t) — a(27)]

11(I)I=X(t, 20, V1)

Iz“) I=X(t, 20, V2)
It is easy to see that

zlsxl , and 225x;

Therefore,
C xlaC 21
Esz
Esz

Since (2.3) is left-invertible, the last equality implies that

Vl 5 V2
which in turn implies that
Jrl'='1‘2

Therefore, by (2.6)

21522

(2.6)

by (2.5)

(2.7)

(2.8)

(2.9)

Going back to the definition of VI, and v2, it is easy to see that (2.7), (2.9),

together with invertibility of B(.) imply that

“1'5“2

which is a contradiction. This concludes the proof of (i).

ii) System (2.4) is minimum-phase.

Proof of (ii): To prove (ii), we transform system (2.3) into the special coordinate
basis of Saberi-Sannuti (1987) (For more information on this transformation, refer to
section 4.2). It has been proved by Saberi-Sannuti (1987) that, due to left-
invertibility of (2.3), there exist nonsingular transformations 1‘ , I‘M , I‘in , integers

K, qa, qb, q,-, r; i=1,...,K such that the transformation

 

 

L"
1,, if

z=r 2,,y=r,,,,, y ,v=r,.,,v(.) (2.10)
~ 8
2
f

u'ansforms system (2.3) into the following form:

f .
in = Add Ea +Aaf7f “FA“S’S

f 2:1) = Abbib +Abfyf (2.11)

 

if =Af2f +Mf5’f +Bf [Da§a+Dbib+Dfif+i-’]

W=Q%
5': = C: 2b

 

L

K K
where the dimensions of 2", 2b, ‘2}, if,and 5", are q... 4b: 2%: 24,-,and
i=1 i=1
K o
r - Zqi, respectively. Moreover, invariant zeros of ( C, A , B ) are the eigen-
i=1

values of A“, (C,,Abb) is observable, and Af, 3,, Cf, Aw, and C, have

the following canonical structure,

A, := Block Diag (Alf, . . . ,Agf)
Bf := Block Diag (Blf, . . . ,BKf)
Cf := Block Diag (le, . . . ,CKf)
Au, := Block Diag (A1,,,...,Am)

C, := Block Diag (C1,, . . . ,CK,)

where Alf =Oqlqu, Blf=lqp le =Iql’ Albb =0’1W1’ Cur-1,1, if i=1,

while
A 010“?! II: B 01M: C I O ]
if — 0‘1:qu 04:“: ’ If — [qt ’ If _ [ q; thl;
00190: [Mt
Aibb = 0"”! Ohm , Ci: - [17; Ohm]
ll = (i-I)X‘Ii t mi = “-1)”?
for 1' >1.
Now,

9 =13} v
=1‘.:.‘B"(z) [u -a(z)]

=I‘.-;‘ [3“(17) [u -.a(f‘z')]

 

10

:= M2) [u — am]

Therefore, transformation (2.10) transforms the nonlinear system (2.4) into the fol-

lowing form:
2.0 = Add 2a +Aaf5’f +Aasys
( 2‘, =Abb2, ”1:17; (2.12)

if =Af§f +Mf5’f +3, [Da‘z'a+Db‘ib+DfEf+B—l(§) [u - a(§)] ]

 

b

W=Q@
5’: = C: 2b

N0 := span in
Due to the canonical structure of 11,, Cf, A», and C,,

N0 = Ker Cz

u(‘z') := a(i) - 3(2) D, 2, (2.13)

On N 0, the closed-loop system (2.12) and (2.13) is

 

iéa =Aaaia
1 2‘, = o (2.14)
if = 0

Therefore, the direction of the vector field on N o is tangent to it. Moreover, N o is

clearly maximal. Therefore, system (2.14) defines the zero dynamics of system (2.4).

 

11

Since system (2.3) is minimum-phase and detectable, the invariant zeros of
(C , A , B) ( which are the eigenvalues of A“ ) are in the Open left-half complex
plane. Therefore, system (2.14) is globally asymptotically stable, which implies that
system (2.4) is minimum-phase. This concludes the proof of (ii). The necessity

proof is very similar to the sufficiency proof, and hence is deleted.

Assumption 2.Gl: System (2.2), i.e., the nominal system, is fully-linearizable
via state feedback, left-invertible, and minimum-phase on S c R” , where S is an

open connected set containing the origin.

It may be argued that Assumption 2.61 is restrictive. However, techniques of
exact linearization have been applied to a number of interesting physical problems in
robotics, control of electric power system, and flight control (refer to references
given later in this section). Most of these problems satisfy Assumption 2.61 for a

meaningful choice of output variables.

Example 2.1: Motion of a robotic arm may be described by the following
dynamic equation [Brady, et.al. (1982)], '

él=§2

. 2.
§2=D-l(§1) [u -E(§1,§2)] ( 15)

where 5,15 R", and Q e R" are the angular position and speed of the joints,
respectively. u e R" denotes the driving torques of the joints, E( . , . )
represents coriolis, centrifugal, and gravitational forces, and D( . ) is the inertia
matrix. Assuming that all the states of the system are available for feedback, the
nonlinear terms in (2.15) can be canceled by

12

u = 3&1 £2) + D(§1)V (2J6)

Cancelling the nonlinear terms as is done by (2.16) is known in the robotics litera-
ture as the method of computed torque. To control the system without using meas-

urements of angular speeds of the joints, define the output vector

y = £1 (2.17)

Applying (2.16) to system (2.15) and (2.17) results in the linear system

§1=§2
$2 = v (2.18)
y = §1

Therefore, system (2.15) and (2.17) is fully-linearizable via state feedback. More-
over, system (2.18) is invertible and has no zero dynamics, which implies that the
nonlinear system (2.15) and (2.17) is invertible and has no zero dynamics (by Propo-

sition 2.1).

Example 2.2: As another example, consider the following model for nonlinear
excitation control of two interconnected synchronous generators, studied by Ilic-Mak

(1989).

 

sk-mk-mo

'-m° E" 21‘--+T 219
amt-fil’ allot-00(03): 030) arts] (-)
° 1 1 r .

341: = T,“ [‘qu —(Ldk‘Ldk')ldk+Efdk]

L

 

for Ic=l, 2, where 5,, is the rotor angle, to,‘ is the rotor speed, Eqk’ is a vol-

tage proportional to damper winding flux linkage, and the currents iq,‘ and id,‘ are

 

13

defined by nonlinear functions of 8,; and eq’. The control variables in (2.19) are
1332f,”c ’s, which are the field voltages. Refer to [Ilic-Mak (1989)] for the details. It is
shown in [llic-Mak (1989)] that the change of variables 23:5,, z,,,=8,,, and
23,560,: transforms (2.19) into the form

ilk = 2 2k

2,, = 2,, (2.20)

is]: = PHI) + 3142) “I:
where Bk (.)’s are invertible. Therefore, assuming that all the states of the system
are available for feedback, the nonlinear terms can be canceled by the control
up = B[1(z)[-pk(z)+vk ]. This requires measurement of the rotors’ angles,
speeds, and accelerations. To control the system without using acceleration measure-

ments, define the output vector

)’ = [211» 221’ 212, 2221'

The dynamics of the system can be represented by

y = Cx (2.21)

{i =Ax +B [p(x)+\v(x)u]
where x = [ 211, 221. 231. 212s 222. Z32 ]’ r A =diag [AI’AZ L

B=di08lBlr32L C=diag[C1,C2], P=[Plr92]'t v=diag[l31.flz].

and u = [ ul, uz 1', where the matrices Ah, 8,, and Ck are given by

 

010 0 100
000 1

The 4x2 transfer function C (s16 — A )-13 is block diagonal, with the diagonal

blocks {—153 -l7]'. This transfer function is left-invertible and has no transmission
3 s

zeros; hence it is minimum-phase.

14

Other physical examples which are left-invertible, minimum-phase and fully-
linearizable for a meaningful set of output variables include the helicopter model of
Meyer-Su-Hunt (1984) and the spacecraft with gas-jet actuator model of Dwyer
(1984). In the case of the helicopter model, the 12th order system is fully-
linearizable with measurement of four state variables, three of which define the posi-
tion and the forth is one of the three attitude angles ( r1, r2, r3, and (p3 in the
notation of Meyer-Su-Hunt (1984) ). In the case of the spacecraft model, the 6th
order model is fully-linearizable with measurement of three state variables which
determine the attitude of the body with respect to an inertial reference frame ( 71,

72, and 73 in the notation of Dwyer (1984) ).

3 State Feedback Control

3.1 Introduction

In this chapter we study the problem of state feedback stabilization of the class of
nonlinear systems defined in Chapter 2, i.e., system (2.1) under Assumption 2.01.
To motivate the discussion, let us see how the nominal system (2.3) can be stabil-
ized. System (2.2) is fully-linearizable, i.e., the transformation 2 = T(§) and the

control

u = alt) + B(§)v , (3.1)

transform (2.2) into the linear system (2.3). Since (A , B) is controllable, one can
find a gain matrix K such that A + BK is Hurwitz. Therefore, the following con-

trol

u = (at) + Bag) K Te) (32)

renders the origin of the nominal system (2.2) asymptotically stable.

To design a state feedback controller for system (2.1), let us transform (2.1)
into the z-coordinates via 2 = T(§). This transformation transforms (2.1) into

15

16

f

12 =A 2 +8 B'1(z) [u +Af(z,t)+Ag(z,t) u +Aed(z,t) — a(z)]

 

y=Cz am
where

(1(2) := 6K5.)

13(2) := Be)

Af(z,t) := 8,( g, t)
A,(z.t) := an t. r)

A¢d(2J) 1: and §. I)

Similar to the case of the nominal system, control law (3.1) can be applied in this
case to cancel the nonlinear terms a(.), and B(.). However, in practice, exact can-
cellation of the nonlinear terms are usually either undesirable due to their complex-
ity, or impossible due to parametric uncertainties. Therefore, instead of exact cancel-

lation, the following control may be used
a = 12(2) + 6(a) v (3.4)

where (i(.) and E(.) are nominal or simplified versions of a(.) and B(.). Apply-
ing (3.4) to system (3.3) results in

{i=Az+Bv+B8(z,v,t) (3.5)

y=Cz

where

8(2, v, t) := B'1(z) [(Aa-tAg (HA!) + (Agar: B) v + A“, ] (3.6)

l7

and

An := 5(2) -- (1(2)
As == 6(2) - 13(2)
The effect of the uncertainties and the simplification of a and [3 have appeared in
(3.5) as a disturbance term. Therefore, the control v should be designed to stabil-
ize (3.5) in the presence of 8(2, v, I). Since the disturbance term 8(2 , v, t)
satisfies the matching condition, such a stabilizing control can be designed under an
assumption on the growth of 8(2, v , t),
Assumption 3.62: The following inequalities are satisfied for all z e S ,

teR+

IB’1(Aa+Aga+Af)ISk1lzl
IB‘1(AB+AgB) I Skz, k2<0

'B-l Adl Sk3'2l +k4

where S c RP is an open connected set containing the origin, It; ’s are nonnega-
tive constants, and 0 is a constant that depends on the robust control technique

being used. Later in this chapter, we will say more about 6.

There are several methods in the literature for designing such a stabilizing state
feedback control. In particular, variable structure control [Utkin (1987)], min-max
control [Corless-Leitmann (1981)], or linear high-gain control [Barmish-Corless-
Leitrnann (1983)] can be used to stabilize system (3.5). Using such techniques, one
finds a state feedback control

v _= ¢(z). «0) = o (3.7)

together with a quadratic Lyapunov function

18

W(z) = z’Pz , P symmetric positive definite (3.8)

such that, under Assumption 3.62, the derivative of W along the trajectory of the
closed-loop system (3.5) and (3.7) satisfies the following inequality for all z e S

%S"Yzl2fl2+71“2'+70 (3'9)

where 72>0, 7120, and 7020.

Example 3.1 (Linear High-Gain Control): As an example of the kind of state
feedback control that we are interested in, we quickly go over the linear high-gain
control, introduced in the work of Barmish-Corless—Leitmann (1983). In this tech-
nique, one starts by choosing K such that A := A -BK is Hurwitz. Then the state

feedback control is chosen to be
1 ,
v=—Kz--EB Pz (3.10)

where Q > 0 is a constant to be chosen, and P is the symmetric positive definite
solution of the Lyapunov equation PA + A ’P = - I p. Consider the Lyapunov func-
tion candidate W (2) = z ’Pz . The derivative of W along the trajectory of the
closed-loop system (3.5) and (3.10) is

W =-IzI2--§-IB'P2 I2+2z’PB8(z,v,t) (3.11)
S-Iz l2-%(1-k2) IB’Pz I2

+2lB’le [(t,+t,+t,lxt)tzl +k4] (3.12)

by Assumption 3.62. It can be seen that to preserve the second term on the right-
hand side of (3.12) as a negative quadratic term [:2 should be strictly less than 1 (
0 of Assumption 3.62 is l in this technique). It can be shown fiom (3.12) that

19

° _ -__§__ 2 2
W S [l 2(1—k7)(k1+k3+k2|K ') ] I2 I

’63

k4
+C (k1+k3+k2IKl)lZI+C2(1_k2)

l-k2

 

 

(3.13)

where we have used the fact that

2

—ay2+byS-Z—a- fory 20,1; 20,and a>0

Choose 2; such that

l
1-—5-— + llrtl2 —
204‘?) (1‘1 k3+k2 ) 2 2

Then,
' 1 2 [‘4
w s-— lz I + r, —(k1+k3+k2|K I)lz I
2 l-k2

k}

+§ 2(1-k2)

 

(3.14)

which is inequality (3.9).

In inequality (3.9), if 71 and 72 are zero, then (3.9) implies that the origin of
the closed-loop system (3.5) and (3.7) is asymptotically stable. In general, when 71
and 72 are not zero, inequality (3.9) implies that the trajectory of the closed-loop
system converges to a neighborhood of the origin. This property is known as uni-

form ultimate boundedness, which is defined in the following way:

Definition 3.1:Consider
x = F(x, t) (3.15)

where x e R" , and let Num) denote the u—neighborhood of set 0 ,i.e.,

20

Nu(fl):={z e R" linfyealz—y I <11}

System (3.15) is said to be uniformly ultimately bounded (U.U.B.) with respect to
the set (2 c R" with E c R" inside the region of attraction, if for every x0 6 Z
and u>0 there exists T 20 such that the solution x(.):[to,oo)—)R" of (3.15)

with x(to)=xo satisfies the following for all to e R :

x(t) e Nu(£2), for all t 2t0+T

Remark 3.1: Definition 3.1 is a modified version of the conventional definition
of UUB found in the work of Corless-Leitmann (1981). The modification allows us

to present an stability results in a concise way.

The following proposition gives an estimate of the set with respect to which the
closed-loop system (3.5) and (3.7) is UUB, when inequality (3.9) is satisfied. The
estimate given in this proposition is a special case of the one given by Leitrrrann
(1981). Nevertheless, we have included the proof in Appendix A, since it contains

certain technicalities that arise due to Definition 3.1.

Proposition 3.1: Let
(2,. :={z 6 R” I W(z) S c} (3.16)
71 712 'Yo % 2
o := 2mm 2—72- + a 7—2 (3.17)

and suppose there exists r>o such that 9,. c S . Then, inequality (3.9) implies
that the closed-loop system (3.5) and (3.7) is U.U.B. with respect to tr, with 9,

inside the region of atuaction.

21

Example 3.1 (Continued): In Example 3.1 we found that W satisfies inequal-
ity (3.14). If k4 in Assumption 3.62 is zero (which corresponds to the case when
A“, vanishes at the origin), then (3.14) implies that the origin of the closed-loop
system (3.5) and (3. 10) is asymptotically stable. When k4 is not zero, the closed-
loop system is UUB with respect to the set 90 as given by Proposition 3.1. Note
that 0 goes to zero, as C goes to zero. In other words, the set 0,, can be made
arbitrarily small by increasing the gain of the second term in (3.10).

The state feedback control of this chapter will be used as the state feedback
component of our observer-based control. The analysis of the next chapter (refer to
section 4.4 ) shows that in order to use the state feedback controller in this context,
the control law (3.7) has to satisfy a Lipschitz condition (Assumption 4.63 in
chapter 4). The linear high-gain control of Example 3.1 satisfies a global Lipschitz
condition. However, min-max control and variable structure control are discontinu-
ous control laws that do not satisfy any Lipschitz condition. Therefore, we have to
use continuous approximations of such control laws. Continuous approximations of
min-max control has been introduced by Corless-Leitmann (1981). Following the
development of Corless-Leitmann (1981), one can come up with a continuous con-
trol law (3.7) and a quadratic Lyapunov function (3.8) that satisfy (3.9). Continuous
approximations of variable structure control has been discussed by Slotine-Sastry
(1983), Slotine (1984), and Ryan-Corless (1984). However, There is no result in the
literature on how to obtain a Lyapunov function of the form (3.8) to satisfy inequal-
ity (3.9). Therefore, in the next two sections, we study the problem of finding such
a Lyapunov function. Variable Structure Control can be applied to a very large
class of nonlinear system, namely those systems that can be transformed into the
regular form. Feedback linearizable systems is only a small subset of this class.
Therefore, in section 3.2 we present the stability results in the general fiamework of

 

22

variable structure control, since these results are of interest on their own. Then, in
section 3.3, we specialize the results of section 3.2 to the case of fully-linearizable

systems.

3.2 Continuous Implementation of Variable Structure Control

To design a variable structure control law for system (2.1), first (2.1) is transformed
via a smooth change of coordinates

x
x := Li] =T(§), x16 ire-mt, e R4,T(O) =0 (3.18)

into the following so-called regular form [Utkin (1987)], [DeCarlo—Zak-Mathews
(1988)]:

J61'=fl(x1r7‘2t‘)
1 1.2 =f2(xl’x29t) 4' 331,121)“ (3.19)

+ 331.12,” [@4leer + Ag(xlvx2tt)u 4' A¢d(le2J)]

 

b

where B(xl,x2,t) is nonsingular for all x1 6 RP'V, 12 e R4 and t e R. The
arguments x 1, x2 and t are deleted for the sake of brevity, whenever no confir-
sion is likely to arise. A special case of (3.19) that was treated by Slotine-Sastry
(1983) and Ryan-Corless (1984) is the case when

f1(x19x29 ‘) =A1111+A12x2

and B is a constant matrix. In this case the nominal state equation is linearizable
via state feedback control.
After transforming system (2.1) into the regular form (3.19), a function p(.) is

found to satisfy the following assumption,

23

Assumption 3.1: There exists a continuously difi'erentiable function
p : RP“! --)R9 such that p(O) = O and the system
it = f 1(xr.p(x r).t) (3.20)
has a globally uniformly asymptotically stable equilibrium point at x1 = 0.

We also need to assume that the uncertainty in the input distribution matrix

A8 is small enough,
Assumption 3.2: IB Ag B"1 I s 8“ < 1
where 5,, is a nonnegative constant.

Then, under Assumptions 3.1 and 3.2, the variable structure control is chosen to

be
u = — ¢(x.r) 13'1 ssn (s) (3.21)
where
s =xa - per) (3.22)
[sen (s)].- == sen S.- i =1.....q (3.23)

and' 4) ( . , . , .) is a scalar-valued function that satisfies the following inequality for

all x e Rp,t e R andanyarbitrarypositiveconstant at:

1
1—5,

 

¢(x,t)2 01+ If2+BAf+BAd--§f-f1l, (3.24)
1

The surface s = 0 is known as the sliding surface.

The stability analysis of the closed-loop system (3.19) and (3.21)-(3.24) is done in
the following way: It can be easily shown that, under Assumption 3.2, the derivative

of the function %s ’3 along the trajectory of the closed-loop system satisfies the

following inequality,

% (é—s’s) =s’s S -als I1 (3.25)

Due to (3.25), the trajectory reaches the sliding surface in finite time and on the slid-
ing surface Assumption 3.1 implies uniform asymptotic stability of the origin. Note

that in this argument, no Lyapunov function is obtained for the closed-loop system.

Now let us replace the signum function in (3.21) by a saturation function of the

form [Slotine-Sastry (1983)]:

 

sgn s,- if Isilzt;
[sat (3)]; := i = 1,. . . ,q (3.26)
C 1 Si .
? otherwrse

i.e., we are considering the following continuous approximation of (3.21),
u = -¢(x,t) B-l sat§(s) (3.27)

The control law (3.27) causes the trajectory of the closed-loop system to converge to

a boundary-layer set

Qz={xeRP : lsilsc, i=1,...,q}

in finite time. Since within the boundary layer {22 the trajectory is not necessarily
confined to the sliding surface s =0, we areforeed towork with aperturbedver-
sion of equation (3.20) rather than equation (3.20) itself, namely,

ir=fr(xr.p(x1).t)+[f1(xr.x2.t)-fr(xr.p(xr).t)] (3.28)

To preserve the stability properties of the unperturbed system (3.20), we need to
impose a growth assumption on the perturbation term. To state this assumption we

use standard converse Lyapunov theorems, e.g., [Hahn (1967)], which, under

25

Assumption 3.1, guarantee the existence of a Lyapunov function V (x1 , t) and

functions (11, (ligand cc, of class K... such that forall x16 RP“? and t e R,

a1(|x1|)SV(x1,t)Sa¢(lxll) (3.29)
3V 3V
3+53f1(xr.p(xr).t )S-aaflxr“) (330)

A function 7: R“ -) R+ is said to be of class K -, if it is continuous, strictly
increasing, 7(0) = 0, and 7(r) —> no as r —> oo. Note that if 7 is of class K..,
then 7’1 is of class K..., and if 71 and 72 are of class K.., then no 72 is also
of class K .. The fact that 03 in (3.30) is of class K .. is not shown in [Hahn
(1967)], but has been shown recently by Sontag (1989).

Assumption 3.3: The Lyapunov function V( 11 , t ) of Inequalities (3.29)
and (3.30) satisfies the following inequality for all x 6 RP and all t e R,

EV

.371- [f1(x1,x2.t)-f1(11r P01). 1)] 5

0L4(llfl”(Kuhn-Min“) (3.31)

where 014(r )ot6(r) S ag(r) and a4(.), u5(.) and a6(.) are of class K ..

When the origin of (3.20) is globally exponentially stable, Assumption 3.3 reduces
to the requirement that f 1( . , . , . ) be globally Lipschitzian in 12. This follows
from the well-known result [Hahn (1967)] that in the case of exponential stability
V(xl , t ) can be chosen such that (2.-(r) =K. r2, i=1, 2, 3 and

.3.

3x ISK lel. Then a..(.), 0.5(.), and a6(.) take the form ai(r)=K.- r,
1 .

ifl, 5, 6.

Theorem 3.1: Under Assumptions 3.1-3.3, there exists a class K .. function
B(.) such that the closed-loop system (3.19), (3.24), and (3.26)-(3.27) is G.U.U.B.

26

with respect to the closed ball

Q§={x e R” | lxl 5503}.

Proof of Theorem 3.1:

Equations (3.19), (3.22) and (3.27) imply that

s' =f2-B [Iq+Ag]¢B-lsat§(s)+B A, +3 A“, - 3??“

=> Si 51‘ = ’95} [30163)]; + 5i1f2‘93 Ag 3'1 Mtg“)

8
+3 A, +3 lid—31115].-

s -¢s. [sat§(S)]i + Is..l [ I f2+13A,+19A..,-?gf—f1 I...
l

+ o IBAxB-lsat§(s)l,.]

s -¢s.- [sat;(s )1.- + |s,-l [qr-ct] . by Assumption 3.2 and (3.24)

S-alsil if lsilzc

which implies that 02 is an invariant set and any trajectory starting outside (21

reaches it in finite time.

Let us calculate the derivative of V( x1 , t ) along the trajectory of the closed-loop

system.

-.._a_v_ a_v
V- at + an fl(xrr12»‘)

27

-21 an
- at 4' 8x1 f1(x1» 901),”

3V

+ _
3x1

[fl(x19x2’ t) -fl(x19 13(11): t)]

S-ag(lxll)+a4(lxll)a5(ls I)

by (3.30) and (3.31). Inside (22, Is I SK C, where K depends on the type of

norm. Hence

V s—aaer I)+a.<|er) «:an

S—éafilxl I) — 014(Ix1 I)

 

—;-a6(ler)-as(Kt;)]

which shows that I} S --;-03(le I) for lel 2 a; 1[2(15(K Q]. Let

151-1051012015]. [52411001 and (33:11-55, and define the sets (21, M1, and
M2 by

017-{1 6 RP I V(Xl, t)SBz(KQ}
M1={x 6 RP I lel SB1(K§)}

M2={X 6 RP I lel SB3(KC)}

The set Q, is dependent on t , but using (3.29), it can be verified that
Mlcflchzmniformlyin t.

Now any trajectory starting outside 92 must enter ()9 in finite time and remain

thereafter. Moreover on the set {Ll-M 1, I} S -%a3(lx1 I). Thus, the trajectory

 

28

must enter the set anflz in finite time, and it remains in the set for all t
thereafter, since V is negative on the boundary of 91. Hence, there exists a finite

time T such that
x(t) e M2092 for all t2T

Since Mn) is continuously differentiable for all x1 6 RP“? ,

“p(xlfl Sfllxll), where K.) is ofclass K... Setting

BO) = 133(Kr) + Kr + 705300))
completes the proof of the theorem, since M may c (2;. D

We illustrate, via an example, that a growth condition like Assumption 3.3 is

indeed needed.

Example 3.2: Consider the system

x1=-x1+(x12 +1 )x2
(3.32)

:22 = u + A“, (t)
where |A2(t)| 5 1.0 is a disturbance term. Choose p(xl) =0. It can be easily
verified that Assumptions 3.1 and 3.2 are satisfied. The discontinuous VSC law
it = -2 sgn (x2) yields izxz S — lle. Hence, the trajectory reaches the sliding
surface x2 = 0 in finite time, and on the surface x2 = 0, the motion is governed by
x'l = - x 1, and the origin is globally asymptotically stable. Now consider the con-
tinuous VSC law u = -2 sat; (xz). Taking x2(0)=l, it can be verified that
l
C
that x1(t) -) co as t —) co. Thus, the system is not globally uniformly ultimately

bounded.

x2(t) 2 g- for all t20. Taking x1(0)= and using x20) 2 -§-, it can be verified

29

Assumption 3.3 is dependent on the regular form in which the nonlinear system
has been expressed. If system (2.1) is state-equation-linearizable, we can always
transform (2.1) into a regular form for which Assumption 3.3 is satisfied :1; . For

instance for Example 3.2 the following change of coordinates

21:11
22=-x1+(x12+1)x2 (3.33)
transforms system (3.32) into
2.1: 22
(3.34)

. (212 +22122- 1) 22
22:

 

2 + (212+1) (u + Adm)
21+]

For (3.34), let the sliding surface be 3 = 21 + 22. Following the procedure outlined

earlier, the ideal variable structure control is

 

 

u=-— 21 ¢(z)sgn (s)
21 +1
2|z z 2 +2 I (3'35)
¢(z)= 12:1 2) +2+212
21+]

However, system (3.34) clearly satisfies Assumption 3.3.
If system (2.1) is not state-equation linearizable, it may still be possible to

satisfy Assumption 3.3 by a change of coordinates. For instance, consider the system

 

:tThispointwasmadebyProfessorJJ.SlotineofMlTinapersonaldiscussion.

3O

1.1=-Xl+11212 (336)

1&2 = u +Ad (t)
where A“, (t) is as in Example 3.2. If variable structure control is designed in the
present coordinates, similar to Example 3.2, one can prove that continuous approxi-
mation of variable structure control would not have the global uniform ultimate
boundedness property of the ideal vsc. Moreover, system (3.36) is not linearizable at
the origin (To see this point, check the necessary and sufficient conditions for state-
equation linearization in [Isidori (1989)]). However, we can use approximate lineari-
zation ideas of Hauser-Sastry-Kokotovie (1989) to design a variable structure control
for which Assumption 3.3 is satisfied. To this end, consider the change of coordi-

nates

21:11

3.37
22=-x1+(x12+a)x2 ( )

where a is an arbitrary positive constant. Transformation (3.37) transforms (3.36)

into

1 (3.38)
.,..-..[13_<_2] [.,_a_<w.].o,2..n..1,an

212m

 

31

Let the sliding surface be s=z 1+2 2. Following the procedure outlined earlier, the
variable structure control is designed to be
2212 '3' IZIZZ-al

1+z¥+a+
2124-0 (212+a)2

 

 

sgn (s) (3.39)

It can be easily checked that Assumption 3.3 is satisfied for this system.

Asymptotic Stability in the
Absence of Persistent Disturbance

Theorem 3.1 only shows G.U.U.B. with respect to 9;. Although Q; can be
made arbitrarily small by choosing C small enough, the origin does not have to be
asymptotically stable (refer to Example 3.3, below). In fact the origin might not be
an equilibrium point at all. However, if external disturbances vanish at the origin,
i.e., A“, (0,0,t) = 0, one might expect that the continuous implementation of VSC
would stabilize the origin. It turns out that this is indeed the case, due to the fact
that inside the boundary layer 92, control (3.27) would act as a high-gain feedback
control which stabilizes the origin, provided C is suficiently small. To prove such

a result we make the following assumption.
Assumption 3.4: There exists a Lyapunov function W (x1 , t ) such that the
following inequalities hold for all t e R, and all x in a domain (2 containing the
cllx1|2$W(le)Sczlx1I2 c1>0,c2>0 (3.40)

LEE

811 I SC3'X1I C320 (3.41)

32

is}: + ggf1(xl,p(xl),t)S-c4lx1|2 c4>0 (3.42)
If 10131.1) -f 101,521)“ S 05 '51‘9' 0520 (3-43)
|f2+BAf+BA¢d-—aa-%f1ISc6IxI c620 (3.44)
Ip(x1)I .<_ c7Ix1 ll c720 (3.45)

Assumption 3.4 is a mild one, since it is required to hold only locally. In fact,
Assumption 3.4 is implied by the smoothness assumptions made earlier, together

with the assumption that A“, (0,0,t )=0 and that 2' = A (t) 2 is uniformly asymptot-

ically stable, where A (t) = $1 f 1(x1,p(x1),t) |x1=0'

Theorem 3.2: Under Assumptions 3.1-3.4, there exists C' >0 such that for all
C < C" , the origin is globally uniformly asymptotically stable.

Proof: Choose Cf small enough such that (2; c Q for all §<§f. In the
proof of Theorem 3.1 it was shown that the trajectory enters the set

anflz c QC in finite time and does not leave it thereafter. Inside this set,

u= - o 3'1 1. Therefore

C

I

SS

s’s' =-¢—c-+s' fz-os AgB'li

-329.
§+BAf+BAed axlfl

Isl2

+c6Is I le

9%- Isl2+c5IsI(Ix1I+IsI+c7lx1||)

33

Let v(x1, t) := W(x1,t) + é-s’s , then

-_ia a_w ,.
V— at "I' axlf1(le2,I)+SS

or
S-C4Ix1I2-I-C305I11IISII-fISIIz

+c6IsI[Ix1I+IsI+c7lxll]

(:4 —a '11 I
s —[ lel Isl ] a
—a —-c Is I
c 6
1 . a C40.
where a = —[c3c5 + c6(l+c7)]. Thus v<0 for §<§2=:———-—-2—. Take
2 203466“ )

§*=min {Cf , {3}. For all §<§* , every trajectory enters anflz where v

satisfies v S -c v for some c >0. Since the trajectory can not leave the set

9109.2, it can be easily seen that it approaches the origin as t —-> oo. C]

One important difference between Theorems 3.1 and 3.2 is that the conclusions
of Theorem 3.1 holds for any §>0, while the conclusion of Theorem 3.2 is
guaranteed to hold only for sufficiently small C. The following example shows that
if Q is not small enough the origin may not be asymptotically stable, while the sys-
tem is G.U.U.B. with respect to 9;.

Example 3.3: Consider the system

il=-x1+x2
iz=axl+u, 0(051
Take p(x1)=0 and ¢=a+lx1|. Inside the boundary layer 971,

u= — é-(a + lxll) x2, and the closed-loop system is given by

X.I=—Xl+12

:22 =a x1 -- i— (a+lx1|) 1:2

The Jacobian of the right-hand side at x = 0 is given by

-1 l
A:
a ——

It can be verified that when C>%, one of the eigenvalues of A is in the open

right-half plane. Hence, the origin is unstable. On the other hand the set 91092,
given by

anaz={xe R2 I lxllsC, llesC}

is an invariant set and every trajectory of the closed-loop system reaches it in finite

time, irrespective of the value of C.

3.3 VSC Design for Fully-linearizable System

In this section, we focus on designing a continuous approximation of variable struc-
ture control for stabilization of (3.5). We will closely follow the development of last

35

section, with some modifications in the control design. The first step of the control

design is the choice of a sliding surface. Choose the sliding surface
s := G 2 = 0 (3.46)
where G is a qx(p —q) matrix such that
i) GB is nonsingular.
ii) G (s1 — A )-13 is minimum-phase.
iii) (6 , A) is detectable.
Choosing G to satisfy (i)-(iii) essentially guarantees that Assumption 3.1 of section

3.2 is satisfied. Consider the following continuous approximation of variable struc-

ture control
* v = - (p(z) (GB )-1 sat; (s) (3.47)

where sat; (.) is given by (3.26), and ¢(z) will be chosen later. To obtain a qua-
dratic Lyapunov function for the closed-loop system (3.5) and (3.47), we use the
same idea that was used in the proof of Theorem 3.2, i.e., first system (3.5) is
transformed into a regular form. In the new coordinates, two Lyapunov functions are
defined, one to characterize the motion of the closed-loop system on the sliding sur-
face, and the other to characterize the motion of the closed-loop system towards the
sliding surface. Then, a weighted sum of these two Lyapunov functions is con-
sidered as a Lyapunov function candidate for the overall closed-loop system. The
following lemma formalizes the ideas mentioned above, providing us with a
Lyapunov function candidate for the closed-loop system.

Lemma 3.1: Let
A = A -B(GB )‘16A -tlB (GB )‘16 (3.47)

where u is an arbitrary positive constant. If the sliding surface s =Gx is chosen

36

such that (i)«(iii) are satisfied, then
a) A as given by Equation (3.47) is Hurwitz.
b) There exist symmetric positive definite matrices P and Q 6 RP "P such
that
P A +A ’P = — Q

PB = G '63.

Remark 3.2 The second term in (3.47) is precisely what is known in the litera-
ture as the equivalent control [Utkin (1987)]. Intuitively one can see the reason for
the introduction of the third term in A . The second term of (3.47) places p -q of
the eigenvalues of the closed-loop system at the invariant zeros of (G ,A ,B ) and
the rest at the origin. So the pmpose of the third term is to shift the eigenvalues
which are at the origin into the open left-half plane.

Proof of Lemma 3.1: There exists a similarity transformation T [Young-

Kokotovic-Utkin (1977)] such that

A A
-1_ 11 12 _ 0
TAT -[A21 A ] TB—[I]

q
GT4=(0 GB)

where A 11 is Hurwitz, due to (ii) and (iii). Then

.. A A
TA T71 _____ ll 12
[ 0 —ul.,

which is clearly Hurwitz. Let (2'1 6 RP ""‘P'q be symmetric positive definite, then
there CXiStS F1 symmetric pOSitiVC dCfinitC SUCII that F1A11+A ll’P-l = -Q—1 . at

37

Q- .= a§1_ ’91-’- 1A 12
' —0tA 12'P1 29(GB YGB

Choose 0t>0 small enough such that Q— is positive definite.

0071 0

Let P =T'[ 0 (GBYGB T and Q =T’Q T

 

Then straight-forward computation shows that
PA + A ’P = - Q
PB = G ’63 El

Now suppose that 8(2 , v, t) in (3.5) satisfies Assumption 3.62 with

1
«I; IGBII IGBI-l

 

and choose positive constants pl and p0 such that the following inequality is

satisfied in the domain of interest,

l-‘Iq_k2IGB I I(GB)-1I

 

plIZ' +p02

and let (p(.) in (3.47) be

9(2) 3:91" I +90

The derivative of W (2 )=2 ’Pz along the trajectory of the closed-loop system (3.5)
and (3.47)

38

=> W = z'(PA +A'P )2 + 22’PB(GB)"1 [-¢(z)sat§s +03 8(2,v,t)+6Az +1102]

= -—z’Qz + 22’6’ [-¢(z)sat 62 +63 5(z,v,t)+6A2 +1162]

= —z’Qz — 2¢(z) isilsatgsli + 23'[GB 5(z,v,t)+GA2 +1162]
i=1

1 :={i e N | lsiSq, Is.-|>C}

I’:={i e N l lSi Sq, lay-15C}

=> W =—z’Qz -2¢(z) Erisgn s,- - 2¢(z) E xiii-

iel iel’ C

+ 23’ [636(2,v,t)+6A2 +1162]

.2
=-z'Q2 -2¢(z)lsll+2¢(z) 2 [Isil—iC—J

iel'

+ 23' [GB 8(z,v,t)+6Az +1162]

S-Ammwflz Iz-MZ)IS II + EQSNZ)
+ 2Is I1I635(z,v,t)+6A2+1162 I
S—hmin(Q)Iz I2+ gig-(NZ)

4Cpl QCDO
2 I2I+ 2

 

=-i.m(Q)lzI2+

 

39

which, by Proposition 3.1, implies uniform ultimate boundedness of the closed-loop
system. Note that, similar to the linear high-gain control of Example 3.1, the esti-
mate of the set of uniform ultimate boundedness given by Proposition 3.1 shrinks as

C goes to zero.

4 Observer-Based Control

4.1 Introduction

In this chapter, we propose an observer—based controller for stabilization of system
(2.1), under the assumption that the nominal system (2.2) is fully-linearizable, left-
invertible, and minimum-phase. In chapter 3, we studied the first step of such a
design process, which is design of an appropriate state feedback controller for sys-
tem (2.1). To design the state feedback controller, we first transformed system (2.1)
into system (3.3). Then, we found the state feedback control (3.4) and (3.7), along
with the quadratic Lyapunov flmction (3.8) such that, under Assumption 3.62, the
derivative of W along the trajectory of the closed-loop system (3.3), (3.4), and
(3.7) satisfies inequality (3.9), which in general implies uniform ultimate bounded-
ness of the closed-loop system. The next step of the control design is to design an
observer to estimate the state 2 of system (3.3). We design the observer based on
the linear part of (3.3) independent of the (possibly uncertain) nonlinear terms. Let

us consider the following observer-based control

{i=A2+3 o(2)+L(y-Ci) (41)

u = 3(2) := 12(2) + 6(2) ((2)

where (to, B(.), and (p(.) are given by (3.4) and (3.7), and L is the observer

4o

41

gain to be designed later in section 4.3. Let e := z — 2‘ be the estimation error,
then the error equation is

é =(A —LC )e +B A(z,f,t) (4.2)

where
A(z.i.t) := M2) [am - «(2) + [15(2) — 13(2)] W‘)
+ Af(z) + A...(z) + 218(2) [12(2) + B(2)¢(2)]]

= 8(2 .¢(z).t) + B"(2) (I.+A.) [Fm -F(z)]

+ ¢(Z) - 11(5) (4.3)
Note that the term A(.,.,.) has been created by three different sources:
1) Uncertainties and disturbances in system (2. 1)
2) Simplification of the nonlinear terms 0t(.) and B(.).
3) Estimation error

Therefore, even if there is neither uncertainties in (2.1), nor any simplification in the
cancellation of the nonlinear terms, the disulrbance term (4.3) will still be present in
the error equation (4.2). In other words, output feedbaCk control of the nominal sys-
tem (2.2) is as difficult a problem as that of system (2.1).

It is well known that in the presence of the term A(2 ,2,t) in (4.2), choosing
L to locate the eigenvalues of (A —LC) in the open left-half complex plane does
not ensure stability of the closed-loop system. Instead, the observer should be
designed such that the disturbance term is decoupled ( either exactly or asymptoti-
cally) fiom the error equation. In robust control of linear systems, such robust
observers could be designed via loop transfer recovery techniques (for a survey,

refer to Stein-Athans (1987)),which consist of asymptotic methods that use Riccati

42

equations and transfer function manipulations. Such transfer function manipulation
can not be extended to nonlinear systems. Therefore, we design the observer
through a singular perturbation approach which has been recently developed by
Saberi-Sannuti (1990) and Esfandiari-Khalil (1989). The conceptual idea of this
approach is that to determine the amount of required gain at each element of the
observer gain matrix L , we need to know the finite and infinite zero structure of the
linear system (2.3). Therefore, we first transform system (2.3) into a canonical form
that explicitly shows its finite and infinite zero structure. Then, in this canonical
form, the observer is designed via asymptotic pole placement to reject the effect of
the disturbance term (4.3) on the error equation (4.2). Esfandiari-Khalil (1989) use
this approach to design nonlinear output feedback controllers for uncertain linear
systems, while Saberi-Sannuti (1990) use it to design multiple-time—scale observers

for loop transfer recovery.

4.2. A Special Coordinate Basis

Consider the linear system (2.3) which was obtained by exact linearization of the
nominal system (2.2). In this section, we transform (2.3) into the special coordinate
basis of Saberi-Sannuti (1987), which explicitly shows the finite and infinite zero
structme of (2.3). The infinite zero structrue of a linear system is closely related to
the number of inherent integrations that exist between its inputs and outputs. There-
fore, the idea behind the special coordinate basis of Saberi-Sannuti (1987) is to
linearly combine and partition the input vector v , as well as the output vector y ,
such that the inherent number of integrations between certain parts of v and
corresponding parts of y are exhibited clearly in the coordinate basis. To this end,
it has been proved by Saberi-Sannuti (1987) that, since (2.3) is left-invertible (by

Assumption 2.61 and Proposition 2.1), there exist nonsingular transformations

43

l" , I‘m , I}... integers K, 4... qb, (1., i =1,...,K such that the transformation

3

. =F ..
.Y “y,

a v = I‘m? (4.4)

 

 

N
II
"I
I j
IN!
H. e- a” .

d

transforms system (2.3) into the following form:

F
2a = Aaaza +Aaf5’f +Aasys

1 2b = Abbib “I'Abfy’f (45)
if = Afif +Mfyf +Bf [Da§a+Dbib+Df2f+i-’]

L

W=93
is = C: 2b

A

 

h

K K
where the dimensions of 2a, 2),, if, y],and j", are ac, qb, Ziqi, Egband
i=1 i=1
K I
P " 241. respectively. Moreover, invariant zeros of (C , A , B ) are the eigen-
i=1

values of A“, ( C,, A“, ) is observable, and AI, 8,, and Cf have the follow-

ing canonical structrn‘e:
Af := Block Diag (Alf, . . . ’AKf)
Bf := Block Diag (Blf’ . . . rBKf)
Cf := Block Diag(C1f,...,CKf)

WITCI'C I" Alf =0“qu Blf =Iqt’ le =Iq!, if I=1, while

01‘“; III 01‘ MI
2 B if =

A - =
If 040% 0?: X11 I'll

 

1’ 0mm denotesthe mxn zeronratrix,and I," denotesthe mxm identitymatrix.

44

[1... 0...] . = on.

for i>1.
Partitioning ii and y, into
r ' f ‘

V1 Y1;

<1
ll

)’f=

9K 7K)“

 

 

 

 

where 17,-, and yif are q.- -dimensional vectors, one can see that the variables ‘17.-

controls the output yif through a stack of i integrators.

The vectors in , 2,, , and 2f span some well-known invariant subspaces of

geometric theory of linear systems [Wonham (1979)]:

l) 2., spans the largest (A ,B )-invariant subspace contained in the Kernel of
C .

2) 2,, spans the largest (A ’,C ’)—controllability subspace contained in the Ker-
nel of B ’.

3) 2f spans the smallest (A ,C )-invariant subspace containing the Range of
B.

Remark 4.1: The transformations 1‘, I}, , and F0... are obtained by Saberi-
Sannuti (1987) via a modification of the structural algorithm of Silverman (1969). A
numerical algorithm which is based on the procedure of Saberi-Sannuti (1987) is
available in Linear Algebra and Systems (LAS) package. Note that many physical
problems of interest to us are already in the form (4.5). For instance, Examples 2.1
and 2.2 of Chapter 2 are already in the form (4.5). Also in Appendix B, we give an

45

alternative way of arriving at (4.5) for a subclass of linear systems under study,
namely for linear systems which are square, invertible, minimum-phase, and have a
left diagonal interactor. The advantage of the algorithm given in Appendix B over
that of Saberi-Sannuti ( 1987) is its simplicity.

4.3 Observer Design

The problem of observer design to reject the effect of the disturbances modeled by
A(2 ,2‘ ,t) becomes an asymptotic pole placement problem in the special coordinate
basis (4.5). Let us choose Lb and L“, i=1, . . . , K, such that Abb — LbC, and
Aif-Lifcif are HIII'WIIZ. LCI

Lf :=diag(L1pr2fs ~ ' ° tLKf)

M(e) := diag (M1(c), . . . ,MK(e))

I I I
. 41 qt ‘1:
M - := —, --, . . . , —.
and choose the observer gain to be
r r
Aaf Ans
L(e) := 1‘ A., L, 1);}, (4.6)
th 'I'M (8) Lf O J

 

 

Then it is easy to show that

f

A... o o
1"1(A-LC)I‘= o Aw-LbC, o (4.7)

 

L

Note that (4.7) has a block triangular structure. The eigenvalues of the first two
diagonal blocks are 0(1), while the eigenvalues of the last diagonal block are

0%). Define

Q

[ E; ] := rle (4.7)

A, := Block Diag (Am , Abb—LbC, ), D, := (Dd, Db )
where e is the estimation error, and dimensions of e, , and g, are q,, +43 and

K
Zia,- , respectively. Then the error equation (4.2) is transformed into
i=1

 

é, = Ase,
g, = [A, - M(e)L,C, 15, +3, [D,e_, +D,e, + r5.1A(z,i,r)] (4'8)
L
Scale g, in the following way
a, := N'1(e)g, (4.9)
N(e) := Block Diag (N 1(8), . . . , Ng(e) ) (4.10)
N..(e) := Block Diag (2‘41, .641.“ , . . . ,1.” ) (411)
Then it is easy to show that
N-1(e)3, = 3, (4.12)
N’1(e) [A, -M(e)L,C, ]N(e) = i: [A, —L,C,] (4.13)

Using (4.12) and (4.13), it can be shown that the scaling (4.9) transforms (4.8) into
the following form:

F
é: =As e:

- -1 .. (45114)
88f =[Af -1:fo 18f +83}: [D383 "FBI-”(8)8! +1.3" A(2 ,2 ,I

 

L

47

The desired disturbance rejection property of the observer can be explicitly seen
in (4.14), since the slow part of (4.14) is completely decoupled from A(2 , 2‘), and
the effect of the term A(2 ,2) on the fast part of (4.14) decreases, as 6 tends to
zero. It is also clear from (4.14) that in the course of achieving such a disturbance
rejection property, we have to locate some of the eigenvalues of (A -LC) far in

the left-half complex plane.

4.4 Closed-pr Stability Analysis

By the development in section 4.3, the closed-loop system (3.3) and (4.1), with L

given by (4.6) can be written as

r

2 = A2 +B¢(z) +B 5(z,¢(2),t) +BB‘1(2)(Iq+Ag ) [F(2‘) — F(z)]
1 é: =As es (415)

 

LEéf =(Af -1:fo )ef “FEBf [D383 +D,N(e)ef +r;1A(Z, f,t)]

where 5(z,v,t) is given by (3.6), A(2,2‘,t) by (4.3),
M) = (2(2) + 6(2) 4(2) (4.16)
and by (4.7) and (4.9)-(4.11), 2, which is the estimate of the state 2 , can be written

as

i=z—e=z-F

 

e!
N(e) e, J = z - Fles - FrN (6)42, (4.17)

Note that N (e) is a polynomial matrix in 6; hence it is bounded for small 8. Sys-
tem (4.15) is a standard singularly perturbed system, with (z, e, ) as the slow
variable and e, as the fast variable. The slow and fast subsystems of (4.15) are

respectively

48

 

2: A2 + B¢(z) + B 5(2,¢(2),t)
( +‘B B‘1(2) (Iq+Ag) [F(Z-F1€s) —- [7(2)] (4.18)
e, =A, e,
and
def
where t=t/£.

Uniform Ultimate Boundedness

In this part, we argue that the uniform ultimate boundedness property of the state
feedback control is preserved by our output feedback controller, under certain condi-
tions. Suppose that we have designed the state feedback control (3.4) and (3.7 ) to
ensure uniform ultimate boundedness of the closed-loop system (3.3), (3.4), and
(3.7), i.e., there exists a quadratic Lyapunov function (3.8) such that inequality (3.9)
is satisfied along the trajectory of the closed-loop system (3.3), (3.4), and (3.7). To
state the theorem on the uniform ultimate boundedness of the output feedback con-

trol, we make the following assumption:

Assumption 4.63: For all z, 2‘ e S
l¢(z)-(|)(2‘)ls1c5|z —2‘l
Iii-1(2) (14+A2) [F(z)-F(2)] I Skglz — 2|

where kg, and k6 are nonnegative constants, and S is the set of Assumption

3.62.

49

Theorem 4.1: Under Assumptions 2.61, 3.62, and 4.63, suppose that the state
feedback control (3.4) and (3.7) is designed such that Lyapunov equation (3.8)
satisfies inequality (3.9), and let a and r be the constants of Proposition 3.1.
Consider system (4.15), and let P, and P f be the symmetric positive definite

solutions of the Lyapunov equations

P: A: +A8’PS =_Iqtt+45

"-4.15 '
- (P) . . .
11”" L. Then there exist posrtrve constants af , r, . and E.

ArunnUD) 9

and a continuous function g : (OE—m" such that for all e 6 (0,2) system (4.15)

 

Suppose that o <

is uniformly ultimate bounded with respect to (be, with 2 inside the region of

attraction, where

(pa ={ (z’,e,’,ef')' E R2" I 33:0, W(sz ef’l’fef < 8(8) }. (4.20)

£={(2’,e,’,e,’)’e R2" I e,’P_,e,<r,,
1min?”
W 2 +0. e 'P e <—-— 4.21)
( ) f f f f 9) (p) (

Moreover,

um 8(8) = a (4.22)
e—ro

Remark 4.2: (4.22) shows that the 2 -projection of the set of uniform ultimate
boundedness in the case of our observer-based control approaches (2., of Pmposi-
tion 3.1 as 8-90.

50

Proof of Theorem 4.1: The arguments e and t are dropped whenever it

causes no confusion. Let

 

 

a _ xmwm‘zlz
f P linina’f)
a = Amman),
" 9).,,m,,(10)l1‘1l2

where I] and F2 are given by (4.17). Let
V, := e,'P,e,

V 2: W(z)+afef’Pfef

~ Amino,»

2:: ’, ’, ’ R2"| V , V ——
{(2 e, ef’) e s <a, < 9 (P)

Let 86 (0,1). Then |N(e)| <1, and it can be easily shown that for all
(z, e,, e,) e 33, z, and i belong to the set (2, CS , where S is the set defined
in Assumptions 3.62 and 4.63, and Q, is defined in Proposition 3.1.

Along the trajectory of the closed-loop system, i}, , and V satisfy the following
inequalities on :2:

l

v s———V 4.23
’ 1mm.) ’ ( ’

V 5’72'2 [24'71'2 |+Yo

(I.
+2kélPB I lz l(lI‘le,| + II‘2N(e)ef l)- —ef-Ief I2

+ 2a, IPfoD, I Ie,| lefl 4' 2a, IPfoDfN(€)I lei '2
+ 2a, Ie, I Ipfsfrgll |A(z,f,t)l

51

 

Let
00 I: 72
V .1.
2
a := +21: IPBIIF I ——‘——
l 71 6 l [knurl(Ps)]
02 3: ‘Yo
a3 := 2k6IPB I Irzl + 2a,(k1+k2k5+k3)IP,B,r,;-,1I
.1.
2a lPBDl—— 2“(war-1|
a := .
5 f f f MW—T 4 f f m
i
+(k5-t-k)lI‘IIPBI‘,;1| V‘ 2
h(z,V_,,ef) := —ao|z |2+al lz l+az+a3|z I la, I
—a4|ef |2+aslef|
Then

V Sh(z,V,,ef) on i: andee (0,1) (4.24)

Note that a32—4aoa4 < O for all e 5 (0,81). where 8151 is a positive constant.
For a 6 (0,81) , let c,(e,V,) and c¢(e,V,) be the unique real positive roots of
the following polynomials

(a3 -4aoa4)c. + (2a3asm1anc. + a3+4a2ar - 0 (4.25)

52

(2232—44022992 + (2a la3+4aoa5)c, + a3+4aoa2 = o (4.26)
These roots satisfy

lim£_,o c,2(e,0) = 4111:?) , lime_,o 0, (8,0) = 0. (4.27)

 

We will show that h(z,V,,ef) is negative if lz I > C2 (e,V_,) or Ief | > cc (£,Vs).
Considering h (z ,V,,ef) as a quadratic term in Ief I implies that

(a3IzI+(15)2
404

 

h(z,Vs,ef)S +az+allz| -a0|zl2

=—l—[ 2 l I2
404 (03-40004) 2

+ (20305+40104) lz | + 03-0-402a4] (4.28)
Comparing (4.25) with (4.28) shows that h(z,V,,ef) < 0 if Iz I > C2 (e,V_, ).
Similarly one can prove that h(z,V_,,ef) < 0 if Ief I > c¢(t-:,V,).
2 2
Let 0 (4n) := M<P)[c.(e.V.>] + 1mm, >a, [c. (4V. )] .Thcn

h (z,v,,e,) < o if V > G(e,v,) and «a, (4.29)

1min? )7
97am?) '

€931 such that g(e)<r for all ee (0;). Let K(V_,) :=G(E,V,). By

Let g (a) := G (8,0). By (4.27), lim¢_,og (e) = (K Therefore there exists

differentiating (4.25) and (4.26) with respect to V,, it can be shown that K (.) is
monotonically increasing, hence K '1(.) is well defined. Choose r,>0 to be strictly

less than min{a,,K'l(r)}. Let 2 be as given by (4.21) and e e (0,E). By

(4.23) and (4.29), considering the direction of the vector field on the boundary of 2,

it can be shown that 2‘. is an invariant set of the trajectory.

53

Next we will show uniform ultimate boundedness with respect to (be, as given by
(4.20). Given u>0, suppose, without loss of generality, that Nu(<b€) c )3. By a sim-
ple topological argument it can be shown that there exists 5>0 such that

R5 :={ (z’,e,’,ef’)’ e R?” I V, 55 , V 55+G(e,5) }c Nu(d>£) (4.30)
and R5 isaninvariant set. Let
F :={ (z’,e,’,ef’)’ e R2" IVSSS , 5+G(£,8) S V S r }

By (4.29), there exists a>0 such that VS-a on F. Using this fact, (4.23), and

the fact that R 5 is an invariant set, it is easy to show that
(z (t )’,e, (t )’,ef (t )')’ e R5 for all tZT

where

T :=2.mx(P,) log%+-‘1; [r -5-—G(e,5)]

This, together with (4.30), proves uniform ultimate boundedness with respect to $8.
El

Corollary 4.1: If Assumptions 2.01, 3.62, and 4.63 hold globally, and the
state feedback control (3.3) and (3.7) is designed such that inequality (3.9) holds
globally, then there exist positive constants af and 2', and a continuous function
g : (0;) --)R+ such that for all e s (0,2), system (4.15) is globally uniformly ulti-
mately bounded with respect to (be, given by (4.20).

Asymptotic Stability at

Next, we prove that if the state feedback control (3.4) and (3.7) renders the origin of

system (3.3) asymptotically stable, then so does our observer-based control.

Theorem 4.2: Under Assumptions 2.61, 3.02, and 4.63, suppose that k4=0,
and the state feedback control (3.4) and (3.7) is designed such that Lyapunov func-
tion (3.8) satisfies inequality (3.9) with “if-70:0. Let r, P... Pf, and 2 be as in
Theorem 4.1. Then there exist positive constants af , r,, and H such that the ori-
gin of system (4.15) is an asymptotically stable equilibrium point with set 2‘. inside
the region of attraction.

Proof of Theorem 4.2: The proof follows as a corollary of the proof of Theorem
4.1.

5 Peaking Phenomenon

Theorems 4.1 and 4.2 imply that some of the poles of the observer (namely, those
associated with (2,) have to be placed far in the left-half complex plane, in order to
achieve stabilization of system (4.15). In general, placing poles far in the left-half
complex plane causes an impulsive-like behavior which is known as the peaking
phenomenon. The effect of the peaking phenomenon on stabilization of nonlinear

systems via state feedback has been recently studied by Sussmann-Kokotovic (1989).
The effect of the peaking phenomenon on the output feedback control proposed
in this paper can be explained in the following way: If the observer gain L(.) is

chosen such that the real part of some of the eigenvalues of (A -LC) have 0 (%)

magnitudes, then the state transition matrix e “‘LC)’ contains terms like

lie-“"6 where «>0 and ie N
8

in general. Therefore, if the initial conditions of the error are 0(1), the transient

behavior of the error e (t) contains overshoots of order 0(-1;-). Since the error
8

equation is coupled to the state equation, these excessively large overshoots are

transmitted to the states of the nonlinear system, causing peaking to appear in these

55

56

states as well. The following example illustrates this phenomenon:

Example 5.1: Consider the problem of stabilization of an inverted pendulum,

whose motion is described by the following equation (See Figure 5.1),

ale d0 . V(t)
— — = -— .1
dt2 + dt +b1sm0 u(t) 2. c030 (5 )

where 0 is the angular position of the pendulum measured versus the stable equili-
brium point, a (t) is the control moment applied to the pivot point, v(t) is the
horizontal acceleration of the pivot point, and 2. is the length of the pendulum.
Assume that only the angle 0 is available for measurement, and that the only infor-

mation available about b1 and v(t) are the following bounds,

|b1|< 1, Iv(t)| < 7t forall t (5.2)

 

Figure 5.1- Pendulum of Example 5.1

57

=fl

y :=x1:=0—1t, x2: dt

(5.3)

where x1 and x2 are the state variables and y is the output. Then the state equa-

tion of the pendulum is

*1 =12
:22 =—x2+u(r) + 80:1. 0 (54)
where
5(x1,t):=blsirtx1+it)-cosx1 (5.5)

1

Following the development of section 3.3, we designed the following variable su'uc-

ture control for system (5.4),
u =-x1 ‘12-2.0 sat;(x1+x7) , C=0.0l (5.6)

which renders system (5.4) globally uniformly ultimately bounded with respect to a
small neighborhood of the origin (Refer to Figtn'e 5.2a for phase plane trajectory of
the closed-loop system for the case when 11(0):].0, and x2(0)=0.0). Following the
algorithm of section 4.3, the observer-based conuol was designed as

r

- 2
Jir=£2'*"fi'()’ ‘f1)

_L

£2=u+-El7(y-£1) (5.7)

u = 41 - £2 - 2.0 sat§(£1+£7) , c = 0.01

 

b

It can be easily checked that Assumptions 2.61, 3.62, and 4.63 hold globally in this
case, and hence by Corollary 4.1 the closed-loop system (5.4) and (5.7) is globally
uniformly ultimately bounded, for sufficiently small a. Figure 5.2 shows results of

58

the simulations with
8 = 0.01 , 11(0) = 1.0 , X2(0) = 21(0) = £2(O) = 0.0
Figures 5.2b and 5.2c show the peaking in the input u and the estimation error,

respeCtively. Note that the input peaks to an 0 (i) quantity.

 

 

59

 

 

 

 

 

 

 

 

 

 

 

 

(a)
8"
- .. _ —- state feedback
output feedback
-4 o ; 2 3 ‘4
”r
0
-40—4
3
(b)
-80-
'- - - state feedback
output feedback
- I 20 r r r l
0 1 2 3 4
a!
//
(C)

 

 

 

-120
. 0

‘1

“-4
§

2
‘1

Figure 5.2- Simulation Results of Example 5.1

60

Example 5.1 clearly shows that peaking in the transient behavior of the system
is undesirable. Peaking might even destabilize the closed-loop system as we
decrease e. The following example illustrates the destabilizing effect of the peaking

phenomenon.

Example 5.2: Consider the second order system,

 

rJE1:1‘2
1 22:3 (1+0)x23 +u (5.8)
L)’ =11

where 0 is an unknown parameter whose nominal value is zero. Let the state feed-

back control be

u =-3x% +v

 

which ensures asymptotic stability of the origin. Figure 5.3 shows the phase portrait
of the closed-loop system (5.8) and (5.9), with 0=0.l. The unstable limit cycle is
the boundary of the region of attraction of the closed-loop system. Following the
algorithm of section 4.3 the observer-based control is designed as .

' 2
£1=£2+EU 'fr)

. 1 .-
i2=V+—2 (Y’xr)
e

.1
u=-3223+v (5 0)

V =-2£1-£2

 

Figure 5.4 show the results of simulations for the closed-loop system (5.8) and
(5.10), with 0:01, $0.014, and the following initial conditions,

x1(0)=0.01, x2(0)=£1(0)-—-f2(0)=0.0

61

The closed-loop system is asymptotically stable in this case, but we can see the large
overshoot in the state of the system. Note that x2 reaches a maximum of 2.1, while
the only nonzero initial state is 0.01. Figure 5.5 shows the results of simulation for
the closed-loop system (5.8) and (5.10), with $0.013, and all the other constants
and initial conditions the same as in Figure 5.4. Figure 5.5 shows that the closed-
loop system is unstable in this case. As a matter of fact, results of our simulations
show that for all a less than 0.013 the closed—loop system is unstable. This is due
to the peaking phenomenon which is present in the observer. The impulsive-like
behavior of the observer state variables is passed to the states of the system. After
an 0 (a) time, the estimation error has decayed to a very small value. However, the
initial jump in 1:2 takes the trajectory out of the region of attraction, resulting in
instability.

It should be emphasized that the instability we have seen in this example does not
contradict Theorems 4.1 and 4.2. The theorems estimate the region of attraction by
the set 2. Notice that 2'. is defined using the scaled estimation error ef . For the
initial state of the closed-loop system to belong to 2, cf (0) should be order 0(1).
From the scaling equations (4.9)-(4.11) we can see that for cf (0) to be order of
one, some components of g, (0) must be of order 0 (83) for some B>0 that is
determined by (4.11). In the above example the initial condition of the unsealed esti-

mation error x 1(0) — 191(0) is 0.01. When scaled by e, the initial condition of the

scaled estimation error becomes 92—1, which ( for sufficiently small 8 ) places the

initial state of the closed-loop system outside the estimate 23 of Theorems 4.1 and

4.2.

0.00

62

STRTE FEEDBRCK CONTROL
THETR=O.1. GRMMR=1.0

 

XI2)

‘I.00

‘2.00

 

 

 

 

r3.00

.00 —2T. 00 -1'. 00 0.00 1'. 00 2. 00 3.00

X”)

Figure 5.3- Phase Portrait of state feedback control of
Example 5.2

63

OBSERVER-BASED CONTROL
THETA=0.1
EPSI=0.014

 

LII)

 

 

 

 

 

 

 

 

 

 

 

'38 ' _jh_____.
:O r r f— ! l
8.
'7
D
Q
0' T ‘ ““1
E9? ' i
w E
8 O I O E
v" : 'r : : 4
10.00 0.03 0 0.09 0.12 0.15

.06
TIME

OBSERVER-BASED CONTROL
rHETn=0.1
EP$I=0.014

2.00

 

 

 

 

 

 

 

 

 

536’ t i -‘=r===——— 2
>< L///// a a s =

8 5 5 5

N. 1 9 l

I

8

«i ‘. T : 1 :
-8 m
:5 z r 1* ' z
x \/ E o I

8‘ : . O U 5

$0.00 2 00 ' €00 6.00 10.0

Figure 5.4- Time profiles of state variables for output feedback
control of Example 5.2, with e = 0.014.

OBSERVER-BASED CONTROL
TH TR=0. I
§1P§551=0.013

 

 

 

 

 

 

058 44___._,. E : 5

7° ' fl. 1 : 1

z) I E E i
't'

 

 

 

 

 

 

 

0.00 0.05 030 0.15 0.20 0.25
TIME

oaSERVER-ansso CONTROL
THETA=0.1
g, EPSI=0.013

 

 

 

 

 

 

 

E
«3 E
:5! : E
(‘N ? t 1.—
N E
>< E
E
8 :
o 1
In
I
0
1n
0 . T I T I
i I 1
I O o
i I t t i
8 2 i
A ' :
-'o *1
w 3
X I

 

 

 

°0.50

0.00 0.05 6.12 0.10 0.24 7. 30
TIME

Figure 5.5- Trme profiles of state variables for output feedback
control of Example 5.2, with e = 0.013.

6 Globally Bounded Control

6.1 Introduction

In Chapters 5, we studied the effects of peaking on the behavior of the closed-loop
system. In this chapter, we argue that the if the state feedback component of the
observer-based control is designed to be globally bounded, the states of the nonlinear
system will not exhibit peaking, and consequently the destabilization phenomenon
associated with peaking will not take place. In order to prove this point, we first
present a new singular perturbation result in section 6.2. Then, in section 6.3, as a
corollary of the result of section 6.2, we present a result on the stability of the
closed-loop system, for the case when the control is globally bounded. Finally, in
section 6.4 we apply bounded control to the examples of chapter 5 and present simu-
lations to show that the undesirable effects of peaking are indeed eliminated.

6.2 Singular Perturbation Result

The closed-loop system (4.15) is a standard two-time-scale singularly perturbed sys-

tem which can be written in the following form,

65

{15 =f(1)+f(x.N(6)y). “(D-=10, x e R"

6:9 =Ay ”some”. y<0) =yo. y e R... (61)

where

z
x.= e8 ,yo=efs

A is Hurwitz, f (0) = 0, and f (x,0) = 0 for all x e R". The slow subsystem is
obtained by setting 8:0 in (6.1) and dropping the initial condition, y(0) =yo to

get
i = f (x), x(0) = x0. (6.2)

Assume that f , f and g are smooth enough to ensure existence and uniqueness
of the solution of (6.1) and (6.2). Denote the solution of (6.1) by (1:80) ,y£(t))
and the solution of (6.2) by x, (t ). Moreover assume that the origin is the unique
equilibrium point of (6.1) and (6.2).

In this section, we study the asymptotic behavior of system (6.1). First, we
recall some known results from singular perturbation theory which are relevant to
our problem. Then we study a case that arises in our problem, namely when the ini-
tial condition of the fast variable, yo, is 0 ( 8‘ B ) ( B is a positive integer).

By Tikhonov’s Theorem [Kokotovic, et.al. (1986)], x80) -) x, (t) as e -9 0+
uniformly on compact time intervals. Hoppensteadt (1966) generalized Tikhonov’s
result to infinite time intervals. We quote a result of Saberi-Khalil (1984) which is a

nonlocal version of Hoppensteat’s result. For the sake of clarity, we will closely
follow the notation of Saberi-Khalil (1984).

Assumption 6.1: There exists a Lyapunov function V : R" -> R... such that
the following inequalities hold for all x e (I:

67

%:if(x)s—Bolx I2 (6.3)
13—:1 sBllxl (6.4)
lexIIZSV(x)SB3Ix|2 (6.5)

where [30, I51, B; and B3 are positive constants, and Q c R" is an open con-

nected set containing the origin.

Assumption 6.2: The following inequalities hold for all x e 52, y e 2 and

66 [0,6]:
If(x.N(e)y)I s0. Iy I (6.6)
Izowcml sflslx I +Bclyl (6.7)

where B4, 05 and 06 are nonnegative constants, E is a positive constant, and

E c R'" is an open connected set containing the origin.

Let P be the symmetric positive definite solution of the Lyapunov equation
PA +A’P =-I,,I and W0) :=y’Py. Choose vo>0 and wo>0 such that

LR :={x e R" | V(x)Svo} c Q and (6.8)

LB :={y e R'" I W(y)$wo} c 2 (6.9)

and define the set L as

L :={(x,y)e 11"me I V“) + W0) 51 } (6.10)

 

V0 “’0

68

Theorem 6.1:]: [Saberi-Khalil (1984)]: Under Assumptions 6.1 and 6.2, there
exists 8" > 0 such that for all e 6 (0,8*) the equilibrium point
(x = 0 , y = 0) of (6.1) is asymptotically stable with L inside the region of
attraction. Moreover for every initial condition (xo,yo) e L, xe(t) —> x,(t) as

8 —9 0+ uniformly in t on [0.00).

In Theorem 6.1, the initial condition (xo,yo) is bounded, uniformly in e,
i.e., it is 0(1). However, from (3.8)-(3.10), when the initial condition of the esti—

mation error e (0) is 0(1),the initial conditions of some of the components of ef

are in general 0 (—1—). The following example shows that in such cases Assump-

eIt—r
tions 6.1 and 6.2 are not sufficient for the convergence of 18(1) towards x, (t) as

e-)0.

Example 6.1: Consider the singularly-perturbed system
12 = -x + y x0 = 1

. .11
I 5y =—-y (6 )

 

1
“:2?

L

which is asymptotically stable for all e > 0 and satisfies Assumptions 6.1 and 6.2
globally. The solution of (5.3) is

 

t The statement of this theorem can be strengthened to IE“) -x,(t) =0(8), using
results of Hoppensteadt (1966). However, the extension will take some space that might divert
attentionfromthemainpointofthisthecis.

69

r

1
e(1—e)

 

x,(t) = e" + (c-‘ - ("8)

-tl£

ya“): 3

.1.
£2

 

while the solution of the slow subsystem 12 = -x, x(0) = 1 is x, (t) = c“. It can
be easily seen that on any compact subset of (0, co), x80) - x, (t) diverges as

8—)0”.

Therefore, we need to develop a trajectory approximation result for system
(6.1), in the case when the initial condition of the fast variable, y , is 0 (6‘5).
From the previous example, it is clear that some additional conditions must be
imposed, if a trajectory approximation result is to hold in the case when yo is
0 (6‘5).

Assumption 6.3: The following inequalities hold for all x e (2, y e R’" and

Be [0,E]:
lf(x)l + If(x,~(e)y)l so,” I +0, (6.12)
|§(x.N(e)y)| 539'): l +Bto|y | +611 (6.13)

where Ba, [39, 510 and I311 are nonnegative constants and B7 is a positive con-
stant.

Inequality (6.12) is a restrictive requirement, because the right-hand—side is
independent of y . This is a requirement that one would not expect to hold in a gen-

eral singularly perturbed system, but it holds in our application when the control is
bounded. Let

70

ER :={x 6 LR | V(x)Sv1 } (6.14)
[:3 :={y e R’" I Iy II S 13—} (6.15)
8

where v1 5 (0,v0), ye (0,00) and Be N arechosen arbitrarily.

Theorem 6.2: Under Assumptions 6.1-6.3, there exists 2 > 0 such that for all
as (0,E) the equilibrium point (x =0,y =0) of (6.1) is asymptotically
stable with ER x [3 inside the region of attraction. Moreover for every initial
condition (xo,yo) e [R x53, x80) —)x,(t) as a -)0+ uniformly in t on

[0,”).

Proof of Theorem 6.2: Since A is a Hurwitz matrix, there exist positive con-

stants K and 0.1 such that

I c"”8 I sit (“M for all t e R,. (6.16)

Claim 1: For every n>0, there exist b0 such that for all e 6 (0,2) the ori-
gin of (6.1) is asymptotically stable with ZR XZB inside the region of attraction.

Moreover,

28 1
lye(t)IS11 foralltzal InIEMI

Proof of Claim 1 : Given n>0', let
C == min { fizz-mica”) . 3(1- 21)} (6.17)
2 V0

and

71

S :={(x,y)e L I W(y)=y’Py SC} (6.18)

where P , v0, wo, and L were introduced after Assumption 6.2, and v1 6 (0,v0)

was introduced in (6.14). Then it is easy to see that

Vo-
(x,y)eS : lxIIS E- and IyISn (6.19)

Moreover, for all (x,y) e S and e 6 [0,2]

W = 2y’P[ % y +§(x.N(t-:)y)]

s—ily l2+2IPI Iy “6511:1443,” I)

..___.“1__ 1’10. 2
s ; (P)+285lPln fiz+286lPlln

=_—S?7+2051p In‘\/:—‘:+2B6IP 1112 onH

where

H :={(x.y)e S I W004}

Therefore, W<0 on H for sufliciently small 8. By Theorem 6.1, the set L is
an invariant set for sufficiently small 8 . Therefore, the set S is an invariant set
for sufficiently small 8 . Moreover, by Theorem 6.1, the origin of the closed-loop
system is asymptotically stable with S inside the region of attraction. To conclude
the proof of Claim 1, it suffices to show that

72

28 1 28
I‘“? ‘" :tTII’ ”437‘“

l

 

 

1
WIN] 6 S

(6.20)

for sufficiently small 8. The idea is to show that for all initial states in the set

LR XLB, y£(t) decays rapidly towards S, while during the same time xe(t) can

not grow out of S , due to inequality (6.12). To show this, we start by calculating a
worst-case bound on the growth of 1:80). By (6.1), (6.3), (6.4), (6.5) and (6.12),

we have

‘1 __Z_-__!_§X "
dtW’leV’zth ax If+fI

sEfiW+ 5138

282 2432'

Solving the above differential inequality for J17 yields

 

on LR me

)IV_(}_)S[‘IVW+a]eb‘—a

 

where
a = 58452 b = 51—97
B: ’ 232
which implies that

\IV—(t_)S[\/Vl'+a]eb‘-a

It can be easily seen that

 

‘IV(t)S\I0.5(Vo-I-V1), forall rsw

(6.21)

(6.22)

(6.23)

(6.24)

73

where

a + 0.5(vo+v1)

a+~lv_t

 

1
=—1n
‘V b

By (6.5) and (6.24), we conclude that
Ix (t) I25-—1—(v0+v1) forall t 5w
8 2B2

which by (6.13) implies

lg (xs(t) ,N(8) ye(t)) I 5 I310 lye(t)I + [312 forall t 51);

where

 

112
I312 3: 59 [V0231] + I311

Now, the solution of (6. 1) is given by

l .
y.(r> = emyo + j e‘M" §( x.(o. Noam) ) dc
Therefore, by (6.16). .we obtain

I y,(t) I s K I y0 I c'“*"‘

f

'I' {K 84110-45)“: [BIO 'yECt) I + 512] d’t

for all t 6 [0y]. Multiply (6.30) by cw" and let

z(t) := came I ye(t) I

(6.25)

(6.26)

(6.27)

(6.28)

(6.29)

(6.30)

(6.31)

74

Then,

I

8K
z(t) SK “yo I + 783' (calm: "1)‘I'£I{Blot"-(Ti)d’r (632)
1

Application of the generalized Bellman-Gronwall inequality [Hahn (1967)] to (6.32)

and use of (6.31) imply that

“1
(KB -—)t
IyE(t)I<Kly0|e ’° E

 

a.
K5128 1 (Kate " 'fi)‘

 

a1 _ EKBlo - e (6.33)
for all t E [Day]. let
0‘1
£2 := 2K510
Then
I yea) I 5 Kyle-8 e" was + E—K—Bl—ZE— (6.34)

ml

for all 8 6 (0,64) and t e [0,\|I]. Furthermore, since 8 1n [fi] —> 0 as 8—)0,
8

there exists 83 6 (0,62) such that

 

 

28 1 .
;1— In 31-] 5 W (6.35)
and
1
2K 3123 g 3
K‘fi+ 011 SI} (P) ] (6.36)

for all 8 6 (0,83). Therefore,

75

 

1
_2-KI312E g 7
”4:8 1n [ I ) I SK'YE-l- 011 SI? (P) J (6.37)

for all 8 6 (0,83). By (6.24) and (6.35)

V( is: In [— 8311] ) s 0. 5(vo+v1) (6.38)

for all 8 6 (0,83). Finally, from (6.37) and (6.38), we can see that
28 1 28 1
[1431' In w]).)‘e(';1n[éfi]):l6 S

for all 8 6 (0,83), which concludes the proof of Claim 1.

 

The first part of Theorem 6.2 clearly follows fiom Claim 1. To prove the uni-
form convergence result, let (8,,) be a positive sequence such that 8,' -)0 as

n—9°°. By Claim 1, there exists §>0 such that for all 8 6 (0,8), xe(t)—)0 as

t—)°°, and %[— + -W—] is negative definite along the trajectory uniformly in
V0

8. Therefore, given §>0, there exists M e N and T>0 such that

Ix,_(t) —x,(t) I <§ (6.39)
for all t 2T and n 2M. Next, we will show that x£.(t)—)0 as n—)oo uni-
formly on [0, T].

Claim 2:

l

£f~(xs.(‘t)’N(en)ye.("))d“-90 asn—wo, uniformly on [DJ].

Proof of Claim 2: Given §>0 , by Claim 1, there exists N 1 e N such that for all
n 2N1, the origin is asymptotically stable and

76

 

_5._ Eu. 1
Iy€.(t)| s 2Tl34 forall t2 “1 1n 83+] (6.40)

Therefore,

I To

I£i(x,'(t),1v(c,)y,_(t))dtl sflo, Ix,‘ I +08] d1:

t
+1] B4ly£.(’t)l at

_ T
5 Tu [EVE 4' I38] + 77?:

where

22,, 1
Tu .— '31— ln [$1.]
and we used (6.12) on [0, Tn] and (6.6) on [Tm t], to arrive at the above inequal-

ity.
There exists N 2 e N such that

28.. 1 7
$111 [2:53] [mV‘B—g‘hfls] <§ forall "ZNZ

Therefore,

I

I {f ( x40. ”(En)y£_(‘t) ) at I (g

for all n2 max{N1, N2} and all t e [0, T], which concludes the proof of Claim

2.

77

By Claim 2 and continuous dependence of solutions of differential equations [Hahn
(1967)] we can show that

xe_(t) —) x,(t) uniformly on [0,T]

This fact together with (6.39) show that x80) —) x, (t) as 8 —> 0" uniformly in
t on [0, co). CI

6.3 Stability Result

In this section we apply Theorem 6.2 to prove that global boundedness of the state
feedback component of the observer-based control prevents the destabilizing effect
of the peaking phenomenon.

Theorem 6.3: Suppose that Assumptions 2.61, 3.62, and 4.63 are satisfied,
and that a(.), B(.), and (p(.) are globally bounded. Then there exist positive con-
stants d1, d2, and 8 such that for all 8 e (0, 8) the origin of the closed-loop
system (3.3) and (4.1), with L (.) given by (4.6), is asymptotically stable with the

set
R :={(z,e)e R2" I W(z)Sd1, Ie I Sdz} (6.41)

inside the region of attraction.
Proof of Theorem 6 .3: The proof follows as a corollary of Theorem 6.2. Since
W (2) is a quadratic Lyapunov function,

nllzIIZSW(z)Sn2|zI2

3W
'5' Sfl3l2l

78

for some positive constants n1, 112, and 113. The set S of Assumptions 3.62, and

4.63 is an open set containing the origin. Therefore, there exist r>0 such that

{zeRpI IlzII Sr}cS (6.42)
Let
,2
21:: z e RP I W(z)$n1-9— (6.43)
o- , r2
2Q .-{e, e R1 I es P38, 5 1min(Ps) m} (6.44)
V(z, c,) := W(z) + d e,'P,e, d>0 (6.45)

There exist vo>0 and 2? >0 such that for all d 2 a?

Q :=‘{(z,e,) 6 RP” I V(z,e,)Svo}c21qu (6.46)
The derivative of V along the trajectory of the slow subsystem (4. 18) is
% = %[A2 +B¢(z) +BB‘l(z) [A01+ AB 0(2)]

+BB‘1(2) [F(z-I‘1e,)-F(z)]] -d '8, I2 (6.47)

By (3.9), Assumption 4.63, and (6.46)

£15-72” l2+n3kngl II‘II lzl |e_,|
-dle,l2 0110
as d
S-T'Z lz-Eles'z

for sufficiently large d . Therefore, for sufficiently large d

79

fl -- 31 z 2
dis m{2’2}'[e,]l on!) (6.48)

which is inequality (6.3) of Assumption 6.1. It can be easily seen that inequalities
(6.4) and (6.5) of Assumption 6.1 are also satisfied. To prove that Assumption 6.2 is
also satisfied, let

2 i={ef E Rn-l I ef’Pfef S (00} (6.49)
that!" ) ’2
m“ P f 9 I I‘2 I 2

The sets 21, 212, and 2}, are chosen such that for all (2, 8,, cf) 6 Elxrqxfg, 2‘
satisfies the bound:

lfl = '2 -1383 -1'2N(8)ef I
s lzl+ 11:11 Ie,|+ II‘zl Ilefl
r r r _
$23-4- 3 + 3 —f
where we have assumed that 851, so that IN (8) I 51. Then, by Assumption
4.03, (6.46), and (6.49)
I 334(2) ([4 + A8) [F(z -I‘,e, -F2N(8)ef)
—F(z—I'1e,)]|Sk6IBI lr2N(8)ef I

which implies inequality (6.6) of Assumption 6.2. To prove inequality (6.7), note

that g of (6.1) is given by

§( (2 ,e, ), N (8)8f ) = B, [0,8, + DfN (8)8, + 1‘5,le 3)] (6.50)

80

By Assumptions 3.62 and 4.63, (6.46), and (6.49)
I A(Z,£) I S(k1+k3+k2k5) I Z I

+ (ks-Pk6) I I‘le, + F2N(8)ef I (6.51)

for all (z , es) 8 Q and cf 8 2. This, together with (6.50), proves inequality
(6.7). Therefore, Assumptions (6.1) and (6.2) are satisfied. It remains to show that
Assumption 6.3 is also satisfied for system (5.1), under the assumption of global
boundedness of (IL), B(.), and (p(.). To show that Assumption 6.3 is satisfied, note
that by Assumptions 2.61 and 63

I A2 +B¢(z) +B§(z, (11(2), t) +

BB-1(z)(lq +Ag) [F(z-I‘1e_,) —F(z)] I s B7 I [Z] II
and by boundedness of F (.)
IBB‘1(z) (14 + As) [F(z-I‘1e,-I‘2N(8)ef) —F(z-I‘le,)] I 5 B8

for all (z, (2,) e 9, and of e R"", for some positive constants B7 and B3,
which implies inequality (6.12). Inequality (6.13) follows similarly from Assumption
3.62, (6.50), (4.3), and boundedness of F (.) and (p(.). Therefore, all the conditions
of Theorem 6.2 are satisfied. Choose v1 6 (0, v0), 7>0, and B=K-1, where K is
the integer introduced along with transformation (4.4) (K can be viewed as the
relative degree of system (2.2)). Then, by Theorem (6.2), there exists 8>0 such that
for all 8 e (0, 8) the origin is an asymptotically stable equilibrium point with the

SC!

81

Pr-

2

ZRXEB =1 e, 6R2” I V(Zres)svlr Ief ' Si?
cf

5 n d

 

 

 

inside the region of attraction. By (4.7), (4.9)-(4.11), and (6.45), it can be easily seen
that there exist d1>0, d2>0 such that

R CZRXZB

which completes the proof. El

Example 5.1 (Continued): Let us apply the following globally bounded control
to the pendulum example of Chapter 5,

 

L u = - 2.0 sat§(£1+£7) , §= 0.01

Figure 6.1 shows results of the simulations ' with e=0.01, and all the initial condi-
tions the same as that of Figure 5.2. Note that although the peaking is present in the
estimation error (Figure 6.1c), there is no peaking in input u or the states of the

plant.

Example 5.2 (Continued): Now let us apply a globally bounded control to
Example 5.2 of Chapter 5. We use a saturation nonlinearity to bound u and v.

The observer-based control in this case is

_JL

82

z. 2
xl=f2+'8"()’ ‘f1)

- 1
J? =v+— —f
2 62.0 1)

u =sato[-3f§ ]+v

 

flutter-22,42]

(6.52)

Figure 6.2 and 6.3 shows the results of simulation for the closed-loop system with

0:01 , 8:0.001, 0:10, and the same initial conditions as in Frgure 5.5. The

closed-loop system is stabilized in this case, due to the fact that the saturation non-

linearity acts as a buffer protecting the plant from the impulsive-like behavior of the

observer. Figure 6.2 shows the behavior of the closed-pr system within the boun-
dary layer.

(a)

(b)

(C)

 

 

o-
-1...
-2.
8”
-3-
- - - state feedback '
1 output feedback
4 0 Ir 2 i l
a, '
I-
O-t
-1‘V)
I
- - — state feedback
output feedback
0 i 2 5 1
3
, .
i i 3
6

 

83

 

 

 

 

 

 

 

 

.4

Figure 6.1- Example 5.1 with bounded control

84

BOUNUED OBSERVER-89550 couraot
THETR=0.l. SAT-1.0
EPSl-0.001

 

 

 

 

 

 

“Ml!

 

 

 

 

 

 

 

 

 

I
I
I
I
I
I
I
I
2
O :
. a
N8 -r 1
v I :
LIJ .
O
O J
Y
I
O
—
O . T 1'
I o .
I o '
o o .
I I I
I I I
I I I
I o .
I | .
I o .
I o .
__ 8 . . .
e ' I g
#96 M Y I I
a : :
I“ l l g
' I
I
o
c
o
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o
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O
I

 

 

0.1!)

 

 

 

 

‘t‘ E
23 E E 5
.—0. k I ' o
‘" l : :
‘3 i . :
.. ; :
'7'0.00 0.20 '40 0.00 *1} 00

IthuneILZHThuuapmofihaioftheinaurvarhdflesfbr
Example 5.2 with bounded control in the boundary layer

85

BUUNDED OBSERVER- BASED CONTROL
THETH=0. 1 591:1. o
8 EPSI= 0. 001

 

 

 

 

 

 

 

 

 

 

 

 

T i T
I I I
I I I
I I I
I I I
i 3 i
I I I
I I I
I I I
I I
8 . .
. I 0
2° ' I I
i i
I I
I I
u i i
I
8 i
I
“'- L J.
I
d *w ea:
I
a
N8 I r 1 l 1
v I
LU
O
$.15 J
n
9.
61 ..... T #-

 

 

 

 

 

 

EE \\\\‘—’//V//”r i 5 g

 

 

 

 

 

0.00 2'.oo 4.00 5.00 8.00 10.0

Figure 6.3- Time profile of the state variables
for Example 5.2 with bounded control

7 Future Work

There are a number of performance and robustness issues that could be the subject

of future work on this problem. We briefly go over them.

Robustness to Unmatched Uncertainties

Our output feedback control scheme is robust with respect to parametric uncertain-
ties that satisfy the matching condition. An important robustness issue is assessing
the performance of the closed-loop system, in the presence of other kinds of model-
ing uncertainties, such as unmatched parametric uncertainties, and unmodeled high-
frequency dynamics. Since the closed-loop stability results of this work were proved
using Lyapunov theory, it is clear that unmatched uncertainties and unmodeled
high-frequency dynamics would not destroy closed-loop stability as long as they are
sufficiently small. Therefore the purpose of such a performance assessment should

be to find quantitative bounds on how small such uncertainties should be.

Semiglobality

The notion of semiglobality was first inuoduced in Sussmann-Kokotovic (1989) in

connection with stabilization of nonlinear systems via state feedback control. In the

86

87

work of Sussmann-Kokotovic (1989), the definition of semiglobality is motivated by
the fact that the conditions that are needed for global stabilization are usually very
stringent. On the other hand, when a nonlinear system is locally stabilized, the
designer in general has no control on the region of attraction of the system. The
notion of semiglobality is a compromise between these two extremes: Given an arbi-
trary bounded set B in the state space, under what conditions is it possible to find
a controller that renders the origin of the closed-loop system asymptotically stable,

with the set B inside the region of attraction?

In the case of our problem, the question of semiglobality can be raised in the
following context: If Assumptions 2.61, 3.62, and 4.63 are globally satisfied, then
Theorem 4.2 ensures global asymptotic stability of the origin. However, It is very
restrictive to assume that functions F (.) and 4)(.) are globally Lipschitzian. On the
other hand, by assuming that Assumption 4.63 is only satisfied on a compact set
around the origin (which boils down to assuming sufficient smoothness of F (.) and

(p(.) ), Theorem 4.3 gives an estimate of the region of attraction which shrinks as
8 tends to zero, due to the peaking phenomenon. In Theorem 6.3, we isolated the
peaking form the plant, and hence, were able to obtain a region of attraction which
was 0(1) large. Now, suppose that we have a static state feedback control that
renders the origin of the closed-loop system globally asymptotically stable. Given
any bounded set B c R2”, is it possible to find an observer-based control that
renders the origin of the closed-loop system asymptotically stable, with B inside
the region of attraction? We believe that the singular perturbation result of section
6.2 can give conditions under which semiglobality can be obtained, at least in the

case when there is no zero dynamics.

88

Input/output Linearizable Systems

Most of the ideas of this work can be generalized to the class of input-output linear-
izable systems. At the moment, the main obstacle to such results seems to be the

lack of appropriate normal forms for input-output linearizable systems.

APPENDICES

Appendix A

Proof of Proposition 3.1

Given 8, without loss of generality, assume that
Ne(Qo) C 9r

Claim : There exists 5>0 such that 120,, c 115620).

Proof of Claim: Suppose not. Then there exists 8 > 0 such that for all n e N,

there exist 2,, e 90“," such that

2,, d N490) : d(z,,,Q°)Ze , for all n (A.l)

Moreover 2,, 6 904.1," C 90+? 904-1 is compact : there exists a subse-

quence of (2,. ). say (2,“), such that 2,“ -9 z, for some 2.
Now since 2,, 6 am," ~90, it follows that o<W(zn)So+% . Hence
W(z ) = limW(z,,‘) = o :- d(z,flo) = 0.

k-yoo
But by (A.l) eSklim d(z,u,flo) = d(z .00) = 0; contradiction.

—).o

By Assumption 2
W s—yzlz |2+71Izl+yo:=g(z)

89

Considering g (2) as a quadratic term in I z I , it is easy to see that

1

2 ‘2—
g(z)<0if I2 I > .Y_‘+ 7—12,.3’.
272 472 72

:- g(z)<0 for allz e {z |W(z)>0'}=Q§ (A-Z)

Let E 2: ac -' 904,5.
If E =E, then 2(t) 6 am for all tZto, since QM is an invariant set of the tra-

jectory.
So suppose Esta Let F :={g(z) |z e 5}. Since 5:93, supF SO.

Claim : sup F < 0.
Proof of Claim: Suppose not. Then sup F=0. i2- is compact and g (.) is continu-
ous an F is compact a 05F :- there exists 2 e ECG; such that

g (z ) =0 which contradicts (A.2).

So W S-a for some a>0. Let T +20 be the first time the trajectory enters (204,5.

Then

0+8 T'Ho

{dWS-Iadt ,TSS;::_5
to .

Moreover since QM is an invariant set of the trajectory

z(t) e QM c N400) c NMQO) for all t 2T+to CI

Appendix B

More on Special Coordinate Basis of Section 4.2

In this appendix, we give a simple explanation of how to arrive at (4.5) for a special
class of systems. Suppose that the transfer function matrix P (s) := C (s! —A )‘l B

is square, invertible, minimum-phase, and has a left diagonal interactor
D(s) := diag (sa‘, . . . , sa"), i.e.,

lims_...D(s)P(S) =L (B.1)
where a,- ’s are nonnegative integers and L is nonsingular. Write P‘1(s) as
P‘1(s) = Q(s) + R(s), where Q(s) is a polynomial matrix and R(s) is a strictly
proper transfer function matrix. P(s) can be written as

P = Q‘1 (1 +RQ‘1)‘1

which implies that P (s) can be represented by the negative feedback connection of
Q'1(s) and R(s) with trim in the feedforward path and R(s) in the feed-
back path.

By (13.1), 1im,_,., Q(s)D-1(s) = 1:1. Hence, Q (s) is column-reduced, <1,-s
are the column degrees of Q(s) and Q (s) can be written in the following form
Q(S) = £7le + thK(S)

91

92

where
K(s)=BIock Diag [(1,s,...,s°“")’]

Therefore, a controllable-form realization of Q’l(s) can be obtained by the
coprime fraction method [Chen (1984)], in the following form

{if = (Aw - BcoLQk. )x, + BcoLu
y = waf

where

0 IGH
A00 := BlOCk Diag [ o 0 ]
B“, := Block Diag [ (0, . . . , O, l)’]
Cw := Block Diag [ (1, O, . . . , 0)]

Let (C,,A,,B,) be a minimal realization of R(s) (Since P(s) is minimum-

phase, A, is Hurwitz). Then P (s) has the following realization,
f
&=&&+%y

1 if = (Aw -BcoLQlc)xf + BcoLu -BeoLCsxs

I

 

y = Coo xf
Now it is easy to see that if the components of x, are interchanged such that

integrator chains of the same length appear in the same block, this realization takes

the form of (4.5), as a special case where 52,, does not exist and M f = 0. El

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