Asymptotic Output Tracking for a Large Class of Uncertain SISO Nonlinear Systems: Stability†

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

ASYMPTOTIC OUTPUT TRACKING FOR A LARGE CLASS OF UNCERTAIN SISO NONLINEAR SYSTEMS: STABILITYt Z. Retchkiman Celltro de IlIvestigatioll y de ESlUdios Avallzados deIIPN. . DeplO. de IlIg. Electrica. Sectioll de COlltrol AII/omatico. Apdo. Postal /4·740. M exico. D.F. M exico

ABSTRACT. The problem of asymptotic output tracking in the presence of uncertainties is reexamined . A characterization of the class of uncertainties dealt with in this study is given . A study of the asymptotic output problem plus stability for the class of uncertainties allowed is also discussed . Then we proceed to make a generalization to the case when there are singular points. Finally an example that illustrates the concepts proposed is presented. KEYWORDS. Asymptotic output tracking • uncertainties • non linear systems • robustness . control. stability. singular points. Remarks:

INTRODUCTION.

I.

!l The existence condition for T such that cS f and

The asymptotic output tracking problem consists mainly on the design of a control law that causes the output of a given system converge to a certain desired trajectory . In this paper. we study the asymptotic output tracking problem with stability of a desired trajectory for SISO non-linear systems when the model of the system is not precisely known i.e. . contains uncertain elements . Under such imperfect knowledge of the model we try to design a control law such that the output of the model will converge to the desired trajectory and the state variables will remain bounded. The purpose of this paper is to present in a unified way the asymptotic output problem with stability for a broad class of uncertainties . This paper is organized as follows . In section 11 we characterize the class of uncertainties dealt with in this study. In III we solve the asymptotic output problem plus stability. In section IV the singular points' case is treated . Section V gives an example and section VI gives some concluding remarks. SYSTEM

H.

cS

xeto)=x

Ill.

f

I

4)

ZI

Z

Z

Z

2

ZU-I Z

Z

Osk';u-l and all

XE

't'xER

n (3)

qe/;.1/) ; /;=(ZI ... · .ZU ) Z

where :

I

aex)=L LU-I hex ) , r

b(x)=L>ex).

and

qe/; .1/)={L t eT-I(Z))}n

LcS L~

,

1

1

1=(l+1

Now by applying the feedback control law

gex) }.

and/or

3 d

Y

r

hex)

2

1/

,

f

(2b) 0 as t .. ",.

b(x) + aex)u

Cf

,

LcS L~

I ..

e2a)

By dropping the stars for sake of simplicity and applying the change of coordinates given by T ={ I U-I dh.dLfh . ... . dL h.tU+I . . .. t }(x). (2) can be written f n Osidori 1989)

ti!:t . o k 2) LcS L hex)= 0 and LcS L~ hex)= 0 for Osk';u- l and all f

, E span{

is

,

y-et)=h-exet)) we will assume that y-et)-yet)

,

f

that It

all ti!:t . o Then (I) can be written as x-et)=f-exet) )+g-exe t) )uet).

1l 3 T=Tex) state transformation and Mex.Tex)) >O such that : cS and cS EKed dT } and 1/ T(x)-x 1/,; M for all

XE Rn . 3) cS and cS

assuming

involutive.

CASE (I) 3 T=T(x) state transformation and Mex.Tex))>O such that: cS and cS EKed dT } and 1/ Tex)-x I/s M for

where toi!:O. xcR. f . g. cSf=f-f. cSg=g-g. ewith f and g

f

by

and

ASYMPTOTIC OUTPUT TRACKING AND STABILITY.

denoting the estimates of the nominal values of f and g respectively) . are smooth vector fields on Rn. hand u are smooth real functions; for uncertainties .e cS f and cS ). that satisfy one of the following conditions:

f

be assured

nonsingular

Consider the system (I), ewith u
(lb)

o

} can is

also important to point out that the norm 1/ Tex)-x 1/ n2 is computed on tt. where Tex)clR - is immersed in the natural way on to Rn. W (3),*e2) eup to u-2) This follows from the definition of the relative degree u. but (2) (up to u-2) does not imply (3). unless u=n.

In this study we present the robustness and stability of a SISO non-linear system. ewith relative degree equal to u). of the form xet)=fexet))+gexet))uet)+ cS fexet))+cS , exet))uet) (la) n

dT

A=span{cS f'cS /

DESCRIPTION AND CHARACTERIZATION OF UNCERTAINTIES.

yet)=hexet)).

,

EKed

hex)

are

not

zero for

U

(U)et)-bex)- L C

IRn.

uet.x)=

t This work was done whlJe the author was at the Dept . of Control of Dulsbur, ', University. (Germany ) ,

Yd

27

being

yd

I =1

ez _yll-H) I-I

d

't'xER

a(xl the

desired

trajectory

to

be

n

(4)

tracked.

substituting (4) into (3) we get a closed loop system of the form : 0 .... 0 0 Z I

l!J{U.

+ I

-ca-I

11

q(~.lI);

y

Z

Z

C a

a-I 1; C 0) (a) OYd+1 =1 IYd +Yd

., .Za ) for all xcRn.

~=(ZI"

(5)

.

I define

e (t)=Z _yO-1) i=I •...• a If we where e is I I d I the so called tracking error the system given by (5) can be rewritten as e e

0

0

I

0

2

0

0

e

0

e

1)

L(~.1)+M(~.ll)+K(~

y

Z

n VxelR US.ll)+M(s.ll)+K(s .ll)U .

I

Defining -Z _ 0-1) - I Yd'

I 2

i=I . ...• a

e -C

a

-C

0

-C

I

e

a-I

a

e

where l;=coHy .y' •. . . Ja-I) & e(t)=col(e .e •• ••• e_) n d d d ..... l 2 We can pick {Co'CI •.. . •C in such a way that the a roots of the polynomial slT.+C sa-·+ ... +C =0 all have Cl-I 0 negative real part. this assures to have el"O as t ......

e

_.>

X

n

solution. denoted by lI (t). of R

it = q(l; (t).lI) t~O.

satisfying lI (O)=O is defined for all R

of

~

=

plus

stability

for

the

of

1.

Suppose

• (a-!l yd' yd.· ... yd

q(l; (t),lI).

satisfying

original

are

11 (0)=0. R

!Z.'tO)_y~I-I)(to)1 < a

Suppose

I

2

UlI(t )-lI (t )II<,s .. o R o

VxelR

AX

I

VxelR

n

Yd,y~ •. .. •y~a)

and following the same philosophy as

cs,

e span{ g(x) }. that is S.t. ,s,=gd ;d"-I .(l2a) (12b)

get Z Z

2 J

Zd b(x)+ Ea(x)+ (d+lla(x)u is

VxelR

n (13)

; S=(ZI. ··· .Za )

y = ZI

Now consider the following control law Cl y(a)(t)-b(x)-Ea(x) -1; C (Z _yO-I) u(t.x)= _d_ _ _ _ __ _ --=I_=..:I--=I_-:..I--=I_.:.d_ _ (d+1 )a(x)

CASE (2) L,s L~ h(x)= 0 and L,s L~ h(x)= 0 for Osksa:-I r and all XE Rn. With TI as in case (ll and taking into account the

,

Then by O_~)ubs~ituting e (t)=ZI-yd 1=1 •.. . a. l

(14) we

(14)

into (13) and defining arrive to (6). Thus

proceeding in the same way as was done in case (I) and (2) we are led to asymptotic output tracking plus boundedness on the state variables .

previous condition. system (I) can be written

k

k

CASE (4) L,s L h(x) and/or L,s L h(x) are not zero for f f r c n Osksa:-I and all xelR Then applying TI we get:

VX~n

~ Z

(10)

n

Then applying TI plus condition (12) to system (I) we

l!ll(t)-1)R(t)!!
ZI = Z 2 Z2 .., Z J

a

3 E=E(,sr) real valued function S. t. ,sr=gE .

this

(5)

e

S=(ZI' ... • Za)

I

q(s.1) system

-Ca-I

I

3 d=d(,s,) real valued function

defined

IZI(tO)_y~I-I)(to)I<,s .. IZI(t)_y~I-I)(t)l
(8) d

b(x)+a(x)u

11 = L(~,1)+M(~.lI)+K(~.lI)u

Z1

~=(ZI· · ···Za)

Z2

y .. ZI

where:

Y

q(S.ll)

I,; i sa:

1I 11 (t o)-lI R(t o)1I < a The corresponding response of the bounded. More precisely Vc>O 3 ,s>O 3

Za

q(s.1)

Z

negative real part. Then for sufficiently small a>O if

a-I

e

in case (1) we are led to asymptotic output tracking plus boundedness on the state variables.

solution is defined for all t~O, bounded and uniformly asymptotically stable. Fjrally ~Rpose that the roots of the polynomial s +Ca_IS +... +Co=O all have

Z

e

where A is a matrix whose eigenvalues can be arbitrarilly assigned by a proper selection of CO.C I•. . .• Ca +/3_.. Thus for sufficiently small values

and bounded. Let llR(t) denote the solution n

-C

0

Z

CASE (3) ,sr and

t~O

-C II

y

bounded and

uniformly asymptotically stable. we can employ the following proposition (Isidori 1989). in order to get asymptotic tracking plus boundedness on the state variables of the system given by (5). but by assumption ! y·(t)-y(t)! .. 0 as t ..... and 3 M(x. T(x»~O such that 11 T(x)-x lis M for all t~to so we obtain tracking

a

II (7)

n

PROPOSITION

o

2

o o

o

o o

I

or. equivalently.

i.e. asymptotic output tracking. Now by assuming that l; Is defined for all t~O and bounded and that the

for all

I

X(t)=col{e (t) •. . . •e a (t)). l

and

q (e(t)+l; .11)

11

asymptotic system.

(9)

e(t)

the system (9) can be written as:

(6)

e

.1l)U

b(x) a(x) L(~.lI)={Lt • I (T-i(Z))}n I' 1 l =Cl.

are

as

in

(3)

and

Za-I 28

Z2+ L,sh(x)+L,sh(x)u r c Z3+ L,sL r h(x)+L,sL r h(x)u VxelR n r a _2 & Cl-2 ZCl+L,sL h(x)+L,sL h(x)u r r r

I

(15)

ZCl

b(x)+a(x)u+L.sL

a-I

r

Cl-1

h(x)+L.sL r

r L(~.lI)+M(~.lI)+K(~.lI)u

11

(

cn:cns ... . ,cn. ; C·,CS, ... C' where are real o I Cl-I 0 I Cl./3-1 coefficients and the change of coordinates (21) has been used (Retchkiman 1993). The conditions on the uncertainties remain the same. (cases 1 and 3). for the cases 2 and 4. the k index runs from to Cl+/3-/.

)

h x U

, ~=(Zl' .. .• ZCl)

°

Y ZI where L(~.lI).M(~.lI) and K(~.lI) are as in (8). Equation (JS) can be rewritten as

rll~IJ+~

°

Z'J=l1

°

zoo l Cl

Z

y

Z

'ixeR

I

Cl

I

V.

In order to illustrate the concepts proposed in this work we consider the following example. Consider the

":::l;:::':':l~:'~[ ~ }",Xl. ',' ""{: ]

J+t(X'U)(J6)

b(x)+a(x)u

n

with

Now substituting the feedback control law given by (~) into (J6) and considering that el(t)=zl-y:l-I. i=I •.. . •Cl.

/;;d =col(y d'y~ •. . .• y~Cl-tl)

(where

are

defined

Then applying the feedback control law (4) u= cos(t)-CO(x -sin(t)); C/O. we are led to asymptotic 2 output tracking plus stability. This is illustrated in the following simulations where p={ xl.exp(-t)) .

ZI Y Now assuming that the zero dynamics of (J7) is exponentially stable and Lipschitz in both of its arguments and that the term t(x.u(t.x)) is of order O(X.u)2 we can conclude asymptotic output tracking plus stability thanks to the next result . 1 1.

PROPOSITION

S uppose

Fig la:

are

10.E-&

sufficiently small. that the zero dynamics of (17) is exponentially stable and Lipschitz in both of its arguments and that the term t(x.u(t.x)) is of order O(x.u)2 then the states of (J6) in closed loop will remain bounded and asymptotic output tracking will be achieved .

o

•

The proof of this follows by adapting the one given in theorem 3 .4. (Hauser et.al!.1989) . IV.

I

Z2

_=yd-y ~

A

"

{\

-10. E-&

o

THE SINGULAR POINT CASE.

The main purpose of this section is to present a brief discussion of how all the results previously obtained can be generalized to the case when there are singular points (Hirschorn. Davis 1987). In case of having singular points it is still possible to get asymptotic output tracking plus stability by proFrly a.eWying the following change of coordinates I/I:R xR+~ + which takes (X.t)H(Z./;;.t) given by Z

and

1.5

\

V 10

Fig lb:

20

V 30

xl

= hex) = ~rh(x) = LCl-lh(x) r

= a (x)+b (x)u o

0.5

0

= a (x.u)+bo(x)u·+bl(x.u)u l = a (x.u.u· )+b (x.u.u· )u+2b (x.u)u·+b (x)u" 2 2 l o

0'~~

o

__- .______, -____- .

10

20

. (/3-2) . (/3-2) =a/3_I(x.u.-.u )+b/3_I(x.u. - .u )u +- ·· +b (X)U(/3-1) o {t I

(18)

(x)}n I =Cl

+

I

and substituting (4) by

u(t.x)-

Cl (Cl)(t)-a(x)- r Cn. (Z _ (I-I)) Yd I = I I -I I Yd b(x)

Fig I:p=xl

X;Ox

a(x)=l and

1I =t =X . 1 2 I It is easily verified that condition (2) of section II holds; moreover all the assumptions of case (2) are also satisfied for p smooth and u as given bellow.

(J7)

• (Cl) yd.yd •. . .• yd

conditions

One can easily verify that <1=1.

b(x)=l; that is. there is internal dynamics

AX + t(x.u(t.x)) ~=(ZI""'ZCl)

initial

Yd=sin(t).

for all t .. O and bounded). and proceeding entirely in the same way as in case (2) we get

q(~.lI);

EXAMPLE.

•

(J9)

29

30

Fig 2&:

_-=yd-y

2.E-"

o

10

20

30

1 Fig 2b:xl

0.S

0.&

0." 0.2

0t-~--r-----'---~

o

VI.

10

20

30

CONCLUSIONS.

This paper tackles the problem of asymptotic output tracking with stability for uncertain SISO non-linear systems. A large class of uncertainties is considered. Once this is done we proceed to propose a control law which allows us to obtain asymptotic convergence to the desired function to be followed and in addition boundedness on the state variables. The singular point case is also treated. The reader interested in other approaches is referred to the bibliography listed in the references . REFERENCES

Isidori

A. (1989). Nonlinear Control Systems: An Introduction. Springer Verlag, New York . Hahn W. (1967). Stability of Motion . Springer Verlag,

Berlin. Vidyasagar M (1980). Decomposition techniques for large scale systems with nonadditive interactions: stability and stabilizability. IEEE Trans on Autom . Control. 2S 773-779. Selliger,R., and Frank,P,M (1991) Fault-Diagnosis by Disturbance Decoupled Nonlinear Observers. Proc.3Oth IEEE CDC .

Hauser,J.,Sastry,S.,Kokotovic,P (1989). Nonlinear Control via Approximate Input-Output Linearization: The ball and beam example. Proc.28th IEEE CDC . S.Behtash (1990) Robust output tracking for nonlinear systems. Internat.J.Control 51. Liao,T,L., Fu,C., and Hsu,C,F (1992) . Output tracking control of nonlinear systems with mismatched uncertainties. Systems and Control Lett . 18 39-47. Hirschorn,R., and Davis,J (1987) . Output tracking for nonlinear systems with singular points. SIAM J. Control and Optimization. 2S 547-557. Retchkiman,Z (1993). The problem of output tracking for non linear systems in the presence of singular points. Proc ACC.

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