Robustness of Uncertain Dynamical Systems with Delay without Matching Assumptions1

Copyright © IF AC Identification and System Parameter Estimation , Beijing, PRC 1988

ROBUSTNESS OF UNCERTAIN DYNAMICAL SYSTEMS WITH DELAY WITHOUT MATCHING ASSUMPTIONS l Xiong Zhong Kai 2 *, G. Leitmann** and F. Garofalo 3 *** *Ch ongqillg Communication Institute , Chongqing, Sic/wan , PRC ** L'nh'ersity of Califomia, Berkelf)', CA 94720 , ('SA ***Dipartimwto di Irifomwtica e Sistemistica, l 'nj,'ersitd di Xapoli , 80127 Xapoli. Italy

Abstract. The theory of uncertain dynamical systems (Cor less and Leit~ann (1981) and Lei tmann (1978), (1979), (1981)) provides a useful tool for the design of controllers for systems subject to parameter and disturbance uncertainties when only a range of variation of these uncertainties is known. The performance guaranteed by these controllers is the ultimate houndedness of the controlled variables. Moreover, increasing the loop gain, the set of ultimate boundedness can be made arbitrarily small. For the design of these controllers, beside the above mentioned information about the possible size of the uncertainties affecting the system, some structural conditions regarding the way in which the uncertainties enter into the system are required. They are the so-called "matching conditions" . Matching of the uncertainties was assumed in an earlier design of the controller of a plant containing delay in the state variables. The aim of this paper is to show that the contro ller designed in Thowsen (1983) is robust with respect to unmatched uncertainties, provided that they do not exceed an ~ Pciiorl determined threshold value. The technique for doing so is similar to the one eve oped in Chen and Leitmann (1987) for the case of plants without time delays. Keywords. Uncertain systems; delay systems; mismatched systems; deterministic control; Lyapunov theory approach. INTROOUCTION

Two main avenues are open to the analyst seeking to cont ro 1 an uncerta in dynami ca 1 system. He may choose a stochastic approach in which information about the uncertain elements as well as about the system response is statistical in nature; for example, see ~strom (1970 ) and Kushner (1966). Loosely speaking, when model 1 ing via random variables, one is content with desirable behavior on the average. The other approach to the control of uncertain systems, and the one for which we shall opt in the present discussion, is deterministic. Available, or assumed, information about uncertain elements is deterministic in nature. Here, one seek s controllers which assure the desired response of the dynamical system.

Analysis and control of linear or non-linear dynamical systems with uncertainty provide theoretical as well as practically significant problems. In order to control the behavior of a system in the real world, be it physical, biological, or socioeconomic, the system analyst seeks to capture the system's salient features in a mathematical model. This abstraction of the real system always contains uncertain elements; these may be parameters, constant or varying, which are unknown or imperfectly known, or they may be unknown or imperfectly known inputs into the system, as well as state measurement error. Despite such imperfect knowledge about the chosen mathematical model, one often seeks to devi se controll ers which will steer the system in so~e desired fashion, for example so that the system response will approach or track a desired reference reSponse; by suitable definition of system (state) variables, such problems can always be cast into the form of stability problems.

In this paper the main goal is to study the robustnesS of a controlled system with delay when the uncertainties do not satisfy the so-called matching conditions. Since no unique definition of robustness of systems currently exists, it is used here to imply satisfactory performance in the presence of unst ructured uncertainty. We shall decompose the uncertainty into two parts --the matched part and the mismatched part. With the previously obtained knowledge that arbitrarily large matched uncertainty can be coped with by certain classes of controllers, the bound of mismatched uncerta inty and its maximum allowable size will be studied such that satisfactory system behavior can still be guaranteed. Our principal result is to give a general method for the estimation of this bound and its upper limit. We emphasiZe here that, even though this bound and its upper limit are dependent on the specific

1Based, in part, on research supported by the NSF and AFOSR. 2Vis1ting Scholar, IJniversity of California, Berkeley. 3Vis1t1ng Scholar, University of California, Berkeley.

269

Xiong Zhong Kai, G. Leitmann and F. Garofalo

270

controller employed, this general result is applicable to any admissible control.

6f(t,x t ) 6= 6f o(t,x t ,p(t,x t )) 6B(t,x ) 6= 6B (t,x ,q(t,x )) t o t t

SYSTEM DESCRIPTION

-

Let C = C([- r,O], Rn) be the space of all continuous functions which map the real interval [- r,O] into Rn. The function segment x ~ {x(t+S), SE [- r,O]} can be chosen as the t state of dynamic systems governed by functional differential equations with delays in the argument of the dependent variable bounded by r > 0; C is then a state space for the system. Let 1·1 denote the euc1 idean norm and 1 • 1 the sup norm on C given by 1 x~ ~ sup IX(t+S)I. The solution at SE[- r,O] time t of the system equation with initial condition Xto =
1s > O}

•

6B(t,x t ) Q 6B o(t,x ,q(t,x )) t t Substitution from (2) and (3) into (1) yields

where

The related system

obtained from (4) when e(t,x t ) is called the nominal system.

=0

and e(t,x t )

=0

The related system

The non-linear, time-varying delay system being studied is governed by x(t) = f(t,x t ) + 6f (t,x t ,p(t,x t )) + [B(t,x t ) l (1)

+ 6B l (t,x t ,q(t,x t )1 u(t) + Cl(t,x t ) v(t) where XtE Cis the system state and u( t) E Rm is the control input. The n-dimensiona1 vector 6f l (t,x t ,p) models the uncertainty in the homogeneous part of the system which depends on the uncertainty vector pEP; the n x n matrix 6B l (t,x t ,q) models the uncertainty in the system's control input matrix which depends on uncertainty vector q E 0, and Cl (t,x t ) v(t) model s the effect of the input disturbance v(t)ER c on x(t). We assume the following decomposition of 6f l , 6B l and Cl:

+ 6f (t,x t ,p(t,x t )) o 6B l (t,x t ,q(t,x t )) = B(t,x t )6B o(t,x t ,q(t,x t )) (2)

obtained from (4) when e(t,x t ) = 0 will be referred to as the system with matched uncertainties. We now introduce the following definition. Definition 1 A feedback control h(·): R xC + Rm renders the uncertain dynamical system (1) practically stable (p.s.) iff there exist .;!E[O,-) and ro> 0 such that the following properties hold: (i) Existence of solutions: Given (to,
possesses a solution x(·): [to,t l )

+

Rn ,

Xto = • , tl > to • (ii) Uniform boundedness: there exists a positive d(6) < solutions x(·): [to.t l ) + Rn,

1.1" In the general case, the components of the parameter vectors p(t,x t ) and q(t,x t ) are allowed to be continuously dependent on both xt and t. Then define

4When no confusion is likely to arise we will shorten x(to,
(3)

6-

6f(t,x t ) = 6f o(t,x t ,p(t,x t }}

6 ~ IX(t)1 .. d(6)

(iii)

l1iven 6E[O,r o)' - such that for all

xt =. ,of (7), o "tE [to,t l }

Extension of solutions:

Every solution

x(·): [to,t l } + Rn, xt =. , of (7) with 1.1 .. 6, o of (7) with 6, can be extended over [to' -}, . (iv) lIniform ultimate boundedness: Given any t > d and any 6E [O,r o )' there is a T(t,6)E [0,-) such that for every sol ution x(·): [t o '-) + Rn,

1.1"

Robustness of Uncertain Dynamical Systems

I I

= + , of (7), + < 6 .. I x( t) I , £ o V t ~ to + T(£,6). (v) Uniform stability: Given any £ > d , there is a positive 6(£) < ~ such that for every solution x(·): [to'~)" Rn, xt = +, of (7), o '+1' 6(£) .. lX(t)1 , £ 1ft ~ to If ro = ~ then h(') renders system (1) globally practically stable (g.p.s.)

xt

Definition 2 A functional f: [t ,t ) xC" Rn is said to be 1 2 quasi-bounded iff if is bounded on every set [t 3 ,t 4 ] x CB ' where t1 ' t3 < t4 < t2 ' and CB ~ C([- r,O]; B), when B is a compact subset of Rn. The notation of quasi-boundedness is necessary since continuous functionals are not necessarily bounded on closed bounded subsets of R x C.

Lm(t,x(t)) ~ av(~{x(t) + 9!V(t,x(t))[f(t,x t ) + B(t,xt)h(t,x t ) + B(t,xt)e(t,x t )]· Assumption 3.

There exist KoE

[O,~)

and M > 0

(possibly infinite) such that 11(M) > 12(K O) and 1 5(IXI) ~ 1 3 (IXI) - 1 4 (IXI) is non-negative and strictly increasing for IXI E[KO,M) •

MAIN RESULT Now we are ready to state the following theorem. Theorem: Consider the dynamical system (4) subject to feedback control h(t,x t ) and satisfying Assumptions 1-3. Then h(t,x t ) ensures practical stability of the mismatched delay system with r o ' i, d(6) and T(£,o) given by (see Fig. 1) ro =

(Y2 10

1 1 )(M), d(o) = (111 0 r2 )(H) where

6

,i

( -1

11 ° 1 2 )(K O) , 6-1 T(£ ,0) N(-1-)+(N-1)Y where 1 = (y5°12 011 )(£) , and N is the smallest positive integer such that 1 (£) + Na ~ 1 2(M), a ~ min [p(s) - s]. 1 s H = max{Ko'o}

=

1 2 (M)

Definition 3 A functional f: [t ,t ) xC .. Rn is said to be 1 2 locally Lipschitzian iff, for each (t,1jI) E [t 1 ,t 2 ) xC, there exist positive numbers a and B such that on the set ([t-Cl, t+a] tI [t ,t 2 )) 1 x {-1jI1 for some K> O. Concerning the system (1) we introduce the following assumptions. Assumption 1.

271

The functionals

Proof : (i) (Existence of solutions) Given any toER and 4>E C, there exists (Driver (1977)) a unique solution x(t ,4» of the system o

= f(t'Xt)+B(t,xt)[h(t,xt)+e(t,xt)]+e(t,xt) x =+ to

x(t) f: B:

R xC" Rn, R xC" Rnxm

e:

R xC" Rm , e:

R xC" Rn

are continuous, quasi-bounded and locally Lipschitzian on R xC. Assumption 2. There exist a feedbacK control h(t,x t ), which is continuous, quasi-bounded and locally Lipschitzian on R xC, a continuously differentiable function V: R x Rn .. R+ , continuous, monotonely increasing functions 1 1 ,1 2 ,1 3 ,P: R+ .. R+ ,and a continuous function 14: R+ .. R with the properties 11 (0) =

13 (0)

= 0 ,

1i m 11 ( s) 5 +

= ~ ,

GO

= 1 2(0)

1i m 12 ( s ) S + co

and along the solutions of (7) •

satisfy 14> I < 0 and define H ~ max {6, KO} • , we can choose M > H such Since lim 1 1(S) s ..

and

p(s) > s for s > 0 , such that

V(t,x(t))

on an internal [to,t o + 6) where 6 > O. We will show next that the existence and uniqueness of every solution x(t o ,4» can be extended to the internal [to'~)' For this purpose, it is sufficient to prove that the solutions are uniformly bounded. (ii) (llniform boundedness) Now let the initial function 4>EC at time to

6

= Lm (t,x(t)) + 9Tx V(t,x(t)) e(t,x t )

, - 13(IX(t)l) + 1 4 (IX(t)l) and Lm(t,x(t)) , - 13 (lX(t)l) if V(t+9,x ( t+9)) 'p(V(t,x(t)) IJ 9<= [- r,O] where

~

that 11 (M ) > 1 2 (H). We shall show that lX(t)1 is uniformly bounded by M on [t o ,t 1] where t 1> to' Suppose thi s were not true; then I x( t) I > tl for some tE [t o ,t 1]. We define t2 ~ inf {t E [t o ,t 1 ~ I x(t). = M} and t3 ~ sup {t E [t o ,t 2] Ilx(t)1 = H}. By continuity of x(·) and IX(tO)1 , H , t2 and t3 are well-defined and, by definition of t , x(t ) ~ 2 2 x(t +9) ,9 E [-r,O]. Hence 2

272

Xiong Zhong Kai, G. Leitmann and F. Garofalo

Y1 Ix (t 2 )1 ( V(t 2 ,x(t 2 ) = V(t 3 ,x(t 3 ) +

Jti2 t3

V(t,x(t))dt ( V(t ,x(t ) 3 3

+ /.2[-Y 3 (H)+Y4(H)]dt ( V(t ,x(t )) ( y (H)< y (M) 3 3 2 1 t3 from which we obtain the contradiction Ix(t~)1 < M. Consequently IX(t)1 ( M on any internal [t o ,t 1] and, (iii) by the extended existence theorem in Driver (1977) the unique solution can be continued on all of [to' .. ). (iv) (Uniform ultimate boundedness) Let x(·): [t ,.. ) + Rn, x = 4>, be a solution with t0 o given 14> 1 < 6. Then, as shown above, IX(t)1 ( M and V(t,x(t)) ( y2(H) for t ) to - r. Choose £ such that 0 < hi 1 0 y2 )(k o ) < £ • We want to show that lX(t)_ ( £ for all t ) to + T , where T is to be determined (T is independent of to' but may depend on £ and 6). If £ ) H , I x( t) I ( £ for t ) to - r follows directly from the uniform houndedness result (ii). So we consider £ < H Define a ~ min [pis) - s] • ll(£) ( s ( y2(H). s

Let N be the smallest positive integer such that y 1(£) + Na ) y2(H). and 1et y ~ h 5oy 21 oy 1 ) (£) > O. We want to show V(t,x(t)) ( Y1(£) + (N-1)a for Y2(H) (t o .t o+T 1] • T1 = -Y- • Suppose, on the contrary. that Y1(£) + (N-1)a < V(t,x(t)) for any t > to; then p(V(t.x(t)) ) V(t.x(t) + a > Y1 (£) V(t+e.x(t+e)). eE(-r.O] and hence + Na) y (H) 2 • 1 V(t.x(t)) ( - Y5(IX(t)l) ( - h5° Y2 0 Y1 )(£))) ( - Y at any such t > to' Note that IX(t)1 < (y 21o Y1)(£) leads to the contradiction Y2(lx(t)1) < Y1(£) < V(t.x(t)) (Y2(IX(t)l) 6

Since V(t.x(t)) ( V(to.x(t o )) - y(t-to) ( Y2(M) - y(t-to) and V(t.x(t)) ) O. we must have V(t.x(t)) ( Yl(£ ) + (N-l)a for some [to.to+Tl1. 6 Y2(H) where Tl = - y - ' However. if V(t.x(t)) = Yl(£) + (N-l)a then p(V (t .x(t))) ) V(t.x(t)) + a

) Yl (£) + Na ) y2(H) ) V(ue.x(t+e)). eE (- r.O); so V ( - Y < O. Hence. V(t.x(t)) ( Yl(£) + (N-l)a for all t) to+Tl' If N = 1. the desired

.

result IX(t)1 (£ for all t ) to+T (with T ~ Tl ) then follows directly from Yl(lx(t)l) ( V(t.x(t)) ( Y1(&) + (N-I)a. For the case N > 1. we proceed by mathematical induction. For k = 1.2 ..... N • 6 Y2(H) define Tk = k(-y-) • and let V(t.x(t)) ( YI(£) + (N-k)a for all t ) to + Tk + (k-1)y. If

Yl(&) + (N-(k+I)]a < V(t.x(t)) for t ) to+Tk+ky • then p(V(t.x(t)) ) V(t.x(t))+a ) Y1(£) + (N-k)a ) V(t+e,x(t+e)), eE(- r,O]; so V (- y . Arguments similar to those employed in the interval [t o .t +T ] above then prove that V(t,x(t)) o 1 ( y 1(£) + (N- ( k+ 1) ) a fo r all t ) to + Tk+ 1+ ky Hence, Yl (IX(t)l) ( V(t,x(t)) ( Yl (e) for all t ) t + T. where T ~ TN + (N-l)y • o (v) (Uniform stabil ity) Let £ > ~ ~ (YiIo Y2)(k ) be specified. We take o 6(£) = h21 0 YI)(&) for £ E(~. H) and 6(£) ( h 10 Yl)(M) for £ ) M. Then. given any solution 2 x(·): (to''') + Rn with 14> 1 (6(£) , it follows from (ii) and (iv) that IX(t)1 ( £ for all t ) to' EXAMPLE To illustrate the preceding result, consider the mismatched delay system x(t) = a(t)x(t)+b(t)x(t-t)+c(t)U(t)+6C(t)+d(t) with a(t)=12cos t-0.5,b(t)=(1+0.5si~ t).c(t)=0.5+cos t, 6C(t) = (0.5 + cos t)x(t) 6c(t) O £ = 8 > (v - 10 y )(k ) = k = 5 • 2. 0 0 1 a = min (p(s)-s] = min (1.44s - s] 14.08, s)32 S)Y (£) l Y = (Y5° y210 Yl )(8) = y 5 (8) = 4.R. If we let

273

Robustness 'o f Uncertain Dynamical Systems

y?(M)

H

= 10, the N = 2, T( I: ,6 ) = N(-y-) -1

= ('(1 °Y2)(H) = 5, ro = ('(llo y 2 )(k o ) = 5

0(6)

d

-1

(Y2

0

+ (N-1)y ) 26,

Y1)(M)

= 10

CONCLUSIONS It has been shown in this paper that practical stability of mismatched non-linear delay systems can be obtained under certain specific conditions. REFERENCES

1.

2. 3. 4. 5. 6.

7.

8.

Astrlim, K.J. (1970). Introduction to Stochastic Control Theory. Academlc Press, New York. Chen, Y.H. (1987). ASME J. Dynamic Systems Meas. Control, Vol. 109, 29-35. Chen, Y.H., and Leitmann, G. (1987). Int. J. Control, Vol. 45, No. 5, 1527-1542. -----1:'ii"i'IeSS, M.J. and Leitmann, G. (1981). IEEE Trans. Autom. Control, Vol. 26, 1139-11~ Oriver, R.D. (1977). OrdinarY-and Delay Differential Equations. Sprlnger-Verlag, New York. Kushner, H.J. (1966). On the status of optimal control and stability for stochastic systems. IEEE International Convention Rec., Vol. 14, 143-151. Leitmann, G. (1978). IEEE Trans. Autom. Control, Vol.~, 1109-1110, (1979). J. Dynamic Sftem Meas. Control, Vol. 101, 212-216, ( 981). J. Dynamlc System HeaS. Control, Vol. 103, 95-102. ihowsen, A. (1983). Int. J. Control, Vol. 1, No. 5, 1135-1143.

Yl (d + Na Y2(M)

- - - - - - -=--;...-_-:---)2 ~------r,

Y, (M)

I I I

I

I

r

o·

Fig. 1 Several Related Quantities for Theorem

M