Invariance Under Bounded Time-Varying Perturbations

Copyright @ IFAC Control Applications of Optimization, St. Petersburg, Russia, 2000

INVARIANCE UNDER BOUNDED TIME-VARYING PERTURBATIONS F. Colonius and W. Kliemann Institute for Mathematics, University of Augsburg, 86150 A ugsburg, Germany, [email protected]; Department of Mathematics , Iowa State University, Ames lA 50011jU.S.A., [email protected]

Abstract: For non linear systems governed by ordinary differential systems with time-varying perturbations, the limit behavior is determined by the invariant control sets. They are obtained by considering the perturbations as controls and determining the maximal invariant subsets of complete controllability. For deterministic perturbations these sets describe the generic limit behavior; for stochastic perturbations, they are the supports of the invariant measures. In this paper, we concentrate on nth-order equations and describe how the invariant control sets change, when the amplitudes of the perturbations increase. Copyright © 2000 IFAC Keywords: nonlinear systems, controllability, perturbation.

m

1. INTRODUCTION :i;

= fo(x) + L

Ui(t)!i(X),

(1)

i=l

The problem when a nominal system loses its stability under perturbations has found considerable interest in the past 15 years. In particular, the notion of stability radius which was introduced by D. Hinrichsen and A.J. Pritchard in Colonius and Kliemann (2000a), Colonius and Kliemann (2000b) has proved to be a useful tool for the analysis of perturbed linear systems. In the present paper we discuss a certain nonlinear analogue of this notion extending the results of Colonius and Kliemann (2000b).

U(t) = (Ui(t)) E U, with smooth vector fields h, ... , fm; the perturbations u(t) take values in a given convex and compact subset U of lRm with 0 E int U. It is convenient to introduce a parameter p ~ 0, which indicates the size of the perturbations and to replace the pointwise constraint u{t) E U by u{t) E UP := p. U. In particular, for p = 0, one recovers the nominal, unperturbed system. We are interested in the supremal p-value, such that the stability behavior of the nominal system is preserved. If the equilibrium does not remain fixed under the disturbances, we have to specify the relevant notion of stability. Considering the perturbation as a control, one finds that for positive p > 0 there is a control set D{p) containing Xo; for p small enough, this control set is invariant.

Consider a nominal nonlinear system :i; = fo(x) in lR d given by a smooth vector field fo with an asymptotically stable equilibrium xo, hence fo(xo) = O. We are interested in the behavior near this equilibrium under time-varying, and bounded perturbations. Thus we consider a system of the form 433

holds. A control set C is an invariant control set if

We will define the corresponding invariance radius as the maximal p-value, for which the control set D(p) containing Xo remains invariant. As our analysis shows, this p-value, where the control set becomes variant, is connected with a merging of control sets and hence with a discontinuity (in the Hausdorff' metric) of the map p t----? D(p). These results are valid under appropriate innerpair conditions. We will show that these conditions are always satisfied for nth order equations.

clC

Invariant control sets determine the generic stability behavior of the system (see Colon ius and Kliemann, 2000a) . They are also of interest in the analysis of Markovian systems, where they are associated to the supports of the invariant measures (see e.g. Colon ius and Kliemann, 1999).

A(D) = {x E JRd , cl 0+ (x)

f: 0,

The following inner-pair condition guarantees that an equilibrium Xo of the nominal system is in the interior of a control set for p > O. For every p

> 0 one has Xo

E intOP'+ (xo) .

(2)

In this case we call the pair (0, xo) E UP x JRd an inner pair. Consider the following maps defined on [0,00) with values in the set of compact subsets of JRd, p I-t clD(p) and p 1--)0 E(p), where D(p) and E(p) are the control sets and chain control sets containing Xo. Here the existence of E(p) is clear, the existence of the control sets D(p) follows from the next theorem (see Chapter 4, Colonius and Kliemann,2000a).

In order to describe the global controllability behavior of the control system (1), we assume that K is positively invariant for all controls u E UP, P ~ 0, and recall the following concepts and results. A subset D with nonvoid interior of the state space JRd is a control set if it is a maximal subset with the property that for all x E D the inclusion ~

n D f: 0} ,

d(cp(Ti, Xi,Ui ),xi+d < c. A chain control set E C IRd is a maximal subset such that (i) for all x, y E E and all e, T > 0 there is a controlled (e, T)-chain from x to Yi and (ii) for all x E E there is u E U with cp(t, x, u) E E for all t E JR. Chain control sets are closed and every control set is contained in a chain control set. Furthermore, if two control sets Dl and D2 satisfy cl Dl n cl D2 f: 0, then there is a chain control set E with Dl U D2 C E.

where O~;(x) = {y E JRd , Y = cp(t,x,u) with 0::; t ::; T and u E UP} and O~T(x) = {y E JRd , X = cp(t,y , u) with 0 ::; t-::; T and u E UP} are the positive and negative reachable sets (or orbits) from x. We discuss the behavior of these control systems in relation to the behavior of the unperturbed system :i; = fo(x). Fix a compact positively invariant set K c M for the unperturbed system and suppose that it has an asymptotically stable equilibrium Xo E int K.

Dc cl{cp(t , x,u), u E U and t

O}.

Invariant control sets may be considered as a generalization of asymptotically stable equilibria for the perturbed system. We will also need the following weaker concept, chain control sets, which allows for arbitrarily small jumps in the trajectories. For e, T > 0, a controlled (e, T)-chain from x to y is given by n E N, times To, ... , T n- 1 ~ T, points Xo = x, ... , Xn-l, Xn = y, and controls uo , ..., Un-l E U such that for all i

We consider the control system (1) with u E UP = {u : JR-+JR m, u(t) E UP for all t E JR, measurable} . We assume that the control range U c JRm is nonvoid, convex and compact with 0 E int U, that p ~ 0, and that for every initial state x E JRd and every control function u E UP there exists a unique trajectory cp(t, x, u), t E JR, satisfying cp(O , x , u) = x . If the dependence on p does not play a role, we simply write U . We assume local accessibility for all p > 0, i.e., for all x E JRd and all T > 0 one has intO~T(x)

~

and for an invariant control set C the invariant domain of attraction is if cl 0+ (x) n C' f: 0 } A intl (C) = x E JRd , for an invariant control { set C' , then C' = C

In this section, we cite the pertinent results from Colonius and Kliemann (2000a) on the global controllability structure of control systems.

f: 0 and

u E U and t

The domain of attraction of a control set D is

2. BACKGROUND ON INVARIANT CONTROL SETS AND CHAIN CONTROL SETS

intO~;(x)

= cl{cp(t,x,u),

If the dependence of D on p plays a role, we indicate this by the argument p in D(p). The same remark applies also to all other notions introduced below. The local accessibility assumption guarantees that the invariant control sets are the closed control sets and that they have nonvoid interior. Recall that every point in K can be steered into an invariant control set.

Theorem 1. Assume that the inner-pair condition (2) is satisfied. Then for every p > 0 there is a control set D(p) with Xo E int D(p) and

{xo}

O}

=

n

p>o 434

D(p)

=

n

p>o

E(p).

Thus L is a nonvoid open set consisting of the points in A(Cd nint K which can be steered to at least one other invariant control set C i • In order to avoid certain degeneracies, we have to be careful about the behavior at the boundary oK of K. We require that trajectories in L remain bounded away from oK. More precisely, assume that the following strong invariance condition is satisfied:

It is clear, that clD(p) C E(p). Under a strength-

ened inner-pair condition, the relation between the control sets and chain control sets above is much closer. Let

1J(p)

= cl {{u, x) E UP X IRd ,

cp(t, X, u) E intD(p) for t E IR} .

We will require that for p' > p every (u, x) E 1J(p) is an inner pair for the p' -system.

For all x E L there is ex > 0 with d(cp(t, x, u), oK) ~ ex, u E U, t ~ 0, and there is eo > 0 such that for all x E cl L and u E U we have that y = limk-+oo cp(tk' x, u) E L for tk -+ 00 then d(y, oK) ~ eo .

Theorem 2. Assume that for all p' > p > 0 and all (u, x) E 1J(p) there is T > 0 with cp(T, x, u) E int 0+ ,p (x) . Then the map p f-t cl D (p) is lower semicontinuous and the map p f-t E(p) is upper semicontinuous (with respect to HausdorfJ metric). The sets of continuity points for both maps coincide, and p. E (0,00) is a continuity point if and only if cl D(p·) = E(p·). There are at most countably many point of discontinuity.

The next theorem indicates the controllability situation when an invariant control set touches the boundary of its invariant domain of attraction. In this case, the chain control set containing the invariant control set must contain another control set and at the invariance radius r the invariant control set merges with another variant control set and itself becomes variant.

Thus the strengthened inner-pair condition guarantees that "almost always" the chain control sets coincide with the closures of control sets. The exceptional points are of particular interest.

Theorem 4. Let the inner-pair condition from Theorem 2 be satisfied and consider an asymptotically stable equilibrium Xo of the nominal system. Suppose that for an increasing control set family D(p) p> 0, containing xo, which is invariant for p :::; Po, one has

In order to discuss the generic limit behavior of trajectories we endow the set of control functions U which is a subset of Loo(lR, IRffi), with the corresponding weak· topology. Then U becomes a compact metric space (cp. Colonius and Kliemann (2000a)) and for the invariant control sets Ci one has that for their domains of attraction U:=l A(Ci ) is open and dense in U x K.

and assume that the strong invariance condition is satisfied with Cl = D(pO). Assume that the chain control set E(pO) containing D(pO) does not contain another invariant control set. Then po coincides with the invariance radius r, the map p f-t D(p) is discontinuous at p = rand D(r) =j:. E(r). Furthermore, there is a control set D' =j:. D(r) for the control range ur such that for p> r the control sets D(p) satisfy

3. A INVARIANCE RADIUS FOR NONLINEAR SYSTEMS We define the invariance radius as follows. Definition 3. Let Xo be an asymptotically stable equilibrium of the nominal system. Then the invariance radius of Xo for the perturbed system (1) is r = sup{p > 0, D(p') 3 Xo is invariant for all 0< p' < p}.

D(p) :J D(r) U D'.

According to Theorem 1, this is well defined provided that the inner pair condition (2) holds. We will describe what happens at p = r . First we note that also for p = r the control set D (r) is invariant. Let K = cl int K C M be a compact, connected, positively invariant (for all u E U) set containing at least two invariant control sets. Then K contains finitely many invariant control sets Cl, ... , Ck, k ~ 2. In order to study the behavior of a (distinguished) invariant control set, say of Cl, define

Next we discuss in some more detail the question, when an invariant control set C touches the boundary of its invariant domain of attraction. A moment's reflection shows that the occurrence of an equilibrium at the boundary of C with nontrivial unstable manifold directed out of C does not necessarily lead to the loss of invariance. It does happen, if the unstable manifold approaches for t -+ 00 another invariant control set. If it returns to C, the invariance radius is not attained. If the unstable manifold approaches for t -+ 00 a variant control set, then, depending on its (global) properties, the invariance radius mayor may not be attained.

k

L=U[A(CdnA(C;})nintK.

(3)

i=2

435

4. THE INNER-PAIR CONDITION

Remark 7. The results above have been proved under a priori boundedness assumptions. Similar methods can also been applied if unbounded trajectories are allowed; compare Colonius and Kliemann (2000b).

In the preceding discussion, the behavior of the reachable sets in dependence on the control range plays a decisive role. We need that the reachable sets are strictly increasing for increasing control range. We will show that this condition is automatically satisfied for nth order equations with additive control Y (n) (t) + an-lY (n-l) + ... + aly(t) + ao = b(y(n)(t), ... , y(t))u(t), u(t) E p. U,

5. CONCLUSIONS We have seen that for nth order systems the following holds: for small amplitudes invariant control sets appear around a stable equilibrium of the unperturbed system. For increasing amplitudes, one observes that the number of invariant control sets decreases. They merge with another (variant) control set and lose their invariance. When the amplitudes are further increased, the resulting control set may again merge with another invariant control set or not. In the first case the merged control set may be invariant or not, depending on global controllability properties; in the second one, invariance cannot be obtained again.

(4)

where ai E IR for i = 0, ... , n - 1, b : IRn -+ IR is a Cl-function, and U is a closed interval in IR containing the origin. Proposition 5. Consider the family (4)P , p 2: 0, of control systems and assume that Ib(Yl , .. ., Yn) I :::; a for some constant a > O. Let Y E OP·+(x) for some T > O. Then for all p' > p ther~ exists a neighborhood N ofy with N c O~ ·+(x). Proof. There exists a control u with values in p. U such that cp(T, x, u) = y. Let a > O. Then for every z in the a-ball B(O, a) around the origin in IRn there exists a Cn-function , : [0, T] -+ IRn with ,(T) = z and

1,(j)(s)1 :::;

~ for all s E [O ,T] and all j

REFERENCES Colonius, F. and W. Kliemann (1999). Continuous, smooth, and control techniques for stochastic dynamics. In: Stochastic Dynamics. (H. Crauel and M . Gundlach, Eds.). SpringerVerlag. Colonius, F. and W. Kliemann (2000a) . The Dynamics of Control. Birkhauser. Colonius, F. and W. Kliemann (2000b). An invariance radius for nonlinear systems. In: Advances in Mathematical Systems. A Volume in Honor of D. Hinrichsen. (F. Colonius, U. Helmke, D. Pratzel-Wolters and F. Wirth, Eds.). Birkhauser. to appear. Hinrichsen, D. and A. J. Pritchard (1986a). Stability radius for structured perturbations and the algebraic Riccati equation. Sys. Control Lett., 8, 105-113. Hinrichsen, D. and A. J. Pritchard (1986b). Stability radius of linear systems. Sys. Control Lett., 7, 1-10.

= 0, ... ,n.

Now fix p' > p. For every z with Iz - yl < a there is a function, as above. Denoting yet) = cp(t, x, u), s E [0, TJ, we find for almost all t E

[O,T] I {yen) + ,en) + an-l (y(n-l) + ,(n-l)) + ... + al (y + ,) + ao} (n) + (n-l) - {Y an-lY + ... + alY + ao} I = I,(n)

+ an_l,(n-l) + '" + al,1

a

:::; 11 + an-l + ... + all T' Choose

a > 0 small enough such that a

a

11 + an-l + ... + all T < p' -

p.

Then there is a control function u ' with values in p' . U such that cp(T, x, u' ) = y + z, as desired. _ In particular, this result can be applied to pairs (u, x) in V(p) as needed in the theory developed above.

Remark 6. Similar results can be shown for coupled higher order systems. For example, consider for x = (Xl, ... , Xn) a system of the form Xi

+ 9i(X) + hi (x)

= Ui(t), i = 1, ...,n.

Then the same arguments can be used to verify that the inner-pair condition holds. 436