Structural stability of linear dynamically varying (LDV) controllers

Systems & Control Letters 44 (2001) 177–187

www.elsevier.com/locate/sysconle

Structural stability of linear dynamically varying (LDV) controllers Stephan Bohacek, Edmond Jonckheere ∗ Department of Electrical Engineering Systems, University of Southern California, Los Angeles, CA 90089-2563, USA Received 4 August 1998; received in revised form 24 April 2001

Abstract LDV systems are linear systems with parameters varying according to a nonlinear dynamical system. This paper examines the robust stability of such systems in the face of perturbations of the nonlinear system. Three classes of perturbations are examined: di3erentiable functions, Lipschitz continuous functions and continuous functions. It is found that in the 6rst two cases the system remains stable. Whereas, if the perturbations are among continuous functions, the closed-loop may not be asymptotically stable, but, instead, is asymptotically bounded with the diameter of the residual set bounded by a function that is continuous in the size of the perturbation. It is also shown that in the case of di3erential perturbations, the resulting optimal LDV controller is continuous in the size of the perturbation. An example is presented that illustrates c 2001 Elsevier the continuity of the variation of the controller in the case of a nonstructurally stable dynamical system.
Science B.V. All rights reserved.

1. Introduction Linear parametrically varying (LPV) systems have been the focus of extensive research [8,1,14]. Essentially, an LPV system is a linear system with parameters that may vary over some set. This paper examines the speci6c case where the variation of the parameters is described by a given dynamical system. Such systems are known as linear dynamically varying (LDV) systems and have applications in nonlinear tracking. An LDV system can be decomposed into two subsystems; a linear system and a nonlinear system, where the nonlinear system drives the parameters of the linear system. While both linear quadratic [2] and H ∞ [3] controllers have been developed for LDV systems, some questions regarding the robustness of the closed loop system remain unanswered. In the case of time-invariant linear systems, robust stability refers to the stability in the face of some uncertain parameters. While such concerns are valid for LDV systems, they can be handled in much the same way as they are in the case of time-invariant linear systems. However, stability in the face of uncertainty in the nonlinear subsystem is a unique concern to LDV systems and is the subject of this paper. Speci6cally, the variation of the parameters of the linear system is given by the nonlinear system (k + 1) = f((k)). It will be shown that a stable LDV system remains stable in the face of small perturbation of f. Three cases are examined: the perturbations are over C 1 functions, Lipschitz continuous functions and C 0 functions. In the 6rst two cases, it is shown that stability is maintained. In the case of continuous perturbations, it is not ∗ Corresponding author. Tel.: +1-213-740-4457. E-mail address: [email protected] (E. Jonckheere).

c 2001 Elsevier Science B.V. All rights reserved. 0167-6911/01/$ - see front matter  PII: S 0 1 6 7 - 6 9 1 1 ( 0 1 ) 0 0 1 4 0 - 2

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possible to guarantee asymptotic stability. Instead, it is shown that the system is asymptotically bounded, i.e., the state of the linear system converges to a small neighborhood of the origin. A related issue is the variation of the optimal LDV controller due to variation of the dynamical system f. This issue is the structural stability of the LDV controller. It will be shown that the linear quadratic controller is structurally stable. This type of stability has applications in the development of computational methods for LDV controllers. As discussed in [2], an eFcient method to compute the controller relies on making a small perturbation in the dynamical system f and 6nding the controller for this perturbed system. The controller for this nearby system is easily found. The question as to whether the controller of the nearby system is an approximation of the controller for the original system is answered aFrmatively in this paper. The paper proceeds as follows: Section 2 formalizes LDV systems, brieGy reviews previous results necessary for this paper and states some required de6nitions. Section 3 discusses conjugacy. Section 4 presents the main result of the paper. Finally, Section 5 presents an example of a structurally unstable system with a structurally stable LDV controller. 2. Background An LDV system is de6ned as x(k + 1) = A(k) x(k) + B(k) u(k); 
C(k) x(k) ; z(k) = D(k) u(k) (k + 1) = f((k))

(1)

with x(0) = x0 ; (0) = 0 ;

where f : Rn → Rn is a continuous map with f(S) = S, with S a compact set, A : Rn → Rn×n , B : Rn → Rn×m , C : Rn → Rp×n , and D : Rn → Rp×m . A continuous LDV is an LDV where the maps A; B; C and D are continuous. This paper only considers continuous LDV systems. The pair (A; f) is exponentially stable if system (1) is exponentially stable. That is, for u = 0 and 0 ∈ S there exist an (0 ) ¡ 1 and a (0 ) ¡ ∞ such that if (0) = 0 , then x(k) ¡ (0 )(0 )k x(0). Similarly, the pair (A; f) is uniformly exponentially stable if the pair (A; f) is exponentially stable and  and  can be chosen independent of (0). The triple (A; B; f) is stabilizable if there exists a bounded feedback F : S → Rm×n such that (A+BF; f) is exponentially stable. The triple (A; C; f) is uniformly detectable if there is a uniformly bounded map H : S → Rn×p such that (A+HC; f) is uniformly exponentially stable, that is, there exist d ¡ 1 and d ¡ ∞ such that x(k) ¡ k x(0) where x(k + 1) = (Afk (0 ) + Hfk (0 ) Cfk (0 ) )x(k). LDV systems naturally arise when controlling nonlinear dynamical systems. Let f : Rn ×Rm → Rn ; f(S; 0) ⊂ S; f ∈ C 1 and let S be compact. Consider the nonlinear tracking problem ’(k + 1) = f(’(k); u(k)); ’(0) = ’0 ; (k + 1) = f((k); 0);

(0) = 0 ∈ S:

(2)

The objective is to 6nd a control u such that ’(k) − (k) → 0. In this context (k) is the desired trajectory and ’(k) is the state of the system under control. De6ne the tracking error x(k) = ’(k) − (k). Then system (2) becomes x(k + 1) = f(’(k); u(k)) − f((k); 0)) = A(k) x(k) + B(k) u(k) + x (x(k); u(k); (k))x(k) + u (x(k); u(k); (k))u(k);

(3)

where (A )i; j = @fi =@j (; 0); (B )i; j = @fi =@uj (; 0) and x (x; u; )x + u (x; u; )u accounts for the higher order terms. Since f ∈ C 1 , if x and u are small, then x (x; u; ) and u (x; u; ) are small. Thus, when x and u are small, system (3) is well approximated by system (1) with f() :=f(; 0). In this case the LDV is said to be induced by f. It was shown in [2] and [3] that if the LDV system (1) induced by f with control u is

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uniformly exponentially stable, then the nonlinear system (2), with control u, is locally uniformly exponentially stable. By de6nition locally uniformly exponentially stable means that there exist  ¡ 1;  ¡ ∞ and # ¿ 0 such that if x(0) = ’(0) − (0) ¡ # then x(k) ¡ k x(0) where ;  and # can be taken independent of the initial condition 0 , i.e. uniformly in 0 and locally in x. Thus, if the LDV system induced by the nonlinear dynamic system f is LDV stabilizable, then the LDV system (@f=@; @f=@u; f) is stabilizable. The following theorems are needed in the sequel: Theorem 1. Suppose (A; C; f) is uniformly detectable. Then (A; f) is uniformly exponentially stable if and only if there exists a bounded function X : S → Rn×n then X = X ¿ 0 such that A Xf() A − X 6 − C C . Moreover;  and  in the de5nition of uniformly exponentially stable can be chosen to depend only on the bound on X and d and d in the de5nition of detectability. Proof. This is a simple extension of Theorem 7:1 in [4]. Theorem 2. Suppose (1) is a continuous LDV system. If (A; B; f) is stabilizable; (A; C; f) is uniformly detectable and D D ¿ 0 for  ∈ S; then there exists a bounded and continuous function X : S → Rn×n with X = X ¿ 0 and satisfying the functional discrete time algebraic Riccati equation X = A Xf() A + C C − A Xf() B (D D + B Xf() B )−1 B Xf() A

(4)

   D(k) + B(k) Xf((k)) B(k) )−1 B(k) Xf((k)) A(k) x(k) is optimal in the The control uLQ (k) = F(k) x(k) := − (D(k) sense that it minimizes the quadratic cost ∞  x(k) Cf k (0 ) Cfk (0 ) x(k) + u(k) Df k (0 ) Dfk (0 ) u(k): V (0 ; u; x0 ) = k=0

Furthermore; this control uniformly exponentially stabilizes the system and x0 X0 x0 = minu V (0 ; u; x0 ). Proof. See [2]. Given a stabilizable map f, we will study maps close to f in the following topologies: Denition 1. Let f; fˆ : Rn × Rm → Rn with f; fˆ ∈ C 1 . The C 1 topology is generated by the metric     @f ˆ @ f   ˆ = sup f(x; u) − f(x; ˆ u) + sup  (x; u) − (x; u) dC 1 (f; f)  @x x∈Rn ; u∈Rm x∈Rn ;u∈Rm  @x    @f  @fˆ   + sup  (x; u) − (x; u)  @u x∈Rn ; u∈Rm  @u where @f=@x(x; u) is the Jacobian matrix of f with respect to x and  ·  is the l2 induced matrix norm. Denition 2. Let f; fˆ : Rn × Rm → Rn , with f; fˆ ∈ LC, where LC denotes the set of Lipschitz continuous functions. The LC topology is generated by the metric   ˆ i (x; u) − fˆ i (y; v))| |f (x; u) − f (y; v) − ( f i i ˆ = sup f(x; u) − f(x; ˆ u) + sup  : dCL (f; f) x − y2 + u − v2 x∈Rn ; u∈Rm x;y∈Rn u;v∈Rm i∈[1; n]

Denition 3. Let f; fˆ : Rn × Rm → Rn , with f; fˆ ∈ C 0 . The C 0 topology is generated by the metric ˆ = dC 0 (f; f)

sup

x∈Rn ; u∈Rm

ˆ u): f(x; u) − f(x;

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Remark 1. The supremums in the last three de6nitions are over x ∈ Rn and u ∈ Rm . This can be eased to the supremum over N where N ⊂ Rn × Rm is a tubular neighborhood of S × {0}. This modi6cation has no e3ect on the development that follows if N is large enough, i.e. for all x ∈ S, supy∈Ex y − x is large enough, where Ex := {y ∈ N: x := arg minv∈S y − v}. In the de6nition of system (1), the set S is invariant, i.e. f(S) = S. As the map f varies it is likely that S is no longer invariant. Indeed, it is possible that arbitrarily small variations in the map f lead to drastic changes in invariant sets. This is problematic since X , the solution to the Riccati equation (4), is only de6ned on S where S is invariant. It is diFcult to discuss the dependence of X on f when as f varies the domain of X greatly varies. Thus we will restrict our attention to variations in f such that S only varies slightly. That is, we will require ˆ + H (S; S) ˆ ¡ ,; d:(f; f)

ˆ S) ˆ = Sˆ with f(S) = S; and f(

(5)

where H (·; ·) is the Hausdor3 metric, i.e.  ˆ ˆ ∈ S}; ˆ  ∈ S} : ˆ := max sup inf { − : ˆ sup inf { − : H (S; S) ˆ Sˆ ∈

∈Sˆ

ˆ In the sequel it will be understood that S is an invariant set of f and Sˆ is an invariant set of f. Next we extend the feedback F : S → Rn×m de6ned by Theorem 2 to all of Rn by F˜ ˆ := F()ˆ , where ˆ = arg min{ − : ˆ  ∈ S}: ()

(6)

ˆ is not necessarily well de6ned The quadratic cost X can be extended to X˜ in the same fashion. Note that () and X˜ might not be continuous. However, by perhaps invoking the axiom of choice, one can properly de6ne ˆ Furthermore, X˜ is continuous on S and if ˆ − ’ ˆ ’ˆ ∈ Sˆ ∪ S with H (S; S) ˆ small, then (). ˆ is small and ; X˜ ˆ − X˜ ’ˆ  is small. Finally, let X : S → Rn and Xˆ : Sˆ → Rn ; de6ne  ˆ dS; Sˆ (X; Xˆ ) := max sup X − Xˆ () ˆ − X ˆ : ˆ ; sup X() ∈S

ˆ Sˆ ∈

ˆ is small [7]. ˆ is small whenever dC 1 (f; f) Now, if f is a hyperbolic on S (see Section 3), then H (S; S) Furthermore, if f : S → S is hyperbolic on S and S is a manifold (i.e. f is an Anosov di3eomorphism) ˆ is small enough, fˆ is hyperbolic on S and S is invariant. Therefore, if f is an Anosov and if dC 1 (f; f) ˆ = 0 and d ˆ (X; Xˆ ) = dC 0 (X; Xˆ ). On the other hand, suppose that S is an attractor for di3eomorphism, H (S; S) S; S f, that is for all ’ ∈ N(S), we have limk→∞ fk (’) ⊂ S, where N(S) is any small enough neighborhood of ˆ is small ˆ is small whenever dC 0 (f; f) S. Thus it is a generic property for such di3eomorphisms that H (S; S) ˆ ˆ ˆ ˆ [9], where S is an attractor for f. Hence in these three cases dC 1 (f; f) is small implies H (S; S) is small, and therefore condition (5) is repetitious. Nonetheless, to maintain generality, condition (5) will be assumed. ˆ 9 0 as On the other hand, for dynamical systems that are not structurally stable, it is possible that H (S; S) ˆ dC (f; f) → 0. Clearly, the LDV controller is not structurally stable in these situations; therefore it is assumed ˆ → 0 implies that H (S; S) ˆ → 0. that f is such that dC (f; f)

3. Conjugacy Conjugacy provides an equivalence relationship among dynamical systems. In this section C 1 and C 0 conjugacies are examined. It will be shown that C 1 conjugacy preserves LDV stabilizability and the conjugacy maps can be used to transform the controller. However, in the case of C 0 conjugacy, LDV stabilizability is not preserved. Since structural stability of dynamical systems implies C 0 conjugacy with nearby systems [7],

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structural stability of the dynamical system alone does not imply the structural stability of LDV stabilizability or of the LDV controller. Let f be LDV stabilizable and let f and fˆ be C 1 conjugate. That is, there exists di3eomorphisms g : Rn → Rn ;

ˆ g(S) = S;

and

h : Rm → Rm

ˆ such that f(; u) = g−1 (f(g(); h(u))). De6ne G() = @g=@(); H () = @h=@. Since g and h are di3eomorphisms, G and H are invertible. Therefore the following diagram commutes:

ˆ is the LDV system induced by f, ˆ then ˆ B; ˆ f) Thus, if (A; B; f) is the LDV system induced by f and (A; −1 −1 m×n ˆ ˆ Ag() = Gg() A Gg() ; Bg() = Gg() B Hg() . If F : S → R uniformly exponentially stabilizes (A; B; f) then −1 ˆ Thus, LDV stabilizability is preserved under ˆ B; ˆ f). Fˆ g() := H F Gg() uniformly exponentially stabilizes (A; 1 C conjugacy. Similarly, LDV uniform detectability is preserved under C 1 conjugacy. −1 Now, suppose (A; B; f) is stabilizable, (A; C; f) is uniformly detectable, D D ¿ 0 for  ∈ S; Cˆ g() = C Gg() −1 and Dˆ g() = D H . Since f ∈ C 1 , the LDV system induced by f is continuous and Theorem 2 implies that −1  there exists a continuous function X : S → Rn×n that solves Eq. (4). It is clear that Xˆ g() := (Gg() ) X Gg() ˆ ˆ ˆ ˆ ˆ solves the Riccati equation associated with the LDV system (A; B; C; D; f). Therefore, LDV systems and quadratic controllers are well de6ned in a coordinate free approach. The topological and geometric issues associated with LDV systems on a nonorientable or nonparallelizable manifolds S are addressed in [6]. Now, if f is hyperbolic and fˆ is C 1 close to f, the fˆ is hyperbolic and fˆ is topologically conjugate to f (for precise result see [7]). However, f and fˆ are not necessarily C 1 conjugate. If hyperbolicity implied that fˆ and f are C 1 conjugate, then LDV would clearly be structurally stable in the hyperbolic case. Note that in the case of topological conjugacy, the nonlinear control u(k) ˆ = h−1 (F(k) (g−1 (’(k)) − g−1 ((k)))) might ˆ not exponentially stabilize f. Consider, for example, a system with no input: f(’) = 12 ’;

S = [0; 12 ]

and the induced LDV system x(k + 1) = 12 x(k):

(7)

Under homeomorphisms g(’) = − (log(’))−1 and g−1 (’) ˆ = e−1=’ , the conjugate system becomes 

1 ˆ = ˆ Sˆ = 0; −1 ˆ ) f( ; ˆ log(1=2) 1 −  log(1=2) and the induced LDV system is

1 x(k ˆ + 1) = x(k); ˆ ˆ ln 2)2 (1 + (k)

ˆ + 1) = f( ˆ ˆ (k)): (k

(8)

The above system is not uniformly exponentially stable because for every  ¡ ∞ and  ¡ 1; x(k) ˆ k k ˆ ˆ ¿  x(0) ˆ for some k. Of course f () → 0, just not exponentially fast. Thus, f and f are topologically conjugate, yet (8) is LDV stabilizable and (7) is not. Therefore, since hyperbolicity only leads to

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topological conjugacy, hyperbolicity will not help to prove the structural stability of LDV stabilizability. We ˆ must rely on the fact that f and fˆ are C 1 close to infer LDV stabilizability of f. Remark 2. The example above illustrates the weakness of the LDV approach compared to nonlinear methods. The LDV approach implies that fˆ is not stable, when, in fact, it is stable. 4. Structural stability ˆ is In this section it will be shown that if fˆ is near an LDV stabilizable f in the C 1 topology and H (S; S) small, then fˆ is LDV stabilizable (Proposition 4). In fact, the LDV optimal quadratic cost varies continuously ˆ + H (S; S)—that ˆ with dC 1 (f; f) is, the map (f; S) → X is continuous where X is the positive semi-de6nite ˆ function that solves Eq. (4) (Proposition 6). Furthermore, if fˆ is near f in the Lipschitz topology and H (S; S) 0 ˆ ˆ is small, then f is also LDV stabilizable (Proposition 7). Finally, if f is near f in the C topology and ˆ is small, then an LDV controller may only stabilize fˆ in the sense that lim supk x(k) ¡ ,, where H (S; S) the control objective is x(k) → 0 (Proposition 8). Lemma 3. Let system (1) be a continuous LDV. If the pair (A; f) is uniformly exponentially stable; then ˆ + dC 0 (A; A) ˆ is uniformly ˆ + H (S; S) ˆ ¡ ,; then the pair (A; ˆ f) there exists an , ¿ 0 such that if dC 0 (f; f) ˆ ˆ f) exponentially stable. Furthermore; the  and  in the de5nition of uniform exponential stability of (A; can be taken to only depend on A; f and ,. Note that this lemma is only examining LDV systems and therefore does not require that A = @f=@x. Proof. Fix K ⊃ N(S), where K is compact and N(S) a tubular neighborhood of S. De6ne ,1 such that ˆ ¡ ,1 implies that Sˆ ⊂ K. Since (A; f) is uniformly exponentially stable, Theorem 2 implies that there H (S; S) exists a continuous positive semi-de6nite function X such that for ˆ ∈ K, A()ˆ Xf(()) ˆ A() ˆ = X() ˆ − I;

(9)

ˆ is de6ned by (6). Since X is continuous, there exists a 2 ¿ 0 such that ’ −  ¡ 2 implies where () X’ − X  ¡ 18 1=(A2 + 1), where AP = sup{A :  ∈ K}. Furthermore, since f is uniformly continuous over K, ˆ ¡ #, we have f() − f() ˆ ¡ 2=3. Therefore, if there exists a # ¿ 0 such that for , ˆ ∈ K, and  −  ˆ ˆ ˆ ˆ dC 0 (f; f) + H (S; S) ¡ min(2=3; #; ,1 ) := ,2 , then for  ∈ S, we have ˆ − f(()) ˆ 6 (f( ˆ − f( ˆ + f( ˆ − f() ˆ + f() ˆ − f(()) ˆ ¡2 ˆ )) ˆ )) ˆ ) ˆ ) (f(

 1 1 1 2 ˆ ˆ ˆ and thus X(f(ˆ )) ˆ −Xf(()) ˆ  ¡ 8 1=(A +1). Hence, if  ∈ S and dC 0 (A; A) ¡ min( 8 (1=X )(1=A), 8 (1=X ); 1) = : ˆ ˆ 21 and d(f; f) + H (S; S) ¡ ,2 , then with a bit of elementary manipulation it can be shown that    ˆ  ˆ ˆ ˆ Aˆ ˆX(f(ˆ )) ˆ Aˆ = A Xf() A + (A − A ) Xf() (A − A ) − A Xf() (A − A )  −(A − Aˆ  ) Xf() A + Aˆ  (Xf() − Xf() )Aˆ  ˆ

1 6 X()ˆ − I: 2 P  ))=min∈S (3(X )). Since X is bounded and continuous, S Set  = 1 − 1=2 min∈S (3(X )) and  = max∈S (3(X compact and X ¿ 0, and  and  are 6nite. Since X(·) solves Eq. (9), it is not hard to show (for example see [12]) that  ¡ 1 and if x(k + 1) = Aˆ fˆk (ˆ ) x(k); then x(k) ¡ k x(0). 0

Proposition 4. Assume f ∈ C 1 induces a stabilizable LDV system; that is; there exists a continuous map F : S → Rm×n such that (A + BF; f) is uniformly exponentially stable. Then there exists a 2 ¿ 0 such that

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183

ˆ + H (S; S) ˆ ¡ 2; then fˆ is LDV stabilizable and is stabilized by the feedback F. Furthermore; if dC 1 (f; f) with this feedback; the ;  and # in the de5nition of locally uniformly exponential stability can be chosen to depend only on f; F and 2. Thus LDV stabilizability is structurally stable. ˆ ; ˆ 0) + @f=@u( ˆ 0)F ˆ . Then dC 0 (A; ˆ ( ˆ ˆ f) where C is a constant that ˜ A) ¡ CdC 1 (f; Proof. De6ne A˜ ˆ = @f=@ ; () ˆ + dC 0 (A; A) ˜ + H (S; S) ˆ ¡ ,; depends on F. Lemma 3 implies that there exists an , ¿ 0 such that if dC 0 (f; f) ˆ ; ˆ 0); ˆ + H (S; S) ˆ ( ˆ is stable. Hence, there exists a 2 ¿ 0 such that if dC 1 (f; f) ˆ ¡ 2; then (@f=@ ˜ f) then (A; ˆ 0); f) ˆ ˆ is stabilizable. Lemma 3 further states that the parameters of stability,  and ; depend on @f=@u( ; A + BF; f and , only. Since , and A + BF depend on F; and A and B are the partial derivatives of f; we conclude that  and  can be taken to only depend on f; F and ,. Thus, if the feedback F stabilizes f; then F also stabilizes any function fˆ near f in the C 1 topology. A natural question is, how good a controller is F? For instance, if F is the LDV quadratic controller for the ˆ That is, are LDV quadratic LDV system induced by f; how far is it from the LDV quadratic controller for f. controllers structurally stable? First, note that detectability is a structurally stable property. That is: Lemma 5. Assume that A; C and f are continuous and (A; C; f) is uniformly detectable. In this case there ˆ + dC 0 (A; A) ˆ is uniformly deˆ + dC 0 (C; C) ˆ + H (S; S) ˆ ¡ 2; then (A; ˆ C; ˆ f) exists a 2 ¿ 0; such that if dC 0 (f; f) ˆ ˆ ˆ ˆ ¡ 2; 0 0 tectable. That is; there exist ˆd ¡ 1 and ˆd ¡ ∞ such that if dC 0 (f; f)+d (A; A)+d (C; C)+H (S; S) C C ˆ then there exists a feedback Lˆ such that 4(k) ¡ ˆd ˆkd 4(0) where 4(k + 1) = (Aˆ + LˆC)4(k). Furthermore; ˆd and ˆd only depend on A; C; f; and 2. Proof. The proof is nearly identical to the proof of Lemma 3. Proposition 6. Let f ∈ C 1 . Assume that the LDV induced by f is stabilizable; (A; C; f) is uniformly deˆ + tectable and D D ¿ 0 for all  ∈ Rn . Then for all , ¿ 0; there exists a 2 ¿ 0 such that if dC 1 (f; f) ˆ ˆ ˆ ˆ dC 0 (C; C) + dC 0 (D; D) + H (S; S) ¡ 2; then dS; Sˆ (X; X ) ¡ ,; where X is the positive semi-de5nite solution to ˆ ˆ D; ˆ f). the Riccati equation (4) induced by (C; D; f) and Xˆ is the positive semi-de5nite solution induced by (C; ˆ + H (S; S) ˆ ¡ 21 and ˆ0 ∈ S; ˆ Proof. Let , ¿ 0. By Proposition 4, there exist ;  and 21 such that if dC 1 (f; f) k ˆ ˆ then x(k) ˆ 6  x(0) ˆ where x(k ˆ + 1) = (Afˆk (ˆ ) + Bfˆk (ˆ ) F(fˆk (ˆ )) )x(k) ˆ and F is the optimal LQ feedback 0 0 0 ˆ ˆ ˆ ˆ gain for (A; B; C; D; f). Therefore, it is possible to show that if dC 1 (f; f)+dC 0 (C; C)+d C 0 (D; D)+H (S; S) ¡ 21 and x0 Xˆ 0 x0 :=min u∈l2

∞  k=0

  x(k) ˆ  Cˆ fˆk (0 ) Cˆ fˆk ( ) x(k) ˆ + u(k) Dˆ fˆk (0 ) Dˆ fˆk ( ) u(k) 0

0

ˆ + Bˆ fˆk ( ) u(k); subject to x(k ˆ + 1) = Aˆ fˆk ( ) x(k) 0

0

2 then Xˆ ˆ ¡ Xˆ ; where Xˆ : = 2 (1=1 − 2 )((CP + 21 )2 + FP (DP + 21 )2 ); CP : = max∈K C ; DP : = max∈K D , FP : = max∈S F : ˆ + dC 0 (C; C) ˆ is detectable with parameters ˆ + H (S; S) ˆ ¡ 22 then (A; ˆ C; ˆ f) Lemma 5 shows that if dC 1 (f; f) ˆ ˆ and  that depend only on 22 ; C; and f. ˆ f) is uniformly exponentially Theorem 1 can then be applied to show that the closed-loop system (Aˆ + Bˆ F; stable with stability parameters that only depend on 21 ; 22 ; C; D and f. Therefore there eixsts a N ¡ ∞ such that  1 , ˆx(0); (10) ˆx(N + 1) ¡ 16 Xˆ

where xˆ is given by xˆ (k + 1) = (Aˆ fˆk (ˆ ) + Bˆ fˆk (ˆ ) Ffˆk (ˆ ) )ˆx(k). 0

0

0

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Let uˆ∗0 ; x0 denote the optimal control due to initial conditions 0 ; x(0) for the system with parameters ˆ De6ne u∗ ˆ B; ˆ D; ˆ C; ˆ f). (A; 0 ;x(0) similarly, but for the system with parameters (A; B; C; D; f). De6ne   XP ; UN : = u ∈ l2 [0; N ]: u2[0; N ] 6 1=2 inf ∈K 7(D D ) ˆ + where 7(D D ) is the minimum singular value of D D . Therefore, there is a 23 ¿ 0 such that if dC 1 (f; f) ∗ ˆ ˆ ˆ dC 0 (C; C) + dC 0 (D; D) + H (S; S) ¡ 23 ; then {uˆ 0 ; x(0) (k): k 6 N } ∈ UN . Note that UN is compact since N ¡ ∞; where N is such that Eq. (10) holds. Let xu; 0 (k + 1) = Afk (0 ) xuˆ∗ ;x (k + 1) + Bfk (0 ) u(k); and de6ne xˆ u; ˆ0 similarly. Since N ¡ ∞; if we 6x 0 0 ˆ ˆ B; ˆ f; (A; B; f); u ∈ UN ; 0 ∈ S and x(0) with x(0) 6 1; then xu; 0 (N + 1) − xˆ u; ˆ0 (N + 1) is continuous in A; and ˆ0 and since UN ; K and {x(0): x(0) 6 1} are compact, there exists a 24 ¿ 0; which can be taken ˆ + dC 0 (A; A) ˆ + dC 0 (B; B) ˆ ¡ 24 ; then ˆ + H (S; S) independently of u; 0 and x(0) such that if dC 0 (f; f)  1 , : (11) xu; (ˆ0 ) (N + 1) − xu; ˆ0 (N + 1) ¡ 16 XP ˆ + dC 0 (C; C) ˆ + dC 0 (D; D) ˆ ¡ min(20 ; 21 ; 22 ; 23 ; 24 ); and x(0) 6 1; then Eqs. ˆ + H (S; S) Therefore, if dC 1 (f; f) (10) and (11) yield 2 P XP x ∗ ˆ ∗ ˆ (N + 1) + xˆ ∗ ˆ (N + 1)2 ˆ (N + 1) = X x ∗ ˆ (N + 1) − x uˆ

;(0 ) ;ˆ x0

uˆ

;(0 ) ;ˆ x0

= XP xuˆ∗

ˆ

;(0 ) ;ˆ x0

+2XP (xuˆ∗

uˆ

(N + 1) − xˆ uˆ∗

uˆ∗ˆ

; x0

;(ˆ0 ) (N

Note that xuˆ∗ is u∗ˆ ;x(0) . 0 De6ne

ˆ

;(0 ) ;ˆ x0

ˆ

;0 ;ˆ x0

ˆ

;(0 ) ;ˆ x0

Likewise, XP ˆx

;0 ;ˆ x0

uˆ

;0 ;ˆ x0

(N + 1)2 + XP ˆxuˆ∗

(N + 1) − xˆ uˆ∗

ˆ

;0 ;ˆ x0

ˆ

;0 ;ˆ x0

(N + 1)) xˆ uˆ∗

ˆ

;0 ;ˆ x0

(N + 1)2

, (N + 1) 6 : 4

, + 1)2 6 : 4

(12) (13)

(N + 1) is the state after the non-optimal control uˆ∗(ˆ0 );x(0) is applied; the optimal control

x(0) Wf (u; N; 0 )x(0) : =

N  k=0

x(k)Cf k (0 ) Cfk (0 ) x(k) + u(k)Df k (0 ) Dfk (0 ) u(k);

(14)

where x(k + 1) = Afk (0 ) x(k) + Bfk (0 ) u(k)

and

x(0) = x(0):

ˆ x(0) is continuˆ N; ˆ0 )ˆx(0) similarly. Since N ¡ ∞; x(0) Wf (u; N; )x(0) − xˆ (0) Wˆ fˆ(u; N; )ˆ De6ne xˆ (0) Wˆ fˆ(u; ˆ Furthermore, since UN and K are compact, if x(0) 6 1; there exists a 25 ¿ 0 ˆ ) and . ˆ B; ˆ D; ˆ C; ˆ f; ous in (A; ˆ ˆ + dC 0 (D; D) ˆ ¡ 25 ; then ˆ + H (S; S) such that if dC 1 (f; f) + dC 0 (C; C) , ˆ x(0) Wf (u; N; (ˆ0 ))x(0) − x(0) Wf (u; N; ˆ0 )x(0) ¡ (15) for all u ∈ UN and ˆ0 ∈ S: 2 ˆ + dC 0 (C; C) ˆ + dC 0 (D; D) ˆ ¡ min(22 ; 24 ; 25 ). Then, for x(0) 6 1; inequalities (12) ˆ + H (S; S) Let dC 1 (f; f) and (15) yield x(0)X ˆ x(0) − x(0)Xˆ ˆ x(0) (0 )

0

6 x(0)Wf (uˆ∗ˆ0 ; x(0) ; N; (ˆ0 ))x(0)

+ xuˆ∗

ˆ

;(0 ) ˆ0 ; x(0)

−x(0)Wˆ fˆ(uˆ∗ˆ0 ; x(0) ; N; ˆ0 )x(0) − xuˆ∗

ˆ

; 0 ˆ0 ; x(0)

¡ ,:

(N + 1) (XfN +1 ((ˆ0 )) )xuˆ∗

ˆ

; (0 ) ˆ0 ; x(0)

(N + 1) (Xˆ fˆN +1 (ˆ ) )xuˆ∗ 0

ˆ

; 0 ˆ0 ; x(0)

(N + 1)

(N + 1)

S. Bohacek, E. Jonckheere / Systems & Control Letters 44 (2001) 177–187

185

Similarly, (13) and (15) yield x(0)Xˆ ˆ0 x(0) − x(0)X(ˆ0 ) x(0) ¡ ,: ˆ ˆ ˆ Therefore, if x(0) 6 1; then sup∈ ˆ Sˆ |x(0)X ((0 ))x(0) − x(0)X (0 )x(0)| ¡ ,. Similarly, it can be shown that ˆ sup∈S |x(0)X ()x(0) − x(0)Xˆ (())x(0)| ¡ ,; and thus dS; Sˆ (X; Xˆ ) ¡ ,. ˆ ˆ Next we weaken the assumptions in Proposition 4. First, we examine the case where f∈LC with dLC (f; f)¡,; 0 ˆ ˆ and 6nd that the above results still hold. Then the case where f ∈ C with dC 0 (f; f) ¡ , is examined. It is shown that such systems can be made stable, but not necessarily asymptotically stable. Proposition 7. Let f ∈ C 1 induce a stabilizable LDV. Then there exists an , ¿ 0 such that if fˆ ∈ LC and ˆ + H (S; S) ˆ ˆ ¡ 2; then the LDV controller induced by f locally uniformly exponentially stabilizes f. dLC (f; f) ˆ + 1); we see that error dynamics is Proof. Since f ∈ C 1 ; de6ning the tracking error xˆ (k) = ’(k ˆ + 1) − (k given by ˆ + 1) = f( ˆ ˆ ’(k); ˆ (k); ’(k ˆ + 1) − (k ˆ u(k)) ˆ − f( 0) ˆ ˆ + (ˆx(k); u(k); ˆ (k)) = Afˆk (ˆ ) xˆ (k) + Bfˆk (ˆ ) u(k) 0

0

ˆ ˆ ˆ ’(k); ˆ (k); + (f( ˆ u(k)) ˆ − f( 0) − (f(’(k); ˆ u(k)) ˆ − f((k))));

(16)

ˆ where (ˆx(k); u(k); ˆ (k)) accounts for the nonlinear part neglected in linear approximation. By Lemma 3 it ˆ + H (S; S) ˆ is small enough, the LDV system (Aˆ + Bˆ F; ˜ f) is uniformly exponentially is clear that if dLC (f; f) ˆ can be decomposed as stable. It can be shown (see [2]) that (x; u; ) ˆ + u (x; u; )u ˆ ˆ = x (x; u; )x (x; u; ) ˆ + u (x; u; )F ˆ P uP → 0. De6ne 9(x; F x; ) ∈ Rn×n+m by and that x (x; u; ) ˆ  → 0 as x; () 9(x; F x; )ij

 ˆ ˆ f (x + ; F x) − f i (; 0) − (fi (x + ; F x) − fi (; 0)))  xj := i ·  (F x) x2 + F x2  j−n

Thus, ˆ 9(x; F()ˆ x; )



x

F()ˆ x



for j 6 n for n ¡ j ¡ n + m:

ˆ F ˆ x) − f( ˆ 0) − (f(x + ; ˆ F ˆ u) − f()) ˆ ˆ + ; ˆ ; = f(x () ()

(17)

ˆ 6 √ndLC (f; f). ˆ Therefore, we see that (16) can be written as a uniformly and sup x ¡x;P 9(x; F()ˆ x; ) exponentially stable linear system, with a perturbation that can be bounded by a O(x2 ) function. It is well known that such systems are asymptotically stable [13]. ˆ ¡ , then asymptotic stability cannot be guaranteed. For example, consider If we only restrict dC 0 (f; f) the dynamical system f(’; u) = 12 ’. Then f is globally uniformly exponentially stable with |’(k) − (k)| ˆ u) = 1 ’ + , sin(’:=(6,)). Then dC 0 (f; f) ˆ 6 , and fˆ has three 6xed points 6 ( 12 )k |’(0) − (0)|. De6ne f(’; 2 corresponding to the solutions of sin(’:=(6,)) = ’=2,. Hence, ±, are stable 6xed points and zero is an unstable 6xed point. Thus, if ’ ¿ 0 and  ¡ 0, then |’(k) − (k)| 9 0. However, lim sup |’(k) − (k)| ¡ 2,. Therefore, fˆ is stable, but not asymptotically stable, instead asymptotically ’(k) − (k) approaches a small set around zero. This form of stability is often referred to as asymptotically bounded and the attractive set that ’(k) − (k) enters is called a residual set.

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S. Bohacek, E. Jonckheere / Systems & Control Letters 44 (2001) 177–187

Proposition 8. Let f ∈ C 1 be LDV stabilizable via the feedback F. Then there exists an , ¿ 0 such that ˆ + H (S; S) ˆ ¡ ,; then the feedback F makes fˆ asymptotically bounded; with the diameter of the if dC 0 (f; f) residual set continuous in ,. Proof. Since (A + BF; f) is uniformly exponential stable, Lemma 3 implies that there exists an , ¿ 0 such ˆ is uniformly exponentially stable if dC 0 (f; f) ˆ + H (S; S) ˆ ¡ ,, where F˜ ˆ:=F ˆ . Consider the ˜ f) that (A + BF;  () system k k x(k + 1) = f(x(k) + fˆ (0 ); F(fˆk ( )) x(k)) − f(fˆ (0 ); 0) 0

k

= (Af( ˆ (k); fˆ (0 ))x(k) ˆ 0 ) + Bf( ˆ 0 ) F(f( ˆ 0 )) )x(k) + x (x(k); F(fˆk ()) x k

+ u (x(k); F(fˆk ()) x(k); fˆ (0 ))F(fˆk ( )) x(k); 0

(18)

where x and u account for the error due to linearization. Since (A + BF; f) is uniformly exponentially stable, Lemma 3, x  and u  → 0 and x and u go to zero, and the fact that uniformly exponentially stable linear systems remain stable under small gain perturbation, we conclude that the nonlinear system (18) is locally uniformly stable. Now, consider the system ˆ ˆ ˆ + 1) = f( ˆ (k); ˆ ’(k); (’(k) ˆ − (k))) − f( 0) ’(k ˆ + 1) − (k ˆ F((k)) ˆ ˆ ˆ (’(k) ˆ − (k))) − f((k); 0) = f(’(k); ˆ F((k)) ˆ ˆ ˆ ˆ (k); ˆ ’(k); (’(k) ˆ − (k))) − f( 0) + (f( ˆ F((k)) ˆ ˆ ˆ (’(k) ˆ − (k))) − f((k); 0)): −f(’(k); ˆ F((k)) ˆ

(19)

We see that (19) is a locally uniformly stable nonlinear system with a extraneous noise input. It is not diFcult to show that such systems are asymptotically bounded. Remark 3. Typically one would presume that control methods of nonlinear maps based on linear approximation are only applicable to di3erentiable maps. However, the last two propositions show that this restriction is not necessary. 5. Structural stability of the optimal LDV controller of the H,enon map The HSenon map is de6ned as
 1 − (a + u(k))(’1 (k))2 + ’2 (k) ’(k + 1) = : b’1 (k) The LDV approximation of this system is

 −2a1 (k) 1 (’1 (k))2 x(k + 1) = x(k) + u(k); b 0 0

(k + 1) = f((k); 0):

The HSenon map has been studied for a wide variety of parameters. It was 6rst introduced with a = 1:4 and b = 0:3 [5]. With these parameters it is not yet known if this system is chaotic. However, computer simulations show that the system has an attractor. An LDV controller for the HSenon map was found in [2]. Since the HSenon map has an attractor [11], one can expect that the attractor does not change for small perturbation of the parameters [9]. In this case, Proposition 6 implies that the optimal controller should not change too drastically for small changes in the parameters if the attractor does not change too much. However, the HSenon map is not structurally stable. For example, if a = 1:392 and b = 0:3, then the attractor S is non-trivial with aperiodic orbits and the map appears to be chaotic. However, there are parameters aˆ arbitrarily close to 1.392

S. Bohacek, E. Jonckheere / Systems & Control Letters 44 (2001) 177–187

187

Fig. 1. Optimal quadratic cost versus system perturbation.

such that the attractor is simply an in6nite set of periodic orbits [10]. Thus, for arbitrarily small changes in the parameters, the dynamics of the system drastically change. However, according to the results presented here, for small changes in the parameters, the closed loop system will remain stable and the optimal controller ˆ + H (S; S)) ˆ should only slightly vary. Fig. 1 con6rms this fact and shows log(dS; Sˆ (X; Xˆ )) versus log(dC 1 (f; f) ˆ where X is the solution to the functional algebraic Riccati equation (4) with a = 1:4 and b = 0:3 and X is the solution for other values of a and b. 6. Conclusion It has been shown that LDV systems are well behaved under small perturbations of the nonlinear subsystem. In particular, for C 1 or Lipschitz perturbations, a stabilized system remains stable. For continuous perturbations, the system merely remains ultimately bounded. However, the size of the residual set is continuous in the perturbation. Therefore, it is not required to have perfect knowledge of the nonlinear system when designing the controller. Indeed, the actual system may not even be di3erentiable and yet methods based on linear approximation will be successful. An important result is that in the case of C 1 perturbations the optimal LDV controller is continuous in the perturbation. This feature of LDV controllers is utilized in eFcient schemes to compute the solution to the functional algebraic Riccati equation [2]. References [1] P. Apkarian, Advanced gain-scheduling techniques for uncertain systems, IEEE Trans. Control Systems Technol. 6 (1) (1998) 21–32. [2] S. Bohacek, E. Jonckheere, Linear dynamically varying LQ control of nonlinear systems over compact sets, IEEE Trans. Automatic Control 46 (2001) 840–852. [3] S. Bohacek, E. Jonckheere, Nonlinear tracking over compact sets with linear dynamically varying H ∞ control, SIAM J. Control Optim., to appear. [4] A. Halanay, V. Ionescu, Time-Varying Discrete Linear Systems, Birkhauser, Basel, 1994. [5] M. Henon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976) 69–77. [6] E.A. Jonckheere, S.K. Bohacek, Ergodic and topological aspects of linear dynamically varying (LDV) control, MTNS98, 1998. [7] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. [8] A. Packard, Gain scheduling via linear fractional transformations, Systems Control Lett. 22 (1994) 79–92. [9] S.Y. Pilyugin, The Space of Dynamical Systems with the C 0 -Topology, Springer, Berlin, 1991. [10] C. Robinson, Bifurcation to in6nitely many sinks, Comm. Math. Phys. 90 (1983) 433–459. [11] C. Robinson, Dynamical Systems, CRC Press, Boca Raton, 1995. [12] W. Rugh, Linear System Theory, Prentice-Hall, Englewood Cli3s, NJ, 1996. [13] M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cli3s, NJ, 1993. [14] F. Wu, X.H. Yang, A. Packard, G. Becker, Induced 1–2 norm control for LPV systems with bounded parameter variation rates, Internat. J. Robust Nonlinear Control 6 (1996) 983–998.