On feedback stabilizability of time-delay systems in Banach spaces

Transcription

1 On feedback stabilizability of time-delay systems in Banach spaces S. Hadd and Q.-C. Zhong Dept. of Electrical Eng. & Electronics The University of Liverpool United Kingdom

2 Outline Background and motivation Hautus criterion Stabilizability of systems with state delays Olbrot s rank condition for systems with state+input delays Stabilizability of state input delay systems A rank condition Two examples Summary S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 2/30

3 Hautus criterion for distributed parameter syst. ẋ(t) = Ax(t) + Bu(t), x(0) = z, t 0 (1) A is the generator of a C 0 -semigroup (T(t)) t 0 on a Banach space X B : U X is linear bounded U is another Banach space The system (1) is called feedback stabilizable if there exists K L(U, X) such that the semigroup generated by A + BK is exponentially stable. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 3/30

4 If T(t) is compact for t t 0 > 0, then the unstable set is finite. σ + (A) := {λ σ(a) : Reλ 0} Theorem 1: (Bhat & Wonham 78) Assume that T(t) is eventually compact. The system (1) is feedback stabilizable if and only if for any λ σ + (A). Im(λ A) + ImB = X (2) S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 4/30

5 Stabilizability of state-delay systems What if there is a delay in the state? {ẋ(t) = Ax(t) + Lxt + Bu(t), t 0, x(0) = z, x 0 = ϕ. (3) A generates a C 0 -semigroup (T(t)) t 0 on a Banach space X, L : W 1,p ([ r, 0],X) X, p > 1, r > 0, linear bounded, history function of x : [ r, ) X is defined as x t : [ r, 0] X, x t (s) = x(t + s), t 0, B : U X is linear bounded, initial values: z X and ϕ L p ([ r, 0],X). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 5/30

6 Transformation (3) into (1) Take the new state variable ( x w(t) = x t ), the system (3) can be transformed into (1) as ẇ(t) = A L w(t) + Bu(t), w(0) = ( z ϕ ), t 0, (4) where the operators are defined on the next slide. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 6/30

7 The new state space is X := X L p ([ r, 0],X). The operators are: A L : D(A L ) X X, ( ) A L A L := D(A L ) := and 0 d dσ which is bounded. { ( z ϕ ) D(A) W 1,p ([ r, 0],X) : f(0) = x B : U X, B = ( B 0 ), } S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 7/30

8 Let S X be the left semigroup on L p ([ r, 0],X) generated by Q X := d dσ, D(Q X ) := { ϕ W 1,p ([ r, 0],X) : ϕ(0) = 0 }. Assumption: Assume that L is an admissible observation operator for S X, i.e., τ 0 LS X (t)f p dt κ p f p, f D(Q X ), (5) where τ > 0 and κ > 0 are constants. Then, A L generates a C 0 -semigroup (T L (t)) t 0 on X (Hadd 05). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 8/30

9 If T(t) is compact for t > 0 then T L (t) is compact for t > r (Matrai 04). λ σ(a) if and only if λ σ(a + Le λ ) with (e λ x)(θ) = e λθ x for x X, θ [ r, 0]. The unstable set is finite. σ + (A L ) = {λ σ(a + Le λ ) : Reλ 0} For each λ C, define (λ) := λ A Le λ, D( (λ)) = D(A). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 9/30

10 Theorem 2: (Nakagiri & Yamamoto 01) Assume that L satisfies the condition (5) and T(t) is compact for t > 0. The system (3) is feedback stabilizable if and only if for any λ σ + (A L ), where Im (λ) + ImB = X (6) (λ) := λ A Le λ, D( (λ)) = D(A). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 10/30

11 Result on state input delay systems What if there are input delays as well? Olbrot (IEEE-AC 78) showed that the feedback stabilizability of the system ẋ(t) = A 0 x(t) + A 1 x(t 1) + Pu(t) + P 1 u(t 1), of which the dimension of the delay-free system is n, is equivalent to the condition Rank [ (λ) P + e λ P 1 ] = n, for λ C with Reλ 0, where (λ) := λi A 0 A 1 e λ. Only partial results available. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 11/30

12 Objective of the research To extend the Olbrot s result to a large class of linear systems with state and input delays in Banach spaces To introduce an equivalent and compact rank condition for the stabilizability of state input delay systems. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 12/30

13 Notation Let (Z, ) be a Banach space and G : D(G) Z Z be a generator of a C 0 -semigroup (V (t)) t 0 on Z. Denote by Z 1 the completion of Z with respect to the norm z 1 = R(λ,G)z for some λ ρ(g). The continuous injection Z Z 1 holds. (V (t)) t 0 can be naturally extended to a strongly continuous semigroup (V 1 (t)) t 0 on Z 1, of which the generator G 1 : Z Z 1 is the extension of G to Z. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 13/30

14 System under consideration ẋ(t) = Ax(t) + Lx t + Bu t, t 0, x(0) = z, x 0 = ϕ, u 0 = ψ (7) A : D(A) X X generates a C 0 -semigroup (T(t)) t 0 on a Banach space X, L : W 1,p ([ r,0],x) X linear bounded, B = (B 1 B 2 B m ) : ( W 1,p ([ r,0], C)) m X linear bounded, z X, ϕ L p ([ r,0],x) and ψ L p ([ r,0], C m ). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 14/30

15 Left shift semigroups The operator Q X f = θ f, D(Q X ) = {f W 1,p ([ r, 0], X) : f(0) = 0}. generates the left semigroup (S X (t)ϕ)(θ) = 0, t + θ 0, ϕ(t + θ), t + θ 0, for t 0, θ [ r, 0] and ϕ L p ([ r, 0], X). The pair (S X,Φ X ) with x(t + θ), t + θ 0, (Φ X (t)x)(θ) = 0, t + θ 0, for the control function x L p loc (R +, X) is a control system on L p ([ r,0], X) and X, which is represented by the unbounded admissible control operator B X := (λ (Q X ) 1 )e λ, λ C, where (Q X ) 1 is the generator of the extrapolation semigroup associated with S X (t). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 15/30

16 In fact, B X is the delta function at zero. For the control function x L p loc [ r, ) of the control system (S X, Φ X ) with x(θ) = ϕ(θ) for a.e. θ [ r, 0], the state trajectory of (S X, Φ X ) is the history function of x given by x t = S X (t)ϕ + Φ X (t)x, t 0. Similarly, we can define Q C, S C, Φ C and B C := (λ (Q C ) 1 )e λ. For the control system (S C, Φ C ) represented by B C, we have u t = S C (t)ψ + Φ C (t)u, t 0 with u(θ) = ψ(θ) for a.e. θ [ r, 0]. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 16/30

17 Assumptions Consider the following assumptions: (A1) L is an admissible observation operator for S X and (Q X, B X,L) generates a regular system on the state space L p ([ r, 0],X), the control space X and the observation space X. (A2) B k is an admissible observation operator for S C and (Q C, B C,B k ) generates a regular system on the state space L p ([ r, 0], C), the control space C and the observation space X for all k = 1, 2,,m. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 17/30

18 Define Z = X L p ([ r, 0],X) L 2 ([ r, 0],U) and take a new state variable ξ(t) = (x(t),x t,u t ). Using conditions (A1) (A2), the delay system (7) can be rewritten as ξ(t) = A L,B ξ(t) + Bu(t),t 0, (8) ξ(0) = (x,ϕ,ψ) X, S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 18/30

19 the generator A L,B : D(A L,B ) Z Z, B A L A L,B = 0, with A L = 0 0 Q C m D(A L,B ) = D(A L ) D(Q C m), A L 0 d dσ (9) the control operator is Bu = ( 0 0 B C mu), u C m, (10) The open-loop (A L,B, B) is well-posed in the sense that B is an admissible control operator for A L,B. (Hadd & Idriss, IMA J. Control Inform. 05) S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 19/30

20 Feedback stabilizability: Definition Assume that (A1) and (A2) hold. We say that the delay system (7) is feedback stabilizable if the open loop (A L,B, B) is feedback stabilizable. That is, there exists C L(D(A L,B ), C m ) such that the triple (A L,B, B, C) generates a regular linear system Σ on Z, C m, C m, the identity matrix K = I C m : C m C m is an admissible feedback for Σ, and the closed-loop system associated with Σ and K is internally stable. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 20/30

21 N&S condition Theorem 3: Assume the conditions (A1) and (A2) are satisfied and T(t) is compact for t > 0. Then (A L,B, B) (or the delay system (7)) is feedback stabilizable if and only if Im (λ) + Im(Be λ ) = X (11) holds for all λ σ + (A L ), where (λ) = (λ A) Le λ, D( (λ)) = D(A). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 21/30

22 Key of the proof The proof of this theorem is based on a generalized Hautus criterion and the following expression ( ( )) R(µ, A L ) Bu = (µ (A L,B ) 1 ) for u C m, µ ρ(a L ). e µ u Be µ u 0, S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 22/30

23 A rank condition: Reflexive X Since T(t) is assumed to be compact for t > 0, the unstable set σ + (A L ) is finite and can be denoted as σ + (A L ) = {λ 1,λ 2,,λ l }. The adjoint of the operator (λ) is given by (λ) = λ A (Le λ ). Set the dimension of the kernel Ker (λ i ) as d i = dim Ker (λ i ) i = 1, 2,..., l and the basis of Ker (λ i ) by (ϕ i 1,ϕi 2,,ϕi d i ). S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 23/30

24 Theorem 4: Assume that (A1) (A2) are satisfied, the space X is reflexive and T(t) is compact for t > 0. Then (A L,B, B) is feedback stabilizable if and only if RankB λi = d i, for i = 1, 2,...,l, where B λi = B 1 e λi 1, ϕ i 1 B 1e λi 1, ϕ i 2 B 1e λi 1, ϕ i d i B 2 e λi 1, ϕ i 1 B 2e λi 1, ϕ i 2 B 2e λi 1, ϕ i d i B m e λi 1, ϕ i 1 B me λi 1, ϕ i 2 B me λi 1, ϕ i d i, : the duality pairing between X and X. The proof is mainly based on the invariance of admissibility of observation operators. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 24/30

25 Back to the Olbrot s result ẋ(t) = Ax(t)+A 1 x(t r)+pu(t)+p 1 u(t r) (12) Then, (λ) = λi A e rλ A 1, λ C, with σ + = {λ 1,λ 2,,λ l } = {λ C : det (λ) = 0 and Reλ 0}. The dimension of Ker (λ i ) is d i for i = 1, 2,,l and the basis of Ker (λ i ) is ϕ i 1,ϕi 2,,ϕi d i. Denote the n d i matrix formed by the basis as ϕ i = (ϕ i 1 ϕ i 2 ϕ i d i ), i = 1, 2,,l. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 25/30

26 According to Theorem 4, we have the following: Corollary The system is feedback stabilizable if and only if Rank [ (P + e rλ i P 1 ) ϕ i] = d i, i = 1, 2,,l. (13) It can be approved that this is actually equivalent to ] Rank [ (λ i ) P + e rλ i P 1 = n for all unstable λ i. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 26/30

27 A more general result When the control space U is not finite dimensional, a similar necessary condition holds. See S. Hadd and Q.-C. Zhong, On feedback stabilizability of linear systems with state and input delays in Banach spaces, provisionally accepted for publication in IEEE Trans. on AC. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 27/30

28 Example 1 Consider the system (12) with A = 1 1, A 1 = 0,P = 0 1 p 11, P 1 = p 21 p1 11 p Hence, (λ) = λi A, σ(a) = { 1, 1}, σ + = {1} Ker (1) = span { ϕ 1 1} with ϕ 1 1 = 2, d 1 = 1 1 The rank condition is Rank ( ( p 11+e r p 1 11 p 21 +e r p 1 21 ) ( 2 1 )) = Rank(2p 11 +p 21 +e r (2p 1 11+p 1 21)) = 1. i.e. 2p 11 + p 21 + e r (2p p 1 21) 0. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 28/30

29 Example 2 Consider the system (12) with A = 0 0, A 1 = e r 1 0 0, P = 1 0 1, P 1 = 0. 0 Here (λ) = λ 0, σ + = {0,1}, e r + e λr λ 1 and Ker (0) = span{( 1 0 )} and Ker (1) = span{( 0 1 )}. Now we have Rank ( ( 1 0 ) ( 1 0 )) = 1 and Rank ( ( 1 0 ) ( 0 1 )) = 0. Thus, λ 1 = 0 is a stabilizable eigenvalue but λ 2 = 1 is not. S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 29/30

30 Summary Background and motivation Hautus criterion Stabilizability of systems with state delays Olbrot s rank condition for systems with state+input delays Stabilizability of state input delay systems A rank condition Two examples S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS p. 30/30