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On feedback stabilizability of time-delay systems in Banach spaces S. Hadd and Q.-C. Zhong q.zhong@liv.ac.uk Dept. of Electrical Eng. & Electronics The University of Liverpool United Kingdom Outline Background and motivation Hautus

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On feedback stabilizability of time-delay systems in Banach spaces

  • S. Hadd and Q.-C. Zhong

q.zhong@liv.ac.uk

  • Dept. of Electrical Eng. & Electronics

The University of Liverpool United Kingdom

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SLIDE 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
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Hautus criterion for distributed parameter syst.

˙ x(t) = Ax(t) + Bu(t), x(0) = z, t ≥ 0 (1) A is the generator of a C0-semigroup (T(t))t≥0

  • n 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
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If T(t) is compact for t ≥ t0 > 0, then the unstable set σ+(A) := {λ ∈ σ(A) : Reλ ≥ 0} is finite. Theorem 1: (Bhat & Wonham ’78) Assume that T(t) is eventually compact. The system (1) is feedback stabilizable if and only if Im(λ − A) + ImB = X (2) for any λ ∈ σ+(A).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 4/30
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SLIDE 5

Stabilizability of state-delay systems

What if there is a delay in the state? ˙ x(t) = Ax(t) + Lxt + Bu(t), t ≥ 0, x(0) = z, x0 = ϕ. (3) A generates a C0-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 xt : [−r, 0] → X, xt(s) = x(t + s), t ≥ 0, B : U → X is linear bounded, initial values: z ∈ X and ϕ ∈ Lp([−r, 0], X).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 5/30
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Transformation (3) into (1)

Take the new state variable w(t) = x xt

  • ,

the system (3) can be transformed into (1) as ˙ w(t) = ALw(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
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SLIDE 7

The new state space is X := X × Lp([−r, 0], X). The operators are: AL : D(AL) ⊂ X → X, AL := A L

d dσ

  • D(AL) :=
  • ( z

ϕ ) ∈ D(A) × W 1,p([−r, 0], X) : f(0) = x

  • and

B : U → X, B = ( B

0 ),

which is bounded.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 7/30
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SLIDE 8

Let SX be the left semigroup on Lp([−r, 0], X) gener- ated by QX := d dσ, D(QX) :=

  • ϕ ∈ W 1,p([−r, 0], X) : ϕ(0) = 0
  • .

Assumption: Assume that L is an admissible observa- tion operator for SX, i.e., τ LSX(t)fp dt ≤ κpfp, ∀ f ∈ D(QX), (5) where τ > 0 and κ > 0 are constants. Then, AL gen- erates a C0-semigroup (TL(t))t≥0 on X (Hadd ’05).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 8/30
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If T(t) is compact for t > 0 then TL(t) is compact for t > r (Matrai ’04). λ ∈ σ(A) if and only if λ ∈ σ(A + Leλ) with (eλx)(θ) = eλθx for x ∈ X, θ ∈ [−r, 0]. The unstable set σ+(AL) = {λ ∈ σ(A + Leλ) : Reλ ≥ 0} is finite. For each λ ∈ C, define ∆(λ) := λ − A − Leλ, D(∆(λ)) = D(A).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 9/30
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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 stabi- lizable if and only if Im∆(λ) + ImB = X (6) for any λ ∈ σ+(AL), where ∆(λ) := λ − A − Leλ, D(∆(λ)) = D(A).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 10/30
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SLIDE 11

Result on state–input delay systems

What if there are input delays as well? Olbrot (IEEE-AC ’78) showed that the feedback stabi- lizability of the system ˙ x(t) = A0x(t) + A1x(t − 1) + Pu(t) + P1u(t − 1),

  • f which the dimension of the delay-free system is n,

is equivalent to the condition Rank

  • ∆(λ) P + e−λP1
  • = n,

for λ ∈ C with Reλ ≥ 0, where ∆(λ) := λI − A0 − A1e−λ. Only partial results available.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 11/30
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Objective of the research

To extend the Olbrot’s result to a large class of lin- ear systems with state and input delays in Banach spaces To introduce an equivalent and compact rank con- dition for the stabilizability of state–input delay systems.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 12/30
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Notation

Let (Z, · ) be a Banach space and G : D(G) ⊂ Z → Z be a generator of a C0-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
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System under consideration

   ˙ x(t) = Ax(t) + Lxt + But, t ≥ 0, x(0) = z, x0 = ϕ, u0 = ψ (7) A : D(A) ⊂ X → X generates a C0-semigroup (T(t))t≥0 on a Banach space X, L : W 1,p([−r, 0], X) → X linear bounded, B = (B1 B2 · · · Bm) :

  • W 1,p([−r, 0], C))m → X linear

bounded, z ∈ X, ϕ ∈ Lp([−r, 0], X) and ψ ∈ Lp([−r, 0], Cm).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 14/30
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SLIDE 15

Left shift semigroups

The operator QXf = ∂ ∂θ f, D(QX) = {f ∈ W 1,p([−r, 0], X) : f(0) = 0}. generates the left semigroup (SX(t)ϕ)(θ) =    0, t + θ ≥ 0, ϕ(t + θ), t + θ ≤ 0, for t ≥ 0, θ ∈ [−r, 0] and ϕ ∈ Lp([−r, 0], X). The pair (SX, ΦX) with (ΦX(t)x)(θ) =    x(t + θ), t + θ ≥ 0, 0, t + θ ≤ 0, for the control function x ∈ Lp

loc(R+, X) is a control system on Lp([−r, 0], X) and X, which

is represented by the unbounded admissible control operator BX := (λ − (QX)−1)eλ, λ ∈ C, where (QX)−1 is the generator of the extrapolation semigroup associated with SX(t).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 15/30
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In fact, BX is the delta function at zero. For the con- trol function x ∈ Lp

loc[−r, ∞) of the control system

(SX, ΦX) with x(θ) = ϕ(θ) for a.e. θ ∈ [−r, 0], the state trajectory of (SX, ΦX) is the history function of x given by xt = SX(t)ϕ + ΦX(t)x, t ≥ 0. Similarly, we can define QC, SC, ΦC and BC := (λ − (QC)−1)eλ. For the control system (SC, ΦC) represented by BC, we have ut = SC(t)ψ + ΦC(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
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Assumptions

Consider the following assumptions: (A1) L is an admissible observation operator for SX and (QX, BX, L) generates a regular sys- tem on the state space Lp([−r, 0], X), the control space X and the observation space X. (A2) Bk is an admissible observation operator for SC and (QC, BC, Bk) generates a regular sys- tem on the state space Lp([−r, 0], C), the con- trol 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
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Define Z = X × Lp([−r, 0], X) × L2([−r, 0], U) and take a new state variable ξ(t) = (x(t), xt, ut)⊤. Using conditions (A1)–(A2), the delay system (7) can be rewritten as    ˙ ξ(t) = AL,Bξ(t) + Bu(t), t ≥ 0, ξ(0) = (x, ϕ, ψ)⊤ ∈ X, (8)

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 18/30
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SLIDE 19

the generator AL,B : D(AL,B) ⊂ Z → Z, AL,B =      AL B QCm      , with AL =  A L

d dσ

  D(AL,B) = D(AL) × D(QCm), (9) the control operator is Bu =

  • BCmu

⊤ , u ∈ Cm, (10) The open-loop (AL,B, B) is well-posed in the sense that B is an admissible control operator for AL,B. (Hadd & Idriss, IMA J. Control Inform. ’05)

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 19/30
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Feedback stabilizability: Definition

Assume that (A1) and (A2) hold. We say that the de- lay system (7) is feedback stabilizable if the open–loop (AL,B, B) is feedback stabilizable. That is, there exists C ∈ L(D(AL,B), Cm) such that the triple (AL,B, B, C) generates a regular linear system Σ on Z, Cm, Cm, the identity matrix K = ICm : Cm → Cm 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
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N&S condition

Theorem 3: Assume the conditions (A1) and (A2) are satisfied and T(t) is compact for t > 0. Then (AL,B, B) (or the delay system (7)) is feedback stabi- lizable if and only if Im∆(λ) + Im(Beλ) = X (11) holds for all λ ∈ σ+(AL), where ∆(λ) = (λ − A) − Leλ, D(∆(λ)) = D(A).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 21/30
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SLIDE 22

Key of the proof

The proof of this theorem is based on a generalized Hautus criterion and the following expression Bu = (µ − (AL,B)−1)

  • R(µ, AL)
  • Beµu
  • eµu
  • ,

for u ∈ Cm, µ ∈ ρ(AL).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 22/30
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A rank condition: Reflexive X

Since T(t) is assumed to be compact for t > 0, the unstable set σ+(AL) is finite and can be denoted as σ+(AL) = {λ1, λ2, · · · , λl}. The adjoint of the operator ∆(λ) is given by ∆(λ)∗ = λ − A∗ − (Leλ)∗. Set the dimension of the kernel Ker∆(λi)∗ as di = dim Ker∆(λi)∗ i = 1, 2, ..., l and the basis of Ker∆(λi)∗ by (ϕi

1, ϕi 2, · · · , ϕi di).

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 23/30
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Theorem 4: Assume that (A1)–(A2) are satisfied, the space X is reflexive and T(t) is compact for t > 0. Then (AL,B, B) is feedback stabilizable if and only if RankBλi = di, for i = 1, 2, . . . , l,

where Bλi =             B1eλi1, ϕi

1

B1eλi1, ϕi

2

· · · · · · B1eλi1, ϕi

di

B2eλi1, ϕi

1

B2eλi1, ϕi

2

· · · · · · B2eλi1, ϕi

di

. . . . . . . . . . . . . . . . . . Bmeλi1, ϕi

1

Bmeλi1, ϕi

2

· · · · · · Bmeλi1, ϕi

di

            .

·, ·: the duality pairing between X and X∗. The proof is mainly based on the invariance of admis- sibility of observation operators.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 24/30
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SLIDE 25

Back to the Olbrot’s result

˙ x(t) = Ax(t)+A1x(t−r)+Pu(t)+P1u(t−r) (12) Then, ∆(λ) = λI − A − e−rλA1, λ ∈ C, with σ+ = {λ1, λ2, · · · , λl} = {λ ∈ C : det ∆(λ) = 0 and Reλ ≥ 0}. The dimension of Ker∆(λi)∗ is di for i = 1, 2, · · · , l and the basis of Ker∆(λi)∗ is ϕi

1, ϕi 2, · · · , ϕi

  • di. Denote the n × di matrix formed by

the basis as ϕi = (ϕi

1 ϕi 2 · · · ϕi di),

i = 1, 2, · · · , l.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 25/30
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SLIDE 26

According to Theorem 4, we have the following: Corollary The system is feedback stabilizable if and only if Rank

  • (P + e−rλiP1)∗ · ϕi

= di, i = 1, 2, · · · , l. (13) It can be approved that this is actually equivalent to Rank

  • ∆(λi) P + e−rλiP1
  • = n

for all unstable λi.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 26/30
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SLIDE 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, provision- ally accepted for publication in IEEE Trans. on AC.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 27/30
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SLIDE 28

Example 1

Consider the system (12) with A =  1 1 −1   , A1 = 0, P =  p11 p21   , P1 =  p1

11

p1

21

  . Hence, ∆(λ) = λI − A, σ(A) = {−1, 1}, σ+ = {1} Ker∆(1)∗ = span

  • ϕ1

1

  • with ϕ1

1 =

 2 1   , d1 = 1 The rank condition is Rank

  • ( p11+e−rp1

11

p21+e−rp1

21 )∗ ( 2

1 )

  • = Rank(2p11+p21+e−r(2p1

11+p1 21)) = 1.

i.e. 2p11 + p21 + e−r(2p1

11 + p1 21) = 0.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 28/30
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SLIDE 29

Example 2

Consider the system (12) with A =   0 e−r 1   , A1 =   0 −1   , P =  1   , P1 = 0. Here ∆(λ) =   λ −e−r + e−λr λ − 1   , σ+ = {0, 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, λ1 = 0 is a stabilizable eigenvalue but λ2 = 1 is not.

  • S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 29/30
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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