on feedback stabilizability of time-delay systems in...

On feedback stabilizability oftime-delay systems in Banach spaces

S. Hadd and Q.-C. [email protected]

Dept. of Electrical Eng. & Electronics

The University of Liverpool

United Kingdom

Outline

Background and motivation

Hautus criterionStabilizability of systems with state delaysOlbrot’s rank condition for systems withstate+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

Hautus criterion for distributed parameter syst.

x(t) = Ax(t) +Bu(t), x(0) = z, t ≥ 0 (1)

A is the generator of aC0-semigroup(T (t))t≥0

on a Banach spaceX

B : U → X is linear bounded

U is another Banach space

The system (1) is calledfeedback stabilizableif thereexistsK ∈ L(U,X) such that the semigroup generatedbyA+BK is exponentially stable.


If T (t) is compact fort ≥ t0 > 0, then the unstable set

σ+(A) := {λ ∈ σ(A) : Reλ ≥ 0}

is finite.

Theorem 1: (Bhat & Wonham ’78)Assume thatT (t) is eventually compact. The system(1) is feedback stabilizable if and only if

Im(λ− A) + ImB = X (2)

for anyλ ∈ σ+(A).


Stabilizability of state-delay systemsWhat 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 aC0-semigroup(T (t))t≥0 on aBanach spaceX,

L : W 1,p([−r, 0], X) → X, p > 1, r > 0, linearbounded,

history function ofx : [−r,∞) → X is definedasxt : [−r, 0] → X, xt(s) = x(t+ s), t ≥ 0,

B : U → X is linear bounded,

initial values:z ∈ X andϕ ∈ Lp([−r, 0], X).S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 5/30

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) = ( zϕ ), t ≥ 0, (4)

where the operators are defined on the next slide.


The new state space is

X := X × Lp([−r, 0], X).

The operators are:AL : D(AL) ⊂ X → X ,

AL :=

(

A L

0 ddσ

)

D(AL) :={

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

andB : U → X , B = ( B0 ),

which is bounded.S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 7/30

Let SX be the left semigroup onLp([−r, 0], X) gener-ated by

QX :=d

dσ,

D(QX) :={

ϕ ∈ W 1,p([−r, 0], X) : ϕ(0) = 0}

.

Assumption: Assume thatL is an admissible observa-tion operator forSX , i.e.,∫ τ

0

‖LSX(t)f‖p dt ≤ κp‖f‖p, ∀ f ∈ D(QX), (5)

whereτ > 0 andκ > 0 are constants. Then,AL gen-erates aC0-semigroup(TL(t))t≥0 onX (Hadd ’05).


If T (t) is compact fort > 0 thenTL(t) iscompact fort > 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 eachλ ∈ C, define

∆(λ) := λ− A− Leλ, D(∆(λ)) = D(A).


Theorem 2: (Nakagiri & Yamamoto ’01)

Assume thatL satisfies the condition (5) andT (t) iscompact fort > 0. The system (3) is feedback stabi-lizable if and only if

Im∆(λ) + ImB = X (6)

for anyλ ∈ σ+(AL), where

∆(λ) := λ− A− Leλ, D(∆(λ)) = D(A).


Result on state–input delay systemsWhat 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),

of which the dimension of the delay-free system isn,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

Objective of the research

To extend the Olbrot’s result to a large class of lin-ear systems with state and input delays in Banachspaces

To introduce an equivalent and compact rank con-dition for the stabilizability of state–input delaysystems.


NotationLet (Z, ‖ · ‖) be a Banach space andG : D(G) ⊂ Z → Z be a

generator of aC0-semigroup(V (t))t≥0 onZ.

Denote byZ−1 the completion ofZ with respect to the norm

‖z‖−1 = ‖R(λ,G)z‖ for someλ ∈ ρ(G).

The continuous injectionZ → Z−1 holds.

(V (t))t≥0 can be naturally extended to a strongly continuous

semigroup(V−1(t))t≥0 onZ−1, of which the generator

G−1 : Z → Z−1 is the extension ofG toZ.


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 aC0-semigroup(T (t))t≥0 on a

Banach spaceX,

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) andψ ∈ Lp([−r, 0],Cm).


Left shift semigroupsThe operator

QXf =∂

∂θf,

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] andϕ ∈ 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 functionx ∈ Lp

loc(R+, X) is a control system onLp([−r, 0], X) andX, 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 associatedwith SX(t).S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 15/30

In fact, BX is the delta function at zero. For the con-trol function x ∈ L

ploc[−r,∞) of the control system

(SX ,ΦX) with x(θ) = ϕ(θ) for a.e. θ ∈ [−r, 0], thestate trajectory of(SX ,ΦX) is the history function ofxgiven by

xt = SX(t)ϕ+ ΦX(t)x, t ≥ 0.

Similarly, we can defineQC, SC,ΦC andBC := (λ − (QC)−1)eλ. For the control system(SC,ΦC) represented byBC, 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

Assumptions

Consider the following assumptions:

(A1) L is an admissible observation operator forSX and (QX ,BX , L) generates a regular sys-tem on the state spaceLp([−r, 0], X), the controlspaceX and the observation spaceX.

(A2) Bk is an admissible observation operatorfor SC and(QC,BC, Bk) generates a regular sys-tem on the state spaceLp([−r, 0],C), the con-trol spaceC and the observation spaceX for allk = 1, 2, · · · ,m.


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) canbe rewritten as

ξ(t) = AL,Bξ(t) + Bu(t), t ≥ 0,

ξ(0) = (x, ϕ, ψ)⊤ ∈ X ,

(8)


the generatorAL,B : D(AL,B) ⊂ Z → Z,

AL,B =

AL

B

0

0 0 QCm

, with AL =

A L

0 ddσ

D(AL,B) = D(AL) ×D(QCm),

(9)

the control operator is

Bu =(

0 0 BCmu

)⊤, u ∈ C

m, (10)

The open-loop(AL,B,B) is well-posed in the sense thatB is an

admissible control operator forAL,B.

(Hadd & Idriss, IMA J. Control Inform. ’05)S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 19/30

Feedback stabilizability: DefinitionAssume 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 existsC ∈ L(D(AL,B),Cm) such that

the triple(AL,B,B, C) generates a regular linearsystemΣ onZ ,Cm,Cm,

the identity matrixK = ICm : Cm → C

m is anadmissible feedback forΣ, and

the closed-loop system associated withΣ andKis internally stable.


N&S condition

Theorem 3: Assume the conditions (A1) and (A2)are satisfied andT (t) is compact fort > 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 allλ ∈ σ+(AL), where

∆(λ) = (λ− A) − Leλ, D(∆(λ)) = D(A).


Key of the proof

The proof of this theorem is based on a generalizedHautus criterion and the following expression

Bu = (µ− (AL,B)−1)

(

R(µ,AL)(

Beµu0

)

eµu

)

,

for u ∈ Cm, µ ∈ ρ(AL).


A rank condition: Reflexive XSinceT (t) is assumed to be compact fort > 0, theunstable setσ+(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 kernelKer∆(λi)∗ as

di = dim Ker∆(λi)∗ i = 1, 2, ..., l

and the basis ofKer∆(λi)∗ by (ϕi1, ϕ

i2, · · · , ϕ

idi).


Theorem 4: Assume that (A1)–(A2) are satisfied,thespaceX is reflexiveandT (t) is compact fort > 0.Then(AL,B,B) is feedback stabilizable if and only if

RankBλi= di, for i = 1, 2, . . . , l,

where

Bλi=

〈B1eλi1, ϕi

1〉〈B1eλi

1, ϕi2〉 · · · · · · 〈B1eλi

1, ϕidi〉

〈B2eλi1, ϕi

1〉〈B2eλi

1, ϕi2〉 · · · · · · 〈B2eλi

1, ϕidi〉

......

......

......

〈Bmeλi1, ϕi

1〉〈Bmeλi

1, ϕi2〉 · · · · · · 〈Bmeλi

1, ϕidi〉

.

〈·, ·〉: the duality pairing betweenX andX∗.The proof is mainly based on the invariance of admis-sibility of observation operators.


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 ofKer∆(λi)

∗ isdi for i = 1, 2, · · · , l and the basis ofKer∆(λi)

∗ isϕi1, ϕ

i2, · · · , ϕ

idi

. Denote then × di matrix formed bythe basis as

ϕi = (ϕi1 ϕi2 · · · ϕidi

), i = 1, 2, · · · , l.


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 unstableλi.


A more general result

When the control spaceU is not finite dimensional, asimilar necessary condition holds.

SeeS. Hadd and Q.-C. Zhong, On feedback stabilizability of linearsystems with state and input delays in Banach spaces, provision-ally accepted for publication in IEEE Trans. on AC.


Example 1Consider the system (12) with

A =

1 1

0 −1

, A1 = 0, P =

p11

p21

, P1 =

p111

p121

.

Hence,∆(λ) = λI − A, σ(A) = {−1, 1}, σ+ = {1}

Ker∆(1)∗ = span{

ϕ11

}

with ϕ11 =

2

1

, d1 = 1

The rank condition is

Rank(

(p11+e−rp1

11

p21+e−rp121

)∗ ( 21 ))

= Rank(2p11+p21+e−r(2p1

11+p121)) = 1.

i.e. 2p11 + p21 + e−r(2p111 + p1

21) 6= 0.


Example 2Consider the system (12) with

A =

0 0

e−r 1

, A1 =

0 0

−1 0

, P =

1

0

, P1 = 0.

Here

∆(λ) =

λ 0

−e−r + e−λr λ − 1

, σ+ = {0, 1},

and

Ker∆(0)∗ = span{( 10)} and Ker∆(1)∗ = span{( 0

1)}.

Now we have

Rank(

( 10)∗ ( 1

0))

= 1 and Rank(

( 10)∗ ( 0

1))

= 0.

Thus,λ1 = 0 is a stabilizable eigenvalue butλ2 = 1 is not.


Summary

Background and motivation

Hautus criterionStabilizability of systems with state delaysOlbrot’s rank condition for systems withstate+input delays

Stabilizability of state–input delay systems

A rank condition

Two examples


on feedback stabilizability of time-delay systems in...

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