on feedback stabilizability of time-delay systems in...
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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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 3/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 4/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 6/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 8/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 9/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 10/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 12/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 13/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 14/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 17/30
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)
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 18/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 20/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 21/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 22/30
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).
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 23/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 24/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 25/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 26/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 27/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 28/30
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.
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 29/30
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
S. HADD & Q.-C. ZHONG: FEEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 30/30