Passivity of infinite-dimensional linear systems with state, input - - PowerPoint PPT Presentation

Passivity of infinite-dimensional linear systems with state, input and

• utput delays
• S. Hadd and Q.-C. Zhong

q.zhong@liv.ac.uk

• Dept. of Electrical Eng. & Electronics

The University of Liverpool United Kingdom

Outline

Introduction to passivity The case for systems with discrete delays The case for systems with distributed delays Conclusion

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 2/27

Introduction to passivity

Roughly speaking, passivity means that the system does not have internal energy sources. Importance: closely related to stability and can be used to solve stabilization problems, e.g., a passive system is stable. It is mainly introduced for finite dimensional systems and has, recently, been extended to infinite dimensional ones. Infinite-dimensional systems: the place of the spectrum in the left half plane is not sufficient for stability. Passivity for state-delay systems is extensively studied, but not for general state-input delay systems. Tools: standard estimation of certain quadratic functions, also called Lyapunov functions.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 3/27

Objective

To investigate the passivity of linear systems with state, input and output systems in Hilbert spaces. To extend the existing theory for passivity of delay systems using a semigroup approach.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 4/27

The case with discrete delays

Consider the system ˙ x(t) = Ax(t) + A1x(t − r) + Bu(t) + B1u(t − r), y(t) = Cx(t), t ≥ 0, (1) together with the initial conditions x(0) = x, x(t) = ϕ(t), u(t) = ψ(t), t ∈ [−r, 0]. A : D(A) ⊂ X → X is the generator of a C0-semigroup on a Hilbert space X, A1 ∈ L(X) and C ∈ L(X, U), B ∈ L(U, X) and B1 ∈ L(U, X), with a Hilbert space U, ϕ : [−r, 0] → X and ψ : [−r, 0] → U are square integrable functions.

Here L(E, F) is the space of all linear bounded operators from E to F with L(E) = L(E, E).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 5/27

Notation

Let (Z, ·, ·) be a Hilbert space with norm z =

• z, z. Let

G : D(G) ⊂ Z → Z be a densely defined linear operator. The adjoint operator G∗ of G is defined as D(G∗) =

• z ∈ Z : ∃ γz ≥ 0,

|Gx, z| ≤ γzx, ∀ x ∈ D(G)

• ,

Gx, z = x, G∗z, ∀ x ∈ D(G), ∀ z ∈ D(G∗). If G : D(G) → Z is a generator then G∗ is so as well. Denote by [D(G∗)]′ the strong dual of D(G∗) then D(G) ⊂ Z ⊂ [D(G∗)]′ with continuous embedding.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 6/27

Define QZf = ∂ ∂θf, D(QZ) = {f ∈ W 1,2([−r, 0], Z) : f(0) = 0}. Then QZ generates a C0-semigroup on L2([−r, 0], Z). For any t ≥ 0 and a function g : [−r, 0] → Z, denote g(t + ·) : [−r, 0] → Z, θ → g(t + θ).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 7/27

Reformulation

Define the space X0 := X × L2([−r, 0], X) and A0 = A A1δ−r

∂ ∂θ

• ,

D(A0) = x

ϕ

• ∈ D(A) × W 1,p([−r, 0], X) :

ϕ(0) = x

• ,

Then A0 generates a C0-semigroup on X and it is closely related to the state-delay equation associated with the system (1).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 8/27

Define X = X0 × L2([−r, 0], U), and ξ : [0, ∞) − → X, t → (x(t), x(t + ·), u(t + ·))⊤. Then the delay system (1) can be rewritten as    ˙ ξ(t) = Aξ(t) + Bu(t), t ≥ 0, y(t) = Cξ(t), t ≥ 0, ξ(0) = (x, ϕ, ψ)⊤ ∈ X, (2)

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 9/27

the generator A =     A A1δ−r B1δ−r

d dσ

QU     , D(A) = D(A0) × D(QU), the control operator B =

• B

BU ⊤ , where BU ∈ L

• U,
• D((QU)∗)

′ satisfies B∗

Uf = f(0) for

f ∈ W 1,2([−r, 0], U), the observation operator C : X → U, C =

• C
• .
• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 10/27

Passivity of system (1): Definition

The operator A generates a C0-semigroup T (t) on X and the state trajectory of the system (2) is ξ(t) = T (t)ξ(0) + t T−1(t − s)Bu(s) ds (3) for t ≥ 0 and ξ(0) ∈ X. Definition 1: Let P ∈ L(X) be a self-adjoint posi- tive operator. The delay system (1) (or (2)) is called impedance P-passive if, for all t > 0, t y(s), u(s) ds ≥ 1

2Pξ(t), ξ(t) − 1 2Pξ(0), ξ(0).

(4)

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 11/27

Interpretation

This definition corresponds to regarding E(t) = 1

2Pξ(t), ξ(t)

as the energy stored in the system at time t, and y(s), u(s) as the incoming energy at time t. Then Definition 1 says that the net increment of energy in the system is not greater than the total incoming energy (some en- ergy is dissipated and, hence, the concept of passiv- ity). In other words, no energy is generated inside the system.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 12/27

Definition 2: Let P ∈ L(X) be a self-adjoint positive

• perator. The delay system (1) (or (2)) is called output-

strictly impedance P-passive if there exists ε > 0 such that for all t ≥ 0, the solution (ξ, y) of the system (2) satisfies 2 t y(s), u(s) ds ≥ Pξ(t), ξ(t) − Pξ(0), ξ(0) +ε t y(τ)2dτ. (5)

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 13/27

Passivity of system (1): Conditions

If there exist positive, self-adjoint operators P, S ∈ L(X) and R ∈ L(U), and ε > 0 such that A∗P + PA + PA1S−1A∗

1P + PB1R−1B∗ 1P + S < εC∗C

C∗ = PB (6) then the delay system (1) is output-strictly impedance P-passive with the self-adjoint positive

• perator P ∈ L(X) defined by

P =     P S R     . (7) Here S ∈ L(L2([−r, 0], X)) and R ∈ L(L2([−r, 0], U)) are positive and self-adjoint multi- plicative operators defined by S : L2([−r, 0], X) → L2([−r, 0], X), (Sf)(s) = Sf(s) R : L2([−r, 0], U) → L2([−r, 0], U), (Rg)(s) = Rg(s).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 14/27

Sketch of the proof

Using the inequality (5) we have, for ∀ x

ϕ ψ

• ∈ D(A),

P x

ϕ ψ

• ∈ D(A∗),

(8) and

• PA

x

ϕ ψ

• ,

x

ϕ ψ

+

• A∗P

x

ϕ ψ

• ,

x

ϕ ψ

≤ εC∗Cx, x. (9)

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 15/27

The adjoint operator of A can be found as A∗ =     A∗ δ0 − d

dσ

− d

dσ

    (10) D(A∗) = x

ϕ ψ

• ∈ D(A∗) × W 1,2([−r, 0], X) × W 1,2([−r, 0], U) :

ϕ(−r) = A∗

1x, ψ(−r) = B∗ 1x

• .

Now the proof follows by combining (8), (9) and (10).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 16/27

Remarks

The system (1) with B1 = 0 (without input delay) was studied by Niculescu and Lozano. In this case, the calculus are simplified since the state-delay system can be reformulated as a distributed-parameter system with bounded control and observation operators. The result of Niculescu and Lozano follows from our result by setting B1 = 0 in (6). In the presence of input delays, the state-input delay system (1) can be only transformed into a distributed-parameter system with unbounded control operator. This is why we have to use the properties of the adjoint operators associated with the system.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 17/27

The case with distributed delays

Consider the system ˙ x(t) = Ax(t) + Lx(t + ·) + Bu(t) + B1u(t − r), y(t) = Cx(t) + Nx(t + ·), t ≥ 0, (11) where the operators L and N are defined as Lϕ = A1ϕ(−r) +

−r

A2ϕ(θ) dθ, Nϕ =

−r

C1ϕ(θ) dθ, with A2 ∈ L(X) and C1 ∈ L(X, U).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 18/27

Reformulation

Similarly, as in the previous section, the delay system (11) can be reformulated as system (2) but with different generator and

• bservation operator given, respectively, by

A =     A L B1δ−r

d dσ

QU     , D(A) = x

ϕ

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

(12) C : X − → U, C = (C N 0). (13) The control operator B remains the same.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 19/27

Definition

We note that the system defined by A, B, C (i.e. the system (2)) is regular in the Salamon–Weiss sense. Definition 3: The delay system (11) is called output strictly impedance passive if there exists a function V : X → [0, ∞) such that the following inequality

2 t y(s), u(s) ds ≥ V (ξ(t)) − V (ξ(0)) + ε t y(τ)2dτ. (14)

for any t ≥ 0 and for some ε > 0. It is called impedance passive if the ε in (14) is zero. Here, the energy storage function is not given explic- itly.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 20/27

Conditions for the passivity

If there exist four positive and self-adjoint operators P, S, K ∈ L(X), R ∈ L(U), and an opera- tor M ∈ L(X) with Mx, y ≥ 0, ∀ x, y ≥ 0, and ε > 0 such that  ∆ H   < ε  C∗C N∗N   C = B∗P, C1 = B∗M, (15) where ∆ = A∗P + PA + (PA1 − M)(S−1A∗

1P + S−1M − M)

+ PB1R−1B∗

1P + S + M ∗ + r

1 4 Σ∗K−1Σ + K

• ,

H = 1(·)M ∗(A1S−1PM + A2)Ψ, with Σ : = M ∗ A + A1S−1(A∗

1P − M ∗) + B1R−1B∗ 1P

• + 2(A∗

1P − 2εC∗ 1 C),

Ψϕ =

−r

ϕ(θ) dθ, 1(θ) = 1 ∀θ ∈ [−r, 0], then (11) is output-strictly impedance passive.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 21/27

Sketch of the proof

Let S and R as in the previous case. Now let M : L2([−r, 0], X) → X be defined by Mϕ =

−r

Mϕ(θ) dθ. Define K ∈ L(X) by K =   P M M∗ S R   . The condition on M shows that K is positive.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 22/27

Let K ∈ L(X) be positive and self-adjoint. Define the following Lyapunov function V (ξ(t)) = Kξ(t), ξ(t) +

−r

t

t+θ

Kx(s), x(s)ds dθ. Here ξ(t) = (x(t), x(t + ·), u(t + ·))⊤, t ≥ 0.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 23/27

Using (14) we obtain, for ∀ x

ϕ ψ

• ∈ D(A),

K x

ϕ ψ

• ∈ D(A∗),

(16) and

• KA

x

ϕ ψ

• ,

x

ϕ ψ

+

• A∗K

x

ϕ ψ

• ,

x

ϕ ψ

+ rKx, x −

−r

Kϕ(θ), ϕ(θ) dθ ≤ 2εCx, Nϕ + εCx, Cx + εNϕ, Nϕ. (17)

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 24/27

The adjoint operator of A is

A∗ =     A∗ δ0 Γ − d

dσ

− d

dσ

    (18) D(A∗) = x

ϕ ψ

• ∈ D(A∗) × W 1,2([−r, 0], X) × W 1,2([−r, 0], U) :

ϕ(−r) = A∗

1x, ψ(−r) = B∗ 1x

• ,

where Γ : X → L2([−r, 0], X) is such that (Γx)(θ) = A∗

2x for all x ∈ X, θ ∈ [−r, 0]. Now the proof follows

by combining (16), (17), (18).

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 25/27

Remark

In the case of no input–output delays, R disappears and M = 0. Then we obtain the passivity condition

• f infinite dimensional state delay systems, which is

an extension of the result of Niculescu and Lozano, IEEE-AC 46: 460–464, 2001.

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 26/27

Summary

Passivity of systems with discrete delays Passivity of systems with distributed delays

• S. HADD & Q.-C. ZHONG: PASSIVITY OF DELAY SYSTEMS – p. 27/27