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

Passivity of infinite-dimensional linear systems with state, input and output 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. H ADD & Q.-C. Z HONG : P ASSIVITY 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. H ADD & Q.-C. Z HONG : P ASSIVITY 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. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 4/27

The case with discrete delays Consider the system x ( t ) = Ax ( t ) + A 1 x ( t − r ) + Bu ( t ) + B 1 u ( t − r ) , ˙ (1) y ( t ) = Cx ( t ) , t ≥ 0 , 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 C 0 -semigroup on a Hilbert space X , A 1 ∈ L ( X ) and C ∈ L ( X, U ) , B ∈ L ( U, X ) and B 1 ∈ 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. H ADD & Q.-C. Z HONG : P ASSIVITY 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 : ∃ γ z ≥ 0 , � |� Gx, z �| ≤ γ z � x � , ∀ x ∈ D ( G ) , � Gx, z � = � x, G ∗ z � , ∀ z ∈ D ( G ∗ ) . ∀ x ∈ 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. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 6/27

Define Q Z f = ∂ ∂θf, D ( Q Z ) = { f ∈ W 1 , 2 ([ − r, 0] , Z ) : f (0) = 0 } . Then Q Z generates a C 0 -semigroup on L 2 ([ − 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. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 7/27

Reformulation Define the space X 0 := X × L 2 ([ − r, 0] , X ) and � A A 1 δ − r � A 0 = , ∂ 0 ∂θ �� x ∈ D ( A ) × W 1 ,p ([ − r, 0] , X ) : � D ( A 0 ) = ϕ � ϕ (0) = x , Then A 0 generates a C 0 -semigroup on X and it is closely related to the state-delay equation associated with the system (1). S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 8/27

Define X = X 0 × L 2 ([ − r, 0] , U ) , and → X , t �→ ( x ( t ) , x ( t + · ) , u ( t + · )) ⊤ . ξ : [0 , ∞ ) − Then the delay system (1) can be rewritten as ˙  ξ ( t ) = A ξ ( t ) + B u ( t ) , t ≥ 0 ,  y ( t ) = C ξ ( t ) , t ≥ 0 , (2) ξ (0) = ( x, ϕ, ψ ) ⊤ ∈ X ,  S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 9/27

the generator   A A 1 δ − r B 1 δ − r   d A = D ( A ) = D ( A 0 ) × D ( Q U ) ,  , 0 0   dσ  0 0 Q U the control operator � ⊤ � B = , 0 B B U � ′ � � � D (( Q U ) ∗ ) satisfies B ∗ where B U ∈ L U f = f (0) for U, f ∈ W 1 , 2 ([ − r, 0] , U ) , the observation operator � � C : X → U, C = . 0 0 C S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 10/27

Passivity of system (1) : Definition The operator A generates a C 0 -semigroup T ( t ) on X and the state trajectory of the system (2) is � t ξ ( t ) = T ( t ) ξ (0) + T − 1 ( t − s ) B u ( s ) ds (3) 0 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 2 �P ξ ( t ) , ξ ( t ) � − 1 2 �P ξ (0) , ξ (0) � . 0 (4) S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 11/27

Interpretation This definition corresponds to regarding E ( t ) = 1 2 �P ξ ( 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. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 12/27

Definition 2 : Let P ∈ L ( X ) be a self-adjoint positive operator. 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 � t 2 � y ( s ) , u ( s ) � ds ≥ �P ξ ( t ) , ξ ( t ) � − �P ξ (0) , ξ (0) � 0 � t � y ( τ ) � 2 dτ. + ε 0 (5) S. H ADD & Q.-C. Z HONG : P ASSIVITY 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 + PA 1 S − 1 A ∗ 1 P + PB 1 R − 1 B ∗ 1 P + S < εC ∗ C (6) C ∗ = PB then the delay system (1) is output-strictly impedance P -passive with the self-adjoint positive operator P ∈ L ( X ) defined by   P 0 0   P =  . (7) 0 S 0    0 0 R Here S ∈ L ( L 2 ([ − r, 0] , X )) and R ∈ L ( L 2 ([ − r, 0] , U )) are positive and self-adjoint multi- plicative operators defined by S : L 2 ([ − r, 0] , X ) → L 2 ([ − r, 0] , X ) , ( S f )( s ) = Sf ( s ) R : L 2 ([ − r, 0] , U ) → L 2 ([ − r, 0] , U ) , ( R g )( s ) = Rg ( s ) . S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 14/27

Sketch of the proof � x � Using the inequality (5) we have, for ∀ ϕ ∈ D ( A ) , ψ � x � ∈ D ( A ∗ ) , ϕ P (8) ψ and � x � x � x � x � � � � � � � � A ∗ P ≤ ε � C ∗ Cx, x � . PA ϕ ϕ + ϕ ϕ , , ψ ψ ψ ψ (9) S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 15/27

The adjoint operator of A can be found as   A ∗ δ 0 0 A ∗ =   − d (10) 0 0   dσ   − d 0 0 dσ �� x � D ( A ∗ ) = ∈ D ( A ∗ ) × W 1 , 2 ([ − r, 0] , X ) × W 1 , 2 ([ − r, 0] , U ) : ϕ ψ � ϕ ( − r ) = A ∗ 1 x, ψ ( − r ) = B ∗ 1 x . Now the proof follows by combining (8), (9) and (10). S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 16/27

Remarks The system (1) with B 1 = 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 B 1 = 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. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 17/27

The case with distributed delays Consider the system x ( t ) = Ax ( t ) + Lx ( t + · ) + Bu ( t ) + B 1 u ( t − r ) , ˙ y ( t ) = Cx ( t ) + Nx ( t + · ) , t ≥ 0 , (11) where the operators L and N are defined as � 0 � 0 Lϕ = A 1 ϕ ( − r ) + A 2 ϕ ( θ ) dθ, Nϕ = C 1 ϕ ( θ ) dθ, − r − r with A 2 ∈ L ( X ) and C 1 ∈ L ( X, U ) . S. H ADD & Q.-C. Z HONG : P ASSIVITY 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 observation operator given, respectively, by   A L B 1 δ − r   A = d  , 0 0   dσ  0 0 Q U �� x � ∈ D ( A ) × W 1 ,p ([ − r, 0] , X ) : ϕ (0) = x � D ( A ) = × D ( Q U ) , ϕ (12) C : X − → U, C = ( C N 0) . (13) The control operator B remains the same. S. H ADD & Q.-C. Z HONG : P ASSIVITY OF DELAY SYSTEMS – p. 19/27