On feedback stabilizability of time-delay systems in Banach spaces - PowerPoint PPT Presentation

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 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. H ADD & Q.-C. Z HONG : F EEDBACK 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 a C 0 -semigroup ( T ( t )) t ≥ 0 on 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. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 3/30

If T ( t ) is compact for t ≥ t 0 > 0 , then the unstable set σ + ( A ) := { λ ∈ σ ( A ) : Re λ ≥ 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 ) + Im B = X (2) for any λ ∈ σ + ( A ) . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 4/30

Stabilizability of state-delay systems What if there is a delay in the state? � ˙ x ( t ) = Ax ( t ) + Lx t + Bu ( t ) , t ≥ 0 , (3) x (0) = z, x 0 = ϕ. A generates a C 0 -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 x t : [ − r, 0] → X , x t ( s ) = x ( t + s ) , t ≥ 0 , B : U → X is linear bounded, initial values: z ∈ X and ϕ ∈ L p ([ − r, 0] , X ) . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 5/30

Transformation (3) into (1) Take the new state variable � x � w ( t ) = , x t the system (3) can be transformed into (1) as w ( t ) = A L w ( t ) + B u ( t ) , w (0) = ( z ˙ ϕ ) , t ≥ 0 , (4) where the operators are defined on the next slide. S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 6/30

The new state space is X := X × L p ([ − r, 0] , X ) . The operators are: A L : D ( A L ) ⊂ X → X , � A � L A L := d 0 dσ � � ϕ ) ∈ D ( A ) × W 1 ,p ([ − r, 0] , X ) : f (0) = x ( z D ( A L ) := and B = ( B B : U → X , 0 ) , which is bounded. S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 7/30

Let S X be the left semigroup on L p ([ − r, 0] , X ) gener- ated by Q X := d dσ, ϕ ∈ W 1 ,p ([ − r, 0] , X ) : ϕ (0) = 0 � � D ( Q X ) := . Assumption : Assume that L is an admissible observa- tion operator for S X , i.e., � τ � LS X ( t ) f � p dt ≤ κ p � f � p , ∀ f ∈ D ( Q X ) , (5) 0 where τ > 0 and κ > 0 are constants. Then, A L gen- erates a C 0 -semigroup ( T L ( t )) t ≥ 0 on X (Hadd ’05). S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 8/30

If T ( t ) is compact for t > 0 then T L ( t ) is compact for t > r (Matrai ’04). λ ∈ σ ( A ) if and only if λ ∈ σ ( A + Le λ ) with ( e λ x )( θ ) = e λθ x for x ∈ X, θ ∈ [ − r, 0] . The unstable set σ + ( A L ) = { λ ∈ σ ( A + Le λ ) : Re λ ≥ 0 } is finite. For each λ ∈ C , define ∆ ( λ ) := λ − A − Le λ , D ( ∆ ( λ )) = D ( A ) . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 9/30

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 ) + Im B = X (6) for any λ ∈ σ + ( A L ) , where ∆ ( λ ) := λ − A − Le λ , D ( ∆ ( λ )) = D ( A ) . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 10/30

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 ) = A 0 x ( t ) + A 1 x ( t − 1) + Pu ( t ) + P 1 u ( t − 1) , ˙ of which the dimension of the delay-free system is n , is equivalent to the condition ∆ ( λ ) P + e − λ P 1 � � = n, Rank for λ ∈ C with Re λ ≥ 0 , where ∆ ( λ ) := λI − A 0 − A 1 e − λ . Only partial results available. S. H ADD & Q.-C. Z HONG : F EEDBACK 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 Banach spaces To introduce an equivalent and compact rank con- dition for the stabilizability of state–input delay systems. S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 12/30

Notation Let ( Z, � · � ) be a Banach space and G : D ( G ) ⊂ Z → Z be a generator of a C 0 -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 ) . → Z − 1 holds. The continuous injection Z ֒ ( 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. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 13/30

System under consideration  x ( t ) = Ax ( t ) + Lx t + Bu t , ˙ t ≥ 0 ,  (7) x (0) = z, x 0 = ϕ, u 0 = ψ  A : D ( A ) ⊂ X → X generates a C 0 -semigroup ( T ( t )) t ≥ 0 on a Banach space X , L : W 1 ,p ([ − r, 0] , X ) → X linear bounded, W 1 ,p ([ − r, 0] , C )) m → X linear � B = ( B 1 B 2 · · · B m ) : bounded, z ∈ X, ϕ ∈ L p ([ − r, 0] , X ) and ψ ∈ L p ([ − r, 0] , C m ) . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 14/30

Left shift semigroups The operator Q X f = ∂ ∂θ f, D ( Q X ) = { f ∈ W 1 ,p ([ − r, 0] , X ) : f (0) = 0 } . generates the left semigroup  0 , t + θ ≥ 0 ,  ( S X ( t ) ϕ )( θ ) = ϕ ( t + θ ) , t + θ ≤ 0 ,  for t ≥ 0 , θ ∈ [ − r, 0] and ϕ ∈ L p ([ − r, 0] , X ) . The pair ( S X , Φ X ) with  x ( t + θ ) , t + θ ≥ 0 ,  (Φ X ( t ) x )( θ ) = 0 , t + θ ≤ 0 ,  for the control function x ∈ L p loc ( R + , X ) is a control system on L p ([ − r, 0] , X ) and X , which is represented by the unbounded admissible control operator B X := ( λ − ( Q X ) − 1 ) e λ , λ ∈ C , where ( Q X ) − 1 is the generator of the extrapolation semigroup associated with S X ( t ) . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 15/30

In fact, B X is the delta function at zero. For the con- trol function x ∈ L p loc [ − r, ∞ ) of the control system ( S X , Φ X ) with x ( θ ) = ϕ ( θ ) for a.e. θ ∈ [ − r, 0] , the state trajectory of ( S X , Φ X ) is the history function of x given by x t = S X ( t ) ϕ + Φ X ( t ) x, t ≥ 0 . Similarly, we can define Q C , S C , Φ C and := ( λ − ( Q C ) − 1 ) e λ . B C For the control system ( S C , Φ C ) represented by B C , we have u t = S C ( t ) ψ + Φ C ( t ) u, t ≥ 0 with u ( θ ) = ψ ( θ ) for a.e. θ ∈ [ − r, 0] . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 16/30

Assumptions Consider the following assumptions: (A1) L is an admissible observation operator for S X and ( Q X , B X , L ) generates a regular sys- tem on the state space L p ([ − r, 0] , X ) , the control space X and the observation space X . (A2) B k is an admissible observation operator for S C and ( Q C , B C , B k ) generates a regular sys- tem on the state space L p ([ − r, 0] , C ) , the con- trol space C and the observation space X for all k = 1 , 2 , · · · , m . S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 17/30

Define Z = X × L p ([ − r, 0] , X ) × L 2 ([ − r, 0] , U ) and take a new state variable ξ ( t ) = ( x ( t ) , x t , u t ) ⊤ . Using conditions (A1)–(A2), the delay system (7) can be rewritten as ˙  ξ ( t ) = A L,B ξ ( t ) + B u ( t ) , t ≥ 0 ,  (8) ξ (0) = ( x, ϕ, ψ ) ⊤ ∈ X ,  S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 18/30

the generator A L,B : D ( A L,B ) ⊂ Z → Z ,   B   A L  A L   A L,B = with A L = 0  ,      d 0  dσ Q C m 0 0 D ( A L,B ) = D ( A L ) × D ( Q C m ) , (9) the control operator is � ⊤ � u ∈ C m , B u = , (10) 0 0 B C m u The open-loop ( A L,B , B ) is well-posed in the sense that B is an admissible control operator for A L,B . (Hadd & Idriss, IMA J. Control Inform. ’05) S. H ADD & Q.-C. Z HONG : F EEDBACK STABILIZABILITY OF DELAY SYSTEMS – p. 19/30