Stability Criteria for Asynchronous Sampled-data Systems - A - PowerPoint PPT Presentation

Introduction Problem Statement Stability analysis Conclusion Stability Criteria for Asynchronous Sampled-data Systems - A Fragmentation Approach C. Briat and A. Seuret KTH, Stockholm, Sweden GIPSA-Lab, Grenoble, France IFAC World Congres 2011, Milano, Italy C. Briat [KTH / ] 1/18

Introduction Problem Statement Stability analysis Conclusion Outline ◮ Introduction ◮ Problem statement and Preliminaries ◮ Stability analysis ◮ Conclusion and Future Works C. Briat [KTH / ] 2/18

Introduction Problem Statement Stability analysis Conclusion Aperiodic sampled-data systems ◮ Discrete-time systems with varying sampling period ◮ Several frameworks ◮ Time-delay systems [Yu et al.], [Fridman et al.] ◮ Impulsive systems [Naghshtabrizi et al.], [Seuret] ◮ Sampled-data systems [Mirkin] ◮ Robust techniques [Fujioka], [Oishi et al.], [Ariba et al.] ◮ Functional-based approaches [Seuret] C. Briat [KTH / ] 3/18

Introduction Problem Statement Stability analysis Conclusion Problem statement C. Briat [KTH / ] 4/18

Introduction Problem Statement Stability analysis Conclusion System and Problem definition ◮ Continuous-time LTI system x ( t ) ˙ = Ax ( t ) + Bu ( t ) (1) x (0) = x 0 with state x and control input u . ◮ Sampled-data control law u ( t ) = Kx ( t k ) , t ∈ [ t k , t k +1 ) (2) where T k := t k +1 − t k ∈ T := [ T min , T max ] , k ∈ N . ◮ Stability analysis problem: given K , find the set T for which for all T k ∈ T stability holds. C. Briat [KTH / ] 5/18

Introduction Problem Statement Stability analysis Conclusion Quadratic stability result ◮ It is straightforward to show that the system is asymptotically stable if there exists P = P T ≻ 0 such that the LMI Φ( T ) T P Φ( T ) − P ≺ 0 for all T ∈ T and where � T Φ( T ) = e AT + e A ( T − s ) dsBK (3) 0 ◮ LMI difficult to check (although possible [Fujioka]) ◮ Difficult to extend to uncertain systems or nonlinear systems ◮ Alternative way ? C. Briat [KTH / ] 6/18

Introduction Problem Statement Stability analysis Conclusion Alternative discrete-time stability condition Theorem Let V ( x ) = x T Px , P = P T ≻ 0 , P finite and define κ k ( τ ) := x ( t k + τ ) , χ k ∈ C ([0 , T ] , R n ) , k ∈ N . Then the two following statements are equivalent: (i) The LMI Φ( T ) T P Φ( T ) − P ≺ 0 holds for all T ∈ T . (ii) There exists a continuous functional V 1 : R × C ([0 , T ] , R ) → R , differentiable over [ t k t k +1 ) satisfying V 1 ( T k , κ k ) = V 1 (0 , κ k ) (4) for all k ∈ N and such that the functional W ( τ ( t ) , κ k ) := V ( x ( t )) + V 1 ( τ ( t ) , κ k ( τ ( t ))) satisfies W ( τ ( t ) , κ k ) = d ˙ dt W ( τ ( t ) , κ k ) < 0 (5) for all τ ∈ [0 , T k ] , T k ∈ T , k ∈ N − { 0 } . Moreover, if one of these two statements is satisfied, the solutions of the sampled-data are asymptotically stable. C. Briat [KTH / ] 7/18

Introduction Problem Statement Stability analysis Conclusion Illustration of the result C. Briat [KTH / ] 8/18

Introduction Problem Statement Stability analysis Conclusion Connection with impulsive approach ◮ In the impulsive framework [Naghshtabrizi et al.], [Seuret], the functional may be considered x ( t ) T Px ( t ) + ( T k − τ )( x ( t ) − x ( t k )) T S ( x ( t ) − x ( t k )) = V � t x ( s ) T R ˙ +( T k − τ ) ˙ x ( s ) ds, t ∈ [ t k , t k +1 ] t k where τ = t − t k and P, S, R symmetric positive definite. ◮ When τ = 0 and τ = T k , the two last terms are 0 ◮ Functional satisfies the boundary conditions → S and R do not need to be positive definite. C. Briat [KTH / ] 9/18

Introduction Problem Statement Stability analysis Conclusion Stability analysis C. Briat [KTH / ] 10/18

Introduction Problem Statement Stability analysis Conclusion Proposed functional ◮ Starting point [Seuret] x ( t ) T Px ( t ) + V 1 ( t ) = V � t ( T k − τ ) ζ ( t ) T [ Sζ ( t ) + 2 Qx ( t k )] + ( T k − τ ) x ( s ) T R ˙ = ˙ x ( s ) ds V 1 t k +( T k − τ ) τx ( t k ) T Ux ( t k ) ζ ( t ) = x ( t ) − x ( t k ) ◮ Fragmentation (Discretization) N pieces ( N + 1 points) k ( t ) = t k + N − i t i ( t − t k ) N � t � t i N − 1 k ( t ) x ( s ) T R ˙ x ( s ) T R i ˙ � ˙ x ( s ) ds → ˙ x ( s ) ds t i − 1 t k ( t ) i =0 k k ( t )) − x ( t i − 1 i =0 ,...,N − 1 { x ( t i ζ ( t ) → ζ k ( t ) = col ( t )) } k C. Briat [KTH / ] 11/18

Introduction Problem Statement Stability analysis Conclusion Stability condition Theorem The sampled-data system is asymptotically stable for any time-varying sampling period in [ T MIN , T MAX ] if there exist constant matrices P = P T ≻ 0 , R i = R T i ≻ 0 , i = 0 , . . . , N − 1 and U = U T ∈ R n × n , S = S T ∈ R nN × nN , Q ∈ R nN × n and Y ∈ R n ( N +1) × nN such that the LMIs Ψ 1 + T MIN (Ψ 2 + Ψ 3 ) ≺ 0 , Ψ 1 + T MAX (Ψ 2 + Ψ 3 ) ≺ 0 � Ψ 1 − T MIN Ψ 3 � Ψ 1 − T MAX Ψ 3 T MIN Y � T MAX Y � 0 , 0 − α − ¯ ≺ − α + ¯ ≺ ⋆ R ⋆ R (6) hold where α − = NT MIN , α + = NT MAX . ◮ Affine in T → semi-infiniteness easy to handle ◮ Affine in the system matrices → easy to extend to uncertain system ◮ No state-transition matrix involved → nonlinear systems C. Briat [KTH / ] 12/18

Introduction Problem Statement Stability analysis Conclusion Example 1 ◮ Let us consider the system x ( t ) = Ax ( t ) + BKx ( t k ) ˙ with � 0 � � � 1 0 0 A = , BK = 0 − 0 . 1 − 0 . 375 − 1 . 15 ◮ Constant sampling-period T = [0 , 1 . 7294] . Theorems Ex.1 [Fridman et al., 04] [0 , 0 . 869] [Naghshtabrizi et al., 08] [0 , 1 . 113] [Fridman et al.,10] [0 , 1 . 695] [Liu et al., 09] [0 , 1 . 695] Proposed result, N = 1 [0 , 1 . 721] Proposed result, N = 3 [0 , 1 . 727] Proposed result, N = 5 [0 , 1 . 728] C. Briat [KTH / ] 13/18

Introduction Problem Statement Stability analysis Conclusion Example 2 ◮ Let us consider the system x ( t ) = Ax ( t ) + BKx ( t k ) ˙ with � − 2 � − 1 � � 0 0 A = , BK = 0 − 0 . 9 − 1 − 1 ◮ Constant sampling-period T = [0 , 3 . 2716] Theorems Ex.2 [Fridman et al., 04] [0 , 0 . 99] [Naghshtabrizi et al., 08] [0 , 1 . 99] [Fridman et al.,10] [0 , 2 . 03] [Liu et al., 09] [0 , 2 . 53] Proposed result, N = 1 [0 , 2 . 51] Proposed result, N = 3 [0 , 2 . 62] Proposed result, N = 5 [0 , 2 . 64] C. Briat [KTH / ] 14/18

Introduction Problem Statement Stability analysis Conclusion Example 3 ◮ Let us consider the system x ( t ) = Ax ( t ) + BKx ( t k ) ˙ with � 0 � 0 1 � 0 � A = BK = , − 2 0 . 1 1 0 ◮ Constant sampling-period T = [0 . 2007 , 2 . 0207] Theorems Ex.3 [Fridman et al., 04] - [Naghshtabrizi et al., 08] - [Fridman et al.,10] - [Liu et al., 09] - Proposed result, N = 1 [0 . 40 , 1 . 11] Proposed result, N = 3 [0 . 40 , 1 . 28] Proposed result, N = 5 [0 . 40 , 1 . 31] C. Briat [KTH / ] 15/18

Introduction Problem Statement Stability analysis Conclusion Conclusion C. Briat [KTH / ] 16/18

Introduction Problem Statement Stability analysis Conclusion Conclusion ◮ Functional-based approach suitable for stability analysis ◮ Fragmentation improves results ◮ Possible extension to uncertain and nonlinear systems C. Briat [KTH / ] 17/18

Introduction Problem Statement Stability analysis Conclusion Thank you for your attention ! C. Briat [KTH / ] 18/18