Dynamic equations on time-scale: application to stability analysis - PowerPoint PPT Presentation

Introduction Problem Statement Stability analysis Stabilization Dynamic equations on time-scale: application to stability analysis and stabilization of aperiodic sampled-data systems C. Briat and U. T. Jönsson KTH, Stockholm, Sweden IFAC World Congres 2011, Milano, Italy C. Briat [KTH / ] 1/22

Introduction Problem Statement Stability analysis Stabilization Outline ◮ Introduction ◮ Problem statement and Preliminaries ◮ Stability analysis ◮ Stabilization ◮ Conclusion and Future Works C. Briat [KTH / ] 2/22

Introduction Problem Statement Stability analysis Stabilization 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], [Hetel et al.], [Oishi et al.], [Ariba et al.] ◮ Functional-based approaches [Seuret] C. Briat [KTH / ] 3/22

Introduction Problem Statement Stability analysis Stabilization Problem statement C. Briat [KTH / ] 4/22

Introduction Problem Statement Stability analysis Stabilization 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 , k ∈ N . ◮ Stability analysis problem: given K , find the set T of admissible T > 0 for which stability still holds. ◮ Find controller gain K maximizing the maximal sampling period. C. Briat [KTH / ] 5/22

Introduction Problem Statement Stability analysis Stabilization Systems over time-scales [Bohner] ◮ Unification/generalization of continuous-time and discrete-time systems � ◮ Examples: T = R + , T = Z + , T = { 0 } ∪ { 1 /k } k ∈ N , T = [ t 2 k , t 2 k +1 ] k ∈ N ◮ Forward jump operator: σ ( t ) = { inf s ∈ T : t < s } ◮ Graininess: µ ( t ) = σ ( t ) − t (distance) ◮ Dynamical system on time-scale: x ∆ ( t ) = Ax ( t ) + Bu ( t ) (3) x (0) = x 0 ◮ ∆ operator [Goodwin]:  f ( t ) − f ( s ) lim if µ ( t ) = 0   t − s  f ∆ ( t ) := s → t,s ∈ T f ( σ ( t )) − f ( t ) if µ ( t ) > 0 .   µ ( t )  C. Briat [KTH / ] 6/22

Introduction Problem Statement Stability analysis Stabilization Stability analysis via Lyapunov functions ◮ Linear systems → V ( x ) = x T Px ◮ Stability condition A T P + PA + µA T PA ≺ 0 , P = P T ≻ 0 (4) ◮ µ fixed: equivalent to a DT criterion ◮ µ → 0 : equivalent to a CT criterion ◮ Spectrum condition: λ ( A ) ⊂ D ( − 1 /µ, 1 /µ ) ◮ µ → 0 : D ( − 1 /µ, 1 /µ ) → C − ◮ µ = µ 0 � = 0 : D (1 /µ 0 , 1 /µ 0 ) analogous to the unit disc. C. Briat [KTH / ] 7/22

Introduction Problem Statement Stability analysis Stabilization Representation of sampled-data systems ◮ DT system: � t � � e A ( t − t k ) + e A ( t − s ) dsBK ( t k ) x ( t ) = x ( t k ) (5) t k ◮ System on TS: z ∆ ( t k ) = A ∆ ( µ ( t k )) z ( t k ) (6) where the new state z coincide with x at sampling instants and e Aµ ( tk ) + Φ( µ ( t k )) BK ( t k ) − I µ ( t k ) − 1 � � A ∆ ( µ ( t k )) = � tk (7) e A ( µ ( tk ) − s ) ds Φ( µ ( t k )) = 0 C. Briat [KTH / ] 8/22

Introduction Problem Statement Stability analysis Stabilization Stability analysis C. Briat [KTH / ] 9/22

Introduction Problem Statement Stability analysis Stabilization A general stability result Theorem The dynamical system z ∆ ( t k ) = A ∆ ( t k ) z ( t k ) , z ( t 0 ) = z 0 , ( t 0 , t k ) ∈ T 2 , t k ≥ t 0 is robustly exponentially stable for µ ( t k ) ∈ µ if the following statements hold: 1. A ( t k ) is rd-continuous and regressive, i.e. det( I + µ ( t k ) A ( t k )) � = 0 for all t k ∈ T and µ ( t k ) ∈ µ . 2. There exist P : T → S n ++ verifying θ 1 I � P ( t k ) � θ 2 I for some 0 < θ 1 < θ 2 < + ∞ and β ∈ (0 , 1 / sup { µ } ) such that M µ ( t k ) ( P ( σ ( t k )) , A ∆ ( t k ) , P ∆ ( t k ) + βP ( t k )) � 0 (8) holds for all µ ( t k ) ∈ µ and all t k ∈ T . C. Briat [KTH / ] 10/22

Introduction Problem Statement Stability analysis Stabilization Graininess dependent Lyapunov function Theorem The dynamical system z ∆ ( t k ) = A ∆ ( µ ( t k )) z ( t k ) , z ( t 0 ) = z 0 , ( t 0 , t k ) ∈ T 2 , t k ≥ t 0 is robustly exponentially stable for µ ( t k ) ∈ µ , inf { µ } > 0 , if the following statements hold: 1. A ( µ ( t k )) is rd-continuous and regressive, i.e. det[ I + µA ( µ )] � = 0 for all µ ∈ µ . 2. There exist a bounded matrix function P : µ → S n ++ such that M µ ( P ( µ n ) , A ∆ ( µ c ) , S ( µ c , µ n )) ≺ 0 (9) holds for all ( µ n , µ c ) ∈ µ 2 and where S ( µ c , µ n )) = µ − 1 ( P ( µ n ) − P ( µ c )) . c C. Briat [KTH / ] 11/22

Introduction Problem Statement Stability analysis Stabilization Example 1 ◮ Scalar system ([Mirkin], [Fridman], [Fujioka]) x ( t ) = − 2 x ( t ) + x ( t k ) . ˙ (10) ◮ TS formalism A ∆ ( µ ) = 3 e − 2 µ − 1 � � . (11) 2 µ ◮ Lyapunov condition � � �� 3 1 + 3 e − 2 µ − 1 e − 2 µ − 1 � � < 0 µ 4 ◮ True for µ > 0 ◮ When µ → 0 , the LHS tends to -6. C. Briat [KTH / ] 12/22

Introduction Problem Statement Stability analysis Stabilization Example 2 Let us consider the system � 0 � � � 1 0 0 A = , BK = (12) 0 − 0 . 1 − 0 . 375 − 1 . 15 Ref. Maximal varying sampling period [Fridman,04] 0.8696 [Yue,05] 0.8871 [Ariba,07 1.009 [Naghshtabrizi,08] 1.1137 [Mirkin,07] 1.3659 [Seuret,09b] 1.6894 [Oishi,09] 1.7294 Proposed result 1.72941 Theoretical 1.7294194 (constant case) C. Briat [KTH / ] 13/22

Introduction Problem Statement Stability analysis Stabilization Example 3 � 0 � � � 0 1 0 A = , BK = − 2 0 . 1 1 0 ◮ Pathological sampling periods: { 2 . 2228 , 4 . 4457 , 6 . 6685 , 8 . 8913 , 11 . 1142 , . . . } , ◮ Constant sampling period ◮ P ( µ ) = P 0 → µ = [0 . 5004 , 1 . 9203] ◮ P ( µ ) = P 0 + P 1 µ → µ = [0 . 2013 , 2 . 0204] ◮ Other disjoint intervals ∈ { [2 . 4706 , 3 . 6963] , [5 . 4307 , 6 . 3447] , [7 . 0277 , 7 . 7249] , µ [10 . 3916 , 10 . 7559] , [11 . 4973 , 11 . 7179] } . ◮ Aperiodic case: ∈ { [0 . 2187 , 1 . 0031] , [0 . 500 , 1 . 9256] , [2 . 47 , 3 . 6] , µ [2 . 77 , 3 . 6963] , [5 . 4584 , 5 . 8004] , [5 . 8172 , 6 . 3447] , (13) [7 . 0339 , 7 . 5009] , [7 . 5000 , 7 . 7070] } . C. Briat [KTH / ] 14/22

Introduction Problem Statement Stability analysis Stabilization Stabilization C. Briat [KTH / ] 15/22

Introduction Problem Statement Stability analysis Stabilization Robust state-feedback design Theorem There exists a quadratically stabilizing switching sampling-period-dependent state-feedback control law if there exist ++ and a bounded continuous matrix function U : µ → R n × m such that the LMI X ∈ S n � Ξ 11 ( µ ) Ξ 12 ( µ ) � ≺ 0 (14) − µ − 1 X ⋆ holds for all µ ∈ µ with µ − 1 [ A e ( µ ) X + Φ( µ ) BU ( µ )] T Ξ 12 ( µ ) = Ξ 12 ( µ ) + Ξ 12 ( µ ) T (15) Ξ 11 ( µ ) = A e ( µ ) = exp( Aµ ) − I In such a case, the controller matrix is given by K ( µ ) = U ( µ ) X − 1 . C. Briat [KTH / ] 16/22

Introduction Problem Statement Stability analysis Stabilization Sampling-period dependent controller Theorem There exists a robustly stabilizing sampling-period-dependent state-feedback control law if there exist a matrix Z ∈ R n × n , bounded continuous matrix functions ++ , U : µ → R n × m and a sufficiently large positive scalar function P : µ → S n ǫ : µ 2 → R ++ such that the matrix inequality   Ξ 11 ( µ c , µ n ) Ξ 12 ( µ c , µ n ) Z  ≺ 0 ⋆ Ξ 22 ( µ c , µ n ) 0 (16)  ⋆ ⋆ Ξ 33 ( µ c , µ n ) holds for all µ ∈ µ , inf { µ } > 0 , with S ( µ c , µ n ) = µ − 1 ( P ( µ n ) − P ( µ c )) and c − Z − Z T + µ c P ( µ n ) Ξ 11 ( µ c , µ n ) = µ − 1 Ξ 12 ( µ c , µ n ) = [ A e ( µ c ) X + Φ( µ c ) BU ( µ c )] + P ( µ n ) c (17) Ξ 22 ( µ c , µ n ) = − ǫ ( µ c , µ n ) P ( µ n ) + S ( µ c , µ n ) Ξ 33 ( µ c , µ n ) = − P ( µ n ) /ǫ ( µ c , µ n ) A e ( µ c ) = exp( Aµ c ) − I In such a case, the controller matrix is given by K ( µ c ) = U ( µ c ) Z − 1 . C. Briat [KTH / ] 17/22

Introduction Problem Statement Stability analysis Stabilization Examples ◮ System 1: � 0 � � � 0 1 A = and B = . (18) − 2 0 . 1 1 ◮ System 2: � 7 � 1 � � 4 A = and B = . (19) 5 11 1 C. Briat [KTH / ] 18/22

Introduction Problem Statement Stability analysis Stabilization Examples µ + for System 1 µ + for System 2 degree of K ( µ ) 0 1.8938 0.1566 1 2.2072 0.3299 2 2.2121 0.5020 3 2.2170 0.6717 4 2.2182 0.8353 5 2.2206 0.9817 6 2.2206 1.0196 C. Briat [KTH / ] 19/22

Introduction Problem Statement Stability analysis Stabilization Conclusion C. Briat [KTH / ] 20/22