Simple conditions for L 2 stability and stabilization of networked - PowerPoint PPT Presentation

Introduction Problem statement Stability Analysis Stabilization Conclusion Simple conditions for L 2 stability and stabilization of networked control systems Y. Ariba, C. Briat, K.H. Johansson KTH, Stockholm, Sweden IFAC World Congres 2011, Milano, Italy C. Briat [KTH / ] 1/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Outline ◮ Introduction ◮ Problem statement ◮ Stability analysis ◮ Control ◮ Conclusion and Future Works C. Briat [KTH / ] 2/20

Introduction Problem statement Stability Analysis Stabilization Conclusion NCS v (t) y (t) Exogenous performance input output t k u (t) x (t) Hold System Network ◮ Remote control ^ x k ◮ Wireless network Controller ◮ Data loss ◮ Time-varying propagation delays ◮ Varying sampling period C. Briat [KTH / ] 3/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Existing approaches ◮ 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.] ◮ Functional-based approaches [Seuret] C. Briat [KTH / ] 4/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Problem statement C. Briat [KTH / ] 5/20

Introduction Problem statement Stability Analysis Stabilization Conclusion NCS model ◮ Process model x ( t ) ˙ = Ax ( t ) + Bu ( t ) + Ev ( t ) y ( t ) = Cx ( t ) + Du ( t ) + Fv ( t ) (1) x (0) = x 0 ◮ Control-law model u ( t ) = Kx ( t k ) t ∈ [ t k , t k +1 ) (2) t k +1 − t k ≤ (1 + m ) T max + τ k +1 τ k ∈ [0 , τ max ] ◮ t k : arrival instants of a new control input ◮ τ k = τ ( t k ) , k ∈ N ◮ m is the number of consecutive dropouts ◮ Varying sampling period → Actual (varying) sampling period+data loss+varying propagation delays C. Briat [KTH / ] 6/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Problem ◮ Find a sampled-data state-feedback control law such that the closed-loop system is asymptotically (exponentially) stable and objective 1: maximize the Maximal Allowable Transfer Interval (MATI) under L 2 disturbance attenuation constraints; or objective 2: minimize L 2 disturbance attenuation gain under a MATI constraint. S MATI := { ( m, τ, T ) ∈ N × R + × R ++ : (1 + m ) T + τ ≤ MATI } . (3) ◮ m : number of consecutive dropouts ◮ T : sampling period ◮ τ : propagation delay C. Briat [KTH / ] 7/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Stability Analysis C. Briat [KTH / ] 8/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Quadratic separation Theorem The interconnected system above is well-posed if there exists a symmetric matrix Θ satisfying the conditions � E � E −A � T −A � ⊥ Θ ⊥ ≻ 0 (4) and � � � � 1 1 � u T , Θ u T � ≤ 0 (5) P T ∇ P T ∇ for all u ∈ L 2 e and all T > 0 . C. Briat [KTH / ] 9/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Alternative system representation       x ( t ) I 1 n x ( t ) ˙ δ ( t ) ∆ sh 1 n x ( t ) ˙ = , (6)       v ( t ) ∆ γ y ( t ) � �� � � �� � � �� � w ( t ) ∇ z ( t )         x ( t ) ˙ A + BK − BK E x ( t ) 1 0 0 − 1 x ( t ) ˙ δ ( t ) 1 0 = 0 0 0 (7)         y ( t ) C + DK − DK F v ( t ) 0 0 1 � �� � � �� � � �� � � �� � E z ( t ) A w ( t ) ◮ I : integral operator, � t ◮ ∆ sh : θ → θ ( s ) ds , t ≤ t k +1 ⇒ δ ( t ) = x ( t ) − x ( t k ) . t k ◮ ∆ γ : virtual operator characterizing the L 2 gain of the transfer v → y . C. Briat [KTH / ] 10/20

Introduction Problem statement Stability Analysis Stabilization Conclusion IQC for the integral operator Lemma The integration operator I is characterized by the IQC: � � � � � � 1 n 0 − P 1 n � x T , x T � ≤ 0 . I 1 n − P I 1 n 0 for all x ∈ L n 2 e and for any matrix P ∈ S n ++ . ◮ Lyapunov condition for stability ◮ Frequency domain I → s − 1 � � ∗ � � � � 1 n − P 1 n 0 � 0 , ∀ℜ [ s ] ≥ 0 s − 1 1 n s − 1 1 n − P 0 Pre- and post-multiply by s ∗ X ( s ) ∗ and sX ( s ) − ( s + s ∗ ) X ( s ) ∗ PX ( s ) � 0 , ∀ℜ [ s ] ≥ 0 (8) Characterization of all ℜ [ s ] ≥ 0 : pick any P = P T ≻ 0 C. Briat [KTH / ] 11/20

Introduction Problem statement Stability Analysis Stabilization Conclusion IQC for ∆ sh Lemma The operator ∆ sh can be characterized by the IQC: � − 4 � � � � � π 2 µ 2 S 1 1 n − S 2 1 n � x T , x T � ≤ 0 . ∆ sh 1 n ∆ sh 1 n − S 2 S 1 for all x ∈ L n 2 e and for any matrices S 1 , S 2 ∈ S n ++ . ◮ S 1 : Bound on the L 2 -gain of 2 µ/π , t k +1 − t k ≤ µ , k ∈ N [Mirkin] ◮ S 2 : Passivity of ∆ sh [Fujioka] C. Briat [KTH / ] 12/20

Introduction Problem statement Stability Analysis Stabilization Conclusion IQC for ∆ γ Lemma The operator ∆ γ , which has an L 2 -induced norm equal to γ − 1 , is characterized by the IQC: � � � � � � − γ − 2 1 q 1 r 0 1 q � x T , x T � ≤ 0 , ∆ γ ∆ γ 0 1 r for all x ∈ L q 2 e . ◮ Upper bound γ on L 2 -gain of v → y C. Briat [KTH / ] 13/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Stability result Theorem The NCS system is asymptotically stable for all ( m, τ, T ) ∈ S µ if there exist matrices P, S 1 , S 2 ∈ S n ++ and a scalar η > 0 such that the LMI � E � E −A � T −A � ⊥ Θ ⊥ ≺ 0 (9) holds where E , A are defined in (7) and   − P 0 0 0 0 0 − 4  π 2 µ 2 S 1  − S 2 0 0 0 0      0 0 − η 1 q 0 0 0  Θ = . (10)     0 0 0    ∗ S 1  0 0 0 0 1 r � Moreover, the closed-loop system satisfies || y || L 2 ≤ 1 /η || v || L 2 . C. Briat [KTH / ] 14/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Example � 0 � 0 � � � − 1 . 006 1 − 1 . 006 � x ( t ) = ˙ x ( t ) + x ( t k ) . (11) 1 0 1 ◮ Maximal constant sampling period: 5 . 8117 . MATI nb. of vars. for n = 2 4 n ( n +1) [Yu, 04] unfeasible 12 2 2 n ( n +1) + 6 n 2 [Yue, 04] 0.970 30 2 4 n ( n +1) + 16 n 2 [Tan, 08] 0.995 76 2 7 n ( n +1) + 16 n 2 [Naghshtabrizi, 06](without delay) 1.272 85 2 3 n ( n +1) Proposed result 1.561 + 1 10 2 C. Briat [KTH / ] 15/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Stabilization C. Briat [KTH / ] 16/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Main result Theorem There exists a matrix K ∈ R m × n such that the NCS is asymptotically stable for all ++ , X ∈ R n × n , U ∈ R m × n and a ( m, τ, T ) ∈ S µ if there exist matrices P, S 1 ∈ S n scalar γ > 0 such that the LMI µ π  − ( X + X T )  P + A ′ − BU E X 2 S 1 0 cl   T C ′ ⋆ − P  0 0 0 0  cl   − ( DU ) T  ⋆ ⋆ − S 1 0 0 0    F T ≺ 0 (12)  ⋆ ⋆ ⋆ − γI 0 0     ⋆ ⋆ ⋆ ⋆ − γI  0 0   − µ π   ⋆ ⋆ ⋆ ⋆ ⋆ − P 2 S 1   ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ − S 1 holds with A ′ cl = AX + BU and C ′ cl = CX + DU . Furthermore, the closed-loop system controlled with gain K = UX − 1 satisfies || y || L 2 ≤ γ || v || L 2 . C. Briat [KTH / ] 17/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Example ◮ Let us consider the open-loop system � − 0 . 8 � 0 . 4 � � − 0 . 01 x ( t ) = ˙ x ( t ) + u ( t ) (13) 1 0 . 1 0 . 1 ◮ [Yu, 04]: system stabilizable for µ ≤ 0 . 6011 . ◮ Proposed result: system stabilizable for µ ≤ 3 . 64826 with the controller gain � − 0 . 3482 − 0 . 3097 � K = . C. Briat [KTH / ] 18/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Conclusion ◮ Approach based on well-posedness and IQCs ◮ LMI results for both stability and stabilization ◮ Tradeoff: L 2 -gain minimization vs. MATI maximization ◮ Low numerical complexity ◮ Can be extended to robust stability and stabilization C. Briat [KTH / ] 19/20

Introduction Problem statement Stability Analysis Stabilization Conclusion Thank you for your attention C. Briat [KTH / ] 20/20