Stabilization of interconnected switched control-affine systems via - PowerPoint PPT Presentation

Stabilization of interconnected switched control-affine systems via a Lyapunov-based small-gain approach Guosong Yang 1 Daniel Liberzon 1 Zhong-Ping Jiang 2 1 Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 2 Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, Brooklyn, NY 11201 May 26, 2017 1 / 12

Introduction Switched system Switching is ubiquitous in realistic system models, such as – Thermostat – Gear tranmission – Power supply Structure of a switched system – A family of dynamics, called modes – A sequence of events, called switches In this work: time-dependent, uncontrolled switching 2 / 12

Introduction Presentation outline Preliminaries Interconnected switched systems and small-gain theorem Stabilization via a small-gain approach 3 / 12

Preliminaries Nonlinear switched system with input x = f σ ( x, w ) , ˙ x (0) = x 0 State x ∈ R n , disturbance w ∈ R m A family of modes f p , p ∈ P , with an index set P A right-continuous, piecewise constant switching signal σ : R + → P that indicates the active mode σ ( t ) Solution x ( · ) is absolutely continuous (no state jump) x 2 x 2 x 2 x 1 x 1 x 1 0 0 0 x = A 1 x ˙ x = A 2 x ˙ x = A σ x ˙ 4 / 12

Preliminaries Stability notions Definition (GAS) A continuous-time system is globally asymptotically stable (GAS) if there is a function β ∈ KL s.t. for all initial state x 0 , | x ( t ) | ≤ β ( | x 0 | , t ) ∀ t ≥ 0 . A function α : R + → R + is of class K if it is continuous, positive definite and strictly increasing; α ∈ K is of class K ∞ if lim r →∞ α ( r ) = ∞ , such as α ( r ) = r 2 or | r | A function γ : R + → R + is of class L if it is continuous, strictly decreasing and lim t →∞ γ ( t ) = 0 , such as γ ( t ) = e − t A function β : R + × R + → R + is of class KL if β ( · , t ) ∈ K for each fixed t , and β ( r, · ) ∈ L for each fixed r > 0 , such as β ( r, t ) = r 2 e − t 5 / 12

Preliminaries Stability notions Definition (GAS) A continuous-time system is globally asymptotically stable (GAS) if there is a function β ∈ KL s.t. for all initial state x 0 , | x ( t ) | ≤ β ( | x 0 | , t ) ∀ t ≥ 0 . The switched system may be unstable even if all individual modes are GAS x 2 x 2 x 2 x 1 x 1 x 1 0 0 0 x = A 1 x ˙ x = A 2 x ˙ x = A σ x ˙ 5 / 12

Preliminaries Stability notions Definition (GAS) A continuous-time system is globally asymptotically stable (GAS) if there is a function β ∈ KL s.t. for all initial state x 0 , | x ( t ) | ≤ β ( | x 0 | , t ) ∀ t ≥ 0 . Definition (ISpS [JTP94]) A continuous-time system is input-to-state practically stable (ISpS) if there are functions β ∈ KL , γ ∈ K ∞ and a constant ε ≥ 0 s.t. for all x 0 and disturbance w , | x ( t ) | ≤ β ( | x 0 | , t ) + γ ( � w � ) + ε ∀ t ≥ 0 . When ε = 0 , ISpS becomes input-to-state stability (ISS) [Son89] When ε = 0 and γ ≡ 0 , ISpS becomes GAS [JTP94] Z.-P. Jiang, A. R. Teel, and L. Praly, Mathematics of Control, Signals, and Systems , 1994 [Son89] E. D. Sontag, IEEE Transactions on Automatic Control , 1989 5 / 12

Preliminaries Lyapunov characterizations x = f σ ( x, w ) , ˙ x (0) = x 0 A common Lyapunov function The switched system is GAS if it admits a Lyapunov function V which decreases along the solution in all modes: D f p V ( x, w ) ≤ − λV ( x ) V with a constant λ > 0 . 0 t 0 t 1 t 2 t 3 t [PW96] L. Praly and Y. Wang, Mathematics of Control, Signals, and Systems , 1996 6 / 12

Preliminaries Lyapunov characterizations x = f σ ( x, w ) , ˙ x (0) = x 0 A common Lyapunov function The switched system is GAS if Multiple Lyapunov functions each mode admits a Lyapunov function V p which decreases along the solution when that mode is active: D f p V p ( x, w ) ≤ − λV p ( x ) , V < and their values at switches are V decreasing: V σ ( t k ) ( x ( t k )) ≤ V σ ( t l ) ( x ( t l )) for all switches t k > t l . 0 t 0 t 1 t 2 t 3 t [PD91] P. Peleties and R. DeCarlo, in 1991 American Control Conference , 1991 [Bra98] M. S. Branicky, IEEE Transactions on Automatic Control , 1998 6 / 12

Preliminaries Lyapunov characterizations x = f σ ( x, w ) , ˙ x (0) = x 0 A common Lyapunov function The switched system is GAS if Multiple Lyapunov functions each mode admits a Lyapunov function V p which decreases along the Dwell-time [Mor96], Average solution when that mode is active: dwell-time (ADT) [HM99] D f p V p ( x, w ) ≤ − λV p ( x ) , V < their values after each switch is V bounded in ratio: ∃ µ ≥ 1 s.t. V p ( x ) ≤ µV q ( x ) , there is an ADT τ a > ln( µ ) /λ with an integer N 0 ≥ 1 : N σ ( t, τ ) ≤ N 0 + ( t − τ ) /τ a . 0 t 0 t 1 t 2 t 3 t [Mor96] A. S. Morse, IEEE Transactions on Automatic Control , 1996 [HM99] J. P. Hespanha and A. S. Morse, in 38th IEEE Conference on Decision and Control , 1999 6 / 12

Preliminaries Lyapunov characterizations x = f σ ( x, w ) , ˙ x (0) = x 0 A common Lyapunov function The switched system is GAS if Multiple Lyapunov functions each mode admits a Lyapunov function V p which decreases along the Dwell-time, ADT solution when that mode is active: Further results under slow switching: D f p V p ( x, w ) ≤ − λV p ( x ) , – ISS with dwell-time [XWL01] their values after each switch is – ISS and integral-ISS with ADT bounded in ratio: ∃ µ ≥ 1 s.t. [VCL07] V p ( x ) ≤ µV q ( x ) , – ISS and IOSS with ADT [ML12] there is an ADT τ a > ln( µ ) /λ with an integer N 0 ≥ 1 : N σ ( t, τ ) ≤ N 0 + ( t − τ ) /τ a . [XWL01] W. Xie, C. Wen, and Z. Li, IEEE Transactions on Automatic Control , 2001 [VCL07] L. Vu, D. Chatterjee, and D. Liberzon, Automatica , 2007 [ML12] M. A. M¨ uller and D. Liberzon, Automatica , 2012 6 / 12

Interconnection and small-gain theorem Interconnected switched systems An interconnection of switched systems with state x = ( x 1 , x 2 ) and external disturbance w x 1 = f 1 ,σ 1 ( x 1 , x 2 , w ) , ˙ x 2 = f 2 ,σ 2 ( x 1 , x 2 , w ) . ˙ Each x i -subsystem regards x j as internal disturbance The switchings σ 1 , σ 2 are independent Each x i -subsystem has stabilizing modes in P s,i and destabilizing ones in P u,i Objective: establish ISpS of the interconnection using – Generalized ISpS-Lyapunov functions – Average dwell-times (ADT) – Time-ratios – A small-gain condition 7 / 12

= i # i Interconnection and small-gain theorem Assumptions (Generalized ISpS-Lyapunov) For each x i -subsystem – Each p s ∈ P s,i admits an ISpS-Lya function V i,p s that decreases when active and V i,p s ( x i ) ≥ max { χ i ( V j,p j ( x j )) , χ w i ( | w | ) , δ i } ; – Each p u ∈ P u,i admits a function V i,p u that may increase when active; – V i,p ( x i ) ≤ µ i V i,q ( x i ) for all p, q ∈ P i . (ADT) There is a large enough ADT τ a,i . (Time-ratio) There is a small enough time-ratio ρ i ∈ [0 , 1) . V i; < i 0 t 0 t 1 t 2 t 3 t 4 t 8 / 12

Interconnection and small-gain theorem Lyapunov-based small-gain theorem (Generalized ISpS-Lyapunov) For each x i -subsystem – Each p s ∈ P s,i admits an ISpS-Lya function V i,p s that decreases when active and V i,p s ( x i ) ≥ max { χ i ( V j,p j ( x j )) , χ w i ( | w | ) , δ i } ; – Each p u ∈ P u,i admits a function V i,p u that may increase when active; – V i,p ( x i ) ≤ µ i V i,q ( x i ) for all p, q ∈ P i . (ADT) There is a large enough ADT τ a,i . (Time-ratio) There is a small enough time-ratio ρ i ∈ [0 , 1) . Theorem (Small-Gain) The interconnection is ISpS provided that the small-gain condition χ ∗ 1 ◦ χ ∗ 2 < Id i := e Θ i χ i . holds with χ ∗ 9 / 12

Interconnection and small-gain theorem Proof: Hybrid ISpS-Lyapunov function (Generalized ISpS-Lyapunov) For each x i -subsystem – Each p s ∈ P s,i admits an ISpS-Lya function V i,p s that decreases when active and V i,p s ( x i ) ≥ max { χ i ( V j,p j ( x j )) , χ w i ( | w | ) , δ i } ; – Each p u ∈ P u,i admits a function V i,p u that may increase when active; – V i,p ( x i ) ≤ µ i V i,q ( x i ) for all p, q ∈ P i . (ADT) There is a large enough ADT τ a,i . (Time-ratio) There is a small enough time-ratio ρ i ∈ [0 , 1) . Auxiliary timer τ i ∈ [0 , Θ i ] = i V i; < i Hybrid ISpS-Lya for # i V i subsystem V i := V i,σ i e τ i i := e Θ i χ i with χ ∗ 0 V i; < i 0 0 t 0 t 1 t 2 t 3 t 4 t t 0 t 1 t 2 t 3 t 4 t 10 / 12

Interconnection and small-gain theorem Proof: Hybrid ISpS-Lyapunov function (Generalized ISpS-Lyapunov) For each x i -subsystem – Each p s ∈ P s,i admits an ISpS-Lya function V i,p s that decreases when active and V i,p s ( x i ) ≥ max { χ i ( V j,p j ( x j )) , χ w i ( | w | ) , δ i } ; – Each p u ∈ P u,i admits a function V i,p u that may increase when active; – V i,p ( x i ) ≤ µ i V i,q ( x i ) for all p, q ∈ P i . (ADT) There is a large enough ADT τ a,i . (Time-ratio) There is a small enough time-ratio ρ i ∈ [0 , 1) . Auxiliary timer τ i ∈ [0 , Θ i ] V 1 Hybrid ISpS-Lya for subsystem V i := V i,σ i e τ i ( @ $ 1 ) ! 1 i := e Θ i χ i with χ ∗ A Small-gain χ ∗ 1 ◦ χ ∗ 2 < Id @ $ 2 Hybrid ISpS-Lya function V := max { ψ ( V 1 ) , V 2 } 0 V 2 [JMW96] Z.-P. Jiang, I. M. Y. Mareels, and Y. Wang, Automatica , 1996 10 / 12