Lyapunov small-gain theorems for not necessarily ISS hybrid systems

Transcription

1 Lyapunov small-gain theorems for not necessarily ISS hybrid systems Andrii Mironchenko, Guosong Yang and Daniel Liberzon Institute of Mathematics University of Würzburg Coordinated Science Laboratory University of Illinois at Urbana-Champaign MTNS 2014 Groningen, 9 July 2014

2 Class of systems ẋ f (x, u), (x, u) C, x + g(x, u), (x, u) D. (Σ) E R + N is a compact hybrid time domain if E = J ([t j, t j+1 ], j) j=0 for some 0 = t 0 t 1 t J+1. E R + N is a hybrid time domain if (T, J) E, E ([0, T ] {0, 1,..., J}) is a compact hybrid time domain. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

3 Comparison functions K := {γ { : R + R + γ(0) = 0, γ is continuous, increasing and unbounded } L := γ : R + R + γ is continuous, strictly decreasing and lim γ(t) = 0 t KL := {β : R + R + R + β(, t) K, t 0, β(r, ) L, r > 0} A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

4 Input-to-state stability Definition (ISS) A set of solution pairs S is pre-iss w.r.t. A : β KL, γ K : (x, u) S, (t, j) dom x x(t, j) A max { β( x(0, 0) A, t + j), γ( u (t,j) ) }. Σ is pre-iss w.r.t. A if S = { all solution pairs (x, u) of Σ} is pre-iss w.r.t. A. Σ is ISS w.r.t. A if Σ is pre-iss w.r.t. A and all solution pairs are complete. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

5 Lyapunov functions Definition V is exponential ISS-LF w.r.t. A X : ψ 1, ψ 2 K, c, d R: ψ 1 ( x A ) V (x) ψ 2 ( x A ) x X { V (x; y) cv (x) (x, u) C, y f (x, u), V (x) χ( u ) V (y) e d V (x) (x, u) D, y g(x, u). A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

6 Lyapunov functions Definition V is exponential ISS-LF w.r.t. A X : ψ 1, ψ 2 K, c, d R: ψ 1 ( x A ) V (x) ψ 2 ( x A ) x X { V (x; y) cv (x) (x, u) C, y f (x, u), V (x) χ( u ) Proposition V (y) e d V (x) (x, u) D, y g(x, u). Let V be exp. ISS-LF for Σ w.r.t. A with d 0. µ 1, η, λ > 0, S[η, λ, µ] := set of solution pairs (x, u) satisfying (d η)(j i) (c λ)(t s) µ (t, j), (s, i) dom x. Then S[η, λ, µ] is pre-iss w.r.t. A. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

7 Interconnected systems Definition ẋ i f i (x, u), (x, u) C, Σ Σ : i : x + i g i (x, u), (x, u) D, i = 1,..., n V i : X i R + is exponential ISS LF for Σ i w.r.t. A i X i : 1) χ ij, χ i K, c i, d i R: (x, u) C, y i f i (x, u), { } n V i (x i ) max max χ ij(v j (x j )), χ i ( u ) V i (x i ; y i ) c i (V i (x i )). j=1,j i 2) d i R: (x, u) D, y i g i (x, u) { } V i (y i ) max e d n i V i (x i ), max χ ij(v j (x j )), χ i ( u ). j=1,j i A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

8 Literature overview D. Liberzon, D. Nesic, A. Teel. Lyapunov-Based Small-Gain Theorems for Hybrid Systems, IEEE TAC, D. Liberzon, D. Nesic, A. Teel. Small-gain theorems of LaSalle type for hybrid systems, CDC Small-gain theorems for interconnections of 2 ISS systems Modification method for interconnections with not necessarily ISS subsystems. LaSalle-Krasovskii type theorem for hybrid systems A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

9 Literature overview D. Liberzon, D. Nesic, A. Teel. Lyapunov-Based Small-Gain Theorems for Hybrid Systems, IEEE TAC, D. Liberzon, D. Nesic, A. Teel. Small-gain theorems of LaSalle type for hybrid systems, CDC Small-gain theorems for interconnections of 2 ISS systems Modification method for interconnections with not necessarily ISS subsystems. LaSalle-Krasovskii type theorem for hybrid systems S. Dashkovskiy, A. Mironchenko. Input-to-State Stability of Nonlinear Impulsive Systems, SICON, S. Dashkovskiy, M. Kosmykov, A. Mironchenko, L. Naujok. Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods, Nonlinear Analysis: Hybrid Systems, Small-gain theorems for interconnections of n impulsive systems with matched instabilities. Stability conditions for nonexponential Lyapunov functions A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

10 Literature overview D. Liberzon, D. Nesic, A. Teel. Lyapunov-Based Small-Gain Theorems for Hybrid Systems, IEEE TAC, D. Liberzon, D. Nesic, A. Teel. Small-gain theorems of LaSalle type for hybrid systems, CDC Small-gain theorems for interconnections of 2 ISS systems Modification method for interconnections with not necessarily ISS subsystems. LaSalle-Krasovskii type theorem for hybrid systems S. Dashkovskiy, A. Mironchenko. Input-to-State Stability of Nonlinear Impulsive Systems, SICON, S. Dashkovskiy, M. Kosmykov, A. Mironchenko, L. Naujok. Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods, Nonlinear Analysis: Hybrid Systems, Small-gain theorems for interconnections of n impulsive systems with matched instabilities. Stability conditions for nonexponential Lyapunov functions A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

11 Interconnections of 2 systems ISS-LF for Σ i V i : X i R + is ISS-Lyapunov functions for Σ i, i = 1, 2 iff (x, u) C, y 1 f 1 (x, u) V 1 (x 1 ) max { χ 12 (V 2 (x 2 )), χ 1 ( u U ) } (x, u) D, y 1 g 1 (x, u) V 1 (y 1 ) max { e d 1V 1 (x 1 ), χ 12 (V 2 (x 2 )), χ 1 ( u ) }. V 1 (x 1 ; y 1 ) c 1 V 1 (x 1 ), (x, u) C, y 2 f 2 (x, u) V 2 (x 2 ) max { χ 21 (V 1 (x 1 )), χ 2 ( u U ) } (x, u) D, y 2 g 2 (x, u) V 2 (y 2 ) max { e d 2V 2 (x 2 ), χ 21 (V 1 (x 1 )), χ 2 ( u ) }. V 2 (x 2 ) c 2 V 2 (x 2 ), A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

12 Interconnections of 2 systems ISS-LF for Σ i V i : X i R + is ISS-Lyapunov functions for Σ i, i = 1, 2 iff (x, u) C, y 1 f 1 (x, u) V 1 (x 1 ) max { χ 12 (V 2 (x 2 )), χ 1 ( u U ) } (x, u) D, y 1 g 1 (x, u) V 1 (y 1 ) max { e d 1V 1 (x 1 ), χ 12 (V 2 (x 2 )), χ 1 ( u ) }. V 1 (x 1 ; y 1 ) c 1 V 1 (x 1 ), (x, u) C, y 2 f 2 (x, u) V 2 (x 2 ) max { χ 21 (V 1 (x 1 )), χ 2 ( u U ) } (x, u) D, y 2 g 2 (x, u) V 2 (y 2 ) max { e d 2V 2 (x 2 ), χ 21 (V 1 (x 1 )), χ 2 ( u ) }. V 2 (x 2 ) c 2 V 2 (x 2 ), How to use these two LFs to study ISS of the interconnection? A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

13 Interconnections of 2 systems ISS-LF for Σ i V i : X i R + is ISS-Lyapunov functions for Σ i, i = 1, 2 iff (x, u) C, y 1 f 1 (x, u) V 1 (x 1 ) max { χ 12 (V 2 (x 2 )), χ 1 ( u U ) } (x, u) D, y 1 g 1 (x, u) V 1 (y 1 ) max { e d 1V 1 (x 1 ), χ 12 (V 2 (x 2 )), χ 1 ( u ) }. V 1 (x 1 ; y 1 ) c 1 V 1 (x 1 ), (x, u) C, y 2 f 2 (x, u) V 2 (x 2 ) max { χ 21 (V 1 (x 1 )), χ 2 ( u U ) } (x, u) D, y 2 g 2 (x, u) V 2 (y 2 ) max { e d 2V 2 (x 2 ), χ 21 (V 1 (x 1 )), χ 2 ( u ) }. V 2 (x 2 ) c 2 V 2 (x 2 ), This depends on χ 12, χ 21 and coefficients c i, d i! A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

14 The case when c i > 0, d i > 0, i = 1, 2 Theorem (Liberzon, Nesic, Teel, IEEE TAC, 2014) Let V 1, V 2 be ISS-Lyapunov function for Σ 1, Σ 2 with gains χ 12, χ 21. Let also c 1, c 2, d 1, d 2 > 0. Then χ 12 χ 21 < id (SGC) Σ is ISS. V (x) := max{v 1 (x 1 ), ρ(v 2 (x 2 ))} is an ISS Lyapunov function for Σ. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

15 The case when c i > 0, d i > 0, i = 1, 2 Theorem (Liberzon, Nesic, Teel, IEEE TAC, 2014) Let V 1, V 2 be ISS-Lyapunov function for Σ 1, Σ 2 with gains χ 12, χ 21. Let also c 1, c 2, d 1, d 2 > 0. Then χ 12 χ 21 < id (SGC) Σ is ISS. V (x) := max{v 1 (x 1 ), ρ(v 2 (x 2 ))} is an ISS Lyapunov function for Σ. If χ ij are linear, ρ can be chosen linear and s.t. V is an exponential LF. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

16 The case when c i > 0, d i > 0, i = 1, 2 Theorem (Liberzon, Nesic, Teel, IEEE TAC, 2014) Let V 1, V 2 be ISS-Lyapunov function for Σ 1, Σ 2 with gains χ 12, χ 21. Let also c 1, c 2, d 1, d 2 > 0. Then χ 12 χ 21 < id (SGC) Σ is ISS. V (x) := max{v 1 (x 1 ), ρ(v 2 (x 2 ))} is an ISS Lyapunov function for Σ. If χ ij are linear, ρ can be chosen linear and s.t. V is an exponential LF. What to do if some of c i, d i are < 0? A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

17 Modification of "bad" subsystems when χ ij are linear Method due to Liberzon, Nesic, Teel, IEEE TAC, V i = LF for Σ i gains χ ij Restrict jumping rate of Σ i i : c i < 0 and i : d i < 0 W i = LF for Σ i gains χ ij The gains increase exponentially with chatter bound! SGT (for systems with ISS subsystems) W = LF for Σ Conditions for ISS of Σ A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

18 A new small-gain theorem Define Γ M := (χ ij ) n n. Theorem Let V i be exp. ISS LF for Σ i w.r.t. A i with d i 0 and linear gains χ ij. ρ(γ M ) < 1 V (x) := max n 1 i=1 s i V i (x i ) is an exp. ISS LF for Σ w.r.t. A = A 1... A n with rate coefficients { ( )} c := min n c sj i, d := min d i, ln χ ij. i=1 i,j:j i s i This LF can be used to prove ISS if all c i > 0 or all d i > 0. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

19 Modification of "bad" subsystems when χ ij are linear Improved modification method V i = LF for Σ i gains χ ij Restrict jumping rate of Σ i i : c i < 0 or i : d i < 0 W i = LF for Σ i gains χ ij The gains increase exponentially with chatter bound! SGT (for systems with matched instabilities) W = LF for Σ Conditions for ISS of Σ A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

20 Making discrete dynamics ISS Define I d := {i {1,..., n} : d i < 0}. i I d restrict the frequency of jumps of Σ i by j k δ i (t s) + N i 0, (ADT) where δ i, N0 i > 0 and (t, j), (s, k) dom x. It can be modeled by the clock τ i [0, δ i ], τ i [0, N i 0 ], τ + i = τ i 1, τ i [1, N i 0 ]. LF for Σ i : W i (z i ) := { Vi (x i ), i / I d, e L i τ i V i (x i ), i I d. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

21 Making discrete dynamics ISS Proposition: ISS LF for modified subsystems 1) (z, u) C and y i f i (z, u), { } n W i (z i ) max max χ ijw j (z j ), χ i u Ẇi(z i ; y i ) c i W i (z i ), j=1 where c i = c i for i / I d ; c i = c i L i δ i for i I d and χ i := χ i, χ ij := χ ij, i / I d, χ i := e L i N i 0 χi, χ ij := e L i N i 0 χij, i I d. 2) (z, u) D and y i g i (z, u), { } W i (y i ) max e d n i W i (z i ), max χ ijw j (z j ), χ i u, j=1 where d i = d i for i / I d and d i = d i + L i for i I d. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

22 Example Let c 1 > 0, d 1 < 0, c 2 > 0 and d 2 < 0. Instabilities are matched no modification. SGT LF V for interconnection with c > 0 and d < 0. Let c 1 > 0, d 1 < 0, c 2 < 0 and d 2 > 0. Instabilities are not matched modify Σ 2. Γ = [ 0 χ12 χ 21 0 ] = [ 0 e L 1 N0 1 χ 12 χ 21 0 ]. SGT Since L 1 = d 1 + ε; χ 12 χ 21 < e L 1N 1 0. N χ 12 χ 21 e d 1. A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

23 Summary and Outlook Main results Outlook New small-gain theorem for hybrid systems. If instabilities are matched no modification is needed. Less restrictive modification method Extension to nonlinear χ ij, using ideas from S. Dashkovskiy and A.M. Input-to-State Stability of Nonlinear Impulsive Systems, SICON, A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

24 Summary and Outlook Main results Outlook New small-gain theorem for hybrid systems. If instabilities are matched no modification is needed. Less restrictive modification method Extension to nonlinear χ ij, using ideas from S. Dashkovskiy and A.M. Input-to-State Stability of Nonlinear Impulsive Systems, SICON, Can one avoid modifications if the instabilities are not matched? A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18

25 Summary and Outlook Main results Outlook New small-gain theorem for hybrid systems. If instabilities are matched no modification is needed. Less restrictive modification method Extension to nonlinear χ ij, using ideas from S. Dashkovskiy and A.M. Input-to-State Stability of Nonlinear Impulsive Systems, SICON, Can one avoid modifications if the instabilities are not matched? Thank you for attention! A. Mironchenko, G. Yang, D. Liberzon ISS of hybrid systems MTNS / 18