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Lecture #11 Stability of switched system: Arbitrary switching

Hybrid Control and Switched Systems. Lecture #11 Stability of switched system: Arbitrary switching. Jo ã o P. Hespanha University of California at Santa Barbara. Summary. Stability under arbitrary switching Instability caused by switching Common Lyapunov function Converse results

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Lecture #11 Stability of switched system: Arbitrary switching

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  1. Hybrid Control and Switched Systems Lecture #11Stability of switched system:Arbitrary switching João P. Hespanha University of Californiaat Santa Barbara

  2. Summary • Stability under arbitrary switching • Instability caused by switching • Common Lyapunov function • Converse results • Algebraic conditions

  3. Switched system parameterized family of vector fields ´fp : Rn!Rnp2 Q parameter set switching signal ´ piecewise constant signal s : [0,1) !Q S´ set of admissible pairs (s, x) with s a switching signal and x a signal in Rn switching times s = 2 s = 1 s = 1 s = 3 t • A solution to the switched system is a pair (x, s) 2 S for which • on every open interval on which s is constant, x is a solution to • at every switching time t, x(t) =r(s(t), s–(t), x–(t) ) time-varying ODE

  4. Time-varying systems vs. Hybrid systems vs. Switched systems Time-varying system´ for each initial condition x(0) there is only one solution (all fp locally Lipschitz) Hybrid system´ for each initial condition q(0), x(0) there is only one solution Switched system´ for each x(0) there may be several solutions, one for each admissible s the notions of stability, convergence, etc.must address “uniformity” over all solutions

  5. Three notions of stability a is independentof x(t0) and s Definition (class K function definition): The equilibrium point xeq is stable if 9a 2 K: ||x(t) – xeq|| ·a(||x(t0) – xeq||) 8t¸t0¸ 0, ||x(t0) – xeq||· c along any solution (x, s) 2 S to the switched system Definition: The equilibrium point xeq2 Rn is asymptotically stable if it is Lyapunov stable and for every solution that exists on [0,1) x(t) !xeq as t!1. Definition (class KL function definition): The equilibrium point xeq2 Rn is uniformly asymptotically stable if 9b2KL: ||x(t) – xeq|| ·b(||x(t0) – xeq||,t – t0) 8t¸t0¸ 0 along any solution (x, s) 2 S to the switched system b is independentof x(t0) and s exponential stability when b(s,t) = c e-lts with c,l > 0

  6. Stability under arbitrary switching Sall´ set of all pairs (s, x) with s piecewise constant and x piecewise continuous r(p, q, x) = x8p,q 2 Q, x 2 Rn no resets any switching signal is admissible If one of the vector fields fq, q2 Q is unstable then the switched system is unstable • Why? • because the switching signals s(t) = q8t is admissible • for this s we cannot find a2K such that ||x(t) – xeq|| ·a(||x(t0) – xeq||) 8t¸t0¸ 0, ||x(t0) – xeq||· c(must less for all s) But even if all fq, q2 Q are stable the switched system may still be unstable …

  7. Stability under arbitrary switching unstable.m

  8. Stability under arbitrary switching for some admissible switching signals the trajectories grow to infinity ) switched system is unstable unstable.m

  9. Lyapunov’s stability theorem (ODEs) Definition (class K function definition):The equilibrium point xeq2 Rn is (Lyapunov) stable if 9a 2 K: ||x(t) – xeq|| ·a(||x(t0) – xeq||) 8t¸t0¸ 0, ||x(t0) – xeq||· c Theorem (Lyapunov): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that Then xeq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, if = 0 only for z = xeq then xeq is a (globally) asymptotically stable equilibrium. Why? V can only stop decreasing when x(t) reaches xeq but V must stop decreasing because it cannot become negative Thus, x(t) must converge to xeq

  10. Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Then • the equilibrium point xeq is Lyapunov stable • if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable. The same V could be used to prove stability for all the unswitched systems

  11. Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Then • the equilibrium point xeq is Lyapunov stable • if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable. Why? (for simplicity consider xeq = 0) 1st Take an arbitrary solution (s, x) and define v(t) ›V( x(t) ) 8t¸ 0 2nd Therefore V( x(t) ) is always bounded

  12. Some facts about functions of class K, KL a(s) 2 K1 class K´ set of functions a : [0,1)![0,1) that are1. continuous2. strictly increasing3. a(0)=0 a(s) 2 K s class K1´ subset of K containing those functions that are unbounded Lemma 1: If a1,a22 K then a(s) ›a1(a2(s))2 K (same for K1) a(s) 2 K1 Lemma 2: If a 2 K1 then a is invertible and a-1 2 K1 y = a(s) s s = a–1(y) Lemma 3: If V: Rn!R is positive definite and radially unbounded function then 9a1,a22K1: a1(||x||) ·V(x) ·a2(||x||) 8x2 Rn ||x|| ·a1–1(V(x) ) a2–1 (V(x))· ||x||

  13. Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Then • the equilibrium point xeq is Lyapunov stable • if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable. Why? (for simplicity consider xeq = 0) 2nd Therefore 3rd Since 9a1,a22K1: a1(||x||) ·V(x) ·a2(||x||) ||x|| ·a1–1(V(x) ) V( x(t) ) is always bounded a1-1 monotone

  14. Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Then • the equilibrium point xeq is Lyapunov stable • if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable. Why? (for simplicity consider xeq = 0) 3rd Since 9a1,a22K1: a1(||x||) ·V(x) ·a2(||x||) 4th Defining a(s) ›a1– 1(a2(s)) 2 K then stability ! (1. is proved)

  15. Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Then • the equilibrium point xeq is Lyapunov stable • if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable. a2–1 (V(x))· ||x|| Why? (for simplicity consider xeq = 0, W(z)9 0 as z !1) 1st As long as W 9 0 as z !1, 9a3 2 K 2nd Take an arbitrary solution (s, x) and define v(t) ›V( x(t) ) 8t¸ 0

  16. Some more facts about functions of class K, KL b(s,t) class KL´ set of functions b:[0,1)£[0,1)![0,1) s.t.1. for each fixed t, b(¢,t) 2K2. for each fixed s, b(s,¢) is monotone decreasing and b(s,t) ! 0 as t!1 (for each fixed t) s b(s,t) (for each fixed s) Lemma 3: Given a 2 K, in case then 9b 2 KL such that After some work …

  17. Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Then • the equilibrium point xeq is Lyapunov stable • if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable. Why? (for simplicity consider xeq = 0, W(z)9 0 as z !1) 2nd Take an arbitrary solution (s, x) and define v(t) ›V( x(t) ) 8t¸ 0 3rd Then 4th Going back to x… class KL function (check !!!) independent of s

  18. Example Defining V(x1,x2) ›x12+ x22 common Lyapunov function uniform asymptotic stability stable.m

  19. Example Defining V(x1,x2) ›x12+ x22 common Lyapunov function stability (not asymptotic) (problems, e.g., close to the x2=0 axis) stable.m

  20. Converse result Theorem: Assume Q is finite. The switched system is uniformly asymptotically stable (on Sall) if and only if there exists a common Lyapunov function, i.e., continuously differentiable, positive definite, radially unbounded function V: Rn!R such that • Note that… • This result generalized for infinite Q but one needs extra technical assumptions • The sufficiency was already established. It turns out that the existence of a common Lyapunov function is also necessary. • Finding a common Lyapunov function may be difficult.E.g., even for linear systems V may not be quadratic

  21. Example The switched system is uniformly exponentially stable for arbitrary switching but there is no common quadratic Lyapunov function stable.m

  22. Algebraic conditions for stability under arbitrary switching linear switched system Suppose 9m ¸n, M 2 Rm£n full rank & { Bq 2 Rm£m : q2 Q }: M Aq = Bq M8q2 Q Defining z›M x Theorem: If V(z) = z’ z is a common Lyapunov function for i.e., then the original switched system is uniformly (exponentially) asymptotically stable • Why? • If V(z) = z’ z is a common Lyapunov function then z converges to zero exponentially fast • z›M x )M’ z = M’ M x ) (M’ M)–1 M’ z = x)x converges to zero exponentially fast

  23. Algebraic conditions for stability under arbitrary switching linear switched system Suppose 9m ¸n, M 2 Rm£n full rank & { Bq 2 Rm£m : q2 Q }: M Aq = Bq M8q2 Q Defining z›M x Theorem: If V(z) = z’ z is a common Lyapunov function for i.e., then the original switched system is uniformly (exponentially) asymptotically stable It turns out that … If the original switched system is uniformly asymptotically stable then such an M always exists (for some m¸ n) but may be difficult to find…

  24. Commuting matrices linear switched system state-transition matrix (s-dependent) t1, t2, t3,…, tk´ switching times of s in the interval [t,t) Recall: in general eM eN¹eM + N¹eN eMunlessM N = N M Suppose that for all p,q2 Q, Ap Aq= Aq Ap Tq´ total time s = q on (t, t) Assuming all Aq are asymptotically stable: 9c,l0> 0 ||eAq t|| ·ce–l0t |Q| ´ # elements in Q

  25. Commuting matrices linear switched system state-transition matrix (s-dependent) t1, t2, t3,…, tk´ switching times of s in the interval [t,t) Recall: in general eM eN¹eM + N¹eN eMunlessM N = N M Theorem: If Q is finite all Aq, q2 Q are asymptotically stable and Ap Aq= Aq Ap 8p,q2 Qthen the switched system is uniformly (exponentially) asymptotically stable

  26. Triangular structures linear switched system Theorem: If all the matrices Aq, q2 Q are asymptotically stable and upper triangular or all lower triangular then the switched system is uniformly (exponentially) asymptotically stable Why? One can find a common quadratic Lyapunov function of the form V(x) = x’ P x with P diagonal… check!

  27. Triangular structures linear switched system Theorem: If all the matrices Aq, q2 Q are asymptotically stable and upper triangular or all lower triangular then the switched system is uniformly (exponentially) asymptotically stable Theorem: If there is a nonsingular matrix T 2 Rn£n such that all the matrices Bq = T Aq T– 1 (T–1BqT = Aq) are upper triangular or all lower triangular then the switched system is uniformly (exponentially) asymptotically stable common similarity transformation Why? 1st The Bq have a common quadratic Lyapunov function V(x) = x’ P x, i.e., P Bq + Bq’ P < 0 2nd Therefore P T T–1 Bq + Bq’ T–1’ T’ P < 0 )T’ P T T–1 BqT+ T’ Bq’ T–1’ T’ PT < 0 Q Aq Aq’ Q Q ›T’PT is a common Lyapunov function

  28. Commuting matrices linear switched system Theorem: If Q is finite all Aq, q2 Q are asymptotically stable and Ap Aq= Aq Ap 8p,q2 Qthen the switched system is uniformly (exponentially) asymptotically stable Another way of proving this result… From Lie Theorem if a set of matrices commute then there exists a common similarity transformation that upper triangularizes all of them There are weaker conditions for simultaneous triangularization (Lie Theorem actually provides the necessary and sufficient condition ´ Lie algebra generated by the matrices must be solvable)

  29. Commuting vector fields nonlinear switched system Theorem: If all unswitched systems are asymptotically stable and then the switched system is uniformly asymptotically stable For linear vector fields becomes: Ap Aq x = Aq Ap x8x 2 Rn, p,q2 Q

  30. Next lecture… • Controller realization for stable switching • Stability under slow switching • Dwell-time switching • Average dwell-time • Stability under brief instabilities

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