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

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

Switched system parameterized family of vector fields ´ f p : R n ! R n p 2  parameter set switching signal ´ piecewise constant signal  : [0, 1 ) !   ´ set of admissible pairs ( , x) with  a switching signal and x a signal in R n t  = 1  = 3  = 2  = 1 switching times A solution to the switched system is a pair (x,  ) 2  for which 1.on every open interval on which  is constant, x is a solution to 2.at every switching time t, x(t) =  (  (t),  – (t), x – (t) ) time-varying ODE

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

Three notions of stability Definition (class  function definition): The equilibrium point x eq is stable if 9  2  : ||x(t) – x eq || ·  (||x(t 0 ) – x eq ||) 8 t ¸ t 0 ¸ 0, ||x(t 0 ) – x eq || · c along any solution (x,  ) 2  to the switched system Definition: The equilibrium point x eq 2 R n is asymptotically stable if it is Lyapunov stable and for every solution that exists on [0, 1 ) x(t) ! x eq as t !1. Definition (class  function definition): The equilibrium point x eq 2 R n is uniformly asymptotically stable if 9  2  : ||x(t) – x eq || ·  (||x(t 0 ) – x eq ||,t – t 0 ) 8 t ¸ t 0 ¸ 0 along any solution (x,  ) 2  to the switched system  is independent of x(t 0 ) and   is independent of x(t 0 ) and  exponential stability when  (s,t) = c e - t s with c, > 0

Stability under arbitrary switching  all ´ set of all pairs ( , x) with  piecewise constant and x piecewise continuous  (p, q, x) = x 8 p,q 2 , x 2 R n no resets any switching signal is admissible If one of the vector fields f q, q 2  is unstable then the switched system is unstable Why? 1.because the switching signals  (t) = q 8 t is admissible 2.for this  we cannot find  2  such that ||x(t) – x eq || ·  (||x(t 0 ) – x eq ||) 8 t ¸ t 0 ¸ 0, ||x(t 0 ) – x eq || · c (must less for all  ) But even if all f q, q 2  are stable the switched system may still be unstable …

Stability under arbitrary switching unstable.m

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

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

Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. The same V could be used to prove stability for all the unswitched systems

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

Some facts about functions of class ,  class  ´ set of functions  : [0, 1 ) ! [0, 1 ) that are 1. continuous 2. strictly increasing 3.  (0)=0 s  (s) 2  1 class  1 ´ subset of  containing those functions that are unbounded  (s) 2  Lemma 2: If  2  1 then  is invertible and  -1 2  1 s  (s) 2  1 s =  –1 (y) y =  (s) Lemma 3:If V: R n ! R is positive definite and radially unbounded function then 9  1,  2 2  1 :  1 (||x||) · V(x) ·  2 (||x||) 8 x 2 R n ||x|| ·  1 –1 ( V(x) )  2 –1 ( V(x) ) · ||x|| Lemma 1: If    2  2  then  (s) ›  1 (  2 (s)) 2  (same for  1 )

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

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

Why? (for simplicity consider x eq = 0, W(z) 9 0 as z ! 1 ) 1 st As long as W 9 0 as z ! 1, 9  3 2  2 nd Take an arbitrary solution ( , x) and define v(t) › V( x(t) ) 8 t ¸ 0 Common Lyapunov function Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable.  2 –1 ( V(x) ) · ||x||

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

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

Example DefiningV(x 1,x 2 ) › x x 2 2 common Lyapunov function uniform asymptotic stability stable.m

Example DefiningV(x 1,x 2 ) › x x 2 2 common Lyapunov function stability (not asymptotic) (problems, e.g., close to the x 2 =0 axis) stable.m

Converse result Theorem: Assume  is finite. The switched system is uniformly asymptotically stable (on  all ) if and only if there exists a common Lyapunov function, i.e., continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Note that… 1.This result generalized for infinite  but one needs extra technical assumptions 2.The sufficiency was already established. It turns out that the existence of a common Lyapunov function is also necessary. 3.Finding a common Lyapunov function may be difficult. E.g., even for linear systems V may not be quadratic

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

Algebraic conditions for stability under arbitrary switching linear switched system Suppose 9 m ¸ n, M 2 R m £ n full rank & { B q 2 R m £ m : q 2  }: M A q = B q M 8 q 2  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? 1.If V(z) = z’ z is a common Lyapunov function then z converges to zero exponentially fast 2. z › M x ) M’ z = M’ M x ) (M’ M) –1 M’ z = x ) x converges to zero exponentially fast

Algebraic conditions for stability under arbitrary switching linear switched system M A q = B q M 8 q 2  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… Suppose 9 m ¸ n, M 2 R m £ n full rank & { B q 2 R m £ m : q 2  }:

Commuting matrices linear switched system Suppose that for all p,q 2 , A p A q = A q A p state-transition matrix (  -dependent) t 1, t 2, t 3, …, t k ´ switching times of  in the interval [t,  ) Recall: in general e M e N  e M + N  e N e M unless M N = N M T q ´ total time  = q on (t,  ) Assuming all A q are asymptotically stable: 9 c,   > 0 ||e A q t || · c e –  t |Q| ´ # elements in 

Commuting matrices linear switched system Theorem: If  is finite all A q, q 2  are asymptotically stable and A p A q = A q A p 8 p,q 2  then the switched system is uniformly (exponentially) asymptotically stable state-transition matrix (  -dependent) t 1, t 2, t 3, …, t k ´ switching times of  in the interval [t,  ) Recall: in general e M e N  e M + N  e N e M unless M N = N M

Triangular structures linear switched system Theorem: If all the matrices A q, q 2  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!

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

Commuting matrices linear switched system Theorem: If  is finite all A q, q 2  are asymptotically stable and A p A q = A q A p 8 p,q 2  then 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)

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: A p A q x = A q A p x 8 x 2 R n, p,q 2 

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