Hybrid Control and Switched Systems Lecture 10 Switched
Hybrid Control and Switched Systems Lecture #10 Switched systems João P. Hespanha University of California at Santa Barbara
Summary Switched systems • Linear switched systems • Lyapunov stability, asymptotic stability, exponential stability Using switched systems to analyze complex hybrid systems
Switched system parameterized family of vector fields ´ fp : Rn ! Rn p 2 Q switching signal ´ piecewise constant signal s : [0, 1) ! Q parameter set S ´ set of admissible switching signals E. g. , S { s : Ns( , t)· 1 + (t – ), 8 t > ¸ 0 } # of discontinuities of s in the interval ( , t) switching times (discontinuities of s) s=1 s=3 s=2 s=1 t A solution to the switched system is any pair (s, x) with s 2 S and x a solution to time-varying ODE
Switched system with state-dependent switching parameterized family of vector fields ´ fp : Rn ! Rn p 2 Q switching signal ´ piecewise constant signal s : [0, 1) ! Q parameter set S ´ set of admissible pairs (s, x) with s a switching signal and x a signal in Rn E. g. , S {(s, x) : Ns( , t)· 1+ sups 2( , t) ||x(s)|| (t – ), 8 t > ¸ 0 } for each x only some s may be admissible switching times s=1 s=3 s=2 s=1 t A solution to the switched system is a pair (s, x) 2 S for which x is a solution to time-varying ODE
Switched system with resets parameterized family of vector fields ´ fp : Rn ! Rn p 2 Q switching signal ´ piecewise constant signal s : [0, 1) ! Q parameter set S ´ set of admissible pairs (s, x) with s a switching signal and x a signal in Rn switching times s=1 s=2 s=3 s=1 t A solution to the switched system is a pair (s, x) 2 S for which 1. on every open interval on which s is constant, x is a solution to 2. at every switching time t, x(t) = r(s(t), s–(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 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
Stability of ODEs equilibrium point ´ xeq 2 Rn for which f(xeq) = 0 class K ´ set of functions : [0, 1)![0, 1) that are 1. continuous 2. strictly increasing 3. (0)=0 (s) ||x(t 0) – xeq|| (||x(t 0) – xeq||) Definition (class K function definition): The equilibrium point xeq is (Lyapunov) stable if 9 2 K: ||x(t) – xeq|| · (||x(t 0) – xeq||) 8 t¸ t 0¸ 0, ||x(t 0) – xeq||· c x(t) xeq t s
Stability of switched systems equilibrium point ´ xeq 2 Rn for which fq(xeq) = 0 8 q 2 Q class K ´ set of functions : [0, 1)![0, 1) that are 1. continuous 2. strictly increasing 3. (0)=0 (s) Definition (class K function definition): The equilibrium point xeq is (Lyapunov) stable if 9 2 K: s of x(t 0) and s ||x(t 0) – xeq|| (||x(t 0) – xeq||) ||x(t) – xeq|| · (||x(t 0) – xeq||) 8 t¸ t 0¸ 0, ||x(t 0) – xeq||· c along any solution (s, x) 2 S to the switched system is independent x(t) xeq t in switched systems one is only concerned about boundedness or convergence of the continuous state
Asymptotic stability of ODEs equilibrium point ´ xeq 2 Rn for which f(xeq) = 0 (s) class K ´ set of functions : [0, 1)![0, 1) that are 1. continuous 2. strictly increasing 3. (0)=0 s ||x(t 0) – xeq|| (||x(t 0) – xeq||) Definition: The equilibrium point xeq is (globally) asymptotically stable if it is Lyapunov stable and for every initial state the solution exists on [0, 1) and x(t) ! xeq as t!1. x(t) xeq t
Asymptotic stability of switched systems equilibrium point ´ xeq 2 Rn for which fq(xeq) = 0 8 q 2 Q class K ´ set of functions : [0, 1)![0, 1) that are 1. continuous 2. strictly increasing 3. (0)=0 (s) ||x(t 0) – xeq|| (||x(t 0) – xeq||) Definition: The equilibrium point xeq is (globally) asymptotically stable if it is Lyapunov stable and for every solution that exists on [0, 1) x(t) ! xeq as t!1. x(t) xeq t s
Asymptotic stability of ODEs (s, t) equilibrium point ´ xeq 2 Rn for which f(xeq) = 0 (for each fixed t) s class KL ´ set of functions : [0, 1)£[0, 1)![0, 1) s. t. 1. for each fixed t, (¢, t) 2 K (s, t) 2. for each fixed s, (s, ¢) is monotone (for each fixed s) decreasing and (s, t) ! 0 as t!1 ||x(t 0) – xeq|| (||x(t 0) – xeq||, 0) Definition (class KL function definition): The equilibrium point xeq is (globally) asymptotically stable if 9 2 KL: ||x(t) – xeq|| · (||x(t 0) – xeq||, t – t 0) 8 t¸ t 0¸ 0 t (||x(t 0) – xeq||, t) xeq x(t) t We have exponential stability when (s, t) = c e-l t s with c, l > 0
Uniform asymptotic stability of switched systems (s, t) equilibrium point ´ xeq 2 Rn for which f(xeq) = 0 (for each fixed t) s class KL ´ set of functions : [0, 1)£[0, 1)![0, 1) s. t. 1. for each fixed t, (¢, t) 2 K (s, t) 2. for each fixed s, (s, ¢) is monotone (for each fixed s) decreasing and (s, t) ! 0 as t!1 ||x(t 0) – xeq|| (||x(t 0) – xeq||, 0) t Definition (class KL function definition): The equilibrium point xeq is uniformly asymptotically stable if 9 2 KL: ||x(t) – xeq|| · (||x(t 0) – xeq||, t – t 0) 8 t¸ t 0¸ 0 along any solution (s, x) 2 S to the switched system is independent of x(t 0) and s (||x(t 0) – xeq||, t) xeq x(t) t We have exponential stability when (s, t) = c e-l t s with c, l > 0
Three notions of stability Definition (class K function definition): The equilibrium point xeq is stable if 9 2 K: is independent of x(t 0) and s ||x(t) – xeq|| · (||x(t 0) – xeq||) 8 t¸ t 0¸ 0, ||x(t 0) – xeq||· c along any solution (x, s) 2 S to the switched system Definition: The equilibrium point xeq 2 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 xeq 2 Rn is uniformly asymptotically stable if 9 2 KL: ||x(t) – xeq|| · (||x(t 0) – xeq||, t – t 0) 8 t¸ t 0¸ 0 is independent along any solution (s, x) 2 S to the switched system of x(t 0) and s exponential stability when (s, t) = c e-l t s with c, l > 0
Example S ´ set of piecewise constant switching signals taking values in Q {– 1, +1} S ´ set of piecewise constant switching signals taking values in Q {– 1, 0} with infinitely many switches and interval between consecutive discontinuities bounded below by 1 S ´ set of piecewise constant switching signals taking values in Q {– 1, 0} with infinitely many switches and interval between consecutive discontinuities below by 1 and above by 2
Example S ´ set of piecewise constant switching signals taking values in Q {– 1, +1} unstable S ´ set of piecewise constant switching signals taking values in Q {– 1, 0} stable but not asympt. S ´ set of piecewise constant switching signals taking values in Q {– 1, 0} with infinitely many switches and interval between consecutive discontinuities bounded below by 1 asympt. stable S ´ set of piecewise constant switching signals taking values in Q {– 1, 0} with infinitely many switches and interval between consecutive discontinuities below by 1 and above by 2 uniformly asympt. stable
Linear switched systems Aq, Rq, q’ 2 Rn£ n q, q’ 2 Q vector fields and reset maps linear on x s=2 s=1 t 0 s=1 s=3 t 1 t 2 t 3 t
Linear switched systems Aq, Rq, q’ 2 Rn£ n q, q’ 2 Q vector fields and reset maps linear on x s=2 s=1 t 0 s=1 s=3 t 1 t 2 t 3 t state-transition matrix for the switched system (s-dependent) t 1, t 2, t 3, …, tk ´ switching times of s in the interval [t, t)
Linear switched systems Aq, Rq, q’ 2 Rn£ n q, q’ 2 Q state-transition matrix (s-dependent) t 1, t 2, t 3, …, tk ´ switching times of s in the interval [t, t) Analogous to what happens for (unswitched) linear systems: 1. Fs( , ) = I 8 2. Fs(t, s) Fs(s, ) = Fs (t, ) 8 t ¸ s ¸ (semi-group property) 3. if t is not a switching time, Fs (t, ) is differentiable at t and 4. if t is a switching time, 5. variation of constants formula holds for systems with inputs for a given s, Fs is a “solution” to the switched system with resets but now Fs may not be nonsingular (will be singular if one of the Rq q’ are)
Uniform vs. exponential stability Aq, Rq, q’ 2 Rn£ n q, q’ 2 Q state-independent switching ´ S is such that (s, x) 2 S ) (s, z) 2 S for any other piecewise continuous z only s determines whether or not (s, x) is admissible Theorem: For switched linear systems with state-independent switching, uniform asymptotic stability implies exponential stability (two notions are equivalent) Outline… 1 st By uniform asymptotic stability 9 2 KL: ||x(t)|| · (||x(t 0)||, t – t 0) 8 t¸ t 0¸ 0 2 nd Choose T sufficiently large so that (1, T) = = e–l < 1 ( l > 0) 3 rd Pick arbitrary solution (s, x ) 2 S 4 th Consider another solution (s, x*) starting at x*( 1) = z x(t 1)/||x(t 1)||. Then x( 2) = Fs( 2, 1) x( 1) = ||x(t 1)|| Fs( 2, 1) z = ||x(t 1)|| x*( 2) ||x*( 2)|| · (||z||, t 2 – 1) = (1, t 2 – 1) exponential decrease of k ) || x( 2) || · (1, t 2 – 1) ||x(t 1)|| any interval of length ¸ k T
Uniform vs. exponential stability Aq, Rq, q’ 2 Rn£ n q, q’ 2 Q state-independent switching ´ S is such that (s, x) 2 S ) (s, z) 2 S for any other piecewise continuous z only s determines whether or not (s, x) is admissible Theorem: For switched linear systems with state-independent switching, uniform asymptotic stability implies exponential stability (two notions are equivalent) Outline… 4 th … || x( 2) || = (1, t 2 – 1) ||x(t 1)|| 5 th Given an arbitrary interval [t 0, t], break it into k floor((t – t 0)/T) intervals of length T plus one interval of length smaller than T …
Example #2: Thermostat y ´ mean temperature room off on heater turn heater off y turn heater on t The state of the system remains bounded as t ! 1:
Example #2: Thermostat y ´ mean temperature room off on heater turn heater off y turn heater on t A 0, A 1 asymptotically stable (all eigenvalues with negative real part) 1. if system would stay in off mode forever then eq. state xeq = A 0– 1 b 0 is asymptotically stable & y ! yoff c 0 A 0– 1 b 0 · y*-h 2. if system would stay in on mode forever then eq. state xeq = A 1– 1 b 1 is asymptotically stable & y ! yon c 1 A 1– 1 b 1 ¸ y* With switching, does the overall state x of the system remains bounded as t ! 1?
Example #2: Thermostat y ´ mean temperature room off on heater One option to prove that the state remains bounded: 1 st Establish a bound of how fast switching can occur: on an interval (t, t) the maximum number of switchings N(t, t) is bounded by Why? maximum derivative of y is proportional to ||x|| and between two consecutive switchings y must have a variation of h a (sequence) property of the discrete -component of the state
Example #2: Thermostat y ´ mean temperature room off on heater One option to prove that the state remains bounded: 1 st Establish a bound of how fast switching can occur: on an interval (t, t) the maximum number of switchings N(t, t) is bounded by x is a solution to the following (state-dependent) switching system: with Why? maximum derivative of y is proportional to ||x|| and between two consecutive switchings y must have a variation of h a (sequence) property of the discrete (tough to analyze directly…) -component of the state
Example #2: Thermostat y ´ mean temperature room off on heater One option to prove the state remains bounded: a (sequence) property of the continuous-dynamics 2 nd Estimate how large x can be from y: For the following (state independent) switching systems there exist constants ¸ 1, , > 0 such that · y* • constants , , depend on N 0 & D • to prove this one needs the system to be observable from y
Example #2: Thermostat 1 st On an interval ( , t) the maximum number of switchings N( , t) is bounded by 2 nd Assuming that the max. number of switchings N( , t) on ( , t) is bounded by Then there exist constants ¸ 1, , > 0 such that 3 rd For any choice of D and h such that x must be bounded for any solution compatible with 1 & 2 above. Hint: prove by contradiction that
Proof… We will show that (*) 1 nd For s = 0, (*) holds because … ¸ 1 2 nd By contradiction suppose that (*) holds strictly for t 2 [0, t*) and with equality at t = t*. Then Therefore, we conclude that ||x(t*)|| can never reach h / (c D) !
Discrete/continuous decoupling 1 st x is a solution to the following (state-dependent) switching system: 2 nd For the following (stateindependent) switching system: There exist constants , , such that property of the discrete evolution property of the interconnection property of a (state-independent) switching systems
Next lecture… Stability under arbitrary switching • Instability caused by switching • Common Lyapunov function • Converse results • Algebraic conditions
- Slides: 29