Hybrid Control and Switched Systems: Stable Switching Controller Realizations

1. Hybrid Control and Switched Systems Lecture #12Controller realizations for stable switching João P. Hespanha University of Californiaat Santa Barbara

2. Summary Controller realization for stable switching

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. 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

5. 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 Can we change the switching system to make it stable?

6. Example #11: Roll-angle control roll-angle q qis uniquely determined by u and the initial conditions input-output model u q process state-space realization AP bP cP

7. Example #11: Roll-angle control roll-angle q measurement noise set-point control ´ drive the roll angle q to a desired value qreference n etrack + + u q qreference set-point controller process + –

8. Example #11: Roll-angle control measurement noise set-point control ´ drive the roll angle q to a desired value qreference n etrack + + u q qreference set-point controller process + – controller 1 controller 2 slow but not very sensitive to noise (low-gain) fast but very sensitive to noise (high-gain)

9. Switching controller measurement noise switching signal taking values inQ›{1,2} s n etrack + + u q qreference switchingcontroller process + – How to build the switching controller to avoid instability ? s = 2 s = 1 s = 2

10. Realization theory (SISO) u, y 2 R nth order input-output model for short a(y) = b(u) state-space model x 2 Rm´ state Definition: (A, b, c) is called a realization of the input-output model if the two models have the same solution y for every given u and zero initial conditions. Theorem: 1. (A, b, c) is a realization of the IO model if and only if 2. Any nth order IO model has a realization with x 2 Rn 3. If all roots of a(s) have negative real part, A can be chosen asymptotically stable 4. For any nonsingular matrix T 2 Rm£m, if (A, b, c) is a realization of an IO model then (TAT–1, Tb, cT–1) is also a realization of the same model

11. Realization theory (SISO) u, y 2 R nth order input-output model for short a(y) = b(u) state-space realization of the IO model x 2 Rm´ state Suppose A is asymptotically stable: 9P > 0, P A + A’ P = – I (P1/2 AP–1/2, P1/2b, cP–1/2) is also a realization of the IO model Theorem: Given any nth order input-output model for which all roots of a(s) have negative real parts, it is always possible to find a realization for it, for which A + A’ = Q < 0

12. Switching between input-output models M› { aq(y) = bq(u) : q2 Q } ´ finite family of nth order input-output models, with all roots of all aq(s) with negative real parts Theorem: There exists a family of realizations for M R› { (Aq, bq, cq) : q2 Q } such that the switched system is exponentially stable for arbitrary switching Why? 1st Choose realizations such that Aq + Aq’ = – Qq < 0 8q2 Q 2nd The function V(x) = x’ x is a common Lyapunov function for the switched system: continuously differentiable, positive definite, radially unbounded, system is uniformly asymptotically stable ) exponentially stable

13. Back to switching controllers… controller 1 controller 2 realization: realization: measurement noise s n etrack + + u q qreference + –

14. Back to switching controllers… measurement noise switching signal taking values inQ›{1,2} s n etrack + + u q qreference + – overall system: Assuming each controller was properly designed, each Aq is asymptotically stable but the overall switched systems could still be unstable This can be avoided by proper choice of the controller realizations

15. Youla parameterization (non-switched systems) Assume process is asymptotically stable u v real process Q(asympt. stable) – + process copy 1. If the real process and its copy have the same initial conditions )v = 0 8t otherwise v converges to zero exponentially fast 2. Since the Q system is asymptotically stable, u converges to zero exponentially fast No matter what we choose for Q, as long as it is asymptotically stable, the overall system is asymptotically stable

16. Youla parameterization (non-switched systems) Assume process is asymptotically stable u v real process Q(asympt. stable) – + process copy stabilizing controller 1. If the real process and its copy have the same initial conditions )v = 0 8t otherwise v converges to zero exponentially fast 2. Since the Q system is asymptotically stable, u converges to zero exponentially fast No matter what we choose for Q, as long as it is asymptotically stable, the overall system is asymptotically stable

17. Youla parameterization (non-switched) Assume process is asymptotically stable v u real process Q(asympt. stable) – + process copy Theorem [Youla-Bongiorno]: 1. For any asymptotically stable Q, this controller asymptotically stabilizes the overall system 2. Any controller that asymptotically stabilizes the overall system is of this form, for an asymptotically stable Q with the same IO model as: controller – a similar parameterization also exists when the process is not asymptotically stable… process Why?

18. “Youla” realizations e v u Q(asympt. stable) + process copy realization for the process copy realization for Q realization for the controller In general these realizations are not minimal

19. and back again to multiple controllers… v u e u Q1 controller 1 – process copy v e u u controller 2 Q2 – process copy realization for Qq

20. Switching controller s e v u switched Q + process copy realization for the process copy Switched Q switched controller

21. Switched closed-loop u v real process – + process copy 1. If the real process and its copy have the same initial conditions )v = 0 8t otherwise v converges to zero exponentially fast 2. If the switched Q system is asymptotically stable, u converges to zero exponentially fast and the overall system is asymptotically stable Always possible by appropriate choice of realizations for each Qq (e.g., by choosing realizations so that V(z) = z’ z is a common Lyapunov function)

22. Switched closed-loop v u real process – + the construction in this slide is only valid for stable processes Theorem: For every family of input-output controller models, there always exist a family a controller realizations such that the switched closed-loop systems is exponentially stable for arbitrary switching. One can actually show that there exists a common quadratic Lyapunov function for the closed-loop. In general the realizations are not minimal

23. Non-asymptotically stable processes 1st Pick one stabilizing “nominal” controller u real process – controller 0 asymptotically stable

24. Non-asymptotically stable processes 2nd repeat previous construction u v Q(asympt. stable) real process – – + controller 0 closed-loopcopy Theorem [Youla-Bongiorno]: 1. For any asymptotically stable Q, this controller asymptotically stabilizes the overall system 2. Any controller that asymptotically stabilizes the overall system is this form, for an appropriately chosen Q: desired controller – controller 0 – closed-loop copy Why?

25. Non-asymptotically stable processes 2nd repeat previous construction u v Q(asympt. stable) real process – – + controller 0 • Q will be stable as long as the controller 0 is stable • other more complicated constructions exist when one cannot find a stable controller 0 closed-loopcopy Theorem [Youla-Bongiorno]: 1. For any asymptotically stable Q, this controller asymptotically stabilizes the overall system 2. Any controller that asymptotically stabilizes the overall system is this form, for an appropriately chosen Q: desired controller – controller 0 – closed-loop copy Why?

26. Switching controller measurement noise switching signal taking values inQ›{1,2} s n etrack + + u q qreference switchingcontroller process + – By proper choice of the controllers realization we can have stability for arbitrary switching. s = 2 s = 1 s = 2

27. Next lecture… • Stability under slow switching • Dwell-time switching • Average dwell-time • Stability under brief instabilities