1 / 31

Lecture #8 Stability and convergence of hybrid systems (topological view)

Hybrid Control and Switched Systems. Lecture #8 Stability and convergence of hybrid systems (topological view). Jo ã o P. Hespanha University of California at Santa Barbara. Summary. Lyapunov stability of hybrid systems. Properties of hybrid systems.

xandy
Download Presentation

Lecture #8 Stability and convergence of hybrid systems (topological view)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Hybrid Control and Switched Systems Lecture #8Stability and convergence of hybrid systems(topological view) João P. Hespanha University of Californiaat Santa Barbara

  2. Summary Lyapunov stability of hybrid systems

  3. Properties of hybrid systems Xsig´ set of all piecewise continuous signals x:[0,T) !Rn, T2(0,1] Qsig´ set of all piecewise constant signals q:[0,T)!Q, T2(0,1] Sequence property´p : Qsig£Xsig! {false,true} E.g., A pair of signals (q, x) 2Qsig£Xsigsatisfiesp if p(q, x) = true A hybrid automatonHsatisfiesp ( write H²p ) if p(q, x) = true, for every solution (q, x) of H “ensemble properties” ´ property of the whole family of solutions(cannot be checked just by looking at isolated solutions)e.g., continuity with respect to initial conditions…

  4. Lyapunov stability of ODEs (recall) Xsig´ set of all piecewise continuous signals taking values in Rn Given a signal x2Xsig, ||x||sig› supt¸0 ||x(t)|| signal norm ODE can be seen as an operator T : Rn!Xsig that maps x02 Rn into the solution that starts at x(0) = x0 Definition (continuity definition): A solution x* is (Lyapunov) stable if T is continuous at x*0›x*(0), i.e., 8e > 0 9d >0 : ||x0 – x*0|| ·d) ||T(x0) – T(x*0)||sig·e supt¸0 ||x(t) – x*(t)|| ·e d e x(t) x*(t) pend.m

  5. Lyapunov stability of hybrid systems mode q1 mode q2 F1(q1,x–) = q2 ? x› F2(q1,x–) Xsig´ set of all piecewise continuous signals x:[0,T) !Rn, T2(0,1] Qsig´ set of all piecewise constant signals q:[0,T)!Q, T2(0,1] Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*,x*) is (Lyapunov) stable if T is continuous at (q*(0), x*(0)). To make sense of continuity we need ways to measure “distances” in Q £Rn and Qsig£ Xsig

  6. Lyapunov stability of hybrid systems A few possible “metrics” one cares very much about the discrete states matching one does not care at all about the discrete states matching Definition (continuity definition): A solution (q*,x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., 8e > 0 9d >0 : d( (q*(0), x*(0)) , (q(0), x(0)) )·d ß supt¸ 0d ( (q*(t), x*(t)) , (q(t), x(t)) )·e • e can be made arbitrarily small by choosing d sufficiently small • If the solution starts close to (q*,x*) it will remain close to it forever Note: may actually not be metrics on Q £ Rn because one may want “zero-distance” between points. However, still define a topology on Q £ Rn, which is what is really needed to make sense of continuity…

  7. Topological spaces Given a set X and a collection TX of subsets of X • (X, TX) is a topological space if • ;, X2TX • A, B2TX)AÅB2TX • A1, A2, …, An2 TX)[i=1nAi2 TX (1·n · 1) T is called a topology and the sets in T are called open and their complements are called closed Intuitively: two elements of X are “arbitrarily close” if for every open set one belongs to, the other also belongs to Examples: Q› { q1, q2, …, qn } (finite) TQ› {; , Q } (trivial topology – all points are close to each other) TQ› {;} [ {all subsets of Q } (discrete topology – no two distinct points are close to each other) TQ› {;, {1}, {1,2} } of {1,2} X›Rn TX› { (possibly infinite) union of all open balls } (norm-induced topology) open ball ´ { x2 Rn : ||x – x0|| < e } A point is only “arbitarily close” to itself(Hausdorff space) How to prove that 2. holds? (Hint: [ and Å are distributive & intersection of two open balls is in T)

  8. Continuity in Topological spaces Given a set X and a collection TX of subsets of X • (X, TX) is a topological space if • ;, X2TX • A, B2TX)AÅB2TX • A1, A2, …, An2 TX)[i=1nAi2 TX (1·n · 1) T is called a topology and the sets in T are called open and their complements are called closed Given a function f: X!Y with X, Y topological space f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½V. 8 V U f(U) Intuitively: “arbitrarily close” points in X are transformed into “arbitrarily close” points in Y f f(x0) x0 For norm-induced topologies we need only consider ballsV› { y : || y – f(x0) || < e } and U› { x : || x – x0 || < d }

  9. Continuity in Topological spaces Given a function f: X!Y with X, Y topological space f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½V. Examples: X›Rn, Y›Rm TX, TY› { (possibly infinite) union of all open balls } (norm-induced top.) open ball ´ { x2 Rn : ||x – x0|| < e } leads to the usual definition of continuity in Rn: f continuous at x0 if 8e > 09d > 0 : ||x – x0|| < d) || f(x) – f(x0)|| < e Could be restated as: for every ball V› { y : || y – f(x0) || <e } there is a ball U› { x : || x – x0 || <d } such that x2U)f(x) 2 V or equivalently f(U) ½V 8 V U f(U) f f(x0) x0

  10. Continuity in Topological spaces f(x) f(x) f(x) f(x) x x x x Given a function f: X!Y with X, Y topological space f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½V. Examples: Q› { q1, q2, …, qn } (finite) 1. TQ› {; , Q } (trivial topology – all points are close to each other) 2. TQ› {;} [ {all subsets of Q } (discrete topology – no two distinct points are close to each other) 3. TQ› {;, {1}, {1,2} } of {1,2} Is any of these functions f : R!Q continuous? (usual norm-topology in R)

  11. Continuity in Topological spaces f(x) f(x) f(x) f(x) x x x x Given a function f: X!Y with X, Y topological space f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½V. Examples: Q› { q1, q2, …, qn } (finite) 1. TQ› {; , Q } (trivial topology – all points are close to each other) 2. TQ› {;} [ {all subsets of Q } (discrete topology – no two points are close to each other) 3. TQ› {;, {1}, {1,2} } of {1,2} (2 is ?close? to 1 but 1 is not ?close? to 2) Is any of these functions f : R!Q continuous? (usual norm-topology in R) continuousfor 1., 2., 3. continuousfor 1., 3. continuousonly for 1. continuousonly for 1.

  12. (for those that don’t want to leave anything to the imagination…) Given a sets Q, X with topologies TQ and TX One can construct a topology TQ£X on Q£X: TQ£X› { A£B : A2Q, B2X } Example: Q› {1, 2}, TQ› {;, {1}, {1,2} }X›R, TR´ norm-induced topology some open sets: ; , { (1,x): x2(1,2) }, { (q,x): q=1,2, x2(1,1) } not open sets: { (1,x): x2(1,2] }, { (2,x): x2(1,2) } One can construct a topology TQsig on the set Qsig of signals q:[0,T)!Q, T2(0,1]: TQsig› sets of the form A› { q2Qsig : q(t) 2A(t) 8t} where the A(t) are a collection of open sets Example: Q› {1, 2}, TQ› {;, {1}, {1,2} }some open sets: { q : q(t) = 1 8t· 1 }, { q : q(t) = 1 8t 2 Q } not open sets: {q : q(t) = 2 8t· 1 } X›R, TR´ norm-induced topology some open sets: { x : x(t) < 0 8t }, { x : |x(t)| < 1 8t } non open sets: { x : s01x(t) dt <1 }

  13. Back to hybrid systems… Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½V Case 1: domain of T: TQ´ trivial topology (all points close to each other) T Rn ´ usual topology induced from Euclidean norm co-domain of T: TQsig´ trivial topology (all signals close to each other) TXsig´ usual topology induced from sup-norm 8e > 0 9d >0 : 8q(0), || x*(0) – x(0))|| < d ) 8t || x*(t) – x(t)|| < e U V one does not care at all about the discrete states matching

  14. Back to hybrid systems… Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½V Case 2: domain of T: TQ´ discrete topology (all points far from each other) T Rn ´ usual topology induced from Euclidean norm co-domain of T: TQsig´ discrete topology (all signals far from each other) TXsig´ usual topology induced from sup-norm 8e > 0 9d >0 : q*(0) = q(0), || x*(0) – x(0))|| < d ) 8t q*(t) = q(t), || x*(t) – x(t)|| <e U V one cares very much about the discrete states matching

  15. Back to hybrid systems… Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½V Case 3: domain of T: TQ´ discrete topology (all points far from each other) T Rn ´ usual topology induced from Euclidean norm co-domain of T: TQsig´ trivial topology (all signals close to each other) TXsig´ usual topology induced from sup-norm 8e > 0 9d >0 : q*(0) = q(0), || x*(0) – x(0))|| < d ) 8t || x*(t) – x(t)|| < e U V one cares very much about the discrete states matching

  16. Back to hybrid systems… Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½V Case 4: domain of T: TQ´ {;, {1}, {1,2} }, Q› {1,2} T Rn ´ usual topology induced from Euclidean norm co-domain of T: TQsig´ trivial topology (all signals close to each other) TXsig´ usual topology induced from sup-norm for q*(0) = 1: 8e > 0 9d >0 : q(0) = 1, || x*(0) – x(0))|| < d ) 8t || x*(t) – x(t)|| < e for q*(0) = 2: 8e > 0 9d >0 : 8q(0), || x*(0) – x(0))|| < d ) 8t || x*(t) – x(t)|| < e small perturbation in x (but no perturbation in q) leads to small change in x small perturbation in x leads to small change in x, regardless of q(0)

  17. Back to hybrid systems… Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½V Case 4: domain of T: TQ´ discrete topology (all points far from each other) T Rn ´ usual topology induced from Euclidean norm co-domain of T: TQ´ {;, {1}, {1,2} }, Q› {1,2} (signal version…) TXsig´ usual topology induced from sup-norm 8e > 0 9d >0 : q*(0) = q(0), || x*(0) – x(0))|| < d ) 8tq*(t) = 2 or q*(t) = q(t), || x*(t) – x(t)|| < e small perturbation in x (but no perturbation in q) leads to small change in x, q* and q may differ only when q* = 2

  18. Example #2: Thermostat x´ mean temperature x· 73 ? q = 2 q = 1 room heater x¸ 77 ? turn heater off 77 x 73 turn heater on 1 1 1 q = 2 2 2 2 Why? t some trajectories are stable others unstable no trajectory is stable all trajectories are stable for discrete topology on Q for the domain and the trivial topology for the codomain for discrete topology on Q (all points far from each other) for trivial topology on Q (all points close to each other)

  19. Example #5: Tank system pump goal ´ prevent the tank from emptying or filling up pump-on inflow ´l = 3 d = .5 ´ delay between command is sent to pump and the time it is executed y constant outflow ´m = 1 t¸.5 ? pump off (q = 1) wait to off (q = 4) y· 1 ? t› 0 wait to on (q = 2) pump on (q = 3) t› 0 y¸ 2 ? t¸.5 ? this topology only distinguishes between modes based on the state of the pump A possible topology for Q: TQ› {;, {1,2}, {3,4}, {1,2,3,4} }

  20. Example #4: Inverted pendulum swing-up q Hybrid controller: 1st pump/remove energy into/from the system by applying maximum force, until E ¼ 0 (energy control) 2nd wait until pendulum is close to the upright position 3th next to upright position use feedback linearization controller u2 [-1,1] E2 [-e,e] ? remove energy stabilize wait E < – e E > e pump energy |w| + |q| ·d ? E2 [-e,e] ?

  21. Example #4: Inverted pendulum swing-up q u2 [-1,1] A possible topology for Q › {r,p,w,s} : TQ› {;, {s}, {r,p,w,s} } 1. for solutions (q*, x*) that start in “s,” we only consider perturbations that also start in “s” 2. for solutions (q*, x*) that start outside “s,” the perturbations can start in any state E2 [-e,e] ? remove energy stabilize wait E < – e E > e pump energy |w| + |q| ·d ? E2 [-e,e] ?

  22. Asymptotic stability for hybrid systems Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Definition (continuity definition): A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½V Definition: A solution (q*, x*) is asymptotically stable if it is stable, every solution (q,x) exists globally, and q!q*, x!x* as t!1 in the sense of the topology on Qsig

  23. Example #7: Server system with congestion control r q¸qmax ? incoming rate r›m r – qmax q server B every solution exists globally and converges to a periodic solution that is stable. Do we have asymptotic stability? rate of service (bandwidth)

  24. Example #7: Server system with congestion control r q¸qmax ? incoming rate r›m r – qmax q server every solution exists globally and converges to a periodic solution that is stable. Do we have asymptotic stability? B rate of service (bandwidth) No, its difficult to have asymptotic stability for non-constant solutions due to the “synchronization” requirement. (not even stability… Always?)

  25. Example #2: Thermostat x´ mean temperature x· 73 ? q = 2 q = 1 room heater x¸ 77 ? turn heater off 77 x 73 turn heater on 1 1 1 q = 2 2 2 2 t all trajectories are stable but not asymptotically no trajectory is stable for trivial topology on Q (all points close to each other) for discrete topology on Q (all points far from each other) Why?

  26. Stability of sets Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Poincaré distance between (q, x), (q*,x*)2 Qsig£ Xsig after t0 distance at the point t where the (q(t), x(t)) is the furthest apart from (q*, x*) can also be viewed as the distance from the trajectory (q, x) to the set {(q*(t), x*(t)) :t¸ t0} (q*(t), x*(t)) (q(t), x(t)) For constant trajectories (q*,x*) its just the sup-norm:

  27. Stability of sets Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Poincaré distance between (q, x), (q*,x*)2 Qsig£ Xsig after t0 Definition: A solution (q*, x*) is Poincaré stable if T is continuous at (q0*, x0*) › (q*(0), x*(0)) for the topology on Xsig induced by the Poincaré distance, 8e > 0 9d >0 : d0((q(0), x(0)), (q*(0), x*(0)))·d ß dP((q*,x*), (q,x); 0) = supt¸ 0inf¸ 0 dT ((q(t), x(t)), (q*(), x*()))·e in more modern terminology one would say that the following set is stable { (q*(t), x*(t)) : t ¸ 0 } ½Q£X (open sets are unions of open Poincaré balls { x2 Xsig : dP(x – x0) < e }. Show this is a topology…)

  28. Stability of sets Hybrid automaton can be seen as an operator T : Q £Rn!Qsig£ Xsig that maps (q0, x0) 2 Q £Rn into the solution that starts at q(0) = q0, x(0) = x0 Poincaré distance between (q, x), (q*,x*)2 Qsig£ Xsig after t0 Definition: A solution (q*, x*) is Poincaré asymptotically stable if it is Poincaré stable, every solution (q, x) exists globally, and dP((q,x), (q*,x*); t )!0 as t!1 in more modern terminology one would say that the following set is asymptotically stable: { (q*(t), x*(t)) : t ¸ 0 } ½Q£X

  29. Example #7: Server system with congestion control r q¸qmax ? incoming rate r›m r – qmax q server r B all trajectories are Poincaré asymptotically stable rate of service (bandwidth) qmax q

  30. To think about … • With hybrid systems there are many possible notions of stability.(especially due to the topology imposed on the discrete state)WHICH ONE IS THE BEST? • (engineering question, not a mathematical one) • What type of perturbations do you want to consider on the initial conditions? • (this will define the topology on the initial conditions) • What type of changes are you willing to accept in the solution? • (this will define the topology on the signals) • Even with ODEs there are several alternatives: e.g., • 8e > 0 9d >0 : ||x0 – xeq|| ·d) supt¸ 0||x(t) – xeq|| ·eor • 8e > 0 9d >0 : ||x0 – xeq|| ·d)s01 ||x(t) – xeq|| dt ·eor • 8e > 0 9d >0 : ||x0 – x0*|| ·d) dP( x, x*; 0 ) ·e • (even for linear systems these definitions may differ: Why?) Lyapunov integral Poincaré

  31. Next lecture… • Analysis tools for hybrid systems • Impact maps • Fixed-point theorem • Stability of periodic solutions • Decoupling • Switched systems • Supervisors

More Related