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Stability and Convergence of Hybrid Systems (Topological View)

This lecture discusses the stability and convergence properties of hybrid systems, with a focus on topological aspects. Topics covered include Lyapunov stability of hybrid systems, properties of hybrid systems, and the continuity definition in topological spaces.

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Stability and Convergence of Hybrid Systems (Topological View)

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  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

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