Hybrid Control and Switched Systems Lecture #8 Stability and convergence of hybrid systems (topological view) João P. Hespanha University of California at Santa Barbara Summary Lyapunov stability of hybrid systems 1
Properties of hybrid systems sig set of all piecewise continuous signals :[0,T) R n, T (0, ] sig set of all piecewise constant signals q:[0,t), T (0, ] Sequence property p : sig sig {false,true} E.g., A pair of signals (q, ) sig sig satisfies p if p(q, ) = true A hybrid automaton H satisfies p ( write H ² p ) if p(q, ) = true, for every solution (q, ) 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 Lyapunov stability of ODEs (recall) sig set of all piecewise continuous signals taking values in R n Given a signal sig, sig ú sup t 0 (t) signal norm ODE can be seen as an operator T : R n sig that maps 0 R n into the solution that starts at (0) = 0 A solution * is (Lyapunov) stable if T is continuous at * 0 ú * (0), i.e., e > 0 δ >0 : 0 * 0 δ T( 0 ) T(* 0 ) sig e δ sup t 0 (t) * (t) e (t) * (t) e pend.m 2
Lyapunov stability of hybrid systems ú Φ 2 (q 1, ) Φ 1 (q 1, ) = q 2? mode q 2 mode q 1 sig set of all piecewise continuous signals :[0,T) R n, T (0, ] sig set of all piecewise constant signals q:[0,t), T (0, ] A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q * (0), * (0)). To make sense of continuity we need ways to measure distances in R n and sig sig 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 A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., e > 0 δ >0 : d( (q * (0), * (0)), (q(0), (0)) ) δ sup t 0 d ( (q * (t), * (t)), (q(t), (t)) ) e 1. If the solution starts close to (q *, * ) it will remain close to it forever 2. e can be made arbitrarily small by choosing δ sufficiently small Note: may actually not be metrics on R n because one may want zero-distance between points. However, still define a topology on R n, which is what is really needed to make sense of continuity 3
Topological spaces Given a set and a collection of subsets of (, ) is a topological space if 1., 2., 3. 1, 2,, n i=1n i (1 n ) Eamples: ú { q 1, q 2,, q n } (finite) ú {, } (trivial topology all points are close to each other) ú { } {all subsets of } (discrete topology no two distinct points are close to each other) ú {, {1}, {1,2} } of {1,2} ú R n Intuitively: two elements of are arbitrarily close if for every open set one belongs to, the other also belongs to is called a topology and the sets in are called open and their complements are called closed A point is only arbitarily close to itself (Hausdorff space) ú { (possibly infinite) union of all open balls } (norm-induced topology) open ball { R n : 0 < e } How to prove that 2. holds? (Hint: and are distributive & intersection of two open balls is in ) Continuity in Topological spaces Given a set and a collection of subsets of (, ) is a topological space if 1., 2., 3. 1, 2,, n i=1n i (1 n ) Given a function f: with, topological space is called a topology and the sets in are called open and their complements are called closed f is continuous at a point 0 in if for every neighborhood (i.e., set containing an open set) of f( 0 ) there is a neighborhood of 0 such that f( ). f( ) f( 0 ) f 0 Intuitively: arbitrarily close points in are transformed into arbitrarily close points in For norm-induced topologies we need only consider balls ú { y : y f( 0 ) < e } and ú { : 0 < δ } 4
Continuity in Topological spaces Given a function f: with, topological space f is continuous at a point 0 in if for every neighborhood (i.e., set containing an open set) of f( 0 ) there is a neighborhood of 0 such that f( ). Eamples: ú R n, ú R m, ú { (possibly infinite) union of all open balls } (norm-induced top.) open ball { R n : 0 < e } leads to the usual definition of continuity in R n : f continuous at 0 if e > 0 δ > 0 : 0 < δ f() f( 0 ) < e f( ) f( 0 ) f 0 Could be restated as: for every ball ú { y : y f( 0 ) < e } there is a ball ú { : 0 < δ } such that f() or equivalently f( ) Continuity in Topological spaces Given a function f: with, topological space f is continuous at a point 0 in if for every neighborhood (i.e., set containing an open set) of f( 0 ) there is a neighborhood of 0 such that f( ). Eamples: ú { q 1, q 2,, q n } (finite) 1. ú {, } (trivial topology all points are close to each other) 2. ú { } {all subsets of } (discrete topology no two distinct points are close to each other) 3. ú {, {1}, {1,2} } of {1,2} Is any of these functions f : R continuous? (usual norm-topology in R) f() f() f() f() 5
Continuity in Topological spaces Given a function f: with, topological space f is continuous at a point 0 in if for every neighborhood (i.e., set containing an open set) of f( 0 ) there is a neighborhood of 0 such that f( ). Eamples: ú { q 1, q 2,, q n } (finite) 1. ú {, } (trivial topology all points are close to each other) 2. ú { } {all subsets of } (discrete topology no two points are close to each other) 3. ú {, {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 continuous? (usual norm-topology in R) f() f() f() f() continuous for 1., 2., 3. continuous for 1., 3. continuous only for 1. continuous only for 1. (for those that don t want to leave anything to the imagination ) Given a sets, with topologies and One can construct a topology on : ú { :, } Eample: ú {1, 2}, ú {, {1}, {1,2} } ú R, R norm-induced topology some open sets:, { (1,): (1,2) }, { (q,): q=1,2, (1, ) } not open sets: { (1,): (1,2] }, { (2,): (1,2) } One can construct a topology sig on the set sig of signals q:[0,t), T (0, ]: sig ú sets of the form ú { q sig : q(t) (t) t} where the (t) are a collection of open sets Eample: ú {1, 2}, ú {, {1}, {1,2} } some open sets: { q : q(t) = 1 t 1 }, { q : q(t) = 1 t Q } not open sets: {q : q(t) = 2 t 1 } ú R, R norm-induced topology some open sets: { : (t) < 0 t }, { : (t) < 1 t } non open sets: { : 0 (t) dt <1 } 6
Back to hybrid systems A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., for every neighborhood of T(q 0*, 0* ) there is a neighborhood of (q 0*, 0* ) such that T( ) Case 1: domain of T: trivial topology (all points close to each other) R n usual topology induced from Euclidean norm co-domain of T: sig trivial topology (all signals close to each other) sig usual topology induced from sup-norm e > 0 δ >0 : q(0), * (0) (0)) < δ t * (t) (t) < e one does not care at all about the discrete states matching Back to hybrid systems A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., for every neighborhood of T(q 0*, 0* ) there is a neighborhood of (q 0*, 0* ) such that T( ) Case 2: domain of T: discrete topology (all points far from each other) R n usual topology induced from Euclidean norm co-domain of T: sig discrete topology (all signals far from each other) sig usual topology induced from sup-norm e > 0 δ >0 : q * (0) = q(0), * (0) (0)) < δ t q * (t) = q(t), * (t) (t) < e one cares very much about the discrete states matching 7
Back to hybrid systems A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., for every neighborhood of T(q 0*, 0* ) there is a neighborhood of (q 0*, 0* ) such that T( ) Case 3: domain of T: Q discrete topology (all points far from each other) R n usual topology induced from Euclidean norm co-domain of T: sig trivial topology (all signals close to each other) sig usual topology induced from sup-norm e > 0 δ >0 : q * (0) = q(0), * (0) (0)) < δ t * (t) (t) < e one cares very much about the discrete states matching Back to hybrid systems A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., for every neighborhood of T(q 0*, 0* ) there is a neighborhood of (q 0*, 0* ) such that T( ) Case 4: domain of T: {, {1}, {1,2} }, ú {1,2} R n usual topology induced from Euclidean norm co-domain of T: sig trivial topology (all signals close to each other) sig usual topology induced from sup-norm small perturbation in (but no perturbation in q) leads to small change in for q * (0) = 1: e > 0 δ >0 : q(0) = 1, * (0) (0)) < δ t * (t) (t) < e for q * (0) = 2: e > 0 δ >0 : q(0), * (0) (0)) < δ t * (t) (t) < e small perturbation in leads to small change in, regardless of q(0) 8
Back to hybrid systems A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., for every neighborhood of T(q 0*, 0* ) there is a neighborhood of (q 0*, 0* ) such that T( ) Case 4: domain of T: discrete topology (all points far from each other) R n usual topology induced from Euclidean norm co-domain of T: {, {1}, {1,2} }, ú {1,2} (signal version ) sig usual topology induced from sup-norm e > 0 δ >0 : q * (0) = q(0), * (0) (0)) < δ tq * (t) = 2 or q * (t) = q(t), * (t) (t) < e small perturbation in (but no perturbation in q) leads to small change in, q * and q may differ only when q * = 2 Eample #2: Thermostat mean temperature room 73? q = 1 q = 2 heater 77? 77 73 q = 2 no trajectory is stable 1 1 1 2 2 2 some trajectories are stable others unstable turn heater off turn heater on t Why? all trajectories are stable for discrete topology on (all points far from each other) for trivial topology on (all points close to each other) for discrete topology on for the domain and the trivial topology for the codomain 9
pump pump-on inflow λ = 3 constant outflow μ = 1 y 1? pump off (q = 1) Eample #5: Tank system y τ.5? goal prevent the tank from emptying or filling up δ =.5 delay between command is sent to pump and the time it is eecuted wait to off (q = 4) τ ú 0 τ ú 0 wait to on (q = 2) τ.5? pump on (q = 3) A possible topology for : ú {, {1,2}, {3,4}, {1,2,3,4} } y 2? this topology only distinguishes between modes based on the state of the pump Eample #4: Inverted pendulum swing-up u [-1,1] θ Hybrid controller: 1 st pump/remove energy into/from the system by applying maimum force, until E 0 (energy control) 2 nd wait until pendulum is close to the upright position 3 th net to upright position use feedback linearization controller remove energy E [-ε,ε]? E < ε E > ε pump energy wait E [-ε,ε]? ω + θ δ? stabilize 10
Eample #4: Inverted pendulum swing-up u [-1,1] θ A possible topology for ú {r,p,w,s} : ú {, {s}, {r,p,w,s} } 1. for solutions (q *, * ) that start in s, we only consider perturbations that also start in s 2. for solutions (q *, * ) that start outside s, the perturbations can start in any state remove energy E [-ε,ε]? E < ε E > ε pump energy wait E [-ε,ε]? ω + θ δ? stabilize Asymptotic stability for hybrid systems A solution (q *, * ) is (Lyapunov) stable if T is continuous at (q 0*, 0* )ú(q * (0), * (0)), i.e., for every neighborhood of T(q 0*, 0* ) there is a neighborhood of (q 0*, 0* ) such that T( ) Definition: A solution (q *, * ) is asymptotically stable if it is stable, every solution (q,) eists globally, and q q *, * as t in the sense of the topology on sig 11
Eample #7: Server system with congestion control incoming rate r q q ma? r ú m r q ma q server B rate of service (bandwidth) every solution eists globally and converges to a periodic solution that is stable. Do we have asymptotic stability? Eample #7: Server system with congestion control incoming rate r q q ma? r ú m r q ma q server B rate of service (bandwidth) every solution eists globally and converges to a periodic solution that is stable. Do we have asymptotic stability? No, its difficult to have asymptotic stability for non-constant solutions due to the synchronization requirement. (not even stability Always?) 12
Eample #2: Thermostat mean temperature room 73? q = 1 q = 2 heater 77? 77 73 q = 2 no trajectory is stable 1 1 1 2 2 2 turn heater off turn heater on t all trajectories are stable but not asymptotically for discrete topology on (all points far from each other) Why? for trivial topology on (all points close to each other) Stability of sets Poincaré distance between (q, ), (q *, * ) sig sig after t 0 (q * (t), * (t)) (q(t), (t)) distance at the point t where the (q(t), (t)) is the furthest apart from (q *, * ) can also be viewed as the distance from the trajectory (q, ) to the set {(q * (t), * (t)) :t t 0 } For constant trajectories (q *, * ) its just the sup-norm: 13
Stability of sets Poincaré distance between (q, ), (q *, * ) sig sig after t 0 Definition: A solution (q *, * ) is Poincaré stable if T is continuous at (q 0*, 0* ) ú (q * (0), * (0)) for the topology on sig induced by the Poincaré distance, e > 0 δ >0 : d 0 ((q(0), (0)), (q * (0), * (0))) δ d P ((q *, * ), (q,); 0) = sup t 0 inf τ 0 d T ((q(t), (t)), (q * (τ), * (τ))) e in more modern terminology one would say that the following set is stable { (q * (t), * (t)) : t 0 } (open sets are unions of open Poincaré balls { sig : d P ( 0 ) < e }. Show this is a topology ) Stability of sets Poincaré distance between (q, ), (q *, * ) sig sig after t 0 Definition: A solution (q *, * ) is Poincaré asymptotically stable if it is Poincaré stable, every solution (q, ) eists globally, and d P ((q,), (q *, * ); t ) 0 as t in more modern terminology one would say that the following set is asymptotically stable: { (q * (t), * (t)) : t 0 } 14
Eample #7: Server system with congestion control incoming rate r q q ma? r ú m r q ma q server B rate of service (bandwidth) r all trajectories are Poincaré asymptotically stable q ma q To think about 1. 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) 2. Even with ODEs there are several alternatives: e.g., e > 0 δ >0 : 0 eq δ sup t 0 (t) eq e or e > 0 δ >0 : 0 eq δ 0 (t) eq dt e or e > 0 δ >0 : 0 0* δ d P (, * ; 0 ) e (even for linear systems these definitions may differ: Why?) Lyapunov integral Poincaré 15
Net lecture Analysis tools for hybrid systems 1. Impact maps Fied-point theorem Stability of periodic solutions 2. Decoupling Switched systems Supervisors 16