Hybrid Systems Course Lyapunov stability

Hybrid Systems Course Lyapunov stability OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata

OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation is a mathematical model of a continuous state continuous time system: X = < n state space f: < n! < n vector field (assigns a velocity vector to each x) Given an initial value x 0 2 X, an execution (solution in the sense of Caratheodory) over the time interval [0,T) is a function x: [0,T)! < n such that: x(0) = x 0 x is continuous and piecewise differentiable

ODE SOLUTION: WELL-POSEDNESS Theorem [global existence and uniqueness non-blocking, deterministic, non-zeno] If f: < n! < n is globally Lipschitz continuous, then 8 x 0 there exists a single solution with x(0)=x 0 defined on [0,1). STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous Definition (equilibrium): x e 2 < n for which f(x e )=0 Remark: {x e } is an invariant set

Definition (stable equilibrium): Graphically: perturbed motion execution starting from x(0)=x 0 d x e equilibrium motion small perturbations lead to small changes in behavior Definition (asymptotically stable equilibrium): and d can be chosen so that Graphically: perturbed motion d x e equilibrium motion small perturbations lead to small changes in behavior and are re-absorbed, in the long run

Definition (asymptotically stable equilibrium): and d can be chosen so that Graphically: d xe small perturbations lead to small changes in behavior and are re-absorbed, in the long run EXAMPLE: PENDULUM l friction coefficient (a) m

m EXAMPLE: PENDULUM unstable equilibrium EXAMPLE: PENDULUM m as. stable equilibrium

EXAMPLE: PENDULUM l m Let x e be asymptotically stable. Definition (domain of attraction): The domain of attraction of x e is the set of x 0 such that execution starting from x(0)=x 0 Definition (globally asymptotically stable equilibrium): x e is globally asymptotically stable (GAS) if its domain of attraction is the whole state space < n

m EXAMPLE: PENDULUM m as. stable equilibrium small perturbations are absorbed, not all perturbations not GAS Let x e be asymptotically stable. Definition (exponential stability): x e is exponentially stable if 9 a, d, >0 such that

STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous Definition (equilibrium): x e 2 < n for which f(x e )=0 Without loss of generality we suppose that x e = 0 if not, then z := x -x e! dz/dt = g(z), g(z) := f(z+x e ) (g(0) = 0) STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous How to prove stability? find a function V: < n! < such that V(0) = 0 and V(x) >0, for all x 0 V(x) is decreasing along the executions of the system V(x) = 3 V(x) = 2 x(t)

STABILITY OF CONTINUOUS SYSTEMS execution x(t) behavior of V along the execution x(t): V(t): = V(x(t)) candidate function V(x) Advantage with respect to exhaustive check of all executions? STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous V: < n! < continuously differentiable (C 1 ) function Rate of change of V along the execution of the ODE system: (Lie derivative of V with respect to f) gradient vector

STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous V: < n! < continuously differentiable (C 1 ) function Rate of change of V along the execution of the ODE system: (Lie derivative of V with respect to f) gradient vector No need to solve the ODE for evaluating if V(x) decreases along the executions of the system LYAPUNOV STABILITY Theorem (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for the system and D½ < n an open set containing x e = 0. If V: D! < is a C 1 function such that V positive definite on D Then, x e is stable. V non increasing along system executions in D (negative semidefinite)

EXAMPLE: PENDULUM l friction coefficient (a) m energy function x e stable LYAPUNOV STABILITY Theorem (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for the system and D½ < n an open set containing x e = 0. If V: D! < is a C 1 function such that Then, x e is stable. If it holds also that Then, x e is asymptotically stable (AS)

LYAPUNOV GAS THEOREM Theorem (Barbashin-Krasovski Theorem): Let x e = 0 be an equilibrium for the system. If V: < n! < is a C 1 function such that V positive definite on < n V decreasing along system executions in < n (negative definite) V radially unbounded Then, x e is globally asymptotically stable (GAS). STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system the elements of matrix e At are linear combinations of e i(a)t, i=1,2,,n

STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system x e =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0) asymptotic stability GAS STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system x e =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0) asymptotic stability GAS Alternative characterization

STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Lyapunov equation STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Lyapunov equation Remarks: Q positive definite (Q>0) iff x T Qx >0 for all x 0 Q positive semidefinite (Q 0) iff x T Qx 0 for all x and x T Q x = 0 for some x 0

STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Proof. (if) V(x) =x T P x is a Lyapunov function Lyapunov equation STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Proof. (only if) Consider Lyapunov equation

STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Proof. (only if) Consider P = P T and P>0 easy to show P unique can be proven by contradiction Lyapunov equation STABILITY OF LINEAR CONTINUOUS SYSTEMS Remarks: for a linear system existence of a (quadratic) Lyapunov function V(x) =x T P x is a necessary and sufficient condition for asymptotic stability it is easy to compute a Lyapunov function since the Lyapunov equation is a linear algebraic equation A T P+PA = -Q

STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential stability): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: for all x(0) = x 0 2 < n, where 0 2 (0, min i Re{ i (A)} ) and >0 is an appropriate constant. STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential stability): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: for all x(0) = x 0 2 < n, where 0 2 (0, min i Re{ i (A)} ) and >0 is an appropriate constant. Im eigenvalues of A o o o o Re

STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential stability): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: STABILITY OF LINEAR CONTINUOUS SYSTEMS Proof (exponential stability): A + 0 I is Hurwitz (eigenvalues are equal to (A) + 0 ) Then, there exists P = P T >0 such that (A + 0 I) T P + P (A + 0 I) <0 which leads to x(t) T [A T P + P A]x(t) < - 2 0 x(t) T P x(t) Define V(x) = x T P x, then from which

STABILITY OF LINEAR CONTINUOUS SYSTEMS (cont d) Proof (exponential stability): thus finally leading to STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential stability): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: for all x(0) = x 0 2 < n, where 0 2 (0, min i Re{ i (A)} ) and >0 is an appropriate constant. Remark:

STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system x e =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0) asymptotic stability GAS exponential stability GES OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata

HYBRID AUTOMATA: FORMAL DEFINITION A hybrid automaton H is a collection H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1,q 2, } is a set of discrete states (modes) X = < n is the continuous state space f: Q X! < n is a set of vector fields on X Init µ Q X is a set of initial states Dom: Q! 2 X assigns to each q2 Q a domain Dom(q) of X E µ Q Q is a set of transitions (edges) G: E! 2 X is a set of guards (guard condition) R: E X! 2 X is a set of reset maps q = q 1 q = q 2

HYBRID TIME SET A hybrid time set is a finite or infinite sequence of intervals = {I i, i=0,1,, M } such that I i = [ i, i ] for i < M I M = [ M, M ] or I M = [ M, M ) if M<1 i = i+1 i i I = 2 2 2 [ I 1 1 ] t 2 1 I t 1 [ 0 ] 0 0 t 1 Á t 2 Á t 3 Á t 4 t 4 I time instants in are 3 3 linearly ordered [ ] 3 t 3 HYBRID TIME SET: LENGTH Two notions of length for a hybrid time set = {I i, i=0,1,, M }: Discrete extent: < > = M+1 Continuous extent: = i=0,1,..,m i - i number of discrete transitions total duration of intervals in < > = 4 = 3-0 [ ] 0 I 0 0 I 3 [ ] 3 3 I = 2 2 2 1 [ I 1 ] 1

HYBRID TIME SET: CLASSIFICATION A hybrid time set = {I i, i=0,1,, M } is Finite: if < > is finite and I M = [ M, M ] Infinite: if < > is infinite or is infinite Zeno: if < > is infinite but is finite finite infinite infinite Zeno Zeno HYBRID TRAJECTORY A hybrid trajectory (, q, x) consists of: A hybrid time set = {I i, i=0,1,, M } Two sequences of functions q = {q i ( ), i=0,1,, M } and x = {x i ( ), i=0,1,, M } such that q i : I i! Q x i : I i! X

HYBRID AUTOMATA: EXECUTION A hybrid trajectory (, q, x) is an execution (solution) of the hybrid automaton H = (Q,X,f,Init,Dom,E,G,R) if it satisfies the following conditions: Initial condition: (q 0 ( 0 ), x 0 ( 0 )) 2 Init Continuous evolution: for all i such that i < i q i : I i! Q is constant x i :I i! X is the solution to the ODE associated with q i ( i ) x i (t) 2 Dom(q i ( i )), t2 [ i, i ) Discrete evolution: (q i ( i ),q i+1 ( i+1 )) 2 E transition is feasible x i ( i ) 2 G((q i ( i ),q i+1 ( i+1 ))) guard condition satisfied x i+1 ( i+1 ) 2 R((q i ( i ),q i+1 ( i+1 )),x i ( i )) reset condition satisfied Well-posedness? HYBRID AUTOMATA: EXECUTION Problems due the hybrid nature: for some initial state (q,x) infinite execution of finite duration Zeno no infinite execution blocking multiple executions non-deterministic We denote by H (q,x) the set of (maximal) executions of H starting from (q,x) H (q,x) 1 the set of infinite executions of H starting from (q,x)

STABILITY OF HYBRID AUTOMATA Definition (equilibrium): H = (Q,X,f,Init,Dom,E,G,R) x e =0 2 X is an equilibrium point of H if: f(q,0) = 0 for all q 2 Q if ((q,q )2 E) Æ (02 G((q,q )) ) R((q,q ),0) = {0} Remarks: discrete transitions are allowed out of (q,0) but only to (q,0) if (q,0) 2 Init and (, q, x) is an execution of H starting from (q,0), then x(t) = 0 for all t2 EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: x 1 x 2 0} Dom(q 2 ) = {x2 X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: x 1 x 2 0} G((q 2,q 1 )) = {x2 X: x 1 x 2 0} R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x}

EXAMPLE: SWITCHED LINEAR SYSTEM x 2 x 1 EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: x 1 x 2 0} Dom(q 2 ) = {x2 X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: x 1 x 2 0} G((q 2,q 1 )) = {x2 X: x 1 x 2 0} R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x} x e = 0 is an equilibrium: f(q,0) = 0 & R((q,q ),0) = {0}

STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if set of (maximal) executions starting from (q 0, x 0 ) 2 Init STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if set of (maximal) executions starting from (q 0, x 0 ) 2 Init Remark: Stability does not imply convergence To analyse convergence, only infinite executions should be considered

STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if Definition (asymptotically stable equilibrium): set of (maximal) executions starting from (q 0, x 0 ) 2 Init Let x e = 0 2 X be an equilibrium point of H. x e = 0 is asymptotically stable if it is stable and d>0 that can be chosen so that set of infinite executions starting from (q 0, x 0 ) 2 Init 1 := i ( i - i ) continuous extent 1 < 1 if Zeno STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if Question: x e = 0 stable equilibrium for each continuous system dx/dt = f(q,x) implies that x e = 0 stable equilibrium for H?

EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: x 1 x 2 0} Dom(q 2 ) = {x2 X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: x 1 x 2 0} G((q 2,q 1 )) = {x2 X: x 1 x 2 0} R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x} x e = 0 is an equilibrium: f(q,0) = 0 & R((q,q ),0) = {0} EXAMPLE: SWITCHED LINEAR SYSTEM x 2 x 1

EXAMPLE: SWITCHED LINEAR SYSTEM Swiching between asymptotically stable linear systems. EXAMPLE: SWITCHED LINEAR SYSTEM q 1 q 2 q 2 q 1 q 1 : quadrants 2 and 4 q 2 : quadrants 1 and 3 Switching between asymptotically stable linear systems, but x e = 0 unstable equilibrium of H

EXAMPLE: SWITCHED LINEAR SYSTEM x( i+1 ) > x( i ) overshoots sum up EXAMPLE: SWITCHED LINEAR SYSTEM q 2 q 1 q 1 q 2 q 1 : quadrants 1 and 3 q 2 : quadrants 2 and 4

EXAMPLE: SWITCHED LINEAR SYSTEM x( i+1 ) < x( i ) Theorem (Lyapunov stability): LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. Consider V: Q D! < is a C 1 function in x such that for all q 2 Q: If for all (, q, x) 2 H (q0,x 0 ) with (q 0,x 0 ) 2 Init Å (Q D), and all q 2 Q, the sequence {V(q( i ),x( i )): q( i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H.

Theorem (Lyapunov stability): LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. Consider V: Q D! < is a C 1 function in x such that for all q 2 Q: V(q,x) Lyapunov function for continuous system q ) x e =0 is stable equilibrium for system q If for all (, q, x) 2 H (q0,x 0 ) with (q 0,x 0 ) 2 Init Å (Q D), and all q 2 Q, the sequence {V(q( i ),x( i )): q( i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H. Theorem (Lyapunov stability): LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. Consider V: Q D! < is a C 1 function in x such that for all q 2 Q: V(q,x) Lyapunov function for continuous system q ) x e =0 is stable equilibrium for system q If for all (, q, x) 2 H (q0,x 0 ) with (q 0,x 0 ) 2 Init Å (Q D), and all q 2 Q, the sequence {V(q( i ),x( i )): q( i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H.

Sketch of the proof. LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) V(q(t),x(t)) V(q 1,x(t)) Lyapunov function for system q 1! decreases when q(t) = q 1, but can increase when q(t) q 1 V(q 2,x(t)) q(t)= q 1 q(t)= q 1 [ ][ ][ ][ 0 0 = 1 1 = 2 2 = 3 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Sketch of the proof. V(q(t),x(t)) V(q 1,x(t)) {V(q 1,x( i ))} non-increasing q(t)= q 1 q(t)= q 1 [ ][ ][ ][ 0 0 = 1 1 = 2 2 = 3

Sketch of the proof. LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) V(q(t),x(t)) V(q 1,x(t)) q(t)= q 1 q(t)= q 1 [ ][ ][ ][ 0 0 = 1 1 = 2 2 = 3 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) V(q(t),x(t)) Lyapunov-like function

EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: Cx 0} Dom(q 2 ) = {x2 X: Cx 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: Cx 0} G((q 2,q 1 )) = {x2 X: Cx 0}, C T 2 < 2 R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x} EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) q 1 q 2

EXAMPLE: SWITCHED LINEAR SYSTEM x 2 Cx = 0 x 1 EXAMPLE: SWITCHED LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T 2 <2 : x e = 0 is an equilibrium: f(q 1,0) = f(q 2,0) = 0 R((q 1,q 2 ),0) = R((q 2,q 1 ),0) = {0}

EXAMPLE: SWITCHED LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T 2 <2 : x e = 0 is an equilibrium: f(q 1,0) = f(q 2,0) = 0 x e = 0 is stable: R((q 1,q 2 ),0) = R((q 2,q 1 ),0) = {0} consider the candidate Lyapunov-like function: V(q i,x) = x T P i x, where P i =P i T >0 solution to A i T P i + P i A i = - I (V(q i,x) is a Lyapunov function for the asymptotically stable linear system q i ) In each discrete state, the continuous system is as. stable. EXAMPLE: SWITCHED LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T 2 <2: x e = 0 is an equilibrium: f(q 1,0) = f(q 2,0) = 0 R((q 1,q 2 ),0) = R((q 2,q 1 ),0) = {0} x e = 0 is stable: consider the candidate Lyapunov-like function: V(q i,x) = x T P i x, where P i =P T i >0 solution to A T i P i + P i A i = - I

EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition Let q( i )=q 1 and x( i )=z. EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition Since V(q 1,x(t)) is not increasing during [ i, i ], then, when x(t) intersects the switching line at i, it does at a z with a 2 (0,1], hence x( i+1 ) = x( i ) x( i ). Let q( i+1 )=q 2 -z Cx = 0 i = i+1 i z

EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition Since V(q 2,x(t)) is decreasing during [ i+1, i+1 ], then, when x(t) intersects the switching line at i+1, x( i+2 ) = x( i+1 ) x( i+1 ) x( i ) -z Cx = 0 i = i+1 i z i+2 EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition From this, it follows that V(q 1,x( i+2 )) V(q 1,x( i )) -z Cx = 0 i = i+1 i z i+2

LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Drawbacks of the approach based on Lyapunov-like functions: In general, it is hard to find a Lyapunov-like function The sequence {V(q( i ),x( i )): q( i ) =q } must be evaluated, which may require solving the ODEs LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: then, x e = 0 is a stable equilibrium of H.

LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: independent of q then, x e = 0 is a stable equilibrium of H. LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: V(x) common Lyapunov function for all systems q then, x e = 0 is a stable equilibrium of H.

LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: V(x) common Lyapunov function for all systems q then, x e = 0 is a stable equilibrium of H. Proof: Define W(q,x) = V(x), 8 q 2 Q and apply the previous theorem V ( x( t)) same V function + identity reset map t

COMPUTATIONAL LYAPUNOV METHODS H PL = (Q,X,f,Init,Dom,E,G,R) non-zeno and such that for all q k 2 Q: f(q k,x) = A k x (linear vector fields) Init ½ [ q2 Q {q } Dom(q) (initialization within the domain) for all x2 X, the set Jump(q k,x):= {(q,x ): (q k,q )2 E, x2g((q k,q )), x 2R((q k,q ),x)} has cardinality 1 if x 2 Dom(q k ), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) 2 Jump(q k,x)! x 2 Dom(q ) and x = x (trivial reset for x) COMPUTATIONAL LYAPUNOV METHODS H PL = (Q,X,f,Init,Dom,E,G,R) non-zeno and such that for all q k 2 Q: f(q k,x) = A k x (linear vector fields) Init ½ [ q2 Q {q } Dom(q) (initialization within the domain) for all x2 X, the set Jump(q k,x):= {(q,x ): (q k,q )2 E, x2g((q k,q )), x 2R((q k,q ),x)} has cardinality 1 if x 2 Dom(q k ), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) 2 Jump(q k,x)! x 2 Dom(q ) and x = x (trivial reset for x) For this class of Piecewise Linear hybrid automata computationally attractive methods exist to construct the Lyapunov-like function

GLOBALLY QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (globally quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists P=P T >0 such that A T k P+ PA k < 0, 8 k Then, x e = 0 is asymptotically stable. Remark: V(x)=x T Px is a common Lyapunov function x e = 0 is stable GLOBALLY QUADRATIC LYAPUNOV FUNCTION Proof (showing exponential stability): There exists >0 such that A k T P+ PA k + I 0, 8 k There exists a unique, infinite, non-zeno execution (,q,x) for every initial state with x:! < n satisfying where k :! [0,1] is such that k k (t)=1, t2 [ i, i ]. Let V(x) = x T Px. Then, for t2 [ i, i ).

GLOBALLY QUADRATIC LYAPUNOV FUNCTION Proof. (cont d) min and max eigenvalues of P Since min x 2 V(x) max x 2, then and, hence, which leads to Then, Since 1 =1 (non-zeno), then x(t) goes to zero exponentially as t! 1 GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 conditions of the theorem satisfied with P = I

GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 GLOBALLY QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (globally quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists P=P T >0 such that A T k P+ PA k < 0, 8 k Then, x e = 0 is asymptotically stable. Remark: A set of LMIs to solve. This problem can be reformulated as a convex optimization problem. Efficient solvers exist.

GLOBALLY QUADRATIC LYAPUNOV FUNCTION Suppose that A k, k=1,2,,n, are Hurwitz matrices. Then, the set of linear matrix inequalities A k T P+ PA k < 0, k=1,2,,n, where P is positive definite symmetric is not feasible if and only if there exist positive definite symmetric matrices R k, k=1,2,,n, such that Proof. Based on results in convex analysis GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 stable node stable focus

GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 stable node stable focus no globally quadratic Lyapunov function exists although x e = 0 stable equilibrium PIECEWISE QUADRATIC LYAPUNOV FUNCTION Idea: consider different quadratic Lyapunov functions on each domain and glue them together so as to provide a (nonquadratic) Lyapunov function for H that is continuous at the switching times

PIECEWISE QUADRATIC LYAPUNOV FUNCTION Idea: consider different quadratic Lyapunov functions on each domain and glue them together so as to provide a (nonquadratic) Lyapunov function for H that is continuous at the switching times Developed for piecewise linear systems with structured domain and reset LMIs characterization COMPUTATIONAL LYAPUNOV METHODS H PL = (Q,X,f,Init,Dom,E,G,R) non-zeno and such that for all q k 2 Q: f(q k,x) = A k x (linear vector fields) Init ½ [ q2 Q {q } Dom(q) (initialization within the domain) for all x2 X, the set Jump(q k,x):= {(q,x ): (q k,q )2 E, x2g((q k,q )), x 2R((q k,q ),x)} has cardinality 1 if x 2 Dom(q), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) 2 Jump(q k,x)! x 2 Dom(q ) and x = x (trivial reset for x)

PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) satisfies the following additional assumptions: Dom(q) = {x 2 X: E q1 x 0, E q2 x 0,, E qn x 0} (each domain is a polygon) E q = [E q1 T E q2 T E qnt ] T 2 < n n defines the domain. (q,x ) 2 Jump(q,x) F q x = F q x, q q, x =x (matching condition at the boundaries of dom(q) and dom(q )) PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 P k E kt W k E k > 0 Then, x e = 0 is asymptotically stable.

PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 P k E kt W k E k > 0 Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 P k E kt W k E k > 0 P k positive definite within Dom(k) Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions

PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 A T k P k + PA k < 0 within Dom(k) P k E kt W k E k > 0 P k positive definite within Dom(k) Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 continuitity of V(x) P k E kt W k E k > 0 Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions

GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 level curves of the piecewise quadratic Lyapunov function (red lines) phase plot of some continuous state trajectories (blue lines) REFERENCES H.K. Khalil. Nonlinear Systems. Prentice Hall, 1996. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. M. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. on Automatic Control, 43(4):475-482, 1998. H. Ye, A. Michel, and L. Hou. Stability theory for hybrid dynamical systems. IEEE Transactions on Automatic Control, 43(4):461-474, 1998. M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov function for hybrid systems. IEEE Transactions on Automatic Control, 43(4):555-559, 1998. R.A. Decarlo, M.S. Branicky, S. Petterson, and B. Lennartson. Perspectives and results on the stability and stabilization of hybrid systems. Proceedings of the IEEE, 88(7):1069-1082, 2000.

Hybrid Systems Course Switched systems & stability OUTLINE Switched Systems Stability of Switched Systems

OUTLINE Switched Systems Stability of Switched Systems SWITCHED SYSTEMS a family of systems

SWITCHED SYSTEMS a family of systems a signal that orchestrates the switching between them s Note: The value of x is preserved when a switching occurs SWITCHED SYSTEMS AS HYBRID SYSTEMS

SWITCHED SYSTEMS vs. HYBRID AUTOMATA switched systems can be seen as a higher-level abstraction of hybrid automata (details of the discrete behavior neglected) simpler to describe but with more solutions than the original hybrid automata (conservative analysis results) SWITCHED SYSTEMS vs. HYBRID AUTOMATA switched systems can be seen as a higher-level abstraction of hybrid automata (details of the discrete behavior neglected) simpler to describe but with more solutions than the original hybrid automata (conservative analysis results) Switched systems are of interest in their own right

SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching

TIME-DEPENDENT SWITCHING s s (exogenous) switching signal piecewise constant function of time s(t) specifies the system that is active at time t SWITCHED LINEAR SYSTEMS family of systems time-dependent switching rule

STATE-DEPENDENT SWITCHING s(x) s (endogenous) switching signal the state space X is partitioned into operating regions, each one associated to a system s(x) specifies the system that is active when the state is x STATE-DEPENDENT SWITCHING

SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching AUTONOMOUS SWITCHING Switching events are triggered by an external mechanism over which we do not have control Examples: unpredictable environmental factors component failures

CONTROLLED SWITCHING Switching are imposed so as to achieve a desired behavior of the resulting system switched control systems SWITCHING CONTROL

SWITCHING CONTROL The closed-loop system is a switched system Reasons for switching: SWITCHING CONTROL nature of the control problem (system with different operation phases)

Reasons for switching: SWITCHING CONTROL nature of the control problem (system with different operation phases) Example: flight control system Reasons for switching: large modeling uncertainty SWITCHING CONTROL

SWITCHING CONTROL Reasons for switching: large modeling uncertainty Example: adaptive switching control P = admissible model set C 1 C 2 C 3 C 4 controller cover Reasons for switching: sensor/actuator limitations SWITCHING CONTROL

SWITCHING CONTROL Reasons for switching: sensor/actuator limitations Example: quantized control u PLANT x q(x) QUANTIZER q(x) x CONTROLLER EXAMPLE: THERMOSTAT Temperature in a room controlled by a thermostat switching a heater on and off Dynamics of the temperature x (in C): heater on: heater off: Goal: regulate the temperature around 20 C Strategy: turn the heater from OFF to ON as soon as x 18 turn the heater from ON to OFF as soon as x 22 ON 18 22 x OFF hysteretic behavior

EXAMPLE: THERMOSTAT Continuous dynamics linear ODEs describing the temperature evolution Discrete dynamics finite automaton describing the behavior of the thermostat Q = {ON,OFF} ON = (OFF,e 1 ) e 1 = [x 18] OFF = (ON,e 2 ) e 2 = [x 22] EXAMPLE: THERMOSTAT interface ODE dx/dt = - 0.2 x + u quantized control input (ON heating power u = 6 OFF heating power u= 0) continuous systems controlled by a discrete logic

CONTROLLED SWITCHING Switching are imposed so as to achieve a desired behavior of the resulting system switched control systems Reasons for switching: large modeling uncertainty nature of the control problem (phase systems) sensor/actuator limitations OUTLINE Switched Systems Stability of Switched Systems

SWITCHED SYSTEMS: TIME-DEPENDENT SWITCHING family of systems with piecewise constant switching signal SWITCHED SYSTEMS: TIME-DEPENDENT SWITCHING family of systems with piecewise constant switching signal Stability of the equilibrium x e =0?

SWITCHING BETWEEN AS. STABLE SYSTEMS SWITCHING BETWEEN AS. STABLE SYSTEMS unstable!

SWITCHING BETWEEN AS. STABLE SYSTEMS unstable! Problem: find conditions that guarantee asymptotic stability under arbitrary switching SWITCHING BETWEEN UNSTABLE SYSTEMS

SWITCHING BETWEEN UNSTABLE SYSTEMS stable! SWITCHING BETWEEN UNSTABLE SYSTEMS stable! Problem: identify those switching signals that ensure asymptotic stability

Stability for arbitrary switching Stability for constrained switching Stability for arbitrary switching Stability for constrained switching

GLOBAL UNIFORM ASYMPTOTIC STABILITY (GUAS) The equilibrium x e =0 is GUAS if it is globally asymptotically stable, uniformly with respect to the switching signals s GLOBAL UNIFORM ASYMPTOTIC STABILITY (GUAS) Assumption: family of systems with GAS equilibrium in x=0 Remark: if the equilibrium x e =0 is not GAS for one of the systems, then it cannot be GUAS for the switched system

COMMON LYAPUNOV FUNCTION The family of systems share a radially unbounded common Lyapunov function at x e =0 if there exists a continuously differentiable function V such that COMMON LYAPUNOV FUNCTION If all systems in the family share a radially unbounded common Lyapunov function at x e =0, then, the equilibrium x e =0 is GUAS. Proof. Same reasoning as for more general hybrid systems

GLOBALLY QUADRATIC LYAPUNOV FUNCTION If there exists such that then, the equilibrium x e =0 is GUAS. Proof. is a radially unbounded common Lyapunov function at x e =0. GLOBALLY QUADRATIC LYAPUNOV FUNCTION The existence of a globally quadratic Lyapunov function is not necessary for x e =0 to be GUAS Example: x e =0 is GUAS but there is no common quadratic Lyapunov function

SWITCHED SYSTEMS WITH A SPECIAL STRUCTURE Hurwitz matrices commute are upper (or lower) triangular COMMUTING HURWITZ MATRICES => GUAS A 1 A 2 A 2 A 1 s 1 s 2 s 1 s 2 s t s1 t1 2 2 t A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A e t t A s 2 ( k... 1 ) 1 (... 1 ) e k x(0) s 0

COMMUTING HURWITZ MATRICES => GUAS 1 A 2 A 2 A 1 quadratic common Lyapunov function: A COMMUTING HURWITZ MATRICES => GUAS 1 A 2 A 2 A 1 quadratic common Lyapunov function: A

TRIANGULAR HURWITZ MATRICES => GUAS TRIANGULAR HURWITZ MATRICES => GUAS

TRIANGULAR HURWITZ MATRICES => GUAS exponentially stable system exponentially decaying perturbation TRIANGULAR HURWITZ MATRICES => GUAS quadratic common Lyapunov function with P diagonal

SWITCHED SYSTEMS WITH A SPECIAL STRUCTURE Hurwitz matrices commute are upper (or lower) triangular can be transformed to upper (or lower) triangular form by a common similarity transformation Stability for arbitrary switching Stability for constrained switching

STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t t i, s i dwell time D STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) t D

STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential stability): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: for all x(0) = x 0 2 < n, where 0 2 (0, min i Re{ i (A)} ) and >0 is an appropriate constant. Remark: STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D 0 0 e i t t e e e D D t log slowest decay rate so that the inequality holds 8 i

STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D log e i t e t D STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D log e i t e D t log D 0 (0, 0)

STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D log D e i t e e 1 D t log D (0, log 0) D 0 D 0 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) t 0

STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 A t e 2 k A s e 1 k e A 2 t 1 A1s1 x (t) e x(0) t DWELL TIME: EXTENSIONS adaptive version: the dwell time is selected based on matrix A i so as to make the dynamics of the system contract by some 2 (0,1) during the dwell time average dwell time N s ( T, t) N 0 T t AD ( t, T # of switches on ) average dwell time N0 0 no switching: cannot switch if T t AD N 1 dwell time: cannot switch twice if T t AD 0 Same bound on AD as in the dwell time case. Larger values of x(t) in finite time because of N 0

STABILITY UNDER STATE-DEPENDENT SWITCHING s: X Q : s(x) = i if x 2 X i STATE-DEPENDENT COMMON LYAPUNOV FUNCTIONS If V: < n! < is a C 1 radially unbounded function such that V(x) state-dependent common Lyapunov function then, x e = 0 is GAS for Remarks: need that only when s is equal to q, i.e. on X q matrices A q are not required to be Hurwitz

STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 A T P PA 0

STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) 0 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x a( A P PA1 ) (1 a)( A2 P PA2 ) T T T 1 x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0

STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x T T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0 Region where system 1 is active for the system Region where system 2 is active is a Lyapunov function at x e =0 => GAS STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x T T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0 Region where system 1 is active Region where system 2 is active

STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x T T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Theorem: If the matrices A 1 and A 2 have a Hurwitz combination, then, there exists a state dependent switching strategy such that the switching system x = A σ x is GAS

STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Theorem: If the matrices A 1 and A 2 have a Hurwitz combination, then, there exists a state dependent switching strategy such that the switching system x = A σ x is GAS Extensions to the m>2 matrices case: two matrices A i and A j have a Hurwitz combination more than 2 matrices have a Hurwitz combination Main source: Switching in Systems and Control Daniel Liberzon, Birkhauser, 2003.