Hybrid Systems - Lecture n. 3 Lyapunov stability

Transcription

1 OUTLINE Focus: stability of equilibrium point Hybrid Systems - Lecture n. 3 Lyapunov stability Maria Prandini DEI - Politecnico di Milano prandini@elet.polimi.it continuous systems decribed by ordinary differential equations (brief review) hybrid automata OUTLINE Focus: stability of 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)

2 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 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 [local existence non-blocking] If f: < n < n is continuous, then x 0 there exists at least a solution with x(0)=x 0 defined on some [0,δ). Theorem [local existence and uniqueness non-blocking, deterministic] If f: < n < n is Lipschitz continuous, then x 0 there exists a single solution with x(0)=x 0 defined on some [0,δ). Theorem [global existence and uniqueness non-blocking, deterministic, non-zeno] If f: < n < n is globally Lipschitz continuous, then x 0 there exists a single solution with x(0)=x 0 defined on [0, ). STABILITY OF CONTINUOUS SYSTEMS Definition (stable equilibrium): with f: < n < n globally Lipschitz continuous Definition (equilibrium): x e < n for which f(x e )=0 Remark: {x e } is an invariant set Definition (stable equilibrium): The equilibrium point x e < n is stable (in the sense of Lyapunov) if Graphically: ε δ xe perturbed motion equilibrium motion execution starting from x(0)=x 0 small perturbations lead to small changes in behavior

3 Definition (stable equilibrium): Definition (asymptotically stable equilibrium): Graphically: and δ can be chosen so that Graphically: perturbed motion ε δ xe phase plot ε δ xe equilibrium motion small perturbations lead to small changes in behavior small perturbations lead to small changes in behavior and are re-absorbed, in the long run Definition (asymptotically stable equilibrium): EXAMPLE: PENDULUM and δ can be chosen so that Graphically: l ε δ xe friction coefficient (α) m small perturbations lead to small changes in behavior and are re-absorbed, in the long run

4 m EXAMPLE: PENDULUM EXAMPLE: PENDULUM unstable equilibrium m as. stable equilibrium EXAMPLE: PENDULUM m EXAMPLE: PENDULUM l m m as. stable equilibrium small perturbations are absorbed, not all perturbations

5 STABILITY OF CONTINUOUS SYSTEMS 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 More definitions: exponentially stable, globally exponentially stable,... with f: < n < n globally Lipschitz continuous Definition (equilibrium): x e < 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 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 execution x(t) behavior of V along the execution x(t): V(t): = V(x(t)) V(x) = 2 x(t) candidate function V(x) Advantage with respect to exhaustive check of all executions?

6 STABILITY OF CONTINUOUS SYSTEMS with f: < n < n globally Lipschitz continuous V: < n < continuously differentiable (C 1 ) function LYAPUNOV STABILITY Theorem (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for the system and D < n an open set containing x e = 0. If V: D < is a C 1 function such that Rate of change of V along the execution of the ODE system: gradient vector Then, x e is stable. V positive definite on D V non increasing along system executions in D (negative semidefinite) 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 D < n an open set containing x e = 0. If V: D < is a C 1 function such that Proof: Then, x e is stable. Finding Lyapunov functions is HARD in general Sufficient condition Lyapunov function for the system and the equilibrium x e Given ε >0, choose r (0,ε) such that B r = {x < n : x r} D Set α : = min{v(x): x = r} > 0 and choose c (0,α). Then, Ω c := {x: V(x) c} B r Since then V(x(t)) V(x(0)), t 0. Hence, all executions starting in Ω c stays in Ω c. V(x) is continuous and V(0) = 0. Then, there is δ >0 such that B δ = {x < n : x δ} Ω c. Therefore, x(0) < δ x(t) < ε, t 0

7 EXAMPLE: PENDULUM LYAPUNOV STABILITY l Theorem (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for the system and D < n an open set containing x e = 0. If V: D < is a C 1 function such that friction coefficient (α) m energy function Then, x e is stable. If it holds also that x e stable Then, x e is asymptotically stable (AS) LYAPUNOV GAS THEOREM Theorem (Barbashin-Krasovski Theorem): Let x e = 0 be an equilibrium for the system. STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is 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) the elements of matrix e At are linear combinations of e λ i (A)t, i=1,2,,n V radially unbounded Then, x e is globally asymptotically stable (GAS).

8 STABILITY OF LINEAR CONTINUOUS SYSTEMS STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system 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 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 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 Lyapunov equation has a unique solution P=P T >0. 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 Lyapunov equation has a unique solution P=P T >0. 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

9 STABILITY OF LINEAR CONTINUOUS SYSTEMS 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 Lyapunov equation has a unique solution P=P T >0. Proof. (if) V(x) =x T P x is a Lyapunov function 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 Lyapunov equation has a unique solution P=P T >0. Proof. (only if) Consider STABILITY OF LINEAR CONTINUOUS SYSTEMS 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 Lyapunov equation has a unique solution P=P T >0. Proof. (only if) Consider P = P T and P>0 easy to show P unique by contradiction 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 Lyapunov equation has a unique solution P=P T >0. Remarks: for a linear system existence of a (quadratic) Lyapunov function V(x) =x T P x is a necessary and sufficient condition it is easy to compute a Lyapunov function since the Lyapunov equation is a linear algebraic equation

10 STABILITY OF LINEAR CONTINUOUS SYSTEMS STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential convergence): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: Theorem (exponential convergence): 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 < n, where -λ 0 (max i Re{λ i (A)}, 0) and µ>0 is an appropriate constant. for all x(0) = x 0 < n, where -λ 0 (max i Re{λ i (A)}, 0) and µ >0 is an appropriate constant. Im eigenvalues of A o o o o Re STABILITY OF LINEAR CONTINUOUS SYSTEMS STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential convergence): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: Proof (exponential convergence): A + λ 0 I is Hurwitz (eigenvalues are equal to λ(a) + λ 0 ) Then, there exists P = P T >0 such that (A + λ 0 I) T P + P (A + λ 0 I) <0 which leads to x(t) T [A T P + P A]x(t) < - 2 λ 0 x(t) T P x(t) Define V(x) = x T P x, then from which

11 STABILITY OF LINEAR CONTINUOUS SYSTEMS STABILITY OF LINEAR CONTINUOUS SYSTEMS (cont d) Proof (exponential convergence): (cont d) Proof (exponential convergence): thus finally leading to thus finally leading to OUTLINE Focus: stability of equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata HYBRID AUTOMATA: FORMAL DEFINITION A hybrid automaton H is a collection 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 Q X is a set of initial states Dom: Q 2 X assigns to each q Q a domain Dom(q) of X E 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

12 HYBRID TIME SET A hybrid time set is a finite or infinite sequence of intervals τ = {I i, i=0,1,, M } such that I i = [τ i, τ i ] for i < M I M = [τ M, τ M ] or I M = [τ M, τ M ) if M< q = q 1 τ i = τ i+1 τ i τ i q = q 2 [ ] τ 0 I 0 τ 0 I 3 [ ] τ 3 τ3 I τ = τ τ 1 [ I 1 ] τ 1 HYBRID TIME SET A hybrid time set is a finite or infinite sequence of intervals τ = {I i, i=0,1,, M } such that I i = [τ i, τ i ] for i < M I M = [τ M, τ M ] or I M = [τ M, τ M ) if M< τ i = τ i+1 τ i τ i I τ = τ τ 1 [ I 1 ] t 2 t τ 1 1 [ ] τ 0 I 0 τ 0 t 1 t 2 t 3 t 4 t 4 I the elements of τ are τ 3 3 linearly ordered [ ] τ3 t 3 τ := i (τ i -τ i ) (continuous extent) HYBRID TRAJECTORY A hybrid trajectory (τ, q, x) consists of: A hybrid time set τ = {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

13 HYBRID AUTOMATA: EXECUTION A hybrid trajectory (τ, q, x) is an execution (solution) of the hybrid automaton if it satisfies the following conditions: Initial condition: (q 0 (τ 0 ), x 0 (τ 0 )) Init Continuous evolution: for all i such that τ i < τ i q i : I i Q is constant x i :I i X is the solution to the ODE associated with q i (τ i ) x i (t) Dom(q i (τ i )), t [τ i,τ i ) Discrete evolution: (q i (τ i ),q i+1 (τ i+1 )) E transition is feasible x i (τ i ) G((q i (τ i ),q i+1 (τ i+1 ))) guard condition satisfied x i (τ i+1 ) R((q i (τ i ),q i+1 (τ i+1 )),x i (τ i )) reset condition satisfied HYBRID AUTOMATA: EXECUTION Well-posedness? 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) the set of infinite executions of H starting from (q,x) STABILITY OF HYBRID AUTOMATA Definition (equilibrium): x e =0 X is an equilibrium point of H if: f(q,0) = 0 for all q Q ((q,q ) E) (0 G((q,q )) R((q,q ),0) = {0} Remarks: discrete transitions are allowed out of (q,0) but only to (q,0) for all (q,0) Init and (τ, q, x) is an execution of H starting from (q,0), then x(t) = 0 for all t τ EXAMPLE: SWITCHING LINEAR SYSTEM 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 {x X: x >0} Dom(q 1 ) = {x X: x 1 x 2 0} Dom(q 2 ) = {x X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x X: x 1 x 2 0} G((q 2,q 1 )) = {x 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}

14 STABILITY OF HYBRID AUTOMATA Definition (stable equilibrium): Let x e = 0 X be an equilibrium point of H. x e = 0 is stable if STABILITY OF HYBRID AUTOMATA Definition (stable equilibrium): Let x e = 0 X be an equilibrium point of H. x e = 0 is stable if set of (maximal) executions starting from (q 0, x 0 ) Init Remark: Stability does not imply convergence To analyse convergence, only infinite executions should be considered set of (maximal) executions starting from (q 0, x 0 ) Init Definition (asymptotically stable equilibrium): Let x e = 0 X be an equilibrium point of H. x e = 0 is asymptotically stable if δ>0 that can be chosen so that set of infinite executions starting from (q 0, x 0 ) Init τ < if Zeno STABILITY OF HYBRID AUTOMATA Definition (stable equilibrium): Let x e = 0 X be an equilibrium point of H. x e = 0 is stable if EXAMPLE: SWITCHING LINEAR SYSTEM 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 {x X: x >0} Dom(q 1 ) = {x X: x 1 x 2 0} Dom(q 2 ) = {x X: x 1 x 2 0} Question: x e = 0 stable equilibrium for each continuous system dx/dt = f(q,x) implies that x e = 0 stable equilibrium for H? E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x X: x 1 x 2 0} G((q 2,q 1 )) = {x 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}

15 EXAMPLE: SWITCHING LINEAR SYSTEM EXAMPLE: SWITCHING LINEAR SYSTEM q 1 : quadrants 2 and 4 q 2 : quadrants 1 and 3 Asymptotically stable linear systems. Asymptotically stable linear systems, but x e = 0 unstable equilibrium of H EXAMPLE: SWITCHING LINEAR SYSTEM EXAMPLE: SWITCHING LINEAR SYSTEM q 1 : quadrants 1 and 3 q 2 : quadrants 2 and 4 x(τ i ) < x(τ i+1 ) x(τ i ) > x(τ i+1 ) overshoots sum up

16 LYAPUNOV STABILITY Theorem (Lyapunov stability): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, (q,q ) E, and D 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 Q: V(q,x) Lyapunov function for continuous system q x e =0 is stable equilibrium for system q LYAPUNOV STABILITY Theorem (Lyapunov stability): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, (q,q ) E, and D 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 Q: V(q,x) Lyapunov function for continuous system q x e =0 is stable equilibrium for system q If for all (τ, q, x) H (q0,x 0 ) with (q 0,x 0 ) Init (Q D), and all q Q, the sequence {V(q(τ i ),x(τ i )): q(τ i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H. If for all (τ, q, x) H (q0,x 0 ) with (q 0,x 0 ) Init (Q D), and all q Q, the sequence {V(q(τ i ),x(τ i )): q(τ i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H. Sketch of the proof. LYAPUNOV STABILITY Sketch of the proof. LYAPUNOV STABILITY 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(t),x(t)) V(q 1,x(t)) {V(q 1,x(τ i ))} non-increasing V(q 2,x(t)) q(t)= q 1 q(t)= q 1 [ ][ ][ ][ τ 0 τ 0 =τ 1 τ 1 =τ 2 τ 2 =τ 3 q(t)= q 1 q(t)= q 1 [ ][ ][ ][ τ 0 τ 0 =τ 1 τ 1 =τ 2 τ 2 =τ 3

17 LYAPUNOV STABILITY LYAPUNOV STABILITY Sketch of the proof. V(q(t),x(t)) V(q 1,x(t)) V(q(t),x(t)) Lyapunov-like function q(t)= q 1 q(t)= q 1 [ ][ ][ ][ τ 0 τ 0 =τ 1 τ 1 =τ 2 τ 2 =τ 3 EXAMPLE: SWITCHING LINEAR SYSTEM EXAMPLE: SWITCHING LINEAR SYSTEM Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: q 1 q 2 Init = Q {x X: x >0} Dom(q 1 ) = {x X: Cx 0} Dom(q 2 ) = {x X: Cx 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x X: Cx 0} G((q 2,q 1 )) = {x X: Cx 0}, C T < 2 R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x}

18 EXAMPLE: SWITCHING LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T <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 it >0 solution to A it 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: SWITCHING LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T <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 it >0 solution to A it P i + P i A i = - I EXAMPLE: SWITCHING LINEAR SYSTEM Test for non-increasing sequence condition The level sets of V(q i,x) = x T P i x are ellipses centered at the origin. A level set intersects the switching line C T x =0 at z and -z. τ i z τ i+2 -z τ i =τ i+1 C T x= 0 EXAMPLE: SWITCHING LINEAR SYSTEM Test for non-increasing sequence condition The level sets of V(q i,x) = x T P i x are ellipses centered at the origin. A level set intersects the switching line C T x =0 at z and -z. Let q(τ i )=q 1 and x(τ i )=z. Since V(q 1,x(t)) is not increasing during [τ i,τ i ], then, when x(t) intersects the switching line at τ i, it does at α z with α (0,1], hence x(τ i+1 ) = x(τ i ) x(τ i ). Let q(τ i+1 )=q 2 Since V(q 2,x(t)) is decreasing during [τ i+1,τ i+1 ], then, when x(t) intersects the switching line at τ i+1, x(τ i+2 ) = x(τ i+1 ) x(τ i+1 ) x(τ i ) From this, it follows that V(q 1,x(τ i+2 )) V(q 1,x(τ i ))

19 Drawbacks: LYAPUNOV STABILITY The sequence {V(q(τ i ),x(τ i )): q(τ i ) =q } must be evaluated, which may require solving the ODEs LYAPUNOV STABILITY Corollary (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, (q,q ) E, and D X=< n an open set containing x e = 0. If V: D < is a C 1 function such that for all q Q: In general, it is hard to find a Lyapunov-like function 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), q Q and apply the 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 Q: f(q k,x) = A k x (linear vector fields) Init q Q {q } Dom(q) (admissible initialization) for all x X, the set Jump(q k,x):= {(q,x ): (q k,q ) E, x G((q k,q )), x R((q,q ),x)} has cardinality 1 if x Dom(q), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) Jump(q k,x) x 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

20 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 kt P+ PA k < 0, k Then, x e = 0 is asymptotically stable. GLOBALLY QUADRATIC LYAPUNOV FUNCTION Alternative proof (showing exponential convergence): There exists γ >0 such that A kt P+ PA k +γ I 0, k There exists a unique, infinite, non-zeno execution (τ,q,x) for every initial state with x: τ < n satisfying where µ k : τ [0,1] is such that k µ k (t)=1, t [τ i,τ i ]. Proof. V(x) = x T Px is a common Lyapunov function Let V(x) = x T Px. Then, for t [τ i,τ i ). GLOBALLY QUADRATIC LYAPUNOV FUNCTION Sketch of the proof. (cont d) Since λ min (P) x 2 V(x) λ max (P) x 2, then and, hence, GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 which leads to Then, Since τ = (non-zeno), then x(t) goes to zero exponentially as t τ conditions of the theorem satisfied with P = I

21 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 kt P+ PA k < 0, k Then, x e = 0 is asymptotically stable. GLOBALLY QUADRATIC LYAPUNOV FUNCTION A globally quadratic Lyapunov function does not exist if and only if there exist positive definite symmetric matrices R k such that Remark: A set of LMIs to solve. This problem can be reformulated as a convex optimization problem. Efficient solvers exist. GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 PIECWISE QUADRATIC LYAPUNOV FUNCTION Developed for piecewise linear systems with structured domain (set of convex polyhedra) and reset Resulting Lyapunov function is continuous at the switching times stable node stable focus LMIs characterization conditions of the theorem NOT satisfied for any P but x e = 0 stable equilibrium piecewise quadratic Lyapunov function

22 REFERENCES H.K. Khalil. Nonlinear Systems. Prentice Hall, S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, M. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. on Automatic Control, 43(4): , H. Ye, A. Michel, and L. Hou. Stability theory for hybrid dynamical systems. IEEE Transactions on Automatic Control, 43(4): , M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov function for hybrid systems. IEEE Transactions on Automatic Control, 43(4): , 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): , 2000.