Hybrid Control and Switched Systems Lecture #7 Stability and convergence of ODEs João P. Hespanha University of California at Santa Barbara Summary Lyapunov stability of ODEs epsilon-delta and beta-function definitions Lyapunov s stability theorem LaSalle s invariance principle Stability of linear systems 1
Properties of hybrid systems sig set of all piecewise continuous signals x:[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, x) sig sig satisfies p if p(q, x) = true A hybrid automaton H satisfies p ( write H ² p ) 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 Lyapunov stability (ODEs) equilibrium point x eq R n for which f(x eq ) = 0 E.g., pendulum equation thus x(t) = x eq t 0 is a solution to the ODE l two equilibrium points: x 1 = 0, x 2 = 0 (down) and x 1 = π, x 2 = 0 (up) θ m 2
Lyapunov stability (ODEs) equilibrium point x eq R n for which f(x eq ) = 0 thus x(t) = x eq t 0 is a solution to the ODE Definition (e δ definition): The equilibrium point x eq R n is (Lyapunov) stable if e > 0 δ >0 : x(t 0 ) x eq δ x(t) x eq e t t 0 0 e x eq δ x(t) 1. if the solution starts close to x eq it will remain close to it forever 2. e can be made arbitrarily small by choosing δ sufficiently small Example #1: Pendulum l θ m x eq =(0,0) stable x eq =(π,0) unstable pend.m 3
Lyapunov stability continuity definition sig set of all piecewise continuous signals taking values in R n Given a signal x sig, x sig ú sup t 0 x(t) signal norm ODE can be seen as an operator T : R n sig that maps x 0 R n into the solution that starts at x(0) = x 0 Definition (continuity definition): The equilibrium point x eq R n is (Lyapunov) stable if T is continuous at x eq : e > 0 δ >0 : x 0 x eq δ T(x 0 ) T(x eq ) sig e x eq e δ x(t) sup t 0 x(t) x eq e can be extended to nonequilibrium solutions Stability of arbitrary solutions sig set of all piecewise continuous signals taking values in R n Given a signal x sig, x sig ú sup t 0 x(t) signal norm ODE can be seen as an operator T : R n sig that maps x 0 R n into the solution that starts at x(0) = x 0 Definition (continuity definition): A solution x * :[0,T) R n is (Lyapunov) stable if T is continuous at x * 0 ú x* (0), i.e., e > 0 δ >0 : x 0 x * 0 δ T(x 0 ) T(x* 0 ) sig e δ sup t 0 x(t) x * (t) e x(t) x * (t) e pend.m 4
Example #2: Van der Pol oscillator x * Lyapunov stable vdp.m E.g., Van der Pol oscillator Stability of arbitrary solutions x * unstable vdp.m 5
Lyapunov stability equilibrium point x eq R n for which f(x eq ) = 0 class set of functions α:[0, ) [0, ) that are 1. continuous 2. strictly increasing 3. α(0)=0 Definition (class function definition): The equilibrium point x eq R n is (Lyapunov) stable if α : x(t) x eq α( x(t 0 ) x eq ) t t 0 0, x(t 0 ) x eq c α(s) s α( x(t 0 ) x eq ) x(t 0 ) x eq x eq x(t) t the function α can be constructed directly from the δ(e) in the e δ (or continuity) definitions Asymptotic stability equilibrium point x eq R n for which f(x eq ) = 0 class set of functions α:[0, ) [0, ) that are 1. continuous 2. strictly increasing 3. α(0)=0 α(s) s Definition: The equilibrium point x eq R n is (globally) asymptotically stable if it is Lyapunov stable and for every initial state the solution exists on [0, ) and x(t) x eq as t. α( x(t 0 ) x eq ) x(t 0 ) x eq x eq x(t) t 6
Asymptotic stability β(s,t) equilibrium point x eq R n for which f(x eq ) = 0 class set of functions β:[0, ) [0, ) [0, ) s.t. 1. for each fixed t, β(,t) 2. for each fixed s, β(s, ) is monotone decreasing and β(s,t) 0 as t (for each fixed t) β(s,t) (for each fixed s) Definition (class function definition): t The equilibrium point x eq R n is (globally) asymptotically stable if β : x(t) x eq β( x(t 0 ) x eq,t t 0 ) t t 0 0 s β( x(t 0 ) x eq,0) x(t 0 ) x eq x eq x(t) β( x(t 0 ) x eq,t) t We have exponential stability when β(s,t) = c e -λ t s with c,λ > 0 linear in s and negative exponential in t Example #1: Pendulum k > 0 (with friction) k = 0 (no friction) x 2 x 1 x eq =(0,0) asymptotically stable x eq =(π,0) unstable x eq =(0,0) stable but not asymptotically x eq =(π,0) unstable pend.m 7
Example #3: Butterfly Convergence by itself does not imply stability, e.g., Why was Mr. Lyapunov so picky? Why shouldn t boundedness and convergence to zero suffice? equilibrium point (0,0) all solutions converge to zero but x eq = (0,0) system is not stable converge.m Lyapunov s stability theorem Definition (class function definition): The equilibrium point x eq R n is (Lyapunov) stable if α : x(t) x eq α( x(t 0 ) x eq ) t t 0 0, x(t 0 ) x eq c Suppose we could show that x(t) x eq always decreases along solutions to the ODE. Then x(t) x eq x(t 0 ) x eq t t 0 0 we could pick α(s) = s Lyapunov stability We can draw the same conclusion by using other measures of how far the solution is from x eq : V: R n R positive definite V(x) 0 x R n with = 0 only for x = 0 V: R n R radially unbounded x V(x) provides a measure of how far x is from x eq (not necessarily a metric may not satisfy triangular inequality) 8
Lyapunov s stability theorem V: R n R positive definite V(x) 0 x R n with = 0 only for x = 0 Q: How to check if V(x(t) x eq ) decreases along solutions? provides a measure of how far x is from x eq (not necessarily a metric may not satisfy triangular inequality) A: V(x(t) x eq ) will decrease if gradient of V can be computed without actually computing x(t) (i.e., solving the ODE) Lyapunov s stability theorem Definition (class function definition): The equilibrium point x eq R n is (Lyapunov) stable if α : x(t) x eq α( x(t 0 ) x eq ) t t 0 0, x(t 0 ) x eq c Lyapunov function Theorem (Lyapunov): Suppose there exists a continuously differentiable, positive definite function V: R n R such that Then x eq is a Lyapunov stable equilibrium. (cup-like function) V(z x eq ) Why? V non increasing V(x(t) x eq ) V(x(t 0 ) x eq ) t t 0 Thus, by making x(t 0 ) x eq small we can make V(x(t) x eq ) arbitrarily small t t 0 So, by making x(t 0 ) x eq small we can make x(t) x eq arbitrarily small t t 0 (we can actually compute α from V explicitly and take c = + ). z 9
Example #1: Pendulum l θ m For x eq = (0,0) positive definite because V(x) = 0 only for x 1 = 2kπ k Z & x 2 = 0 (all these points are really the same because x 1 is an angle) Therefore x eq =(0,0) is Lyapunov stable pend.m Example #1: Pendulum l θ m For x eq = (π,0) positive definite because V(x) = 0 only for x 1 = 2kπ k Z & x 2 = 0 (all these points are really the same because x 1 is an angle) Cannot conclude that x eq =(π,0) is Lyapunov stable (in fact it is not!) pend.m 10
Lyapunov s stability theorem Definition (class function definition): The equilibrium point x eq R n is (Lyapunov) stable if α : x(t) x eq α( x(t 0 ) x eq ) t t 0 0, x(t 0 ) x eq c Theorem (Lyapunov): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n R such that Then x eq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, if = 0 only for z = x eq then x eq is a (globally) asymptotically stable equilibrium. Why? V can only stop decreasing when x(t) reaches x eq but V must stop decreasing because it cannot become negative Thus, x(t) must converge to x eq Lyapunov s stability theorem Definition (class function definition): The equilibrium point x eq R n is (Lyapunov) stable if α : x(t) x eq α( x(t 0 ) x eq ) t t 0 0, x(t 0 ) x eq c Theorem (Lyapunov): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n R such that Then x eq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, if = 0 only for z = x eq then x eq is a (globally) asymptotically stable equilibrium. What if for other z then x eq? Can we still claim some form of convergence? 11
Example #1: Pendulum l θ m For x eq = (0,0) not strict for (x 1 0, x 2 =0!) pend.m LaSalle s Invariance Principle M R n is an invariant set x(t 0 ) M x(t) M t t 0 (in the context of hybrid systems: Reach(M) M ) Theorem (LaSalle Invariance Principle): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n R such that Then x eq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, x(t) converges to the largest invariant set M contained in E ú { z R n : W(z) = 0 } Note that: 1. When W(z) = 0 only for z = x eq then E = {x eq }. Since M E, M = {x eq } and therefore x(t) x eq asympt. stability 2. Even when E is larger then {x eq } we often have M = {x eq } and can conclude asymptotic stability. Lyapunov theorem 12
Example #1: Pendulum l θ m For x eq = (0,0) Inside E, the ODE becomes E ú { (x 1,x 2 ): x 1 R, x 2 =0} define set M for which system remains inside E Therefore x converges to M ú { (x 1,x 2 ): x 1 = k π Z, x 2 =0} However, the equilibrium point x eq =(0,0) is not (globally) asymptotically stable because if the system starts, e.g., at (π,0) it remains there forever. pend.m Linear systems Solution to a linear ODE: Theorem: The origin x eq = 0 is an equilibrium point. It is 1. Lyapunov stable if and only if all eigenvalues of A have negative or zero real parts and for each eigenvalue with zero real part there is an independent eigenvector. 2. Asymptotically stable if and only if all eigenvalues of A have negative real parts. In this case the origin is actually exponentially stable 13
Linear systems linear.m Lyapunov equation Solution to a linear ODE: Theorem: The origin x eq = 0 is an equilibrium point. It is asymptotically stable if and only if for every positive symmetric definite matrix Q the equation A P + P A = Q Lyapunov equation has a unique solutions P that is symmetric and positive definite Recall: given a symmetric matrix P P is positive definite all eigenvalues are positive P positive definite x P x > 0 x 0 P is positive semi-definite all eigenvalues are positive or zero P positive semi-definite x P x 0 x 14
Lyapunov equation Solution to a linear ODE: Theorem: The origin x eq = 0 is an equilibrium point. It is asymptotically stable if and only if for every positive symmetric definite matrix Q the equation A P + P A = Q Lyapunov equation has a unique solutions P that is symmetric and positive definite Why? 1. asympt. stable P exists and is unique (constructive proof) A is asympt. stable e At decreases to zero exponentiall fast P is well defined (limit exists and is finite) change of integration variable τ = T s Lyapunov equation Solution to a linear ODE: Theorem: The origin x eq = 0 is an equilibrium point. It is asymptotically stable if and only if for every positive symmetric definite matrix Q the equation A P + P A = Q Lyapunov equation has a unique solutions P that is symmetric and positive definite Why? 2. P exists asymp. stable Consider the quadratic Lyapunov equation: V(x) = x P x V is positive definite & radially unbounded because P is positive definite V is continuously differentiable: thus system is asymptotically stable by Lyapunov Theorem 15
Next lecture Lyapunov stability of hybrid systems 16