Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 6: Storage Functions And Stability Analysis 1 This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems. 6.1 Stability of an equilibria In this section we consider ODE models x˙(t) = a(x(t)), (6.1) where a : X ?? R n is a continuous function deflned on an open subset X of R n . Remem- ber that a point ? x 0 ) = 0, i.e. if x(t) · ?x 0 ≤ X is an equilibrium of (6.1) if a(? x 0 is a solution of (6.1). Depending on the behavior of other solutions of (6.1) (they may stay close to x 0 , or converge to ?? x 0 as t ? →, or satisfy some other speciflcations) the equilibrium may be called stable, asymptotically stable, etc. Various types of stability of equilibria can be derived using storage functions. On the other hand, in many cases existence of storage functions with certain properties is impled by stability of equilibria. 6.1.1 Locally stable equilibria Remember that a point ?x 0 ≤ X is called a (locally) stable equilibrium of ODE (6.1) if for every ? > 0 there exists – > 0 such that all maximal solutions x = x(t) of (6.1) with x 0 |? – are deinfed for all t ? 0, and satisfy |x(t) ? ? 0 | < ? for all t ? 0.|x(0) ? ? x The statement below uses the notion of a lower semicontinuity: a function f : Y ?? R, deflned on a subset Y of R n , is called lower semicontinuous if lim inf f(?) ? f(? xx x ? ) ? ? ? ≤ Y. r?0,r>0 x→Y : |? x ? |<r? x?? 1 Version of September 24, 2003 ‰ 2 Theorem 6.1 ?x 0 ≤ X is a locally stable equilibrium of (6.1) if and only if there exist c > 0 and a lower semicontinuous function V : B c (?x 0 ) ?? R, deflned on x 0 ) = {? x 0 ∞ < c}B c (? x : ∞x? ? and continuous at x 0 , such that V (x(t)) is monotonically non-increasing along the solu-? tions of (6.1), and V (? x) ? ? x 0 )/{?x 0 ) < V (? x ≤ B c (? x 0 }. Proof To prove that (ii) implies (i), deflne x) ? V (? x? ?V ? (r) = inf{V (? x 0 ) : |? x 0 | = r for r ≤ (0,c). Since V is assumed lower semicontinuous, the inflmum is actually a min- imum, and hence is strictly positive for all r ≤ (0,c). On the other hand, since V is continuous at ?x 0 , V ? (r) converges to zero as r ? 0. Hence, for a given ? > 0, one can flnd – > 0 such that x) ? ? x? ?V ? (min{?,c/2}) > V (? x : |? x 0 | < –. Hence a solution x = x(t) of (6.1) with an initial condition such that |x(0) ? ?x 0 | < – (and hence V (x(0)) < V ? (min{?,c/2}) cannot cross the sphere |x? ?? x 0 | = min{?,c/2}. To prove that (i) implies (ii), deflne V by V (? x 0 ∞ : t ? 0, x(0) = ?x) = sup{∞x(t) ? ? x, x(￠) satisfles (6.1) }. (6.2) Since, by assumption, solutions starting close enough to ?x 0 never leave a given disc cen- tered at x 0 , V is well deflned in a neigborhood X 0 of x 0 . Then, by its very deflnition, ? V (x(t)) is not increasing for every solution of (6.1) starting in X 0 . Since V is a supremum, it is lower semicontinuous (actually, here we use the fact, not mentioned before, that if 0 ? and x k (t 1 ) ? ?x 1 ? then there exists = x k (t) are solutions of (6.1) such that x k (t 0 ) ? ?xx k 0 ? and x(t 1 ) = x? 1 ? a solution of (6.1) with x(t 0 ) = x? ). Moreover, V is continuous at x 0 , because of stability of the equilibrium x 0 . One can ask whether existence of a Lyapunov function from a better class (say, con- tinuous functions) is possible. The answer, in general, is negative, as demonstrated by the following example. Example 6.1 The equilibrium ? x 0 = 0 of the flrst order ODE Let a : R ?? R be deflned by x 2 )sgn(? x), x ≡exp(?1/? x) sin 2 (? ? = 0, a(?x) = ?0, x = 0. Then a is arbitrary number of times difierentialble and the equilibrium ?x 0 = 0 of (6.1) is locally stable. However, every continuous function V : R ?? R which does not increase along system trajectories will achieve a maximum at ?x 0 = 0. 3 For the case of a linear system, however, local stability of equilibrium ?x 0 = 0 implies existence of a Lyapunov function which is a positive deflnite quadratic form. Theorem 6.2 If a : R n ?? R n is deflned by a(?) = Axx ? where A is a given n- by-n matrix, then equilibrium ?x 0 = 0 of (6.1) is locally stable if and ? only if there exists a matrix Q = Q ? > 0 such that V (x(t)) = x(t) Qx(t) is monotonically non-increasing along the solutions of (6.1). The proof of this theorem, which can be based on considering a Jordan form of A, is usually a part of a standard linear systems class. 6.1.2 Locally asymptotically stable equilibria A point ?x 0 is called a (locally) asymptotically stable equilibrium of (6.1) if it is a stable equilibria, and, in addition, there exists e 0 > 0 such that every solution of (6.1) with x 0 | < ? 0 converges to ?|x(0) ? ? x 0 as t ? →. Theorem 6.3 If V : X ?? R is a continuous function such that V (? x) ? ? xx 0 ) < V (? x ≤ X/{? 0 }, and V (x(t)) is strictly monotonically decreasing for every solution of (6.1) except x(t) · x 0 then ?? x 0 is a locally asymptotically stable equilibrium of (6.1). Proof From Theorem 6.1, ?x 0 is a locally stable equilibrium. It is su–cient to show that every solution x = x(t) of (6.1) starting su–ciently close to ?x 0 will converge to ?x 0 as t ? →. Assume the contrary. Then x(t) is bounded, and hence will have at ? x ? which is not xleast one limit point ? ? 0 . In addition, the limit V of V (x(t)) will exist. Consider a solution x ? = x ? (t) starting from that point. By continuous dependence on ? initial conditions we conclude that V (x ? (t)) = V is constant along this solution, which contradicts the assumptions. A similar theorem deriving existence of a smooth Lyapunov function is also valid. Theorem 6.4 If ? 0 is an asymptotically stable equilibrium of system (6.1) where a : X ??x R n is a continuously difierentiable function deflned on an open subset X of R n then there x x xexists a continuously difierentiable function V : B ? (? 0 ) ?? R such that V (? 0 ) < V (?) for all ? = ?x ≡ x 0 and ∈V (? x) < 0 ? ? x 0 )/{? x)a(? x ≤ B ? (? x 0 }. 4 Proof Deflne V by ? V (x(0)) = ‰(|x(t)| 2 )dt, 0 where ‰ : [0, →) ?? [0, →) is positive for positive arguments and continuously difieren- tiable. If V is correctly deflned and difierentiable, difierentiation of V (x(t)) with respect to t at t = 0 yields ∈V (x(0))a(x(0)) = ?‰(|x(0)| 2 ), which proves the theorem. To make the integral convergent and continuously difieren- tiable, it is su–cient to make ‰(y) converging to zero quickly enough as y ? 0. For the case of a linear system, a classical Lyapunov theorem shows that local stability of equilibrium ?x 0 = 0 implies existence of a strict Lyapunov function which is a positive deflnite quadratic form. Theorem 6.5 If a : R n ?? R n is deflned by a(? xx) = A? where A is a given n- by-n matrix, then equilibrium ?x 0 = 0 of (6.1) is locally asymptotically stable if and only if there exists a matrix Q = Q ? > 0 such that, for V (? x x,x) = ? ? Q? ∈V (? x = ?|?x)A? x| 2 . 6.1.3 Globally asymptotically stable equilibria Here we consider the case when a : R n ?? R n in deflned for all vectors. An equilibrium ?x 0 of (6.1) is called globally asymptotically stable if it is locally stable and every solution of (6.1) converges to ?x 0 as t ? →. ?Theorem 6.6 If function V : R n ?? R has a unique minimum at x 0 , is strictly mono- tonically decreasing along every trajectory of (6.1)except x(t) · ?x 0 , and has bounded level sets then ?x 0 is a globally asymptotically stable equilibrium of (6.1). The proof of the theorem follows the lines of the proof of Theorem 6.4. Note that the assumption that the level sets of V are bounded is critically important: without it, some solutions of (6.1) may converge to inflnity instead of ?x 0 .