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Unformatted text preview: Lyapunov stability In mathematics , the notion of Lyapunov stability occurs in the study of dynamical systems . In simple terms, if all solutions of the dynamical system that start out near an equilibrium point x e stay near x e forever, then x e is Lyapunov stable . More strongly, if all solutions that start out near x e converge to x e , then x e is asymptotically stable . The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability , which concerns the behaviour of different but "nearby" solutions to differential equations. Contents 1 Definition for continuous-time systems 2 Definition for iterated systems 3 Lyapunov stability theorems o 3.1 Lyapunov second theorem on stability 4 Stability for linear state space models 5 Stability for systems with inputs 6 Example 7 Barbalat's lemma and stability of time-varying systems 8 References 9 See also Definition for continuous-time systems Consider an autonomous nonlinear dynamical system , where denotes the system state vector, an open set containing the origin, and continuous on . Without loss of generality, we may assume that the origin is an equilibrium. 1. The origin of the above system is said to be Lyapunov stable , if, for every > 0 , there exists a = ( ) > 0 such that, if , then , for every . 2. The origin of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists > 0 such that if , then . 3. The origin of the above system is said to be exponentially stable if it is asymptotically stable and if there exist , , > 0 such that if , then , for . Conceptually, the meanings of the above terms are the following: 1. Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a...
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This note was uploaded on 10/16/2011 for the course MECHATRONI 111 taught by Professor Jung during the Spring '11 term at Hanyang University.
- Spring '11