Lyapunov stability theory

Lyapunov stability theory - Excerpted from"A Mathematical...

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4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We present a survey of the results that we shall need in the sequel, with no proofs. The interested reader should consult a standard text, such as Vidyasagar [ ? ] or Khalil [ ? ], for details. 4.1 Basic definitions Consider a dynamical system which satisfies ˙ x = f ( x, t ) x ( t 0 )= x 0 x R n . (4.31) We will assume that f ( x, t ) satisfies the standard conditions for the exis- tence and uniqueness of solutions. Such conditions are, for instance, that f ( x, t ) is Lipschitz continuous with respect to x , uniformly in t , and piece- wise continuous in t .Apo in t x R n is an equilibrium point of (4.31) if f ( x ,t ) 0. Intuitively and somewhat crudely speaking, we say an equi- librium point is locally stable if all solutions which start near x (meaning that the initial conditions are in a neighborhood of x ) remain near x for all time. The equilibrium point x is said to be locally asymptotically stable if x is locally stable and, furthermore, all solutions starting near x tend towards x as t →∞ . We say somewhat crude because the time-varying nature of equation (4.31) introduces all kinds of additional subtleties. Nonetheless, it is intuitive that a pendulum has a locally sta- ble equilibrium point when the pendulum is hanging straight down and an unstable equilibrium point when it is pointing straight up. If the pen- dulum is damped, the stable equilibrium point is locally asymptotically stable. By shifting the origin of the system, we may assume that the equi- librium point of interest occurs at x = 0. If multiple equilibrium points exist, we will need to study the stability of each by appropriately shifting the origin. 43
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Defnition 4.1. Stability in the sense oF Lyapunov The equilibrium point x = 0 of (4.31) is stable (in the sense of Lyapunov) at t = t 0 if for any ±> 0thereex istsa δ ( t 0 ) > 0 such that k x ( t 0 ) k = ⇒k x ( t ) k <±, t t 0 . (4.32) Lyapunov stability is a very mild requirement on equilibrium points. In particular, it does not require that trajectories starting close to the origin tend to the origin asymptotically. Also, stability is deFned at a time instant t 0 . Uniform stability is a concept which guarantees that the equilibrium point is not losing stability. We insist that for a uniformly stable equilibrium point x , δ in the DeFnition 4.1 not be a function of t 0 , so that equation (4.32) may hold for all t 0 . Asymptotic stability is made precise in the following deFnition: Defnition 4.2. Asymptotic stability An equilibrium point x = 0 of (4.31) is asymptotically stable at t = t 0 if 1. x = 0 is stable, and 2. x = 0 is locally attractive; i.e., there exists δ ( t 0 ) such that k x ( t 0 ) k = lim t →∞ x ( t )=0 . (4.33) As in the previous deFnition, asymptotic stability is deFned at t 0 .
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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.

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Lyapunov stability theory - Excerpted from"A Mathematical...

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