Liapunov stability

Excerpted from "A Mathematical Introduction to Robotic Manipulation" by R. M. Murray, Z. Li and S. S. Sastry

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 noproofs. 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(t0 ) = x0 x ∈ Rn . (4.31)

We will assume that f (x, t) satisfies the standard conditions for the existence and uniqueness of solutions. Such conditions are, for instance, that f (x, t) is Lipschitzcontinuous with respect to x, uniformly in t, and piecewise continuous in t. A point x∗ ∈ Rn is an equilibrium point of (4.31) if f (x∗ , t) ≡ 0. Intuitively and somewhat crudely speaking, we say an equilibrium 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 tobe 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 stable equilibrium point when the pendulum is hanging straight down and an unstableequilibrium point when it is pointing straight up. If the pendulum is damped, the stable equilibrium point is locally asymptotically stable. By shifting the origin of the system, we may assume that the equilibrium 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

Definition 4.1. Stabilityin the sense of Lyapunov The equilibrium point x∗ = 0 of (4.31) is stable (in the sense of Lyapunov) at t = t0 if for any > 0 there exists a δ(t0 , ) > 0 such that x(t0 ) < δ =⇒ x(t) < , ∀t ≥ t0 . (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 defined at a time instant t0 . 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 Definition 4.1 not be a function of t0 , so that equation (4.32) may hold for all t0 . Asymptotic stability is made precise in the following definition: Definition 4.2. Asymptotic stabilityAn equilibrium point x∗ = 0 of (4.31) is asymptotically stable at t = t0 if 1. x∗ = 0 is stable, and 2. x∗ = 0 is locally attractive; i.e., there exists δ(t0 ) such that x(t0 ) < δ =⇒
t→∞

lim x(t) = 0.

(4.33)

As in the previous definition, asymptotic stability is defined at t0 . Uniform asymptotic stability requires: 1. x∗ = 0 is uniformly stable, and 2. x∗ = 0 is uniformly locallyattractive; i.e., there exists δ independent of t0 for which equation (4.33) holds. Further, it is required that the convergence in equation (4.33) is uniform. Finally, we say that an equilibrium point is unstable if it is not stable. This is less of a tautology than it sounds and the reader should be sure he or she can negate the definition of stability in the sense of Lyapunov to get a definition ofinstability. In robotics, we are almost always interested in uniformly asymptotically stable equilibria. If we wish to move the robot to a point, we would like to actually converge to that point, not merely remain nearby. Figure 4.7 illustrates the difference between stability in the sense of Lyapunov and asymptotic stability. Definitions 4.1 and 4.2 are local definitions; they describe the behavior of...