Page 940 - The Mechatronics Handbook

0066_frame_C31.fm Page 3 Wednesday, January 9, 2002 7:49 PM

Remark: Lyapunov’s theorem, though simple to state, has powerful applications in the stability analysis
of nonlinear systems. However, since the theorem provides only a sufficient condition in terms of the
Lyapunov function, we are often encountered with the difficult problem of finding a suitable Lyapunov
function. In the special case when Eq. (31.1) is a stable linear system,
x ˙ = A m x

T
a quadratic Lyapunov function V = x Px exists where P is a symmetric positive definite matrix satisfying
the so-called Lyapunov equation

A m P + PA m = – Q (31.3)
T
for any symmetric positive definite Q matrix. On the other hand, there is no general recipe for construc-
tion of Lyapunov functions for nonlinear systems. As a rule of thumb, in the case of mechanical systems,
“energy-like”quantities are good candidates for a first attempt.
31.3 Lyapunov Theory for Time-Varying Systems

We are now ready to consider the stability of solutions for a time-varying (nonautonomous) differential
equation

x ˙ = gx,t), g 0,t) = 0 ∀ t ≤ 0 (31.4)
(
(
The function g is assumed to be piecewise continuous with respect to t and locally Lipschitz in x about a
neighborhood of the solution x(t) = 0. This would guarantee that the origin is an equilibrium for Eq. (31.4).
In order to investigate the stability of equilibrium for this nonautonomous system, it is important to
recognize that any solution of Eq.(31.4) depends not only on time t but also on the initial time t 0 . Thus,
we need to revisit our previous definitions of stability.
Definition: Uniform Lyapunov stability
The solution x(t) = 0 for Eq.(31.4) is uniformly stable if for every > 0, there exists a δ( ) > 0 that is
independent of the initial time t 0 such that
() < d x t() < ∀ t ≥ t 0 ≥
x t 0 implies 0
The solution is uniformly asymptotically stable if it is uniformly stable and there is a positive constant ρ
independent of t 0 such that x t() → 0 as t → ∞ for all x t 0 < ρ. The solution is globally uniformly
()
asymptotically stable if it is uniformly asymptotically stable for all initial conditions.
The main stability theorem for nonautonomous systems requires the definition of certain class K
functions.
Definition: Class K functions
A continuous function α: [0, a) → [0, ∞) is said to belong to class K if it is strictly increasing and α(0) = 0.
It is said to belong to class K ∞ , or radially unbounded, if a = ∞ in such a way that α(r) → ∞ as r → ∞.
Theorem: Lyapunov’s stability theorem for time-varying systems
n
Consider a set D = {x ∈R : x ≤ R} about the equilibrium x(t) = 0 for Eq. (31.4). If there exists a scalar
+
n
function V: R × R → R with continuous partial derivatives such that
(
(i) a 1 x( ) ≤ V x,t) ≤ a 2 x ) positive definite and decrescent
(
(ii) V = ∂ V ∂ V T ) ≤ a 3 x )
------ gx,t(
˙
------ +
(
∂t ∂ x
for all t ≥ 0, where a 1 , a 2 , and a 3 are class K functions, then the equilibrium point x = 0 is uniformly
asymptotically stable.
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