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2.7 Lyapunov Stability Theorems 79

EXAMPLE 2.7–10: Krasovkii’s Theorem

Consider the nonlinear system described by

where f(0)=g(0)=0. Let us find the Jacobian matrix

Let P=I, Q= I and check the conditions on the system which would look
like

which should be 0.

On the other hand, suppose we have the linear time-invariant system

The transfer function is then given by

-1
P(s)=C(sI-A) B
Note that P(s) is strictly-proper. The following stability result then holds
[Desoer and Vidyasagar 1975].

THEOREM 2.7–5: Suppose P(s) is a stable transfer function then

1. If u(t)∈L ∞ i.e. u(t) is bounded, then so is y(t) and .
2. If lim t→∞ u(t)=0 then lim t→∞ y(t)=0.

Copyright © 2004 by Marcel Dekker, Inc.