Nonlinear Control Sy.. - Free

100 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
Neglecting the higher-order terms (HOTs) and recalling that, by assumption, f (xe) = 0,
we have that
r
d (x) = (Xe)(x - xe). (3.24)
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Now defining
x = x - xe1 A =f (Xe) (3.25)
we have that X = i, and moreover
Ate. (3.26)
We now whether it is possible to investigate the local stability of the nonlinear system
(3.23) about the equilibrium point xe by analyzing the properties of the linear time-invariant
system (3.26). The following theorem, known as Lyapunov's indirect method, shows that if
the linearized system (3.26) is exponentially stable, then it is indeed the case that for the
original system (3.23) the equilibrium xe is locally exponentially stable. To simplify our
notation, we assume that the equilibrium point is the origin.
Theorem 3.11 Let x = 0 be an equilibrium point for the system (3.23). Assume that f is
continuously differentiable in D, and let A be defined as in (3.25). Then if the eigenvalues
) of the matrix A satisfy ate(.\a) < 0, the origin is an exponentially stable equilibrium point
for the system (3.23).
The proof is omitted since it is a special case of Theorem 4.7 in the next chapter (see Section
4.5 for the proof of the time-varying equivalent of this result).
3.11 Instability
So far we have investigated the problem of stability. All the results seen so far are, however,
sufficient conditions for stability. Thus the usefulness of these results is limited by our
ability to find a function V(.) that satisfies the conditions of one of the stability theorems
seen so far. If our attempt to find this function fails, then no conclusions can be drawn with
respect to the stability properties of the particular equilibrium point under study. In these
circumstances it is useful to study the opposite problem, namely; whether it is possible to
show that the origin is actually unstable. The literature on instability is almost as extensive
as that on stability. Perhaps the most famous and useful result is a theorem due to Chetaev
given next.
Theorem 3.12 (Chetaev) Consider the autonomous dynamical systems (3.1) and assume
that x = 0 is an equilibrium point. Let V : D -+ IR have the following properties: