Nonlinear Control Sy.. - Free

4.7. CONVERSE THEOREMS
that is
Amin(Q) > 2ry Amax(P)
or, equivalently
min (Q)
ti
(4.35)
2Amax(P)
From equation (4.35) we see that the origin is exponentially stable provided that the value
of ry satisfies the bound (4.35)
4.7 Converse Theorems
All the stability theorems seen so far provide sufficient conditions for stability. Indeed, all
of these theorems read more or less as follows:
if there exists a function (or that satisfies ... , then the equilibrium
point xe satisfies one of the stability definitions.
None of these theorems, however, provides a systematic way of finding the Lyapunov
function. Thus, unless one can "guess" a suitable Lyapunov function, one can never conclude
anything about the stability properties of the equilibrium point.
An important question clearly arises here, namely, suppose that an equilibrium point
satisfies one of the forms of stability. Does this imply the existence of a Lyapunov function
that satisfies the conditions of the corresponding stability theorem? If so, then the search
for the suitable Lyapunov function is not in vain. In all cases, the question above can be
answered affirmatively, and the theorems related to this questions are known as converse
theorems. The main shortcoming of these theorems is that their proof invariably relies on the
construction of a Lyapunov function that is based on knowledge of the state trajectory (and
thus on the solution of the nonlinear differential equation). This fact makes the converse
theorems not very useful in applications since, as discussed earlier, few nonlinear equation
can be solved analytically. In fact, the whole point of the Lyapunov theory is to provide an
answer to the stability analysis without solving the differential equation. Nevertheless, the
theorems have at least conceptual value, and we now state them for completeness.
Theorem 4.9 Consider the dynamical system i = f (x, t). Assume that f satisfies a Lipschitz
continuity condition in D C Rn, and that 0 E D is an equilibrium state. Then we
have
If the equilibrium is uniformly stable, then there exists a function W(-, ) : D x [0, oo) -a
IIt that satisfies the conditions of Theorem 4.1 .
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