Biomedical Engineering Reference
In-Depth Information
C.
All the properties of the multiple equilibria
dynamical systems are analyzed using Lyapunov-
like results. Before introducing them let us present
briefly the basic idea of the Lyapunov stability
tool, together with its strengths and weakness.
The so-called method of the Lyapunov functions
has its background originating from Mechanics:
the energy
E
of an isolated dissipative system de-
creases in time until the system reach the resting
state, where
dE / dt =
0 (the Lagrange-Dirichlet
theorem); in fact, at that point the energy
E
reach
its minimum and the corresponding state is an
equilibrium state for the system.
In 1892, A.M. Lyapunov—a Russian math-
ematician, proposed some generalized state
functions, which are energy-like: they are of the
constant sign and decrease in time along the tra-
jectories of the system of differential equations.
The importance of the Lyapunov method resides
in reducing the number of the equations needed
in order to analyze the behavior of a system: from
a number of equations equal to the number of the
system's state variables to only one function—the
Lyapunov function. A Lyapunov function de-
scribes completely the system behavior; at each
moment a single real number gives information
on the entire system. Moreover, since these func-
tions may have not any connection with physics
and the system's energy, they can be used for a
wide class of systems to determine the stability
of an equilibrium point.
Let us mention here that the method gives
only sufficient conditions for the stability of an
equilibrium what means that on one hand, if one
can not find a Lyapunov function for a system then
one can not say anything about the qualitative be-
havior of its equilibrium (if it is stable or unstable);
on the other hand, it is a quasi-permanent task to
find sharper, less restrictive sufficient conditions
(i.e. closer to the necessary ones) on system's
parameters (in the case of neural networks—on
synaptic weights) by improving the Lyapunov
function. For an easy-to-understand background
on Lyapunov stability see for instance, Chapter III
of the course by Khalil (1992); for applications of
Lyapunov methods in stability the reader is sent
to Chapter V (
op. cit.
) and the topic of Halanay
and Răsvan (1993).
For a linear system, if the equilibrium at the
origin
=
x
is globally asymptotically stable
(hence exponentially stable), then that equilibrium
represents the global minimum for the associated
Lyapunov function. A vertical section on the
Lyapunov surface for a globally asymptotically
stable fixed point may have the allure in Figure 2.
For a nonlinear system with multiple equilibria,
the Lyapunov surface will have multiple local
minima, as in Figure 3.
In general, there are no specific procedures
for constructing Lyapunov functions for a given
system. For a linear
m
-dimensional system, a
quadratic Lyapunov function
0
m
de-
V
:
→
scribed by
)(
, with
P
a symmetric
positive definite matrix—may provide necessary
and sufficient conditions for exponential stabil-
ity. Unfortunately, for nonlinear systems or even
more—for time-delay systems, the sharpest most
general quadratic Lyapunov function(al) is rather
difficult to manipulate. In these cases, an approach
is to associate the Lyapunov function in a natural
way i.e. as an energy of a certain kind, but “guess-
ing” a Lyapunov function(al) which gives sharper
sufficient criteria for a given system “remains an
art and a challenge” (Răsvan, 1998).
For systems with several equilibria in the time
invariant case (and this is exactly the case in the
AI devices, in particular for the case of neural
networks) the following Lyapunov-like results
are basic (Leonov, Reitmann & Smirnova, 1992:
statements in Chapter I, proofs in Chapter IV):
V
x
=
x
T
Px
Lemma 1.
Consider the nonlinear system (1)
and assume existence of a continuous function
m
V
:
D
⊂
→
such that:
i)
for any solution x of (1) V
(
x
(
t
))
is nonincreas-
ing;
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