Closed-Loop Control Learning - Neural Networks: Methodology and Applications - page 293

applications. It was first invented in the framework of single variable sys-
tems with the intensive use of Laplace transform and it was easily extended
to multivariable systems. Nevertheless, those basic methodologies have no
straightforward extensions for nonlinear controlled dynamical systems. We
just mention them here for the sake of completeness.
In the asymptotically stable case, the stability of equilibrium of nonlinear
dynamical systems can be derived from the stability of the linearized system
around that equilibrium. If x ∗ is a fixed point of the dynamical system x ( k +
1) = f [ x ( k )], the linearized dynamical system around x ∗ is the following
dynamical system x ( k +1) = x ∗ ]+ x ∗ . That system is linear

∇ f x ∗ [ x ( k ) −

with respect to the fixed point x ∗ ; f x ∗ is the Jacobian matrix of the partial

derivatives of f in x ∗ . Then the following result holds:

∇

Linearization Theorem. Provided that the linearized system around x ∗ is
asymptotically stable, x ∗ is a stable and asymptotically stable fixed point of
the nonlinear dynamical system.

With the linearization, the transfer function of the linearized system be-
comes a convenient tool for the analysis and synthesis of control laws for
nonlinear systems [Slotine et al. 1991]. Specifically, a linearization theorem of
controlled dynamical system states that if the linearized system is control-
lable, that system is locally stabilized when the closed loop control law of the
linearized system is applied to the original nonlinear system [Sontag 1990].
The Liapunov function method [Slotine et al. 1991], which is a straight-
forward generalization of stability concept of dissipative physical systems, is
a general method of investigation of the stability of equilibrium of nonlinear
dynamical systems.
In spite of the important linearization theorem we have just mentioned,
numerous di culties must be overcome when studying the stability of nonlin-
ear systems:

• The dynamics of a nonlinear system may exhibit several fixed points with

different stabilities: the linearization theorem is a local theorem which
gives no information on the size of attraction basins of asymptotically
stable fixed points;

• Dynamical attractors may exist, which confer a global stability to the

nonlinear system even if there is no stable fixed point: the simpler example
of such attractors is the stable limit cycle. Such an attractor exists in the
Van der Pol oscillator, which was described in the previous chapter.

When noise is considered, the study of stability of systems is completely
changed. In the previous chapter, in the section devoted to the modeling of dy-
namical systems, that the stochastic equivalent of a dynamical deterministic
system is a Markov process, and that the stochastic equivalent of equilibrium
is the invariant measure of this process (see definition in Chap. 4). When a
linear dynamical system is disturbed by gaussian white noise, that probability
describes the asymptotic fluctuations of the state around the zero fixed point