78 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
type controller is used in missile and spacecraft control systems. Other types of
nonlinearities have to be dealt with in the controller design as well.
There is no closed-form solution summarizing the response characteristics of
nonlinear systems. These systems display a variety of behaviors that have to
be studied quite specifically using some iterative solution technique. Typically,
the solution of nonlinear systems requires simulation of the dynamic behavior
for a variety of inputs and initial conditions. The most obvious departure from
linear system behavior is the dependence of the response on the amplitude of
the excitation. This excitation can be either initial conditions or the forcing
input, or both. A common situation is a system that responds to some small
initial conditions by returning to rest in a well-behaved stable manner, while
diverging in an unstable manner for some other initial conditions. This type of
behavior is classified assaddle-point behavior. In some cases, the response to
certain initial conditions may lead to continuous oscillation, the characteristics
of which are a property of the system. Such oscillations give rise to two classes
of systems, namely, systems that exhibitlimit cycle behaviorand systems
that arechaotic. A limit cycle is the trajectory of the system in its state space
where the trajectory closes in on itself and will not result in a steady state. A
chaotic system on the other hand, is a system for which trajectories in the state
space are unique and never follow the same path twice. This phenomenon is
not possible in linear systems.
2.9 Linearization
A study and design of controllers for nonlinear systems can be greatly simplified
by approximating it about a desired operating point by a linear system. The re-
sulting linear system allows one to say a great deal about the performance of the
nonlinear system for small departures around the operating point. Any response
that carries variables through a range that exceeds the limits of reasonable lin-
ear approximation does not reflect the behavior of the nonlinear system. This
requires repeated linearizations about new operating points, and the resulting
solutions being ìpatchedî together. In fact, this technique of ìpatchingî has
significant connotation in the ability of fuzzy logic-based systems being ìuni-
versal approximators.î A discussion of universal approximation is provided in
Chapter 4 on fuzzy control. Consider a nonlinear system represented by the
equation
x ̇(t)=f(x(t),u(t)) (2.41)Linearization can be performed by expanding the nonlinear equations into a
Taylor series about an operating point and neglecting all terms higher than
first-order terms. Let the nominal operating point be denotedx 0 (t),which
corresponds to the nominal inputu 0 (t)and somefixed initial states. Expanding
Equation (2.41) into a Taylor series aboutx(t)=x 0 (t), and neglecting all terms