A First Course in FUZZY and NEURAL CONTROL

20 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

Let us further pursue our problem of controlling an inverted pendulum. The
linearized model in Equation (2.2) could be perturbed by some disturbancee,
say, resulting in
®φ(t)−φ(t)=u(t)+e (2.12)
To see if our PD control law is sufficient to handle this new situation, we
put (2.8) into (2.12), resulting in

®φ(t)+βφ ̇(t)+(α−1)φ(t)=e (2.13)

and examine the behavior of the solutions of (2.13) fortlarge. It can be shown,
unfortunately, that no solutions of (2.13) converge to zero ast→+∞.Sowe
need to modify (2.8) further to arrive at an acceptable control law. Without
going into details here (but see examples in Section 2.7), the additional term to
add to our previous PD control law is of the form

−γ

Zt

0

φ(s)ds

This term is used to offset a nonzero error in the PD control. Thus, our new
control law takes the form

u(t)=−αφ(t)−βφ ̇(t)−γ

Zt

0

φ(s)ds (2.14)

A control law of the form (2.14) is called aproportional-integral-derivative
(PID) control. PID control is very popular in designing controllers for linear sys-
tems. It is important to note that, while PID controls are derived heuristically,
stability analysis requires the existence of mathematical models of the dynamics
of the systems, and stability analysis is crucial for designing controllers.
In our control example, we started out with a nonlinear system. But since
our control objective was local in nature, we were able to linearize the system and
then apply powerful techniques in linear systems. For global control problems,
as well as for highly nonlinear systems, one should look for nonlinear control
methods. In view of the complex behaviors of nonlinear systems, there are no
systematic tools and procedures for designing nonlinear control systems. The
existing design tools are applicable to particular classes of control problems.
However, stability analysis of nonlinear systems can be based on Lyapunovís
stability theory.

2.1.2 Example:invertedpendulumonacart...........

We look at a standard approach for controlling an inverted pendulum, which we
will contrast later with fuzzy control methods. The following mechanical system
is referred to as aninverted pendulum system.In this system, illustrated
in Figure 2.3, a rod is hinged on top of a cart. The cart is free to move in the
horizontal plane, and the objective is to balance the rod in the vertical position.
Without any control actions on the cart, if the rod were initially in the vertical