A First Course in FUZZY and NEURAL CONTROL

16 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

form, the mathematical model of the motion of a pendulum, which is derived
from mechanics, is
®θ(t)+sinθ(t)=u(t) (2.1)

whereθ(t)denotes the angle at timet,®θ(t)is the second derivative ofθ(t),k
is a constant, andu(t)is the torque applied at timet. See Figure 2.1. Note

Figure 2.1. Motion of pendulum

that Equation (2.1) is anonlineardifferential equation.
The vertical positionθ=πis anequilibrium pointwhenθ ̇=0andu=0,but
it is unstable. We can make a change of variable to denote this equilibrium point
as zero: Letφ=θ−π, then this equilibrium point is(φ=0,φ ̇=0,u=0).
Suppose we would like to keep the pendulum upright, as shown in Figure
2.2, by manipulating the torqueu(t). The appropriateu(t)that does the job

Figure 2.2. Upright pendulum

is called thecontrol lawof this system. It is clear that in order to achieve our
control objective, we need to answer two questions:

1. How do we derive a control law from Equation (2.1)?


2. If such a control law exists, how do we implement it?