158 CHAPTER 4. FUZZY CONTROL
(d) Compare the results from the PID controller and the fuzzy controller.
- The open-loop transfer function of a plant is given by
GP(s)=
1
s(s+T 1 )(s+T 2 )(s+T 3 )
ForT 1 =1,T 2 =2,andT 3 =3, the open-loop response is unstable, while
a closed-loop unity feedback systemis stable. However, the response of
the system is slow for a unit step input. The objective is to obtain a faster
closed-loop response.
(a) Develop a Simulink model of the system and obtain a satisfactory set
of PID control parameters that will improve the settling time by at
least 75%. Use a less than 5% overshoot criteria.
(b) Develop a fuzzy control model that can match the performance of
the PID control system.
(c) How well do the PID and fuzzy control systems perform if the para-
metersT 1 ,T 2 ,andT 3 vary±10%from their nominal values.
- The ball and beam problem is a nonlinear system for which we wish to
examine the effectiveness of using fuzzy control. Thefigure below illus-
trates the system.
A ball of massMplaced on a beam of lengthLis allowed to roll along the
length of the beam. A lever arm of negligible mass mounted onto a gear
and driven by a servomotor is used to tilt the beam in either direction.
The beam angleαis controlled by a rotational motion of the servo, shown
asθ.Withαinitially zero, the ball is in a stationary position. Whenα
is positive (in relation to the horizontal) the ball moves to the left due to
gravity, and whenαis negative the ball moves to the right. The objective
in this exercise is to develop a fuzzy controller so that the ball position
can be controlled to any positionRalong the beam.
For convenience, we derive the equation of motion for this nonlinear prob-
lem. AssumingJto represent the moment of inertia of the ball,rthe
