A First Course in FUZZY and NEURAL CONTROL

46 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

Figure 2.16. Controller in forward or feedback path

controller either in the forward path or in the feedback path as illustrated in
Figure 2.16. Once the structure of the control system is decided upon, suitable
parameters can then be chosen for the controller model to obtain the desired
plant response characteristics.
Consider for example, the closed-loop transfer function of an unstable plant
given by the control ratio

Gp(s)=

C(s)

R(s)

=

1

(s+1)(s−2)

whereC(s)is the plant output andR(s)is the plant input. Upon examining the
roots, it is clear that the system is indeedunstableñthatis,therootsatisfying
s−2=0lies in the right-halfs-plane. Suppose we wish to develop a controller
such that the system becomes stable and has specific response characteristics.
LetGc(s)andGp(s)represent the controller and plant transfer functions, re-
spectively. If the controller is in the forward path, then the closed-loop transfer
function would be given as

G 1 (s)=

Gc(s)Gp(s)

1+Gc(s)Gp(s)

Substituting the given plant transfer function and rearranging yields

G 1 (s)=

Gc(s)

(s+1)(s−2) +Gc(s)

The denominator of this new control ratio represents the characteristic equation
of the system that includes both the controller and the plant. We are now in a