A First Course in FUZZY and NEURAL CONTROL

2.5. CONTROLLER DESIGN 47

position to choose any desired set of roots that yield satisfactory performance.
Suppose we choose the desired characteristic equation as

(s+2)(s+3)=0

Equating the denominator ofG 1 (s)with the desired characteristic equation
yields
(s+1)(s−2) +Gc(s)=(s+2)(s+3)

Expanding and simplifying yields the controller mathematical form

Gc(s)=(6s+8)
Suppose we choose the feedback path for the controller design. In this case
the resulting closed-loop transfer function would be given as

G 2 (s)=

Gp(s)

1+Gp(s)Gc(s)

Substituting the given plant transfer function and rearranging yields

G 2 (s)=

1

(s+1)(s−2) +Gc(s)

Once again, if we choose the desired characteristic equation as(s+2)(s+3) = 0,
we obtain the same result as before, namely,

Gc(s)=(6s+8)

In either case, the controller is of the form

Gc(s)=KDs+KP

This clearly has meaning in terms of derivative and proportional gain control.
More will be discussed along these lines in later sections. The point we wish to
make here is that given the mathematical model of a plant, it is entirely possible
to obtain a suitable controller design. What we have seen in this example is
the addition of terms to the overall closed-loop transfer function such that some
desired performance criteria can be met. In thefirst case where the controller
is placed in the forward path, the resulting closed-loop transfer function of the
control system is

G 1 (s)=

6 s+8

s^2 +5s+6

where the numerator term 6 s+8represents the addition of a ìzeroî to the
transfer function. Such manipulations of the plant mathematical models to
yield desired plant responses is characterized in classical control techniques as
root locus analysisandthecriterionusedtoadjusttheopen-loopsensitivity
gain is called theRouth-Hurwitz stability criterion. Adjusting the open-
loop sensitivity gain, and placing limits on the range of this gain will assure
stable operation of the system within prescribed limits. We urge the reader
to refer to one of many excellent control systems books in the literature for a
complete treatise on this topic. See [16] for example.