2.5. CONTROLLER DESIGN 45
Figure 2.15. Closed-loop feedback system- The error between the input valueE(s)and the outputC(s)of the system
amplified or reduced by the feedback transfer functionH(s)is
E(s)=R(s)−H(s)C(s)- Substituting the errorE(s)into the equation for output of the plant, we
get
C(s)=G(s)[R(s)−H(s)C(s)] - Collecting terms and rearranging, we get
C(s)
R(s)=
G(s)
1+G(s)H(s)that we can write as
G(s)
1+G(s)H(s)=
N(s)
D(s)
whereN(s)andD(s)are polynomials in the variables.FactoringN(s)displays all the zeros of the plant transfer function; factoring
D(s)displays all the poles of the plant transfer function. The roots ofD(s)are
called thecharacteristic rootsof the system;D(s)=0is thecharacteristic
equation.
A knowledge of the poles and zeros of the plant transfer function provides
the means to examine the transient behavior of the system, and to obtain limits
on the loop gain to guarantee stability of the system. If a system is inherently
unstable, such as the case of an inverted pendulum, the plant transfer function
provides the designer with information on how to compensate the pole-zero
behavior of the plant with that of a controller. In this case, the controller poles
and zeros are selected in combination with the plant poles and zeros to produce
a transient response that meets the necessary design requirements.
Knowledge of the plant mathematical model clearly provides the means to
obtain the mathematical form of the controller. We can choose to place the