With the following method a control system according to
Figure 10.1 using the controller given by
Eq. (10.6) will be designed for a plant described by
Eq. (10.5) such that the closed loop behaves like the
desired transfer function Eq. (10.1). Hereby the orders of the
controller numerator and denominator
polynomials are equal, i.e.
.
The closed-loop poles are the roots of
the characteristic equation, which one obtains from
 |
, the coefficients depend
linearly on the
plant and controller parameters.
Comparing both equations, the first coefficient is
and
for 
and 
for and 
. The coefficients 
are obtained from the
poles. For the first, second last and last one gets
according to
Eqs. (10.19), (10.20) and (10.21) are directly given by the closed-loop poles, the coefficients of
Eq. (10.16), (10.17) and (10.18) contain the required
controller parameters. Comparing
both sides of the latter equation one obtains the synthesis equation, which is a system of linear
equations for 
unknown controller coefficients
. The number of equations is 
. A
unique solution exists if 
.
A detailed analysis shows, however, that a controller obtained in
this way does not usually achieve the desired goals. Because of
its small gain a finite steady-state error may occur. This must be taken into
consideration during the design. For plants with
integral behaviour an order for the controller is sufficient; for
proportional behaviour or when
disturbances at the input of an integral
plant are taken into consideration, the gain must be influenced so
that an integral behaviour of the controller can be obtained. This
happens if the order of the controller
is increased by one, i.e. 
, such that the system of equations
is of lower rank. This gives an additional degree of freedom and
allows one to choose the controller gain
, which is usually introduced as a reciprocal gain
factor:
