Next: Zeros of the closed Up: Generalised compensator design method Previous: Generalised compensator design method   Contents

## The basic idea

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

With respect to the polynomials defined in Eqs. (10.5) and (10.6) this gives

 (10.14)

On the other hand it follows from Eq. (10.1) that

 (10.15)

This polynomial has order , the coefficients depend linearly on the plant and controller parameters. Comparing both equations, the first coefficient is

 (10.16)

and the last because of and

 (10.17)

A general representation is given by

 (10.18)

whereby

for     and

for     and

and . The coefficients are obtained from the poles. For the first, second last and last one gets

 (10.19) (10.20) (10.21)

While the coefficients 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:

 (10.22)

Indeed, the order of the closed loop will be increased; it is now double that of the plant order.

Next: Zeros of the closed Up: Generalised compensator design method Previous: Generalised compensator design method   Contents
Christian Schmid 2005-05-09