# 10

# Controller Realization

A control algorithm which takes the form of a *z*-transform polynomial must be realized in the computer in the form of a program containing unit delays, constant multipliers, and adders.

A given controller transfer function can be realized in many different ways. Mathematically the alternative realizations are all equivalent, differing only in the way they are implemented. Different realizations have different computational efficiencies, different sensitivities to parameter errors, and different programming efforts are needed in each case. Only some of the important realizations, such as the direct structure, cascaded structure and parallel structure, as well as the second-order structures, are described in this chapter.

## 10.1 DIRECT STRUCTURE

The transfer function *D*(*z*) of a digital controller can be represented in general by a ratio of two polynomials

In direct structure the coefficients *a*_{j} and *b*_{j} appear as multipliers. There are several forms of direct structure, and we shall look at two of the most popular ones: the direct canonical structure and the direct noncanonical structure.

### 10.1.1 Direct Canonical Structure

Remembering that *b*_{0} = 1, we can rewrite (10.1) as

Let us now introduce a variable *R*(*z*) such that

or

and

Assume that the transfer function of ...