Basic Principles of Modern Methods for Design of Digital Controllers

The notation -will be specially employed when an interpretation in the frequency domain is needed(in this case ).
The digital PI controller is characterized by the polynomials (see Equations 2.5.6 and 2.5.7):


(2.5.20)
(2.5.21)

The closed loop system transfer function (with respect to the reference r(t)) in the general form is given by Equation 2.5.12.

The characteristic polynomial , whose roots are the desired closed loop system poles, essentially defines the performances. As a general rule, it is chosen as a second-order polynomial corresponding to the discretization of a second-order continuous-time system with a specified natural frequency and damping and ,for example, and can be obtained on the basis of the diagrams given in Figures 1.10 or 1.11) starting from specifications in the time domain. The coefficients corresponding to the polynomial are obtained either by conversion tables mentioned in Table 2.4, or by Scilab and MATLAB® functions given in Section 2.3. In this case, sampling period T_s, natural frequency and damping must be specified.
We recall that the relation between and T_s must be respected (see Section 2.2.2, Equation 2.2.7):
(2.5.22)
For a plant having an equivalent discrete-time transfer operator (function) given by Equation 2.5.19, and the use of a digital PI controller, the closed loop system poles are given by Equation 2.5.13, and they are


(2.5.23)
By rearranging the terms in Equation 2.5.23 in ascending powers, we get
(2.5.24)

For the polynomial Equation 2.5.24 to be verified, it is necessary that the coefficients of the same powers must be equal on both sides. Thus the following system is obtained:

(2.5.25)
which gives for and the results
(2.5.26)

One can see that the parameters of the controller depend upon the performance specifications (the desired closed loop poles) and the plant model parameters.

By using Equation 2.5.7, one can obtain the parameters of the continuous-time
PI controller: