Synthesis of Standard Regulators

by Means of Normalized Diagrams

M.E. Penati, M. Zanzi

University of Bologna

Viale Risor­gimento 2, 40136 Bologna, Italy

e-mail: gbertoni@deis.unibo.it

 

In this paper we will present some procedures for the analytic synthe­sis of some SISO linear con­trol systems when the order of the system itself doesn't go beyond the third. The synthesis is made by means of standard regulators placed in the feedfor­ward or in the feedback loop of the system.

As is well known, in cases in which, in order to obtain a required behavior of a SISO linear sta­tionary system, it is possible to assign the system poles without hav­ing to vary or add on ze­ros, it is sufficient to use a suitable algebraic feedback of the states. On the contrary, when it is necessary to vary or add on zeros, or when not all states are available and for some rea­sons it is not convenient to utilize an ob­server, the synthesis can be made by means of the so-called regulators (standard regulators or compensating networks); with these systems, it is possible to assign not only some of the required po­les, but also some of the required zeros.

As we also know, standard regulators are algebraic or dynamic systems of the first or se­cond order and their output is proportional to the input, or to the derivative (or to the inte­gral) of the input itself, or to a linear combination of these functions.

 As to the design method of standard regulators, if the feedback system is a system for which it is possible to determine the relations between the system parame­ters and the parameters of the system out­puts to canonical in­puts, we can determine the regu­lator parameters analytically by imposing that the parameters of the overall system lead to (at least approximately) the required behavior.

In particular the rela­tions between the parameters of the above systems and their outputs to canonical inputs are supplied  by some normalized diagrams presented by the authors in Appendix. Moreover, of the relations referred to the design procedures, it is possible to supply normalized diagrams which, therefore, can be utilized whatever the numerical values of the pa­rameters are. Consequently, it is very easy to utilize these design procedures.

    In the following paragraphs we will examine the analytic synthesis procedures of feed­forward and feedback standard regulators. As to the feedforward ones, we will examine pro­portional derivative (PD) regulators and proportional integral (PI) regulators while, as to the feedback regulators, we will examine PD and the so called hybrid regulators which are made up by the parallel of a proportional regulator and of a PD. In order to simplify, we will only examine systems with algebraic sensors and, moreover, the plant transfer function G_s(s) has no zeros, but only real poles.