A note on the parameterization of stabilizing controllers for SISO systems

The technique proposed in this paper gives a simple and useful procedure for solving the Bezout identity in the domain of stable rational transfer functions for finding the parameterization of all stabilizing controllers for single-input single-output control systems. This procedure leads to a solution of linear simultaneous equations involving the parameters of the solution-functions of the Bezout identity. One important consequence of this procedure is that it leads to a relatively lower-order for the controller and the compensated closed loop transfer functions. The procedure is illustrated by desi ning the longitudinal mode control systems for the aircra t YF-16 satisfying its flying qualities. Rf Introduction With the pioneering work of Bongiorno and Youla 11 in I which they parameterized the set of all stabi izing controllers, this area of research has exploded during the past decade. Using this parameterization, Zames and Francis [2] solved the problem of finding an optimal controller (for SISO System) so that the H~ norm of the sensitivity function is minimum. These results have since been generalized to multi-input-multi-output (MIMO) control systems [3]. The recent book of Vidyasagar [5] is an excellent comprehensive reference for this area. Those who have used this technique for nontrivial numerical examples would agree that this elegant approach of designing control systems for stability and performance leads to high orders for the controller and the resultant closed loop transfer functions. Recall that this approach is based on solving the Bezout identity [5]. Usually, the Bezout identity is solved either by solving a pair of Riccati equations or by resorting to some pole-placement technique. As far as we know, there are no known precise guidelines for selecting the eigenvalues of appropriate matrices involved in this process. We strongly feel that the lack of these guidelines for selecting the eigenvalues in the above procedure is one of the main reasons for large order controllers. In this paper, we propose a systematic procedure for solving the Bezout identity by representing the given proper transfer function of a plant in a suitable way. The solution of the Bezout identity turns out to be very simple but extremely useful in as much as it leads to low order controller and the resulting closed loop transfer function. In addition to this, from the form of the resulting closed loop transfer function, the choice of the free parameter becomes quite transparent for meeting performance specifications. This proposed procedure is subject to one technical constraint viz. the number of unstable poles for the plant does not exceed the number of minimum phase zeros by one. But this does not seem to be a serious constraint as most practical control systems satisfy this condition. Assistant Professor, Dept. of Electrical Systems Engineering, "Copyright o 1988 by L.R. Pujara. Published The Algorithm We follow the notation of Vidyasagar [5 . S denotes the set of proper rational functions with rea coefficients and Hurwitz denominator. 1 Consider a typical unity feedback control system represented by the following figure: Figure 1 Suppose the plant transfer function P(s) is represented as where the polynomial ni(s) corresponds to all nonminimum phase zeros of P(s), the polynomial di(s) corresponds to all unstable poles of P(s), and ni(s), di(s) are Hurwitz polynomials.