Tuning the controller for object models with one zero and one to four poles and astatism using the polynomial method

Abstract

This paper presents the procedure for tuning a controller for process models with one zero, one to four poles, and astatism using the polynomial method, by imposing the damping ratio and settling time of the synthesized system. The numerator and denominator polynomials of the model's transfer function are decomposed into components containing the zeros in the left and right half of the complex plane. Based on the object models order and the conditions for solving the algebraic equation system, physical realizability of the control algorithm, and system robustness, the desired polynomial of the synthesized system is constructed. This polynomial is composed of two polynomials with unknown coefficients, and the degrees and unknown polynomials, as well as the degree of the desired polynomial, are calculated. The dominant poles of the synthesized system are determined from the imposed damping ratio and settling time, and based on them, the closed-loop system's characteristic polynomial is constructed. If necessary, additional real poles are added, placed as far as possible from the dominant poles to meet system performance requirements. From the equality of polynomials, by matching the coefficients of like powers of the variable s from both sides of the equation, an algebraic equation system is obtained, from which the unknown coefficients and polynomials are determined. Based on the stable components of the object model and the unknown polynomials, the transfer function of the control algorithm is constructed. Examples of controller tuning for one zero and one to fourth poles models using the polynomial method are analyzed. The synthesized systems exhibit high performance and good robustness.