New Tools for Tuning PID Controllers Using Orthogonal Polynomials

This paper presents an algorithm to determine PID controller tuning regions that guarantee the stability of closed-loop control systems using the pole placement method, assigning a known robustly stable polynomial that depends on a vector of uncertain parameters as the desired closed-loop characteristic polynomial. The algorithm generates PID gains contingent on uncertain parameters. This robustly stable polynomial is constructed using modified classical weights and relies on well-known properties of the theory of orthogonal polynomials, including the recurrence relation. In addition, it considers linear combinations of two orthogonal polynomials with consecutive degrees. An advantage of the proposed approach is the considerable flexibility in selecting both the closed-loop characteristic polynomial and number of uncertain parameters. As this problem arises from the challenge of assigning n closed-loop poles with only three free gains, the methodology uses the Moore-Penrose generalized inverse of a certain matrix to obtain expressions of the PID gains. Therefore, the proposed methodology does not work in certain cases of linear time-invariant systems. Three examples were provided to illustrate the design algorithm.