Design of PID controllers with maximum degree of stability

• Main Authors: Szu-yuan Lin, 林思源
• Other Authors: Chyi Hwang
• Format: Others
• Language: zh-TW
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/97951365887610062042

碩士 === 義守大學 === 生物技術與化學工程研究所碩士班 === 97 === For a stable continuous-time control system, the degree of stability is the distance between the rightmost pole of the closed-loop system and the imaginary axis of the complex plane. The goal of designing PID controller with maximum degree of stability is to adjust the controller parameters so that the distance between the rightmost pole and imaginary axis is maximized. This is min-max optimization problem. Due to the fact that the maximum real part of the polynomial root is continuous but not differentiable function of the parameters entering the polynomial coefficients. Solving the above-mentioned min-max problem is not an easy task. In this thesis, we propose an efficient numerical procedure for choosing PID controller parameters so that the degree of stability of the closed-loop is maximized. It is based on testing the existence of stabilizing PID controller parameter domain. More precisely, let p(s;k) be the characteristic polynomial of the closed-loop system, where k=(kp,ki,kd) is the parameter vector of the PID controller. The stabilizing PID controller parameter domain is defined to KS(σ)={k∈R3:q(s;k,σ)=0, Re{s}<0}, based on the theory given by KS(σ) stable domain, we present a computer algorithm to test if KS(σ) is empty or not. This algorithm is caused along with the bisection of σ, we can obtain the desired PID controller parameter vector k*which achieves the maximum degree of stability σ*of the closed-loop system. That is σ*=max σ(k)=min{-max Re{si(k)}}, where si are the roots of the characteristic polynomial p(s;k*).