Design of maximum-stability PID controllers for LTI systems based on a stabilizing-set construction method

Background: The PID algorithm has been widely used in control of chemical processes. A maximum-stability PID controller can provide superior stability robustness towards plant's variations and maximum exponential decay rate for disturbance rejection. However, design of maximum-stability PID controller is a min-max optimization problem and an effective method is still lack in the literature. Methods: Based on the characterization of stabilizing PID controller set, an efficient algorithm is developed to test if a plant is sigma-stabilizable, where sigma is abscissa or stability degree of the Hurwitz stable closed-loop characteristic polynomial. This algorithm is then used along with a bisection strategy to find a sigma-interval [sigma(epsilon)*, sigma epsilon* + epsilon] which contains the maximum stability degree sigma* for a specified epsilon, and the PID controller parameter set for achieving the stability degree sigma(epsilon)*. Significant findings: This paper has presented a systematic and efficient approach to design PID controllers with maximum degree of stability. The principal results include: (i) an improved theorem is presented for identifying stabilizing k(p)-intervals such that unnecessary computations are avoided; (ii) a simple yet effective method has been adopted to provide a non-conservative interval of sigma which facilitates the bisectional branch-and-bound operation; (iii) the design procedure does not involve the actual construction of stabilizing PID controller sets thus renders its efficiency. (C) 2022 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.