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

•An improved characterization of the entire stabilizing PID controller set for a given process is presented which can reduce the computation effort in testing the existence of controller set.•We develop an efficient algorithm for testing the existence of σ-stabilizable PID controller set without actually constructing the set.•An algorithm is offered to identify a non-conservative interval of stability degree within which the achievable maximum stability degree lies.•A bisection strategy is applied along with algorithms of testing the existence of σ-stabilizable PID controller set and estimating the interval of stability degree to design maximum-stability PID controllers. 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. Based on the characterization of stabilizing PID controller set, an efficient algorithm is developed to test if a plant is σ-stabilizable, where σ 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 σ-interval [σε*,σε*+ε] which contains the maximum stability degree σ* for a specified ε, and the PID controller parameter set for achieving the stability degree σε*. 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 kp-intervals such that unnecessary computations are avoided; (ii) a simple yet effective method has been adopted to provide a non-conservative interval of σ 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. [Display omitted]