Stability Analysis of Second‐Order Systems With Digital PID Controller: An Impulsive System Conversion Method

This paper investigates the stability analysis of second‐order systems with digital proportional‐integral‐derivative (PID) controller based on the impulsive system conversion. The evaluation method of digital PID controlled systems is optimised to more precisely determine its maximum acceptable sampling period. This approach reduces the communication burden under the expected performance. Firstly, since the sampled‐data PID (SDPID) controller represents a standard form of the digital PID controller, the second‐order system using SDPID controller is converted to impulsive form through the introduction of a virtual variable. The system conversion method is further applied to the second‐order system with polynomial uncertainty to demonstrate its effectiveness. Next, an impulsive‐type looped‐functional is established for the aforementioned impulsive system to lose the functional's decreasing constraint. Based on the aforementioned modifications, several less conservative criteria are developed. Based on these criteria, a more precise evaluation of system's admissible range of sampling periods is achieved. This results in a reduction in the utilisation of communication resources. Finally, the numerical example and experimental test are performed to validate the superiority and the effectiveness of the developed method.