Low-order digital controller design based on l₂ parametric stability margin

The uncertainties inherent in engineering control systems are generally intricate and multifaceted. Sometimes, the model parameters of the plants may even fluctuate significantly. To address this undesirable scenario and ensure system stability under such conditions, the controller parameters with the largest parametric stability margin in the stabilizing set should be selected. In industrial processes, the proportional-integral-derivative (PID) controller is the most widely used control approach. However, an excessive focus on the system's dynamic performance may result in PID controller tuning parameters that are dangerously close to the stability boundary, even with widely used tuning methods. In this paper, a novel design strategy for digital PID controllers is proposed based on a class of second-order uncertain discrete-time control systems. First, the stabilizing set of PID parameters, which consists of a family of parallel convex polygons in three-dimensional space, is derived from the stability condition of the closed-loop system. Then, the coordinate axes are rotated to make the convex polygons perpendicular to one of the axes. Next, an algorithm based on linear programming is developed to determine the coordinates of the Chebyshev center of the stabilizing set. Finally, the coordinates of the Chebyshev center in the original space are obtained via the inverse coordinate-axes rotation transformation and then adopted as the controller parameters. The main contribution of this paper is the development of a tuning method for digital PID controllers, which provides the maximum l₂ parametric stability margin and makes the controller non-fragile in the PID parameter space. In addition, this paper explicitly presents the method for determining the PID parameter stabilizing set for second-order discrete-time systems.