Singular Frequencies based Robust PID Controller Design and Analysis in Parameter Space

Finding the complete set of stabilizing parameters for a Proportional–Integral–Derivative (PID) controller is a long-standing challenge in control engineering, especially when dealing with high-order systems. This study proposes a practical computational framework to map the entire stability region in the (Kp,Ki,Kd) parameter space utilizing the singular frequency decoupling method. By fixing the proportional gain (Kp), the boundaries of the stability region in the (Ki,Kd) plane are analytically derived using linear equations. This approach allows for the automatic identification of stabilizing polygons without the need for complex and time-consuming grid-based searches. A key focus of this work is the extension of this methodology to handle parametric uncertainties within interval plants. Instead of relying solely on traditional methods, this study employs a robust analysis based on vertex plant configurations. "By evaluating the stabilizing proportional gain intervals for the extreme corners of the uncertainty box and finding their mathematical intersection, a common solution region is identified. Furthermore, robust stability within the (Ki, Kd) plane is ensured by intersecting the stabilising polygons of the individual vertex plants and conducting a robust stability analysis of the parameter space, taking into account the uncertain plant parameters." To validate the approach, various representative system models are examined and analyzed. The results demonstrate that the proposed method provides a reliable and efficient tool for designers to determine robust controller gains with guaranteed stability, visualized through 2D robust polygons and 3D stabilizing solids.