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

Area of Science:

• Control Systems Engineering
• Systems Theory
• Applied Mathematics

Background:

• Engineering control systems face complex uncertainties, often leading to significant plant parameter fluctuations.
• Proportional-Integral-Derivative (PID) controllers are widely used but can be tuned near stability boundaries, risking system integrity.
• Existing tuning methods may not adequately address parametric uncertainties, compromising controller robustness.

Purpose of the Study:

• To propose a novel design strategy for digital PID controllers for second-order uncertain discrete-time systems.
• To develop a tuning method that maximizes the parametric stability margin, ensuring system robustness.
• To create a non-fragile PID controller design resilient to parameter variations.

Main Methods:

• Derived the stabilizing set of PID parameters as convex polygons in 3D space based on closed-loop stability conditions.
• Employed coordinate-axes rotation to simplify the geometric analysis of the stabilizing set.
• Developed a linear programming algorithm to find the Chebyshev center of the stabilizing set for optimal parameter selection.

Main Results:

• Identified the stabilizing set as a family of parallel convex polygons.
• Successfully determined the Chebyshev center coordinates, representing optimal PID controller parameters.
• Achieved the maximum l2 parametric stability margin for the digital PID controller.

Conclusions:

• The proposed method provides a robust tuning strategy for digital PID controllers in uncertain systems.
• The developed technique ensures controller non-fragility by maximizing the parametric stability margin.
• This work offers a systematic approach to determining PID parameter stabilizing sets for second-order discrete-time systems.