Computation of All Robustly Stabilizing PID Controllers Based on H-∞ Robust Stability Condition

Abstract

In this paper, the computation of all robustly stabilizing Proportional-Integral-Derivative (PID) controllers for Single-Input-Single-Output (SISO) Linear Time Invariant (LTI) systems with/without time delay and unstructured uncertainty is presented. The study proposes a graphical technique to plot the regions of all robustly stabilizing PID controllers that satisfy the H- $\infty $ based robust stability condition. These regions are formed by the Real Root Boundary (RRB), the Complex Root Boundary (CRB), and the Infinite Root Boundary (IRB), which represent the transitions of the roots of the characteristic equation from the left half-plane to the right half-plane (or vice versa), based on Hurwitz stability criteria. The regions are depicted in the ( $k_{i}$ - $k_{d}$ ), ( $k_{p}$ - $k_{i}$ ), and ( $k_{p}$ - $k_{d}$ ) 2-D planes for fixed values of $k_{p}$ , $k_{d}$ , and $k_{i}$ , respectively. The methodology is detailed step by step and demonstrated through various examples. Additionally, stability analyses are visually performed using Nyquist envelopes and uncertainty discs.

Keywords

• H-infinity norm
• PID controller
• robust control
• robust stability
• stability boundary locus
• unstructured uncertainty