A numerical approach to the co-design of PID controllers and low-pass filters for time-delay systems

TL;DR

This paper introduces a numerical co-design method for PID controllers and low-pass filters in time-delay systems, optimizing spectral properties to enhance stability and performance.

Contribution

It presents a systematic framework for simultaneous optimization of PID gains and filter constants, considering the filter as an integral part of the control design.

Findings

Numerical examples demonstrate improved stability and robustness.

The approach applies to both SISO and MIMO systems with delays.

Optimizing spectral abscissa enhances control performance.

Abstract

This paper addresses the numerical optimization of proportional-integral-derivative (PID) controllers for linear time-invariant systems with delays, where the derivative action is implemented using a low-pass filter. While performance assessment is often based on the spectral abscissa of the ideal PID-controlled system, the inclusion of a derivative filter fundamentally alters the closed-loop spectral properties and cannot be treated as a post-processing step. In particular, the spectral abscissa of the filtered closed-loop system may differ significantly from that of its unfiltered counterpart, potentially affecting both stability and performance. We propose a systematic numerical design framework in which the PID gains and the filter constant are optimized simultaneously by directly minimizing the spectral abscissa of the filtered closed-loop system. Treating the filter as an integral part of the control design allows us to reconcile robustness at high frequencies, in the sense of mitigating fragility issues due to approximate identities, with performance at low frequencies, in addition to counter measurement noise amplification. At the end of the presentation, numerical examples illustrate the proposed approach and highlight the benefits of controller-filter co-design. The results apply to general linear systems with input and/or state delays and are valid for both single-input single-output (SISO) and multi-input multi-output (MIMO) configurations.