Bode Integral Limitation For Irrational Systems

TL;DR

This paper explores the limitations of Bode integrals in feedback control systems, especially when using fractional and fractal PID controllers on irrational systems, extending classical results to these complex cases.

Contribution

It extends Bode integral analysis to fractional and fractal PID controllers applied to irrational systems, providing conditions under which classical limitations hold or can be mitigated.

Findings

Bode integral structure remains similar under certain convergence conditions.

Sufficient conditions identified for controllers to reduce Bode sensitivity integral.

Analysis covers cases with and without limit points of open-loop poles.

Abstract

Bode integrals of sensitivity and sensitivity-like functions along with complementary sensitivity and complementary sensitivity-like functions are conventionally used for describing performance limitations of a feedback control system. In this paper, we investigate the Bode integral and evaluate what happens when a fractional order Proportional-Integral-Derivative (PID) controller is used in a feedback control system. We extend our analysis to when fractal PID controllers are applied to irrational systems. We split this into two cases: when the sequence of infinitely many right half plane open-loop poles doesn't have any limit points and when it does have a limit point. In both cases, we prove that the structure of the Bode Integral is similar to the classical version under certain conditions of convergence. We also provide a sufficient condition for the controller to lower the Bode sensitivity integral.