Periodic Regulation of Linear Time-Delay Systems via Youla-Kučera Parametrization

TL;DR

The paper tackles periodic regulation of time-delay (infinite-dimensional) systems by decoupling stabilization from signal-model embedding using Youla-Kučera parametrization. It proposes a structured Youla parameter, $Q_M(s)=\sum_{k=0}^N a_k e^{-s\tau_k}$, to impose zeros in the closed-loop sensitivity at targeted harmonic frequencies $\omega_l$, turning the design into solving a linear system $A x = B$ with $x=[a_0,\ldots,a_N]^T$. This yields a practical method to augment any stabilizing controller without compromising stability, validated through three unstable time-delay examples where periodic regulation at $f=4$ Hz (and more harmonics) is achieved. The approach leverages coprime factorization concepts and IMC connections, offering a scalable route for periodic regulation while highlighting robustness considerations due to added zeros; extensions to MIMO settings and experimental validation are proposed for future work.

Abstract

The paper proposes an alternative way to achieve the Internal Model Principle (IMP) in contrast to the standard way, where a model of the signal one wishes to track/reject is directly substituted into the closed-loop. The proposed alternative approach relies on an already-existing stabilizing controller, which can be further augmented with a Youla-Kučera parameter to let it implicitly admit a model of a signal without altering its stabilizing feature. Thus, with the proposed method, the standard design steps of realizing IMP are reversed. The validity and potential of the proposed approach are demonstrated by considering three different types of time-delay systems. It is shown through simulations that all considered unstable systems, despite the infinite-dimensional closed-loop, can be straightforwardly periodically regulated by augmenting their stabilizing PI controllers. Thanks to a specifically chosen structure for the Youla-Kučera parameter, the required tuning can be done by solving a set of linear equations.

Figures (8)

• Figure 1: Classical Feedback Control.
• Figure 2: Internal Model Control.
• Figure 3: Classical Feedback Control in Youla-Kučera form.
• Figure 4: (Top) Stabilization and Rejection Performance of the PI Controller for the Unstable System with Input Delay in \ref{['eq:G_unstable']}(Below) Improved Rejection Performance of the Augmented PI Controller with Youla-Kučera parameter.
• Figure 5: Pole and Zero Spectrum of the Sensitivity for the Retarded Closed-loop System from Example 2.