Linear Quadratic Regulators: A New Look

TL;DR

This paper presents a novel algebraic perspective on linear quadratic regulators, linking control theory with module theory over differential operators, leading to new methods for optimal control and robust performance.

Contribution

It introduces an algebraic framework for LQR design using module theory, enabling explicit solutions and improved control strategies.

Findings

Algebraic characterization of controllability as module freeness.

Explicit open-loop control strategies from flat outputs.

Enhanced robustness through model-free control integration.

Abstract

Linear time-invariant control systems can be considered as finitely generated modules over the commutative principal ideal ring $\mathbb{R}[\frac{d}{dt}]$ of linear differential operators with respect to the time derivative. The Kalman controllability in this algebraic language is translated as the freeness of the system module. Linear quadratic regulators rely on quadratic Lagrangians, or cost functions. Any flat output, i.e., any basis of the corresponding free module leads to an open-loop control strategy via an Euler-Lagrange equation, which becomes here a linear ordinary differential equation with constant coefficients. In this approach, the two-point boundary value problem, including the control variables, becomes tractable. It yields notions of optimal time horizon, optimal parameter design and optimal rest-to-rest trajectories. The loop is closed via an intelligent controller derived from model-free control, which is known to exhibit excellent performance concerning model mismatches and disturbances.