Time-Invariant Linear Quadratic Regulators: A New Look

Abstract

This paper demonstrates that linear ordinary differential equations with constant coefficients serve as the main building block for time-invariant linear quadratic regulators (LQRs). By applying the Euler-Lagrange equations to a flat output, i.e., to a basis of the free finitely-generated module over the commutative principal ideal ring of time-invariant linear differential operators with respect to the time derivation, this result emerges directly. In module-theoretic terms. The freeness of this structure corresponds to the classical Kalman controllability condition (M. Fliess, Some basic structural properties of generalized linear systems, Syst. Contr. Lett., 15 (1990) 391-396). Within this framework, the two-point boundary value problem, including the control variables, becomes analytically tractable, enabling the derivation of optimal time horizons, optimal parameter design, and rest-to-rest trajectories. Furthermore, a novel turnpike-like phenomenon is established via nonstandard analysis. The feedback loop is closed thanks to the HEOL approach (C. Join, E. Delaleau, M. Fliess, Flatness-based control revisited: The HEOL setting, C.R. Math., 362 (2024) 1693-1706), an extension of model-free control (MFC) that leverages intelligent controllers. Our viewpoint, which significantly simplifies the presentation by avoiding the conventional Kalman state-space representation, relies on foundational algebraic concepts that, while elementary, are rarely used in today's control theory. To aid intuition, some rudimentary examples are presented, most of which include computer simulations.