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Linear Quadratic Regulators: A New Look

Cédric Join, Emmanuel Delaleau, Michel Fliess

TL;DR

The paper reframes LQR within a module-theoretic framework over the differential operator ring, linking Kalman controllability to freeness and expressing all system variables via flat outputs. It develops a comprehensive algebraic toolkit—rings, presentation matrices, the Laplace functor, Lagrangians, and variational systems—and uses Euler–Lagrange equations to obtain tractable open-loop optimal trajectories on finite horizons, with a clear path to two-point boundary-value problems. It further discusses how to close the loop with model-free intelligent control to handle disturbances, and illustrates the theory with several examples including optimal time horizons and turnpike-like considerations. The approach provides an intrinsic, representation-free lens on controllability, observability, and optimality that connects differential-algebraic methods with classical LQR concepts, enabling parameter design and robust trajectory synthesis in a principled way.

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.

Linear Quadratic Regulators: A New Look

TL;DR

The paper reframes LQR within a module-theoretic framework over the differential operator ring, linking Kalman controllability to freeness and expressing all system variables via flat outputs. It develops a comprehensive algebraic toolkit—rings, presentation matrices, the Laplace functor, Lagrangians, and variational systems—and uses Euler–Lagrange equations to obtain tractable open-loop optimal trajectories on finite horizons, with a clear path to two-point boundary-value problems. It further discusses how to close the loop with model-free intelligent control to handle disturbances, and illustrates the theory with several examples including optimal time horizons and turnpike-like considerations. The approach provides an intrinsic, representation-free lens on controllability, observability, and optimality that connects differential-algebraic methods with classical LQR concepts, enabling parameter design and robust trajectory synthesis in a principled way.

Abstract

Linear time-invariant control systems can be considered as finitely generated modules over the commutative principal ideal ring 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.

Paper Structure

This paper contains 27 sections, 16 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Module-theoretic preliminaries
  3. Rings and modules
  4. Systems, dynamics, control and output variables
  5. Equations
  6. State-variable representation
  7. Controllability and observability
  8. Controllability
  9. Observability
  10. Well-formed dynamics
  11. Presentation
  12. An elementary example: Controllable canonical form
  13. Laplace functor
  14. Definition
  15. Transfer matrix
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\Lambda$ be an input-output system where ${\bf u} = \{u_1, \dots, u_m\}$ and ${\bf y} = \{y_1, \dots, y_p\}$ are respectively the control and output variables. There exists for the dynamics a Kalman representation kalman where the dimension of the Kalman state $\{x_1, \dots, x_n\}$ is equal to

Figures (1)

  • Figure :

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 2
  • Definition 1
  • Definition 2
  • Proposition 3
  • Proposition 4
  • proof
  • Corollary 5
  • Lemma 6
  • Proposition 7
  • ...and 13 more