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Tracking controllability for finite-dimensional linear systems

Sebastián Zamorano, Enrique Zuazua

TL;DR

This work develops a functional-analytic framework for exact and approximate output tracking in finite-dimensional linear systems by recasting tracking as the surjectivity of the control-to-output map $\Lambda_C$ and introducing the tracking Gramian $G_C=\Lambda_C\Lambda_C^*$. An adjoint-based observability inequality provides a dual condition for exact tracking, enabling a Hilbert-Uniqueness-Method (HUM) construction of minimum-norm tracking controls with $u_{\min}=\Lambda_C^*G_C^{-1}f$, and clarifies intrinsic regularity constraints on reference trajectories. The authors extend the HUM to a trajectory-space setting, derive a dual formulation for approximate tracking via a convex functional, and present explicit scalar-input formulas together with compatibility and projection phenomena for multiple outputs. Numerical experiments on ODEs and semi-discretized PDEs (including a semi-discrete wave equation) demonstrate the method’s effectiveness for smooth and non-smooth targets, while highlighting the practical impact of relative-degree and Ran$(C)$-based limitations on exact tracking. The results provide a principled energy-minimization approach to tracking, reveal fundamental regularity and structural restrictions, and guide robust tracking strategies in discretized or measurement-derived settings.

Abstract

We develop a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a nonstandard observability inequality for the adjoint dynamics. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report numerical experiments for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and non-smooth targets.

Tracking controllability for finite-dimensional linear systems

TL;DR

This work develops a functional-analytic framework for exact and approximate output tracking in finite-dimensional linear systems by recasting tracking as the surjectivity of the control-to-output map and introducing the tracking Gramian . An adjoint-based observability inequality provides a dual condition for exact tracking, enabling a Hilbert-Uniqueness-Method (HUM) construction of minimum-norm tracking controls with , and clarifies intrinsic regularity constraints on reference trajectories. The authors extend the HUM to a trajectory-space setting, derive a dual formulation for approximate tracking via a convex functional, and present explicit scalar-input formulas together with compatibility and projection phenomena for multiple outputs. Numerical experiments on ODEs and semi-discretized PDEs (including a semi-discrete wave equation) demonstrate the method’s effectiveness for smooth and non-smooth targets, while highlighting the practical impact of relative-degree and Ran-based limitations on exact tracking. The results provide a principled energy-minimization approach to tracking, reveal fundamental regularity and structural restrictions, and guide robust tracking strategies in discretized or measurement-derived settings.

Abstract

We develop a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a nonstandard observability inequality for the adjoint dynamics. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report numerical experiments for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and non-smooth targets.
Paper Structure (16 sections, 8 theorems, 45 equations, 1 figure)

This paper contains 16 sections, 8 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction, problem formulation and main results
  2. Notation
  3. Tracking controllability: first results
  4. The input–output operator
  5. The tracking Gramian operator
  6. Hilbert-Uniqueness-Method for tracking control
  7. $C$-tracking observability
  8. Tracking observability inequality: preliminary results
  9. Approximate output tracking controllability
  10. Scalar-input case: explicit formulas and compatibility conditions
  11. Controller form and exact tracking
  12. Compatibility conditions for multiple outputs
  13. When exact tracking is impossible: optimal projection
  14. Numerical examples
  15. Semi-discrete wave equation
  16. ...and 1 more sections

Key Result

Lemma 2.2

The adjoint operator $\Lambda_C^*: H^{-1}(0,T;\mathbb{R}^p) \to L^2(0,T;\mathbb{R}^m)$ is given by for every $\psi\in H^{-1}(0,T;\mathbb{R}^p)$.

Figures (1)

  • Figure 1: Output $y(t)=x_1(t)$ (blue) and non-smooth target $f(t)$ (red dashed). Despite insufficient regularity for classical tracking formulas, accurate tracking is achieved via the variational approach.

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Remark 3.2
  • ...and 20 more