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Flows of vector fields and the Kalman Theorem

Fabio Bagagiolo, Cristina Giannotti, Andrea Spiro, Marta Zoppello

TL;DR

This paper presents two elementary proofs of the local Kalman controllability theorem for the linear system $\dot q = A q + B u$, connecting reachability to Lie-algebraic growth. The first proof derives the Kalman result as a corollary of the Chow-Rashevskiıı theorem by exploiting convexity properties of the controllable set when the control set $\mathscr{K}$ contains a neighborhood of the origin. The second proof avoids convexity and instead uses flows in an extended space-time $\mathscr{M}$ and the geometry of stepped graphs to relate controllability to the orbit structure of suitable vector fields. Together, these proofs illuminate how local controllability results can be obtained from flow and orbit theory and pave the way for nonlinear generalizations, as evidenced by applications to real-analytic nonlinear control systems. The work thus strengthens the bridge between linear controllability criteria and geometric control theory, with potential for broad generalizations.

Abstract

We give two proofs of the Kalman Theorem, alternative to the most common ones, which infer such a classical result of Control Theory using just very basic facts on flows of vector fields. These proofs are apt to be generalised in diverse directions -- in fact one of them has been already generalised, yielding new criteria for local controllability of non-linear real analytic controlled systems.

Flows of vector fields and the Kalman Theorem

TL;DR

This paper presents two elementary proofs of the local Kalman controllability theorem for the linear system , connecting reachability to Lie-algebraic growth. The first proof derives the Kalman result as a corollary of the Chow-Rashevskiıı theorem by exploiting convexity properties of the controllable set when the control set contains a neighborhood of the origin. The second proof avoids convexity and instead uses flows in an extended space-time and the geometry of stepped graphs to relate controllability to the orbit structure of suitable vector fields. Together, these proofs illuminate how local controllability results can be obtained from flow and orbit theory and pave the way for nonlinear generalizations, as evidenced by applications to real-analytic nonlinear control systems. The work thus strengthens the bridge between linear controllability criteria and geometric control theory, with potential for broad generalizations.

Abstract

We give two proofs of the Kalman Theorem, alternative to the most common ones, which infer such a classical result of Control Theory using just very basic facts on flows of vector fields. These proofs are apt to be generalised in diverse directions -- in fact one of them has been already generalised, yielding new criteria for local controllability of non-linear real analytic controlled systems.

Paper Structure

This paper contains 14 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vector fields, Lie brackets and flows
  4. Pushed-forward vector fields and their flows
  5. Piecewise regular curves and their compositions
  6. The Chow-Rashevskiı Theorem and the Kalman Theorem
  7. Orbits of sets of vector fields and the Chow-Rashevskiı Theorem
  8. The Kalman Theorem
  9. The Kalman Theorem implies the corollary of the Chow-Rashevskiı Theorem for sets of vector fields with either constant or linear components
  10. A proof of the Kalman Theorem, based on the Chow-Rashewskiı Theorem
  11. Another proof of the Kalman Theorem, based on general facts on flows of vector fields
  12. Stepped graphs in the extended space-time
  13. Final points of stepped graphs as points in orbits of flows of vector fields
  14. Our second proof of the sufficiency part of the Kalman Theorem

Key Result

Theorem 3.1

For any family $\mathscr{F}$ of $\mathcal{C}^\infty$ vector fields on an open subset $\mathscr{U} \subset \mathbb{R}^n$ and for any $x_o \in \mathscr{U}$, the corresponding set $\operatorname{Orb}^\mathscr{F}(x_o)$ is a connected immersed submanifold of $\mathbb{R}^n$ of dimension greater than or eq

Theorems & Definitions (10)

  • Theorem 3.1: Chow-Rashevskiı-Sussmann ChRaSu0Su
  • Theorem 3.2: Kalman Theorem
  • Proposition 5.1
  • proof
  • Remark 5.2
  • Remark 6.1
  • Lemma 6.2
  • Lemma 6.3
  • Lemma 6.4
  • Theorem 6.5