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Dynamic feedback linearization of two-input control systems via successive one-fold prolongations

Florentina Nicolau, Witold Respondek, Shunjie Li

TL;DR

This work addresses dynamic linearization of nonlinear two-input control systems via successive one-fold prolongations, introducing the LSOPI framework that relies on replacing the first noninvolutive linearizability distribution with an involutive corank-one subdistribution to gain new involutive directions. It provides a constructive algorithm that, under constant-rank assumptions, either yields a static feedback linearizable prolongation or proves the system is not LSOPI, and it characterizes the LSOPI-distributions $\mathcal{H}^k$ (unique in some subcases, non-unique in Case III). The paper establishes precise verifiable criteria linking flatness, LSOPI, LSOP, and L$\ell$P notions, relates LSOPI to existing dynamic linearization results, and illustrates the framework on the chained form. Overall, the results offer a practical, geometry-grounded procedure to achieve dynamic linearization for two-input systems and clarify the boundaries between flatness, LSOPI, and LSOP.

Abstract

In this paper, we propose a constructive algorithm to dynamically linearize two-input control systems via successive one-fold prolongations of a control that has to be suitably chosen at each step of the algorithm. Linearization via successive one-fold prolongations requires special properties of the linearizability distributions $\mathcal{D}^0 \subset \mathcal{D}^1 \subset\mathcal{D}^2 \subset \cdots$. Contrary to the case of static feedback linearizability, they need not be involutive but the first noninvolutive one has to contain an involutive subdistribution of corank one. The main idea of the proposed algorithm is to replace, at each step, the first noninvolutive distribution by its involutive subdistribution of corank one, thus for the prolonged system we gain at least one new involutive distribution. Our algorithm is constructive, gives sufficient conditions for flatness, and can be seen as the dual of the dynamic feedback linearization algorithm of Battilotti and Califano [2003, 2005].

Dynamic feedback linearization of two-input control systems via successive one-fold prolongations

TL;DR

This work addresses dynamic linearization of nonlinear two-input control systems via successive one-fold prolongations, introducing the LSOPI framework that relies on replacing the first noninvolutive linearizability distribution with an involutive corank-one subdistribution to gain new involutive directions. It provides a constructive algorithm that, under constant-rank assumptions, either yields a static feedback linearizable prolongation or proves the system is not LSOPI, and it characterizes the LSOPI-distributions (unique in some subcases, non-unique in Case III). The paper establishes precise verifiable criteria linking flatness, LSOPI, LSOP, and LP notions, relates LSOPI to existing dynamic linearization results, and illustrates the framework on the chained form. Overall, the results offer a practical, geometry-grounded procedure to achieve dynamic linearization for two-input systems and clarify the boundaries between flatness, LSOPI, and LSOP.

Abstract

In this paper, we propose a constructive algorithm to dynamically linearize two-input control systems via successive one-fold prolongations of a control that has to be suitably chosen at each step of the algorithm. Linearization via successive one-fold prolongations requires special properties of the linearizability distributions . Contrary to the case of static feedback linearizability, they need not be involutive but the first noninvolutive one has to contain an involutive subdistribution of corank one. The main idea of the proposed algorithm is to replace, at each step, the first noninvolutive distribution by its involutive subdistribution of corank one, thus for the prolonged system we gain at least one new involutive distribution. Our algorithm is constructive, gives sufficient conditions for flatness, and can be seen as the dual of the dynamic feedback linearization algorithm of Battilotti and Califano [2003, 2005].
Paper Structure (17 sections, 84 equations, 3 figures, 3 algorithms)

This paper contains 17 sections, 84 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Flatness
  3. Main results
  4. LSOPI-definition
  5. Algorithm for dynamic linearization of LSOPI-systems
  6. Construction of the LSOPI-distribution $\mathcal{H}^{k}$
  7. Constructive LSOPI-algorithm
  8. Interplay between flatness, LSOPI, LSOP and L$\ell$P properties for two-input control systems
  9. Example: the chained form
  10. Proofs
  11. Proof of Proposition \ref{['prop: nec cond flatness']}
  12. Proof of Proposition \ref{['prop: equiv H feedback']}
  13. Proof of Proposition \ref{['prop Dk noninvolutive']}
  14. Proof of Proposition \ref{['prop: H case II']}
  15. Proof of Proposition \ref{['prop: H case III']}
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Construction of $\mathcal{H}^k$ by Proposition \ref{['prop: H case II']}.
  • Figure 2: Construction of $\mathcal{H}^k$ by Proposition \ref{['prop: H case III']} and relation \ref{['eq: H case C5']}.
  • Figure 3: Construction of $\mathcal{H}^k$ by Proposition \ref{['prop: C-iii']}.

Theorems & Definitions (12)

  • Remark 3.1: LSOPI versus LSOP
  • Remark 3.2
  • Remark 3.3
  • proof
  • Remark 3.4
  • Remark 4.1
  • proof
  • Remark 5.1
  • Remark A.1
  • proof
  • ...and 2 more