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On the Post-Lie Structure in SISO Affine Feedback Control Systems

Kurusch Ebrahimi-Fard, W. Steven Gray, Venkatesh G. S.

Abstract

The main objective of this work is to show that the single-input, single-output (SISO) affine feedback group, a transformation group in the context of the affine feedback interconnection of Chen-Fliess series, is a post-group in the sense of Bai, Guo, Sheng and Tang.

On the Post-Lie Structure in SISO Affine Feedback Control Systems

Abstract

The main objective of this work is to show that the single-input, single-output (SISO) affine feedback group, a transformation group in the context of the affine feedback interconnection of Chen-Fliess series, is a post-group in the sense of Bai, Guo, Sheng and Tang.
Paper Structure (8 sections, 10 theorems, 47 equations, 3 figures)

This paper contains 8 sections, 10 theorems, 47 equations, 3 figures.

Table of Contents

  1. Post-Groups: An Overview
  2. Preliminaries on Chen--Fliess Series
  3. Chen--Fliess Series
  4. (Feed-forward) Chen--Fliess Series
  5. Affine Feedback Group: Grossman--Larson Group of a Post-Group
  6. From post-Lie Group to post-Lie Algebra
  7. Post-Lie Algebra of the Affine Feedback
  8. $\mathcal{H}$ and the Cointeraction

Key Result

Theorem 1.1

Bai2022postgroup If $\left(G, \cdot, \triangleleft\right)$ is a post-group, then $\left(G, \star \right)$ forms a group called the Grossman--Larson group and shares the unit with $(G, \cdot)$. The inverse with respect to (eqn:GLprod) is and is a group homomorphism.Note that in Bai2022postgroup, $\left(G, \star \right)$ was called the sub–adjacent group. The authors believe, however, that the na

Figures (3)

  • Figure 1: Composition of Chen--Fliess series
  • Figure 2: Composition of Chen--Fliess series $F_c[u]$ and $uF_{d_1}[u] + F_{d_2}[u]$.
  • Figure 3: Affine Feedback Interconnection of $F_{c}$ with $F_{\mathbf{d}}$

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 5 more