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On Structural Non-commutativity in Affine Feedback of SISO Nonlinear Systems

Venkatesh G. S.

Abstract

The affine feedback connection of SISO nonlinear systems modeled by Chen--Fliess series is shown to be a group action on the plant which is isomorphic to the semi-direct product of shuffle and additive group of non-commutative formal power series. The additive and multiplicative feedback loops in an affine feedback connection are thus proven to be structurally non-commutative. A flip in the order of these loops results in a net additive feedback loop.

On Structural Non-commutativity in Affine Feedback of SISO Nonlinear Systems

Abstract

The affine feedback connection of SISO nonlinear systems modeled by Chen--Fliess series is shown to be a group action on the plant which is isomorphic to the semi-direct product of shuffle and additive group of non-commutative formal power series. The additive and multiplicative feedback loops in an affine feedback connection are thus proven to be structurally non-commutative. A flip in the order of these loops results in a net additive feedback loop.
Paper Structure (6 sections, 15 theorems, 50 equations, 7 figures)

This paper contains 6 sections, 15 theorems, 50 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Formal Power Series over ${\mathbb R}$
  3. Chen--Fliess Series
  4. Affine Feedback Interconnection of Chen--Fliess Series
  5. Affine Feedback Group
  6. Affine Feedback Interconnection

Key Result

Theorem 3.1

Ferfera_80 Let $c,d$ and $e \in \hbox{${\mathbb R}^{}\langle\langle X \rangle\rangle$}$, then $\left(c{ \;\sqcup \space\sqcup\;} d\right) \circ e = \left(c \circ e\right) { \;\sqcup \space\sqcup\;} \left(d \circ e\right)$.

Figures (7)

  • Figure 1: Affine Feedback Interconnection
  • Figure 2: Composition of Chen--Fliess series
  • Figure 3: Composition of Chen--Fliess series: $F_c$ and $uF_{d_1} + F_{d_2}$
  • Figure 4: Affine Feedback Interconnection of $F_{c}$ with $F_{\mathbf{d}}$
  • Figure 5: Non-commutativity of Additive and Multiplicative Feedback Loops.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 3.1
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.1
  • Lemma 4.1
  • Theorem 4.2
  • Definition 4.3
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • ...and 9 more