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A Generalization of Input-Output Linearization via Dynamic Switching Between Melds of Output Functions

Mirko Mizzoni, Pieter van Goor, Barbara Bazzana, Antonio Franchi

TL;DR

The paper addresses control of nonlinear systems with more candidate outputs than inputs by introducing a meld framework that selects square output subsets from a larger deck to preserve exact feedback linearization. It develops compatibility conditions and a dwell-time-based switching scheme to guarantee uniform boundedness and exponentially stable error dynamics within each interval, while ensuring seamless tracking of outputs shared across melds during transitions. A rigorous theorem guarantees stability under prescribed switching and reference alignment, and a numerical 3R manipulator example demonstrates practical viability with multiple melds. This approach advances robust, modular control for redundant actuators in robotics and related domains, with potential for broader application and extension to uncertain models.

Abstract

This letter presents a systematic framework for switching between different sets of outputs for the control of nonlinear systems via feedback linearization. We introduce the concept of a meld to formally define a valid, feedback-linearizable subset of outputs that can be selected from a larger deck of possible outputs. The main contribution is a formal proof establishing that under suitable dwell-time and compatibility conditions, it is possible to switch between different melds while guaranteeing the uniform boundedness of the system state. We further show that the error dynamics of the active outputs remain exponentially stable within each switching interval and that outputs common to consecutive melds are tracked seamlessly through transitions. The proposed theory is valid for any feedback linearizable nonlinear system, such as, e.g., robots, aerial and terrestrial vehicles, etc.. We demonstrate it on a simple numerical simulation of a robotic manipulator.

A Generalization of Input-Output Linearization via Dynamic Switching Between Melds of Output Functions

TL;DR

The paper addresses control of nonlinear systems with more candidate outputs than inputs by introducing a meld framework that selects square output subsets from a larger deck to preserve exact feedback linearization. It develops compatibility conditions and a dwell-time-based switching scheme to guarantee uniform boundedness and exponentially stable error dynamics within each interval, while ensuring seamless tracking of outputs shared across melds during transitions. A rigorous theorem guarantees stability under prescribed switching and reference alignment, and a numerical 3R manipulator example demonstrates practical viability with multiple melds. This approach advances robust, modular control for redundant actuators in robotics and related domains, with potential for broader application and extension to uncertain models.

Abstract

This letter presents a systematic framework for switching between different sets of outputs for the control of nonlinear systems via feedback linearization. We introduce the concept of a meld to formally define a valid, feedback-linearizable subset of outputs that can be selected from a larger deck of possible outputs. The main contribution is a formal proof establishing that under suitable dwell-time and compatibility conditions, it is possible to switch between different melds while guaranteeing the uniform boundedness of the system state. We further show that the error dynamics of the active outputs remain exponentially stable within each switching interval and that outputs common to consecutive melds are tracked seamlessly through transitions. The proposed theory is valid for any feedback linearizable nonlinear system, such as, e.g., robots, aerial and terrestrial vehicles, etc.. We demonstrate it on a simple numerical simulation of a robotic manipulator.
Paper Structure (10 sections, 4 theorems, 46 equations, 2 figures)

This paper contains 10 sections, 4 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exact state–space linearization (via output function)
  4. Melds of a Control System
  5. Output Tracking of Switching Melds
  6. Recall of Standard Single Output Array Tracking
  7. Extension to Output Tracking of Switching Melds
  8. Numerical Example with a 3R Robot and a Multi-dimensional Deck of Outputs
  9. Description of the task
  10. Conclusions and Future works

Key Result

Proposition 1

Let be given the pair $(\Sigma,\Delta)$ and assume that $|\mathbb{M}| =\mu \geq 2$. Let the switching signal $\bm{\sigma}(t)$ be piecewise constant. Fix $k \geq 0$ and $l \in \mathbb{N}_0$, and consider the set $\mathcal{S}_{k,l}$. Then, under the control law eq:control, for every $i\in\mathcal{S}_{

Figures (2)

  • Figure 1: Three-degree-of-freedom planar robot with two vacuum grippers mounted at the end-effector and on the second link. The robot is tasked with aspirating items at specified locations.
  • Figure 2: In red the outputs of a meld selected in that interval. The five different colored regions represent the five different melds. When an output is selected, the reference trajectory is the one displayed in dotted.

Theorems & Definitions (12)

  • Definition 1: Meld
  • Definition 2: Validity Set
  • Remark III.1
  • Definition 3
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 2 more