Input Influence Matrix Design for MIMO Discrete-Time Ultra-Local Model

TL;DR

This paper provides guidelines for designing input influence matrices in MIMO systems using Ultra-Local Models and model-free control, analyzing stability and demonstrating effectiveness through simulations.

Contribution

It introduces a method to design input influence matrices for MIMO ULM-based control, ensuring local asymptotic stability of the error dynamics.

Findings

Wide range of influence matrices can ensure stability.

Stability depends on eigenvalues of the difference matrix.

Simulation verifies theoretical stability conditions.

Abstract

Ultra-Local Models (ULM) have been applied to perform model-free control of nonlinear systems with unknown or partially known dynamics. Unfortunately, extending these methods to MIMO systems requires designing a dense input influence matrix which is challenging. This paper presents guidelines for designing an input influence matrix for discrete-time, control-affine MIMO systems using an ULM-based controller. This paper analyzes the case that uses ULM and a model-free control scheme: the H\"older-continuous Finite-Time Stable (FTS) control. By comparing the ULM with the actual system dynamics, the paper describes how to extract the input-dependent part from the lumped ULM dynamics and finds that the tracking and state estimation error are coupled. The stability of the ULM-FTS error dynamics is affected by the eigenvalues of the difference (defined by matrix multiplication) between the actual and designed input influence matrix. Finally, the paper shows that a wide range of input influence matrix designs can keep the ULM-FTS error dynamics (at least locally) asymptotically stable. A numerical simulation is included to verify the result. The analysis can also be extended to other ULM-based controllers.