Error Bound for Non-Zero Initial Condition Using the Singular Perturbation Approximation Method

Adnan Daraghmeh,Naji Qatanani

doi:10.3390/math6110232

Adnan Daraghmeh, Naji Qatanani

Open Access

https://doi.org/10.3390/math6110232

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Journal: Mathematics	Publication Date: Oct 30, 2018
Citations: 1	License type: CC BY 4.0

Affiliation: An-Najah National University

Abstract

In this article, we present up to date results on the balanced model reduction techniques for linear control systems, in particular the singular perturbation approximation. One of the most important features of this method is it allows for an a priori L 2 and H ∞ bounds for the approximation error. This method has been successfully applied for systems with homogeneous initial conditions, however, the main focus in this work is to derive an L 2 error bound for singular perturbation approximation for system with inhomogeneous initial conditions, extending the work by Antoulas et al. The theoretical results are validated numerically.

Highlights

Linear systems have been under investigation for quite long time due to their wide range of applications in physics, mathematics and engineering
If we reduced the original system using the singular perturbation approximation, there is an error bound available for the reduced system of transfer function Ḡr of the stable and balanced system ( A, B, C, D )
We introduce an approach to find the error bound between the outputs of the original and the reduced systems with non-zero initial condition using the method of singular perturbation approximation (SPA)

Summary

Introduction

Linear systems have been under investigation for quite long time due to their wide range of applications in physics, mathematics and engineering. The subject is such a fundamental and deep one that there is no doubt that linear systems will continue to be a main focus of study for long time to come. A common feature of the model used is that it is high-dimensional and displays a variety of time scales. If the time scales in the system are well separated, it is possible to eliminate the fast degrees of freedom and to derive low-ordered reduced models, using averaging and homogenization techniques. Homogenization of linear control systems has been widely studied by various authors [1,2,3,4]

Related Work

Preliminaries

The Reciprocal System of a Linear Continuous Dynamical System

Conclusions