Model Order Reduction: A Comparison between Integer and Non-Integer Order Systems Approaches

Riccardo Caponetto,Emanuele Murgano,José Tenreiro Machado,Maria Gabriella Xibilia

doi:10.3390/e21090876

Riccardo Caponetto, Emanuele Murgano + Show 2 more

Open Access

https://doi.org/10.3390/e21090876

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Abstract

In this paper, classical and non-integer model order reduction methodologies are compared. Non integer order calculus has been used to generalize many classical control strategies. The property of compressing information in modelling systems, distributed in time and space, and the capability of describing long-term memory effects in dynamical systems are two features suggesting also the application of fractional calculus in model order reduction. In the paper, an open loop balanced realization is compared with three approaches based on a non-integer representation of the reduced system. Several case studies are considered and compared. The results confirm the capability of fractional order systems to capture and compress the dynamics of high order systems.

Highlights

A relevant topic in automatic control is the Model Order Reduction (MOR)
Different strategies are presented in the literature, among which, in the case of asymptotically stable Linear Time-Invariant (LTI) systems, one of the most relevant is based on the open loop balanced realization [1]
Looking at the approximation error values, it can be observed that the techniques based on optimization algorithms are able to fit the Interger Order Transfer Function (IOTF) with good results

Summary

Introduction

A relevant topic in automatic control is the Model Order Reduction (MOR). The MOR approximates a high-order Linear Time-Invariant (LTI) system with a low-order one, neglecting the less significant state-space variables, decreasing the number of variables and parameters needed for its representation, and simplifying the controller design procedure. Different strategies are presented in the literature, among which, in the case of asymptotically stable LTI systems, one of the most relevant is based on the open loop balanced realization [1]. Other techniques have been defined using either optimization algorithms such as Genetic Algorithm (GA) [2] and Particle Swarm Optimization [3], or using Artificial Neural Networks [4]. In parallel to these methods, new MOR strategies, see [5], have been investigated applying. The open loop balanced realization is compared with three other procedures that provide fractional order reduced systems. MOR techniques are discussed; in Section 3, comparative examples are given and, in Section 4, conclusions are drawn

Some Notes on Fractional Calculus

Open-Loop Balancing Reduction

Implicit Model Order Reduction via Fractional Order Calculus

Fractional Order Transfer Function Fitting

Numerical Examples

Conclusions