Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems

Marek Rydel,Rafał Stanisławski,Krzysztof Latawiec

doi:10.3390/sym11020258

Abstract

In this paper we investigate an implementation of new model order reduction techniques to linear time-invariant discrete-time commensurate fractional-order state space systems to obtain lower dimensional fractional-order models. Since the models of physical systems correctly approximate the physical phenomena of the modeled systems for restricted time and frequency ranges only, a special attention is given to time- and frequency-limited balanced truncation and frequency-weighted methods. Mathematical formulas for calculation of the time- and frequency-limited, as well as frequency-weighted controllability and observability Gramians, are extended to fractional-order systems. An instructive simulation experiment corroborates the potential of the introduced methodology.

Highlights

We focus on the generalization of such approaches to reduction and an accurate approximation in given frequency and time intervals for the discrete-time commensurate fractional-order systems
Given that the input-to-state map and the state-to-output map of the fractional-order system are modified by the connected weighting functions, it is possible to generalize the definitions of the infinite controllability and observability Gramians for the fractional-order system (Lemma 2) to the frequency-weighted Gramians
This paper presents new results in Balanced Truncation (BT) model order reduction in limited time- and frequencyintervals for discrete-time commensurate fractional-order (DTCFO) systems

Summary

Introduction

In the field of modeling and simulation of fractional-order systems there are two different approaches to application of model order reduction (MOR) techniques: (1) Approximation of fractional-order systems by high integer-order models and their reduction to the low integer-order ones, and (2) reduction of the fractional-order systems without changing the class of the model, i.e., the reduced model is the fractional-order one. The first approach can be implemented by either determination of the fractional-order derivative/difference approximators involved in a fractional-order system [1,2] or by selection of integer-order approximators to the whole fractional-order systems [3,4,5,6,7,8] In both approaches, a very high integer-order model is usually obtained, which is not effective from the computational point of view due to large memory requirements and long simulation times. Likewise, when the reduced model is used to carry out a simulation in the determined time interval, an appropriate approximation accuracy of the output signal y(t) is required only for t lower than a specified final time of simulation For these reasons, the reduction aims to determine such a reduced model which is accurate in the given frequency range [ωmin , ωmax ] and/or time interval [tmin , tmax ].

System Representation

Model Order Reduction

Gramians in the Time Domain

Gramians in the Frequency Domain

Frequency Weighted Gramians

Simulation Examples

Conclusions