Computational Intelligence and Neuroscience

Volume 2015, Article ID 615079, 9 pages

http://dx.doi.org/10.1155/2015/615079

## A Robust Computational Technique for Model Order Reduction of Two-Time-Scale Discrete Systems via Genetic Algorithms

^{1}Department of Electrical Engineering, The University of Jordan, Amman 11942, Jordan^{2}Department of Mechatronics Engineering, The University of Jordan, Amman 11942, Jordan

Received 17 November 2014; Accepted 26 February 2015

Academic Editor: Saeid Sanei

Copyright © 2015 Othman M. K. Alsmadi and Zaer S. Abo-Hammour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A robust computational technique for model order reduction (MOR) of multi-time-scale discrete systems (single input single output (SISO) and multi-input multioutput (MIMO)) is presented in this paper. This work is motivated by the singular perturbation of multi-time-scale systems where some specific dynamics may not have significant influence on the overall system behavior. The new approach is proposed using genetic algorithms (GA) with the advantage of obtaining a reduced order model, maintaining the exact dominant dynamics in the reduced order, and minimizing the steady state error. The reduction process is performed by obtaining an upper triangular transformed matrix of the system state matrix defined in state space representation along with the elements of *B*, *C*, and *D* matrices. The GA computational procedure is based on maximizing the fitness function corresponding to the response deviation between the full and reduced order models. The proposed computational intelligence MOR method is compared to recently published work on MOR techniques where simulation results show the potential and advantages of the new approach.

#### 1. Introduction

Model order reduction (MOR) of multi-time-scale systems has been an important subject area in control engineering for many years [1, 2]. In many industrial control systems, simple controllers are preferable. However, derivation of the mathematical model often leads to detailed description of a complex model in the form of high order differential equations [2]. Due to this point of view along with other different design objectives, model order reduction has been an active research area in the control society since the 1960s where a large number of model order reduction methods have been introduced in literature for single input single output (SISO) as well as MIMO type systems. The reduction operation is to search for the possibility of finding some lower order equations of the same type that may be considered to adequately reflect the dominant characteristics of the original system. The objective of simplification is to obtain a low order model of the existing high order model such that both are equivalent in terms of system response and being close to each other in some physical representation means. Model reduction problems have attracted much attention in recent years; for example, the model reduction problem has been investigated using artificial neural networks [3], genetic algorithms [4], and invasive weed optimization [5]. It was also used in nonlinear systems [6], gain scheduling [7], linear time-varying systems [8, 9], and linear parameter-varying systems [10].

To obtain a model of lower order, a significant number of methods have been proposed in recent and earlier years, some for continuous time systems [3–5] and some for discrete-time systems [1, 11–15]. Some methods, such as model order reduction by matching Markov parameters [16], were introduced to ensure stability of the reduced order model. A popular technique for obtaining reduced order models is the Krylov subspace [17]; however, stability of the reduced model is not guaranteed. Another important group of reduction algorithm is the eigenvalue preservation technique [3–5, 11] where important eigenvalues of the system are retained to find suitable lower order models.

A numerous number of MOR methods are available for continuous systems, but very few have been devoted to the discrete-time systems MOR. The discrete-time system MOR may be performed in two different ways. The first one is performed based on transforming a continuous time model into another form using different types of transformation as seen in [6, 18]. In this group of MOR, the reduction process is completely performed in the continuous time form. The discrete reduced order model is then obtained by the corresponding inverse transformation of the continuous time reduced model. The second method for obtaining a discrete reduced order model, which is known as a direct method [14], is deriving the discrete reduced order model directly without using any type of transformation. Some of these methods perform the MOR using canonical expansion of -transfer function and stable optimal methods [13, 19], power decomposition and system identification [20], and multipoint step response matching [21]. New optimization techniques, particle swarm optimization [22], and artificial neural networks [11] have also been introduced for MOR of discrete-time systems.

GA-based MOR, on the other hand, has received some of the researchers’ attention as well. Recently, Ponda et al. [23] employed a particular swarm optimization technique to obtain a reduced order model of SISO large scale linear systems. Their technique is based on the integral square error (ISE). Vishwakarma and Prasad [24] proposed a mixed method for reducing the order of large-scale linear systems. They have synthesized the denominator of the reduced order transfer function using modified pole clustering while the coefficients of the numerator elements are computed using GA. Parmar et al. [25] presented a technique for model order reduction using GA for SISO linear time systems. They have focused on obtaining a reduced order model that maintains stability and retains the steady state value. In spite of the methods available in literature, each method has advantages and disadvantages when tried on a particular system. In addition to that, no approach always gives the best results for all systems. It is important to mention that GAs have also been used for model system identification, where order and parameters are set to be determined, as we have investigated in [26, 27]. In this paper, however, and as motivated by the singular perturbation method which has the characterization of multi-time-scale systems, the GA procedure is performed with the advantages of retaining the exact dominant dynamics in the designed model, obtaining a new robust model with a lower order, and maintaining a minimum steady state response error.

The work in this paper is organized as follows: Section 2 presents problem formulation of the discrete full and reduced order models. In Section 3, the genetic algorithm approach for MOR of multi-time-scale discrete systems is presented. Illustrative examples utilizing the new approach along with simulation comparative results of different MOR techniques are presented in Section 4. Section 5 presents an overall conclusion of the proposed MOR method.

#### 2. Problem Formulation

In this paper, MOR is investigated for discrete LTI systems of both SISO and MIMO type models. For SISO systems, a transfer function model is used, while the state space representation is used for MIMO systems.

For the SISO systems, consider the discrete-time system described by where is the input and is the output of the system at the th sampling instant. Equation (1) can be written in the form of a pulse transfer function aswith . The characteristic polynomial contains the system dominant and nondominant poles (distinct, repeated, or complex) where their number, , is referred to as the model order. The corresponding desired reduced th order model is given bywhere some of the coefficients and may be zeros as long as . For the MIMO systems, consider the following th order discrete-time system:where is the time index, is the state vector, and are the input and output vectors, respectively, and , , , are matrices of appropriate dimensions with , , and being the system order, number of inputs, and number of outputs, respectively. The corresponding desired reduced th order model is obtained as follows: where is -state vector, is the reduced order model output, and , , , and are matrices with appropriate dimensions.

#### 3. Genetic Algorithms with MOR

GAs are based on principles inspired from the genetic and evolution mechanisms observed in natural systems. Their basic principle is the maintenance of a population of solutions to the problem that evolves in time. They are based on the triangle of genetic reproduction, evaluation, and selection [12]. Genetic reproduction is performed by means of two basic genetic operators: crossover and mutation. Evaluation is performed by means of the fitness function that depends on the specific problem. Selection is the mechanism that selects parent individuals with probability proportional to their relative fitness.

In this paper, using the computational intelligence of GA, we will obtain the reduced order model based on only the dominant dynamics of the system. For dynamic decoupling, the GA will set the reduced order model state matrix in the modal form where all of the selected eigenvalues (system dynamics), real and/or complex, are placed on the diagonal. Thus, the reduced order model state matrix , in (6), is designed to have the following decoupling format:where the original system dominant poles (real and/or complex) are preserved in the diagonal, seen as , (real) and , (complex). Notice that, for this reduced order model, . To insure that the dominant poles are preserved in the reduced order model and for further order reduction, the following condition is satisfied:Take into account that if , then (9) is to be redefined accordingly if necessary. For simplicity, the modal form is chosen, which implies that all elements seen in (8) as ( and ) are set to zero.

The GA will determine the parameters of the and (and if necessary) in (6) and (7). Hence, the total number of elements that the GA will need to find is given by

It is to be noted that all of the parameters that the GA will have to find are restricted to be real values. Now, based on the number of unknown parameters (), the GA creates a population of individuals, where each parameter is basically an individual in this population. The population consists of different “sets” of solutions. Each solution set is called a chromosome, which contains individuals. Given a population size (), a matrix consisting of rows is formed with each row containing one set of solutions for the unknown parameter values. This would result in a matrix containing × elements. Each element in this matrix contains a value pertaining to one unknown parameter, where each row presents one set of solutions.

The genetic algorithm used in this work will operate as follows.

##### 3.1. Initialization

An initial population comprising individuals is randomly generated. The GA type used in this paper is a binary genetic algorithm, where each value in the solution set consists of a number of bits (genes). The number of bits used to encode each numerical value depends on three variables, lower parameter bound (), upper parameter bound (), and accuracy (). Hence, the number of bits used is defined as follows [30]:Given that each parameter value will consist of number of bits, each solution set will consist of bits. This represents one row of the entire × matrix.

The GA starts by randomly initializing a binary matrix with rows and columns. Each row (set of solutions) is made up of multiple values decoded into binary and placed next to each other as illustrated in Figure 1 for one chromosome. This chromosome consists of three parameters (individuals) with each individual being made of three genes. These genes (bits) can be later decoded back into decimal values, which in return provide the desired parameters’ values.