Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3102845, 13 pages

http://dx.doi.org/10.1155/2016/3102845

## An Improved Global Harmony Search Algorithm for the Identification of Nonlinear Discrete-Time Systems Based on Volterra Filter Modeling

^{1}School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China^{2}Xuzhou College of Industrial Technology, Xuzhou, Jiangsu 221140, China

Received 6 November 2015; Revised 9 January 2016; Accepted 17 January 2016

Academic Editor: Erik Cuevas

Copyright © 2016 Zongyan Li and Deliang Li. 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

This paper describes an improved global harmony search (IGHS) algorithm for identifying the nonlinear discrete-time systems based on second-order Volterra model. The IGHS is an improved version of the novel global harmony search (NGHS) algorithm, and it makes two significant improvements on the NGHS. First, the genetic mutation operation is modified by combining normal distribution and Cauchy distribution, which enables the IGHS to fully explore and exploit the solution space. Second, an opposition-based learning (OBL) is introduced and modified to improve the quality of harmony vectors. The IGHS algorithm is implemented on two numerical examples, and they are nonlinear discrete-time rational system and the real heat exchanger, respectively. The results of the IGHS are compared with those of the other three methods, and it has been verified to be more effective than the other three methods on solving the above two problems with different input signals and system memory sizes.

#### 1. Introduction

The Volterra model is a kind of nonlinear filter model, which is usually employed to track and identify plenty of complex nonlinear systems. In order to enhance the quality of the system identification, it is very crucial issue to select optimum model coefficient called the kernel. Therefore, the Volterra model is essentially an extension of linear filter model to nonlinear case. During the past decade, there have been many research works on the Volterra model. Campello et al. [1] tackled the problem of expanding Volterra models using Laguerre functions. The global optimal solution is obtained when each multidimensional kernel of the model is decomposed into a number of independent orthonormal bases. Furthermore, the solution obtained is able to minimize the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of functions. Masugi and Takuma [2] described a Volterra system-based nonlinear study of video-packet transmission over IP networks. Based on the Volterra system, the authors performed a time-series analysis of measured data for network response evaluation. The novel method can reproduce the time-series responses observed in video-packet transmission over the Internet, characterizing nonlinear dynamic behaviors such that the obtained results gave an appropriate depiction of network conditions at different times. Gruber et al. [3] presented a nonlinear model predictive control (NMPC) method based on a second-order Volterra series model for greenhouse temperature control using natural ventilation. These models, denoting the simple and logical extension of convolution models, are capable of describing the nonlinear dynamic feature of the ventilation and other environmental conditions on the greenhouse temperature. Many applications of Volterra series modeling were executed in the frequency-domain based on the Generalized Frequency Response Functions (GFRF) [4, 5]. In the light of the wide applications of Volterra series, Li and Billings [6] presented an approach to estimate the GFRF in a piecewise manner for duffing type oscillators for the underrepresented weakly nonlinear region. Additionally, they devised a new method to obtain the energy contributions of each order kernel. The new nonparametric method can not only construct the amplitude-invariant GFRF over a certain excitation range, but also avoid building large sets of time-domain models. In addition, the Volterra filter model can be found in some other application areas [7–14].

Chang [7] devised an improved particle swarm optimization (IPSO) to implement system identification based on Volterra filter model. In this paper, we develop an improved global harmony search (IGHS) algorithm and try the IGHS as an efficient candidate for system identification based on Volterra filter model. The harmony search (HS) algorithm was firstly proposed by [15]. The HS is a simple but efficient algorithm, and its many improved versions have been applied into many problems including reliability problems [16], reactor core fuel management optimization [17], and sizing optimization of truss structures [18].

The paper is organized as follows. In Section 2, the pruned second-order Volterra filter model is simply presented. In Section 3, a novel global harmony search algorithm is introduced. In Section 4, an improved global harmony search algorithm is proposed, and its procedure is fully explained. In Section 5, four harmony search algorithms are used for two examples with different signal inputs and memory sizes. We end this paper with some conclusions in Section 6.

#### 2. Volterra Filter Model and Its Pruned Second-Order Form

The Volterra filter model is an efficient method for the identification of nonlinear discrete systems, and it has come into researchers’ notice in recent decades. The discrete form of Volterra filter model of the th order [7] is given bywhere denotes the system memory size. Equation (1) denotes the Volterra filter model with the infinite series. However, this model is hard to compute and master due to its complex and expatiatory formula. In this paper, we only study its simplified and approximate form called the truncated second-order Volterra model [7, 19], which is stated as follows:

In order to facilitate expression, (2) can also be expressed as the following vector form:

Here, the superscript represents the transpose of a vector and stands for the Volterra kernel vector given by

In addition, denotes the Volterra input vector given by

In the light of (3), the vector lengths of both and are the same and are calculated as follows [19]:

To achieve the nearest approximation of the actual system output, appropriate kernel vector should be determined under the input vector . In this paper, an improved global harmony search (IGHS) algorithm is proposed to determine kernel vector . The IGHS is an improved version of novel global harmony search (NGHS) algorithm [20]; thus, both the NGHS and the IGHS will be presented in the following sections.

#### 3. Novel Global Harmony Search (NGHS) Algorithm

Novel global harmony search (NGHS) algorithm [20] is a variant of harmony search (HS) algorithm [15], and it is superior to the HS for solving unconstrained optimization problems. The NGHS improvises new harmony vectors by combining position updating and mutation. Concretely, the steps of NGHS are explained as follows.

*Step 1 (initialize the NGHS parameters and the problem parameters). *The NGHS parameters consist of harmony memory size , the number of improvisations , and mutation rate . In addition, the problem parameters include the number of problem variables , the lower bound , and upper bound of the th () problem variable. Furthermore, the number of improvisations () is actually the total number of generations for adjusting the parameters related to the identification problem. In each generation, only one new candidate solution including parameters is generated, and this solution is accepted if and only if it is better than the worst one of the previous solutions.

*Step 2 (initialize harmony memory (HM)). *The initial harmony memory () can be expressed in the following matrix form:where stands for the th () variable of the th () harmony vector. Moreover, it is randomly produced from a uniform distribution in the ranges . In HM, any vector () represents a candidate solution of the parameters (as in (4)) needed for solving the identification problem. Furthermore, the length of is equal to , which is exactly the number of the parameters in the Volterra kernel vector (as in (6)).

*Step 3 (improvise a new harmony). *Improvisation is actually the operation of producing a new harmony vector. For the NGHS, its improvisation mainly includes two steps, and they are position updating and genetic mutation, respectively. More specifically, the improvisation can be presented in Table 1.

Here, “best" and “worst” stand for the indexes of the best harmony and the worst harmony in , respectively. , , and denote three uniformly generated random numbers in . With respect to position updating, a new harmony vector is improvised near the best harmony vector, which can facilitate the convergence rate of the NGHS. On the other hand, it is worth noticing that genetic mutation is an event of small probability, and it is utilized to avoid the premature convergence of the NGHS.