Continuous neural identifier for uncertain nonlinear systems with time delays in the input signal

Abstract

Time-delay systems have been successfully used to represent the complexity of some dynamic systems. Time-delay is often used for modeling many real systems. Among others, biological and chemical plants have been described using time-delay terms with better results than those models that have not consider them. However, getting those models represented a challenge and sometimes the results were not so satisfactory. Non-parametric modeling offered an alternative to obtain suitable and usable models. Continuous neural networks (CNN) have been considered as a real alternative to provide models over uncertain non-parametric systems. This article introduces the design of a specific class of non-parametric model for uncertain time-delay system based on CNN considering the so-called delayed learning laws analysis. The convergence analysis as well as the learning laws were produced by means of a Lyapunov–Krasovskii functional. Three examples were developed to demonstrate the effectiveness of the modeling process forced by the identifier proposed in this study. The first example was a simple nonlinear model used as benchmark example. The second example regarded the human immunodeficiency virus dynamic behavior is used to show the performance of the suggested non-parametric identifier based on CNN for no fictitious neither academic models. Finally, a third example describing the evolution of hepatitis B virus served to test the identifier presented in this study and was also useful to provide evidence of its superior performance against a non-delayed identifier based on CNN.

Introduction

Time-delays are common in biological and chemical systems. Some processes in biological and chemical systems (Xu et al., 2009, Zhang et al., 2005), such as the gene expression (Chueh & Lu, 2012), course of an infection (Gourley, Kuang, & Nagy, 2008), biosignal response to a stimulus and feedback control in signal transduction networks involve time-delays (Jeong-Woo & Jun-Hoo, 1997).

A time delay in input signal appears in models of real systems due to different reasons. Usually its presence is forced by the physical nature of the system (Hale, 1977). Transport delays (like in chemical or pneumatic systems) or computational delay (e.g. in digital controllers or communication networks (Kruszewski, Jiang, Fridman, Richard, & Toguyeni, 2012)) are regular sources of delayed input signal. Input delay can also be introduced artificially  for including the sampling effect in mathematical models (see, for example,  Fridman, 2010, Fridman et al., 2004, Polyakov, 2012).

For a long time, mathematical modeling has been applied to the study and analysis of this class of real plants. These systems are formally represented with delayed differential equation (DDE) models (Bocharova & Rihan, 2000).

Complicated dynamical behavior arises as a consequence of time-delays in certain systems (Kaifa, Wendi, Haiyan, & Xianning, 2007). Biological and chemical systems with significant time-delays may exhibit limit cycle oscillations and chaotic behavior (Mackey & Glass, 1977). In addition, incorporating time-delays in those models is often essential to capture the whole dynamic behavior for such class of systems. Ignoring the time-delays in biological and chemical models yields to conclusions that do not contain the complete information of the system (Mier-y Terán-Romero, Silber, & Hatzimanikatis, 2010).

Today, the number of time-delay models for biological and chemical systems is growing more and more (Hethcote and Levin, 1989, Kruszewski et al., 2012). However, the complexities associated with these models did not come only by the own nature of the model but also from its uncertainties and complex interaction between the states (Sen, Dodla, Johnston, & Sethia, 2010).

Hence, some alternatives must be proposed to generate adequate representation of such time-delay systems. A feasible option has been the neural network framework to produce a non-parametric model (dynamic system whose trajectories reproduce the same behavior of the uncertain system without the knowledge of the mathematical model) of the uncertain time-delay system (Yoo & Park, 2009). This task can be done using the so-called neural network identifier (NNI) (Poznyak, Sánchez, & Yu, 2001). Nevertheless, this NNI must be designed in such a way that time-delay effect must be considered in its structure.

To obtain a better approach for the class of biological and chemical systems that contain delays, the topology of the NNI should also considered delays. The existence of time-delays on the NNI may cause oscillations and instability (Xua, Lamb, Hoc, & Zoua, 2005). For this reason, the stability of time-delay neural networks (TDNN) has long been investigated (Faydasicok and Arik, 2013, Zhao et al., 2013). Designing of TDNN demanded not only the global stability of the identifier but also the incorporation of some structural properties coming from the uncertain system to be identified. Also, it is often desirable to have a TDNN that converges fast enough to the trajectories of the uncertain system. The exponential stability analysis problem for TDNN can provide such behavior. Indeed, there is a large number of results on this topic (Arik, 2004, Liu and Liao, 2004, Marcus and Westervelt, 1989, Xu et al., 2003).

Most of the literature (Arik, 2000, Cao, 2000, Joy, 2000) made reference to two kinds of TDNN, according to how the delay affect their stability. One is referred to as delay independent stability and the other delay dependent stability (Liao, Chen, & Sanchez, 2002). For this paper, we deal with the delay dependent stability case.

Today, the available results regarding time-delay systems only consider the stability of the TDNN. However, the non-parametric identifier problem has been poorly explored (Ge, Du, Qian, & Liang, 2009). In this sense, the remarkable properties of time-delay stability analysis have been wasted for solving the problem of the non-parametric identifier based on CNN.

Since the last couple of decades, CNN have been deeply investigated for applications in identification, control and state estimation for systems without time-delay (Chairez, 2009, Poznyak et al., 2001). TDNN have been actively studied but they have not been (to the authors knowledge) proposed as identifier of continuous uncertain systems (Arik, 2004, Liao et al., 2002, Souza et al., 2007). This paper introduces a novel structure for TDNN used as identifier for uncertain time-delay nonlinear systems (UTDS). The method for adjusting the weights associated to the TDNN (the so-called learning laws) as well as the convergence regime are based on a special class of Lyapunov–Krasovskii functional. The convergence region is characterized by the same convergence analysis. Three numerical examples were developed to show the performance of the proposed NNI. All the three examples illustrated the convergence performance achieved by the non-parametric identifier introduced in this paper and its superior performance compared to a non-delayed identifier based on CNN.

The paper is organized as follows: The Section  2 presents the class of uncertain time-delay systems to be identified by the TDNN. Section  3 describes the class of TDNN used to construct the TDNN identifier. Section  4 describes all the details regarding the identifier structure including the learning laws and the characterization of the convergence region. Section  5 describes the numerical simulations to test the identifier performance. Finally, the last section of this paper gives some conclusions about the novel identifier developed in this study.