Complexity

Volume 2018, Article ID 8731048, 10 pages

https://doi.org/10.1155/2018/8731048

## Observer-Based Fuzzy Control for Memristive Circuit Systems

^{1}Wuxi Institute of Technology, Wuxi 214121, China^{2}Institute of System Engineering, Jiangnan University, Wuxi 214122, China

Correspondence should be addressed to Qian Ye; moc.621@eynaiqq

Received 1 April 2018; Revised 27 May 2018; Accepted 13 June 2018; Published 24 September 2018

Academic Editor: Jing Na

Copyright © 2018 Qian Ye and Xuyang Lou. 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 proposes an observer-based fuzzy control scheme for a class of memristive chaotic circuit systems. First, the Takagi-Sugeno fuzzy model is adopted to reconstruct the nonlinear chaotic circuit system. Next, based on the proposed fuzzy model, an observer-based fuzzy controller is developed to estimate the states and stabilize the origin. Third, the results are extended to explore the -gain observer-based fuzzy control for the chaotic system with disturbances. Finally, simulation results are also addressed to show the effectiveness of the proposed control scheme.

#### 1. Introduction

In 1971, Chua postulated the existence of a fourth circuit element [1], called memristor, which was realized by Williams’s group of HP Labs only 37 years later [2]. In recent years, the memristor has attracted much attention due to its potential application in associative memory [3], image processing [4], filter [5], programmable analog circuits [6], and so on. In particular, Pershin and Ventra experimentally demonstrated the formation of associative memory in a simple neural network, which consists of three electronic neurons using memristor-emulator synapses [3]. A new image encryption algorithm was presented in [4] based on chaos with the piecewise-linear memristor in Chua’s circuit. The authors in [5] experimentally demonstrated an adaptive filter by introducing a memristor and using the memristive properties of vanadium dioxide. In [6], a memristor was designed for a pulse-programmable midband differential gain amplifier with fine resolution.

The HP memristor is described by a nonlinear constitutive relation as introduced in [7]
between the device terminal voltage and terminal current where , and , where and are the memristance and memductance. Memristor-based systems may exhibit complex behaviors, such as chaotic and hyperchaotic dynamics. Recently, chaos control, hybrid control, and synchronization of memristor-based or memristive chaotic systems have received intensive investigation [8–14]. However, the “piecewise-linear” nonlinearity characterization by introducing a memristor may lead to challenges in dealing with the chaotic systems. Fuzzy modeling approaches result in a way that the original systems can be decomposed into a number of linear subsystems. In the context of the Takagi-Sugeno fuzzy models, Zhong et al*.* [8] addressed fuzzy modeling and impulsive control of the memristor-based Chua chaotic system. Cafagna and Grassi presented a novel fractional-order memristor-based chaotic system and carried out the theoretical analysis of the system dynamics [9]. The Takagi-Sugeno fuzzy method emerged as a promising approach for approximating nonlinear systems [8, 15, 16]. More recently, a new fuzzy model of the memristor-based Lorenz circuit, which was employed to synchronize with the memristor-based Chua circuit, was explored in [14].

Despite the rich achievements, most of the above results mainly focused on stability or synchronization of memristive chaotic circuit systems rather than state estimation [17, 18]. However, in real chaotic circuits, it is often the case that only partial information about the states (for instance, voltage) is available in the system outputs. Therefore, in order to utilize the memristive chaotic circuit systems, one often needs to estimate the system state through available measurement, and then use the estimated system to achieve synchronization, optimal control [19], or tracking performances. In addition, general results on state estimation and observer-based control for such memristive systems do not seem to have received much attention so far. To the best of our knowledge, there is a lack of effort in the observer-based control and synchronization of memristive systems [18].

Inspired by [16], this paper aims at investigating the observer-based fuzzy control for the stabilization of the memristive Chua circuit systems with or without external disturbances. An observer-based fuzzy control scheme based on the Takagi-Sugeno fuzzy model of the Chua systems is proposed. The controller design based on linear matrix inequality (LMI) conditions is developed. The results are extended to explore the -gain control problem for the chaotic system with disturbances using the observer-based fuzzy control approach. In addition, the nonlinear -gain control problem is transformed into a suboptimal control problem, that is, to minimize the upper bound of the -gain of the closed-loop system subject to LMI constraints.

*Notations. * denotes the -dimensional Euclidian space. Given a vector , denotes its transpose. denotes the Euclidean norm. For a function , and represent the transpose of matrix and the inverse of matrix , respectively. We use to denote a positive- (negative-) definite matrix , and (resp., ) denotes the identity matrix (resp., zero matrix) of appropriate dimension. denotes a block diagonal matrix. The symbol “” within a matrix represents the symmetric term of the matrix. and represent the maximum and minimum eigenvalue of the real symmetric matrix , respectively. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

#### 2. Modeling and Control of Memristive Chua Systems

##### 2.1. Memristive System without Disturbances

Consider the memristive Chua circuit system [7]: with the output , and

When taking the parameters , and , (2) exhibits the chaotic behavior in [8] as shown in Figure 1.