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.

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(b)

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Figure 1 

Memristive Chua system.

Our aim is to estimate the states of (2) and stabilize the origin of the system. To do this, by imposing a controller into the system, one gets where denotes the vector of the states, denotes the vector of the control inputs, , , and

According to [19], we construct the Takagi-Sugeno fuzzy model for (2) as follows:

Rule 1. If is , then

Rule 2. If is , then where is , and is ,

Through a center-average defuzzifier, the overall fuzzy system is represented as where the membership functions are

Suppose that the state variables are not fully measurable and only the flux of the capacitor (i.e., ) is measurable (when choosing other partial states, the results are also applicable), we propose the following fuzzy state observer for fuzzy model (9).

Observer rule : if is , , then where is an estimate of and is the observer gain for the th observer rule.

Note that is applied for the observer rule as it is available from the system (2). Similarly, the overall fuzzy observer is represented as follows:

To stabilize the memristive chaotic system (2), the observer-based fuzzy controller is given by where is the observer rule index, is the control gain for the th controller rule.

The proposed observer-based fuzzy control scheme is depicted in Figure 2. The fuzzy observer estimates the states of the memristive chaotic systems based on the measurable output. Then, the estimated states are utilized in the observer-based fuzzy controller and the output of the controller will be imposed on the chaotic systems.

Figure 2 

Observer-based fuzzy control structure diagram.

We denote the estimation error as . Hence, (9) is equivalent to

By differentiating , one can readily obtain where we used the fact .

Combining with (14)-(15) yields where

Let us define

then the augmented system in (16) can be rewritten as

Proposition 1. For the augmented system (16), if there exists a symmetric positive-definite matrix and a positive scalar such that the following matrix inequality holds for all , then the augmented system (16) is asymptotically stable.

Proof. Consider the Lyapunov function candidate where , is symmetric. After calculating the derivative of along the trajectories of (18), we have where is an arbitrary scalar. Using (19), we get

It is obvious that is positive definite and , and . According to the Lyapunov stability theorem [20], Theorem 5.16, the origin of (16) is globally asymptotically stable. This completes the proof.

Remark 1. It is worth pointing out that this matrix inequality (19) is hardly tractable numerically. Moreover, linearizing the matrix inequality is also a very difficult task due to the presence of the coupling term in .

For the convenience of design, let .

Hence, the matrix inequalities in (19) are equivalent to the following matrix inequalities: where for all .

Since there are no effective algorithms for solving simultaneously, we use the separation method to solve the problem. Note that (23) can be decoupled as follows: where is a positive scalar. It is obvious that if and then (23) holds.

Note that (26) is related to the controller part (the parameters are and ) and (27) is related to the observer part (the parameters are and ), respectively. Now we can determine simultaneously by the following arrangement.

It follows by pre- and postmultiplying the inequality (26) by the matrix where

By the well-known Schur complement [21], the inequality (28) is equivalent to

Based on the above analysis, we can obtain the following observer-based control theorem.

Theorem 1. System (2) is asymptotically stabilizable by (13) if for fixed scalars and , there exist two positive-definite matrices and and two matrices and , so that the following two LMI conditions are feasible: for all , where

Moreover, the stabilizing observer-based control gains are given by , .

According to the analysis above, the design procedure is summarized as follows:

Design procedure:

Step 1. Construct the fuzzy plant rules in (9) and the observer rules in (12).

Step 2. Take a set of positive scalars and iteratively.

Step 3. Solve the following linear matrix inequality problem to obtain (thus , can also be derived).

Step 4. If and cannot be found, try another set of iteratively and repeat Steps 3-4.

Step 5. Construct the fuzzy observer (12).

Step 6. Construct the fuzzy controller (13).

2.2. Memristive System with Disturbances

In this subsection, we will extend the above results to deal with the -gain control problem for the memristive chaotic system with external disturbances.

Consider the following the memristive Chua system with external disturbances where and denotes the vector of the bounded external disturbances.

Applying the same fuzzy observer (12) and the observer-based fuzzy control law (13), we have where

Denote

Then (36) can be expressed as

The objective is to determine a fuzzy controller for the closed-loop system (38) with the following -gain performance as small as possible. Given a disturbance attenuation , we need to achieve where and are some positive scalars.

Following [16], to extend the results in the previous subsection, the -gain control performance in (39) is guaranteed for an attenuated level , and .

The control problem can be transformed to solve the following minimization problem: for all , where

Note that instead of is formulated in the constraint conditions, thus, the term should be modified as follows to solve the minimization problem. It is observed that if , that is, which is equivalent to and

Therefore, we can summarize the theorem for the -gain performance as follows.

Theorem 2. The closed-loop system (38) has the -gain performance if for fixed scalars and there exist two positive-definite matrices and two matrices and , the following minimization problem is solved,