New Results on the Control for a Kind of Uncertain Chaotic Systems Based on Fuzzy Logic

Abstract

In this paper, the problem of the control for an uncertain nonlinear chaotic system has been studied; based on fuzzy logic, a kind of single-dimensional controller is constructed for the control of the chaotic systems in the situation that uncertainties and unknowns exist; at last some typical numerical simulations are carried out, and corresponding results illuminate the effectiveness of the controller.

1. Introduction

Nonlinear systems exist in real engineering widely. Since the pioneering work from Lurie in 1944, the research on nonlinear system control has become the challenging issue, and many techniques, such as differential geometry technique [1, 2], sliding mode technique [3–6] and so on, have been proposed to deal with this problem. It can be noted that these approaches are based on multidimensional control. However, in some cases, the single-dimensional controller is more cherished for its simpler structure and more convenient application in practice.

As an important branch of nonlinear systems, chaotic system and its control received many attentions, and a lot of related results have been reported so far [7–14]. For instance, in [7], based on output feedback control strategy, a method was presented to realize the control for unified chaotic systems; in [8], the synchronization control for Lü systems with unknown parameters was investigated; in [9], the adaptive control for the synchronization of hyperchaotic systems was studied; in [10], the fuzzy control for Arneodo chaotic system is discussed. However most of these researches focused on just one typical chaotic system. In addition, it is well known that there exist many kinds of uncertainties in practical control system, and the following chaotic system model is studied. where , , are the known system parameters and satisfy , where and are the positive scalars, and are the unknown terms, and are the uncertainties, is the known term, is the control parameter, is the system output, and is the single-dimensional control input. A lot of chaotic systems can be transformed into the system with the form (1) through topological mapping.

As an important technique, fuzzy techniques are very suitable for the research of nonlinear and complex systems (see [15–23] and references therein), and they will be introduced to design the single-dimensional controller for system (1) in this paper. Some simulations will be included to illuminate the effectiveness of the constructed controller.

2. Model Description and Preliminaries

It is well known that fuzzy logic system can approximate the nonlinear function. Let denote the smooth function and denote the fuzzy logic system. There exists the optimal parameter for the least approximation error, where and are bounded sets of and x.

Define fuzzy rules as

Define the following fuzzy logic system [16]where , is the fuzzy membership function, .

Let and ; one can get .

Hence, if is the continuous function from a compact set, can approximate , which means that there exist and , such that

where .

In the paper, the following lemmas are concerned.

Lemma 1 (see [24]). If and , one has

3. Main Results

For convenience, let , and .

Step 1. Define the tracking error ; is the desired trajectory.
For the first subsystem of system (1), the virtual variable is introduced, such thatwhere .

Step 2. For the second subsystem of system (1), the virtual variable is introduced, such thatwhere .

Step k (). For k-th subsystem of system (1), the virtual variable is introduced, such thatwhere

Step n. For the n-th subsystem of system (1), one can getwhere

Then, the following tracking error dynamic system can be derivedwhere

The object of this paper is to design a controller, such that

Choose the first Lyapunov function asthenwhere

Let , , where is used to approximate the nonlinear function , then

Choose the second Lyapunov function as

Let , , where is used to approximate the nonlinear function , thenwhere

Let , , where is used to approximate the nonlinear function , then

Choose the k-th Lyapunov function () as

Hencewhere

Let , , where is used to approximate the nonlinear function , then

It is consistent with our notation that , , where is used to approximate the nonlinear function , then

Choose the n-th Lyapunov function as

Hencewhere

Suppose that approximate the nonlinear function and that is based on Lyapunov theory, then the following theoretical result can be obtained.

Theorem 2. For , and , based on the controllerand the adaptive lawthen the output of chaotic system (10) can track the desired trajectory.

Proof. Based on (25), construct Lyapunov function aswhere .
Combined with (28), it can be concluded that DefineSupposing that , it can be derivedHenceConsiderand with adaptive law (29), one can getConsiderThenOne can deriveConsiderThen One can deriveChoosing , one can obtain .
Let ThenThe solution of differential equation isConsidering (44), it can be derived thatDefine From (42), one can obtainHence, when , one can get , which means .
Integrating both sides of inequality (48) from 0 to T, one can getConsider One can getHencewhich means . From error dynamic system (10), it can be concluded that . Accordingly based on Lemma 1, one can get , which means the achievement of the track control. The proof of Theorem 2 is thus completed.

4. Numerical Simulation

First the following uncertain Arneodo system is considered.where

Let desired trajectory , initial value , and the simulation results are displayed in Figures 1–7.

Figure 1 

Chaotic attractor of Arneodo system.

Figure 2 

State response of x₁ of Arneodo system.

Figure 3 

State response of x₁ of uncertain Arneodo system.

Figure 4 

Fuzzy membership function.

Figure 5 

State response of control input.

Figure 6 

State response of y_d and y.

Figure 7 

State response of position tracking error.

Remark 3. Figure 1 displays the chaotic attractor of Arneodo system. Figure 2 displays the state response of x₁ of Arneodo system. From Figures 1 and 2, it can be seen that Arneodo system has the complicated dynamical behavior. Figure 3 displays the state response of variable x₁ of uncertain Arneodo system. It can be seen that the existence of unknowns and uncertainties makes Arneodo system unstable.

Remark 4. Figure 4 displays the fuzzy membership function. Figure 5 displays the state response of control input. Figure 6 displays the state response of y_d and y. Figure 7 displays the state response of position tracking error. From Figures 4–7, it can be seen that for uncertain Arneodo system, the position tracking can be achieved during 0.5 second based on the designed controller.

5. Conclusion

In this paper, based on fuzzy logic, a single-dimensional controller has been constructed for the control of a kind of uncertain chaotic systems. Some typical examples have been employed and corresponding simulation results have illuminated the effectiveness of proposed controller.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work is supported by National Natural Science Foundation of China (11472297, 51475453) and Key Laboratory of Fluid and Power Machinery of the Ministry of Education.