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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 734754, 9 pages
http://dx.doi.org/10.1155/2012/734754
Research Article

Guaranteed Cost Control Design of 4D Lorenz-Stenflo Chaotic System via T-S Fuzzy Approach

Department of Electrical Engineering, Far East University, Tainan 74448, Taiwan

Received 12 January 2012; Accepted 12 March 2012

Academic Editor: Recai Kilic

Copyright © 2012 Yi-You Hou et al. 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 investigates the guaranteed cost control of chaos problem in 4D Lorenz-Stenflo (LS) system via Takagi-Sugeno (T-S) fuzzy method approach. Based on Lyapunov stability theory and linear matrix inequality (LMI) technique, a state feedback controller is proposed to stabilize the 4D Lorenz-Stenflo chaotic system. An illustrative example is provided to verify the validity of the results developed in this paper.

1. Introduction

Chaos phenomenon which is a deterministic nonlinear dynamical system has been generally developed over the past two decades, based on its particular properties, such as broadband noise-like waveform, and depending sensitively on the system’s precise initial conditions, and so forth. Due to its powerful applications in engineering systems, both control and synchronization/stability problems have extensively been studied in the past decades for chaotic systems. Recently, many papers studied the hyperchaotic system, and some dynamical behaviors are studied, such as Chen’s system [1], Lorenz-Stenflo system [2], Josephson junctions [3], cell neural network [4], Lü system [5, 6], and Genesio System [7]. Several control schemes for the stability/synchronization/solution problem of nonlinear systems have been studied extensively, such as backstepping design [8], feedback control [9], adaptive control [10], intermittent control [11], fuzzy model based [12], and multistep differential transform [13]. On the other hand, Takagi-Sugeno (T-S) fuzzy concept was introduced by the pioneering work of Takagi and Sugeno and has been successfully and effectively used in complex nonlinear systems [14]. The main feature of T-S fuzzy model is that a nonlinear system can be approximated by a set of T-S linear models. The overall fuzzy model of complex nonlinear systems is achieved by fuzzy “blending” of the set of T-S linear models. Therefore, the controller design and the stability analysis of nonlinear systems can be analyzed via T-S fuzzy models and the so-called parallel distributed compensation (PDC) scheme [1518].

Inspired by the researches mentioned above, this paper examines the problem of stability for the 4D Lorenz-Stenflo systems. To achieve this goal, based on the Lyapunov stability theory, PDC scheme, and the LMI optimization technique, a controller is derived to guarantee stability of the 4D Lorenz-Stenflo system. Finally, an example is given to illustrate the usefulness of the obtained results.

2. Problem Formulation and Main Results

A 4D Lorenz-Stenflo chaotic system is expressed by the following differential equation [2]:̇𝑥1𝑥(𝑡)=𝑎2(𝑡)𝑥1(𝑡)+𝑏𝑥4(𝑡),̇𝑥2(𝑡)=𝑐𝑥1(𝑡)𝑥1(𝑡)𝑥3(𝑡)𝑥2(𝑡),̇𝑥3(𝑡)=𝑥1(𝑡)𝑥2(𝑡)𝑑𝑥3(𝑡),̇𝑥4(𝑡)=𝑥1(𝑡)𝑎𝑥4(𝑡),(2.1) where 𝑥1, 𝑥2,𝑥3, 𝑥4 are state variables and 𝑎, 𝑏, 𝑐, 𝑑 are called the Prandel number, rotation number, Rayleigh number, and geometric parameter of the system, respectively [2]. To investigate the control design of system (2.5), let the system’s state vector 𝑥(𝑡)=[𝑥1(𝑡)𝑥2(𝑡)𝑥3(𝑡)𝑥4(𝑡)]𝑇 and the control input vector 𝑢(𝑡). Then, the state equations of 4D Lorenz-Stenflo chaotic system (2.1) can be represented as follows:̇𝑥(𝑡)=𝐴(𝑥(𝑡))𝑥(𝑡)+𝐵𝑢(𝑡),(2.2) where 𝐴(𝑥(𝑡))=𝑎𝑎0𝑏𝑐1𝑥1(𝑡)00𝑥1(𝑡)𝑑0100𝑎(2.3) and 𝐵 is known constant matrix with appropriate dimensions.

The aim of this paper is to stabilize 4D Lorenz-Stenflo chaotic systems using T-S fuzzy controller. The continuous fuzzy system was proposed to represent a nonlinear system [14]. The system dynamics (2.2) can be captured by a set of fuzzy rules which characterize local correlations in the state space. Each local dynamic described by the fuzzy IF-THEN rule has the property of linear input-output relation. Based on the T-S fuzzy model concept, a general class of T-S fuzzy 4D Lorenz-Stenflo chaotic systems is considered as follows

Model Rule i
If 𝑧1(𝑡) is 𝑀𝑖1 and 𝑧𝑟(𝑡) is 𝑀𝑖𝑟, then ̇𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐵𝑖𝑢(𝑡),(2.4) where 𝑧1(𝑡),𝑧2(𝑡),,𝑧𝑟(𝑡) are known premise variables, 𝑀𝑖𝑗, 𝑖{1,2,,𝑚}, 𝑗{1,2,,𝑟} is the fuzzy set, and 𝑚 is the number of model rules; 𝑥(𝑡) is the state vector and 𝑢(𝑡)is input vector. The matrices 𝐴𝑖 and 𝐵𝑖 are known constant matrices with appropriate dimensions. Given a pair of (𝑥(𝑡),𝑢(𝑡)), the final outputs of the fuzzy system are inferred as follows: ̇𝑥(𝑡)=𝑚𝑖=1𝑤𝑖𝐴(𝑧(𝑡))𝑖𝑥(𝑡)𝐵𝑖𝑢(𝑡)𝑚𝑖=1𝑤𝑖=(𝑧(𝑡))𝑚𝑖=1𝜂𝑖𝐴(𝑧(𝑡))𝑖𝑥(𝑡)+𝐵𝑖,𝑢(𝑡)(2.5) where𝑧(𝑡)=[𝑧1(𝑡)𝑧2(𝑡)𝑧𝑟(𝑡)], 𝑤𝑖(𝑧(𝑡))=𝑟𝑗=1𝑀𝑖𝑗(𝑧𝑗(𝑡)), 𝜂𝑖(𝑧(𝑡))=𝑤𝑖(𝑧(𝑡))/𝑚𝑖=1𝑤𝑖(𝑧(𝑡)). The term 𝑀𝑖𝑗(𝑧𝑗(𝑡)) is the grade of membership of 𝑧𝑗(𝑡) in 𝑀𝑖𝑗. In this paper, we assume that 𝑤𝑖(𝑧(𝑡))0, 𝑖{1,2,,𝑚}, and 𝑚𝑖=1𝑤𝑖(𝑧(𝑡))>0. Therefore, we have 𝜂𝑖(𝑧(𝑡))0, 𝑖{1,2,,𝑚} and 𝑚𝑖=1𝜂𝑖(𝑧(𝑡))=1, for all 𝑡0.To derive the main results, we first introduce the cost function of system (2.4) as follows: 𝐽=0𝑥𝑇(𝑠)𝑄𝑥(𝑠)+𝑢𝑇(𝑠)𝑅𝑢(𝑠)𝑑𝑠,(2.6) where 𝑄 and 𝑅 are two given positive definite symmetric matrices. Associated with cost function (2.6), the fuzzy guaranteed cost control is defined as follows.

Definition 2.1. Consider the T-S fuzzy system (2.4); if there exist a control law 𝑢(𝑡) and a positive scalar 𝐽 such that the closed-loop system is stable and the value of cost function (2.6) satisfies 𝐽𝐽, then 𝐽 is said to be a guaranteed cost and 𝑢(𝑡) is said to be a guaranteed cost control law for the T-S fuzzy 4D Lorenz-Stenflo chaotic systems (2.4).
This paper aims at designing a guaranteed cost control law for the asymptotic stabilization of the T-S fuzzy 4D Lorenz-Stenflo chaotic systems (2.4). To achieve this control goal, we utilize the concept of PDC [14] scheme and select the fuzzy guaranteed cost controller via state feedback as follows.

Control Rule 𝑗
If 𝑧1(𝑡) is 𝑀𝑗1 and 𝑧𝑟(𝑡) is 𝑀𝑗𝑟, then 𝑢(𝑡)=𝐾𝑗𝑥(𝑡),𝑡0,(2.7) where 𝐾𝑗, 𝑗{1,2,,𝑚} are the state feedback gains. Hence, the overall state feedback control law is represented as follows: 𝑢(𝑡)=𝑚𝑗=1𝜂𝑗(𝑧(𝑡))𝐾𝑗𝑥(𝑡),𝑡0.(2.8) Before proposing the main theorem for determining the feedback gains𝐾𝑗(𝑗=1,2,,𝑚), a lemma is introduced.

Lemma 2.2 (see [19] (Schur complement)). For a given matrix 𝑆=𝑆11𝑆12𝑆𝑇12𝑆22 with 𝑆11=𝑆𝑇11, 𝑆22=𝑆𝑇22, then the following conditions are equivalent:(1)𝑆<0, (2)𝑆22<0, 𝑆11𝑆12𝑆122𝑆𝑇12<0.Now we present an asymptotic stabilization condition for T-S fuzzy 4D Lorenz-Stenflo chaotic systems (2.4).

Theorem 2.3. If there exist some positive definite symmetric matrices 𝑃 and matrices 𝐾𝑗, 𝑗{1,2,,𝑚} such that the following LMI condition holds for all 𝑖,𝑗{1,2,,𝑚}: Φ𝑖𝑗=𝐴𝑖𝑃+𝑃𝐴𝑇𝑖𝐵𝑖𝐾𝑗𝐾𝑇𝑗𝐵𝑇𝑖𝑃𝐾𝑇𝑗𝑄10𝑅1<0.(2.9) Then system (2.4) is asymptotically stabilizable by controller (2.8). The stabilizing feedback control gain is given by 𝐾𝑗=𝐾𝑗𝑃1, and the system performance (2.6) is bounded by 𝐽𝐽=𝑥𝑇(0)𝑃𝑥(0),(2.10) where 𝑃𝑃=1.

Proof. Define the Lyapunov functional: 𝑉(𝑥(𝑡))=𝑥𝑇(𝑡)𝑃𝑥(𝑡),(2.11) where 𝑉(𝑥(𝑡)) is a legitimate Lyapunov functional candidate and 𝑃 is positive definite symmetric matrices. By the system (2.4) with 𝑚𝑖=1𝜂𝑖(𝑧(𝑡))=1, the time derivatives of 𝑉(𝑥(𝑡)), along the trajectories of system (2.4) with (2.6) and (2.8), satisfy ̇𝑉(𝑥(𝑡))𝑚𝑚𝑖=1𝑗=1𝜂𝑖(𝑧(𝑡))𝜂𝑗𝑥(𝑧(𝑡))𝑇(𝑡)𝑄+𝐾𝑇𝑗𝑅𝐾𝑗=𝑥(𝑡)𝑚𝑚𝑖=1𝑗=1𝜂𝑖(𝑧(𝑡))𝜂𝑗𝑥(𝑧(𝑡))𝑇(𝑡)𝑃𝐴𝑖+𝐴𝑇𝑖𝑃𝐾𝑇𝑗𝐵𝑇𝑖𝑃𝑃𝐵𝑖𝐾𝑗𝑄𝐾𝑇𝑗𝑅𝐾𝑗𝑥(𝑡)𝑚𝑚𝑖=1𝑗=1𝜂𝑖(𝑧(𝑡))𝜂𝑗(𝑧(𝑡))𝑥𝑇(𝑡)Φ𝑖𝑗𝑥(𝑡).(2.12) In order to guarantee ̇𝑉(𝑥(𝑡))𝑚𝑖=1𝑚𝑗=1𝜂𝑖(𝑧(𝑡))𝜂𝑗(𝑧(𝑡)){𝑥𝑇(𝑡)(𝑄+𝐾𝑇𝑗𝑅𝐾𝑗)𝑥(𝑡)}<0, we need to satisfy Φ𝑖𝑗<0. By Lemma 2.2 (Schur complement) [19], and premultiplying and postmultiplying the Φ𝑖𝑗 in (2.12) by 𝑃1>0, Φ𝑖𝑗<0 are equivalent to Φ𝑖𝑗<0 in (2.9), then we can obtain the following: ̇𝑉(𝑥(𝑡))𝑚𝑚𝑖=1𝑗=1𝜂𝑖(𝑧(𝑡))𝜂𝑗(𝑧(𝑡))𝑥𝑇(𝑡)𝑄+𝐾𝑇𝑗𝑅𝐾𝑗𝑥𝑥(𝑡)=𝑇(𝑡)𝑄𝑥(𝑡)+𝑢(𝑡)𝑅𝑢(𝑡)<0.(2.13)
From the inequality (2.13), ̇𝑉(𝑥(𝑡))<0, we conclude that system (2.4) with (2.6) is asymptotically stable. Integrating (2.13) from 0 to , we have 0̇𝑉(𝑥(𝑠))𝑑𝑠=lim𝑡𝑉(𝑥(𝑡))𝑉(𝑥(0))0𝑥𝑇(𝑠)𝑄𝑥(𝑠)+𝑢𝑇(𝑠)𝑅𝑢(𝑠)𝑑𝑠.(2.14) Since that the system (2.4) with (2.6) is asymptotically stable, we can obtain the following results: lim𝑡𝑉(𝑥(𝑡))=0.(2.15) Consequently, 𝐽=0[𝑥𝑇(𝑠)𝑄𝑥(𝑠)+𝑢𝑇(𝑠)𝑅𝑢(𝑠)]𝑑𝑠𝑥𝑇(0)𝑃𝑥(0)=𝑉(𝑥(0))=𝐽. This completes the proof.

3. Numerical Simulation and Analysis

In this section, a numerical example is presented to demonstrate and verify the performance of the proposed results. Consider a 4D Lorenz-Stenflo as given in (2.1) with the following parameters [2]: 𝑎=1.0, 𝑏=1.5, 𝑐=26, and 𝑑=0.7.

From the simulation result, we can get that 𝑥1(𝑡) is bounded in interval [77]. By solving the equation, 𝑀1 and 𝑀2 are obtained as follows:𝑀1𝑥1=1(𝑡)2𝑥1+1(𝑡)7,𝑀2𝑥1(𝑡)=1𝑀1𝑥1=1(𝑡)2𝑥11(𝑡)7.(3.1)𝑀1and 𝑀2 can be interpreted as membership functions of fuzzy sets. Using these fuzzy sets, the nonlinear system with time-varying delays can be expressed by the following T-S fuzzy models.

Rule 1. If 𝑥1(𝑡) is 𝑀1, then ̇𝑥(𝑡)=𝐴1𝑥(𝑡)+𝐵1𝑢(𝑡),(3.2)

Rule 2. IF 𝑥1(𝑡) is 𝑀2, then ̇𝑥(𝑡)=𝐴2𝑥(𝑡)+𝐵2𝑢(𝑡),(3.3) where 𝑥𝑥(𝑡)=1(𝑡)𝑥2(𝑡)𝑥3(𝑡)𝑥4(𝑡)𝑇,𝐴1=,𝐴1101.526170070.7010012=1101.526170070.701001,𝐵1=1000,𝐵2=1001.(3.4) By the theorem, the stabilizing fuzzy control gains are given by 𝐾1=𝐾2=[61.3924.8570.1374.026].
Consequently, the minimal guaranteed cost is 𝐽=6.26×1011. The simulation results with initial conditions 𝑥(0)=[0.10.1300.1]𝑇 are shown in Figures 1 and 2. The chaotic attractor of 4D Lorenz-Stenflo system is given in Figure 1. The system state responses trajectory of controller design is shown in Figure 2. When 𝑡=20 sec, it is obvious that the feedback control gain can guarantee stable of 4D Lorenz-Stenflo systems. From the simulation results, it is shown that the proposed controller works well to guarantee stable.

fig1
Figure 1: The chaotic attractor of the 4D Lorenz-Stenflo chaotic system.
fig2
Figure 2: The state responses of the controlled 4D Lorenz-Stenflo chaotic system.

4. Conclusion

This paper has presented the solutions to the guaranteed cost control of chaos problem via the Takagi-Sugeno fuzzy control for 4D Lorenz-Stenflo system. Based on Lyapunov stability theory and LMI technique, the guaranteed cost control gains can be easily obtained through a convex optimization problem. Finally, a numerical example shows the validity and superiority of the developed result.

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