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

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.


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 15-18 .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.

Problem Formulation and Main Results
A 4D Lorenz-Stenflo chaotic system is expressed by the following differential equation 2 : where x 1 , x 2 , x 3 , x 4 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 x 1 t x 2 t x 3 t x 4 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: where 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.

Model Rule i
If where z 1 t , z 2 t , . . ., z r t are known premise variables, M ij , 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 A i and B i 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: In this paper, we assume that w i z t ≥ 0, i ∈ {1, 2, . . ., m}, and m i 1 w i z t > 0. Therefore, we have η i z t ≥ 0, i ∈ {1, 2, . . ., m} and m i 1 η i z t 1, for all t ≥ 0.To derive the main results, we first introduce the cost function of system 2.4 as follows: 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 J ≤ J * , 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 z 1 t is M j1 and . . .z r t is M jr , then where K j , j ∈ {1, 2, . . ., m} are the state feedback gains.Hence, the overall state feedback control law is represented as follows: Then system 2.4 is asymptotically stabilizable by controller 2.8 .The stabilizing feedback control gain is given by K j K j P −1 , and the system performance 2.6 is bounded by
Proof.Define the Lyapunov functional: where V x t is a legitimate Lyapunov functional candidate and P is positive definite symmetric matrices.By the system 2.4 with m i 1 η i z t 1, the time derivatives of V x t , along the trajectories of system 2.4 with 2.6 and 2.8 , satisfy

2.12
In order to guarantee V x t − m i 1 m j 1 η i z t η j z t {x T t Q K T j RK j x t } < 0, we need to satisfy Φ ij < 0. By Lemma 2.2 Schur complement 19 , and premultiplying and postmultiplying the Φ ij in 2.12 by P −1 > 0, Φ ij < 0 are equivalent to Φ ij < 0 in 2.9 , then we can obtain the following:

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

2.14
Since that the system 2.4 with 2.6 is asymptotically stable, we can obtain the following results: This completes the proof.

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 x 1 t is bounded in interval −7 7 .By solving the equation, M 1 and M 2 are obtained as follows: 3.1 M 1 and M 2 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.where

3.4
By the theorem, the stabilizing fuzzy control gains are given by K 1 K 2 61.392 4.857 − 0.137 4.026 .
Consequently, the minimal guaranteed cost is J * 6.26 × 10 −11 .The simulation results with initial conditions x 0 0.1 0.1 30 0.1 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.

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.
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

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