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

Authors: Yi-You Hou; Meei-Ling Hung; Jui-Sheng Lin
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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.


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## Body

## 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 [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.


## 2. Problem Formulation and Main Results

A 4D Lorenz-Stenflo chaotic system is expressed by the following differential equation [2]:(2.1)ẋ1(t)=a(x2(t)-x1(t))+bx4(t),ẋ2(t)=cx1(t)-x1(t)x3(t)-x2(t),ẋ3(t)=x1(t)x2(t)-dx3(t),ẋ4(t)=-x1(t)-ax4(t),
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:(2.2)ẋ(t)=A(x(t))x(t)+Bu(t),
where
(2.3)A(x(t))=[-aa0bc-1-x1(t)00x1(t)-d0-100-a]
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 followsModel Rulei
Ifz1(t) is Mi1 and …zr(t) is Mir, then
(2.4)ẋ(t)=Aix(t)+Biu(t),
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:
(2.5)ẋ(t)=∑i=1mwi(z(t))⋅{Aix(t)Biu(t)}∑i=1mwi(z(t))=∑i=1mηi(z(t))⋅{Aix(t)+Biu(t)},
wherez(t)=[z1(t)z2(t)⋯zr(t)], wi(z(t))=∏j=1rMij(zj(t)), ηi(z(t))=wi(z(t))/∑i=1mwi(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 ∑i=1mwi(z(t))>0. Therefore, we have ηi(z(t))≥0, i∈{1,2,…,m} and ∑i=1mηi(z(t))=1, for all t≥0.To derive the main results, we first introduce the cost function of system (2.4) as follows:
(2.6)J=∫0∞[xT(s)⋅Q⋅x(s)+uT(s)⋅R⋅u(s)]ds,
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 Rulej
Ifz1(t) is Mj1 and …zr(t) is Mjr, then
(2.7)u(t)=-Kjx(t),t≥0,
where Kj, j∈{1,2,…,m} are the state feedback gains. Hence, the overall state feedback control law is represented as follows:
(2.8)u(t)=-∑j=1mηj(z(t))⋅Kjx(t),t≥0.
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 matrixS=[S11S12S12TS22] with S11=S11T, S22=S22T, then the following conditions are equivalent:(1)
S<0,(2)
S22<0, S11-S12S22-1S12T<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 matricesP̂ and matrices K̂j, j∈{1,2,…,m} such that the following LMI condition holds for all i,j∈{1,2,…,m}:
(2.9)Φ̃ij=[AiP̂+P̂AiT-BiK̂j-K̂jTBiTP̂K̂jT*-Q-10**-R-1]<0.
Then system (2.4) is asymptotically stabilizable by controller (2.8). The stabilizing feedback control gain is given by Kj=K̂jP̂-1, and the system performance (2.6) is bounded by
(2.10)J≤J*=xT(0)Px(0),
where P=P̂-1.Proof.
Define the Lyapunov functional:(2.11)V(x(t))=xT(t)Px(t),
where V(x(t)) is a legitimate Lyapunov functional candidate and P is positive definite symmetric matrices. By the system (2.4) with ∑i=1mηi(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)V̇(x(t))-∑i=1m∑j=1mηi(z(t))ηj(z(t)){xT(t)(Q+KjTRKj)x(t)}=∑i=1m∑j=1mηi(z(t))ηj(z(t)){xT(t)(PAi+AiTP-KjTBiTP-PBiKj-Q-KjTRKj)x(t)}≤∑i=1m∑j=1mηi(z(t))ηj(z(t))xT(t)Φijx(t).
In order to guarantee V̇(x(t))-∑i=1m∑j=1mηi(z(t))ηj(z(t)){xT(t)(Q+KjTRKj)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 P-1>0, Φij<0 are equivalent to Φ̃ij<0 in (2.9), then we can obtain the following:
(2.13)V̇(x(t))≤-∑i=1m∑j=1mηi(z(t))ηj(z(t))xT(t)(Q+KjTRKj)x(t)=-(xT(t)⋅Q⋅x(t)+u(t)⋅R⋅u(t))<0.
From the inequality (2.13), V̇(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)∫0∞V̇(x(s))ds=lim⁡t→∞V(x(t))-V(x(0))≤-∫0∞[xT(s)⋅Q⋅x(s)+uT(s)⋅R⋅u(s)]ds.
Since that the system (2.4) with (2.6) is asymptotically stable, we can obtain the following results:
(2.15)lim⁡t→∞V(x(t))=0.
Consequently, J=∫0∞[xT(s)·Q·x(s)+uT(s)·R·u(s)]ds≤xT(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 thatx1(t) is bounded in interval [-77]. By solving the equation, M1 and M2 are obtained as follows:(3.1)M1(x1(t))=12(1+x1(t)7),M2(x1(t))=1-M1(x1(t))=12(1-x1(t)7).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.
Ifx1(t) is M1, then
(3.2)ẋ(t)=A1x(t)+B1u(t),Rule 2.
IFx1(t) is M2, then
(3.3)ẋ(t)=A2x(t)+B2u(t),
where
(3.4)x(t)=[x1(t)x2(t)x3(t)x4(t)]T,A1=[-1101.526-1700-7-0.70-100-1],A2=[-1101.526-1-7007-0.70-100-1],B1=[1000],B2=[1001].
By the theorem, the stabilizing fuzzy control gains are given by K1=K2=[61.3924.857-0.1374.026].
Consequently, the minimal guaranteed cost isJ*=6.26×10-11. 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.The chaotic attractor of the 4D Lorenz-Stenflo chaotic system.
(a)(b)(c)(d)The state responses of the controlled 4D Lorenz-Stenflo chaotic system.
(a)(b)(c)(d)


## 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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