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

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 [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]:𝑥𝑥̇̇₁₂(𝑡)=𝑎𝑥₂(𝑡)−𝑥₁(𝑡)+𝑏𝑥₄(𝑡),𝑥̇₃(𝑡)=𝑐𝑥₁(𝑡)−𝑥₁(𝑡)𝑥₃(𝑡)−𝑥₂(𝑡),𝑥̇₄(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)−𝑑𝑥₃(𝑡),(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:𝑥̇(𝑡)=𝐴(𝑥(𝑡))𝑥(𝑡)+𝐵𝑢(𝑡),(2.2) where 𝐴(𝑥(𝑡))=⎡⎢⎢⎢⎢⎢⎢⎣−𝑐𝑎−𝑎1−𝑥0¹𝑏0𝑥₁(𝑡)0−10(𝑡)−0𝑑−0𝑎⎤⎥⎥⎥⎥⎥⎥⎦(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 𝑀_𝑖1 and …𝑧_𝑟(𝑡) is 𝑀_𝑖𝑟, then 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),(2.4) where 𝑧₁(𝑡),𝑧₂(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: 𝑥̇(𝑡)=∑^𝑚𝑖=1𝑤_𝑖(𝑧(𝑡))⋅𝐴_𝑖𝑥(𝑡)𝐵_𝑖𝑢(𝑡)=∑^𝑚𝑖=1𝑤_𝑖(𝑧(𝑡))_𝑚_𝑖=1𝜂_𝑖(𝑧(𝑡))⋅𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),(2.5) where𝑧(𝑡)=[𝑧₁(𝑡)𝑧₂(𝑡)⋯𝑧_𝑟(𝑡)], 𝑤_𝑖(𝑧(𝑡))=∏^𝑟𝑗=1𝑀_𝑖𝑗(𝑧_𝑗(𝑡)), 𝜂_𝑖(𝑧(𝑡))=𝑤_𝑖(𝑧(𝑡))/∑^𝑚𝑖=1𝑤_𝑖(𝑧(𝑡)). The term 𝑀_𝑖𝑗(𝑧_𝑗(𝑡)) is the grade of membership of 𝑧_𝑗(𝑡) in 𝑀_𝑖𝑗. In this paper, we assume that 𝑤_𝑖(𝑧(𝑡))≥0, 𝑖∈{1,2,…,𝑚}, and ∑^𝑚𝑖=1𝑤_𝑖(𝑧(𝑡))>0. Therefore, we have 𝜂_𝑖(𝑧(𝑡))≥0, 𝑖∈{1,2,…,𝑚} and ∑^𝑚𝑖=1𝜂_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: 𝐽=^∞0𝑥^𝑇(𝑠)⋅𝑄⋅𝑥(𝑠)+𝑢^𝑇(𝑠)⋅𝑅⋅𝑢(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 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 𝑧₁(𝑡) is 𝑀_𝑗1 and …𝑧_𝑟(𝑡) is 𝑀_𝑗𝑟, then 𝑢(𝑡)=−𝐾_𝑗𝑥(𝑡),𝑡≥0,(2.7) where 𝐾_𝑗, 𝑗∈{1,2,…,𝑚} are the state feedback gains. Hence, the overall state feedback control law is represented as follows: 𝑢(𝑡)=−^𝑚_𝑗=1𝜂_𝑗(𝑧(𝑡))⋅𝐾_𝑗𝑥(𝑡),𝑡≥0.(2.8) Before proposing the main theorem for determining the feedback gains𝐾_𝑗(𝑗=1,2,…,𝑚), a lemma is introduced.

Lemma 2.2 (see [19] (Schur complement)). For a given matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22} with 𝑆₁₁=𝑆^𝑇11, 𝑆₂₂=𝑆^𝑇22, then the following conditions are equivalent:(1)𝑆<0, (2)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹²²𝑆^𝑇12<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 𝐾_𝑗, 𝑗∈{1,2,…,𝑚} such that the following LMI condition holds for all 𝑖,𝑗∈{1,2,…,𝑚}: Φ_𝑖𝑗=⎡⎢⎢⎢⎢⎣𝐴^𝑖𝑃+𝑃𝐴^𝑇𝑖−𝐵_𝑖𝐾_𝑗−𝐾^𝑇𝑗𝐵^𝑇𝑖𝑃𝐾^𝑇𝑗∗∗−𝑄∗⁻¹−𝑅0₋₁⎤⎥⎥⎥⎥⎦<0.(2.9) Then system (2.4) is asymptotically stabilizable by controller (2.8). The stabilizing feedback control gain is given by 𝐾_𝑗=𝐾_𝑗𝑃⁻¹, and the system performance (2.6) is bounded by 𝐽≤𝐽^∗=𝑥^𝑇(0)𝑃𝑥(0),(2.10) where 𝑃=𝑃⁻¹.

Proof. Define the Lyapunov functional: 𝑉(𝑥(𝑡))=𝑥^𝑇(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=1𝜂_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 𝑉̇(𝑥(𝑡))−^𝑚_𝑚𝑖=1_𝑗=1=𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑄+𝐾^𝑇𝑗𝑅𝐾_𝑗𝑥(𝑡)_𝑚_𝑚𝑖=1_𝑗=1≤𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑃𝐴_𝑖+𝐴^𝑇𝑖𝑃−𝐾^𝑇𝑗𝐵^𝑇𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑗−𝑄−𝐾^𝑇𝑗𝑅𝐾_𝑗𝑥(𝑡)_𝑚_𝑚𝑖=1_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)Φ_𝑖𝑗𝑥(𝑡).(2.12) In order to guarantee 𝑉̇(𝑥(𝑡))−∑^𝑚𝑖=1∑^𝑚𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡)){𝑥^𝑇(𝑡)(𝑄+𝐾^𝑇𝑗𝑅𝐾_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: 𝑉̇(𝑥(𝑡))≤−^𝑚_𝑚𝑖=1_𝑗=1=−𝑥_𝑇𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑄+𝐾^𝑇𝑗𝑅𝐾_𝑗𝑥(𝑡)(𝑡)⋅𝑄⋅𝑥(𝑡)+𝑢(𝑡)⋅𝑅⋅𝑢(𝑡)<0.(2.13)
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 ^∞0𝑉̇(𝑥(𝑠))𝑑𝑠=lim_𝑡→∞𝑉(𝑥(𝑡))−𝑉(𝑥(0))≤−^∞0𝑥^𝑇(𝑠)⋅𝑄⋅𝑥(𝑠)+𝑢^𝑇(𝑠)⋅𝑅⋅𝑢(𝑠)𝑑𝑠.(2.14) Since that the system (2.4) with (2.6) is asymptotically stable, we can obtain the following results: lim_𝑡→∞𝑉(𝑥(𝑡))=0.(2.15) Consequently, 𝐽=∫^∞0[𝑥^𝑇(𝑠)⋅𝑄⋅𝑥(𝑠)+𝑢^𝑇(𝑠)⋅𝑅⋅𝑢(𝑠)]𝑑𝑠≤𝑥^𝑇(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:𝑀₁𝑥₁(𝑡)=121+𝑥¹(𝑡)7,𝑀²𝑥₁(𝑡)=1−𝑀₁𝑥₁(𝑡)=121−𝑥¹(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 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐵₁𝑢(𝑡),(3.2)

Rule 2. IF 𝑥₁(𝑡) is 𝑀₂, then 𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝐵₂𝑢(𝑡),(3.3) where 𝑥(𝑡)=𝐴𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)𝑥₄(𝑡)^𝑇,𝐴₁=⎡⎢⎢⎢⎢⎢⎢⎣−−20611−−1071−0700.71−00.1₂5⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣−−20611−1071−−0007.71−00.15⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₁=⎡⎢⎢⎢⎢⎢⎢⎣1000⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₂=⎡⎢⎢⎢⎢⎢⎢⎣1001⎤⎥⎥⎥⎥⎥⎥⎦.(3.4) By the theorem, the stabilizing fuzzy control gains are given by 𝐾₁=𝐾₂=[61.3924.857−0.1374.026].
Consequently, the minimal guaranteed cost is 𝐽^∗=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.

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Figure 1 

The chaotic attractor of the 4D Lorenz-Stenflo chaotic system.

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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.