Volume 14 Issue 1 - May 21, 2010 PDF
Controlling a Time-Varying Unified Chaotic System via Interval Type 2 Fuzzy Sliding-Mode Technique
Tzuu-Hseng S. Li1,*, Ming-Ying Hsiao2, Jia-Zhen Lee1, Shun-Hung Tsai3
1Department of Electrical Engineering, National Cheng Kung University, Tainan, 70101, Taiwan
2Department of Electrical Engineering, Fortune Institute of Technology, Kaohsiung 83160, Taiwan
3Graduate Institute of Automation Technology, National Taipei University of Technology Taipei, Taipei 10608, Taiwan
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Chaos in control systems and controlling chaos in dynamical systems have been intensively studied over the past two decades. Nowadays, various techniques and methods have been proposed to achieve controlling chaos, such as synchronization of different chaotic system, adaptive control methods, back-stepping design techniques, and fuzzy sliding-mode control methods. In this study, we propose an interval type-2 fuzzy sliding mode control (IT2-FSMC) to control a time-varying unified chaotic system. This technique is a combination of the interval type-2 fuzzy logical control (IT2-FLC) method and the sliding mode control (SMC) approach and inherits the advantages of these two methods. The main contributions of this paper are: i) the decoupled IT2-FSMC is proposed for controlling the time-varying unified chaotic system; ii) the asymptotical stability can be achieved in the sense of Lyapunov stability criterion.

Consider a unified chaotic system which unites the classical Lorenz chaotic system, the Lü chaotic system, and the Chen’s chaotic system, described below


where α∈[0,1]. The system (1) is the Lorenz system for α∈[0,0.8), the Chen system for α∈(0.8,1], and the Lü system for α=0.8
The controlled unified chaotic system can be expressed as


where are the controllers to be designed. The tracking problem of (2) is to find a suitable control law to let the states follow a desired trajectory asymptotically. The tracking error of the state vector is defined as


where . Since each of the subsystems is controlled by an individual control variable ui , we may design 3 decoupled sliding surfaces for the controllers as:


where . Since (4) is a Hurwitz polynomial, maintaining system states on will result in approaching zero with the initial condition . Consequently, the tracking control is fulfilled.

For each subsystem, a suitable control rule ui has to be found so as to keep the error on the sliding surface . To achieve this purpose, a positive Lyapunov function V is defined as


A sufficient condition for the stability of the system is given by


If we choose η>0, equation (6) states that the system is driven to the sliding surface as the absolute distance from is diminishing.

If the state trajectory has reached the sliding surface , then from (4) one obtains which implies that


An interval type-2 fuzzy set (IT2-FS) can be expressed as


The IT2-FLC contains the following five components: fuzzifier, rule base, fuzzy inference engine, type-reducer, and defuzzifier.

The fuzzifier maps crisp inputs into IT2-FSs. Here, the Gaussian shape MF with uncertain mean is chosen, and then the upper bound of the primary MF with uncertain mean can be expressed as


and the corresponding lower bound MF is


where [m1,m2] is the span of the uncertain mean and σ is the standard deviation.

The rule base in the IT2-FLC is also characterized by IF–THEN rules. We consider an IT2-FLC as having two inputs, and , and a single output, , then, the j-th rule of the IT2FLC for the control input can be written as


Here we select the diagonal-type rule table due to the similarity between the diagonal type FLC and SMC, then one can redefine the diagonal in terms of an SMC with boundary layer, and verify its stability and robustness.

The inference engine combines all the fired rules and gives a non-linear mapping from the input IT2-FS to the output IT2-FS. The fuzzy rule in (12) can be written as


which results in an interval set described by their associated upper bound and lower bound for the j-th rule of control input


The type-reducer is an extension version of the type-1 defuzzifier by applying the Extension Principle to a specific defuzzification method. The COS type-reduction is adopted here and the resulting type-reduced set can be expressed as


where Ucos(e) is an interval output set determined by its left-most point ul and its right-most point ur. In order to compute ul and ur, the Karnik–Mendel iterative procedure  is needed.

We defuzzify the interval set by using the average of ul and ur; hence, the defuzzified crisp output becomes


The main advantage of the FSMC is that the number of rules can be reduced dramatically in comparison with the traditional FLC. Now, the decoupled sliding function is the only input of the proposed IT2FSMC and the j-th rule for i-th control input can be described as


The outputs of the IT2FSMC uifsmc(si), i=1,2,3 satisfy the following condition


which means that the longer the distance of s is, the larger amplitude the control action will have.
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