Abstract
A motivation for using fuzzy systems stems in part from the fact that they are particularly suitable for processes when the physical systems or qualitative criteria are too complex to model and they have provided an efficient and effective way in the control of complex uncertain nonlinear systems. To realize a fuzzy model-based design for chaotic systems, it is mostly preferred to represent them by T–S fuzzy models. In this paper, a new fuzzy modeling method has been introduced for chaotic systems via the interval type-2 Takagi–Sugeno (IT2 T–S) fuzzy model. An IT2 fuzzy model is proposed to represent a chaotic system subjected to parametric uncertainty, covered by the lower and upper membership functions of the interval type-2 fuzzy sets. Investigating many well-known chaotic systems, it is obvious that nonlinear terms have a single common variable or they depend only on one variable. If it is taken as the premise variable of fuzzy rules and another premise variable is defined subject to parametric uncertainties, a simple IT2 T–S fuzzy dynamical model can be obtained and will represent many well-known chaotic systems. This IT2 T–S fuzzy model can be used for physical application, chaotic synchronization, etc. The proposed approach is numerically applied to the well-known Lorenz system and Rossler system in MATLAB environment.
Highlights
A chaotic system is a highly complex dynamic nonlinear system and its response exhibits an excessive sensitivity to the initial conditions
If it is taken as the premise variable of fuzzy rules and another premise variable is defined subject to parametric uncertainties, a simple interval type-2 Takagi–Sugeno (IT2 T–S) fuzzy dynamical model can be obtained and will represent many well-known chaotic systems
To deal with some problems still existing in control of chaotic systems via type-1 T–S fuzzy models, we propose an IT2 T–S fuzzy model based on sector nonlinearity to represent the chaotic system subject to parametric uncertainty covered by the lower and upper membership functions of the interval type-2 fuzzy sets
Summary
A chaotic system is a highly complex dynamic nonlinear system and its response exhibits an excessive sensitivity to the initial conditions. To deal with some problems still existing in control of chaotic systems via type-1 T–S fuzzy models, we propose an IT2 T–S fuzzy model based on sector nonlinearity to represent the chaotic system subject to parametric uncertainty covered by the lower and upper membership functions of the interval type-2 fuzzy sets. Assumption 2 Suppose that there is a chaotic system with uncertain parameters, i.e. in Eq (7), some elements of DFðxÞ is equal to ekdðtÞ Á xj, where 0\ek\ck is the amplitude of the white noise ekdðtÞ This method is general, for simplicity and less computational effort, only the construction of IT2 T–S fuzzy model for Lorenz system and Rossler system subject to parametric uncertainty will be illustrated, while considering chaotic system subject to uncertain parameter of DFðxÞ as assumption 1 with known lower and upper bounds. Ð11Þ the following type-1 fuzzy rule is employed to describe the Lorenz system: Rule i: IF Z1ðxðtÞÞ is M1i ð12Þ
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