**Wavelet Robust Control by Fuzzy Boundary Layer via Time-variant Sliding Surface**

**Majid YARAHMADI **

Faculty of Mathematics, Yazd University,

Yazd, 89195-741, Iran

**Seyed Mehdi KARBASSI**

Faculty of Mathematics, Yazd University

Yazd, 89195-741, Iran

**Ahmad MIRZAEI**

Faculty of Electrical and Computer Engineering, Yazd University

Yazd, 89195-741, Iran

**Abstract**: In this paper, a new wavelet robust control by fuzzy boundary layer via time-variant sliding surface (WRCFBL) for an uncertain nonlinear system is presented. New terminologies, rejection parameter and rejection regulator, for designing a time-variant sliding surface are defined. The time variant sliding surface operates as an adaptive filter. Wavelet network is used to design an indirect controller. An adjustable control gain parameter, rejection parameter and the wavelet network coefficients are on-line tuned. Instead of saturation function a hyperbolic tangent function is used. Also, a fuzzy system that adopts absolute value of sliding surface as input and the boundary layer parameter as output is defined. This fuzzy system tunes the boundary layer width. Control system stability is guaranteed by using the Lyapunov method. The proposed method attenuates efficiently the effects of the system uncertainties and un-modelled frequencies. Also, the chattering phenomenon is completely eliminated. In addition, three theorems and one lemma, which facilitate design of the proposed controller, are proved. Also, a simulation example is presented to illustrate the performances and the advantages of the proposed method.

**Keywords**: Robust control, wavelet networks, time-variant sliding surface, rejection regulator, fuzzy boundary layer.

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**CITE THIS PAPER AS**:

Majid YARAHMADI, Seyed Mehdi KARBASSI, Ahmad MIRZAEI, **Wavelet Robust Control by Fuzzy Boundary Layer via Time-variant Sliding Surface, ***Studies in Informatics and Control*, ISSN 1220-1766, vol. 19 (2), pp. 113-124, 2010.

**1. Introduction**

There are several techniques to control of an uncertain nonlinear system. A typical approach is sliding mode control technique [14]. In the sliding mode technique, the proper transformation of tracking errors to generalize errors is introduced, so that order tracking problem can be transformed into an equivalent first order stabilization problem [16]. The sliding-mode control employs a discontinuous control to derive the system state to reach and maintain its motion on sliding surface. The discontinuity in the control action provides the chattering and the un-modeled frequencies may be activated, which are undesirable in application. To avoid these drawbacks, the boundary layer technique is exploited [14]. For achieving the better tracking performance a varying boundary layer is considered. In [9], the self-tuning laws based on the bounded modeling error, for adjusting the boundary layer width and the other parameters have also been proposed. Furthermore, for calculating the control gain parameter, the difference functions and must be obtained, that is a drawback. The auto-tuning neurons computation for designing the sliding-mode control [3] and the fuzzy adjusting method for finding the suitable boundary layer width [12] are used. Most practical systems are non-linear and complex in nature with uncertain dynamics that may not be easily modeled mathematically. For this purpose, the identification methods are usually exploited [1], [2], [11], [17]. A direct adaptive fuzzy sliding mode control for uncertain nonlinear systems was presented in [13]. The GA-based fuzzy sliding mode controller with modified adaptive laws for robust control of an uncertain nonlinear plant has also been presented [4]. Recently, wavelets have led to advanced tools in many scientific and application research areas [5]. Multiscale analysis, synthesis properties and the learning abilities of neural wavelet networks, for approximation of nonlinear functions are well established [6], [15], [18]. In the literature only time-invariant sliding surface has been studied extensively. Here for the first time, a new case of time-variant sliding equation is presented. For this purpose, the rejection regulator based on a parameter that is called ‘rejection parameter’ is defined.

For objectively choosing the coefficients of error states in sliding equation rejection regulator is used. By tuning the rejection parameter, we can adjust the break frequency bandwidth and also the coefficients of error states in sliding equation. Such sliding equation, as a chain of adaptive first-order low-pass filters, rejects all un-modeled frequencies. The tracking precision is not guaranteed by using the saturation function. Therefore, instead of saturation function, a hyperbolic tangent function is used. Also, a fuzzy system, that adopts absolute value of sliding surface as input and the boundary layer parameter as output, is defined. This fuzzy system tunes the boundary layer width. Control system stability is guaranteed by using the Lyapunov method. For facilitating the design of the proposed controller, three theorems and one lemma are proved. This paper is organized as follows. Problem formulation is presented in section 2. In section 3, sliding mode control with new terminologies as rejection parameter and rejection regulator is presented. Two theorems and one lemma complete this section. Section 4 has two subsections. In the first subsection a wavelet network system is described briefly and a theorem for presenting the tuning laws is explained. This theorem adjusts the wavelet network coefficients, rejection parameter and control gain parameter. In the last subsection of the section 4, structure of a fuzzy system, to design a fuzzy boundary layer, is presented. An example to illustrate the effectiveness of the proposed method is presented in section 5. Finally, the paper is concluded in section 6.

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