In this paper we consider an approximation method of a formal linearization which transform time-varying nonlinear systems into time-varying linear ones and its applications. This linearization is a kind of a coordinate transformation by introducing a linearizing function which consists of the Chebyshev polynomials. The nonlinear time-varying systems are approximately transformed into linear time-varying systems with respect to this linearizing functions using Chebyshev expansion to the state variable and Laguerre expansion to the time variable. As applications, nonlinear observer and filter are synthesized for time-varying nonlinear systems. Numerical experiments are included to demonstrate the validity of the linearization. The results show that the accuracy of the approximation by the linearization improves as the order of the Chebyshev and Laguerre polynomials increases.

In this paper we are concerned with designing an extremum seeking control law for nonlinear systems. This is a modification of a standard extremum seeking controller. It is equipped with an accelerator to the original one aimed at achieving the maximum operating point more rapidly. This accelerator is designed by making use of a polynomial identification of an uncertain output map, the Butterworth filter to smoothen the control, and analog-digital converters. Numerical experiments show how this modified approach can be well in control of the Monod model of bioreactors.

In this paper we propose a formal linearization method which permits us to transform nonlinear systems into linear systems by means of the Chebyshev interpolation. Nonlinear systems are usually represented by nonlinear differential equations. We introduce a linearizing function that consists of a sequence of the Chebyshev polynomials. The nonlinear equations are approximated by the method of Chebyshev interpolation and linearized with respect to the linearizing function. The excellent characteristics of this method are as follows: high accuracy of the approximation, convenient design, simple operation, easy usage of computer, etc. The coefficients of the resulting linear system are obtained by recurrence formula. The paper also have error bounds of this linearization which show that the accuracy of the approximation by the linearization increases as the order of the Chebyshev polynomials increases. A nonlinear filter is synthesized as an application of this method. Numerical computer experiments show that the proposed method is able to linearize a given nonlinear system properly.