Kalman Filter in Control and Modeling

Originaly, a filter is a physical device for removing unwanted components of mixtures (gas, liquid, solid). In the area of telecommunications, signals are mixtures of different frequencies, and the term of filter is used to describe the attenuation of the unwanted frequencies. Since 1940, the concept of a filter was extended to the separation of signals from noise. With Kalman filter, the meaning of filter is well beyond the notion of separation. It also includes the solution of an inversion problem, in which one knows how to represent the measurable variables as functions of variables of principle interest. Least squares method proposed by Carl Friedrich Gauss in 1795 was the first method for forming an optimal estimate from noisy data, and it provides an important connection between the experimental and theoretical sciences. Before Kalman, Norbert Wiener proposed his famous filter called Wiener filter which was restricted only to stationary scalar signals and noises, the solution obtained by this filter is not recursive and needs the storing of the entire pas observed data. Kalman filter is a generalization of Wiener filter. The significance of this filter is in its ability to accommodate vector signals and noises which may be non stationary. The solution is recursive in that each update estimate of the state is computed from the previous estimate and the new input data, so, contrary to Wiener filter, only the previous estimate requires storage, so Kalman filter eliminate the need for storing the entire pas observed data. In this chapter, we present two important applications of Kalman filter. In the first one we show how this filter can be used as an adaptive controller system (Chafaa et al., 2006). Studies proposed in this part illustrate a structure for the control of a positional system towards a mobile target in a three dimensional space (see Fig.1). In the presence of a random disturbances (white noise) or when few system parameters change, the use of an adaptive and optimal controller turns out necessary (Mudi & Nikhil, 1999; Zdzislaw, 2005). In this case we are choosing to use Kalman filter as a controller. This technique is based on the theory of Kalman's filtering (Kalman, 1960; Eubank, 2006), it transforms Kalman's filter into a Kalman controller. In the second application we give the use of such filter in estimating the membership functions of fuzzy sets in order to obtain a fuzzy model (Chafaa et al., 2007). Fuzzy modelling is an effective tool for the approximation of nonlinear systems. Takagi-Sugeno (TS) model is widely used fuzzy modeling technique (Takagi & Sugeno, 1986; Angelov & Filev, 2004). The TS model utilizes the idea of linearization in a fuzzily defined region of the state space. Due to the fuzzy regions, the nonlinear system is decomposed into a multiSource: Kalman Filter, Book edited by: Vedran Kordić, ISBN 978-953-307-094-0, pp. 390, May 2010, INTECH, Croatia, downloaded from SCIYO.COM