Fractional and Kernel Methods with Their Application to Nonlinear Systems

Abstract

Motivated by the recent development in the field of nonlinear filtering, this dissertation presents a new set of nonlinear fractional and kernel algorithms for the solution of bench mark nonlinear problems. Fractional methods have the ability to handle the nonlinear problems more efficiently as compared with the linear methods, because of the fact that the derivative will be taken by a small time step. A new algorithm modified fractional least mean square algorithm is developed that basically is an extension of fractional least mean square algorithm. A scaling factor is introduced that manually adjust the weightage between the least mean square and fractional least mean square parts of the algorithm according to the nature of the problem. The algorithm is then validated on a set of bench mark nonlinear problems. Another algorithm is developed in which all adjustable parameters of the modified fractional least mean square algorithm is made adaptive by using gradient method accordingly by minimizing the mean square error. A method is also introduced adapt the order of fractional derivative. Secondly the formulation of the kernel fractional affine projection algorithm is introduced with the inclusion of Reimann Louisville fractional derivative to Gradient based stochastic Newton recursive method to minimize the cost function of the kernel affine projection algorithm. This approach extends the idea of fractional signal processing in reproducing kernel Hilbert space. The algorithm is then tested on some well known nonlinear problems. A new identification scheme is established for the Hammerstein nonlinear controlled autoregressive( CAR) system using kernel affine projection algorithm. The proposed scheme is validated in comparison with Affine projection algorithm. A new square root extended kernel recursive least squares algorithm is introduced for the unforced dynamical nonlinear state space model. This algorithm utilizes the numerical properties of the matrix computation by the use of an orthonormal triangularization process that is based on numerically stable givens rotation. It gives a considerable reduction of time and computational complexity. The algorithm is illustrated by discussing its application to stationary, as well as non stationary Mackey Glass series and Lorenz series prediction.