Adaptive filtering using Lyapunov theory and artificial intelligent techniques

Adaptive filtering has gained popularity in numerous applications to help cope with time-variations of system parameters, and to compensate for the lack of a priori knowledge of statistical properties of the input data. Therefore, it is an area of research that has important implications for many problems in signal processing, control and estimation, communication and others. Over the last several decades, a wide range of filter structures and algorithms has been developed. Finite Impulsive Response (FIR) and Infinite Impulse Response (UR) transversal filters are two well-established linear models for adaptive filtering. However, there are several circumstances that the performances of these filters are unsatisfactory. Nonlinear polynomial filtering had been first considered by some researchers. More recently, artificial intelligent techniques such as neural networks and fuzzy logic have undergone rapid development and become recognized as powerful nonlinear approximation methods. Hence various nonlinear adaptive filtering techniques using multi-layer perceptron (MLP), radial basis function (RBF) and fuzzy logic have been developed. Adaptive filtering algorithm is another important topic for adaptive filtering. There are two well-studied algorithms for adaptive filtering: recursive least squares (RLS) and least mean square (LMS) algorithms. LMS algorithm attempts to minimize the mean square of the error signal by employing a stochastic gradient technique. It is strongly dependent on the input signal spectral characteristics and its convergence depends on the eigen-value spread of the autocorrelation matrix. In contrast, several advantages of RLS over LMS in terms of tracking behavior and fast convergence are well known. It is independent of input spectral characteristics but it is of high computational complexity. Furthermore, it exhibits unstable performance. Methods of avoiding instability have been proposed in the literature but the stability problems of adaptive filters have not been solved if there are some bounded input disturbances. This thesis has provided a fundamental breakthrough in understanding of the Lyapunov stability-based adaptive filtering mechanism, yielding further conditions and solutions for a number of nonlinear filtering problems using Lyapunov stability theory. The first issue to be addressed is the mathematical foundation of Lyapunov stability theory for adaptive filtering systems. Linear models such as FIR and IIR transversal filters using Lyapunov stability theory are developed and analyzed. A new insight is given into the stabilization problem of the adaptive filtering algorithm. The developed Lyapunov stability-based adaptive filtering can provide stability and high tracking precision for adaptive filtering systems. It can overcome the low tracking precision and instability problems of conventional adaptive filtering systems. The designs of those adaptive filters are independent of stochastic properties of signals. The analysis and design of the Lyapunov sense adaptive filters are significantly simplified compared to existing conventional filtering algorithms. The successful outcome of the thesis will in no doubt make significant contributions to and impacts on research in the field of intelligent signal processing and communications systems. Further investigations presented in this thesis include the theory and design of RBF neural network-based nonlinear adaptive filters with Lyapunov stability, fuzzy adaptive filters with Lyapunov sense fuzzy rules, neural adaptive filters with the backpropagation learning rules in Lyapunov sense with guaranteed stability, polynomial adaptive filters with Lyapunov stability and parallel signal processing using Lyapunov theory. These new adaptive filtering schemes have been implemented to different applications. Simulation examples have been performed to investigate various performances such as tracking precision, stability, and robustness of the developed schemes. In summary, this thesis has provided an advanced understanding of the Lyapunov stability-based adaptive filtering mechanism. Several adaptive filtering schemes using artificial intelligent techniques and Lyapunov theory have been developed.