Solution of Time-varying Singular Nonlinear Systems by Single-term Walsh Series

Singular nonlinear systems has been of interest to some investigators [4, 12], however no closed-form solution was given in [4, 12]. In some analysis of neural networks, both singular systems [8] and bilinear systems [16] have been used. For singular bilinear systems, Lewis et al. [11] applied the Walsh function (WF) approach for time-invariant singular bilinear systems and Hsiao and Wang [9] used the Haar wavelets for the solution of time-varying singular nonlinear systems. Walsh functions (WFs) have received considerable attention in dealing with various problems of dynamic systems. Chen and Hsiao [5, 6, 7] applied the WF technique to the analysis, optimal control, and synthesis of linear systems. WFs have also found wide applications in signal processing, communication, and pattern recognition [13]. Rao et al. [14] presented a method of extending computation beyond the limit of the initial normal interval in Walsh series analysis of dynamical systems. In [14] various time functions in the system were first expanded in terms of their truncated WF with unknown coefficients. Using the Kronecker product [10], the unknown coefficient of the rate variable was obtained by finding the inverse of a square matrix. It was shown that this method involve some numerical difficulties if the dimension of this matrix is large. To remove the inconveniences in WF technique, the single-term Walsh series (STWS) was introduced in [14], and Balachandran and Murugesan [1, 2, 3] applied STWS technique to the analysis of the linear and nonlinear singular systems. The STWS method provides block-pulse and discrete solutions to any length of time. In the present paper, we use the STWS approach for the solution of time-varying singular nonlinear systems. As compared to [9], our method is simpler and consumes less computer time.