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.


Introduction
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.
The paper is organized as follows: in Section 2 we describe the basic properties of the WF and STWS required for our subsequent development.Section 3 is devoted to the formulation of the time-varying singular nonlinear systems.In Section 4 we apply the proposed numerical method to the time-varying singular nonlinear systems and in Section 5, we report our numerical finding and demonstrate the accuracy of the proposed method.

Walsh functions.
A function f (t), integrable in [0,1), may be approximated using WF as where φ i (t) is the ith WF and f i is the corresponding coefficient.In practice, only the first m terms are considered, where m is an integral power of 2. Then from (2.1), we get where (2.3) The coefficients f i are chosen to minimize the mean integral square error and are given by The integration of the vector Φ(t) defined in (2.3) can be approximated by where E is the m × m operational matrix for integration with E 1×1 = 1/2 and is given in [16].

Single-term Walsh series.
With the STWS approach, in the first interval, the given function is expanded as STWS in the normalized interval τ ∈ [0,1), which corresponds to t ∈ [0,1/m) by defining τ = mt, m being any integer.In STWS, the matrix E in (2.6) becomes E = 1/2.

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Let ẋ(τ) and x(τ) be expanded by STWS series in the first interval as and in the kth interval as Integrating (2.7) with E = 1/2, we get where x(0) is the initial condition.According to Sannuti [15], we have In general, for any interval k, k = 1,2,..., we obtain (2.12) In (2.11) and (2.12), X (k) and x(k) give the block-pulse and the discrete values of the state, respectively.

Problem statement
Consider a time-varying singular nonlinear system of the following form: where the singular matrix E(t)∈R n×n , the nonlinear function f ∈ R n , the state x(t) ∈ R n , and the control u(t) ∈ R q .The response x(t) is required to be found.

Solution of time-varying singular nonlinear systems via STWS
Normalizing (3.1) by defining τ = mt, we get Let E(τ) be expressed by STWS in the kth interval as where E (k) ∈ R n×n .By using (2.8) and (2.11), we get To solve (4.1), we first substitute (4.3) in f (τ,x(τ),u(τ)); we then express the resulting equation by STWS as Using (4.1), (4.2), (4.3), and (4.4), we get By solving (4.5), the components of V (k) can be obtained.By substituting V (k) in (2.11) and (2.12), we obtain block-pulse and discrete approximations of the state, respectively.Further, using (2.7), we get Thus, we can obtain a continuous approximation of the state as

Numerical examples
Three examples are given in this section.These examples were considered by Hsiao and Wang [9] by using Haar wavelets.Our method differs from their approach and thus these examples could be used as a basis for comparison.
The comparison between STWS solution with m = 32 and the exact solution for t ∈ [0,4) is shown in Figure 5.1.
Example 5.3.Consider a time-invariant nonlinear singular system of the following form [9]: The results obtained by STWS with m = 24 and m = 32 and those obtained by Haar wavelets with m = 32 and m = 128 are presented in Tables 5.4 and 5.5, respectively.

Conclusion
The properties of STWS are used to solve the time-varying singular nonlinear systems.
The key idea is to transform the time-varying functions into STWS.The method can be implemented using a digital computer.It occupies less memory space and consumes less computer time than the method in [9].Illustrative examples were included to demonstrate the validity and applicability of the technique.