Research Article
A Novel and Efficient Computational Algorithm of STWS for Generalized Linear Non-Singular/Singular Time Varying Systems
Department of Mathematics, National Institute of Technology, Tiruchirapalli-620 015, Tarnilnadu, India
A good amount of work has been done in recent years on the application of Walsh Functions (WFs) to various problems such as non-linear dynamic systems, design and analysis of time varying systems and optimization and identification of systems. Chen and Shih (1978) have applied Walsh series analysis to the optimal control of time varying linear systems. Sivaramakrishnan and Srisailam (1985) have presented the WFs method to obtain the solution of second order state space systems. Sheih et al. (1978) have obtained the solution of state space equations using Block Pulse Function (BPF) techniques. Rao et al. (1980) and Rao (1983) have introduced the Single Term Walsh Series (STWS) approach to remove inconveniences of the WF and BPF methods. Campbell and Rose (1982) have shown that a singular second order system may be decomposed into a linear and a singular subsystem. Using the single term Walsh series method, vibrating mechanical system has been analyzed by Palanisamy and Arunachalam (1987). Thanushkodi et al. (1988) have solved first and third order linear differential equations with constant coefficients, by second order state space formulation via STWS. In many investigations (Palanisamy and Arunachalam, 1987; Thanushkodi et al., 1988; Thanushkodi et al., 1990), second (or third) order state space equations are transformed into their equivalent first (or second) order state space equations while analyzing second (or third) order state space equations. Also, Murugesan et al. (2000) converted the observer design in a generalized state-space transistor circuit into a second order system. By doing this, the sizes of the original matrices (rxr) are doubled (2rx2r) and the matrix to be inverted has its size drastically enhanced which implied that the technique mentioned above is not computationally effective and consequently demands large computing time and storage space as well.
STWS method is being adapted by many investigators to compute numerical solutions of industrially applicable problems. Recently, Sepehrian and Razzaghi (2005) have used STWS method to solve the non-linear Volterra-Hammerstein integral equations in which the problem has been reduced to solving a system of non-linear algebraic equations. They have demonstrated the validity and applicability of the technique with the help of several examples. A general and efficient pole-placement solution for linear time-invariant systems with state-PID feedback is derived (Guo et al., 2006). Their results open a new area for the design and tuning of state-PID feedback types of controller. Boukas (2006) has dealt with the class of linear discrete-time systems with varying time delay. The problems of stability and stabilizability for this class of systems are discissed. A numerical comparison between the variational iteration method and the Adomain decomposition method for solving linear systems of fractional differential equations has been presented by Momani and Odibat (2007).
It is of immense importance to point out that suppose the problem of 2nth order state-space system with r unknown dependent variables is solved by the method of second-order state space formulation via STWS developed by Thanushkodi et al. (1988) and Murugesan et al. (2000), then the size of the matrix to be inverted becomes (nrxnr). To have a method that is computationally effective, a modest effort has been made to develop a new numerical algorithm that computes numerical solution of state space system of any order.
WALSH SERIES
A function f(t) integrable in [0,1) can be approximated using Walsh series as
(1) |
Where, Ψi(t) is ith Walsh function and fi is the corresponding coefficient. In practice, while approximating a function only the first k components are considered where k is an integral power of 2. If the coefficients of the Walsh functions are concisely written as k vectors then
(2) |
Where, T denotes transpose and
(3) |
Then,
(4) |
The coefficients fi are chosen to minimize the mean integral square error,
(5) |
The coefficients are given by
(6) |
It has been proved that
(7) |
Where, E is called the operational matrix of integration in terms of WFs. In STWS, E = 1/2 (Rao et al., 1980).
NUMERICAL ALGORITHM FOR GENERAL1ZED LINEAR STATE SPACE ENGINEERING SYSTEM
Let us consider a generalized (2nth order) non-singular/singular time varying system. The dynamics of this system is described by the 2nth order state space equation as
(8) |
Where, x(t) is an r-state vector, A is an either non-singular or singular rxr matrix, B1, B2, , B2n, are r x r matrices, C is an r x m matrix, u(t) is an m-input vector, x(1)(t) is the first derivative of x(t)(= (t)),x(2) (t) is the second derivative of x (t) (= (t)), , (t)) and x(2n)(t) is the 2nth derivative of x(t). The initial conditions are
(9) |
It is of importance to note that for a given value of 2n and r, systems of the form in Eq. 8 along with Eq. 9 cover a variety of cases in many areas of science and technology such as electrical circuits, network theory, power systems, nuclear reactor plants, robotics, aircraft systems, neurological events, etc.With the STWS approach, the given function is expanded into single term Walsh series in the normalized interval τ ∈ (0, 1), which corresponds to τ ∈ (0, 1/k) by defining t = τ/k, k being any integer. In the normalized interval, Eq. 8 becomes
(10) |
Expanding x(2n)(τ), x(2n-1)(τ), ..., x(1)(τ), x(τ) and u(τ) by STWS at the ith interval
(11) |
R2n(i) is the block pulse value of the 2nth order rate vector, R2n-1(i) is the block pulse value of the (2n-1)th order rate vector, , R1(i) is the block pulse value of the first order rate vector, R(i) is the block pulse value of state vector of x(τ) respectively and P(i) is the block pulse value of input vector u(τ).
The following recursive relationship for both block pulse and discrete values are obtained as
(12) |
Where:
(13) |
and i = 1,2,3, the interval number. The x(i) give the discrete values of the state vector and the R(i) give the block-pulse function (BPF) values of the state for any length of time. Higher accuracy can be obtained for increased value of k. This is the main advantage of the present method.
ILLUSTRATIVE EXAMPLES
To compare the results and advantage of the new method and software discussed in this study, two examples are considered. Finally, the presently computed results are compared with that of first order and second order state formulations.
Example 1
Consider the example taken from Sivaramakrishnan and Srisailam (1985).
(14) |
with x(0) =5, x(1)(0) = -6 and x(2)(0) = 14, excited by an unit step input u(t).
After differentiating the above equation once and defining the state variables as below,
(15) |
the second-order state space form gets transferred to
(16) |
with the following initial conditions:
Taking z1(1) = z2 and z3(1) = z4, the first-order state-space form of Eq. 16 may be obtained as
(17) |
with the following initial conditions
It is of importance to note that the present algorithm computes the results of Eq. 14, the Eq. 16 is numerically solved by the second-order state space STWS (Thanushkodi et al., 1988; Palanisamy and Arunachalam, 1987) and the Eq. 17 has been solved by the first-order state space STWS (Sivaramakrishnan and Srisailam, 1985). And these computed results are tabulated (Table 1 and 2). A comparative study reveals that the results computed by the present algorithm tally exactly with that of Thanushkodi et al. (1990) and Sivaramakrishnan and Srisailam (1985) with the added advantage of considerable reduction in the size of the matrix to be inverted. In the present case it is a mere (1x1) matrix compared to an (2x2) matrix involved in the second-order formulation and a matrix of size (4x4) in the case of first-order state space formulation.
Table 1: | Comparison of solutions using STWS method (Example 1) |
Table 2: | Comparison of solutions using STWS method (Example 1) |
Generalizing the above analysis, it is demonstrated that for a differential equation of (2n-1)th order with r unknown dependent variables, the originally given matrix to be inverted is of dimensions (rxr) in the present analysis whereas the size of matrix to be inverted becomes (nrxnr) in the second-order state space formulation and in the case of the first-order state space formulation, its size becomes (2nrx2nr).
Example 2
Let us consider the fourth order linear ordinary differential equation as
(18) |
with the initial conditions
Let us define the state variables as
(19) |
With the help of Eq. 19, the second order state space system of Eq. 18 becomes
(20) |
with the following initial conditions:
It is of interest to mention that Eq. 20 represents the stiff system.
Taking z1 = y1, z2 = y1(1), z3 = y2 and z4 = y2(1), the first-order state space form of Eq. 20 may be obtained as
(21) |
with the following initial conditions:
The present algorithm performs the numerical computation of Eq. 18 as it is where as the Eq. 20 is numerically solved by the second-order state space STWS (Sekar et al., 2004) and the Eq. 21 has been solved using the first-order state space STWS adopted by Sivaramakrishnan and Srisailam (1985) and Ponalagusamy et al. (2001). The computed results are shown in Table 3 and 4.
Table 3: | Comparison of numerical values of x using STWS (Example 2) |
Table 4: | Comparison of numerical values of x using STWS (Example 2) |
It is observed that the results computed by the present algorithm tally exactly with that of Sivaramakrishnan and Srisailam (1985), Ponalagusamy et al. (2001) and Sekar et al. (2004) with the advantage of considerable reduction in the computational efforts. The matrix to be inverted in the present method has just reduced itself to a scalar (1x1), compared to a (2x2) matrix involved with the second-order state space STWS and an (4x4) matrix in the first-order state space STWS formulation.
Generalizing the above analysis, it is verified that for a differential equation of 2nth order with r unknown dependent variables, the originally given matrix to be inverted is of dimensions (rxr) in the present analysis whereas the size of matrix to be inverted becomes (nrxnr) in the second-order state space formulation and in the case of the first-order state space formulation, its size becomes (2nrx2nr).
The present numerical algorithm given by Eq. 12 has been applied to several practical problems (second and third order state space systems representing mechanical vibrating systems, electrical power systems, dynamic systems, etc.). A comparative study reveals that the results computed by the present numerical algorithm are exactly the same as obtained by Sivaramakrishnan and Srisailam (1985), Thanushkodi et al. (1988), Palanisamy and Arunachalam (1987) and Sekar et al. (2004) in their respective illustrative examples that are solved using the first/second order state space formulation via STWS (Table 1-4), with the significant feature of considerable reduction in the system matrix structure and the size of the matrix to be inverted resulting in lesser computing time and saving in storage space. (Table 5). It is of importance to mention here that the present method does not require the Kronecker product of matrices and there is no need for the operational matrices of integration. Unlike the approaches adapted by Sivaramakrishnan and Srisailam (1985), Thanushkodi et al. (1988), Palanisamy and Arunachalam (1987) and Sekar et al. (2004), the new version of computational algorithm of Single-Term-Walsh-Series(STWS) solves the given state space system with any order without converting into its lower order which in turn, implies that the original size of system matrix and the size of matrix to be inverted are not increased and remained the same. So, the proposed new numerical algorithm is computationally very effective in lesser computing time as well as storage space.
Table 5: | Comparative Analysis of Present and Existing Numerical Techniques |
A new STWS method in 2nth order state-space formulation for the analysis of the generalized linear, non-singular or singular, time varying system governing 2nth order state-space equation has been introduced. The pivotal characteristic of this method is that the sizes of the system matrices and the matrix to be inverted are smaller (the original size of the given system matrices and the size of matrices to be inverted are unaltered) in the present numerical algorithm than those in the existing methods (first and second-order state-space formulations via STWS) as the value of 2n (the order of state space system) increases. It is observed from the research works done by Sivaramakrishnan and Srisailam (1985), Thanushkodi et al. (1988), Palanisamy and Arunachalam (1987) and Sekar et al. (2004) that their approach does inevitably make the given system of equations more complicated in its structure and requires the inversion of very large matrices. Hence, the algorithm presently developed reduces appreciably computational burden and this is applicable to handle large-scale problems. This is simple and recursive in nature and it is easy to be implemented for a digital computer. Further, the recursive relations provide block pulse values, discrete values and continuous approximation for any length of time and there is no restriction on k as in the case with Walsh series approach. It can also be parallelized by following any of the strategies such as ones presented in Chu et al. (1993) and Li et al. (1999) in the case of computing matrix inverse which, in turn, reduces the computational burden.