The Runge-Kutta (RK)-Butcher algorithm is used to study the timevarying singular bilinear systems with the exact solutions. The results (discrete solutions) obtained using the Haar wavelets, Single-Term Walsh series (STWS) and RK-Butcher algorithms are compared with the exact solutions of the time-varying singular bilinear systems. It is found that the solution obtained using the RK-Butcher algorithm is closer to the exact solutions of the time-varying singular bilinear systems. The RK-Butcher algorithm can easily be implemented using a digital computer and the solution can be obtained for any length of time, which is an added advantage of this algorithm

Batri, K.

Murugesh, V.

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International Journal of Computers, Communications & ControlVol. III (2008), No. 1, pp. 103-109State Analysis of Time-Varying Singular Bilinear Systems by RK-ButcherAlgorithmsV. Murugesh, K. BatriAbstract: The Runge-Kutta (RK)-Butcher algorithm is used to study the time-varying singular bilinear systems with the exact solutions. The results (discrete so-lutions) obtained using the Haar wavelets, Single-Term Walsh series (STWS) andRK-Butcher algorithms are compared with the exact solutions of the time-varyingsingular bilinear systems. It is found that the solution obtained using the RK-Butcheralgorithm is closer to the exact solutions of the time-varying singular bilinear sys-tems. The RK-Butcher algorithm can easily be implemented using a digital computerand the solution can be obtained for any length of time, which is an added advantageof this algorithm.Keywords: Time-varying singular bilinear systems, Haar wavelets, Runge-KuttaButcher algorithm, STWS algorithm.1 IntroductionThe development of singular bilinear systems has been studied by some researchers. Campbell [1]had done some preliminary work, but there was no available closed-form solution in that paper. In someanalysis of neural networks, both singular systems [2] and bilinear systems [3] have been used. Themultipliers and algebraic interconnections between singular systems and bilinear systems are allowed indynamical systems. For singular bilinear systems, Lewis et al. [4] have been discussed extensively in theliterature. However, the solution due to Lewis et al. only applies for the time-invariant case. Hsiao andWang [5] applied the Haar wavelets for the solution of time -varying singular bilinear systems. Sepehrianand Razzaghi [6] applied the STWS approach for finding the numerical solution of time-varying bilinearsystems.Runge-Kutta (RK) methods have become very popular both as computational techniques and as atopic for research [7-12]. Butcher [8] derived the best RK pair, together with an error estimate, and in allstatistical measures this approach is known as the RK-Butcher algorithm. This algorithm appears to beof sixth order because it requires six function evaluations, but in practice the ’working order’ is closer tofive (fifth order). However, the accuracy of the results obtained is better than that of all other algorithmsexamined including the RK- Fehlberg, RK-Merson, RK-centroidal mean (RKCeM) and RK-arithmeticmean (RKAM) algorithms.Murugesh and Murugesan [13-15] introduced the RK-Butcher algorithm in Raster and Time-multiplexingCNN simulations. Recently, [16, 18] the RK-Butcher algorithm is used to find the numerical solution ofan industrial application problem. In this article, we present a new approach for solving the time-varyingsingular bilinear systems using the RK-Butcher algorithm with more accuracy.2 The RK-Butcher AlgorithmThe normal order of an RK algorithm is the approximate number of leading terms of an infinite Taylorseries which calculates the trajectory of a moving point [17]. The remainder of the infinite sum, whichis excluded, is referred to as the local truncation error (LTE). These RK algorithms are forward-lookingpredictors, i.e. they do not use any information from preceding steps to predict the future position of apoint. For this reason, they require a minimum of input data and consequently are very simple to programand use.Copyright © 2006-2008 by CCC Publications104 V. Murugesh, K. BatriThe general p-stage Runge-Kutta method for solving an IVP isy′ = f (x,y) (1)with the initial condition y(x0) = y0 is defined byyn+1 = yn+hp∑i=1bikiwhereki = f(xn+ cih, yn+hp∑j=1ai jk j), i = 1,2,3, ......, pandci =p∑j=1ai j, i = i,2,3, ....., pIn the preceding equations c and b are p-dimensional vectors and A(ai j) is the p× p matrix. Thenthe Butcher array takes the formc1 a11c2 a21 a22c3 a31 a32 a33· · · ·· · · ·· · · ·· · · · ·cp ap1 ap2 ap3 appb1 b2 bp−1 bpThe RK-Butcher algorithm of equation (1) is of the formk1 = h f (xn, yn)k2 = h f(xn+h4, yn+k14)k3 = h f(xn+h4, yn+k18+k28)k4 = h f(xn+h2, yn− k22 + k3)(2)k5 = h f(xn+3h4, yn+3k116+9k416)k6 = h f(xn+h, yn− 3k17 +2k27+12k37− 12k47+8k57)5th order predictoryn+1 = yn+190(7k1+32k3+12k4+32k5+7k6)4th order predictory∗n+1 = yn+16(k1+4k4+ k6)The local truncation error estimate (EE) isEE = yn+1− y∗n+1Then the formation of the Butcher array of the above equation (2) takes the following formState Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms 1050141414181812 0 -12 134316 0 09161 372712712787790 032901290329079016 0 046 0163 Analysis of Time-Varying Singular Bilinear SystemsConsider the linear first order time-varying singular systemK.x(t) = Ax(t)+B(t)u(t) (3)with x0 = x(0), where K is an n×n singular matrix, A is an n×n matrix, B is an n× r matrix. x(t) isn-state vectors and u(t) is an r-input vector.The time-varying singular bilinear system is of the formE(t).x(t) = A(t)x(t)+q∑i=1Ni(t)x(t)ui(t)+B(t)u(t) (4)equation (4)is written in the form (3) asE(t).x(t) =(A(t)+q∑i=1Ni(t)ui(t))x(t)+B(t)u(t) (5)where the singular matrix E(t) ∈ Rn×n, the state x(t) ∈ Rn , the control u(t) ∈ Rq, A(t) ∈ Rn×n andB(t) ∈ Rn×q. Ni(t) ∈ Rn×n and ui(t), i = 1,2,3, ....,q, are the components of u(t). The response x(t),0≤t ≤ ti, is required to be found. The time-varying singular bilinear systems are much more difficult to solvethan the time invariant singular bilinear systems. Therefore, many authors have tried various transformmethods to over come these difficulties. In this article, we introduce RK-Butcher algorithms with moreaccuracy to solve these time-varying singular bilinear systems.4 Numerical ExampleConsider the time-varying singular bilinear system of the following form (Hsiao and Wang [5]) andSepehrian and Razzaghi[6]).E(t) =0 −t 01 0 t0 1 0 A(t) =−2 t 10 −4 2−2t 0 1 N1(t) = 1 −t 10 3 −22t 0 −2 (6)B(t) =[2 1 3]T, u(t) = 1,with initial condition x(0) =[12 2 5]T106 V. Murugesh, K. Batriwhen we solve (5), the analytic solution for x(t) can be shown asx(t) = (2− t)(exp(−t2 )+ exp(t))+82exp(−t2 )− exp(t)+1exp(−t2)+ exp(t)+3 (7)The discrete solutions of equation (5) are evaluated using the RK-Butcher algorithms (with step sizet = 0.25 ) represented in equation (2) and the results are compared with the solutions obtained by theHaar wavelets method by Hsiao and Wang [5] and the STWS method by Sepehrian and Razzaghi [6].The results are shown in tables 1 - 3 along with the exact solutions calculated using equation (7). Errorsbetween the exact and discrete solutions are also given in Tables 1 - 3.S.No TimeDiscrete solution x1ValuesExactSolutionsHaarSolutionsHaarErrorSTWSSolutionsSTWSErrorRK-ButcherSolutionsRK-ButcherError1 0 12.000000 12.0000 0.000000 12.00000 0.000000 12.000000 0.0000002 0.25 11.886053 11.8861 0.000047 11.88605 0.000003 11.886053 0.0000003 0.5 11.791414 11.7914 0.000014 11.79142 0.000006 11.791414 0.0000004 0.75 11.711533 11.7115 0.000033 11.71154 0.000007 11.711533 0.0000005 1 11.641283 11.6413 0.000017 11.64128 0.000003 11.641283 0.0000006 1.25 11.574810 11.5748 0.000010 11.57481 0.000000 11.574810 0.0000007 1.5 11.505362 11.5054 0.000038 11.50537 0.000008 11.505362 0.0000008 1.75 11.425089 11.4251 0.000011 11.42510 0.000011 11.425089 0.0000009 2 11.324812 11.3248 0.000012 11.32481 0.000002 11.324812 0.000000Table 1: Solutions for the problem at various values of x1.S.No TimeDiscrete solution x2ValuesExactSolutionsHaarSolutionsHaarErrorSTWSSolutionsSTWSErrorRK-ButcherSolutionsRK-ButcherError1 0 2.000000 2.0000 0.000000 2.00000 0.000000 2.000000 0.0000002 0.25 1.745678 1.7457 0.000022 1.74568 0.000002 1.745678 0.0000003 0.5 1.480968 1.4810 0.000032 1.48097 0.000002 1.480968 0.0000004 0.75 1.203067 1.2031 0.000033 1.20307 0.000003 1.203067 0.0000005 1 0.908880 0.9089 0.000020 0.90888 0.000000 0.908880 0.0000006 1.25 0.594985 0.5950 0.000015 0.59498 0.000005 0.594985 0.0000007 1.5 0.257579 0.2576 0.000021 0.25758 0.000001 0.257579 0.0000008 1.75 -0.107578 -0.1076 0.000022 -0.10758 0.000002 -0.107578 0.0000009 2 -0.505221 -0.5052 0.000021 -0.50522 0.000001 -0.505221 0.000000Table 2: Solutions for the problem at various values of x2.5 ConclusionsThe discrete solutions obtained using the RK-Butcher algorithm gives more accurate values whencompared to the Haar wavelets method discussed by Hsiao and Wang [5] and the STWS method byState Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms 107S.No TimeDiscrete solution x3ValuesExactSolutionsHaarSolutionsHaarErrorSTWSSolutionsSTWSErrorRK-ButcherSolutionsRK-ButcherError1 0 5.000000 5.0000 0.000000 5.00000 0.000000 5.000000 0.0000002 0.25 5.072562 5.0726 0.000038 5.07256 0.000002 5.072562 0.0000003 0.5 5.166522 5.1665 0.000022 5.16652 0.000002 5.166522 0.0000004 0.75 5.284021 5.2840 0.000021 5.28402 0.000001 5.284021 0.0000005 1 5.427522 5.4275 0.000022 5.42752 0.000002 5.427522 0.0000006 1.25 5.599862 5.5999 0.000038 5.59986 0.000002 5.599862 0.0000007 1.5 5.804289 5.8043 0.000011 5.80429 0.000001 5.804289 0.0000008 1.75 6.044524 6.0445 0.000024 6.04451 0.000014 6.044524 0.0000009 2 6.324812 6.3248 0.000012 6.32480 0.000012 6.324812 0.000000Table 3: Solutions for the problem at various values of x3.Sepehrian and Razzaghi [6]. From the tables 1-3, one can observe that the solutions obtained by theRK-Butcher algorithm match well with the exact solutions of the time-varying singular bilinear systems,but the Haar wavelets and STWS methods yields a little error. Hence the RK-Butcher algorithm is moresuitable for studying the time-varying bilinear systems.Bibliography[1] Campell, S.L., 1987, Bilinear nonlinear descriptor control systems, CRSC Technical Report102386-01, Department of Mathematics, N.C. State University, Raleigh, NC 27695.[2] Declaris, N. and Rindos, A., 1984, Semistate analysis of neural networks in Apysia Californica,Proceedings of the 27th MSCS, 686-689.[3] Wiener, N., 1948, Cybernetics (Cambridge, MIT Press).[4] Lewis, F. L., Mertzios, B. C., and Marszalek, W., 1991, Analysis of singular bilinear systems usingWalsh functions, IEE Proceedings Part-D,138, 89-92.[5] Hsiao, C. H., and Wang, W. J., 2000, State Analysis of time-varying Singular Bilinear Systems viaHaar wavelets, Mathematics and Computers in Simulation, 52, 11-20.[6] Sepehrian, B., and Razzaghi, M., 2003, State Analysis of time-varying Singular Bilinear Systmesby Single-Term Walsh Series, International Journal of Computer Mathematics, 80, 413-418.[7] Alexander, R.K. and Coyle, J.J.,1990, Runge-Kutta methods for differential-algebraic systems.SIAM Journal of Numerical Analysis, 27, 736-752.[8] Butcher, J.C., 1987, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta andGeneral Linear Methods (Chichester: JohnWiley).[9] Butcher, J.C., 2003, Numerical Methods for Ordinary Differential Equations (Chichester: JohnWi-ley).[10] Shampine, L.F., 1994, Numerical Solution of Ordinary Differential Equations (NewYork: Chapman& Hall).108 V. Murugesh, K. Batri[11] Yaakub, A.R. and Evans, D.J., 1999, A fourth order Runge-Kutta RK(4,4) method with error con-trol. International Journal of Computer Mathematics, 71, 383-411.[12] Yaakub, A.R. and Evans, D.J., 1999, New Runge-Kutta starters of multi-step methods. InternationalJournal of Computer Mathematics, 71, 99-104.[13] Murugesh, V., and Murugesan, K., 2004, Comparison of Numerical Integration Algorithms inRaster CNN Simulation, Lecture Notes in Computer Science, 3285, 115-122.[14] Murugesh, V. and Murugesan, K., 2005, Simulation of Cellular Neural Networks using the RK-Butcher algorithm, International Journal of Management and Systems, 21, 65-78.[15] Murugesh, V., and Murugesan, K., 2006, Simulation of Time-Multiplexing Cellular Neural Net-works with Numerical Integration Algorithms, Lecture Notes in Computer Science, 3991, 115-122.[16] Devarajan Gopal, Murugesh, V., and Murugesan, K., 2006, Numerical Solution of Second-orderRobot Arm Control Problem using Runge-Kutta Butcher Algorithm, International Journal of Com-puter Mathematics, 83, 345-356.[17] Shampine, L.F. and Gordon, M.K., 1975, Computer Solutions of Ordinary Differential Equations:The Initial Value Problem (San Francisco, CA:W.H. Freeman).[18] V. Murugesh and K. Batri, “An Efficient Numerical Integration Algorithm for Cellular Neural Net-work Based Hole-Filler Template Design”, International Journal of Computers, Communications& Control, Vol. II (2007), No. 4, pp. 367-374.V. MurugeshDepartment of Information and Communication EngineeringHannam University133 Ojung-dong Daeduk-gu, Daejeon 306-791, Republic of KoreaE-mail: murugesh72@gmail.comK. BatriDepartment of Computer Science and EngineeringMuthayammal Engineering CollegeRasipuram 637 408IndiaE-mail: krishnan.batri@gmail.comReceived: November 26, 2007Dr. V. Murugesh obtained his Bachelor of Science in Com-puter Science and Master of Computer Applications degree fromBharathiar University, Coimbatore, India during 1992 and 1995respectively. Completed his PhD in Computer Science fromBharathidasan University, Tiruchirappalli, India during 2006. Hehas held various positions at National Institute of Technology,Tiruchirappalli, India and Sona College of Technology, Salem,India. Currently, he is working as Assistant Professor in theDepartment of Information and Communication Engineering atHannam University, Daejeon, Republic of Korea. His fields ofinterest are in Neural Network based Image Processing and Sci-entific computing. He has published more than 30 technical pa-pers in International, National journals and conferences.State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms 109Krishnan Batri received the M.E. from Madurai Kamarj Uni-versity in 2003. He is a Research Scholar with the Depart-ment of Computer Science and Engineering in the National Insti-tute of Technology Tiruchirapalli, Tamil Nadu, India. Currentlyhe is working as a Assistant Proffessor with the department ofComputer Science and Engineering at Muthayammal Engineer-ing College,Rasipuram, Tamilnadu, India . His research inter-ests include Information Retrieval, Data fusion and Genetic algo-rithms.

State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms

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State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms

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