SECOND ORDER TWO DIMENSIONAL SYSTEMS: COMPUTING THE TRANSFER FUNCTION

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1 Journal of ELECTRICAL ENGINEERING, VOL. 55, NO , 2004, SECOND ORDER TWO DIMENSIONAL SYSTEMS: COMPUTING THE TRANSFER FUNCTION Gege E. Antoniou Marinos T. Michael In this paper the discrete Fourier transfm is used to determine the coefficients of a transfer function of a new twodimensional model of second-der: xi 1 +2, i 2 +2) = A 0 xi 1 +1, i 2 +1)+A 1 xi 1 +1, i 2 )+A 2 xi 1, i 2 +1). The algithm is straightfward and has been implemented using the software package Matlab TM. Two step-by-step examples illustrating the application of the algithm are presented. K e y w o r d s: two-dimensional 2D) systems, second-der systems, transfer function, Fourier transfm 1 INTRODUCTION During the past two decades there has been extensive research in two dimensional 2D) systems. This is due to the extensive range of applications, especially in engineering and computing D systems can be represented with a transfer function in polynomial fm using state space like structures 3. State space based techniques play a very crucial role in the analysis and synthesis of 2D systems. An imptant problem is to determine the coefficients of a transfer function from its model representation and vice versa. Leverrier-Fadeeva, discrete Fourier transfm DFT) algithms and Vanderlmode matrices can be modified to be used f various models. The DFT has been used f the evaluation of the transfer function coefficients f linear, singular, and multidimensional state space systems 4 7. In this paper a computer implementable algithm is proposed f the computation of the 2D transfer function f a new 2D system that is also of second-der. Second-der one-dimensional systems have been used in circuit they, linear control systems, filtering, mechanical system modeling and applied mathematics The proposed algithm determines the coefficients of the determinantal polynomial and the coefficients of the adjoint polynomial matrix, using the DFT. The computational speed of the method can be improved by using fast Fourier transfm techniques. It is noted that the proposed algithm easily can be modified to be used with second-der 2D systems of other types SECOND ORDER TWO DIMENSIONAL SYSTEM A second-der two-dimensional SO2D) system has the following structure 11: xi 1 + 2,i 2 + 2) = A 0 xi 1 + 1,i 2 + 1) + A 1 xi 1 + 1,i 2 ) + A 2 xi 1,i 2 + 1) + B 1 ui 1 + 1,i 2 ) + B 2 ui 1,i 2 + 1), yi 1,i 2 ) = C xi 1,i 2 ) 1), xi 1,i 2 ) R λ, ui 1,i 2 ) R m, yi 1,i 2 ) R p, i 1,i 2 are integer-valued vertical and hizontal codinates, respectively, xi 1,i 2 ) is the local vect at i 1,i 2 ), ui 1,i 2 ) is the input vect and yi 1,i 2 ) is the output vect. A k, f k = 0,1,2 and B 1, B 2, C, are real matrices of appropriate dimensions denoting the characteristics of the SO2D system that can also be represented by a 2D transfer function, as in the regular 2D systems 12. It is noted that this particular SO2D system 1) is an extension of the regular 2D Fnasini-Marchesini model 12 to cover systems of second-der. F me 2D second-der structures the reader can refer to 11. Applying the 2D z i, i = 1,2 transfm to system 1), with zero initial conditions, the transfer function is found to be: Tz 1,z 2 ) = CIz 2 1z 2 2 A 0 z 1 z 2 A 1 z 1 A 2 z 2 1 B 1 z 1 + B 2 z 2. 2) In the following section an interpolative approach is developed f determining the transfer function Tz 1,z 2 ), given the matrices A k, k = 0,1,2 and B 1 B 2, C using the 2D DFT. F the sake of completeness a brief description of the 2D DFT follows. Image Processing and Systems Labaty, Department of Computer Science, Montclair State University, Montclair, N.J ISSN c 2004 FEI STU

2 Journal of ELECTRICAL ENGINEERING VOL. 55, NO , D DISCRETE FOURIER TRANSFORM Consider the finite sequences Xk 1,k 2 ) and Xr 1,r 2 ), k i,r i = 0,...,M i, i = 1,2. In der f the sequences Xk 1,k 2 ) and Xr 1,r 2 ), to constitute a 2D DFT pair the following relations should hold 13: Xr 1,r 2 ) = Xk 1,k 2 ) = 1 R M 1 M 2 k 1=0 k 2=0 M 1 r 1=0 r 2=0 Xk 1,k 2 )W k1r1 1 W k2r2 2, M 2 3) Xr 1,r 2 )W k1r1 1 W k2r2 2 4) R = M 1 + 1)M 2 + 1), 5) W i = e 2πj)/Mi+1), i = 1,2, 6) X, X are discrete argument matrix valued functions, with dimensions p m. In the following section an interpolative approach is developed f determining the transfer function Ts), given the matrices A i, i = 0,1,2 and B 1, B 2 C, using the 2D DFT. 4 ALGORITHM The transfer function of a SO2D system 1) is Tz 1,z 2 ) = Nz 1,z 2 ) dz 1,z 2 ) Nz 1,z 2 ) =Cadj Iz 2 1z 2 2 A 0 z 1 z 2 A 1 z 1 7) A 2 z 2 B1 z 1 + B 2 z 2 ), 8) dz 1,z 2 ) =det Iz 2 1z 2 2 A 0 z 1 z 2 A 1 z 1 A 2 z 2. 9) Equations 8) and 9) can be written in polynomial fm as follows: Nz 1,z 2 ) = n P max n P max z λ1 1 zλ2 2 10) with n P max := max 2λ 1),2λ 1) ). The numerat coefficients are matrices with dimensions p m). dz 1,z 2 ) = n q max n q max z λ1 1 zλ2 2 11) n q max := max2λ,2λ). The denominat coefficients are scalars. The numerat polynomial matrix Nz 1,z 2 ) and the denominat polynomial dz 1,z 2 ) can be numerically computed at R = r + 1) 2 points equally spaced on the unit 2D disc. The R points are chosen as z 1,z 2 ) = vi 1 ),vi 2 ), i 1,i 2 = 0,...,r with r = 2λ, accding to definition as: v 1 i) = v 2 i) = W i, i = 0,...,r 12) W i = e 2πj)/r+1), i = 1,2. 13) The values of the transfer function 7) at the R points are the cresponding 2D DFT coefficients. 4.1 Denominat Polynomial To evaluate the denominat coefficients define = det Iv 2 1i 1 )v 2 2i 2 ) A 0 v 1 i 1 )v 2 i 2 ) A 1 v 1 i 1 ) A 2 v 2 i 2 ). 14) Therefe, using equations 11) and 14), can be defined as = dv 1 i 1 ),v 2 i 2 ). 15) Provided that at least one of 0. Equations 11), 12) and 15) yield r r = W i1λ1+i2λ2) 16) In the above equation 16) it is obvious that and fm a 2D DFT pair. Therefe the coefficients can be computed using the inverse 2D DFT, as follows: = 1 r r a i1,i R 2 W i1λ1+i2λ2) 17) i 1=0 i 2=0 4.2 Numerat Polynomial To evaluate the numerat matrix polynomial, define = CadjIv 2 1i 1 )v 2 2i 2 ) A 0 v 1 i 1 )v 2 i 2 ) A 1 v 1 i 1 ) A 2 v 2 i 2 ) B 1 v 1 i 1 ) + B 2 v 2 i 2 ). 18) Using equations 10) and 18), can be defined as = Nv 1 i 1 ),v 2 i 2 ). 19) Equations 10), 12) and 19) yield = r 1 r 1 W i1λ1+i2λ2). 20) In the above equation 20),, fm a 2D DFT pair. Therefe the coefficients can be computed, using the inverse 2D DFT, as follows = 1 R r 1 r 1 i 1=0 i 2=0 W i1λ1+i2λ2). 21) Two salient examples, simple yet illustrative of the theetical concepts presented in this wk, follow below

3 298 G. E. Antoniou M.T. Michael: SECOND-ORDER TWO-DIMENSIONAL SYSTEMS: COMPUTING THE TRANSFER... 5 NUMERICAL EXAMPLES 5.1 Single-Input Single-Output Consider the following single-input single-output SO2D system xi 1 + 2,i 2 + 2) = A 0 xi 1 + 1,i 2 + 1) + A 1 xi 1 + 1,i 2 ) +A 2 xi 1,i 2 + 1) + B 1 ui 1 + 1,i 2 ) + B 2 ui 1,i 2 + 1), yi 1,i 2 ) = C xi 1,i 2 ) 22) 1 3 A 0 = 0 0 B 1 = 1 0, A 1 =, B 2 = , A 2 =,C =., 1 1 Since λ = 2, the r = 2λ = 4. Therefe R = r + 1) 2 = 25. The direct application of the proposed algithm yields a 00 a 01 a 02 a 03 a 04 a 10 a 11 a 12 a 13 a 14 a 20 a 21 a 22 a 23 a 24 a 30 a 31 a 32 a 33 a 34 a 40 a 41 a 42 a 43 a j j j j j j j j j j j j j j j j j j j j j j j j j and F 00 F 01 F 02 F 03 F 04 F 10 F 11 F 12 F 13 F 14 F 20 F 21 F 22 F 23 F 24 F 30 F 31 F 32 F 33 F 34 F 40 F 41 F 42 F 43 F j j j j j j j j j j j j j j j j j j j j j j j j j Using 17), the denominat coefficients are q 00 q 01 q 02 q 03 q q 10 q 11 q 12 q 13 q q 20 q 21 q 22 q 23 q q 30 q 31 q 32 q 33 q q 40 q 41 q 42 q 43 q Using 21), the numerat matrix polynomials are P 00 P 01 P 02 P 03 P P 10 P 11 P 12 P 13 P P 20 P 21 P 22 P 23 P P 30 P 31 P 32 P 33 P P 40 P 41 P 42 P 43 P Once the denominat and the adjoint matrix coefficients have been computed, the transfer function Tz 1,z 2 ) is determined as Tz 1,z 2 ) = P 23 z 2 1z P 12 z 1 z P 11 z 1 z 2 ) q44 z 4 1z q 33 z 3 1z q 23 z 2 1z q 21 z 2 1z 2 + q 20 z q 12 z 1 z 2 2 Tz 1,z 2 ) = + q 11 z 1 z 2 + q 02 z 2 2 z1z z 1 z2 2 + z 1 z 2 z1 4z4 2 z3 1 z3 2 z2 1 z3 2 + z2 1 z 2 z1 2 2z 1z2 2 2z 1z 2 z2 2. The above result can be verified using 2). 5.2 Multiple-Input Single-Output Consider the following two-input single-output SO2D system: xi 1 + 2,i 2 + 2) = A 0 xi 1 + 1,i 2 + 1) + A 1 xi 1 + 1,i 2 ) +A 2 xi 1,i 2 + 1) + B 1 ui 1 + 1,i 2 ) + B 2 ui 1,i 2 + 1), yi 1,i 2 ) = Cxi 1,i 2 ) 23) 1 3 A 0 = 0 0 B 1 = , A 1 =, B 2 = , A 2 = 1 1, C = 1 0. Since λ = 2, the r = 2λ = 4. Therefe R = r + 1) 2 = 25. The direct application of the proposed algithm yields a 00 a 01 a 02 a 03 a 04 a 10 a 11 a 12 a 13 a 14 a 20 a 21 a 22 a 23 a 24 a 30 a 31 a 32 a 33 a 34 a 40 a 41 a 42 a 43 a 44

4 Journal of ELECTRICAL ENGINEERING VOL. 55, NO , j j j j j j j j j j j j j j j j j j j j j j j j j and F 00 = , F 01 = j j, F 02 = j j, F 03 = j j, F 04 = j j, F 10 = j j, F 11 = j j, F 12 = j j, F 13 = j j, F 14 = j j, F 20 = j j, F 21 = j j, F 22 = j j, F 23 = j j, F 24 = j j, F 30 = j j F 31 = j j, F 32 = j j, F 33 = j j, F 34 = j j, F 40 = j j, F 41 = j j, F 42 = j j, F 43 = j j, F 44 = j j. Using 17), the denominat coefficients are q 00 q 01 q 02 q 03 q q 10 q 11 q 12 q 13 q q 20 q 21 q 22 q 23 q q 30 q 31 q 32 q 33 q q 40 q 41 q 42 q 43 q Using 21), the numerat matrix polynomials are P 00 P 01 P 02 P 03 P 04 P 10 P 11 P 12 P 13 P 14 P 20 P 21 P 22 P 23 P 24 P 30 P 31 P 32 P 33 P 34 P 40 P 41 P 42 P 43 P = Once the denominat and the adjoint matrix coefficients have been computed, the transfer function Tz 1,z 2 ) is determined as Tz 1,z 2 ) = P 02 z P 11 z 1 z 2 + P 12 z 1 z P 20 z 2 1 +P 21 z 2 1z 2 +P 23 z 2 1z 3 2 +P 32 z 3 1z 2 2) q02 z 2 2 +q 11 z 1 z 2 +q 12 z 1 z q 20 z q 21 z 2 1z 2 + q 23 z 2 1z q 33 z 3 1z q 44 z 4 1z 4 2 Tz 1,z 2 ) = z z 1 z z 1 z ) + z z1z z1z z1z z2 2 2z 1 z 2 2z 1 z2 2 z1 2 +z1z 2 2 z1z z1z z1z z 2 Tz 1,z 2 ) = 2 7z 1 z 2 4z1 2 9z1z z1z z1z z 1 z 2 + 3z 1 z2 2 2z1 2 + z1z z1z ) 1. z2 2 2z 1 z 2 2z 1 z2 2 z1+z 2 1z 2 2 z1z z1z 3 2+z 3 1z2) 4 4 The above result can be verified using 2). 6 COMPLEXITY OF THE ALGORITHM The proposed algithm has two parts. In the first part the matrices and the scalars are evaluated with a cost of pmrλ 3 operations. In the second part the coefficients of and are evaluated using the DFT with a cost of pmr 2 + R 2 operations. F me efficient computation, especially f high der systems, fast Fourier methods can be used to implement the DFT 13. Due to the inherent modularity and the algithmic structure of the presented method high parallelism is permitted. In this case the computation of each determinant, 16), and each matrix product, 20), can be distributed over a number of processing elements, considerably reducing the computation time of the algithm.

5 300 G. E. Antoniou M.T. Michael: SECOND-ORDER TWO-DIMENSIONAL SYSTEMS: COMPUTING THE TRANSFER... 7 CONCLUSIONS An algithm was presented f the computation of the transfer function f a new 2D system model of second der. The technique is using the DFT algithm and has been implemented with the software package Matlab TM. To further improve the computational speed of the algithm, fast Fourier techniques and VHDL/FPGA based implementations can be used. The presented algithm can be easily modified to be used f regular, singular and positive SO2D models of other types 11. Also the presented model/algithm can easily be extended to systems with higher der and dimensions. It is noted that known problems such as stability, coefficient sensitivity, filtering etc, can be studied using the SO2D system model. Acknowledgment This wk was suppted by the Margaret and Herman Sokol Faculty Award, Faculty Scholarship Incentive Program of MSU/CS and 2004 Carreer Development Award The auths would like to thank the anonymous reviewers f their valuable comments. Singular Systems Using the DFT, IEEE Trans. Circuit Syst. CAS ), YEUNG, K. S. KUMBI, F.: Symbolic Matrix Inversion with Application to Electronic Circuits, IEEE Trans. Circuit Syst. CAS-35 No ), HENRION, D. SEBEK, M. KUCERA, V.: Robust Pole Placement f Design Second-Order Systems: an LMI Approach, IFAC Symposium on Robust Control, Milan, Italy, CHAHLAOUI, Y. LEMONNIER, D. MEERBERGEN, K. VANDENDORPE, A. Van DOOREN, P. : Model Reduction of Second Order Systems, MTNS2002, Fifteenth International Symposium on Mathematical They of Netwks and Systems, University of Notre Dame, GOODWIN, C.: Real Time Block Recursive Parameter Estimation of Second Order Systems, PhD Thesis, Nottingham Trent University, U.K, ANTONIOU, G. E.: Second-Order 2D Systems: Models and Transfer Functions, International Conference in Circuits Systems and Computers, Cfu, Greece, FORNASINI, E. MARCHESINI, E.: Doubly Indexed Dynamical Systems: State Space Models and Structural Properties, Math. System They 12 No ), MITRA, S. K.: Digital Signal Processing, McGraw-Hill, New Yk, Received 8 September 2004 References 1 KACZOREK, T.: Positive 1D and 2D Systems, Communication and Control Engineering Series, Springer-Verlag, Berlin, BOSE, N. K.: Applied Multidimensional Systems They, Van Nostrand Reinhold, New Yk, WU-SHENG LU ANTONIOU, A.: Two-Dimensional Digital Filters, Marcel-Dekker, New Yk, LEE, T.: IEEE Trans. Circuit Syst.. 5 PACCAGNELLA, L. E.et al : FFT Calculation of a Determinantal Polynomial, IEEE Trans. Autom. Control AC ), ANTONIOU, G. E. GLENTIS, G. O. A. VAROUFAKIS, S. J. KARRAS, D. A.: Transfer Function Determination of Gege Antoniou PhD) Awarded his Doct of Electrical Engineering Infmatics) degree in 1998 from National Technical University of Athens, Athens, Greece. He is currently a profess in the department of Computer Science at Montclair State University, Montclair, New Jersey, USA. His present research interests include system they, multidimensional digital signal processing and FPGA/VHDL applications. He is a Seni member of the IEEE. Marinos T. Michael BSCS) Awarded his Bachel of Science in Computer Science degree Magna Cum Laude) in 2003 from Montclair State University, Montclair, New Jersey, USA. He is currently a graduate student in Computer Science at the same university. He is a student member of the IEEE. SLOVART G.T.G. s.r.o. GmbH of periodicals and of non-periodically printed matters, book s and CD - ROM s Krupinská 4 PO BOX 152, Bratislava 5,Slovakia tel.: , fax.: gtg@internet.sk, SLOVART G.T.G. s.r.o. GmbH