本篇論文主要包括兩部分。第一部分闡述如何運用保結構演算法來求解各種類型的黎卡迪方程式，第二部分主要著眼於週期奇異系統的平衡實現化理論。

在第一部分中，我們分別探討求解週期離散型、連續型以及廣義離散型代數黎卡迪方程式之保結構算法。上述各類算法均以求解離散型代數黎卡迪方程式之保結構演算法為基石，加以推廣而得之。並且我們可在比可穩定化與可偵測化更弱的假設條件下，更進一步證明此一保結構算法的二次收斂性。通過大量Matlab測試集的檢驗，可知此一保結構算法無論在精確度上與執行效率上均優於其他算法。

在第二部分中，我們先針對週期奇異系統的完全可達性與完成可觀性，給出一系列的充分必要條件。由這些數學等價條件中，我們可定義出週期可達性與可觀性之葛雷米矩陣，並且可進一步證明出這些對稱半正定的葛雷米矩陣滿足某些廣義週期離散型李雅普諾夫方程式。此外，我們還提出一套數值上穩定且可行的算法來求解這些李雅普諾夫方程式。最後，我們還提出週期奇異系統的平衡實現化問題並且提供解決方案。

This dissertation is consisted of two parts. The first part treats of applications of the structure-preserving doubling algorithm (SDA) to solve various algebraic Riccati equations, while the second part concerns with the problem of balanced realization for discrete-time periodic descriptor systems.

In the first part, we investigate structure-preserving algorithms for computing the symmetric positive semi-definite solutions to the periodic discrete-time algebraic Riccati equations (P-DAREs), continuous-time algebraic Riccati equations (CAREs) and generalized discrete-time algebraic Riccati equations (G-DAREs), respectively. All are based on the SDA algorithm for solving the discrete-time algebraic Riccati equations (DAREs). In Section 2 of
Chapter 1, we develop the SDA algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability. With several numerical results, the algorithm is shown to be efficient, out-performing other algorithms on a large set of benchmark
problems.

In the second part, necessary and sufficient conditions are
derived for complete reachability and observability of periodic time-varying descriptor systems. Applying these conditions, the symmetric positive semi-definite reachability/observability Gramians are defined and can be shown to satisfy some projected generalized discrete-time periodic Lyapunov equations. We propose a numerical method for solving these projected Lyapunov equations, and an illustrative numerical example is given. As an application
of our results, the balanced realization of periodic descriptor systems is discussed.

[1] J. Abels and P. Benner, DAREX -- a collection of benchmark examples for discrete-time algebraic Riccati equations (version 2.0), Tech. Rep. SLICOT Working Note 1999-16, The Working Group on Software, 1999.

[2] G. Ammar and V. Mehrmann, On {H}amiltonian and symplectic {H}essenberg forms, Lin. Alg. Appl., 149 (1991), pp.~55--72.

[3] B. D. O. Anderson, Second-order convergent algorithms for the steady-state Riccati equation, Int. J. Control, 28 (1978), pp.~295--306.

[4] M. Athans, W. Levine, and A. Levis, A system for the optimal and suboptimal position and velocity control for a string of high-speed vehicles}, in Proc. 5th Int. Analogue Computation Meetings, Lausanne, Switzerland, 1967.

[5] Z. Bai and J. Demmel, On swapping diagonal blocks in real {S}chur form, Lin. Alg. Appl., 186 (1993), pp.~73--95.

[6] Z. Bai and J. Demmel, Using the matrix sign function to compute invariant subspaces, SIAM J. Matrix Analy. Appl., 19 (1998), pp.~205--225.

[7] Z. Bai, J. Demmel, and M. Gu, An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems, Numer. Math., 76 (1997), pp.~279--308.

[8] Z. Bai and Q. Qian, Inverse free parallel method for the numerical solution of algebraic Riccati equations, in Proc. Fifth SIAM Conf. Appl. Lin. Alg., Snowbird, UT, J. G. Lewis, ed., SIAM, Philadelphia, PA, 1994, pp.~167--171.

[9] L. Balzer, Accelerated convergence of the matrix sign function, Int. J. Control, 21 (1980), pp.~1057--1078.

[10] A. Y. Barraud, Investigation autour de la fonction signe d'une matrice, application \'{a} l'\'{e}quation de Riccati}, R.A.I.R.O. Automatique, 13 (1979), pp.~335--368.

[11] A. Y. Barraud, Produit \'{e}toile et fonction signe de matrice. application \'{a} l'\'{e}quation de {R}iccati dans le cas discret, R.A.I.R.O. Automatique}, 14 (1980), pp.~55--85.

[12] M. S. Bazaraa, H. D. Sheraii, and C. M. Shetty, Nonlinear Programming, John Wiley & Sons, 1993.

[13] A. N. Beavers and E. D. Denman, Asymptotic solutions to the matrix Riccati equation, Mathematical Biosciences, 20 (1974), pp.~339--344.

[14] A. N. Beavers and E. D. Denman, A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues, Numer. Math., 21 (1974), pp.~389--396.

[15] A. N. Beavers and E. D. Denman, A new similarity transformation method for eigenvalues and eigenvectors, Mathematical Biosciences, 21 (1974), pp.~143--169.

[16] A. N. Beavers and E. D. Denman, A new solution method for matrix quadratic equations, Mathematical Biosciences, 20 (1974), pp.~135--143.

[17] D. J. Bender, Lyapunov-like equations and reachability/observability gramians for descriptor systems, IEEE Trans. Auto. Control, 32 (1987), pp.~343--348.

[18] P. Benner, Contributions to the numerical solutions of algebraic Riccati equations and related eigenvalue problems, PhD Dissertation, Fakult{\"{a}}t f{\"{u}}r Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany, 1997.

[19] P. Benner and R. Byers, Evaluating products of matrix pencils and collapsing matrix products, Num. Lin. Alg. Appl., 8 (2001), pp.~357--380.

[20] P. Benner, R. Byers, R. Mayo, E. S. Quintana-Orti, and V. Hernandez, Parallel algorithms for LQ optimal control of discrete-time periodic linear systems}, J. Para. Distr. Comp., 62 (2002), pp.~306--325.

[21] P. Benner, A. J. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case, Tech. Rep. SPC 95_23, Fakult\"at f\"ur Mathematik, TU Chem\-nitz--Zwickau, 09107 Chem\-nitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

[22] P. Benner, A. J. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Rep. SPC 95_22, Fakult\"at f\"ur Mathematik, TU Chem\-nitz--Zwickau, 09107 Chem\-nitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

[23] P. Benner, R. Mayo, E. S. Quintana-Orti, and V. Hernandez, A coarse grain parallel solver for periodic Riccati equations, Tech. Rep. 2000-01, Depto. de Informatica, 12080-Castellon, Spain, 2000.

[24] P. Benner, R. Mayo, E. S. Quintana-Orti, and V. Hernandez, Solving diacrete-time periodic Riccati equations on a cluster, in Euro-Par 2000 Parallel Processing, A. Bode, T. Ludwig, W. Karl, and R.~Wismuller, eds., no. 1900 in Lecture Notes in Computer Science, Springer-Verlag, 2000, pp.~824--828.

[25] P. Benner, V. Mehrmann, V. Sima, S. V. Huffel, and A. Varga, SLICOT -- a subroutine library in systems and control theory, Applied and Computational Control, Signals, and Circuits, 1 (1999), pp.~499--539.

[26] M. C. Berg, N. Amit, and J. D. Powell, Multirate digital control system design}, IEEE Trans. Auto. Control, 33 (1988), pp.~1139--1150.

[27] W. Bialkowski, Application of steady-state {K}alman filters --- theory with field results, in Proc. Joint Automat. Cont. Conf., Philadelphia, PA, 1978.

[28] S. Bittanti, Deterministic and stochastic linear periodic systems, in Time Series and Linear Systems, S. Bittanti, ed., Springer Verlag, New York, 1986, pp.~141--182.

[29] S. Bittanti and P. Colaneri, Analysis of discrete-time linear periodic systems, in Control and Dynamic Systems, C.~T. Leondes, ed., vol. 78, Academic Press, New York, 1996.

[30] S. Bittanti and P. Colaneri, Periodic control, in Wiley Encyclopedia of Electrical and Electronic Engineering, J. G. Webster, ed., vol. 16, Wiley, New York, 1999, pp.~59--74.

[31] S. Bittanti, P. Colaneri, and G. D. Nicolao, The difference periodic Riccati equation for the periodic prediction problem, IEEE Trans. Auto. Control, 33 (1988), pp.~706--712.

[32] S. Bittanti, P. Colaneri, and G. D. Nicolao, The periodic Riccati equation, in The Riccati Equation, S. Bittanti, A. Laub, and J. Willems, eds., Springer-Verlag, 1991, pp.~127--162.

[33] A. Bojanczyk, G. H. Golub, and P. Van~Dooren, The periodic Schur decomposition. Algorithms and applications, in Proc. SPIE Conference, vol. 1770, San Diego, 1992, pp.~31--42.

[34] J. H. Brandts, Matlab code for sorting real Schur forms, Num. Lin. Alg. Appl., 9 (2002), pp.~249--261.

[35] R. Bru, C. Coll, and N. Thome, Compensating periodic descriptor systems, Sys. Contr. Lett., 43 (2001), pp.~133--139.

[36] A. Bunse-Gerstner, R. Byers, and V. Mehrmann, A chart of numerical methods for structured eigenvalue problems, SIAM J. Matrix Analy. Appl., 13 (1992), pp.~419--453.

[37] A. Bunse-Gerstner, V. Mehrmann, and N. K. Nichols, Regularization of descriptor systems by derivative and proportional state feedback, SIAM J. Matrix Analy. Appl., 13 (1992), pp.~46--67.

[38] A. Bunse-Gerstner, V. Mehrmann, and Watkins, An SR algorithm for Hamiltonian matrices, based on Gaussian elimination, Methods of Operations Research, 58 (1989), pp.~15--26.

[39] R. Byers, A Hamiltonian QR-algorithm, SIAM J. Sci. Statist. Comput., 7 (1986), pp.~212--229.

[40] R. Byers, Numerical stability and instability in matrix sign function based algorithms, in Computational and Combinatorial Methods in System Theory}, C. Byrnes and A. Lindquist, eds., North-Holland, 1986, pp.~185--200.

[41] R. Byers, Solving the algebraic {R}iccati equation with the matrix sign function, Lin. Alg. Appl., 85 (1987), pp.~267--279.

[42] R. Byers, C. He, and V. Mehrmann, the matrix sign function method and the computation of invariant subspaces, SIAM J. Matrix Analy. Appl., 18 (1997), pp.~615--632.

[43] R. Byers and N. Rhee, Cyclic Schur and Hessenberg Schur numerical methods for solving periodic Lyapunov and Sylvester equations, Technical Report, Dept. of Mathematics, Univ. of Missouri at Kansas City, 1995.

[44] E. K.-W. Chu, H.-Y. Fan, and W.-W. Lin, A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations, preprint 2002-29, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003.

[45] E. K.-W. Chu, H.-Y. Fan, and W.-W. Lin, A structure-preserving doubling algorithm for continuous-time algebraic {R}iccati equations}, preprint 2002-28, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003.

[46] E. K.-W. Chu, H.-Y. Fan, W.-W. Lin, and C.-S. Wang, A structure-preserving doubling algorithm for periodic descrete-time algebraic Riccati equations, preprint 2002-18, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003.

[47] R. E. Crochiere and L. R. Rabiner, Mutirate Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1993.

[48] L. Dai, Singualr Control Systems, Springer-Verlag Berlin, Heidelberg, 1989.

[49] E. J. Davison and W. Gesing, The systematic design of control systems for the multivariable servomechenism problem, in Alternatives for Linear Multivariable Control, M. K. Sain and J. L. Peczkowsky, eds., Nat. Eng. Consortium Inc., Chicago, IL, 1978.

[50] E. Denman and R. Beavers, The matrix sign function and computations in systems, Appl. Math. Comp., 2 (1976), pp.~63--94.

[51] L. Dieci, Some numerical considerations and Newton's method revisited for solving algebraic Riccati equations, IEEE Trans. Auto. Control, 36 (1991), pp.~608--616.

[52] A. Feuer and G. C. Goodwin, Sampling in Digital Signal Processing and Control, Birkhauser, New York, 1996.

[53] D. S. Flamm and A. J. Laub, A new shift-invariant representation of perioidc linear systems, Sys. Contr. Lett., 17 (1991), pp.~9--14.

[54] C. Foulard, S. Gentil, and J. P. Sandraz, Commande et regulation par calculateur numerique: De la theorie aux applications, Eyrolles, Paris, 1977.

[55] B. Francis and T. T. Georgiou, Stability theory for linear time-invariant plants with periodic digital controllers, IEEE Trans. Auto. Control, 33 (1988), pp.~820--832.

[56] J. Gardiner and A. J. Laub, A generalization of the matrix-sign-function solution to the algebraic {R}iccati equations, Int. J. Control, 44 (1986), pp.~823--832.

[57] W. A. Gardner, Cyclostationarity in Communnications and Signal Processing, IEEE Press, New York, 1994.

[58] K. Glover, All optimal {H}ankel-norm approximations of linear multivariable systems and their ${L}^{\infty}$-errors bounds, Int. J. Control, 39 (1984), pp.~1115--1193.

[59] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, 1996.

[60] T. Gudmundsson, C. Kenney, and A. J. Laub, Scaling of the discrete-time algebraic {R}iccati equation to enhance stability of the Schur solution method, IEEE Trans. Auto. Control, 37 (1992), pp.~513--518.

[61] S. Hammarling, Newton's method for solving the algebraic Riccati equation}, NPL Rep. DITC 12/82, Nat. Phys. Lab., Teddington, Middlesex TW11 0LW, U.K., 1982.

[62] J. J. Hench and A. J. Laub, Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Auto. Control, 39 (1994), pp.~1197--1210.

[63] N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, PA, 1996.

[64] J. L. Howland, The sign matrix and the separation of matrix eigenvalues, Lin. Alg. Appl., 49 (1983), pp.~221--332.

[65] P. Hr. Petkov, N. D. Christov, and M. M. Konstantinov, On the numerical properties of the {S}chur approach for solving the matrix Riccati equation, Sys. Contr. Lett., 9 (1987), pp.~197--201.

[66] G. D. Ianculescu, J. Ly, A. J. Laub, and P. M. Papadopoulos, Space station freedom solar array ${H}_{\infty}$ control. Talk at 31st IEEE Conf. on Decision and Control, Tucson, AZ, Dec. 1992.

[67] R. W. Isniewski and M. Blanke, Fully magnetic attitude control for spacecraft subject to gravity gradient, Automatica, 35 (1999), pp.~1201--1214.

[68] W. Johnson, Helicopter Theory, Princeton University Press, Princeton, NJ, 1996.

[69] M. Kimura, Convergence of the doubling algorithm for the discrete-time algebraic Riccati equation, Int. J. Syst. Sci., 19 (1988), pp.~701--711.

[70] M. Kimura, Doubling algorithm for continuous-time algebraic Riccati equation, Int. J. Syst. Sci., 20 (1989), pp.~191--202.

[71] D. Kleinman, On an iterative technique for Riccati equation computations, IEEE Trans. Auto. Control, AC-13 (1968), pp.~114--115.

[72] M. Kono, Eigenvalue assignment in linear discrete-time system, Int. J. Control, 32 (1980), pp.~149--158.

[73] Y.-C. Kuo, W.-W. Lin, and S.-F. Xu, Regularrization of linear discrete-time perioidc descriptor systems by derivative and proportional state feedback. To appear in SIAM J. Matrix Analy. Appl., 2004.

[74] D. Lainiotis, N. Assimakis, and S. Katsikas, New doubling algorithm for the discrete periodic Riccati equation, Appl. Maths. Comp., 60 (1994), pp.~265--283.

[75] A. J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Auto. Control, 24 (1979), pp.~913--921.

[76] A. J. Laub, Algebraic aspects of generalized eigenvalue problems for solving Riccati equations, in Computational and Combinatorial Methods in Systems Theory}, C. I. Byrnes and A. Lindquist, eds., Elsevier (North-Holland), 1986, pp.~213--227.

[77] A. J. Laub, Invariant subspace methods for the numerical solution of Riccati equations, in The Riccati Equation, S. Bittanti, A. J. Laub, and J. C. Willems, eds., Springer-Verlag, Berlin, 1991, pp.~163--196.

[78] A. J. Laub, M. T. Heath, C. C. Paige, and R. C. Ward, Computation of system balancing transformations and other applications of simutaneous diagonalization algorithms, IEEE Trans. Auto. Control, 32 (1987), pp.~115--122.

[79] F. L. Lewis, Fundamental, reachability, and observability matrices for discrete descriptor systems, IEEE Trans. Auto. Control, 30 (1985), pp.~502--505.

[80] W.-W. Lin and J.-G. Sun, Perturbation analysis for eigenproblem of periodic matrix pairs, Lin. Alg. Appl., 337 (2001), pp.~157--187.

[81] W.-W. Lin and J.-G. Sun, Perturbation analysis of the periodic discrete-time algebraic Riccati equation, SIAM J. Matrix Analy. Appl., 24 (2002), pp.~411--438.

[82] W.-W. Lin, C.-S. Wang, and Q.-F. Xu, Numerical computation of the minimum $H_{\infty}\/$ norm of the discrete-time output feedback control problem, SIAM J. Numer. Anal., 38 (2000), pp.~515--547.

[83] M.-L. Liou and Y.-L. Kuo, Exact analysis of switched capacitor circuits with arbitrary inputs, IEEE Trans. Circuits and Systems, 26 (1979), pp.~213--223.

[84] L.-Z. Lu and W.-W. Lin, An iterative algorithm for the solution of the discrete time algebraic Riccati equations, Lin. Alg. Appl., 189 (1993), pp.~465--488.

[85] L.-Z. Lu, W.-W. Lin, and C. E. M. Pearce, An efficient algorithm for the discrete-time algebraic Riccati equation, IEEE Trans. Auto. Control, 44 (1999), pp.~1216--1220.

[86] A. N. Malyshev, Parallel algorithm for solving some spectral problems of linear algebra, Lin. Alg. Appl., 188/189 (1993), pp.~489--520.

[87] A. Marzollo, Periodic Optimization, Springer Verlag, Berlin, 1972.

[88] MathWorks, MATLAB user's guide (for UNIX Workstations), The Math Works, Inc., 1992.

[89] R. McKillip, Periodic model following controller for the control-configured helicopter, Journal of the American Helicopter Society, 36 (1991), pp.~4--12.

[90] V. Mehrmann, A symplectic orthogonal method for single input or single output discrete time optimal linear quadratic control problems, SIAM J. Matrix Analy. Appl., (1988), pp.~221--248.

[91] V. Mehrmann, The Autonomous Linear Quadratic Control Problem, Springer-Verlag, 1991.

[92] V. Mehrmann, A step toward a unified treatment of continuous and discrete time control problems, Lin. Alg. Appl., 241-243 (1996), pp.~749--779.

[93] V. Mehrmann and E. Tan, Defect correction methods for the solution of algebraic Riccati equations, IEEE Trans. Auto. Control, 33 (1988), pp.~695--698.

[94] B. C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Auto. Control, 26 (1981), pp.~17--32.

[95] C. C. Paige and C. F. Van Loan , A Schur decomposition for Hamiltonian matrices, Lin. Alg. Appl., 41 (1981), pp.~11--32.

[96] T. Pappas, A. J. Laub, and N. R. Sandell, On the numerical solution of the discrete-time algebraic Riccati equation, IEEE Trans. Auto. Control, 25 (1980), pp.~631--641.

[97] L. Patnaik, N. Viswanadham, and I. Sarma, Computer control algorithms for a tubular ammonia reactor, IEEE Trans. Auto. Control, 25 (1980), pp.~642--651.

[98] T. Penzl, Numerical solution of generalized Lyapunov equations, Adv. Comput. Math., 8 (1998), pp.~33--48.

[99] P. Petkov, N. Christov, and M. Konstantinov, A posteriori error analysis of the generalized Schur approach for solving the discrete matrix Riccati equation, preprint, Department of Automatics, Higher Institute of Mechenical
and Electrical Engineering, 1756 Sofia, Bulgaria, 1989.

[100] M. E. Pittelkau, Optimal periodic control for spacecraft pointing and attitude dtermination, J. of Guidance, Control, and Dynamics, 16 (1993), pp.~1078--1084.

[101] J. A. Richards, Analysis of Periodically Time-Varying Systems, Springer-Verlag, Berlin, 1983.

[102] J. Roberts, Linear model reduction and solution of the algebraic Riccati equation by the use of the sign function, Int. J. Control, 32 (1980), pp.~667--687.

[103] N. Sandell, On Newton's method for Riccati equation solution, IEEE Trans. Auto. Control, 19 (1974), pp.~254--255.

[104] V. Sima, Algorithms for Linear-Quadratic Optimization, volume 200 of Pure and Applied Mathematics}, Marcel Dekker, Inc., New York, NY, 1996.

[105] J. Sreedhar and P. Van Dooren, Periodic Schur form and some matrix equations, Systems and Networks: Mathematical Theory and Applications, 77 (1994), pp.~339--362.

[106] J. Sreedhar and P. Van Dooren, Forward/backward decomposition of periodic descriptor systems and two point boundary value problems, in European Control Conf., 1997.

[107] J. Sreedhar and P. Van Dooren, Periodic descriptor systems: solvability and conditionability, IEEE Trans. Auto. Control, 44 (1999), pp.~310--313.

[108] G. W. Stewart, HQR3 and EXCHNG: F}ortran subroutines for calculating and ordering the eigenvalues of a real upper Hessenberg matrix, ACM Trans. Math. Software, 2 (1976), pp.~275--280.

[109] G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.

[110] T. Stykel, Model reduction of descriptor systems, Technical Report 720-2001, Institut fur Mathematik, TU Berlin, D-10263 Berlin, Germany, 2001.

[111] T. Stykel, Analysis and numerical solution of generalized Lyapunov equations, PhD Dissertation, Institut fur Mathematik, Tecnische Universitat Berlin, Berlin, 2002.

[112] T. Stykel, Stability and inertia theorems for generalized Lyapunov equations, Lin. Alg. Appl., 355 (2002), pp.~297--314.

[113] T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation, Technical Report 04-2003, Institut fur Mathematik, TU Berlin, D-10263 Berlin, Germany, 2003.

[114] T. Stykel, Input-output invariants for descriptor systems, preprint PIMS-03-1, Pacific Institute for the Mathematical Sciences, Canada, 2003.

[115] J.-G. Sun, Sensitivity analysis of the discrete-time algebraic Riccati equation, Lin. Alg. Appl., 275/276 (1998), pp.~595--615.

[116] L. Tong, G. Xu, and T. Kailath, Blind identification and equalization based on second-order statistics: A time domain approach, IEEE Trans. Information Theory, 40 (1994), pp.~340--349.

[117] P. P. Vaidyanathan, Multirate digital filters, filter banks, polyphase networks, and applications: A tutorial, in Proc. IEEE, vol.~78, 1990, pp.~56--93.

[118] P. P. Vaidyanathan, Mutirate Systems and Filter-Banks, Prentice-Hall, Englewood Cliffs, NJ, 1993.

[119] P. Van Dooren, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. Statist. Comput., 2 (1981), pp.~121--135.

[120] P. Van Dooren, Two point boundary value and periodic eigenvalue problems, in Proc. 1999 IEEE Intel. Symp. CACSD, Kohala Coast-Island, Hawaii, USA, K. Kirchgassner et al., ed., August 1999, pp.~22--27.

[121] P. Van Dooren and J. Sreedhar, When is a periodic discrete-time system equivalent to a time invariant one?, Lin. Alg. Appl., 212/213 (1994), pp.~131--151.

[122] A. Varga, Periodic {L}yapunov equations: some applications and new algorithms, Int. J. Control, 67 (1997), pp.~69--87.

[123] A. Varga, Balancing related methods for minimal realization of perioidc systems, Sys. Contr. Lett., 36 (1999), pp.~339--349.

[124] A. Varga, Balanced truncation model reduction of periodic systems, in Proc. of IEEE Conference on Decision and Control, Sydney, Australia, 2000.

[125] A. Varga, Robust and minimum norm pole assignment with periodic state feedback, IEEE Trans. Auto. Control, 45 (2000), pp.~1017--1022.

[126] A. Varga and S. Pieters, Gradient-based approach to solve optimal periodic output feedback control problems, Automatica, 34 (1998), pp.~477--481.

[127] A. Varga and P. Van Dooren, Computing the zeros of periodic descriptor systems, Sys. Contr. Lett., 50 (2003), pp.~371--381.

[128] J. Vlach, K. Singhai, and M. Vlach, Computer oriented formulation of equations and analysis of switched-capacitor networks, IEEE Trans. Circuits and Systems, 31 (1984), pp.~735--765.

[129] J. Xin, H. Kagiwada, A. Sano, H. Tsuj, and S. Yoshimoto, Regularization approach for detection of cyclostationary signals in antenna array processing}, in IFAC Symposium on System Identification, vol. 2, 1997, pp.~529--534.

[130] V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Perioidc Coefficients, Wiley, New York, 1975.

[131] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1996.