第一部分，我們想利用狀態和微分回饋控制把一個時變週期離散的系統正則化，在這部分我們找到一個可以正則化的充分條件，並利用構造方法證明在這充分條件之下可以構造出狀態和微分回饋控制矩陣，使的這週期閉迴路系統可以被正則化。另外也對於有限特徵值配置的反問題有一些結果。
第二部分，我們是對二次特徵值反問題加以探討，也就是說給定部份特徵值和特徵向量要反求二次 n 維矩陣對，在這部分主要分成兩種不同的類型，第一種類型是給定部份特徵值的個數是不大於 n 個，第二種類型是解決給定部份特徵值的個數是等於 n+1 個。這兩類問題主要都是利用構造的方法構造出二次矩陣對，但解決的技巧是全然不同的。

In Chapter 1, we consider the regularization problem for the linear time-varying
discrete-time periodic descriptor systems by derivative and proportional state feedback
controls. Sufficient conditions are given under which derivative and proportional
state feedback controls can be constructed so that the periodic closed-loop
systems are regular and of index at most one. The construction procedures used to
establish the theory are based on orthogonal and elementary matrix transformations
and can, therefore, be developed to a numerically efficient algorithm. The problem
of finite pole assignment of periodic descriptor systems is also studied.
In Chapter 2, the inverse eigenvalue problem of constructing real and symmetric
square matrices M,C and K of size n × n for the quadratic pencil Q(λ) = λ^2M +
λC + K so that Q(λ) has a prescribed subset of eigenvalues and eigenvectors is
considered. This chapter consists of two parts addressing two related but different
problems.

References
[1] http://me.lsu.edu/~ram/papers/publications.html.
[2] M. C. Berg, N. Amit, and J. D. Powell. Multirate digital control system design. IEEE Trans. Aut. Contr., 33, 1988.
[3] S. Bittanti. Deterministic and stochastic linear periodic systems. Springer-Verlag,
New York, 1986.
[4] S. Bittanti and P. Bolzern. Discrete-time linear periodic systems: Graimian and
modal criteria for reachability and controllability. Int. J. Contr., 41:909–928, 1985.
[5] S. Bittanti, P. Colaneri, and G. de Nicolao. The difference periodic riccati equation
for the periodic prediction problem. IEEE Trans. Aut. Contr., 33:706–712, 1988.
[6] A. Bojanczyk, G. Golub, and P. Van Dooren. The periodic schure decomposition. algorithms
and applications. Proceedings of the SPIE Conference, San Diego, 1770:31–
42, 1992.
[7] A. Bunse-Gerstner, R. Byers, V. Mehrmann, and N.K. Nichols. Feedback design for
regularizing descriptor systems. Linear Algebra Appl., 299:119–151, 1999.
[8] A. Bunse-Gerstner, V. Mehrmann, and N.K. Nichols. Regularization of descriptor
systems by derivative and proportional state feedback. SIAM J. Matrix Anal. Appl.,
13:46–67, 1992.
[9] A. Bunse-Gerstner, V. Mehrmann, and N.K. Nichols. Regularization of descriptor
systems by output feedback. IEEE Trans. Aut. Contr., AC-39:1742–1747, 1994.
[10] R. Byers, T. Greerts, and V. Mehrmann. Descriptor systems without controllability
at infinity. SIAM J. Control Optim., 35:462–479, 1997.
[11] R. Byers, P. Kunkel, and V. Mehrmann. Regularization of linear descriptor systems
with variable coefficients. SIAM J. Control Optim., 35:117–133, 1997.
[12] J. Carvalho, B. N. Datta, W. W. Lin, and C. S. Wang. Eigenvalue embedding in
a quadratical pencil using symmetric low rank updates. NCTS preprint in Math.,
2001.
[13] D.L. Chu, H. C. Chan, and D.W.C. Ho. Regularization of singular systems by
derivative and proportional output feedback. SIAM J. Matrix Anal., 19:21–38, 1998.
[14] D.L. Chu and V. Mehrmann. Disturbance decoupling for descriptor systems by state
feedback. SIAM J. Control Optim., 38:1830–1858, 2000.
[15] D.L. Chu, V. Mehrmann, and N.K. Nichols. Minimum norm regularization of descriptor
systems by mixed output feedback. Linear Algebra Appl., 296:39–77, 1999.
[16] E. K-W Chu and B. N. Datta. Numerically robust pole assignment for second-order
systems. International Journal of Control., 64:1113–1127, 1996.
[17] Moody T. Chu. Inverse eigenvalue problems. SIAM Rev., 40:1–39, 1998.
[18] Moody T. Chu and Gene H. Golub. Structured inverse eigenvalue problems. Acta
Numerica, 11:1–71, 2002.
[19] B. N. Datta. Finite element model updating, eigenstructure assignment and eigenvalue
embedding techniques for vibrating systems, the special issue on vibration
control. Mechanical Systems and Signal Processing, 16:83–96, 2002.
[20] B. N. Datta, S. Elhay, Y. M. Ram, and D. R. Sarkissian. Partial eigenstructure
assignment for the quadratic pencil. Journal of Sound and Vibration, 230:101–110,
2000.
[21] B. N. Datta and D. R. Sarkissian. Theory and computations of some inverse eigenvalue
problems for the quadratic pencil, in contemporary mathematics, volume onstructured
matrices in operator theory, control, and signal and image processing.
American Mathematical Society, pages 221–240, 2001.
[22] P. Van Dooren and J. Sreedhar. When is a periodic discrete-time system equivalent
to a time-invariant one? Linear Algbra Appl., 212/213:131–151, 1994.
[23] W. R. Ferng, W. W. Lin, D. Pierce, and C. S. Wang. Nonequivalence transformation
of λ-matrix eigenproblems and model embedding approach to model tunning. Num.
Lin. alg. appl., 8:53–70, 2001.
[24] D.S. Flamm and A.J. Laub. A new shift-invariant representation of periodic linear
systems. Systems and Control Lett., 17:9–14, 1991.
[25] B. Francis and T. Georgiou. Stability theory for linear time-invariant plants with
periodic digital controllers. IEEE Trans. Aut. Contr., 33:820–832, 1988.
[26] F.R. Gantmacher. The Theory of Matrices. Chelsea, New York, 1959.
[27] T. Geerts. Solvability conditions, consistency, and weak consistency for linear
differential-algebraic equations and time-invariant linear systems: The general case.
Linear Algebra. Appl., 181:111–130, 1993.
[28] I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press,
New York, 1982.
[29] J. J. Hench and A. J. Laub. Numerical solution of the discrete-time periodic riccati
equation. IEEE Trans. Aut. Contr., 39:1197–1210, 1994.
[30] J. Kautsky, N.K. Nichols, and E.K.-W. Chu. Robust pole assignment in singular
control systems. Linear Algebra Appl., 121:9–37, 1989.
[31] M. Kono. Eigenvalue assignment in linear discrete-time system. Int. J. Control.,
32:149–158, 1980.
[32] P. Kunkel, V. Mehrmann, and W. Rath. Analysis and numerical solution of control
problems in desciptor form. Math. Control Signals Systems, 14:29–61, 2001.
[33] P. Lancaster and M. Tismenetsky. The Theory of Matrices. Academic Press, Orlando,
Florida, 1985.
[34] W.W. Lin, P. Van Dooren, and Q.F. Xu. Equivalent characterizations of periodic
invariant subspaces. NCTS-TR-0215, National Tsing Hua Unv., Taiwain, 2002.
[35] W.W. Lin and J.G. Sun. Perturbation analysis for the eigenproblem of periodic
matrix pairs. Linear Algebra Appl., 337:157–187, 2001.
[36] M.L. Liou and Y. L. Kuo. Exact analysis of switched capacitor circuits with arbitrary
inputs. IEEE Trans. Circuits and Systems, 26:213–223, 1979.
[37] N. K. Nichols and J. Kautsky. Robust eigenstructure assignment in quadratic matrix
polynomials: nonsingular case. SIAM J. Matrix Anal. Appl., 23:77–102, 2001.
[38] Y. M. Ram and S. Elhay. An inverse eigenvalue problem for the symmetric tridiagonal
quadratic pencil with application of damped oscillatory systems. SIAM J. Applied
Math., 56:232–244, 1996.
[39] J.A. Richards. Analysis of Periodically Time-Varying Systems. Springer-Verlag,
Berlin, 1983.
[40] D. D. Sivan and Y. M. Ram. Physical modifications to vibratory systems with
assigned eigendata. ASME J. Applied Mechanics, 66:427–432, 1999.
[41] J. Sreedhar and P. Van Dooren. Forward/backward decomposition of periodic desciptor
systems. Proc. 1997 ECC Brussels,Belgium,, pages FR–A–L7, 1997.
[42] J. Sreedhar and P. Van Dooren. Periodic descriptor system: solvability and conditionability.
IEEE Trans. Aut. Contr., AC-44:310–313, 1999.
[43] L. Starek and D. J. Inman. Symmetric inverse eigenvalue vibration problem and its
applications. Mechanical Systems and Signal Processing, 15:11–29, 2001.
[44] Fran¸coise Tisseur and Karl Meerbergen. The quadratic eigenvalue problem. SIAM
Review, 43:235–286, 2001.
[45] A. Varga. Balancing related methods for minimal realization of periodic systems.
Systems and Control Lett., 36:339–349, 1999.
[46] A. Varga. Robust and minimum norm pole assignment with periodic state feedback.
IEEE Trans. Aut. Contr., 45:1017–1022, 2000.
[47] J. Vlach, K. Singhal, and M. Vlach. Computer oriented formulation of equations
and analysis of switched-capacitor networks. IEEE Trans. Circuits and Systems,
31:735–765, 1984.
[48] D. C. Zimmerman and M. Widengren. Correcting finite element models using a
symmetric eigenstructure assignment technique. AIAA J., 28, No. 9:1670–1676, 1990.