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A survey of linear singular systems

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Abstract

This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties. The frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.

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References

  1. Adams, M. B., B. C. Lévy, and A. S. Willsky, “Linear Smoothing for Descriptor Systems,”Proc. 23rd IEEE Conf. Dec. and Control, pp. 1–6, Las Vegas, NV, December 1984.

  2. Aplevich, J. D., “Time Domain Input-Output Representations of Linear Systems,”Automatica, Vol.17, pp. 509–522, 1979.

    Google Scholar 

  3. Armentano, V. A., “Eigenvalue Placement for Generalized Linear Systems,”Systems and Control Letters, Vol.4. pp. 199–202, June 1984a.

    Google Scholar 

  4. Armentano, V. A., “The Pencil (sE-A) and Controllability-Observability for Generalized Linear Systems: A Geometric Approach,”Proc. 23rd IEEE Conf. Dec. and Control, pp. 1507–1510, Las Vegas, NV, December 1984b.

  5. Armentano, V. A., “Exact Disturbance Decoupling by a Proportional-Derivative State Feedback Law,” preprint, 1985.

  6. Bernhard, P., “On Singular Implicit Linear Dynamical Systems,”SIAM J. Control and Optim., Vol.20, No. 5, pp. 612–633, September 1982.

    Google Scholar 

  7. Bhattacharyya, S. P., and V. A. Oliviera, “Simulation and Control of Discrete Generalized State Space Systems via Silverman's Algorithm,” TCSL Research Memorandum 81-10, 1981.

  8. Brayton, R. K., F. G. Gustavson, and G. D. Hachtel, “A New Efficient Algorithm for Solving Differential-Algebraic Systems Using Implicit Backward Differentiation Formulae,”Proc. IEEE, Vol.60, No. 1 pp. 98–114, January 1972.

    Google Scholar 

  9. Brenan, K. E., “Difference Approximation for Higher Index Differential-Algebraic Systems with Applications in Trajectory Control,”Proc. 23rd IEEE Conf. Dec. and Control, pp. 291–292, December 1984.

  10. Brunovsky, P., “A Classification of Linear Controllable Systems,”Kybernetika, Vol.6, pp. 173–188, 1970.

    Google Scholar 

  11. Bryson, A. E., Jr., and Y.-C. Ho,Applied Optimal Control, New York: Hemisphere, 1975.

    Google Scholar 

  12. Campbell, S. L., C. D. Meyer, Jr., and N. J. Rose, “Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Coefficients,”SIAM J. Appl. Math., Vol.10, pp. 542–551, 1976.

    Google Scholar 

  13. Campbell, S. L.,Singular Systems of Differential Equations, San Francisco: Pitman, 1980.

    Google Scholar 

  14. Cobb, D., “Feedback and Pole-Placement in Descriptor Variable Systems,”Int. J. Control, Vol.33, No. 6, pp. 1135–1146, 1981.

    Google Scholar 

  15. Cobb, D., “On the Solution of Linear Differential Equations with Singular Coefficients,”J. Diff. Eq., Vol.46, pp. 310–323, 1982.

    Google Scholar 

  16. Cobb, D., “Descriptor Variable Systems and Optimal State Regulation,”IEEE Trans. Automat. Control, Vol.AC-28, No. 5, pp. 601–611, May 1983.

    Google Scholar 

  17. Cobb, J. D.,Descriptor Variable and Generalized Singularly Perturbed Systems: A Geometric Approach, Ph.D. Thesis, Department of Electrical Engineering, University of Illinois, 1980.

  18. Cobb, J. D., “Controllability, Observability, and Duality in Singular Systems,”IEEE Trans. Automat. Control, Vol.AC-29 pp. 1076–1082, 1984a.

    Google Scholar 

  19. Cobb, J. D., “Slow and Fast Stability in Singular Systems,”Proc. 23rd IEEE Conf. Dec. and Control, pp. 280–282, December 1984b.

  20. DeClaris, N., and A. Rindos, “Semistate Analysis of Neural Networks in Apysia Californica,”Proc. 27th MSCS, Morgantown, WV, pp. 686–689, 1984.

  21. Dziurla, B., and R. Newcomb, “The Drazin Inverse and Semi-State Equations,”Proc. 4th Int. Symp. Math. Theory of Networks and Systems, pp. 283–289, Delft, The Netherlands, 1979.

  22. Dziurla, B., and R. W. Newcomb, “An Example of the Continuation Method of Solving Semistate Equations,”Proc. 23rd IEEE Conf. Dec. and Control, pp. 274–279, December 1984.

  23. Fettweis, A., “On the Algebraic Derivation of State Equations,”IEEE Trans. Circuit Theory, Vol.CT-16, pp. 171–175, 1969.

    Google Scholar 

  24. Gantmacher, F. R.,Theory of Matrices, New York: Chelsea Pub. Co., 1959.

    Google Scholar 

  25. Gear, C. W., “Simultaneous Numerical Solution of Differential-Algebraic Equations,”IEEE Trans. Circuit Theory, Vol.CT-18, pp. 89–95, 1971.

    Google Scholar 

  26. Gohberg, I., and L. Rodman, “On Spectral Analysis of Non-Monic Matrix and Operator Polynomials, I. Reduction to Monic Polynomials,”Israel J. Mathematics, Vol.30, pp. 133–151, 1978.

    Google Scholar 

  27. Hayton, G. E., P. Fretwell, and A. C. Pugh, “Fundamental Equivalence of Generalized State Space Systems,”Proc. 23rd IEEE Conf. Decision and Control, pp. 289–290, Las Vegas, NV, December 1984.

  28. Karcanias, N., and G. E. Hayton, “State-Space and Transfer Function Invariant Infinite Zeros: A Unified Approach,”Proc. JACC, paper TA-4C, Charlottesville, VA, 1981a.

  29. Karcanias, N., and G. E. Hayton, “Generalized Autonomous Dynamical Systems, Algebraic Duality and Geometric Theory,”IFAC VIII Triennial World Congress, Kyoto, Japan, August 1981b.

  30. Khasina, E. N., “Control of Singular Linear Dynamic Systems,”Autom. Rem. Control, Vol.43, No. 4, pp. 448–455, April 1982.

    Google Scholar 

  31. Kokotovic, P. V., R. E. O'Malley, Jr., and P. Sannuti, “Singular Perturbations and Order Reduction in Control Theory—An Overview,”Automatica, Vol.12, pp. 123–132, March 1976.

    Google Scholar 

  32. Kronecker, L., “Algebraishe Reduction der Schaaren Bilinearer Formen,”S.-B. Akad, Berlin, pp. 763–776, 1980.

  33. Lancaster, P., “A Fundamental Theorem on Lambda Matrices with Applications: 1. Ordinary Differential Equations with Constant Coefficients,”Linear Algebra and Its Applications, Vol.18, pp. 189–211, 1977.

    Google Scholar 

  34. Langenhop, C. E., “The Laurent Expansion for a Nearly Singular Matrix,”Linear Algebra and Its Applications, Vol.4, pp. 329–340, 1971.

    Google Scholar 

  35. Langenhop, C. E., “Controllability and Stabilization of Regular Singular Linear Systems with Constant Coefficients,” Dept. of Math., S. Illinois University, December 6, 1979.

  36. Larson, R. E., D. G. Luenberger, and D. N. Stengel, “Descriptor Variable Theory and Spatial Dynamic Programming,” Top. Rep., Systems Control, Inc., 1978.

  37. Lewis, F., “Descriptor Systems: Expanded Descriptor Equation and Markov Parameters,”IEEE Trans. Automat. Control, Vol.AC-28, No. 5, pp. 623–627, May 1983a.

    Google Scholar 

  38. Lewis, F., “Inversion of Descriptor Systems,”Proc. ACC, pp. 1153–1158, San Francisco, CA, June 1983b.

  39. Lewis, F. L., “Adjoint Matrix, Bézout Theorem, Cayley-Hamilton Theorem, and Fadeev's Method for the Matrix Pencil (sE-A),”Proc. 22nd Conf. Dec. and Control, pp. 1282–1288, December 1983c.

  40. Lewis, F. L., “Descriptor Systems: Decomposition into Forward and Backward Subsystems,”IEEE Trans. Automat. Control, Vol.AC-29, No. 2, pp. 167–170, February 1984.

    Google Scholar 

  41. Lewis, F. L. “Fundamental, Reachability, and Observability Matrices for Discrete Descriptor Systems,”IEEE Trans. Automat. Control, Vol.AC-30, pp. 502–505, May 1985a.

    Google Scholar 

  42. Lewis, F. L., “Preliminary Notes on Optimal Control for Singular Systems,”Proc. 24th IEEE Conf. on Dec. and Control, Ft. Lauderdale, FL, December 1985b.

  43. Lewis, F. L., “Optimal Control for Singular Systems,” in preparation, 1986.

  44. Lewis, F. L., and K. Ozcaldiran, “The Relative Eigenstructure Problem and Descriptor Systems,” SIAM National Meeting, Denver, CO, June 1983.

  45. Lewis, F. L., and K. Ozcaldiran, “Reachability and Controllability for Descriptor Systems,”Proc. 27th Midwestern Symp. Circuits and Sys., pp. 690–695, Morgantown, WV, June 1984.

  46. Lewis, F. L., and K. Ozcaldiran, “On the Eigenstructure Assignment of Singular Systems,”Proc. 24th IEEE Conf. on Dec. and Control, Ft. Lauderdale, FL, December 1985.

  47. Lostedt, P., and L. R. Petzold, “Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints,” Sandia National Laboratories, Report SAND83-8877, 1983.

  48. Luenberger, D. G., “Dynamic Equations in Descriptor Form,”IEEE Trans. Automat. Control, Vol.AC-22, pp. 312–321, 1977.

    Google Scholar 

  49. Luenberger, D. G., “Time-Invariant Descriptor Systems,”Automatica, Vol.14, pp. 473–480, 1978.

    Google Scholar 

  50. Luenberger, D. G., “Nonlinear Descriptor Systems,”J. Economic Dynamics and Control, Vol.1, pp. 219–242, 1979.

    Google Scholar 

  51. Luenberger, D. G., and A. Arbel, “Singular Dynamic Leontief Systems,”Econometrica, 1977.

  52. Manke, J. W.,et al., “Solvability of Large-Scale Descriptor Systems,” Boeing Computer Services Co., 1978.

  53. McMillan, B., “Introduction to Formal Realizability Theory,”Bell Syst. Tech. Journal, Vol.31, No. 2, pp. 217–279, March 1952; Vol.31, No. 4, pp. 541–600, May 1952.

    Google Scholar 

  54. Mertzios, B. G., “Leverrier's Algorithm for Singular Systems,”IEEE Trans. Automat. Control, Vol.AC-29, No. 7, pp. 652–653, July 1984.

    Google Scholar 

  55. Molinari, B. P., “Structural Invariants of Linear Multivariable Systems,”Int. J. Control, Vol.28, pp. 525–535, 1979.

    Google Scholar 

  56. Moore, B. C., “On the Flexibility Offered by State Feedback in Multivariable Systems Beyond Closed Loop Eigenvalue Assignment,”IEEE Trans. Automat. Control, pp. 689–692, October 1976.

  57. Mukundan, R., and W. Dayawansa, “Feedback Control of Singular Systems — Proportional and Derivative Feedback of the State,”Int. J. Systems Sci., Vol.14, pp. 615–632, 1983.

    Google Scholar 

  58. Newcomb, R. W.,Linear Multiport Synthesis, New York: McGraw-Hill, 1966.

    Google Scholar 

  59. Newcomb, R. W., “The Semistate Description of Nonlinear Time-Variable Circuits,”IEEE Trans. Circuits and Systems, Vol.CAS-28, No. 1, pp. 62–71, January 1981.

    Google Scholar 

  60. Newcomb, R. W., “Semistate Design Theory: Binary and Swept Hysteresis,”J. Circuits, Sys., Sig. Proc., Vol.1, No. 2, pp. 203–216, 1982.

    Google Scholar 

  61. Ozcaldiran, K.,Control of Descriptor Systems, Ph.D. Thesis, School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA, June 1985.

    Google Scholar 

  62. Ozcaldiran, K., “A Geometric Characterization of the Reachable and the Controllable Subspaces of Descriptor Systems,”Circuits, Sys., Sig. Proc., this issue, 1986.

  63. Ozcaldiran, K., and F. L. Lewis, “A Result on the Placement of Infinite Eigenvalues in Descriptor Systems,” Proc.ACC, pp. 366–371, San Diego, CA, June 1984.

  64. Pandolfi, L., “Controllability and Stabilization for Linear Systems of Algebraic and Differential Equations,”J. Optimization Theory and Applic., Vol.30, pp. 601–620, 1980.

    Google Scholar 

  65. Pandolfi, L., “On the Regulator Problem for Linear Degenerate Control Systems,”JOTA, Vol.33, No. 2, pp. 241–254, February 1981.

    Google Scholar 

  66. Pugh, A. C., and G. E. Hayton, “The Extended State Space and Matrix Pencils,”Proc. 23rd IEEE Conf. Dec. and Control, Las Vegas, NV, December 1984.

  67. Pugh, A. C., P. Fretwell, and G. E. Hayton, “Some Transformations of Matrix Equivalence Arising from Linear Systems Theory,”Proc. ACC, pp. 633–637, San Francisco, CA June 1983.

  68. Pugh, A. C., and P. A. Ratcliffe, “On the Zeros and Poles of a Rational Matrix,”Int. J. Control, Vol.30, pp. 213–226, 1979.

    Google Scholar 

  69. Rose, N. J., “The Laurent Expansion of a Generalized Resolvent with Some Applications,”SIAM J. Math. Anal., Vol.9, pp. 751–758, 1978.

    Google Scholar 

  70. Rosenbrock, H. H.,State-Space and Multivariable Theory, London: Nelson, 1970.

    Google Scholar 

  71. Rosenbrock, H. H., “Structural Properties of Linear Dynamical Systems,”Int. J. Control, Vol.20, 191–202, 1974.

    Google Scholar 

  72. Rosenbrock, H. H., and A. C. Pugh, “Contributions to a Hierarchical Theory of Systems,”Int. J. Control, Vol.19, No. 5, pp. 845–867, 1974.

    Google Scholar 

  73. Saidahmed, M. T., and M. E. Zaghloul, “On the Generalized State-Space Singular Linear Systems,”Proc. IEEE Int. Symp. Circuits and Systems, pp. 653–656, Newport Beach, CA, 1983.

  74. Sastry, S. S., and C. A. Desoer, “Jump Behavior of Circuits and Systems,”IEEE Trans. Circuits and Systems, Vol.CAS-28, No. 12, pp. 1109–1123, December 1981.

    Google Scholar 

  75. Silverman, L. M., “Discrete Riccati Equations: Alternative Algorithms, Asymptotic Properties, and System Theory Interpretations,”Control and Dynamic Systems, Vol.12, C. T. Leondes, ed., pp. 313–386, New York: Academic, 1976.

    Google Scholar 

  76. Sincovec, R. F., A. M. Erisman, E. L. Yip, and M. A. Epton, “Analysis of Descriptor Systems Using Numerical Algorithms,”IEEE Trans. Automat. Control, Vol.AC-26, No. 1, pp. 139–147, February 1981.

    Google Scholar 

  77. Singh, S. P. and R-W Liu, “Existence of State Equation Representation of Linear Large-Scale Dynamical Systems,”IEEE Trans. Circuit Theory, Vol.CT-20, No. 3, pp. 239–246, May 1973.

    Google Scholar 

  78. Spong, M. W., “A Semistate Approach to Feedback Stabilization of Neutral Delay Systems,”Circuits, Sys., Sig. Proc., this issue, 1986.

  79. Stevens, B. L., “Modeling, Simulation, and Analysis with State Variables,” Report LG84RR002, Lockheed-Georgia Co., Marietta, GA, June 1984.

    Google Scholar 

  80. Stott, B., “Power System Response Dynamic Calculations,”Proc. IEEE, Vol.67, No. 2, pp. 219–241, February 1979.

    Google Scholar 

  81. Van Dooren, P., “The Computation of Kronecker's Canonical Form of a Singular Pencil,”Linear Algebra and Its Applications, Vol.27, pp. 103–140, 1979.

    Google Scholar 

  82. Van Dooren, P., “The Generalized Eigenstructure Problem in Linear System Theory,”IEEE Trans. Automat. Control, Vol.AC-26, pp. 111–129, 1981.

    Google Scholar 

  83. Verghese, G. C.,Infinite-Frequency Behavior in Generalized Dynamical Systems, Ph.D. Thesis, Dept. of Electrical Engineering, Stanford University, 1978.

  84. Verghese, G. C., and T. Kailath, “Impulsive Behavior in Dynamical Systems: Structure and Significance,”Proc. 4th Int. Symp, Math. Theory, Networks, Sys., pp. 162–168, Delft, The Netherlands, July 1979a.

  85. Verghese, G. C., and T. Kailath, “Eigenvector Chains for Finite and Infinite Zeros of Rational Matrices,”Proc. 18th Conf. Dec. and Control, pp. 31–32, Ft. Lauderdale, FL, December 1979b.

  86. Verghese, G. C., “Further Notes on Singular Systems,”Proc. JACC, paper TA-4B, Charlottesville, VA, June 1981.

  87. Verghese, G. C., B. C. Lévy, and T. Kailath, “A Generalized State-Space for Singular Systems,”IEEE Trans. Automat. Control, Vol.AC-26, pp. 811–831, 1981.

    Google Scholar 

  88. Verghese, G., P. Van Dooren, and T. Kailath, “Properties of the System Matrix of a Generalized State-Space System,”Int. J. Control, Vol.30, No. 2, pp. 235–243, 1979.

    Google Scholar 

  89. Weierstrass, K., “Zur Theorie der Bilinearen und Quadratischen Formen,”Monatsh. Akad. Wiss. Berlin, pp. 310–338, 1867.

  90. Wilkinson, J. H., “Linear Differential Equations and Kronecker's Canonical Form,”Recent Advances in Numerical Analysis, C. de Boor and G. Golub, ed., pp. 231–265, New York: Academic, 1978.

    Google Scholar 

  91. Wong, K. T., “The Eigenvalue Problem λTx + Sx,”J. Diff. Eq., Vol.16, pp. 270–280, 1974.

    Google Scholar 

  92. Wonham, W. M.,Linear Multivariable Control: A Geometric Approach, 2nd ed., New York: Springer-Verlag, 1979.

    Google Scholar 

  93. Yip, E. L., and R. F. Sincovec, “Solvability, Controllability, and Observability of Continuous Descriptor Systems,”IEEE Trans. Automat. Control, Vol.AC-26, No. 3, pp. 702–707, June 1981.

    Google Scholar 

  94. Zaghloul, M. E., and R. W. Newcomb, “Semistate Implementation: Differentiator Example,”Circuits, Systems, and Sig. Proc., this issue.

  95. Zeeman, E. C., “Duffing's Equation in Brain Modelling,”J. Inst. Math. and Its App., pp. 207–14, July 1976.

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    F. L. Lewis

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Lewis, F.L. A survey of linear singular systems. Circuits Systems and Signal Process 5, 3–36 (1986). https://doi.org/10.1007/BF01600184

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