A survey of linear singular systems. (English) Zbl 0613.93029

This is a historical survey of system theory for singular linear systems, i.e., systems of the form Eẋ\(=Ax+Bu\) with E possibly singular. Topics discussed include: a formula for the solution (extending the variation of constants formula), controllability and observability, stability and feedback, regulator theory, classification and equivalence.
Reviewer: P.Brunovsky


93C05 Linear systems in control theory
15A09 Theory of matrix inversion and generalized inverses
34A99 General theory for ordinary differential equations
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
93B05 Controllability
93B07 Observability
93D15 Stabilization of systems by feedback
Full Text: DOI


[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. · Zbl 0463.93031
[3] Armentano, V. A., ”Eigenvalue Placement for Generalized Linear Systems,”Systems and Control Letters, Vol.4. pp. 199–202, June 1984a. · Zbl 0538.93024
[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. · Zbl 0491.93004
[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.
[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. · Zbl 0199.48202
[11] Bryson, A. E., Jr., and Y.-C. Ho,Applied Optimal Control, New York: Hemisphere, 1975.
[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.
[13] Campbell, S. L.,Singular Systems of Differential Equations, San Francisco: Pitman, 1980. · Zbl 0419.34007
[14] Cobb, D., ”Feedback and Pole-Placement in Descriptor Variable Systems,”Int. J. Control, Vol.33, No. 6, pp. 1135–1146, 1981. · Zbl 0464.93039
[15] Cobb, D., ”On the Solution of Linear Differential Equations with Singular Coefficients,”J. Diff. Eq., Vol.46, pp. 310–323, 1982. · Zbl 0489.34006
[16] Cobb, D., ”Descriptor Variable Systems and Optimal State Regulation,”IEEE Trans. Automat. Control, Vol.AC-28, No. 5, pp. 601–611, May 1983. · Zbl 0522.93036
[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.
[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. · Zbl 0499.94035
[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. · Zbl 0531.93036
[23] Fettweis, A., ”On the Algebraic Derivation of State Equations,”IEEE Trans. Circuit Theory, Vol.CT-16, pp. 171–175, 1969.
[24] Gantmacher, F. R.,Theory of Matrices, New York: Chelsea Pub. Co., 1959. · Zbl 0085.01001
[25] Gear, C. W., ”Simultaneous Numerical Solution of Differential-Algebraic Equations,”IEEE Trans. Circuit Theory, Vol.CT-18, pp. 89–95, 1971.
[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. · Zbl 0396.47009
[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. · Zbl 0616.93002
[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. · Zbl 0525.93015
[30] Khasina, E. N., ”Control of Singular Linear Dynamic Systems,”Autom. Rem. Control, Vol.43, No. 4, pp. 448–455, April 1982. · Zbl 0507.93013
[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. · Zbl 0323.93020
[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. · Zbl 0388.15003
[34] Langenhop, C. E., ”The Laurent Expansion for a Nearly Singular Matrix,”Linear Algebra and Its Applications, Vol.4, pp. 329–340, 1971. · Zbl 0224.15005
[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. · Zbl 0517.93005
[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. · Zbl 0534.93013
[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. · Zbl 0557.93011
[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. · Zbl 0623.93031
[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. · Zbl 0354.93007
[49] Luenberger, D. G., ”Time-Invariant Descriptor Systems,”Automatica, Vol.14, pp. 473–480, 1978. · Zbl 0398.93040
[50] Luenberger, D. G., ”Nonlinear Descriptor Systems,”J. Economic Dynamics and Control, Vol.1, pp. 219–242, 1979.
[51] Luenberger, D. G., and A. Arbel, ”Singular Dynamic Leontief Systems,”Econometrica, 1977. · Zbl 0368.90029
[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.
[54] Mertzios, B. G., ”Leverrier’s Algorithm for Singular Systems,”IEEE Trans. Automat. Control, Vol.AC-29, No. 7, pp. 652–653, July 1984. · Zbl 0541.93018
[55] Molinari, B. P., ”Structural Invariants of Linear Multivariable Systems,”Int. J. Control, Vol.28, pp. 525–535, 1979.
[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. · Zbl 0332.93047
[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. · Zbl 0509.34004
[58] Newcomb, R. W.,Linear Multiport Synthesis, New York: McGraw-Hill, 1966.
[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.
[60] Newcomb, R. W., ”Semistate Design Theory: Binary and Swept Hysteresis,”J. Circuits, Sys., Sig. Proc., Vol.1, No. 2, pp. 203–216, 1982.
[61] Ozcaldiran, K.,Control of Descriptor Systems, Ph.D. Thesis, School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA, June 1985. · Zbl 0606.93017
[62] Ozcaldiran, K., ”A Geometric Characterization of the Reachable and the Controllable Subspaces of Descriptor Systems,”Circuits, Sys., Sig. Proc., this issue, 1986. · Zbl 0606.93017
[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. · Zbl 0397.93006
[65] Pandolfi, L., ”On the Regulator Problem for Linear Degenerate Control Systems,”JOTA, Vol.33, No. 2, pp. 241–254, February 1981. · Zbl 0421.93036
[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. · Zbl 0416.15013
[69] Rose, N. J., ”The Laurent Expansion of a Generalized Resolvent with Some Applications,”SIAM J. Math. Anal., Vol.9, pp. 751–758, 1978. · Zbl 0387.40005
[70] Rosenbrock, H. H.,State-Space and Multivariable Theory, London: Nelson, 1970. · Zbl 0246.93010
[71] Rosenbrock, H. H., ”Structural Properties of Linear Dynamical Systems,”Int. J. Control, Vol.20, 191–202, 1974. · Zbl 0285.93019
[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. · Zbl 0286.93002
[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. · Zbl 0476.93036
[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. · Zbl 0362.49014
[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. · Zbl 0495.93027
[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.
[78] Spong, M. W., ”A Semistate Approach to Feedback Stabilization of Neutral Delay Systems,”Circuits, Sys., Sig. Proc., this issue, 1986. · Zbl 0614.93051
[79] Stevens, B. L., ”Modeling, Simulation, and Analysis with State Variables,” Report LG84RR002, Lockheed-Georgia Co., Marietta, GA, June 1984.
[80] Stott, B., ”Power System Response Dynamic Calculations,”Proc. IEEE, Vol.67, No. 2, pp. 219–241, February 1979.
[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. · Zbl 0416.65026
[82] Van Dooren, P., ”The Generalized Eigenstructure Problem in Linear System Theory,”IEEE Trans. Automat. Control, Vol.AC-26, pp. 111–129, 1981. · Zbl 0462.93013
[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. · Zbl 0505.93013
[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. · Zbl 0541.34040
[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. · Zbl 0418.93016
[89] Weierstrass, K., ”Zur Theorie der Bilinearen und Quadratischen Formen,”Monatsh. Akad. Wiss. Berlin, pp. 310–338, 1867. · JFM 01.0054.04
[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.
[91] Wong, K. T., ”The Eigenvalue Problem {\(\lambda\)}Tx + Sx,”J. Diff. Eq., Vol.16, pp. 270–280, 1974. · Zbl 0327.15015
[92] Wonham, W. M.,Linear Multivariable Control: A Geometric Approach, 2nd ed., New York: Springer-Verlag, 1979. · Zbl 0424.93001
[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. · Zbl 0482.93013
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.