Advertisement

Controllability of Linear Differential-Algebraic Systems—A Survey

  • Thomas BergerEmail author
  • Timo Reis
Chapter
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

Different concepts related to controllability of differential-algebraic equations are described. The class of systems considered consists of linear differential-algebraic equations with constant coefficients. Regularity, which is, loosely speaking, a concept related to existence and uniqueness of solutions for any inhomogeneity, is not required in this article. The concepts of impulse controllability, controllability at infinity, behavioral controllability, and strong and complete controllability are described and defined in the time domain. Equivalent criteria that generalize the Hautus test are presented and proved.

Special emphasis is placed on normal forms under state space transformation and, further, under state space, input and feedback transformations. Special forms generalizing the Kalman decomposition and Brunovský form are presented. Consequences for state feedback design and geometric interpretation of the space of reachable states in terms of invariant subspaces are proved.

Keywords

Differential-algebraic equations Controllability Stabilizability Kalman decomposition Canonical form Feedback Hautus criterion Invariant subspaces 

Mathematics Subject Classification (2010)

34A09 15A22 93B05 15A21 93B25 93B27 93B52 

Notes

Acknowledgements

We are indebted to Harry L. Trentelman (University of Groningen) for providing helpful comments on the behavioral approach.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975) zbMATHGoogle Scholar
  2. 2.
    Anderson, B.D.O., Vongpanitlerd, S.: Network Analysis and Synthesis—A Modern Systems Theory Approach. Prentice-Hall, Englewood Cliffs (1973) Google Scholar
  3. 3.
    Aplevich, J.D.: Minimal representations of implicit linear systems. Automatica 21(3), 259–269 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aplevich, J.D.: Implicit Linear Systems. Lecture Notes in Control and Information Sciences, vol. 152. Springer, Berlin (1991) zbMATHCrossRefGoogle Scholar
  5. 5.
    Armentano, V.A.: Eigenvalue placement for generalized linear systems. Syst. Control Lett. 4, 199–202 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Armentano, V.A.: The pencil (sEA) and controllability-observability for generalized linear systems: a geometric approach. SIAM J. Control Optim. 24, 616–638 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998) zbMATHCrossRefGoogle Scholar
  8. 8.
    Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Grundlehren der mathematischen Wissenschaften, vol. 264. Springer, Berlin (1984) zbMATHCrossRefGoogle Scholar
  9. 9.
    Aubin, J.P., Frankowska, H.: Set Valued Analysis. Birkhäuser, Boston (1990) zbMATHGoogle Scholar
  10. 10.
    Augustin, F., Rentrop, P.: Numerical methods and codes for differential algebraic equations. In: Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, vol. 2. Springer, Berlin (2012) Google Scholar
  11. 11.
    Banaszuk, A., Przyłuski, K.M.: On perturbations of controllable implicit linear systems. IMA J. Math. Control Inf. 16, 91–102 (1999) zbMATHCrossRefGoogle Scholar
  12. 12.
    Banaszuk, A., Kociȩcki, M., Przyłuski, K.M.: On Hautus-type conditions for controllability of implicit linear discrete-time systems. Circuits Syst. Signal Process. 8(3), 289–298 (1989) zbMATHCrossRefGoogle Scholar
  13. 13.
    Banaszuk, A., Kociȩcki, M., Przyłuski, K.M.: Implicit linear discrete-time systems. Math. Control Signals Syst. 3(3), 271–297 (1990) zbMATHCrossRefGoogle Scholar
  14. 14.
    Banaszuk, A., Kociȩcki, M., Przyłuski, K.M.: Kalman-type decomposition for implicit linear discrete-time systems, and its applications. Int. J. Control 52(5), 1263–1271 (1990) zbMATHCrossRefGoogle Scholar
  15. 15.
    Banaszuk, A., Kociȩcki, M., Lewis, F.L.: Kalman decomposition for implicit linear systems. IEEE Trans. Autom. Control 37(10), 1509–1514 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Basile, G., Marro, G.: Controlled and Conditioned Invariants in Linear System Theory. Prentice-Hall, Englewood Cliffs (1992) zbMATHGoogle Scholar
  17. 17.
    Belevitch, V.: Classical Network Theory. Holden-Day, San Francisco (1968) zbMATHGoogle Scholar
  18. 18.
    Belur, M., Trentelman, H.: Stabilization, pole placement and regular implementability. IEEE Trans. Autom. Control 47(5), 735–744 (2002) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bender, D.J., Laub, A.J.: Controllability and observability at infinity of multivariable linear second-order models. IEEE Trans. Autom. Control AC-30, 1234–1237 (1985) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bender, D.J., Laub, A.J.: The linear-quadratic optimal regulator for descriptor systems. In: Proc. 24th IEEE Conf. Decis. Control, Ft. Lauderdale, FL, pp. 957–962 (1985) Google Scholar
  21. 21.
    Bender, D., Laub, A.: The linear quadratic optimal regulator problem for descriptor systems. IEEE Trans. Autom. Control 32, 672–688 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Berger, T., Trenn, S.: The quasi-Kronecker form for matrix pencils. SIAM J. Matrix Anal. Appl. 33(2), 336–368 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Berger, T., Trenn, S.: Addition to: “The quasi-Kronecker form for matrix pencils”. SIAM J. Matrix Anal. Appl. 34(1), 94–101 (2013). doi: 10.1137/120883244 CrossRefGoogle Scholar
  24. 24.
    Berger, T., Ilchmann, A., Reis, T.: Normal forms, high-gain, and funnel control for linear differential-algebraic systems. In: Biegler, L.T., Campbell, S.L., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints. Advances in Design and Control, vol. 23, pp. 127–164. SIAM, Philadelphia (2012) CrossRefGoogle Scholar
  25. 25.
    Berger, T., Ilchmann, A., Reis, T.: Zero dynamics and funnel control of linear differential-algebraic systems. Math. Control Signals Syst. 24(3), 219–263 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Berger, T., Ilchmann, A., Trenn, S.: The quasi-Weierstraß form for regular matrix pencils. Linear Algebra Appl. 436(10), 4052–4069 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Bernhard, P.: On singular implicit linear dynamical systems. SIAM J. Control Optim. 20(5), 612–633 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Birkhoff, G., MacLane, S.: A Survey of Modern Algebra, 4th edn. Macmillan Publishing Co., New York (1977) zbMATHGoogle Scholar
  29. 29.
    Bonilla Estrada, M., Malabre, M.: On the control of linear systems having internal variations. Automatica 39, 1989–1996 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Bonilla, M., Malabre, M., Loiseau, J.J.: Implicit systems reachability: a geometric point of view. In: Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China, pp. 4270–4275 (2009) Google Scholar
  31. 31.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Amsterdam (1989) zbMATHGoogle Scholar
  32. 32.
    Brunovský, P.: A classification of linear controllable systems. Kybernetika 3, 137–187 (1970) Google Scholar
  33. 33.
    Bunse-Gerstner, A., Mehrmann, V., Nichols, N.K.: On derivative and proportional feedback design for descriptor systems. In: Kaashoek, M.A., et al. (eds.) Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, Amsterdam, Netherlands (1989) Google Scholar
  34. 34.
    Bunse-Gerstner, A., Mehrmann, V., Nichols, N.K.: Regularization of descriptor systems by derivative and proportional state feedback. Report, University of Reading, Dept. of Math., Numerical Analysis Group, Reading, UK (1991) Google Scholar
  35. 35.
    Byers, R., Kunkel, P., Mehrmann, V.: Regularization of linear descriptor systems with variable coefficients. SIAM J. Control Optim. 35, 117–133 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Calahan, D.A.: Computer-Aided Network Design. McGraw-Hill, New York (1972). Rev. edn Google Scholar
  37. 37.
    Campbell, S.L.: Singular Systems of Differential Equations I. Pitman, New York (1980) Google Scholar
  38. 38.
    Campbell, S.L.: Singular Systems of Differential Equations II. Pitman, New York (1982) zbMATHGoogle Scholar
  39. 39.
    Campbell, S.L., Carl, D., Meyer, J., Rose, N.J.: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31(3), 411–425 (1976). http://link.aip.org/link/?SMM/31/411/1. doi: 10.1137/0131035 MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Campbell, S.L., Nichols, N.K., Terrell, W.J.: Duality, observability, and controllability for linear time-varying descriptor systems. Circuits Syst. Signal Process. 10(4), 455–470 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Christodoulou, M.A., Paraskevopoulos, P.N.: Solvability, controllability, and observability of singular systems. J. Optim. Theory Appl. 45, 53–72 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Cobb, J.D.: Descriptor Variable and Generalized Singularly Perturbed Systems: A Geometric Approach. Univ. of Illinois, Dept. of Electrical Engineering, Urbana-Champaign (1980) Google Scholar
  43. 43.
    Cobb, J.D.: Feedback and pole placement in descriptor variable systems. Int. J. Control 33(6), 1135–1146 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Cobb, J.D.: On the solution of linear differential equations with singular coefficients. J. Differ. Equ. 46, 310–323 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Cobb, J.D.: Descriptor variable systems and optimal state regulation. IEEE Trans. Autom. Control AC-28, 601–611 (1983) MathSciNetCrossRefGoogle Scholar
  46. 46.
    Cobb, J.D.: Controllability, observability and duality in singular systems. IEEE Trans. Autom. Control AC-29, 1076–1082 (1984) MathSciNetCrossRefGoogle Scholar
  47. 47.
    Crouch, P.E., van der Schaft, A.J.: Variational and Hamiltonian Control Systems. Lecture Notes in Control and Information Sciences, vol. 101. Springer, Berlin (1986) Google Scholar
  48. 48.
    Cuthrell, J.E., Biegler, L.T.: On the optimization of differential-algebraic process systems. AIChE J. 33(8), 1257–1270 (1987) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, Berlin (1989) zbMATHCrossRefGoogle Scholar
  50. 50.
    Daoutidis, P.: DAEs in chemical engineering: a survey. In: Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, vol. 2. Springer, Berlin (2012) Google Scholar
  51. 51.
    Diehla, M., Uslu, I., Findeisen, R., Schwarzkopf, S., Allgöwer, F., Bock, H.G., Bürner, T., Gilles, E.D., Kienle, A., Schlöder, J.P., Stein, E.: Real-time optimization for large scale processes: nonlinear model predictive control of a high purity distillation column. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems: State of the Art, pp. 363–384. Springer, Berlin (2001) CrossRefGoogle Scholar
  52. 52.
    Diehla, M., Bock, H.G., Schlöder, J.P., Findeisen, R., Nagyc, Z., Allgöwer, F.: Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control 12, 577–585 (2002) CrossRefGoogle Scholar
  53. 53.
    Dieudonné, J.: Sur la réduction canonique des couples des matrices. Bull. Soc. Math. Fr. 74, 130–146 (1946) zbMATHGoogle Scholar
  54. 54.
    Dziurla, B., Newcomb, R.W.: Nonregular Semistate Systems: Examples and Input-Output Pairing. IEEE Press, New York (1987) Google Scholar
  55. 55.
    Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics. Teubner, Stuttgart (1998) zbMATHGoogle Scholar
  56. 56.
    Eliopoulou, H., Karcanias, N.: Properties of reachability and almost reachability subspaces of implicit systems: the extension problem. Kybernetika 31(6), 530–540 (1995) MathSciNetzbMATHGoogle Scholar
  57. 57.
    Fletcher, L.R., Kautsky, J., Nichols, N.K.: Eigenstructure assignment in descriptor systems. IEEE Trans. Autom. Control AC-31, 1138–1141 (1986) CrossRefGoogle Scholar
  58. 58.
    Frankowska, H.: On controllability and observability of implicit systems. Syst. Control Lett. 14, 219–225 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Führer, C., Leimkuhler, B.J.: Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math. 59, 55–69 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Gantmacher, F.R.: The Theory of Matrices, vols. I & II. Chelsea, New York (1959) zbMATHGoogle Scholar
  61. 61.
    Geerts, A.H.W.T.: 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) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Geerts, A.H.W.T., Mehrmann, V.: Linear differential equations with constant coefficients: a distributional approach. Tech. Rep. SFB 343 90-073, Bielefeld University, Germany (1990) Google Scholar
  63. 63.
    Glüsing-Lüerßen, H.: Feedback canonical form for singular systems. Int. J. Control 52(2), 347–376 (1990) zbMATHCrossRefGoogle Scholar
  64. 64.
    Glüsing-Lüerßen, H., Hinrichsen, D.: A Jordan control canonical form for singular systems. Int. J. Control 48(5), 1769–1785 (1988) zbMATHCrossRefGoogle Scholar
  65. 65.
    Gresho, P.M.: Incompressible fluid dynamics: some fundamental formulation issues. Annu. Rev. Fluid Mech. 23, 413–453 (1991) MathSciNetCrossRefGoogle Scholar
  66. 66.
    Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Teubner-Texte zur Mathematik, vol. 88. Teubner, Leipzig (1986) zbMATHGoogle Scholar
  67. 67.
    Haug, E.J.: Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989) Google Scholar
  68. 68.
    Hautus, M.L.J.: Controllability and observability condition for linear autonomous systems. Proc. Ned. Akad. Wet., Ser. A 72, 443–448 (1969) MathSciNetzbMATHGoogle Scholar
  69. 69.
    Helmke, U., Shayman, M.A.: A canonical form for controllable singular systems. Syst. Control Lett. 12(2), 111–122 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Texts in Applied Mathematics, vol. 48. Springer, Berlin (2005) zbMATHGoogle Scholar
  71. 71.
    Hou, M.: Controllability and elimination of impulsive modes in descriptor systems. IEEE Trans. Autom. Control 49(10), 1723–1727 (2004) CrossRefGoogle Scholar
  72. 72.
    Ilchmann, A., Mehrmann, V.: A behavioural approach to time-varying linear systems, Part 1: general theory. SIAM J. Control Optim. 44(5), 1725–1747 (2005) MathSciNetCrossRefGoogle Scholar
  73. 73.
    Ilchmann, A., Mehrmann, V.: A behavioural approach to time-varying linear systems, Part 2: descriptor systems. SIAM J. Control Optim. 44(5), 1748–1765 (2005) MathSciNetCrossRefGoogle Scholar
  74. 74.
    Ilchmann, A., Nürnberger, I., Schmale, W.: Time-varying polynomial matrix systems. Int. J. Control 40(2), 329–362 (1984) zbMATHCrossRefGoogle Scholar
  75. 75.
    Ishihara, J.Y., Terra, M.H.: Impulse controllability and observability of rectangular descriptor systems. IEEE Trans. Autom. Control 46(6), 991–994 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Isidori, A.: Nonlinear Control Systems, 3rd edn. Communications and Control Engineering Series. Springer, Berlin (1995) zbMATHGoogle Scholar
  77. 77.
    Isidori, A.: Nonlinear Control Systems II. Communications and Control Engineering Series. Springer, London (1999) zbMATHCrossRefGoogle Scholar
  78. 78.
    Jaffe, S., Karcanias, N.: Matrix pencil characterization of almost (A,B)-invariant subspaces: a classification of geometric concepts. Int. J. Control 33(1), 51–93 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Julius, A., van der Schaft, A.: Compatibility of behavioral interconnections. In: Proc. 7th European Control Conf. 2003, Cambridge, UK (2003) Google Scholar
  80. 80.
    Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs (1980) zbMATHGoogle Scholar
  81. 81.
    Kalman, R.E.: On the general theory of control systems. In: Proceedings of the First International Congress on Automatic Control, Moscow, 1960, pp. 481–493. Butterworth’s, London (1961) Google Scholar
  82. 82.
    Kalman, R.E.: Canonical structure of linear dynamical systems. Proc. Natl. Acad. Sci. USA 48(4), 596–600 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Kalman, R.E.: Mathematical description of linear dynamical systems. SIAM J. Control Optim. 1, 152–192 (1963) MathSciNetzbMATHGoogle Scholar
  84. 84.
    Karcanias, N.: Regular state-space realizations of singular system control problems. In: Proc. 26th IEEE Conf. Decis. Control, Los Angeles, CA, pp. 1144–1146 (1987) CrossRefGoogle Scholar
  85. 85.
    Karcanias, N., Hayton, G.E.: Generalised autonomous dynamical systems, algebraic duality and geometric theory. In: Proc. 8th IFAC World Congress, Kyoto, 1981, vol. III, pp. 13–18 (1981) Google Scholar
  86. 86.
    Karcanias, N., Kalogeropoulos, G.: A matrix pencil approach to the study of singular systems: algebraic and geometric aspects. In: Proc. Int. Symp. on Singular Systems, Atlanta, GA, pp. 29–33 (1987) Google Scholar
  87. 87.
    Karcanias, N., Kalogeropoulos, G.: Geometric theory and feedback invariants of generalized linear systems: a matrix pencil approach. Circuits Syst. Signal Process. 8(3), 375–397 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Karcanias, N., Kouvaritakis, B.: The output zeroing problem and its relationship to the invariant zero structure: a matrix pencil approach. Int. J. Control 30(3), 395–415 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Knobloch, H.W., Kwakernaak, H.: Lineare Kontrolltheorie. Springer, Berlin (1985) zbMATHCrossRefGoogle Scholar
  90. 90.
    Koumboulis, F.N., Mertzios, B.G.: On Kalman’s controllability and observability criteria for singular systems. Circuits Syst. Signal Process. 18(3), 269–290 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Kouvaritakis, B., MacFarlane, A.G.J.: Geometric approach to analysis and synthesis of system zeros Part 1. Square systems. Int. J. Control 23(2), 149–166 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Kouvaritakis, B., MacFarlane, A.G.J.: Geometric approach to analysis and synthesis of system zeros Part 2. Non-square systems. Int. J. Control 23(2), 167–181 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Kronecker, L.: Algebraische Reduction der Schaaren Bilinearer Formen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, pp. 1225–1237 (1890) Google Scholar
  94. 94.
    Kučera, V., Zagalak, P.: Fundamental theorem of state feedback for singular systems. Automatica 24(5), 653–658 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Kuijper, M.: First-Order Representations of Linear Systems. Birkhäuser, Boston (1994) zbMATHCrossRefGoogle Scholar
  96. 96.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich (2006) zbMATHCrossRefGoogle Scholar
  97. 97.
    Kunkel, P., Mehrmann, V., Rath, W.: Analysis and numerical solution of control problems in descriptor form. Math. Control Signals Syst. 14, 29–61 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Lamour, R., März, R., Tischendorf, C.: Differential Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Heidelberg (2012) Google Scholar
  99. 99.
    Lewis, F.L.: A survey of linear singular systems. IEEE Proc., Circuits Syst. Signal Process. 5(1), 3–36 (1986) zbMATHCrossRefGoogle Scholar
  100. 100.
    Lewis, F.L.: A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatica 28(1), 119–137 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Lewis, F.L., Özçaldiran, K.: Reachability and controllability for descriptor systems. In: Proc. 27, Midwest Symp. on Circ. Syst., Morgantown, WV (1984) Google Scholar
  102. 102.
    Lewis, F.L., Özçaldiran, K.: On the Eigenstructure Assignment of Singular Systems. IEEE Press, New York (1985) Google Scholar
  103. 103.
    Lewis, F.L., Özçaldiran, K.: Geometric structure and feedback in singular systems. IEEE Trans. Autom. Control AC-34(4), 450–455 (1989) CrossRefGoogle Scholar
  104. 104.
    Loiseau, J.: Some geometric considerations about the Kronecker normal form. Int. J. Control 42(6), 1411–1431 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    Loiseau, J., Özçaldiran, K., Malabre, M., Karcanias, N.: Feedback canonical forms of singular systems. Kybernetika 27(4), 289–305 (1991) MathSciNetzbMATHGoogle Scholar
  106. 106.
    Lötstedt, P., Petzold, L.R.: Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for backward differentiation formulas. Math. Comput. 46(174), 491–516 (1986) zbMATHGoogle Scholar
  107. 107.
    Luenberger, D.G.: Dynamic equations in descriptor form. EEE Trans. Autom. Control AC-22, 312–321 (1977) MathSciNetCrossRefGoogle Scholar
  108. 108.
    Luenberger, D.G.: Time-invariant descriptor systems. Automatica 14, 473–480 (1978) zbMATHCrossRefGoogle Scholar
  109. 109.
    Luenberger, D.G.: Introduction to Dynamic Systems: Theory, Models and Applications. Wiley, New York (1979) zbMATHGoogle Scholar
  110. 110.
    Luenberger, D.G.: Nonlinear descriptor systems. J. Econ. Dyn. Control 1, 219–242 (1979) MathSciNetGoogle Scholar
  111. 111.
    Luenberger, D.G., Arbel, A.: Singular dynamic Leontief systems. Econometrica 45, 991–995 (1977) zbMATHCrossRefGoogle Scholar
  112. 112.
    Malabre, M.: More Geometry About Singular Systems. IEEE Press, New York (1987) Google Scholar
  113. 113.
    Malabre, M.: Generalized linear systems: geometric and structural approaches. Linear Algebra Appl. 122–124, 591–621 (1989) MathSciNetCrossRefGoogle Scholar
  114. 114.
    Masubuchi, I.: Stability and stabilization of implicit systems. In: Proc. 39th IEEE Conf. Decis. Control, Sydney, Australia, vol. 12, pp. 3636–3641 (2000) Google Scholar
  115. 115.
    Mertzios, B.G., Christodoulou, M.A., Syrmos, B.L., Lewis, F.L.: Direct controllability and observability time domain conditions of singular systems. IEEE Trans. Autom. Control 33(8), 788–791 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Müller, P.C.: Remark on the solution of linear time-invariant descriptor systems. In: PAMM—Proc. Appl. Math. Mech., GAMM Annual Meeting 2005, Luxemburg, vol. 5, pp. 175–176. Wiley-VCH Verlag GmbH, Weinheim (2005). doi: 10.1002/pamm.200510066 Google Scholar
  117. 117.
    Newcomb, R.W.: The semistate description of nonlinear time-variable circuits. IEEE Trans. Circuits Syst. CAS-28, 62–71 (1981) MathSciNetCrossRefGoogle Scholar
  118. 118.
    Özçaldiran, K.: Control of descriptor systems. Ph.D. thesis, Georgia Institute of Technology (1985) Google Scholar
  119. 119.
    Özçaldiran, K.: A geometric characterization of the reachable and controllable subspaces of descriptor systems. IEEE Proc., Circuits Syst. Signal Process. 5, 37–48 (1986) zbMATHCrossRefGoogle Scholar
  120. 120.
    Özçaldiran, K., Haliločlu, L.: Structural properties of singular systems. Kybernetika 29(6), 518–546 (1993) MathSciNetzbMATHGoogle Scholar
  121. 121.
    Özçaldiran, K., Lewis, F.L.: A geometric approach to eigenstructure assignment for singular systems. IEEE Trans. Autom. Control AC-32(7), 629–632 (1987) CrossRefGoogle Scholar
  122. 122.
    Özçaldiran, K., Lewis, F.L.: Generalized reachability subspaces for singular systems. SIAM J. Control Optim. 27, 495–510 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Özçaldiran, K., Lewis, F.L.: On the regularizability of singular systems. IEEE Trans. Autom. Control 35(10), 1156 (1990) zbMATHCrossRefGoogle Scholar
  124. 124.
    Pandolfi, L.: Controllability and stabilization for linear systems of algebraic and differential equations. J. Optim. Theory Appl. 30, 601–620 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    Pandolfi, L.: On the regulator problem for linear degenerate control systems. J. Optim. Theory Appl. 33, 241–254 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9, 213–231 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    Petzold, L.R.: Numerical solution of differential-algebraic equations in mechanical systems simulation. Physica D 60, 269–279 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory. A Behavioral Approach. Springer, New York (1997) zbMATHGoogle Scholar
  129. 129.
    Popov, V.M.: Hyperstability of Control Systems. Springer, Berlin (1973). Translation based on a revised text prepared shortly after the publication of the Romanian ed., 1966 zbMATHCrossRefGoogle Scholar
  130. 130.
    Przyłuski, K.M., Sosnowski, A.M.: Remarks on the theory of implicit linear continuous-time systems. Kybernetika 30(5), 507–515 (1994) MathSciNetzbMATHGoogle Scholar
  131. 131.
    Pugh, A.C., Ratcliffe, P.A.: On the zeros and poles of a rational matrix. Int. J. Control 30, 213–226 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  132. 132.
    Rabier, P.J., Rheinboldt, W.C.: Classical and generalized solutions of time-dependent linear differential-algebraic equations. Linear Algebra Appl. 245, 259–293 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Rath, W.: Feedback design and regularization for linear descriptor systems with variable coefficients. Dissertation, TU Chemnitz, Chemnitz, Germany (1997) Google Scholar
  134. 134.
    Riaza, R.: Differential-Algebraic Systems. Analytical Aspects and Circuit Applications. World Scientific Publishing, Basel (2008) zbMATHCrossRefGoogle Scholar
  135. 135.
    Riaza, R.: DAEs in circuit modelling: a survey. In: Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, vol. 2. Springer, Berlin (2012) Google Scholar
  136. 136.
    Rosenbrock, H.H.: State Space and Multivariable Theory. Wiley, New York (1970) zbMATHGoogle Scholar
  137. 137.
    Rosenbrock, H.H.: Structural properties of linear dynamical systems. Int. J. Control 20, 191–202 (1974) MathSciNetzbMATHCrossRefGoogle Scholar
  138. 138.
    Rugh, W.J.: Linear System Theory, 2nd edn. Information and System Sciences Series. Prentice-Hall, New York (1996) zbMATHGoogle Scholar
  139. 139.
    Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1, 149–188 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Shayman, M.A., Zhou, Z.: Feedback control and classification of generalized linear systems. IEEE Trans. Autom. Control 32(6), 483–490 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    Simeon, B., Führer, C., Rentrop, P.: Differential-algebraic equations in vehicle system dynamics. Surv. Math. Ind. 1, 1–37 (1991) zbMATHGoogle Scholar
  142. 142.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York (1998) zbMATHGoogle Scholar
  143. 143.
    Trenn, S.: Distributional differential algebraic equations. Ph.D. thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Ilmenau, Germany (2009). http://www.db-thueringen.de/servlets/DocumentServlet?id=13581
  144. 144.
    Trenn, S.: Regularity of distributional differential algebraic equations. Math. Control Signals Syst. 21(3), 229–264 (2009). doi: 10.1007/s00498-009-0045-4 MathSciNetzbMATHCrossRefGoogle Scholar
  145. 145.
    Trenn, S.: Solution concepts for linear DAEs: a survey. In: Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, vol. 2. Springer, Berlin (2013) Google Scholar
  146. 146.
    Trentelman, H., Willems, J.: The behavioral approach as a paradigm for modelling interconnected systems. Eur. J. Control 9(2–3), 296–306 (2003) CrossRefGoogle Scholar
  147. 147.
    Trentelman, H.L., Stoorvogel, A.A., Hautus, M.: Control Theory for Linear Systems. Communications and Control Engineering. Springer, London (2001) CrossRefGoogle Scholar
  148. 148.
    van der Schaft, A.J.: System Theoretic Descriptions of Physical Systems. CWI Tract, No. 3. CWI, Amsterdam (1984) Google Scholar
  149. 149.
    van der Schaft, A.J.: Port-Hamiltonian differential-algebraic systems. In: Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, vol. 2. Springer, Berlin (2012) Google Scholar
  150. 150.
    van der Schaft, A.J., Schumacher, J.M.H.: The complementary-slackness class of hybrid systems. Math. Control Signals Syst. 9, 266–301 (1996). doi: 10.1007/BF02551330 zbMATHCrossRefGoogle Scholar
  151. 151.
    Verghese, G.C.: Infinite-frequency behavio in generalized dynamical systems. Ph.D. thesis, Stanford University (1978) Google Scholar
  152. 152.
    Verghese, G.C.: Further notes on singular systems. In: Proc. Joint American Contr. Conf. (1981). Paper TA-4B Google Scholar
  153. 153.
    Verghese, G.C., Kailath, T.: Eigenvector chains for finite and infinite zeros of rational matrices. In: Proc. 18th Conf. Dec. and Control, Ft. Lauderdale, FL, pp. 31–32 (1979) Google Scholar
  154. 154.
    Verghese, G.C., Kailath, T.: Impulsive behavior in dynamical systems: structure and significance. In: Dewilde, P. (ed.) Proc. 4th MTNS, pp. 162–168 (1979) Google Scholar
  155. 155.
    Verghese, G.C., Levy, B.C., Kailath, T.: A generalized state-space for singular systems. IEEE Trans. Autom. Control AC-26(4), 811–831 (1981) MathSciNetCrossRefGoogle Scholar
  156. 156.
    Wang, C.J.: Controllability and observability of linear time-varying singular systems. IEEE Trans. Autom. Control 44(10), 1901–1905 (1999) zbMATHCrossRefGoogle Scholar
  157. 157.
    Wang, C.J., Liao, H.E.: Impulse observability and impulse controllability of linear time-varying singular systems. Automatica 2001(37), 1867–1872 (2001 CrossRefGoogle Scholar
  158. 158.
    Weierstraß, K.: Zur Theorie der bilinearen und quadratischen Formen. Berl. Monatsb., pp. 310–338 (1868) Google Scholar
  159. 159.
    Wilkinson, J.H.: Linear differential equations and Kronecker’s canonical form. In: de Boor, C., Golub, G.H. (eds.) Recent Advances in Numerical Analysis, pp. 231–265. Academic Press, New York (1978) Google Scholar
  160. 160.
    Willems, J.C.: System theoretic models for the analysis of physical systems. Ric. Autom. 10, 71–106 (1979) MathSciNetGoogle Scholar
  161. 161.
    Willems, J.C.: Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Autom. Control AC-36(3), 259–294 (1991) MathSciNetCrossRefGoogle Scholar
  162. 162.
    Willems, J.C.: On interconnections, control, and feedback. IEEE Trans. Autom. Control 42, 326–339 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  163. 163.
    Willems, J.C.: The behavioral approach to open and interconnected systems. IEEE Control Syst. Mag. 27(6), 46–99 (2007) MathSciNetCrossRefGoogle Scholar
  164. 164.
    Wong, K.T.: The eigenvalue problem λTx+Sx. J. Differ. Equ. 16, 270–280 (1974) zbMATHCrossRefGoogle Scholar
  165. 165.
    Wonham, W.M.: On pole assignment in multi–input controllable linear systems. IEEE Trans. Autom. Control AC-12, 660–665 (1967) CrossRefGoogle Scholar
  166. 166.
    Wonham, W.M.: Linear Multivariable Control: A Geometric Approach, 3rd edn. Springer, New York (1985) zbMATHCrossRefGoogle Scholar
  167. 167.
    Wood, J., Zerz, E.: Notes on the definition of behavioural controllability. Syst. Control Lett. 37, 31–37 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  168. 168.
    Yamada, T., Luenberger, D.G.: Generic controllability theorems for descriptor systems. IEEE Trans. Autom. Control 30(2), 144–152 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  169. 169.
    Yip, E.L., Sincovec, R.F.: Solvability, controllability and observability of continuous descriptor systems. IEEE Trans. Autom. Control AC-26, 702–707 (1981) MathSciNetCrossRefGoogle Scholar
  170. 170.
    Zhou, Z., Shayman, M.A., Tarn, T.J.: Singular systems: a new approach in the time domain. IEEE Trans. Autom. Control 32(1), 42–50 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  171. 171.
    Zubova, S.P.: On full controllability criteria of a descriptor system. The polynomial solution of a control problem with checkpoints. Autom. Remote Control 72(1), 23–37 (2011) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität IlmenauIlmenauGermany
  2. 2.Fachbereich MathematikUniversität HamburgHamburgGermany

Personalised recommendations