## On Kalman’s controllability and observability criteria for singular systems.(English)Zbl 0941.93006

Controllability and observability properties of a singular system $$Ex'= Ax+ Bu$$, $$y= Cx$$ are studied. Controllability, R-controllability, and impulse controllability are considered as are equivalent types of observability. Criteria are given as simple rank conditions related to various Markov parameters. Characterizations of the controllability and observability subspaces are derived.

### MSC:

 93B05 Controllability 93B07 Observability 93C35 Multivariable systems, multidimensional control systems
Full Text:

### References:

 [1] V. Belevich,Classical Network Theory, Holden-Day, San Francisco, CA, 1968. [2] S. L. Campbell,Singular Systems of Differential Equations, Pitman, New York and San Francisco, 1980. · Zbl 0419.34007 [3] M. A. Christodoulou and P. N. Paraskevopoulos, Solvability, controllability and observability of singular systems,J. Optim. Theory Appl., vol. 45 (1985), pp. 53-57. · Zbl 0537.93017 [4] J. D. Cobb, Controllability, observability and duality in singular systems,IEEE Trans. Automat. Control, vol. 29 (1984), pp. 1076-1082. [5] L. Dai,Singular Control Systems, Springer-Verlag, Berlin and New York, 1989. · Zbl 0669.93034 [6] G. Doetsch,Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin and New York, 1974. · Zbl 0278.44001 [7] F. R. Gantmacher,The Theory of Matrices, 2nd ed., vol. 2, Chelsea, New York, 1990. · Zbl 0085.01001 [8] M. L. J. Hautus, Controllability and observability conditions of linear autonomous systems,Nederl. Akad. Wetensch., Proc. Ser. A, vol. 72 (1969), pp. 443-448. · Zbl 0188.46801 [9] U. Helmke, J. Rosenthal, and J. M. Schumacher, A controllability test for general first-order representations,Automatica, vol. 33 (1997), pp. 192-301. · Zbl 0874.93018 [10] T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980. [11] R. E. Kalman, On the general theory of control systems,Proc. First IFAC Congress, Moscow, 1960. · Zbl 0112.06303 [12] F. N. Koumboulis, New techniques for analysis and feedback design for regular and singular systems, Ph.D. thesis, National Technical University of Athens, Dept. Electrical Engrg., Greece, 1991. · Zbl 0757.93012 [13] F. L. Lewis, A survey of linear singular systems,Circuits Systems Signal Process., vol. 5 (1986), pp. 3-36. · Zbl 0613.93029 [14] F. L. Lewis, A tutorial on the geometric analysis of linear time-invariant implicit systems,Automatica, vol. 28 (1992), pp. 119-137. · Zbl 0745.93033 [15] F. L. Lewis and B. G. Mertzios, On the analysis of discrete linear time-invariant singular systems,IEEE Trans. Automat. Control, vol. 35 (1990), pp. 506-511. · Zbl 0705.93058 [16] B. G. Mertzios, The fundamental matrix in continuous generalized systems and the direct solution of the generalized state space model, submitted toJ. of Dynamical and Control Systems, 1999. [17] B. G. Mertzios, M. A. Christodoulou, B. L. Syrmos, and F. L. Lewis, Direct controllability and observability time domain conditions of singular systems,IEEE Trans. Automat. Control, vol. 33 (1988), pp. 788-791. · Zbl 0666.93012 [18] B. G. Mertzios and F. L. Lewis, Fundamental matrix of discrete singular systems,Circuits Systems Signal Process., vol. 8 (1989), pp. 341-355. · Zbl 0689.93041 [19] L. Pandolfi, Controllability and stabilizability for linear systems of algebraic and differential equations,J. Optim. Theory Appl., vol. 30 (1980), pp. 601-620. · Zbl 0397.93006 [20] V. M. Popov,Hyperstability of Control Systems, Springer-Verlag, Berlin and New York, 1973. · Zbl 0276.93033 [21] M. A. Shayman and Z. Zhou, Feedback control and classification of generalized linear systems,IEEE Trans. Automat. Control, vol. 32 (1987), pp. 483-494. · Zbl 0624.93028 [22] G. C. Verghese, B. C. Levy, and T. Kailath, A generalized state-space for singular systems,IEEE Trans. Automat. Control, vol. 27 (1981), pp. 811-831. · Zbl 0541.34040 [23] W. M. Wohnam,Linear Multivariable Control?A Geometric Approach, Springer-Verlag, Berlin and New York, 1989. [24] Z. Zhou, M. A. Shayman, and J.-J. Tarn, Singular-systems: A new approach in the time-domain,IEEE Trans. Automat. Control, vol. 32 (1987), pp. 42-50. · Zbl 0612.93030
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.