On generic properties of linear systems: An overview. (English) Zbl 0546.93020

Summary: The topological space of linear, time-invariant systems is considered. A property of linear systems is called generic if the systems equipped with this property occupy an open and dense subspace of the space of systems. Several generic properties of linear systems are reviewed, including controllability, invertibility, structural stability, as well as some topological properties of orbits of the feedback group. A new estimate for the number of orbits of the feedback group is produced, and a differential geometric proof of the existence of generic controllability indices is given.


93B25 Algebraic methods
93C05 Linear systems in control theory
93C35 Multivariable systems, multidimensional control systems
93B05 Controllability
93B07 Observability
93D15 Stabilization of systems by feedback
93C99 Model systems in control theory
Full Text: EuDML


[1] R. W. Brockett: The geometry of the set of controllable linear systems. Res. Rep. Aut. Contr. Lab., Nagoya University 24 (1977), 1 - 7.
[2] R. W. Brockett, M. D. Mesarović: The reproducibility of multivariable systems. J. Math. Anal. Appl. 11 (1965), 548-563. · Zbl 0135.31301 · doi:10.1016/0022-247X(65)90104-6
[3] P. Brunovský: A classification of linear controllable systems. Kybernetika 6 (1970), 3, 173-187. · Zbl 0199.48202
[4] L. E. Dickson: Theory of Numbers. Vol. II. Chelsea, New York 1971.
[5] G. H. Hardy, S. Ramanujan: Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. 17 (1918), 75-115.
[6] R. Hermann, C. Martin: Algebro-Geometric and Lie-Theoretic Techniques in Systems Theory. Part A. Math. Sci. Press, Brookline 1977. · Zbl 0362.93002
[7] R. E. Kalman: Algebraic geometric description of the class of linear systems of constant dimension. Proc. 8th Annual Princeton Conf. on Inform. Sci. and Systems, Princeton 1974.
[8] A. W. Olbrot: Genericity and non-genericity of properties of mathematical models. Arch. Automat. Telemech. 5 (1980), 4, 473-481 · Zbl 0459.93004
[9] J. Palis, Jr., W. de Melo: Geometric Theory of Dynamical Systems. Springer-Verlag, Berlin 1982. · Zbl 0491.58001
[10] M. K. Sain, J. L. Massey: Invertibility of linear time-invariant systems. IEEE Trans. Automat. Control AC-15 (1969), 141-149.
[11] K. Tchoń: On some operations preserving generic properties of systems. Internat. J. Gen. Systems 9 (1983), 89-94.
[12] R. Thom: Structural Stability and Morphogenesis. Benjamin, New York 1975. · Zbl 0303.92002
[13] F. W. Warner: Foundations of Differentiable Manifolds and Lie Groups. Scott-Foresman, Glenview 1971. · Zbl 0241.58001
[14] J. C. Willems: Topological classification and structural stability of linear systems. J. Differential Equations 35 (1980), 306 - 318. · Zbl 0394.93024 · doi:10.1016/0022-0396(80)90031-5
[15] J. C. Willems, W. H. Hesselink: Generic properties of the pole placement problem. Proc. 1978 IFAC Congress, Helsinki, Finland, 1725-1729.
[16] W. M. Wonham: Linear Multivariable Control: A Geometric Approach. Second Edition. Springer-Verlag, Berlin 1979. · Zbl 0424.93001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.