Equivalence of controllability conditions of linear multidimensional systems and solvability of the silvester polynomial matrix equation

In the paper the following controllability criteria of linear multidimensional systems are considered: Kalman rank criterion; Popov-Belevich-Hautus modal (frequency) test, when nondegeneracy of the controllability band matrix is necessary and sufficient for a controllable system. The statements determining the equivalence of band controllability conditions of a linear multidimensional system and the solvability conditions of the Sylvester linear polynomial matrix equation relatively to polynomial matrix of degree are presented. The dual statements for the observability criteria are given. The application of the developed method to linear dynamic systems is shown in practice.

References

[1] Kalman R.E., Falb P.L., Arbib M.A. Topics in mathematical system theory. McGraw-Hill, 1969.

[2] Zhou K. Essentials of robust control. Upper Saddle River. N.J., Prentice Hall, 1998.

[3] Misrikhanov M.Sh., Ryabchenko V.N. The band formula for A.N. Krylov’s problem. Automation and Remote Control, 2007, vol. 68, no. 12, pp. 2142-2157.

[4] Misrikhanov M.Sh., Ryabchenko V.N. Algebraic and matrix methods in the theory of linear MIMO-systems. Vestnik IGEU, 2005, vol. 5, pp. 196-240 (in Russ.).

[5] Antsaklis P.J., Gao Z. Polynomial and rational matrix interpolation: theory and control applications. Int. J. Contr., 1993, vol. 58 (2), pp. 349-404.

[6] Kailath T. Linear Systems. N.J., Prentice Hall. Englewood Cliffs, 1980.

[7] Henrion D., Sebek M. Efficient algorithms for polynomial matrix equation triangularization. Proc. of the IEEE Mediterranean Conference on Contr. and Automat. Alghero, Sardinia, Italy, 1998.