Abstract. The problem of criterion matrices formation for the multidimensional dynamic systems is considered. The criterion matrices can be used for the properties analysis of a multidimensional system in a stationary state. The procedure of criterion matrices formation is considered in relation to the problem of estimating the tendency of multidimensional dynamic systems to degeneration, which is a measure of the robustness of a multidimensional system. The case of multidimensional continuous-time dynamic systems is considered as an example for criterion matrices construction. The problem is solved using the Faddeev — LeVerrier algorithm, which is supplemented by the Cayley — Hamilton theorem. The obtained real-valued construction for the formation of criterion matrices of the input-output relationship of multidimensional dynamical systems is focused on the problem of a priori express control of the degeneracy of dynamical systems of the multidimensional input — multidimensional output type in static.

Keywords: Faddeev — LeVerrier algorithm, condition number, degeneration, criterion matrix, Cayley — Hamilton theorem, degeneration factor

References:

1. Dorf R.C., Bishop R.H. Modern Control Systems, Prentice Hall, 2010.
2. Akunov T.A., Ushakov A.V. Journal of Computer and Systems Sciences International, 2003, no. 4, pp. 503–510. (in Russ.)
3. Slita O.V., Ushakov A.V. Journal of Computer and Systems Sciences International, 2008, no. 4, pp. 518–526. (in Russ.)
4. Dudarenko N.A., Ushakov A.V. Journal of Automation and Information Sciences, 2011, no. 6(43), pp. 30–39.
5. Nikiforov V.O., Slita O.V., Ushakov A.V. Intellektual'noye upravleniye v usloviyakh neopredelennosti (Intelligent Control under Uncertainty), St. Petersburg, 2011, 231 р. (in Russ.)
6. Scogestad S., Havre K. European symposium on computer aided process engineering-6. Part B, 1996, vol. 20, pp. S1005–S1010.
7. Golub G.H., Van Loan Ch.F. Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, 1996, 728 р.
8. Dudarenko N.A., Ushakov A.V. Journal of Automation and Information Sciences, 2013, no. 6(45), pp. 36–47.
9. Dudarenko N.A., Polyakova M.V., Ushakov A.V. Optoelectronics, Instrumentation and Data Processing, 2012, no. 5(48), pp. 483–488.
10. Gantmacher F.R. The Theory of Matrices, AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000, 660 р.
11. Wilkinson J.H. The algebraic eigenvalue problem, Oxford, Clarendon Press, 1965.
12. Moore B.C. IEEE Transactions on Automatic Control, 1981, no. 1(AC-26), pp. 17–31.
13. Birk W., Dudarenko N.A. IEEE Transactions on Control Systems Technology, 2016, no. 2(24), pp. 565–577.