T. A. Akunov, N. A. Dudarenko, N. A. Polinova, A. V. Ushakov

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The paper deals with steady aperiodic continuous system, state matrix of which has a real spectrum of the eigenvalues which absolute value is less than unity. The latest authors’ works show that for such absolute values and multiple structure of eigenvalues on the free motion trajectories of the system by norm of the state vector the significant overshoot is detected, alternated by monotonous motion toward a state of rest. In order to minimize the overshoot value, it is proposed to modify the structure of the eigenvalues, transforming it into a simple one. The result of structure modification is the following: initial eigenvalue and shifted along the real axis of the complex plane to the left by a fixed value relative to the adjacent eigenvalues; each of them has unit multiplicity. Such modification gives the possibility to form the estimation of the proximity degree of eigenvalues simple structure to the multiple one. Moreover, it can be defined in a relative form, which guarantees the reduction of the above overshoot for the free motion trajectory. Results of computer experiments illustrate the issues of the paper.

Keywords: aperiodic system, proximity degree of eigenvalues to multiplicity, norm, trajectory

1.        Akunov T.A., Dudarenko N.A., Polinova N.A., UshakovA.V. Issledovanie kolebatel’nosti protsessov v aperiodicheskikh nepreryvnykh sistemakh, porozhdaemoi faktorom kratnosti sobstvennykh chisel [Process oscillativity study in aperiodic continuous systems, generated by eigenvalues multiplication factor]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics,2013, no. 3 (85), pp. 55–61.
2.        Akunov T., Dudarenko N., Polinova N., Ushakov A. Factor multiplicity of the state matrix in the system dynamics. Proceedings of the 18th WSEAS International Conference on Applied Mathematics (AMATH`13). Budapest, Hungary, 2013, vol. 20, pp. 58­–63­­.
3.        Bhattacharyya S.P., deSouza E. Pole assignment via Sylvester’s equation. System and Control Letters, 1982, vol. 1, no. 4, pp. 261–263.
4.        Kautsky J., Nichols N.K., Chu E.K.-W. Robust pole assignment in singular control systems. Linear Algebra and Its Applications, 1985, vol. 121, pp. 9–37.
5.        Alexandridis A.T., Galanos G.D. Optimal pole placement for linear multi input controllable system. IEEE transactions on Circuit and System, 1987, vol. CAS-34, no. 12, pp. 1602–1604.
6.        Valasek M., Olgac N. Efficient pole placement technique for linear time-variant SISO systems. IEE Proceesings: Control Theory and Applications, 1995, vol. 142, no. 5, pp. 451–458. doi: 10.1049/ip-cta:19951959
7.        Chu E.K. Pole assignment for second-order systems. Mechanical Systems and Signal Processing, 2002, vol. 16, no. 1, pp. 39–59. doi: 10.1006/mssp.2001.1439
8.        De La Sen M. ON pole placement controllers for linear time-delay systems with commensurate points delays. Mathematical Problems in Engineering, 2005, vol. 2005, no 1, pp. 123–140. doi: 10.1155/MPE.2005.123
9.        Hasan N. Design and analysis of pole-placement controller for interconnected power systems. International Journal of Emerging Technology and Advanced Engineering, 2012, vol. 2, no. 8, pp. 212–217.
10.     Zhang L., Wang X.T. Partial eigenvalue assignment for high order system by multi-input control. Mechanical Systems and Signal Processing, 2014, vol. 42, no. 1–2, pp. 129–136. doi: 10.1016/j.ymssp.2013.06.026
11.     Dudarenko N.A., Slita O.V., UshakovA.V. Matematicheskie osnovy sovremennoi teorii upravleniya: apparat metoda prostranstva sostoyanii [Mathematical foundations of modern control theory: the apparatus of the state space method] Ed. A.V. Ushakov. St. Petersburg, SPbSU ITMO Publ., 2008, 323 p.
12.     Andreev Yu.N. Upravlenie konechnomernymi lineinymi ob”ektami [Control of finitelinear objects]. Moscow, Nauka Publ., 1976, 424 p.
13.     Gantmakher F.R. Teoriya matrits [Matrix theory]. Moscow, Nauka Publ., 1973, 575 p.
14.     Akunov T.A., Dudarenko N.A., Polinova N.A., UshakovA.V. Issledovanie protsessov v nepreryvnykh sistemakh s kratnymi kompleksno-sopryazhennymi sobstvennymi chislami ikh matrits sostoyaniya [Research of processes in continuous systems with multiple complex conjugated eigenvalues of their state matrix]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics,2013, no. 4 (86), pp. 25–33.
15.     Golub G.H., Van Loan C.F. Matrix computations.Baltimore, Johns Hopkins University Press, 1996. 728 p.
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