Generalization of Bass --- Gura Formula for Linear Dynamic Systems with Vector Control

Авторы: Lapin A.V., Zubov N.E. Опубликовано: 23.04.2020
Опубликовано в выпуске: #2(89)/2020  
DOI: 10.18698/1812-3368-2020-2-41-64

Раздел: Математика и механика | Рубрика: Математическая физика  
Ключевые слова: automatic control system, modal controller, analytic solution, scalar control, vector control, state-vector feedback, matrix spectrum, characteristic polynomial, block-matrix, similarity transformation, block transposition of a matrix

The compact analytic formula of calculating the feedback law (controller matrix) coefficients is developed for solving the synthesis problem of modal controller providing desired pole placement by means of the fully measured state vector in linear dynamic systems with vector control. This formula represents the generalization of the known Bass --- Gura formula, used for synthesizing modal controllers in systems with scalar control, to systems with vector control. The obtained solution is applicable to systems with state-space dimension divisible by the number of control inputs and the matrix composed of the linearly independent first block columns of the Kalman controllability matrix by a number corresponding to the quantity of the mentioned multiplicity is reversible. To use the mentioned formula, it's not required to additionally transfer the described systems of the indicated class to special canonical forms. This formula may be applied to solve both numeric and analytic problems of modal control in mentioned class, independently on a specific ratio of state-vector and control-vector dimensions as well as on existence and multiplicity of real-value poles and complex-conjugate pairs of poles in original and desirable spectrums of state matrix. The examples are considered that prove the possibility of applying the generalized block-matrix Bass --- Gura formula to calculate modal controllers for the described class of systems with vector control


[1] Zubov N.E., Vorob’eva E.A., Mikrin E.A., et al. Synthesis of stabilizing spacecraft control based on generalized Ackermann’s formula. J. Comput. Syst. Sci. Int., 2011, vol. 50, iss. 1, pp. 93--103. DOI: https://doi.org/10.1134/S1064230711010199

[2] Zubov N.E., Mikrin E.A., Misrikhanov M.Sh., et al. Modification of the exact pole placement method and its application for the control of spacecraft motion. J. Comput. Syst. Sci. Int., 2013, vol. 52, iss. 2, pp. 279--292. DOI: https://doi.org/10.1134/S1064230713020135

[3] Zubov N.E., Mikrin E.A., Misrikhanov M.Sh., et al. Finite eigenvalue assignment for a descriptor system. Dokl. Math., 2015, vol. 91, iss. 1, pp. 64--67. DOI: https://doi.org/10.1134/S1064562415010226

[4] Zubov N.E., Lapin A.V., Mikrin E.A., et al. Output control of the spectrum of a linear dynamic system in terms of the Van der Woude method. Dokl. Math., 2017, vol. 96, iss. 2, pp. 457--460. DOI: https://doi.org/10.1134/S1064562417050179

[5] Zubov N.E., Zybin E.Yu., Mikrin E.A., et al. Output control of a spacecraft motion spectrum. J. Comput. Syst. Sci. Int., 2014, vol. 53, iss. 4, pp. 576--586. DOI: https://doi.org/10.1134/S1064230714040170

[6] Fu M. Pole placement via static output feedback is NP-hard. IEEE Trans. Autom. Control, 2004, vol. 49, iss. 5, pp. 855--857. DOI: https://doi.org/10.1109/TAC.2004.828311

[7] Eremenko A., Gabrielov A. Pole placement by static output feedback for generic linear systems. SIAM J. Control Optim., 2002, vol. 41, iss. 1, pp. 303--312. DOI: https://doi.org/10.1137/S0363012901391913

[8] Franke M. Eigenvalue assignment by static output feedback --- on a new solvability condition and the computation of low gain feedback matrices. Int. J. Control, 2014, vol. 87, iss. 1, pp. 64--75. DOI: https://doi.org/10.1080/00207179.2013.822102

[9] Yang K., Orsi R. Generalized pole placement via static output feedback: a methodology based on projections. Automatica, 2006, vol. 42, iss. 12, pp. 2143--2150. DOI: https://doi.org/10.1016/j.automatica.2006.06.021

[10] Peretz Y. A randomized approximation algorithm for the minimal-norm static-output-feedback problem. Automatica, 2016, vol. 63, pp. 221--234. DOI: https://doi.org/10.1016/j.automatica.2015.10.001

[11] Shimjith S.R., Tiwari A.P., Bandyopadhyay B. Modeling and control of a large nuclear reactor. Berlin, Heidelberg, Springer, 2013.

[12] Bass R.W., Gura I. High order system design via state-space considerations. Proc. JACC, 1965, vol. 3, pp. 311--318.

[13] Kuo B.C. Digital control systems. Oxford Univ. Press, 1995.

[14] Nordstrom K., Norlander H. On the multi input pole placement control problem. Proc. 36 IEEE Conf. Decision Contr., San Diego, CA, USA, 1997, vol. 5, pp. 4288--4293. DOI: https://doi.org/10.1109/CDC.1997.649511

[15] Gantmakher F.R. Teoriya matrits [Matrix theory]. Moscow, Fizmatlit Publ., 2004.