﻿ Generalization of Bass --- Gura Formula for Linear Dynamic Systems with Vector Control | Herald of the Bauman Moscow State Technical University. Natural Sciences | # Generalization of Bass --- Gura Formula for Linear Dynamic Systems with Vector Control

 Authors: Lapin A.V., Zubov N.E. Published: 23.04.2020 Published in issue: #2(89)/2020 DOI: 10.18698/1812-3368-2020-2-41-64 Category: Mathematics | Chapter: Mathematical Physics Keywords: 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

## References

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

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

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

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

 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

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