Generalization of Ackermann Formula for a Certain Class of Multidimensional Dynamic Systems with Vector Input

Authors: Lapin A.V., Zubov N.E., Proletarskii A.V. Published: 24.08.2023
Published in issue: #4(109)/2023  
DOI: 10.18698/1812-3368-2023-4-18-38

Category: Mathematics and Mechanics | Chapter: Computational Mathematics  
Keywords: modal control by state, controller, controllability index, similarity transformation, Bass --- Gura formula, Ackermann formula, multilevel decomposition


A compact analytical formula is obtained that determines the entire set of solutions of the modal control problem for a wide class of multidimensional dynamical systems with vector input, where the number of states is divisible by the number of control inputs, and the controllability index is equal to the quotient of this division. This formula generalizes to systems with the vector input the Ackermann formula applied to multidimensional systems with scalar input. The basis to obtaining the generalized Ackermann formula lies in the original concepts of the Luenberger generalized canonical form and operations of the matrices block transposition. For the most convenient calculation of controller, the original system with vector input is reduced to the generalized Luenberger canonical form using the two successive similarity transformations. A lemma is proved that demonstrates the compact analytical form of the inverse transformation matrix. Transition equivalence makes it possible to obtain a complete countably infinite parametrized set of solutions to the modal control problem under consideration. Its parametrization is provided by selecting block coefficients of the matrix polynomial, which determinant corresponds to the given scalar characteristic polynomial. In cases, where the matrix polynomial involved in parametrization is not reduced to the multipliers, the generalized Ackermann formula contains solutions to the modal control problem that could not be obtained using the existing decomposition method. Examples are presented demonstrating both suitability of the proposed formula for analytical synthesis of modal controllers by state in systems with vector input and its advantages in comparison with the decomposition method

Please cite this article in English as:

Lapin A.V., Zubov N.E., Proletarskii A.V. Generalization of Ackermann formula for a certain class of multidimensional dynamic systems with vector input. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2023, no. 4 (109), pp. 18--38 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2023-4-18-38


[1] Bass R.W., Gura I. High order system design via state-space considerations. Proc. Joint Automatic Control Conf., 1965, vol. 3, pp. 311--318.

[2] Ackermann J. Der Entwurf linearer Regelungssysteme im Zustandsraum. Automatisierungstechnik, 1972, vol. 20, iss. 1-2, pp. 297--300. DOI: https://doi.org/10.1524/auto.1972.20.112.297

[3] Mikrin E.A., Zubov N.E., Lapin A.V., et al. Analytical formula of calculating a controller for linear SIMO-system. Differentsialnye uravneniya i protsessy upravleniya [Differential Equations and Control Processes], 2020, no. 1, pp. 1--11 (in Russ.).

[4] Hasan M., Namin A., Negre C. Toeplitz matrix approach for binary field multiplication using quadrinomials. IEEE Trans. Very Large Scale Integr. VLSI Syst., 2012, vol. 20, iss. 3, pp. 449--458. DOI: https://doi.org/10.1109/TVLSI.2011.2106524

[5] Lapin A.V., Zubov N.E. Generalization of Bass --- Gura formula for linear dynamic systems with vector control. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2020, no. 2 (89), pp. 41--64. DOI: https://doi.org/10.18698/1812-3368-2020-2-41-64

[6] Tian G., Xiaoli L., Shuguang Z., et al. An algorithm to determine the index of structural controllability for network system. ICISCE, 2016, pp. 819--823. DOI: https://doi.org/10.1109/ICISCE.2016.179

[7] Nordstrom K., Norlander H. On the multi input pole placement control problem. Proc. 36th IEEE Conf. on Decision and Control, 1997, vol. 5, pp. 4288--4293. DOI: https://doi.org/10.1109/CDC.1997.649511

[8] 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, no. 1, pp. 93--103. DOI: https://doi.org/10.1134/S1064230711010199

[9] Luenberger D.G. Canonical form for linear multivariable systems. IEEE Trans. Automat. Contr., 1967, vol. 12, iss. 3, pp. 290--293. DOI: https://doi.org/10.1109/TAC.1967.1098584

[10] Gantmacher F.R. The theory of matrices. Chelsea, 2000.

[11] Lapin A.V., Zubov N.E. MATLAB based implementation of analytic algorithms of modal control with state-vector feedback and output-vector feedback. Inzhenernyy zhurnal: nauka i innovatsii [Engineering Journal: Science and Innovation], 2020, no. 1 (in Russ.). DOI: https://doi.org/10.18698/2308-6033-2020-1-1950

[12] Zubov N.E., Lapin A.V., Mikrin E.A. Synthesis of decoupling laws for controlling the angular motion of landing module with solid-fuel landing engine minimizing the transient time. J. Comput. Syst. Sci. Int., 2013, vol. 52, no. 3, pp. 480--490. DOI: https://doi.org/10.1134/S1064230713030179

[13] Zubov N.E., Lapin A.V., Ryabchenko V.N. On relation between modal controllability of dynamic MIMO-system by output and a type of matrices with desirable spectra. Differentsialnye uravneniya i protsessy upravleniya [Differential Equations and Control Processes], 2021, no. 2, pp. 1--12 (in Russ.).

[14] Lapin A.V., Zubov N.E. Parametric synthesis of modal control with output feedback for descent module attitude stabilization. RusAutoCon, 2019. DOI: https://doi.org/10.1109/RusAutoCon.2019.8867744

[15] Zubov N.E., Lapin A.V., Ryabchenko V.N., et al. A robust control algorithm of a descent vehicle angular motion in the Earth’s atmosphere. Appl. Sci., 2022, vol. 12, iss. 2, art. 731. DOI: https://doi.org/10.3390/app12020731

[16] Zubov N.E., Lapin A.V., Ryabchenko V.N. Analytical synthesis of a modal controller by output vector for attitude control of a descent module during its descent in the Earth’s atmosphere. Russ. Aeronaut., 2019, vol. 62, no. 3, pp. 401--416. DOI: https://doi.org/10.3103/S1068799819030073