Abstract
A problem of modal control by output for fourth-order dynamical systems with two inputs and two outputs is presented. For a certain class of such systems, an approach is proposed for reducing the problem under consideration to a control (direct version) or observation (dual version) problem for a system with a single input. The approach is based on two successive similarity transformations of the closed-loop system with a controller by output, which make it possible to reset one of the rows of the controller matrix by state or one of the columns of the observer matrix. The class of systems for which the condition of such zeroing is simultaneously a condition for the existence of an output controller is studied. Theorems on the inequality of the indices of controllability and observability in the system under the conditions presented are proved. A variant of using the well-known Bass–Gura and Ackermann formulas is proposed, which significantly simplifies the symbolic expressions for the controller (observer) in the transformed system. Examples of the application of the proposed approach, both in the direct and in the dual version, are considered. Symbolic calculations in MATLAB validate the results.
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REFERENCES
N. E. Zubov, E. A. Mikrin, and V. N. Ryabchenko, “Controlling a linear MIMO system by a measurement vector using multilevel decomposition,” J. Comput. Syst. Sci. Int. 59 (2), 151–160 (2020). https://doi.org/10.1134/S1064230720020136
N. E. Zubov, V. N. Ryabchenko, and I. V. Sorokin, “Output control of the longitudinal motion spectrum of a single-rotor helicopter,” Russ. Aeronaut. 63 (2), 249–259 (2020).
N. E. Zubov, E. A. Mikrin, V. N. Ryabchenko, and A. V. Fomichev, “Synthesis of control laws for aircraft lateral motion at the lack of data on the slip angle: Analytical solution,” Russ. Aeronaut. 60 (1), 64–73 (2017).
N. E. Zubov, E. A. Mikrin, A. S. Oleinik, V. N. Ryabchenko, and D. E. Efanov, “Estimation of the angular velocity of a spacecraft in the orbital stabilization mode from measurements of the local vertical sensor, " Vestn. Mosk. Gos. Tekh. Univ. im. N. E. Baumana, Ser. "Priborostroenie”, No. 5, 3–15 (2014).
N. E. Zubov, A. V. Lapin, and V. N. Ryabchenko, “Analytical algorithm for constructing the orbital orientation of a spacecraft with an incomplete measurement of the state vector components,” J. Comput. Syst. Sci. Int. 58 (6), 969–979 (2019). https://doi.org/10.1134/S1064230719040178
N. E. Zubov, A. V. Lapin, E. A. Mikrin, and V. N. Ryabchenko, “Output control of the spectrum of a linear dynamic system in terms of the Van der Woude method,” Dokl. Math. 96 (2), 457–460 (2017).
G. A. Leonov and M. M. Shumafov, Methods for the Stabilization of Linear Controllable Systems (S.-Peterb. univ., St. Petersburg, 2005) [in Russian].
A. V. Lapin and N. E. Zubov, “Parametric synthesis of modal control with output feedback for descent module attitude stabilization,” in Int. Russian Automation Conf. (Sochi, 2019), pp. 1–6. https://doi.org/10.1109/RusAutoCon.2019.8867744.
A. V. Lapin and N. E. Zubov, “Generalization of Bass–Gura formula for linear dynamic systems with vector control, Vestn. Mosk. Gos. Tekh. Univ. im. N. E. Baumana, Ser. "Estestv. nauki” 89 (2), 41–64 (2020). https://doi.org/10.18698/1812-3368-2020-2-41-64
A. V. Lapin and N. E. Zubov, “Analytic solution of the problem of stabilizing orbital orientation of a spacecraft with flywheel engines,” AIP Conf. Proc. 2318 (1), 130009 (2021). https://doi.org/10.1063/5.0036155
N. E. Zubov, A. V. Lapin, and V. N. Ryabchenko, “On the relation between modal controllability of a dynamical MIMO-system by output and the type of matrices with desirable spectra,” Diff. Uravn. Protsessy Upr., No. 2, 1–12 (2021).
E. A. Mikrin, V. N. Ryabchenko, N. E. Zubov, and A. V. Lapin, “Analysis and synthesis of dynamical MIMO-systems on the basis of special-type band matrices,” Diff. Uravn. Protsessy Upr., No. 2, 1–14 (2020).
B. A. Skorokhod and V. S. Kolezhuk, “Determination of the observability index of a linear discrete system with a vector output,” in Optimization of Industrial Processes: Collection of Scientific Works, Moscow, 2003, pp. 24–28.
F. R. Gantmakher, Theory of Matrices (Nauka, Moscow, 1966) [in Russian].
A. V. Lapin and N. E. Zubov, “MATLAB implementation of analytical algorithms for modal control by state and output,” Inzh. Zh.: Nauka Innovatsii, No. 1, 1–16 (2020). https://doi.org/10.18698/2308-6033-2020-1-1950
N. E. Zubov, A. V. Lapin, and V. N. Ryabchenko, “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. 62 (3), 401–416 (2019).
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Zubov, N.E., Lapin, A.V. Reducing the Problem of the Modal Control by Output for Stationary Fourth-Order Systems with Two Inputs and Two Outputs to the Control by State for a System with a Single Input. J. Comput. Syst. Sci. Int. 62, 43–60 (2023). https://doi.org/10.1134/S1064230723010124
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DOI: https://doi.org/10.1134/S1064230723010124