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## Archive of Issues

Belarus; Russia Izhevsk; Minsk
Year
2019
Volume
29
Issue
3
Pages
301-311
 Section Mathematics Title On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system Author(-s) Banshchikova I.N.a, Makarov E.K.b, Popova S.N.a Affiliations Udmurt State Universitya, Institute of Mathematics, National Academy of Sciences of Belarusb Abstract We consider a problem of assigning the Lyapunov spectrum for a linear control discrete-time system $$x(m+1)=A(m)x(m)+B(m)u(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n},\ u\in\mathbb R^{k}, \qquad (1)$$ in a small neighborhood of the Lyapunov spectrum of the free system $$x(m+1)=A(m)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n},\qquad (2)$$ by means of linear feedback $u(m)=U(m)x(m)$. We assume that the norm of the feedback matrix $U(\cdot)$ satisfies the Lipschitz estimate with respect to the required shift of the Lyapunov spectrum. This property is called proportional local assignability of the Lyapunov spectrum of the closed-loop system $$x(m+1)=\bigl(A(m)+B(m)U(m)\bigr)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n}. \qquad (3)$$ We previously proved that uniform complete controllability of system (1) and stability of the Lyapunov spectrum of free system (2) are sufficient conditions for proportional local assignability of the Lyapunov spectrum of closed-loop system (3). In this paper we give an example demonstrating that these conditions are not necessary. Keywords linear discrete-time system, Lyapunov exponents, controllability, stabilizability UDC 517.962.22, 517.977 MSC 93B55, 39A06, 39A22 DOI 10.20537/vm190301 Received 22 July 2019 Language Russian Citation Banshchikova I.N., Makarov E.K., Popova S.N. On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 301-311. References Demidovich V.B. A certain criterion for the stability of difference equations, Differ. Uravn., 1969, vol. 5, no. 7, pp. 1247-1255 (in Russian). Gaishun I.V. Sistemy s diskretnym vremenem (Discrete-time systems), Minsk: Institute of Mathematics of the National Academy of Sciences of Belarus, 2001, 400 p. Babiarz A., Banshchikova I., Czornik A., Makarov E., Niezabitowski M., Popova S. On assignability of Lyapunov spectrum of discrete linear time-varying system with control, 2016 21st International Conference on Methods and Models in Automation and Robotics (MMAR), IEEE, 2016, pp. 697-701. https://doi.org/10.1109/MMAR.2016.7575221 Babiarz A., Czornik A., Makarov E., Niezabitowski M., Popova S. Pole placement theorem for discrete time-varying linear systems, SIAM Journal on Control and Optimization, 2017, vol. 55, no. 2, pp. 671-692. https://doi.org/10.1137/15M1033666 Babiarz A., Banshchikova I., Czornik A., Makarov E., Niezabitowski M., Popova S. Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems, IEEE Transactions on Automatic Control, 2018, vol. 63, issue 11, pp. 3825-3837. https://doi.org/10.1109/TAC.2018.2823086 Popova S.N. Assignability of certain Lyapunov invariants for linear discrete-time systems, IFAC-PapersOnLine, 2018, vol. 51, issue 32, pp. 40-45. https://doi.org/10.1016/j.ifacol.2018.11.350 Babiarz A., Banshchikova I., Czornik A., Makarov E., Niezabitowski M., Popova S. Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems, SIAM Journal on Control and Optimization, 2019, vol. 57, no. 2, pp. 1355-1377. https://doi.org/10.1137/17M1141734 Halanay A., Ionescu V. Time-varying discrete linear systems: input-output operators, Riccati equations, disturbance attenuation, Basel: Springer, 1994, 230 p. https://doi.org/10.1007/978-3-0348-8499-0 Banshchikova I.N., Popova S.N. On the spectral set of a linear discrete system with stable Lyapunov exponents, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2016, vol. 26, issue 1, pp. 15-26 (in Russian). https://doi.org/10.20537/vm160102 Bylov B.F., Vinograd R.E., Grobman D.M., Nemytskii V.V. Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov exponents and its application to problems of stability), Moscow: Nauka, 1966, 576 p. Banshchikova I.N. An example of a linear discrete system with unstable Lyapunov exponents, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2016, vol. 26, issue 2, pp. 169-176 (in Russian). https://doi.org/10.20537/vm160203 Full text