Finite spectrum assignment problem for bilinear systems with several delays

V. A. Zaitsev ; I. G. Kim ; V. E. Khartovskii

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 3, pp. 319-331

Résumé

A bilinear control system defined by a linear stationary differential system with several non-commensurate delays in the state variable is considered. A problem of finite spectrum assignment by constant control is studied. One needs to construct constant control vectors such that the characteristic function of the closed-loop system is equal to a polynomial with arbitrary given coefficients. Conditions on coefficients of the system are obtained under which the criterion was found for solvability of the finite spectrum assignment problem. Interconnection of the criterion conditions with the property of consistency for the truncated system without delays is shown. Corollaries on stabilization of bilinear systems with delays are obtained. The similar results are obtained for discrete-time bilinear systems with several delays. An illustrative example is considered.

Détail

Keywords: linear systems with delays, spectrum assignment, stabilization, bilinear system.

Bibliographie

[1] N. N. Krasovskii, Yu. S. Osipov, “On the stabilization of motions of a plant with delay in a control system”, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1963, no. 6, 3–15 (in Russian) | Zbl

[2] Yu. S. Osipov, “On stabilization of control systems with delay”, Differ. Uravn., 1:5 (1965), 605–618 (in Russian) | MR | Zbl

[3] L. Pandolfi, “Stabilization of neutral functional-differential equations”, Journal of Optimization Theory and Applications, 20:2 (1976), 191–204 | DOI | MR | Zbl

[4] A. Z. Manitius, A. W. Olbrot, “Finite spectrum assignment problem for systems with delays”, IEEE Transactions on Automatic Control, 24:4 (1979), 541–553 | DOI | MR | Zbl

[5] W. S. Lu, E. Lee, S. Zak, “On the stabilization of linear neutral delay-difference systems”, IEEE Transactions on Automatic Control, 31:1 (1986), 65–67 | DOI | MR | Zbl

[6] S. I. Minyaev, A. S. Fursov, “Topological approach to the simultaneous stabilization of plants with delay”, Differential Equations, 49:11 (2013), 1423–1431 | DOI | MR | Zbl

[7] K. Watanabe, “Finite spectrum assignment and observer for multivarible systems with commensurate delay”, IEEE Transactions on Automatic Control, 31:6 (1986), 543–550 | DOI | MR | Zbl

[8] Q. G. Wang, T. H. Lee, K. K. Tan, Finite spectrum assignment for time-delay systems, Springer, London, 1998, 124 pp. | MR | Zbl

[9] A. V. Metel'skii, “Finite spectrum assignment problem for a differential system of neutral type”, Differential Equations, 51:1 (2015), 69–82 | DOI | MR | Zbl

[10] A. V. Metel'skii, “Spectral reducibility of delay differential systems by a dynamic controller”, Differential Equations, 47:11 (2011), 1642–1659 | DOI | MR | Zbl

[11] V. E. Khartovskii, “Spectral reduction of linear systems of the neutral type”, Differential Equations, 53:3 (2017), 366–382 | DOI | MR

[12] V. E. Khartovskii, “Finite spectrum assignment for completely regular differential-algebraic systems with aftereffect”, Differential Equations, 54:6 (2018), 823–838 | DOI | MR | Zbl

[13] V. M. Marchenko, “Control of systems with aftereffect in scales of linear controllers with respect to the type of feedback”, Differential Equations, 47:7 (2011), 1014–1028 | DOI | MR | Zbl

[14] A. T. Pavlovskaya, V. E. Khartovskii, “Control of neutral delay linear systems using feedback with dynamic structure”, Journal of Computer and Systems Sciences International, 53:3 (2014), 305–319 | DOI | MR | Zbl

[15] A. V. Metel'skii, V. E. Khartovskii, “Criteria for modal controllability of linear systems of neutral type”, Differential Equations, 52:11 (2016), 1453–1468 | DOI | MR | Zbl

[16] V. E. Khartovskii, “Modal controllability for systems of neutral type in classes of differential-difference controllers”, Automation and Remote Control, 78:11 (2017), 1941–1954 | DOI | MR | Zbl

[17] V. E. Khartovskii, “Criteria for modal controllability of completely regular differential-algebraic systems with aftereffect”, Differential Equations, 54:4 (2018), 509–524 | DOI | MR | Zbl

[18] V. E. Khartovskii, “Synthesis of damping controllers for the solution of completely regular differential-algebraic delay systems”, Differential Equations, 53:4 (2017), 539–550 | DOI | MR | Zbl

[19] V. L. Kharitonov, “Predictor-based management: implementation challenge”, Differential Equations and Control Processes, 2015, no. 4, 51–65 (in Russian) http://diffjournal.spbu.ru/pdf/kharitonov2.pdf | MR

[20] S. Mondie, W. Mihiels, “Finite spectrum assignment of unstable time-delay systems with a safe implementation”, IEEE Transactions on Automatic Control, 48:12 (2003), 2207–2212 | DOI | MR | Zbl

[21] V. B. Kolmanovskii, N. I. Koroleva, “The synthesis of bilinear systems with delayed control”, Journal of Applied Mathematics and Mechanics, 57:1 (1993), 37–45 | DOI | MR

[22] X. Yang, Yu. Stepanenko, “Stabilization of discrete bilinear systems with time delayed feedback”, Proceeding of the American Control Conference, 1993, 1051–1055 | DOI | MR

[23] X. Yang, L. K. Chen, “Stability of discrete bilinear systems with time-delayed feedback functions”, IEEE Transactions on Automatic Control, 38:1 (1993), 158–163 | DOI | MR | Zbl

[24] S. I. Niculescu, S. Tarbouriech, J. M. Dion, L. Dugard, “Stability criteria for bilinear systems with delayed state and saturating actuators”, Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995, 2064–2069 | DOI

[25] D. W. C. Ho, G. Lu, Y. Zheng, “Global stabilisation for bilinear systems with time delay”, IEE Proceedings – Control Theory and Applications, 149:1 (2002), 89–94 | DOI

[26] V. A. Zaitsev, I. G. Kim, “On finite spectrum assignment problem in bilinear systems with state delay”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 29:1 (2019), 19–28 | DOI | MR

[27] V. A. Zaitsev, I. G. Kim, “On assignment of arbitrary finite spectrum for bilinear systems with a delay”, Proc. XIII All-Russia Conf. on Control Problems, RCCP–2014, Institute of Control Problems of the Russian Academy of Sciences, M., 2019, 1058–1062 (in Russian)

[28] V. A. Zaitsev, I. G. Kim, “On spectrum control and stabilization of bilinear systems with several delays”, Proc. XIX International scientific conference on differential equations, Erugin readings-2019, v. 1, Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 2019, 114–116 (in Russian) | MR

[29] V. A. Zaitsev, I. G. Kim, “Finite spectrum assignment problem in linear systems with state delay by static output feedback”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 26:4 (2016), 463–473 (in Russian) | DOI | MR | Zbl

[30] V. A. Zaitsev, “Necessary and sufficient conditions in a spectrum control problem”, Differential Equations, 2010, no. 12, 1789–1793 | DOI | MR | Zbl

[31] V. A. Zaitsev, “Control of spectrum in bilinear systems”, Differential Equations, 2010, no. 7, 1071–1075 | DOI | MR | Zbl

[32] I. G. Kim, V. A. Zaitsev, “Spectrum assignment by static output feedback for linear systems with time delays in states”, 2018 14th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy's Conference) (STAB), IEEE, 2018, 1–4 | DOI

[33] V. A. Zaitsev, E. L. Tonkov, “Attainability, compatibility and V. M. Millionshchikov's method of rotations”, Russian Mathematics, 43:2, 42–52 | MR | Zbl

[34] V. A. Zaitsev, “Consistent systems, pole assignment, I”, Differential Equations, 2012, no. 1, 120–135 | DOI | MR | Zbl

[35] V. A. Zaitsev, “Consistency and eigenvalue assignment for discrete-time bilinear systems: I”, Differential Equations, 50:11 (2014), 1495–1507 | DOI | MR | Zbl

[36] V. A. Zaitsev, “Consistency and eigenvalue assignment for discrete-time bilinear systems: II”, Differential Equations, 51:4 (2015), 510–522 | DOI | MR | Zbl