Abstract
For control systems with deviating argument, we consider basic problems of qualitative control theory such as problems of stabilization and modal control in scales of linear controllers with respect to the type of feedback, from the simplest difference controllers to integral controllers of general form. We analyze the results obtained in this direction. Special attention is paid to the stabilization of two-dimensional systems by feedback in the form of difference controllers. For the case in which the construction of a difference controller is impossible or too difficult, an integral feedback is used. Unlike the well-known Krasovskii-Osipov method for the construction of integral feedback in delay systems, the suggested method is based on the Paley-Wiener theorem for entire functions of exponential type.
Similar content being viewed by others
References
Krasovskii, N.N. and Osipov, Yu.S., On the Stabilization of Motions of a Plant with Delays in the Control System, Izv. Akad. Nauk SSSR Tehn. Kibernet., 1963, no. 6, pp. 3–15.
Osipov, Yu.S., On the Stabilization of Control Systems with Delay, Differ. Uravn., 1965, vol. 1, no. 5, pp. 606–618.
Gabelaya, A.G., Ivanenko, V.I., and Odarich, O.N., Stabilization of Linear Autonomous Control Systems, Kibernetika, 1975, no. 3, pp. 69–72.
Pandolfi, L., Stabilization of Neutral Functional-Differential Equations, J. Optim. Theory Appl., 1976, vol. 20, pp. 191–204.
Olbrot, A.W., Stabilizability, Detectability and Spectrum Assignment for Linear Autonomous Systems with General Time Delays, IEEE Trans. Automat. Control, 1978, vol. AC-23, pp. 887–890.
Watanabe, K., Stabilization of Linear Systems with Non-Commensurate Delays, Internat. J. Control, 1988, vol. 48, no. 81, pp. 333–342.
Borkovskaya, I.M. and Marchenko, V.M., On an Approach to the Problem of the Stabilization of Systems with Delay, Prikl. Mat. Mekh., 1996, vol. 60, no. 4, pp. 531–541.
Rabah, R., Sklyar, G.M., and Rezounenko, A.V., On Strong Stability and Stabilizability of Linear Systems of Neutral Type, Adv. Time-Delay Systems, Berlin, 2004, pp. 257–268.
Marchenko, V.M. and Yakimenko, A.A., On the Stabilizing Controller Design for Delay Systems of Neutral Type, Differ. Uravn., 2007, vol. 43, no. 11, pp. 1480–1486.
Letov, A.M., Analytic Construction of Controllers, Avtomat. i Telemekh., 1960, no. 4, pp. 436–441; no. 5, pp. 561–568; no. 6, pp. 661–665; 1961, no. 4, pp. 425–435.
Rosenbrock, H.H., Distinctive Problems of Process Control, Chem. Engng. Progr., 1962, vol. 58, no. 9, pp. 43–50.
Wonham, W.M., On Pole Assignment in Multi-Input Controllable Systems, IEEE Trans. Automat. Control, 1967, vol. AC-12, no. 6, pp. 660–665.
Porter, B. and Grossley, T.R., Modal Control, London, 1972.
Kuzovkov, N.T., Modal’noe upravlenie i nablyudayushchie ustroistva (Modal Control and Observing Devices), Moscow, 1976.
Andreev, Yu.N., Algebraic State Space Methods in Linear Plant Control Theory (Survey of Foreign Literature), Avtomat. i Telemekh., 1977, no. 3, pp. 5–50.
Bulatov, V.I., Kalyuzhnaya, T.S., and Naumovich, R.F., Control of the Spectrum of Differential Equations, Differ. Uravn., 1974, vol. 10, no. 11, pp. 1946–1952.
Morse, A.S., Ring Models for Delay-Differential Systems, Automatica, 1976, vol. 12, no. 5, pp. 529–531.
Asmykovich, I.K. and Marchenko, V.M., Spectrum Control in Systems with Delay, Avtomat. i Telemekh., 1976, no. 7, pp. 5–14.
Marchenko, V.M., On the Problem of Modal Control in Linear Systems with Delays, Dokl. Akad. Nauk BSSR, 1978, vol. 22, no. 5, pp. 401–404.
Watanabe, K., Finite Spectrum Assignment of Linear Systems with a Class of Noncommensurate Delays, Internat. J. Control, 1987, vol. 47, no. 5, pp. 1277–1289.
Marchenko, V.M., Modal Control in Systems with Aftereffect, Avtomat. i Telemekh., 1988, no. 11, pp. 73–83.
Gorecki, H., Fuksa, S., Grabowski, P., and Korytowski, A., Analysis and Synthesis of Time Delay Systems, Warszawa, 1989.
Borkovskaya, I.M. and Marchenko, V.M., Modal Control of Systems with Distributed Delay under Incomplete Information Conditions, Differ. Uravn., 1993, vol. 29, no. 11, pp. 1928–1936.
Marchenko, V.M. and Yakimenko, A.A., On the Modal Control of Multi-Input Systems with Retarded Argument of Neutral Type, Differ. Uravn., 2008, vol. 44, no. 11, pp. 1534–1543.
Kalman, R., Falb, P., and Arbib, M., Topics in Mathematical System Theory, New York: McGraw-Hill, 1969. Translated under the title Ocherki po matematicheskoi teorii sistem, Moscow: Mir, 1971.
Kamen, E.W., Linear Systems with Commensurate Time Delays: Stability and Stabilization Independent of Delay, IEEE Trans. Automat. Control, 1982, vol. AC-27, no. 2, pp. 367–375.
Datko, R., Remarks Concerning the Asymptotic Stability and Stabilization of Linear Delay Differential Equations, Math. Anal. Appl., 1985, vol. 111, no. 2, pp. 571–584.
Filimonov, A.B., Controlling the Spectrum of Linear Stationary Objects with Lags, Avtomat. i Telemekh., 1983, no. 4, pp. 34–42.
Shilov, G.E., Matematicheskii analiz. Spetsial’nyi Kurs (Mathematical Analysis. Special Course), Moscow, 1960.
Pearson, J.B. and Ding, C.Y., Compensator Design for Multivariable Linear Systems, IEEE Trans. Automat. Control, 1985, vol. AC-14, no. 2, pp. 130–134.
El’sgol’ts, L.E. and Norkin, S.B., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Deviating Argument), Moscow: Nauka, 1971.
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.M. Marchenko, 2011, published in Differentsial’nye Uravneniya, 2011, Vol. 47, No. 7, pp. 1003–1017.
Rights and permissions
About this article
Cite this article
Marchenko, V.M. Control of systems with aftereffect in scales of linear controllers with respect to the type of feedback. Diff Equat 47, 1014–1028 (2011). https://doi.org/10.1134/S0012266111070111
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266111070111