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Mekhatronika, Avtomatizatsiya, Upravlenie, 2017, vol. 18, no. 6, pp. 376382
DOI: 10.17587/mau.18.376-382


Necessary Conditions of Time-Optimality of Linear Dynamical Systems

V. I. Lovchakov, lowi50@mail.ru, Tula State University, Tula, 300012, Russian Federation

Corresponding author: Lovchakov Vladimir I., D. Sc., Full Professor, Tula State University, department of electrical engineering and electrical equipment, Tula, 300012, Russian Federation, e-mail: lovvi50@mail.ru

Received on November 16, 2016
Accepted on December 09, 2016

We formulate and solve the problem of regulator design with minimal transient time (in terms of classical control theory) under specified constraints on control and overregulation. The problem is solved in a class of linear high-order control systems described by a transfer function having a constant in the numerator. The main stage of the proposed design approach consists in the determination of optimal transfer function of closed-loop control system with the structure composed by the sequential connection of specific number of aperiodic and oscillatory links. We prove the theorem establishing that for even order control systems these links must have the same values of time constants to provide the minimum transient time, and time constants must be selected from the condition of satisfying the control constraints. Correspondingly, damping coefficients of oscillatory links must be determined from the condition of providing the specified overregulation value of designed system. The theorem is proved for the special, but important for control practice case of specifying the permissible system overregulation value being equal to the value of the zone, using which the transient time of this control system is determined. This predetermine the possibility of theorem application to the solution (in the linear approximation) of so-called problem of time-optimality for one phase coordinate of an object, which is actual for engineering practice. Conditions of theorem are not only necessary, but also sufficient conditions of time-optimality for several subclasses of control systems. Systems with monotonic transient processes with prohibited overregulation having, correspondingly, only negative real poles are an example of such subclass. After determination of optimal transfer function of closed-loop control system subject to theorem conditions, it is recommended to perform regulator design providing the time-optimality by the synthesis method according to the desired transfer function or by the method of modal control depending on the technical conditions of regulator realization.
Keywords: linear one-dimensional object, time-optimality criterion, overregulation, poles of optimal system


For citation:

Lovchakov V. I. Necessary Conditions of Time-Optimality of Linear Dynamical Systems, Mekhatronika, Avtomatizatsiya, Upravlenie, 2017, vol. 18, no. 6, pp. 376382.
DOI: 10.17587/mau.18.376-382

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