Synthesis of Linear Control Systems with Maximum Speed and Given Overshoot

We study the solution of the so-called maximum speed problem of a linear control system, in which, unlike the classical optimal speed problem with a relay-type control, a linear control algorithm is determined for a linear object that provides the maximum speed of the system. It is of practical importance due to the widespread practical application of linear control laws. The problem is formulated in relation to continuous, one-dimensional high-order control objects described by the corresponding transfer function or an equivalent system of ordinary differential equations in a certain phase space. The time of the transition process t_tp of the designed system is understood in the sense of the classical theory of automatic control and is determined using the zone  D = spr   4,321 %, equal to the given (desired) value of the overshoot of the synthesized system. This overshoot corresponds to an oscillatory link with a damping coefficient   √2/2  — the Butterworth filter of the second order. Accordingly, the maximum performance problem is put in the following formulation: it is required to find a linear feedback algorithm that provides the closed control system with a given astatism order na and transferring the control object from the initial state to the final one, determined by the constant signal of the regulator’s task, with the minimum value of transient time t_tp and set value overshoot s = s _pr when performing a constraint on the control signal |u(t)| m u_max. At present, this problem has been solved by an approximately algebraic method of synthesizing linear control systems in determining the desired transfer function of a projected closed system based on typical (reference) normalized transfer functions (NTF). The approximate nature of the decision is determined by the fact that the NTF used in the synthesis of high-speed systems are established empirically. This paper proposes a mathematically sound solution to the problem of maximum speed using the theory of analytical design of optimal controllers (ADOC). The maximum speed and set limits on the overshoot and the value of the control signal in the synthesized system are provided by the proposed method for selecting the weights of the quadratic quality functional. We emphasize that the proposed method for the synthesis of high-speed systems, in contrast to the algebraic method, is applicable to a wider class of control objects: both minimal-phase and non-minimal-phase, as containing zeroes, and not. The method is illustrated by an example of the synthesis of a high-speed fourth-order control system containing the results of its simulation.

1. Pontriagin L. S., Boltianskii V. G., Gamkrelidze R. V., Mishchenko E. F. Mathematical theory of optimal processes, Moscow, Fizmatlit Publ., 1961, 302 p (in Russian).

2. Atans M., Falb P. L. Optimal Control, An Introduction to the Theory and Its Applications, McGraw-Hill, New York, 1966.

3. Ivanov V. A., Faldin N. V. theory of optimal control systems, Moscow, Nauka Publ., 1981, 336 p (in Russian).

4. Kliuev A. S., Kolesnikov A. A. The optimization of control systems by time-optimality, Moscow, Energoizdat Publ., 1982. 240 p. (in Russian).

5. Filimonov A. B., Filimonov N. B. The hybrid scheme of the task solution of linear time-optimality based on the formalism of the polyhedral optimization, Mekhatronika, Avtomatizatsiia, Upravlenie, 2014, no.7 , pp. 3—9 (in Russian).

6. Kayumov O. R. Globally controlled mechanical systems, Moscow, Fizmatlit Publ., 2007, 168 p (in Russian).

7. Weinberg L. Network Analysis and Synthesis, McGrawHill, New York, 1962.

8. Abdulaev N. D., Petrov Yu. P. Theory and design methods of optimal regulators, Leningrad, Energoatomizdat Publ., 1985. 240 p (in Russian).

9. Aleksandrov A. G., Palenov M. V. Status and development prospects of adaptive PID controllers, AiT Publ., 2014, no. 2, pp. 16—30 (in Russian).

10. Kim D. P. The synthesis of optimal time-optimality continuous linear controller, AiT Publ., 2009, no. 3, pp. 5—16 (in Russian).

11. Kim D. P. The algebraic method of the synthesis of linear continuous control systems. Mekhatronika, avtomatizatsiia, upravlenie, 2011, no. 1, pp. 9—15 (in Russian).

12. Kim D. P. Finding the desirable transfer function in the synthesis of control systems by the algebraic method. Mekhatronika, Avtomatizatsiia, Upravlenie, 2011, no. 5, pp. 15—21 (in Russian).

13. Kim D. P. Algebraic methods for the synthesis of ACS, Moscow, Fizmatlit Publ., 2014. 164 p. (in Russian).

14. Kim D. P. Analytical method for the synthesis of static static control systems, Mekhatronika, Avtomatizatsiia, Upravlenie, 2019, no. 5, pp. 274—279 (in Russian).

15. Gajduk A. R. Theory and methods of analytical synthesis of automatic control systems (polynomial approach), Moscow, Fizmatlit Publ., 2012. 360 p. (in Russian).

16. Pupkov K. A. Methods of classical and modern control theory: 3 volumes, Moscow, MGTU im. N. E. Baumana Publ., 2000. 736 p. (in Russian).

17. Krasovskii A. A., Pospelov G. S. Automatics and technical cybernetics fundamentals, Moscow, Gostekhizdat Publ., 1962 (in Russian).

18. Lovchakov V. I., Mozzhechkov V. A. Synthesis of linear control systems with maximum speed, Izvestiya TulGU. Technical science, iss. 4. Tula, TulSU Publ., 2016, pp. 149—159 (in Russian).

19. Kwakernaak H., Sivan R. Linear optimal control systems. Wilev Interscience, A Division of John Wiley Sons, Inc. New York-London-Sydney-Toronto 1972.

20. Kawasaki N., Kobayashi H., Shimemura E. Relation between pole assignment and LQ-regulator, Int. J. Contr., 1998, vol. 47, no. 4, pp. 947—951.

21. Miroslav D. Lutovac. Filter Design for Signal Processing using MATLAB and Mathematica, New Jersey, USA, Prentice Hall, 2001.

22. Sadovoy A. V., Sukhinin B. V., Sokhina Yu. V. Optimal control systems for precision electric drives, Kiev, ISIMO Publ., 1996, 298 p (in Ukraine).

23. Lovchakov V. I., SHibyakin O. A. The solution of a problem of speed of response on output coordinate for linear dynamic systems, Mekhatronika, Avtomatizatsiia, Upravlenie, 2019, no. 9, pp. 532—541 (in Russian).