Optimal stabilization of operation of liquid-propellant rocket engine

DOI: 10.34759/trd-2022-125-03

Аuthors

Bairamov F. D.^*, Bairamov B. F.^**

1,2Naberezhnye Chelny Institute of Kazan Federal University, Kazan, Russia, Republic of Tatarstan

*e-mail: fdbairamov@mail.ru
**e-mail: bfbairamov@mail.ru

Abstract

The Lyapunov function method is applied to solve the problem on optimal stabilization of the steady-state mode of operation of the two-component liquid-propellant rocket engine with turbo-driven pump assembly by regulating the pressures in the oxidizer and fuel tanks with considering wave processes in the flow lines. Liquid-propellant rocket engine is a complex mechanical system containing two distributed links and finite-dimensional links located at both endpoints of the distributed links. The linearized equations of dynamics of separate links are drawn up. After an exception of some variables from these equations system of dynamic equations of liquid-propellant rocket engine in general have been obtained. To solve the problem on stabilization, first, the Lyapunov function method is used to determine the set of controls (laws for regulating the pressures in tanks) ensuring asymptotic stability of liquid-propellant rocket engine operation. Then, the optimal control is determined on this set by the Lagrange function method from the condition for minimum of the norm at each moment of time. Based on specific equations, the Lyapunov function is constructed as the sum of integral and ordinary quadratic forms, the sign-definiteness of which is checked by the Sylvester criterion. The developed control laws can be implemented quite simply and accurately in practice. It is not possible to ensure the asymptotic stability of liquid-propellant rocket engine operation without regulating the pressures in the tanks. The liquid-propellant rocket engine belongs to the class of systems with distributed and lumped parameters, described by linear equations in partial and ordinary derivatives. Some equations of dynamics of liquid-propellant rocket engine do not contain time derivatives. The methodology of synthesis of optimal controls with the smallest value of the norm at each moment of time in systems with distributed and lumped parameters, some equations of which do not contain time derivatives, has been developed. The need for such control arises, for example, when determining the boost pressure in the hydraulic tanks of the hydraulic system; when determining the boost pressure of the fuel tank, which ensures stable operation of the heating furnace. The developed methodology can also be used to study stability of such systems. For example, when studying the stability of operation of a rotary-type wind turbine with a vertical axis of rotation together with a pump. The shaft that transmits the torque of the wind turbine to the pump has a considerable length, so the problem is solved, taking into account the elasticity of this shaft.

Keywords:

two-component liquid-propellant rocket engine, equations of dynamics, analysis of asymptotic stability by the Lyapunov function method, optimal laws for regulating the pressures in the oxidizer and fuel tanks

References

1. Crocco L., Cheng Sin-I. Theory of combustion instability in liquid propellant rocket motors, London, Butterworths scientific Publ., New York, Interscience Publ. Inc., 1956, 200 p.
2. Glikman B.F. Avtomaticheskoe regulirovanie zhidkostnykh raketnykh dvigatelei (Automatic control of liquid-propellant rocket engines), Moscow, Mashinostroenie, 1974, 336 p.
3. Makhin V.A., Prisnyakov V.F., Belik N.P. Dinamika zhidkostnykh raketnykh dvigatelei (Dynamics of liquid-propellant rocket engines), Moscow, Mashinostroenie, 1969, 334 p.
4. Babkin A.I., Belov S.I., Rutovskii N.B., Solov’ev E.V. Osnovy teorii avtomaticheskogo upravleniya raketnymi dvigatel’nymi ustanovkami (Foundations of the theory of automatic control of rocket propulsion systems), Moscow, Mashinostroenie, 1978, 328 p.
5. Sirazetdinov T.K. Ustoichivost’ sistem s raspredelennymi parametrami (Stability of systems with distributed parameters), Novosibirsk, Nauka, 1987, 230 p.
6. Kolesnikov N.S., Sukhov V.N. Uprugii letatel’nyi apparat kak ob"ekt avtomaticheskogo upravleniya (Elastic aircraft as an object of automatic control), Moscow, Mashinostroenie, 1974, 268 p.
7. Moshkin E.K. Nestatsionarnye rezhimy raboty ZhRD (Non-stationary modes of operation of liquid-propellant rocket engines), Moscow, Mashinostroenie, 1970, 336 p.
8. Bairamov B.F., Bairamov F.D. Prikladnaya matematika i mekhanika, 2018, vol. 82, no. 6, pp. 757-763. DOI: 10.31857/S003282350002739-2
9. Bairamov B.F., Bairamov F.D. Synthesis control in the systems with distributed and concentrated parameters, Helix, 2020, vol. 10, no. 5, pp. 156–162. DOI: 10.29042/2020-10-5-156-162
10. Krapivnykh E.V. Trudy MAI, 2015, no. 80. URL: https://trudymai.ru/eng/published.php?ID=56924
11. Galeev A.V. Trudy MAI, 2016, no. 86. URL: https://trudymai.ru/eng/published.php?ID=67814
12. Timushev S.F., Fedoseev S.Yu. Trudy MAI, 2015, no. 83. URL: https://trudymai.ru/eng/published.php?ID=62080
13. Klimenko D.V., Timushev S.F., Korchinskii V.V. Trudy MAI, 2015, no. 82. URL: https://trudymai.ru/eng/published.php?ID=58687
14. Wang P.K.C. Theory of stability and control for distributed parameter systems (a Bibliography), International Journal of Control, 1968, vol. 7, no. 2. pp. 101-116. DOI: 10.1080/00207176808905588
15. Wang P.K.C. On the stability of equilibrium of mixed distributed and lumped parameter control systems, International Journal of Control, 1966, vol. 3, no. 2, pp. 139-147.
16. Bairamov F.D., Bairamov B.F., Mardamshin I.G. Vestnik Kazanskogo gosudarstvennogo tekhnicheskogo universiteta im. A.N. Tupoleva, 2009, no. 4, pp. 103-106.
17. Bairamov F.D. Izvestiya vuzov. Aviatsionnaya tekhnika, 1978, no. 4, pp. 16-22.
18. Semenov P.K. Izvestiya vuzov. Aviatsionnaya tekhnika, 1972, no. 3, pp. 16-21.
19. Tikhonov A.N., Samarskii A.A. Uravnenya matematicheskoi fiziki (Equations of mathematical physics), Moscow, Nauka, 1977, 736 p.
20. Malkin I.G. Teoriya ustoichivosti dvizheniya (Theory of motion stability), Moscow Gostekhizdat, 1952, 432 p.

Download