Journal of Applied and Computational Mechanics
Larin Parameterization to Solve the Problem of Analytical Construction of the Optimal Regulator of Oscillatory Systems with Liquid Dampers
Document Type : Research Paper
Authors
1 Institute of Applied Mathematics, Baku State University, Z.Khalilov, 23, AZ1148 Baku, Azerbaijan
2 Institute of Information Technologies, ANAS, Baku, Azerbaijan
Abstract
The problem of the analytical construction of the optimal regulator of oscillatory systems with liquid dampers on the complex plane is considered. Since the fractional derivative is included in the differential equation describing the oscillatory systems with liquid dampers movement, the corresponding input-output transfer function also contains fractional rational orders, the general Larin parameterization scheme is modifying for this case. The results are illustrated by numerical examples and it is shown that they coincide with Letov’s А.М. analytical construction of the optimal regulator.
Keywords
- Optimal regulators
- Larin parameterization
- Frequency method
- Transfer function
- Oscillatory systems
- Liquid damper
- Fractional derivative
Main Subjects
[1] Letov, A.M., Analytical Construction of Regulators, Automatics and Telemechanics, 21(4), 1960, 436–441.
[2] Bryson, A., Ho, Yu-Shi, Applied Theory of Optimal Control, Mir., Moscow, 1972.
[3] Letov, A. M., Flight Dynamics and Control, Nauka: Fizmatlit, Moscow, 1969.
[4] Larin, V.B., Statistical Problems of Vibration Protection, Nauk. Dumka, Kiev, 1974.
[5] Sage, A.P., White, C.C., Optimum Systems Control, Englewood Cliffs: Prentice-Hall, New York, 1977.
[6] Aliev, F.A., Mutallimov, M.M., Ismailov, N.A., Radzhabov,, M.A., Algorithms for Constructing Optimal Controllers for Gaslift Operation, Automation and Remote Control, 73(8), 2012, 1279-1289.
[7] Fazli, H., Nieto, J.J., Bahrami, F., On the Existence and Unicueness Results for Nonlinear Sequential Fractional Differential Equations, Applied and Computational Mathematics, 17(1), 2018, 36-47.
[8] Bonilla, B., Rivero, M., Trujillo J.J., On Systems of Linear Fractional Differential Equations with Constant Coefficients, Applied Mathematics and Computation, 187(1), 2007, 68-78.
[9] Aliev, F.A., Aliev, N.A., Safarova, N.A., Gasimova, K.G., Velieva, N.I., Solution of Linear Fractional-Derivative Ordinary Differential Equations with Constant Matrix Coefficients, Applied and Computational Mathematics, 17(3), 2018, 317-322.
[10] Aliev, F.A., Abbasov, A.N., Mutallimov, M.M., Algorithm for the Solution of the Problem Optimization of the Energy Expenses at the Exploitation of Chinks by Subsurface-pump Installations, Applied and Computational Mathematics, 3(1), 2004, 2-9.
[11] Larin, V.B., Control of Walking Apparatus, Nauk. Dumka, Kiev, 1980.
[12] Bordyug, B.A., Larin, V.B., Timoshenko, A.G., Tasks for Controlling Walking Apparatus, Nauk. Dumka, Kiev, 1985.
[13] Aliev, F.A., Aliev, N.A., Safarova, N.A., Transformation of the Mittag-Leffler Function to an Exponential Function and some of its Applications to Problems with a Fractional Derivative, Applied and Computational Mathematics, 18(3), 2019, 316-325.
[14] Ashyralyev, A., Erdogan, A.S., Tekalan S.N., An Investigation on Finite Difference Method for the First Order Partial Differential Equation, Applied and Computational Mathematics, 18(3), 2019, 247-260.
[15] Aliev, F.A., Aliev, N.A., Ismailov, N.A., Analytical Construction of Regulators for Oscillatory Systems with Liquid Dampers, arXiv: 2004.10388, Math, 2020.
[16] Huseynov, F.A., Kazymov, Sh.P., Technology Process of Oil-gas Production of a Production Well, INIGIKAR, Baku, 2009.
[17] Larin, V.B., On a Problem of Analytical Construction of Regulators, Automation and Remote Control, 7, 1966, 30-40.
[18] Larin, V. B., Suntsev, V.N., On the Problem of Analytical Construction of Regulators, Automation and Remote Control, 12, 1968, 142-144.
[19] Larin, V.B., Naumenko, K.I., Suntsev, V.N., Synthesis of Optimal Linear Systems with Feedback, Naukova Dumka, Kiev, 1973.
[20] Aliev, F.A., Larin, V. B., Naumenko, K. I., Suntsev, V. N., Optimization of Linear Time-Invariant Control Systems, Naukova Dumka, Kiev, 1978.
[21] Aliev, F.A., Bordyug, B.A., Larin, V.B., H2-optimization and the State Space Method in the Problem of Synthesis of Optimal Regulators, ELM, Baku, 1991.
[22] Aliev, F.A., Larin, V.B., Optimization of Linear Control Systems: Analytical Methods and Computational Algorithms, Gordon and Breach Science Publishers, London, 1998.
[23] Youla D., Jabr, H.A., Bongiorno, J.J., Modern Winner–Hopf Design of Optimall Controllers. Part II: the Multivariable Case, IEEE Transactions on Automatic Control, 21(3), 1976, 319-338.
[24] Kucera, V., Statistic-Multi Variable Control: a Polynomial Equation Approach, IEEE Transactions on Automatic Control, 25(5), 1980, 913-919.
[25] Desoer, C.A., Liu, R.M., Murran, J., Salens, R., Feedback System Design: the Fraction Respertation Approach to Analysis and Systems, IEEE Transactions on Automatic Control, 25(3), 1980, 399-421.
[26] Aliev, F.A., Larin, V.B., Comment on “Persistent Inputs and the Standart H2-multivariable Control Problem” by K. Park and J.J. Bongiorno Jr., International Journal of Control, 83(6), 2010, 1296-1298.
[27] Aliev, F.A., Aliev, N.A., Safarova, N.A., Gasimova, K.G., Analytical Construction of Regulators for Systems with Fractional Derivatives, Proceedings of the Institute of Applied Mathematics, 6(2), 2017, 252-265.
[28] Ashyralyev, A., Hicdurmaz, B., A Stable Second Order of Accuracy Difference Scheme for a Fractional Schrödinger Differential Equations, Applied and Computational Mathematics, 17(1), 2018, 10-21.
[29] Harikrishnanv, S., Kanagarajan, K., Elsayed, E.M., Existence and Stability Results for Differential Equations with Complex Order Involving Hilfer Fractional Derivatives, TWMS Journal of Pure and Applied Mathematics, 10(1), 2019, 94-101.
[30] Restrepo, J.E., Chinchane, V.L., Agarwal, P., Weighted Reverse Fractional Inequalities of Minkowski’s and Holder’s Type, TWMS Journal of Pure and Applied Mathematics, 10(2), 2019, 188-198.
[31] Panakhov, E., Ercan, A., Bas, E., Ozarslan, R., Hilfer Fractional Spectral Problem via Bessel Operators, TWMS Journal of Pure and Applied Mathematics, 10(2), 2019, 199-211.
[32] Mehdizadeh Khalsaraei, M., Shokri, A., A New Explicit Singularly P-Stable Four-Step Method for the Numerical Solution of Second Order IVPs, Iranian Journal of Mathematical Chemistry, 11(1), 2020, 17-31.
[33] Shokri, A., Mehdizadeh Khalsaraei, M., Atashyar, A., A New Two-Step Hybrid Singularly P-Stable Method for the Numerical Solution of Second-Order IVPs with Oscillating Solutions’, Iranian Journal of Mathematical Chemistry, 11 (2), 2020, 113-132.
[34] Chang Sheldon, S.L., Synthesis of Optimal Automatic Control Systems, Mechanical Engineering, Moscow, 1964.
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