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    Similarity and Analogousness in Dynamical Systems and Their Characteristic Features

    2019, Vol. 15, no. 3, pp.  213-220

    Author(s): Misyurin S. Y., Kreinin G. V., Nosova N. Y.

    Mathematical models describing technically oriented dynamical systems are generally rather complex. Very time-consuming interactive procedures have to be used when selecting the structure and parameters of the system. Direct enumeration of options using such procedures can be avoided by applying a number of means, in particular, dimension methods and similarity theory. The use of dimension and similarity theory along with the general qualitative analysis of the system can serve as an effective theoretical research method. At the same time, these theories are simple. Using dimension and similarity theory, it is possible to draw conclusions when considering phenomena that depend on a large number of parameters, but so that some of them become insignificant in certain cases.
    The combined method of using the theory of similarity, analogousness and methods developed by the authors for testing the drive model provides insight into its dynamics, controllability and other properties. The proposed approach is based on systematization and optimization of the process of forming a dimensionless model and similarity criteria, its focus on solving the formulated problem, as well as on special methods of modeling and processing of simulation results. It improves the efficiency of using similarity properties in solving analysis and synthesis problems. The advantage of this approach manifests itself in the ultimate simplification of the dimensionless model compared to the original model. The reduced (dimensionless) model is characterized by a high versatility and efficiency of finding the optimal and final solution in the selection of parameters of the real device, as it contains a significantly smaller number of parameters, which makes it convenient in solving problems of analysis and, in particular, synthesis of the system.
    Dimension methods and similarity theory are successfully applied in the study of dynamical systems of different classes. The problems that arise are mainly related to the selection of a rational combination of the main units of measurement of physical quantities, the transition to dimensionless models and the formation of basic similarity criteria. The structure and the form of the dimensionless model depend on the adopted units of measurement of the variables appearing in the equations of the model and on the expressions assigned to its coefficients. Specified problems are solved by researchers, as a rule, by appealing to their intuition and experience. Meanwhile, there exist well-known systematized approaches to solving similar problems based on the method of the theory of analogousness.
    Keywords: similarity, analogousness, hydraulic drive, dynamical system, dimensionless parameters
    Citation: Misyurin S. Y., Kreinin G. V., Nosova N. Y., Similarity and Analogousness in Dynamical Systems and Their Characteristic Features, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp.  213-220

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    [1] Merritt, H. E., Hydraulic Control Systems, Wiley, New York, 1991, 368 pp.
    [2] Has, Z. and Rahmat, M. F., “Application of Self-Tuning Fuzzy PID Controller on Industrial Hydraulic Actuator Using System Identification Approach”, Int. J. Smart Sensing Intell. Syst., 2:2 (2009), 246-–261  crossref
    [3] Huang, Y. J., Kuo, T. C., and Lee, H. K., “Fuzzy-PD Controller Design with Stability Equations for Electro-Hydraulic Servo Systems”, ICCAS'2007: Proc. of the Internat. Conf. on Control, Automation and Systems (Seoul, Korea, Oct 2007), 2407–2410
    [4] Alleyne, A. and Liu, R., “A Simplified Approach to Force Control for Electro-Hydraulic Systems”, Control Eng. Pract., 8:12 (2000), 1347–1356  crossref
    [5] Lizarde, C., Loukianov, A. G., and Sanchez, E., “Force Tracking Neural Block Control for an Electro-Hydraulic Actuator via Second-Order Sliding Mode”, Int. J. Robust Nonlinear Control, 18:3 (2008), 319–332  crossref  mathscinet
    [6] Avila, M. A., Loukianov, A. G., and Sanchez, E. N., “Electro-Hydraulic Actuator Trajectory Tracking”, Proc. of the American Control Conference (Boston, Mass., 2004), v. 3, 2603–2608
    [7] Cotsaftis, M. and Keskinen, E., “Smooth High Precision Contact Position Control of Rotating Cylinders with Hydraulic Actuators”, Proc. of the 12th IFToMM World Congress (Besancon, France, 2007), 738–743
    [8] Lu, X., Du, F., Jia, Q., Ren, B., and Wang, X., “Sliding Mode Force Control of an Electrohydraulic Servo System with RBF Neural Network Compensation”, Mechanika, 25:1 (2019), 32–37  crossref
    [9] Guan, C. and Pan, S., “Adaptive Sliding Mode Control of Electro-Hydraulic System with Nonlinear Unknown Parameters”, Control Eng. Pract., 16:11 (2008), 1275–1284  crossref
    [10] Raade, J. W. and Kazerooni, H., “Analysis and Design of a Novel Hydraulic Power Source for Mobile Robots”, IEEE Trans. Autom. Sci. Eng., 2:3 (2005), 226–232  crossref
    [11] Wang, X., “Modeling and Control of a Torque Load System with Servo Actuators Dynamics”, Proc. Inst. Mech. Eng. G J. Aer., 231:9 (2016), 1676–1685  crossref
    [12] Sedov, L. I., Similarity and Dimensional Methods in Mechanics, 10th ed., CRC, Boca Raton, Fla., 1993, 496 pp.  mathscinet
    [13] Bukhgolts, N. N., Basic Course of Theoretical Mechanics: P. 2, Nauka, Moscow, 1972, 328 pp. (Russian)
    [14] Sonin, A. A., The Physical Basis of Dimensional Analysis, 2nd ed., MIT, Cambridge, Mass., 2001, 57 pp.  adsnasa
    [15] Mamontov, M. A., Similarity, Min. Oboron. SSSR, Moscow, 1971, 59 pp. (Russian)
    [16] Yang, G., Yao, J., Le, G., and Ma, D., “Adaptive Integral Robust Control of Hydraulic Systems with Asymptotic Tracking”, Mechatronics, 40 (2016), 78–86  crossref  mathscinet
    [17] Guo, K., Wei, J., Fang, J., Feng, R., and Wang, X., “Position Tracking Control of Electro-Hydraulic Single-Rod Actuator Based on an Extended Disturbance Observer”, Mechatronics, 27 (2015), 47–56  crossref
    [18] Tri, M. N., Nam, C. N. D., Park, G. H., and Ahn, K. K., “Trajectory Control of an Electro Hydraulic Actuator Using an Iterative Backstepping Control Scheme”, Mechatronics, 29 (2015), 96–102  crossref
    [19] Li, L., Huang, H., Zhao, F., Triebe, M. J., and Liu, Z., “Analysis of a Novel Energy-Efficient System with Double-Actuator for Hydraulic Press”, Mechatronics, 47 (2017), 77–87  crossref  elib
    [20] Márton, L., Fodor, S., and Sepehri, N., “A Practical Method for Friction Identification in Hydraulic Actuators”, Mechatronics, 21:1 (2011), 350–-356  crossref  elib
    [21] Misyurin, S. Yu. and Kreinin, G. V., “Power Optimization Criteria of a Mechanical Unit of an Automated Actuator”, Dokl. Phys., 60:1 (2015), 15–18  crossref  adsnasa  elib; Dokl. Akad. Nauk, 460:1 (2015), 39–42 (Russian)
    [22] Misyurin, S. Yu. and Kreinin, G. V., “Dynamics and Design of a Power Unit with a Hydraulic Piston Actuator”, Dokl. Phys., 61:7 (2016), 354–359  crossref  adsnasa  elib; Dokl. Akad. Nauk, 469:3 (2016), 302–307 (Russian)
    [23] Cotsaftis, M. and Keskinen, E., “Smooth High Precision Contact Position Control of Rotating Cylinders with Hydraulic Actuators”, Proc. of the 12th IFToMM World Congress (Besancon, France, 2007), 738–743

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