Optimization of the Electromechanical System by Formation of a Feedback Matrix Based On State Variables

: pp. 18 - 26
Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University

The task of providing the required dynamic performance of technical systems is one of the main tasks of the automatic control theory. The synthesis of such systems is carried out based on certain criteria that characterize control quality. Today the most common criterion of the functioning of a dynamic system is an integral quadratic form, which includes not only the coordinates of the object, and also the control influences. It should be noted that the inclusion of the control component in the integrated quality criterion allows, in the case of its minimization, to receive control influences of limited amplitude, which is especially important during the design of the a control systems for electromechanical objects. Thus, one of the modern approaches to creating optimal linear stationary dynamic systems consist in:

  • writing equations that describe such systems in the state-space form;
  • formation of systems optimality criteria in the form of an integral functional of the quadratic forms of these variables and control influences;
  • minimization of these functionals by constructing regulators as a set of feedback based on state variables and synthesis of coefficients of these connections.

The problem belongs to the class of variational problems and in general, it is reduced to solving Riccati equations, differential or algebraic: differential for nonstationary systems, when the matrix P, which is included in this equation, depends on time and the integral quality criterion has limits of integration from t1 to t2, or algebraic, when we have a stationary system, it is clear that the matrix P does not depend on time and the limit of integration of the quality criterion is from zero to infinity. It is for many electromechanical systems that it is advisable to minimize such a criterion at long intervals. Such systems include tracking systems, stabili- zation systems, etc. Thus, the problem of synthesis of the optimal electromechanical system by finding the control influences of such a system based on the principle of analytical design of regulators, as the problem is called in Ukrainian literature, or as in Western literature — “linear quadratic regulator”.

The article contains problem statement, the research relevance, purpose statement, analysis of the latest research and publications, presentation of the main material, conclusions and bibliography.

  1. Egupov N., Methods of the classical and modern theory of automatic control: Synthesis of regulators and theory of optimization of automatic control systems. Textbook. Moscow, Russia: The Bauman University Publishing House, 2000. 736 p. (Rus)
  2. Willems J. C Least square stationary optimal control and the algebraic Riccati equation// IEEE Trans.Autom.Control.1971. v. 6. P. 621–634
  3. Kim D. Theory of Automatic control, Vol. 2: Multi-dimensional, nonlinear, optimal and adaptive systems. Moscow: FIZMATHLIT. 2004. 464p. (Rus)
  4. Bernsterein D. S. LQG control with an H-performance bound: A.Riccati equation approach /D. S. Bernsterein, W.M.Haddad // IEEE Trans.Autom.Control.1989. Vol. 34. No 3
  5. Dorf R., Bishop R. Modern control systems. Translated by V. Kopylov, Moscow, Russia: Binom.Laboratory of knowledge, 2002. 832 p. (Rus).
  6. Poljak B., Shherbakov P., Robust stability and control. Moscow, Russia: Nauka, 2002. 303 p.(Rus)
  7. Lozynskyj O., Lozynskyj A., Marushhak Ya., Paranchuk Ya., Tcyapa V., Synthesis of linear optimal dynamic systems: Tutorial, Lviv, Ukraine: Publishing House of Lviv Polytechnic National University, 2016. 392 p. (Ukr)
  8. J. Hsu and A. Meyer, Modern Control Principle and Applications. Transl. From Eng. Edited by Yu. I. Topchev. M: Mashinostroyeniye, 1972, 544p
  9. Petersen I. R., Hollot C. V. A Riccati equation approach to the stabilization of uncertain linear systems // Automatica. 1986. Vol. 22, No 4, 397–411
  10. Lozynskyj О. Y., Biletskyi Y. О., Lozynskyj А. О., Moroz V. І. Formation of the fundamental matrix of an open electromechanical system and its application for the calculation of time processes of state variables. Energy engineering and control systems. 2020. Vol. 6. No 2 (Ukr, on review)
  11. Kaleniuk P., Rudavskyi Y., Tatsiyi R., Differential equations: Tutorial, Lviv, Ukraine: Publishing House of Lviv Polytechnic National University, 2014. 380 p. (Ukr)