NUMERSCFL METOD OF CALCULATING SWITCHING MOMENT OF RELAY CONTROL IN TASRS OF MAXIMUM SPEED
DOI:
https://doi.org/10.17721/2519-481X/2025/86-03Keywords:
switching moments, control action, control system, numerical method, optimal control, quasi-optimal control, speed, transient processAbstract
antennas, launchers, etc., are described by high-order differential equations. .The theorem about n-intervals, well-known in the theory of automatic control, proved by O. A. Feldbaum, requires n (or n-1) control actions, which leads to increased wear of relay switching devices and requires a sufficiently long time to calculate the moments of optimal relay control . In this connection, there are limitations to the application of optimal control of low-inertia systems. This means that during the technical implementation of fast automatic targeting, it is necessary to implement a transition process that is optimal in terms of speed in the automatic control system with a large number of control action switches. In practice, it is undesirable to operate the drive with frequent switching. This leads to a large load, reduces its reliability and causes rapid wear of its elements. This, in turn, leads to a decrease in the reliability of the drive and to a decrease in the combat readiness ratio of the weapon. In this case, it is expedient to move from optimal control to quasi-optimal by reducing the order of the differential equation to the third [9]. Optimization in this case is implemented by three switching intervals of the control action and is sufficiently fully investigated in the works of many authors [6, 7].
It is possible to expand the range of use of optimal control systems in the class of low-inertia control objects if you calculate in advance the moments of switching in optimal and quasi-optimal systems and determine the value of the parameter at these moments, and then control the switching either according to the program (time) law or according to the deviation of the actual value parameter from the one calculated at the time of switching.
The task of synthesizing a speed-optimal control system is reduced to the synthesis of a high-speed system control device that would ensure the transfer of the state vector of the controlled object from the initial point to the final point.
The article proposes a method for calculating the switching moments of the control action, which creates the possibility of a rational transition from theoretically optimal to practically realized quasi-optimal control, increasing the speed of the guidance system. The rate of action increases as the speed of the target increases and the distance to it decreases. The guidance drives of the weapons must provide maximum speed in order to increase the possibility of destroying targets that fly at high speeds. The speed-optimal process is described by a high-order differential equation. The number of switching moments of the control action is equal to the order of the differential equation. This is known as the n-interval theorem. Thus, the task of determining the optimal control is reduced to the task of determining the moments of change of the control sign and the final value of time when the object is transferred from one point of the phase space to another. The work provides a method for calculating moments of time by a numerical method directly from the equation of the system. In this case, the solution of the problem is simplified and practically not complicated when the order of the differential equation increases and when there are restrictions on the phase coordinates of the object or in the case of a variable structure of the object. The method also does not depend on the type of roots of the characteristic equation. The objective function is also minimized numerically. The Nelder-Mead method is used to minimize it.[1] The conducted numerical experiment confirms the possibility of representing the control system of the nth order by an equivalent system of a lower order.
References
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