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The relevance of the work is due to the growing requirements for the dynamic characteristics of electric drives. In particular, together with the requirements of ensuring high accuracy and maximum at given speed limits, a typical task of designing such systems is the mandatory formation of transition diagrams in the form of monotonic time functions. The purpose of this study is to develop an adaptive algorithm for the synthesis of the third-order sliding mode control systems based on the method of N-i switching. Changing the shape of transient trajectory depends on the magnitude of the movement, which requires adaptation of the settings of the control system of the electric drive to the features of the current positioning mode. On the basis of the N-i switching method, an algorithm for synthesizing the parameters of a re-lay control system with cascade-subordinated structure, ensures non-oscillatory initiation of a sliding mode at various positioning modes, has been created. It is constructed by integrating the results of a number of previous works, in which the synthesis of relay control systems based on the analysis of the roots of the sliding equation of the position regulator is performed. This algorithm cannot be formally considered as an optimization tool due to the incompatibility of this problem with the aperiodization taken as the purpose, which comes about for certain forms of transient trajectories. But for such cases, the loss of performance relatively optimal one is negligible. Thus, the result of the application of the proposed algorithm in most practically significant cases is an optimal third-order system with aperiodic entry into the sliding mode. When controlling the electric drive, such a system will ensure the monotonous nature of the movement of the working body of the electromechanical system. The developed block diagram is focused on the practical implementation of the algorithm by the software of controllers of precision electric drives.

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The relevance of the work is due to the need to adapt the methods of the theory of optimal control to modern technical capabilities. The complexity of optimization of control systems by variational methods led to the creation of the N–i switching method. It is characterized by the simplicity of the mathematical apparatus, however, certain methodological aspects encourage research aimed at improving this method with a focus on software implementation. The purpose of this study is to develop the mathematical apparatus of the N–i switching method by developing a technique for correcting the settings of cascade structure relay systems for speed control of electric drives, based on the use of empirically obtained data. To achieve this goal, the following tasks were solved in the work: the parameters of the mathematical model of the electric drive were calculated, on which a series of numerical experiments were performed under the conditions of varying the calculated amplitude of the voltage of the power converter, which made it possible to reveal a stable pattern of changes in the speed of the system depending on the coefficient of the calculated voltage; built a calibration diagram, which is designed to determine the correction factor in the entire range of speed control. The revealed regularity is based on the indirect influence of the varied parameters on the conditions of controllers switching, the moments of occurrence and the nature of their sliding modes. The result of the work is a method for correcting the parameters of relay controllers, which brings the duration of the speed control of the electric drive closer to the empirically set optimal value. Its implementation does not require the involvement of large additional resources and provides an improvement in the quality of transient processes. The empirical component of the proposed optimization method consists not in the direct selection of one of the key parameters of the system, but in the preliminary refinement of the initial data, which is effective within the used synthesis method. The prospect of this study is an analytical solution to the problem of determining the correction coefficient, which will contribute to the practical application of the proposed optimization algorithm.