Analysis of the stability of multi-mode systems with approximating nonlinear control laws

The paper analyzes the stability conditions for a multi-mode system with nonlinear links that approximate typical nonlinearities: relay ones with a dead zone and saturation. The problem of stability of systems associated with the implementation of nonlinear control laws based on approximating functions in the form of S-shaped sigma functions has not been considered until now. This problem is especially relevant for multi-mode systems, in which the requirements for the quality of transients in different modes are not the same. Using the example of a nonlinear system with an S-shaped control function obtained based on approximating the combination of functions «relay + relay with a dead band + saturation», averaging straight lines are obtained, the location of which confirms the sufficient conditions for the stability of the system with approximating control. The fulfillment of the obtained conditions has been verified by experimental research of systems with approximating control. Comparative analysis of transient processes in the original nonlinear system and in the system with approximating control made it possible to conclude that the existence of the averaging straight line and its location in the sector specified by the original relay system with a dead zone is a necessary and sufficient condition for the stability of the system with approximating control. This conclusion is based on the equivalence of the results obtained by line arising nonlinear characteristics using the idea of vibrolinearization and by approximating these characteristics by continuous nonlinear functions such as S-shaped sigma functions. Moreover, the performance of systems with approximating control in the transient mode is higher, and self-oscillations in the steady-state are absent in comparison with a system with relay control functions.