Determination of the Adaptation Function Parameters of the Observer of Electric Drive Rotation Speed

Abstract

Several well-known structures of speed observers have been developed using Lyapunov’s direct method, which defines the observer structure but does not provide a way to determine the PI-controller parameters included in it. For one of such observers, the structure of which is defined by the adaptation function as the vector product of the rotor flux linkage and the estimation error of the stator current, this paper addresses the problem of parametric synthesis by employing the results of the stability analysis of the linearized models of the output nonlinear observer. First, the observer containing only the integral component was analyzed, and it was proved that the system is neutrally stable for any large positive gains of the integral component. It was revealed that, in addition to a zero root, the observer has asymptotes for the real parts of other roots of the characteristic equation of the linearized system. Analytical expressions were obtained for determining the limiting values of the roots in terms of the observer equation parameters. Then, a proportional component of the controller from the adaptation function was added, and for this observer a characteristic equation of its linearized model was also derived. Using the Hurwitz algebraic stability criterion, analytical relationships between the values of the controller components were obtained at which the observer remains neutrally stable. These relationships made it possible to construct the stability boundaries of the observer in the parameter plane for different values of the rotor rotation speed of the induction machine (IM). The stability analysis resulted in determining appropriate parameter limits for the controller within the observer: lower boundary for the integral component and an upper boundary for the proportional component. The obtained properties of the local stability of first-approximation were then generalized for the output system of the observer nonlinear equations. The applicability of this approach to the given dynamical system was verified by mathematical modeling.