Frequencies estimation in multisinusoidal time-varying parameter of the first order discrete linear system with the application to indirect adaptive control

The paper addresses the problem of adaptive frequencies estimation for multisinusoidal Time-Varying (TV) parameter of a discrete linear system of the first order. It is assumed that the amplitudes, frequencies, and phases of the harmonics in the TV parameter are unknown, however the number of harmonics is known. The novelty of the proposed approach consists in the fact that the frequencies identification is possible even if the system output crosses zero when the information about the TV parameter is inaccessible. In this case, when the proposed solution is used in a problem of adaptive control of the system considered, the frequencies identification and the work of TV parameter observer are independent, what increases the rate and precision of controller parameters tuning. The problem is solved by transformation of the plant model into a regression model linear with respect to unknown frequencies and used for design of identification algorithms. In the paper, two identification algorithms are applied. The first one is the standard gradient algorithm, while the second one is the algorithm with improved parametric convergence achieved by accumulation of regressor over past period of time and referred to as algorithm with memory regressor extension. The problem of control is solved with the use of: certainty equivalence principle; internal model principle according to which the TV parameter is represented as the output of dynamic autonomous model (exosystem) and involving of this model into the structure of the control law; observer of the exosystem state; one of the proposed frequencies identifier; and a formula of recalculation of the frequencies estimates into the controller adjustable parameters. A procedure of transformation of the TV system into a regression linear with respect to unknown frequencies used for design of identification algorithms is represented. The obtained solution is applied to the problem of indirect (identification-based) adaptive control of the TV system considered in the paper. The main distinguishing feature of the solution proposed consists in independence of the obtained identifiers from the observation property of the TV parameter what increases the transient performance and precision of the indirect adaptive control algorithms designed for the considered class of TV systems. The proposed solution can be used in problems of control of parametric resonance systems.