Algorithms of Adaptive Tracking of Unknown Multi-Sinusoidal Signals in MIMO Linear Systems with Multiple Input Delay

In this paper the output adaptive tracking problem for the class of multi input multi output (MIMO) linear time-invariant plants with known parameters, unmeasurable state and known multiple input delay is considered. The reference signals for each output are represented by multi-sinusoidal functions generated by autonomous linear dynamical system of known order but with unknown parameters. The amplitudes, phase shifts and frequencies of harmonics of these functions are unknown. Two different solutions of the problem are proposed and based on modified augmented error scheme that removes the delays from adaptation loop. The first solution uses standard gradient adaptation algorithm providing convergence properties of the closed-loop system and has relatively slow transients, while the second one uses adaptation algorithm with the regressor recording (and hence “memory” of regressor) and dramatically improved parametric convergence. The algorithms can be applied for arbitrary input delays without loss of stability. The designed adaptive control law ensures boundness of all signals in the closed-loop system and drives tracking error to zero. Besides, the controls are free from requirement of identification of the reference model parameters.