Adaptive Output Tracking for MIMO Linear Systems with Different Control Delays Affected by Unknown External Disturbances

This paper presents the problem of adaptive output tracking for a class of unstable multi-input multi-output linear time-invariant systems affected by unknown external disturbances, taking into account different control delays across the channels. It is assumed that both the reference signal and the external disturbances have a multi-harmonic form with unknown frequencies, phases, biases, and amplitudes. The disturbances may be unmatched in the system and affect both the inputs and the outputs. For such systems, a linear state feedback control law is first designed based on the classical Falb—Wolovich method to decouple the multivariable system into independent control channels. This approach to the channel decoupling allows, in the case of the presence of nonminimum-phase zeros in the transfer function, to exclude them and transform the original system into transfer functions with independent integrators. Then, an observer is constructed to estimate the states of the reference signal and the external disturbances. Finally, an adaptive controller is synthesized to compensate for the external disturbances and ensure accurate output tracking of the reference signal. In this work, the adaptive algorithm with memory regressor extension is employed with the aim of improving the convergence rate of the parameter adaptation. The proposed method guarantees that all signals in the closed-loop system remain bounded and ensures the asymptotic stability of the output. To validate the proposed approach, numerical simulations are carried out in the MATLAB environment. The proposed solution is feasible for "square" systems, when the number of inputs and outputs of a multi-channel system is the same.

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