Adaptive Output Tracking for 1-D Wave Equations Subject to Unknown Harmonic Disturbances

In this article, we are concerned with an adaptive output tracking problem for 1-D antistable wave equations with unknown harmonic disturbance and reference signals which have both unknown frequencies and amplitudes. The output and disturbance are noncollocated to the control input. First, we construct an auxiliary system and internal model dynamic to restructure the measurable tracking error such that the error consists of these available states. Then, an adaptive method is used to estimate the unknown parameters. Using an invertible transformation, we convert the noncollocated configuration into the collocated ones so that the conventional method can be applied. Finally, error-based feedback is proposed to realize output tracking. Numerical simulations validate these theoretical results.