Greedy Pursuit-Based Hierarchical Iterative Algorithm for Multi-Input Systems with Unknown Time Delays and Colored Noise

For multi-input systems with unknown input time delays and colored noise, joint estimation of parameters and time delays is challenging because the over-parameterized model is high-dimensional and the regression vector contains unmeasurable noise terms. This paper investigates greedy-based hierarchical iterative identification algorithms, which require only a practical upper bound lon the time delays instead of their exact values. Using this bound, a sparse over-parameterized pseudo-linear regression model is constructed. To handle the unmeasurable noise terms in the information matrix, the hierarchical identification principle is employed, replacing these terms with their estimates obtained in the previous iteration. A greedy search strategy is then adopted to locate the nonzero key parameters, which effectively reduces the model dimension and improves estimation efficiency. Based on this, the greedy-based hierarchical extended least squares (GH-ELS) and greedy-based hierarchical gradient pursuit iterative (GH-GPI) algorithms are derived for joint estimation of system parameters and time delays with limited sampled data. Simulation results for two examples show that, under the tested configurations, both algorithms effectively handle colored noise and unknown time delays. For the five-input example with L=300 and the noise variances σ2=0.502 and 1.002, the estimation errors of GH-ELS are about 1.69% and 3.26%, respectively. Those of GH-GPI are about 4.38% and 5.19%, respectively. All time delays are recovered exactly. Monte Carlo simulations for Example 2 further demonstrate low estimation errors and high support-set recovery rates across different data lengths and noise variances, and the estimation accuracy achieved with limited sampled data is comparable to that obtained with longer data lengths.