Convergence of Adaptive Estimators of Time-Varying Linear Systems using Basis Functions: Continuous Time Results

The convergence properties of adaptive filtering algorithms are investigated in situations where the optimal filter is modeled as a time-varying linear system whose parameters are expanded over basis functions. This type of model is one approach when parameters cannot be considered as slowly varying, and is appropriate for modeling certain mobile radio channels and in the identification of the dynamics of vascular autoregulation in kidneys. Appropriate adaptive algorithms are developed in a continuous-time setting, and the local convergence of these algorithms is studied. Conditions for convergence are shown to include an excitation condition on the algorithm regressor and a passivity condition on an algorithm operator. The excitation conditions are interpreted in terms of system signals and the parameter basis functions using previously established results in the discrete-time case. A test for the passivity condition is developed whose application is presented via an illustrative example.