A confluence matrix condition for exponential error convergence in overparametrized adaptive systems

Many practical adaptive feedforward systems are overparametrized and, for this reason, will not satisfy persistent excitation (PE) conditions. For these systems, a weaker PE condition is proposed under which it is shown that the while parameters may not converge, the cancellation error (the error between the desired and estimated outputs) still converges exponentially. Bounds are given on the exponential rate of convergence useful for understanding the various tradeoffs and for systematic optimization and design purposes. The convergence rate is determined by properties of the confluence matrix (defined herein) that plays a role similar to that played by the autocorrelation matrix for fully PE systems. As a case study, the structure of the confluence matrix is examined in detail for adaptive systems with a tap delay line (TDL) regressor and sinusoid excitation.