The analysis of the continuous-time LMS algorithm

Abstract

In this correspondence, a continuous-time analog adaptive filter is suggested via the digital prototype. The continuous-time LMS algorithm is then described by a set of simultaneous first-order equations. The adaptive gain is shown to be unbounded theoretically.

Fault reconstruction for delay systems via least squares and time‐shifted sliding mode observers

“…Notice that since the regressors are known and the coefficients are already estimated, the fault reconstruction in (40) is performed by several vector inner products or by a matrix to vector multiplication.…”

“…From (51) it can be seen that the speed of convergence depends on the parameter and the initial conditiond i (0). After the convergence, an estimate of the current fault (t) can be obtained using (40) as with the RLS algorithm.…”

The Trapezoidal Method of Steepest-Descent and its Application to Adaptive Filtering

“…It is established in the signal processing literature [9] that the integration method used by (2) above is known as rectangular Euler integration. It is also known that there exists a whole family of better approximations among which is the Trapezoidal method [9].…”

“…Now the discrete-time version of (1) is normally quoted and can be found by discrete integration. For example we can approximate the derivative of the vector in (1) as dW(t) dt (W k +1 W k ) / T (2) where the sampling interval T is normalised to unity and W k is the discrete weight vector at some sample interval k.…”

“…One possible exception is found in [1] where the continuous-time version is explored. Furthermore a continuous-time LMS algorithm is found in [2]. For a scalar costfunction J which minimises some quadratic error, the steepest descent is described as the vector differential equation dW(t) dt = μ J W(t) (1) W is a vector of unknown weights and μ is a scalar gain.…”