Table 1 Summary of adaptive robust parameter estimation algorithm (10)–(13) with scale factor (25) and EPE-based VFF (27)–(30)

Step 1

Let at stage k, k ≥ N, the parameter vector estimate Ŵ(k − 1), the scale estimates s(k − 1), …, s(k − L + 1), the error signals e(k − 1), …, e(k − L + 1) and the matrix P(k − 1) from the (L − 1) previous stages are known.

Step 2

Take the current input, x(k), and form the regression vector in (3) X ^T(k) = {x(k), x(k − 1), …, x(k − N + 1)} of length N, assuming that the (N − 1) most recent inputs are given.

Step 3

Take the current output, d(k), and calculate the current error signal, e(k), from (11) using X(k) from step 2, and define the current data frame E _L  = {e(k), e(k − 1), …, e(k − L + 1)} of length L < N, assuming that the (L − 1) most recent errors are previously stored.

Step 4

Calculate the normalised error e(k)/s(k − 1) and the winsorised error ψ(e(k)/s(k − 1)) from (4) with σ = 1; then calculate the weight ω(k) in (25) by using (8) with W ₀ = Ŵ(k − 1) and s = s(k − 1); finally, calculate the scale factor s(k) from (25).

Step 5

Define the current data frame of normalised residuals E _NL  = {e(k)/s(k), e(k − 1)/s(k − 1), …, e(k − L + 1)/s(k − L + 1)} from steps 1, 3 and 4; then calculate the robust discrimination function, Q(k), in (28), using the data set E _NL ; finally, calculate the VFF, ρ(k), from (29) and (30).

Step 6

Calculate the winsorised error, ψ(e(k)/s(k)), from (4), with σ = 1 and by using e(k) from step 3 and s(k) from step 4; then calculate the weight, ω(k), in (8) by using (4) with W ₀ = Ŵ(k − 1) and s = s(k).

Step 7

Calculate the matrix, M(k), in (12) with ρ = ρ(k) from step 5; then calculate the matrix, K(k), in (12) by using X(k) from step 2 and ω(k) from step 6.

Step 8

Calculate the parameter vector update, Ŵ(k), in (10), by using d(k) and e(k) from step 3, as well as K(k) from step 7.

Step 9

Calculate the weighting matrix, P(k), in (13) by using M(k) and K(k) from step 7, together with X(k) from step 2.

Step 10

Tune the time counter, that is increase the time index, k ← k + 1, and go back to step 2.