Two-Level Recursive Identification of Hammerstein System by Interaction Prediction Method

Abstract

The paper concerns identification of Hammerstein system under nonparametric prior knowledge about the static nonlinear characteristic. The identification task is decomposed by the prediction of the hidden interaction signal. The standard kernel approach is modified to cope with the problem of constant offset between the regression function and the static characteristic in Hammerstein system, which was not solved in the previous papers. The idea is based on alternate updating of the offset and the estimate of the impulse response of the linear block. Both levels of the algorithm are given in the recursive version.

Appendices

Appendices

1.1 Appendix A. Calculation of ${\mathbf{M}}_{{\left( {\varvec{M} + 1} \right) \times \left( {\varvec{M} + 1} \right)}}$ and $\varvec{L}_{{\varvec{M} + 1}}$

A1. Recursive computing of matrix $\varvec{M}_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}}$

Taking into account definitions of $P_{N}$ and ${\mathbf{1}}_{{N \times \left( {M + 1} \right)}}$, after simple algebra we obtain

$$ {\mathbf{M}}_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}} = \frac{1}{2N}\left[ {P_{N}^{T} {\mathbf{1}}_{{N \times \left( {M + 1} \right)}} + \left( {P_{N}^{T} {\mathbf{1}}_{{N \times \left( {M + 1} \right)}} } \right)^{T} } \right] = \left[ {m_{ij} } \right]_{i,j = 0,1, \ldots ,M} $$

where

$$ m_{ij} = \frac{1}{2}\left[ {\underbrace {{\frac{1}{N}\sum\nolimits_{l = 0}^{N - 1} {\hat{R}^{P} \left( {u_{k + l - i} } \right)} }}_{{m_{iN} }} + \underbrace {{\frac{1}{N}\sum\nolimits_{l = 0}^{N - 1} {\hat{R}^{P} \left( {u_{k + l - i} } \right)} }}_{{m_{jN} }}} \right] $$

(12)

and $m_{ij} = m_{ji}$. Thus, $m_{ij} \triangleq m_{ij,N} = \frac{1}{2}\left[ {m_{iN} + m_{jN} } \right].$

For empirical means in (12) we simply get recursive versions

$$ m_{iN} = \frac{N - 1}{N}\left[ {m_{i,N - 1} + \frac{1}{N - 1}\hat{R}^{P} \left( {u_{{k + \left( {N - 1} \right) - i}} } \right)} \right], $$

(13)

$$ m_{jN} = \frac{N - 1}{N}\left[ {m_{j,N - 1} + \frac{1}{N - 1}\hat{R}^{P} \left( {u_{{k + \left( {N - 1} \right) - j}} } \right)} \right], $$

(14)

which further leads to $m_{ij,N} = \frac{N - 1}{N}\left[ {m_{ij,N - 1} + \frac{1}{N - 1}\bar{R}_{ij}^{P} } \right],$ where

$$ \bar{R}_{ij}^{P} = \frac{{\hat{R}^{P} \left( {u_{{k + \left( {N - 1} \right) - i}} } \right) + \hat{R}^{P} \left( {u_{{k + \left( {N - 1} \right) - j}} } \right)}}{2} $$

or equivalently $m_{ij,N} = \left( {\frac{N - 1}{N}} \right)m_{ij,N - 1} + \frac{1}{N}\bar{R}_{ij}^{P} .$ Hence, for the matrix

$$ {\mathbf{M}}_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}} \triangleq {\mathbf{M}}_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}}^{N} = \left[ {m_{ij,N} } \right]_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}} \triangleq {\mathbf{M}}^{N} $$

we get

$$ {\mathbf{M}}^{N} = \frac{N - 1}{N}{\mathbf{M}}^{N - 1} + \frac{1}{N}{\mathbf{R}}^{N - 1} , $$

(15)

where ${\mathbf{R}}^{N - 1} = \left[ {\bar{R}_{ij}^{P} } \right]_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}}$.

A2. Recursive computing of vector $L_{M + 1}$

Since (cf. (5))

$$ L_{M + 1} = \left( {P_{N}^{T} P_{N} } \right)^{ - 1} P_{N}^{T} Y_{N} , $$

(16)

the vector $L_{M + 1} = L_{M + 1}^{N} \triangleq L^{N}$ is in fact the least squares model of the linear dynamic object with the input $P_{N}$, parameters $L$ and the output $Y_{N}$, i.e. the result of the following optimization task $Y_{N} - P_{N} L_{2}^{2} \to { \hbox{min} }_{L}$, where $_{2}$ is the Euclidean norm. Taking into account definition of the matrix $P_{N}$, and denoting $G_{N} = \left( {P_{N}^{T} P_{N} } \right)^{ - 1}$, after standard steps we get recursive version of (16)

$$ L^{N} = L^{N - 1} + G_{N} \widehat{{\bar{R}^{P} }}^{T} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)\left[ {y_{{k + \left( {N - 1} \right)}} - \hat{\bar{R}}^{P} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)L^{N - 1} } \right], $$

$$ G_{N} = G_{N - 1} - \frac{{G_{N - 1} \widehat{{\bar{R}^{P} }}^{T} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)\hat{\bar{R}}^{P} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)G_{N - 1} }}{{1 + \widehat{{\bar{R}^{P} }}^{T} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)G_{N - 1} \widehat{{\bar{R}^{P} }}^{T} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)}}. $$

(17)

Consequently,

$$ L^{N} = L^{N - 1} + G_{N} \rho_{N - 1} , $$

(18)

where

$$ \rho_{N - 1} = \widehat{{\bar{R}^{P} }}^{T} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)\left[ {y_{{k + \left( {N - 1} \right)}} - \hat{\bar{R}}^{P} \left( {\bar{u}_{{k + \left( {N - 1} \right)}} } \right)L^{N - 1} } \right]. $$

(19)

1.2 Appendix B. Calculating of ${\mathbf{\mathcal{L}}}_{\varvec{N}}$ and ${\mathbf{\mathcal{M}}}_{\varvec{N}}$

B1. Recursive computing of the numerator ${\mathcal{L}}_{N}$

For ${\mathcal{L}}_{N}$, owing to (18), we have

$$ {\mathcal{L}}_{N} = L^{{N^{T} }} {\mathbf{M}}^{N} L^{N} = \left[ {L^{N - 1} + G_{N} \rho_{N - 1} } \right]^{T} {\mathbf{M}}^{N} \left[ {L^{N - 1} + G_{N} \rho_{N - 1} } \right] $$

$$ = L^{{N - 1^{T} }} {\mathbf{M}}^{N} L^{N - 1} + \left( {G_{N} \rho_{N - 1} } \right)^{T} {\mathbf{M}}^{N} \left( {G_{N} \rho_{N - 1} } \right) + 2L^{{N - 1^{T} }} {\mathbf{M}}^{N} \left( {G_{N} \rho_{N - 1} } \right), $$

and further, owing to (15), we get

$$ L^{{N - 1^{T} }} {\mathbf{M}}^{N} L^{N - 1} = \frac{N - 1}{N}L^{{N - 1^{T} }} {\mathbf{M}}^{N - 1} L^{N - 1} + \frac{1}{N}L^{{N - 1^{T} }} {\mathbf{R}}^{N - 1} L^{N - 1} , $$

which leads to ${\mathcal{L}}_{N} = \left( {\frac{N - 1}{N}} \right){\mathcal{L}}_{N - 1} + {\mathcal{K}}_{{\mathcal{L}}}^{N}$, where

$$ {\mathcal{K}}_{{\mathcal{L}}}^{N} = \left( {G_{N} \rho_{N - 1} } \right)^{T} {\mathbf{M}}^{N} \left( {G_{N} \rho_{N - 1} } \right) + 2L^{{N - 1^{T} }} {\mathbf{M}}^{N} \left( {G_{N} \rho_{N - 1} } \right) + \left( {\frac{1}{N}} \right)L^{{N - 1^{T} }} {\mathbf{R}}^{N - 1} L^{N - 1} . $$

B2. Recursive computing of the denominator ${\mathcal{M}}_{N}$

For ${\mathcal{M}}_{N}$ we have ${\mathcal{M}}_{N} = {\mathcal{M}}_{N - 1} + {\mathcal{K}}_{{\mathcal{M}}}^{N}$, where ${\mathcal{M}}_{N - 1} = L^{{N - 1^{T} }} 1_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}} L^{N - 1} ,$ and ${\mathcal{K}}_{{\mathcal{M}}}^{N} = \left( {G_{N} \rho_{N - 1} } \right)^{T} 1_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}} \left( {G_{N} \rho_{N - 1} } \right) + 2L^{{N - 1^{T} }} 1_{{\left( {M + 1} \right) \times \left( {M + 1} \right)}} \left( {G_{N} \rho_{N - 1} } \right).$