Multistep algorithm for single module identification

Identification of a single target module is performed on the basis of a selected predictor model and measurement data of the node signals and external excitation signals that appear in this predictor model. The applied method is the local multistep method:

1. Estimate a high-order ARX model between all relevant external excitation signals and the selected predictor model outputs $w_Y$;
2. Use the estimated ARX model to reconstruct the innovation signals affecting the predictor model outputs;
3. Estimate a direct MISO model including all modules that have a link to the target model output, according to

$w_j(t) = G_{j*}(q) w_{D_j}(t) + [H_{j*}(q) – I_{j*}]\hat \xi (t) + e_j(t)$

with:

• $w_j$ the target module output
• $G_{j*}$ a row vector of (parametrized) transfer functions
• $H_{j*}$ a row vector of parametrized transfer functions, representing the noise model
• $I_{j*}$ the j-th row of the identity matrix of dimension $dim(w_Y)$
• $\hat \xi(t)$ the reconstructed innovation signals from the first step.

This step is performed in an output error model structure. The method is introduced and documented in [1] on the basis of a related algorithm for full network identification presented in [2].
The predictor model outputs need to be chosen in such a way that there are no confounding variables with the nodes that serve as predictor model inputs, and that the parallel-path and loop condition is satisfied. These and other consistency conditions for the predictor model can be checked, prior to identification, in the Predictor Model Window of the SYSDYNET app, or through the m-file predmodel_analysis_multistep.m.

Implementation aspects:

• Modules in the network are assumed to all be strictly proper (no direct feedthrough terms).
• Effective use is made of modules that are known a priori.

References:

1. S.J.M. Fonken, K.R. Ramaswamy and P.M.J. Van den Hof (2023). Local identification in dynamic networks using a multi-step least squares method. Proc. 62nd IEEE Conf. Decision and Control, 13-15 December 2023, Marina Bay Sands, Singapore, pp. 431-436.
2. S.J.M. Fonken, K.R. Ramaswamy and P.M.J. Van den Hof (2022). A scalable multi-step least squares method for network identification with unknown disturbance topology. Automatica, Volume 141 (110295), July 2022.