Multistep method for single module identification

Identification of a single target module is performed on the basis of a selected set of measured node signals $w_S$ and external excitation signals. A three-step procedure is executed as follows:

1. Estimation of a high-order MIMO ARX model between all relevant external excitation signals and the group of signals $w_{Y}$, $Y \subset S$, that share a confounding variable with the target output;
2. Use the estimated ARX model to reconstruct the innovation signals affecting the node signals $w_Y$;
3. Estimate a MISO model including all modules that have a link to the target model output, according to

$w_j(t) = G_{j*}(q,\theta) w_{D_j}(t) + [P_{j*}(q,\theta) – I_{j*}]\hat \xi (t) + T_{j*}(q,\theta) r(t) + e_j(t)$

with:

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

For the final step, a choice can be made between two estimation algorithms:

• (a) a (classical) output error algorithm, where all relevant terms are parametrized with numerator/denominator representations, and a quadratic cost function is optimized, and
• (b) an Empirical Bayes kernel-based method, where only the target module is parametrically modelled, and all other terms are represented by Gaussian processes.

The attractive property of this method is that it avoids to estimate parametric multi-output models.

The method is introduced and documented in [1] on the basis of a related algorithm for full network identification presented in [2]. The Empirical Bayes estimation method is presented in [3]. 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 allowed to have direct feedthrough terms.
• Effective use is made of modules that are known a priori.
• Since in the EBLDM method only the target module is estimated, simulation results and residual tests of identified models have limited value if the predictor model has multiple inputs. Additional inputs are currently discarded in the simulation.

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.
3. V.C. Rajagopal, K.R. Ramaswamy and P.M.J. Van den Hof (2020). A regularized kernel-based method for learning a module in a dynamic network with correlated noise. Proc. 59th IEEE Conf. Decision and Control, Jeju Island, Republic of Korea, 15-18 December 2020, pp. 4348-4353.