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Operations Window

In the Operations window some basic structural operations and tests can be performed on the network. For all operations, a set of relevant nodes can be selected/deselected either by toggling the checkboxes in the Operations panel, or by clicking on the nodes in the network plot.

Invariant modules and immersion​

Immersion is the construction of a new network, where a selected set of nodes are maintained and unselected nodes are removed, while the time series of all maintained nodes remain invariant. This network operation is closely related to a so-called Kron reduction, i.e. Gaussian elimination of the unselected nodes.

The invariant modules test, tests which modules remain invariant after immersing the unselected nodes in the network.  This test can be done separate from, typically before, the actual immersion.

Parallel path and loop test​

A target module can be selected either using the dropdown menu in the Parallel Path panel, or by clicking on the module in the network plot. The selected module with input wiw_i and output wjw_j remains invariant after immersion, if the parallel path and loop test is satisfied. This test verifies whether

  • Every path from wiw_i to wjw_j passes through a node that is in the set of selected nodes;
  • Every loop around wjw_j passes through a node that is in the set of selected nodes;

and presents the result in the plotted graph.

The test is described in A. Dankers, P.M.J. Van den Hof, X. Bombois and P.S.C. Heuberger (2016). Identification of dynamic models in complex networks with predictior error methods - predictor input selection. IEEE Trans. Automatic Control, Vol. 61, no. 4, pp. 937-952, April 2016.

Canonical noise model​

This operation transforms the network to a network where only the selected nodes have a direct contribution from disturbances, and the unselected nodes are disturbance-free. In this transformation the time series of the selected node variables, as well as the modules in network matrix GG, remain invariant. The transformation only changes the noise model HH. 

The transformation is described in S. Shi, X. Cheng and P.M.J. Van den Hof (2023), Single module identifiability in linear dynamic networks with partial excitation and measurement.  IEEE Trans. Automatic Control, Vol. 68, no. 1, pp. 285-300, January 2023.