Results

(under construction)

 

Modelling dynamic networks

There are many different ways of representing interconnected dynamic systems. The format that we have chosen to use as a basis for our developments is the so-called module-framework, where linear dynamic systems are represented by transfer functions (modules) that are interconnected through signals, which are interconnected through summation points.  External signals affect the network in the form of (measured) excitation/probing signals and (unmeasured) disturbance signals. The topological properties of the network are characterized by a graph, in which the signals are represented as nodes, and the modules are (directed) links between the nodes.

The network is slightly different from the often used state-representation of a dynamic network. This is motivated by our focus on data-driven modeling problems. Since in a large-scale system typically not all states can be measured, we have chosen for a framework where the node signals could be considered as those states that can possibly be measured. States that can not be measured are “hidden” in the dynamic transfer functions in the models.

The module representation has been introduced in Van den Hof et al. (2013) and has been adopted by several other research groups in this domain.

Relations between module and state-space representations have been explored in Kivits et al. (2018).

Physical systems are often characterized by network representations with a particular type of couplings, referred to as diffusive couplings, meaning that an effect occuring between two nodes is dependent on the difference between the node variables. Think of a current that flows on a link as a results of a voltage difference between nodes. These physical networks are characterized by a non-directed graph where all links between nodes have some symmetry properties. Identification methods for these physical networks have been developed in Kivits et al. (CDC-2019 and ArXiv-2021).

A prototype dynamic network
Mass-spring-damper network with mass-positions as nodes
Pseudo-tree covering for excitation allocation

Network identifiability

Network identifiability is a property that determines whether different network models can be distinguished from each other on the basis of measured data. It is typically applied to network models with a given topology, and dependent on the following aspects:

  • The presence and location of measured external excitation signals
  • The presence, location and correlation structure of disturbance signals
  • The presence of a priori known modules with fixed dynamics

Network identifiabilty has been defined and analyzed in Weerts et al. (Automatica, March 2018; PhD-Thesis, 2018). It is a necessary property for arriving at a consistent estimate of the network dynamics. A synthesis algorithm for allocating external excitation signals to achieve (generic) network identifiability is presented in Cheng et al., (IEEE-TAC, 2022). It is fully graph-based and determines the location of external excitation signals, for the situation that all node signals are measured, on the basis of a pseudo-tree covering of the network graph.

Single module identifiability

When focusing on the determination of a single module in the dynamic network, network identifiability can be restricted to apply to this single target module only. The resulting question then is: where to excite the network and which node signals to measure in order to arrive at identifiability of the target module. The question is most attractively addressed in a generic sense, leading to analysis and synthesis algorithms that are based on the network graph only.

This problem is addressed in Weerts et al. (CDC 2018) and Shi et al. (IFAC 2020, ArXiv 2020) for the situation of full measurement of all node signals, and in Shi et al. (ArXiv 2021) for the situation of partial measurement and partial excitation.

For (generic) identifiability of a single module G_{ji}, a disconnecting set D determines where to allocate external excitation signals and which node signals to measure.

Full network identification

Single module identification