(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 and Van den Hof  (IFAC SYSYS 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 IEEE Trans. Autom. Control 2023) for the situation of full rank disturbances. It has appeared that for the handling of these diffusively coupled networks, polynomial model representations are more attractive than the transfer function representation that are used in the module framework. The prime argument for this is that the structural symmetry properties of the networks are directly apparent in the chosen polynomial form.

The identification of a single interconnection / component in a diffusively coupled network has also been addressed in a similar prediction error framework, see Kivits and Van den Hof (IEEE CDC 2022). It has been shown that in the situation of a single interconnection, it suffices to measure the node signals that form the single interconnection, and in addition all node signals that are direct neighbours of the first two. This shows that a local identification problem can indeed be kept local in terms of the required measured node signals.

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.

The algorithm has been extended to also effectively deal with modules in the network that are known/fixed a priori, see Dreef et al., (IEEE LCSS 2022).

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, Automatica 2020) for the situation of full measurement of all node signals, and in Shi et al. (IEEE Trans. Autom. Control 2023) 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.
Dynamic network with reduced-rank disturbances

Full network identification

Using standard (prediction error) methods for full network identification will most often lead to non-convex optimization problems that become unpractical in terms of computational complexity and the lack of guarantees of reaching global optima. This particularly holds true if the networks become increasingly complex in terms of the number of nodes involved. Therefore alternative methods have been developed that restrict to computational attractive steps most often relying on convex optimizations. In a teamwork between researchers from KTH Stockholm and TU Eindhoven, a sequential least squares method has been developed for identifying dynamic networks in ARMAX form, for the situation of full rank disturbances (Weerts et al., IFAC SYSID 2018).  A follow up was made in terms of a multi-step algorithm, exploiting the so-called weighted nullspace fitting reasoning, in Fonken, Ramaswamy and Van den Hof (Automatica, 2022), where the step was made to include estimation of the noise rank and noise topology within the algorithm.

Identification methods for single module identification

Switching modules as a way to create identifiability

Case studies

Brain networks


Printed Circuit Board (PCB) testing

Data-driven and model-based control in dynamic networks