Allocation of Excitation Signals for Generic Identifiability of Linear Dynamic Networks

A recent research direction in data-driven modeling is the identification of dynamic networks, in which measured vertex signals are interconnected by dynamic edges represented by causal linear transfer functions. The major question addressed in this paper is where to allocate external excitation signals such that a network model set becomes generically identifiable when measuring all vertex signals. To tackle this synthesis problem, a novel graph structure, referred to as \textit{directed pseudotree}, is introduced, and the generic identifiability of a network model set can be featured by a set of disjoint directed pseudotrees that cover all the parameterized edges of an \textit{extended graph}, which includes the correlation structure of the process noises. Thereby, an algorithmic procedure is devised, aiming to decompose the extended graph into a minimal number of disjoint pseudotrees, whose roots then provide the appropriate locations for excitation signals. Furthermore, the proposed approach can be adapted using the notion of \textit{anti-pseudotrees} to solve a dual problem, that is to select a minimal number of measurement signals for generic identifiability of the overall network, under the assumption that all the vertices are excited

Exploiting unmeasured disturbance signals in identifiability of linear dynamic networks with partial measurement and partial excitation

Identifiability conditions for networks of transfer functions require a sucientnumber of external excitation signals, which are typically measured reference signals. In this abstract, we introduce an equivalent network model structure to address the contribution of unmeasured noises to identifiability analysis in the setting with partial excitation and partial measurement. With this model structure, unmeasured disturbance signals can be exploited as excitation sources, which leads to less conservative identifiability conditions

Local module identification in dynamic networks with correlated noise: the full input case

The identification of local modules in dynamic networks with known topology has recently been addressed by formulating conditions for arriving at consistent estimates of the module dynamics, typically under the assumption of having disturbances that are uncorrelated over the different nodes. The conditions typically reflect the selection of a set of node signals that are taken as predictor inputs in a MISO identification setup. In this paper an extension is made to arrive at an identification setup for the situation that process noises on the different node signals can be correlated with each other. In this situation the local module may need to be embedded in a MIMO identification setup for arriving at a consistent estimate with maximum likelihood properties. This requires the proper treatment of confounding variables. The result is an algorithm that, based on the given network topology and disturbance correlation structure, selects an appropriate set of node signals as predictor inputs and outputs in a MISO or MIMO identification setup. As a first step in the analysis, we restrict attention to the (slightly conservative) situation where the selected output node signals are predicted based on all of their in-neighbor node signals in the network.Comment: Extended version of paper submitted to the 58th IEEE Conf. Decision and Control, Nice, 201

Prediction error identification of linear dynamic networks with rank-reduced noise

Dynamic networks are interconnected dynamic systems with measured node signals and dynamic modules reflecting the links between the nodes. We address the problem of \red{identifying a dynamic network with known topology, on the basis of measured signals}, for the situation of additive process noise on the node signals that is spatially correlated and that is allowed to have a spectral density that is singular. A prediction error approach is followed in which all node signals in the network are jointly predicted. The resulting joint-direct identification method, generalizes the classical direct method for closed-loop identification to handle situations of mutually correlated noise on inputs and outputs. When applied to general dynamic networks with rank-reduced noise, it appears that the natural identification criterion becomes a weighted LS criterion that is subject to a constraint. This constrained criterion is shown to lead to maximum likelihood estimates of the dynamic network and therefore to minimum variance properties, reaching the Cramer-Rao lower bound in the case of Gaussian noise.Comment: 17 pages, 5 figures, revision submitted for publication in Automatica, 4 April 201

Structure Identifiability of an NDS with LFT Parametrized Subsystems

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the NDS is structurally identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.Comment: 16 page

Combinatorial Characterization for Global Identifiability of Separable Networks with Partial Excitation and Measurement

This work focuses on the generic identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by transfer functions according to a known topology, some nodes are excited, some are measured, and only a part of the transfer functions are known. Our goal is to determine whether the unknown transfer functions can be generically recovered based on the input-output data collected from the excited and measured nodes. We introduce the notion of separable networks, for which global and so-called local identifiability are equivalent. A novel approach yields a necessary and sufficient combinatorial characterization for local identifiability for such graphs, in terms of existence of paths and conditions on their parity. Furthermore, this yields a necessary condition not only for separable networks, but for networks of any topology.Comment: 8 pages, 1 figure, article to appear in IEEE Conference on Decision and Control 202

Single module identifiability in linear dynamic networks

A recent development in data-driven modelling addresses the problem of identifying dynamic models of interconnected systems, represented as linear dynamic networks. For these networks the notion network identifiability has been introduced recently, which reflects the property that different network models can be distinguished from each other. Network identifiability is extended to cover the uniqueness of a single module in the network model. Conditions for single module identifiability are derived and formulated in terms of path-based topological properties of the network models.Comment: 6 pages, 2 figures, submitted to Control Systems Letters (L-CSS) and the 57th IEEE Conference on Decision and Control (CDC

Identifiability of undirected dynamical networks:A graph-theoretic approach

This paper deals with identifiability of undirected dynamical networks with single-integrator node dynamics. We assume that the graph structure of such networks is known, and aim to find graph-theoretic conditions under which the state matrix of the network can be uniquely identified. As our main contribution, we present a graph coloring condition that ensures identifiability of the network's state matrix. Additionally, we show how the framework can be used to assess identifiability of dynamical networks with general, higher-order node dynamics. As an interesting corollary of our results, we find that excitation and measurement of all network nodes is not required. In fact, for many network structures, identification is possible with only small fractions of measured and excited nodes.Comment: 6 page

A Necessary Condition for Network Identifiability With Partial Excitation and Measurement

This article considers dynamic networks where vertices and edges represent manifest signals and causal dependencies among the signals, respectively. We address the problem of how to determine if the dynamics of a network can be identified when only partial vertices are measured and excited. A necessary condition for network identifiability is presented, where the analysis is performed based on identifying the dependency of a set of rational functions from excited vertices to measured ones. This condition is further characterized by using an edge-removal procedure on the associated bipartite graph. Moreover, on the basis of necessity analysis, we provide a necessary and sufficient condition for identifiability in circular networks.</p