Prediction error identification with rank-reduced output noise

\u3cp\u3eIn data-driven modelling in dynamic networks, it is commonly assumed that all measured node variables in the network are noise-disturbed and that the network (vector) noise process is full rank. However when the scale of the network increases, this full rank assumption may not be considered as realistic, as noises on different node signals can be strongly correlated. In this paper it is analyzed how a prediction error method can deal with a noise disturbance whose dimension is strictly larger than the number of white noise signals than is required to generate it (rank-reduced noise). Based on maximum likelihood considerations, an appropriate prediction error identification criterion will be derived and consistency will be shown, while variance results will be demonstrated in a simulation example.\u3c/p\u3

Identification of dynamic networks operating in the presence of algebraic loops

When identifying all modules in a dynamic network it is natural to treat all node variables in a symmetric way, i.e. not having pre-assigned roles of `inputs' and `outputs'. In a prediction error setting this implies that every node signal is predicted on the basis of all other nodes. A usual restriction in direct and joint-io methods for dynamic network and closed-loop identification is the need for a delay to be present in every loop (absence of algebraic loops). It is shown that the classical one-step-ahead predictor that incorporates direct feedt-hrough terms in models can not be used in a dynamic network setting. It has to be replaced by a network predictor, for which consistency results are shown when applied in a direct identification method. The result is a one-stage direct/joint-io method that can handle the presence of algebraic loops. It is illustrated that the identified models have improved variance properties over instrumental variable estimation methods

Single module identifiability in linear dynamic networks

\u3cp\u3eA recent development in data-driven modeling addresses the problem of identifying dynamic models of interconnected systems, represented as linear dynamic networks. For these networks the notion of 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, and conditions for single module identifiability are derived and formulated in terms of path-based topological properties of the network models.\u3c/p\u3

From closed-loop identification to dynamic networks:Generalization of the direct method

\u3cp\u3eIdentification methods for identifying (modules in) dynamic cyclic networks, are typically based on the standard methods that are available for identification of dynamic systems in closed-loop. The commonly used direct method for closed-loop prediction error identification is one of the available tools. In this paper we are going to show the consequences when the direct method is used under conditions that are more general than the classical closed-loop case. We will do so by focusing on a simple two-node (feedback) network where we add additional disturbances, excitation signals and sensor noise. The direct method loses consistency when correlated disturbances are present on node signals, or when sensor noises are present. A generalization of the direct method, the joint-direct method, is explored, that is based on a vector predictor and includes a conditioning on external excitation signals. It is shown to be able to cope with the above situations, and to retain consistency of the module estimates.\u3c/p\u3

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

\u3cp\u3eDynamic networks are interconnected dynamic systems with measured node signals and dynamic modules reflecting the links between the nodes. We address the problem of 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 Cramér–Rao lower bound in the case of Gaussian noise. In order to reduce technical complexity, the analysis is restricted to dynamic networks with strictly proper modules.\u3c/p\u3

Identifiability of dynamic networks with part of the nodes noise-free

In dynamic network identification a major goal is to uniquely identify the topology and dynamic links between the measured node variables. It is common practice to assume that process noises affect every output in multivariable system identification, and every node in dynamic networks with a full rank noise process. For many practical situations this assumption might be overly strong. This leads to the question of how to handle situations where the process noise is not full rank, i.e. when the number of white noise processes driving the network is strictly smaller than the number of nodes. In this paper a first step towards answering this question is taken by addressing the case of a dynamic network where some nodes are noise-free, and others are disturbed with a (correlated) process noise. In this situation the predictor filters that generate the one-step-ahead prediction of the node signals are non-unique, and the appropriate identification criterion leads to a constrained optimization problem. It is assessed when it is possible to distinguish between models on the basis of this criterion, leading to new notions of network identifiability. It appears that a sufficient condition for network identifiability is that every node signal in the network is excited by an external excitation signal or a process noise signal that is uncorrelated with other node excitations

Identifiability of linear dynamic networks

\u3cp\u3eDynamic networks are structured interconnections of dynamical systems (modules) driven by external excitation and disturbance signals. In order to identify their dynamical properties and/or their topology consistently from measured data, we need to make sure that the network model set is identifiable. We introduce the notion of network identifiability, as a property of a parametrized model set, that ensures that different network models can be distinguished from each other when performing identification on the basis of measured data. Different from the classical notion of (parameter) identifiability, we focus on the distinction between network models in terms of their transfer functions. For a given structured model set with a pre-chosen topology, identifiability typically requires conditions on the presence and location of excitation signals, and on presence, location and correlation of disturbance signals. Because in a dynamic network, disturbances cannot always be considered to be of full-rank, the reduced-rank situation is also covered, meaning that the number of driving white noise processes can be strictly less than the number of disturbance variables. This includes the situation of having noise-free nodes.\u3c/p\u3