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Identification of dynamic models in complex networks with prediction error methods : basic methods for consistent module estimates.

International audienceThis paper addresses the problem of obtaining an estimate of a particular module of interest that is embedded in a dynamic network with known interconnection structure. In this paper it is shown that there is considerable freedom as to which variables can be included as inputs to the predictor, while still obtaining consistent estimates of the particular module of interest. This freedom is encoded into sufficient conditions on the set of predictor inputs that allow for consistent identification of the module. The conditions can be used to design a sensor placement scheme, or to determine whether it is possible to obtain consistent estimates while refraining from measuring particular variables in the network. As identification methods the Direct and Two Stage Prediction-Error methods are considered. Algorithms are presented for checking the conditions using tools from graph theory

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Abstract-In many areas of signal, system, and control theory, orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space e;. To this end, use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, e.g., the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e., to obtain a good approximation by retaining only a finite number of expansion coefficients. Consequences for identification of expansion coefficients are analyzed, and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients.

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Abstract-In many areas of signal, system, and control theory, orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space e;. To this end, use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, e.g., the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e., to obtain a good approximation by retaining only a finite number of expansion coefficients. Consequences for identification of expansion coefficients are analyzed, and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients.

When identifying a dynamical system on the basis of experimentally measured data records, one of the most important issues to address carefully is the choice of an appropriate model structure or model set. The model set reflects the collection of models among which a best model is sought on the basis of the given data records. The choice of model set directly influences the maximum achievable accuracy of the identified model. On the one hand, the model set should be as large and flexible as possible in order to contain as many candidate models as possible. This reduces the structural or bias error in the model. On the other hand, when parameterizing the model set with (real-valued) parameters, the number of parameters should be kept as small as possible because of the principle of parsimony [34,283]. This principle states that the variability of identified models increases with increasing number of parameters. The conflict between large flexible model sets and parsimoniously parameterized model sets is directly related to the well-known bias/variance trade-off that is present in estimation problems. It is easy to show that (partial) knowledge of the system to be identified can be used to shape the choice of model structure. This is indicated in a simple example.k }. For simplicity, we will represent a system model by its pulse response. If one would know that the shape of the system's pulse response would be like the shape of {g (o) k }, then a very simple parameterization of an appropriate model set could beIn this -trivial -situation of a model set with one parameter, the value α = 1 leads to an exact presentation of the system.