Just-in-Time Models with Applications to Dynamical Systems

Sammanfattning: System identification deals with the problem of estimating models of dynamical systems given observations from the systems. In this thesis we focus on the nonlinear modeling problem, and, in particular, on the situation that occurs when a very large amount of data is available.Traditional treatments of the estimation problem in statistics and system identification have mainly focused on global modeling approaches, i.e., the model has been optimized using the entire data set. However, when the number of samples becomes large, this approach becomes less attractive mainly because of the computational complexity.We instead assume that all observations are stored in a database, and that models are built dynamically as the actual need arises. When a model is really needed in a neighborhood around an operating point, a subset of the data closest to the operating point is retrieved from the database, and a local modeling operationis performed on that subset. For this concept, the name Just-in-Time models has been adopted.It is proposed that the Just-in-Time estimator is formed as a weighted average of the data in the neighborhood, where the weights are optimized such that the pointwise mean square error (MSE) measure is minimized. The number of data retrieved from the database is determined using a local bias/variance error tradeo. This is closely related to the nonparametric kernel estimation concept which is commonly used in statistics. A review of kernel methods is therefore presented in one of the introductory chapters.The asymptotical properties of the method are investigated. It is shown that the Just-in-Time estimator produces consistent estimates, and that the convergence rate as a function of the sample size is of the same order as for the kernel methods. Two important applications for the concept are presented. The first one considers nonlinear time domain identification, which is the problem of predicting the outputs of nonlinear dynamical systems given data sets of past inputs and outputs of the systems. The second one occurs within frequency domain identication whenone is faced with the problem of estimating the frequency response function of alinear system.Compared to global methods, the advantage with Just-in-Time models is that they are optimized locally, which might increase the performance. A possible drawback is the computational complexity, both because we have to search for neighborhoods in a multidimensional regressor space, and because the derived estimator is quite demanding in terms of computational eort.