Nonlinear system identification with kernels: Applications of derivatives in reproducing kernel Hilbert spaces

Abstract : This thesis will focus exclusively on the application of kernel-based nonparametric methods to nonlinear identification problems. As for other nonlinear methods, two key questions in kernel-based identification are the questions of how to define a nonlinear model (kernel selection) and how to tune the complexity of the model (regularisation). The following chapter will discuss how these questions are usually dealt with in the literature. The principal contribution of this thesis is the presentation and investigation of two optimisation criteria (one existing in the literature and one novel proposition) for structural approximation and complexity tuning in kernel-based nonlinear system identification. Both methods are based on the idea of incorporating feature-based complexity constraints into the optimisation criterion, by penalising derivatives of functions. Essentially, such methods offer the user flexibility in the definition of a kernel function and the choice of regularisation term, which opens new possibilities with respect to how nonlinear models can be estimated in practice. Both methods bear strong links with other methods from the literature, which will be examined in detail in Chapters 2 and 3 and will form the basis of the subsequent developments of the thesis. Whilst analogy will be made with parallel frameworks, the discussion will be rooted in the framework of Reproducing Kernel Hilbert Spaces (RKHS). Using RKHS methods will allow analysis of the methods presented from both a theoretical and a practical point-of-view. Furthermore, the methods developed will be applied to several identification ‘case studies’, comprising of both simulation and real-data examples, notably: • Structural detection in static nonlinear systems. • Controlling smoothness in LPV models. • Complexity tuning using structural penalties in NARX systems. • Internet traffic modelling using kernel methods
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Yusuf Bhujwalla. Nonlinear system identification with kernels: Applications of derivatives in reproducing kernel Hilbert spaces. Automatic Control Engineering. Université de Lorraine, 2017. English. ⟨NNT : 2017LORR0315⟩. ⟨tel-01755007⟩

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