Applicazione di Kernel Generalizzati al Learning-Based Non-Linear Model Predictive Control

Gaussian Processes are a powerful tool that is finding successful applications in the Control Systems field due to their high flexibility and ability to identify dynamical systems. In a general context, Gaussian Process Regression is a Bayesian non-parametric method which makes predictions based on the posterior probability density given training data. The behavior of the predictions (regressed function) is completely characterized by the covariance function, whose properties are directly inherited by the regressed function. In many problems, such function is chosen ad hoc and depends on prior knowledge of the underlying phenomena, i.e. depends on human intervention. Learning-Based Non-Linear Model Predictive Control (LB-NMPC) scheme is a data-driven implementation of the MPC used to control non-linear dynamical systems while satisfying plant constraints, it is aimed at improving the nominal model available to the controller by approximating underlying missing dynamics with a Gaussian process using past plant observations as training data. We implement the so-called Generalized Kernels to the Gaussian process within the LB-NMPC, exploiting the spectral representation of a positive-definite function that is modelled as mixture of a basis function. Such basis function can have desired properties such as smoothness or degree of differentiability, generating richer model classes that can be used to identify more complex dynamical systems. This procedure automatizes the selection of the kernel function, while still maintaining an analytic expression of the regressed function, allowing its implementation on the MPC scheme and, in particular, it has a strong link with RKHS framework that is useful to analyze the model class rigorously.