A trajectory-based framework for data-driven system analysis and control

The vector space of all input-output trajectories of a discrete-time linear\ntime-invariant (LTI) system is spanned by time-shifts of a single measured\ntrajectory, given that the respective input signal is persistently exciting.\nThis fact, which was proven in the behavioral control framework, shows that a\nsingle measured trajectory can capture the full behavior of an LTI system and\nmight therefore be used directly for system analysis and controller design,\nwithout explicitly identifying a model. In this paper, we translate the result\nfrom the behavioral context to the classical state-space control framework and\nwe extend it to certain classes of nonlinear systems, which are linear in\nsuitable input-output coordinates. Moreover, we show how this extension can be\napplied to the data-driven simulation problem, where we introduce\nkernel-methods to obtain a rich set of basis functions.\n