Nonlinear Data-Driven Control Part I: Trajectory Representation under quasi-Linear Parameter Varying Embeddings

Abstract

Recent literature has shown how linear time-invariant (LTI) systems can be represented by trajectories features, that is relying on a single input-output (IO) data dictionary to span all possible system trajectories, as long as the input is persistently exciting. The so-called behavioural framework is a promising alternative for controller synthesis without the necessity of system identification. In this paper, we benefit from differential inclusion in order to adapt previous results to the case quasi-Linear Parameter Varying (qLPV) embeddings, which are use to represent nonlinear dynamical systems along suitable IO coordinates. Accordingly, we propose a set of data-driven analysis tools for a wide class of nonlinear systems, which enable nonlinear data-driven simulation and predictions. Furthermore, a parameter-dependent dissipativity analysis verification setup is also presented, which serves to assess stability of the system within a bounded operation region. The major requirement is that there should exist a scheduling function which maps the nonlinear outputs into a finite number of scheduling variables, and this function should be analytically known. The effectiveness of the proposed tools is tested in practice and shown to provide accurate descriptions of the nonlinear dynamics by the means of a linear representation structure. For such, we consider a high-fidelity nonlinear simulator of a rotational pendulum benchmark simulator and an electro-mechanical positioning experimental validation test-bench. We also show that, even if the scheduling function is erroneously selected, the proposed framework is still able to offer a trustworthy representation of the output dynamics.