Nonlinear Data-Driven Control Part I: An Overview of Trajectory Representations

In the literature, a recent debate has been brought up regarding how linear time-invariant systems can be represented by trajectories features. That is, how a single input–output (IO) data dictionary can be exploited to span all possible system trajectories, as long as the input is persistently exciting. Indeed, the so-called behavioural framework is a promising alternative for controller synthesis without the necessity of system identification procedures. In this paper, we provide an overview of the available results. In particular, we focus on how quasi-Linear Parameter Varying (qLPV) embeddings, in the data-driven context, can be used to represent nonlinear dynamical systems along suitable IO coordinates. We debate the topics of nonlinear data-driven simulation and predictions, as proposed in recent works. The effectiveness of the surveyed 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 debate that, even if the qLPV scheduling function is erroneously selected, the framework is still able to offer a reasonably trustworthy representation of the system.