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

## TL;DR

This paper presents a trajectory-based data-driven framework for analyzing and controlling LTI and certain nonlinear systems, enabling system understanding and control without explicit model identification by leveraging measured trajectories and kernel methods.

## Contribution

It extends behavioral system theory to classical state-space and nonlinear systems, introducing kernel methods for data-driven simulation.

## Key findings

- Single measured trajectory can capture full system behavior for LTI systems.
- Extension to nonlinear systems linear in input-output coordinates.
- Kernel methods enable rich basis functions for data-driven simulation.

## Abstract

The vector space of all input-output trajectories of a discrete-time linear time-invariant (LTI) system is spanned by time-shifts of a single measured trajectory, given that the respective input signal is persistently exciting. This fact, which was proven in the behavioral control framework, shows that a single measured trajectory can capture the full behavior of an LTI system and might therefore be used directly for system analysis and controller design, without explicitly identifying a model. In this paper, we translate the result from the behavioral context to the classical state-space control framework and we extend it to certain classes of nonlinear systems, which are linear in suitable input-output coordinates. Moreover, we show how this extension can be applied to the data-driven simulation problem, where we introduce kernel-methods to obtain a rich set of basis functions.