The Intrinsic State Variable in Fundamental Lemma and Its Use in Stability Design for Data-based Control

TL;DR

This paper introduces the coefficient vector in Willems' fundamental lemma as an intrinsic, observable state variable for data-driven control, enabling simplified stability analysis and control design without prior system knowledge.

Contribution

It demonstrates that the coefficient vector serves as an intrinsic state variable, allowing for a new, straightforward stability and control design approach in data-based systems.

Findings

The coefficient vector is an observable state variable.

A Lyapunov-based stability method is developed using the coefficient vector.

Explicit control actions are constructed from the coefficient vector.

Abstract

In the data-based setting, analysis and control design of dynamical systems using measured data are typically based on overlapping trajectory segments of the input and output variables. This could lead to complex designs because the system internal dynamics, which is typically reflected by the system state variable, is unavailable. In this paper, we will show that the coefficient vector in a modified version of Willems' fundamental lemma is an intrinsic and observable state variable for the system behavior. This argument evolves from the behavioral framework without the requirement of prior knowledge on the causality among system variables or any predefined representation structure (e.g., a state space representation). Such a view allows for the construction of a state map based on the fundamental lemma, bridging the trajectory space and the state space. The state property of the coefficient vector allows for a simple stability design approach using memoryless quadratic functions of it as Lyapunov functions, from which the control action for each step can be explicitly constructed. Using the coefficient vector as a state variable could see wide applications in the analysis and control design of dynamical systems including directions beyond the discussions in this paper.