On Linear Time-Invariant Systems Analysis via A Single Trajectory: A Linear Programming Approach

TL;DR

This paper introduces a linear programming-based method to analyze linear time-invariant systems using only a single trajectory, enabling stability and performance assessments without prior system knowledge.

Contribution

It presents a novel data-driven approach that extracts system stability and performance metrics from a single trajectory via LP formulations, without requiring system models.

Findings

Successfully characterizes stability regions from a single trajectory.

Derives bounds on output energy and peak using the proposed method.

Shows close agreement between learned and true system metrics.

Abstract

In this note, a novel methodology that can extract a number of analysis results for linear time-invariant systems (LTI) given only a single trajectory of the considered system is proposed. The superiority of the proposed technique relies on the fact that it provides an automatic and formal way to obtain valuable information about the controlled system by only having access to a single trajectory over a finite period of time (i.e., the system dynamics is assumed to be unknown). At first, we characterize the stability region of LTI systems given only a single trajectory dataset by constructing the associated Lyapunov function of the system. The Lyapunov function is found by formulating and solving a linear programming (LP) problem. Then, we extend the same methodology to a variety of essential analysis results for LTI systems such as deriving bounds on the output energy, deriving bounds on output peak, deriving $\mathbf{L}_2$ and RMS gains. To illustrate the efficacy of the proposed data-driven paradigm, a comparison analysis between the learned LTI system metrics and the true ones is provided.