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.

中文翻译：

---

基于单一轨迹的线性时不变系统分析：一种线性规划方法

在本说明中，提出了一种新颖的方法，该方法可以提取线性时不变系统 (LTI) 的许多分析结果，仅给定所考虑系统的单个轨迹。所提出的技术的优越性取决于这样一个事实，即它提供了一种自动和正式的方式来获取有关受控系统的有价值的信息，只需在有限的时间内访问单个轨迹（即，系统动力学假设为未知）。首先，我们通过构造系统的关联李雅普诺夫函数来表征仅给定单个轨迹数据集的 LTI 系统的稳定区域。Lyapunov 函数是通过制定和求解线性规划 (LP) 问题来找到的。然后，我们将相同的方法扩展到 LTI 系统的各种基本分析结果，例如导出输出能量的边界、导出输出峰值的边界、导出 $\mathbf{L}_2$ 和 RMS 增益。为了说明所提出的数据驱动范式的有效性，提供了学习的 LTI 系统指标与真实指标之间的比较分析。