A Small-data Solution to Data-driven Lyapunov Equations: Data Reduction from <i>O</i>(<i>n</i>²) to <i>O</i>(<i>n</i>)

When a mathematical model is not available for a dynamical system, it is reasonable to use a data-driven approach for analysis and control of the system. With this motivation, the authors have recently developed a data-driven solution to Lyapunov equations, which uses not the model but the data of several state trajectories of the system. However, the number of state trajectories to uniquely determine the solution is O(n2) for the dimension n of the system. This prevents us from applying the method to a case with a large n. Thus, this paper proposes a novel class of data-driven Lyapunov equations, which requires a smaller amount of data. Although the previous method constructs one scalar equation from one state trajectory, the proposed method constructs three scalar equations from any combination of two state trajectories. Based on this idea, we derive data-driven Lyapunov equations such that the number of state trajectories to uniquely determine the solution is O(n).