Data-driven Estimation and Maximization of Controllability Gramians

Controllability Gramians are the most widely used measures of controllability, which quantify the reachable state sets of systems with a control input of unit energy. They are utilized as a performance index in various design problems, such as actuator placement and model reduction. If a mathematical model is available for the system to be considered, the controllability Gramian can be easily calculated by its definition or equivalent conditions. However, it is not always the case due to the insufficiency of data for system modeling. In such a case, it is practical to use a data-driven method for directly estimating the controllability Gramian. This paper establishes a data-driven framework of estimating and maximizing the controllability Gramians of unknown linear systems. We first develop a method to estimate controllability Gramians with the measurement data of system behaviors. Furthermore, we present a data-driven solution to maximize the degree of controllability, measured by the trace of the controllability Gramian, with respect to the input matrix.