Gramians

Gramians are mathematical tools used primarily in systems theory and control engineering to analyze and understand dynamic systems. They help quantify how different aspects of a system interact over time. There are two types: the controllability Gramian, which indicates how easily a system can be controlled from different states, and the observability Gramian, which measures how well a system's internal states can be inferred from its external outputs. By using Gramians, engineers can optimize system performance and design effective control strategies, ensuring systems respond well to changes and disturbances.