Controllability and Observability Gramians
Gramians quantify how strongly states can be reached by inputs and seen by outputs, grading the degree of controllability and observability.
Beyond Yes or No
Controllability and observability, tested by rank conditions, give only a yes-or-no answer. The Gramians refine this into a matter of degree: they measure how much control energy is needed to reach a state, and how much output energy a state produces. Two systems can both be controllable, yet one may require far more effort to steer in some directions than the other.
The controllability Gramian
The controllability Gramian is a symmetric positive matrix computed from the state and input matrices A and B, obtained by integrating the state-transition response driven by the inputs. States that lie in directions where the Gramian is large are easy to reach with little control energy; states in directions where it is small require large energy. A singular Gramian signals an uncontrollable direction.
The observability Gramian
Dually, the observability Gramian is computed from A and C. States in directions where it is large produce strong, easily measured output signals; states in directions where it is small are nearly invisible in the output. A singular observability Gramian signals an unobservable direction. The two Gramians together grade the system's structural strength in every direction.
Computing them
- For a stable system, each Gramian is the unique solution of a Lyapunov matrix equation, solved by standard numerical routines.
- The eigenvalues and eigenvectors of a Gramian reveal the strong and weak directions of control or observation.
- A small but nonzero eigenvalue flags a direction that is technically controllable or observable but practically nearly not.
Balanced realization and model reduction
The most important application is model reduction. By transforming to coordinates where the two Gramians are equal and diagonal, a balanced realization, each state is ranked by how strongly it participates in both control and observation. States with tiny joint participation contribute little to input-output behavior and can be discarded, yielding a lower-order model that closely approximates the full one. This balanced truncation is a workhorse for simplifying high-order models of complex plants.
Gramians thus turn the binary structural properties into quantitative measures that guide sensor and actuator placement, robustness assessment, and the reduction of large simulation models to tractable control-design models.