Metrics for Quantifying Controllability¶
While controllability offers a binary quantification of control (are all state tranformations possible?), there are more finegrained metrics that can be used. The amount of energy used to make a state transformation is contained in the Gramian matrix, and gives an assessment of how practical a control configuration might be, even if it is technically controllable.

netcontrolz.
finite_horizon_gramian
(A, B, T)[source]¶ Returns the finite horizon (finite time) Gramian matrix when the system is driven to the origin:
where stands for transpose
Driving a system from to uses energy:
If (origin) then this simplifies to
Bounds on the energy can be established by the max and min eigenvalues of ():
Note that matrices
A
andB
should be realizations, not sparsity patterns.

netcontrolz.
finite_horizon_discrete_time_gramian
(A, B, T)[source]¶ Returns the finite horizon (finite time,
T
) Gramian matrix when the system is driven from the origin:Driving a system from the origin to uses energy:
Bounds on the energy can be established by the max and min eigenvalues of ():
Note that matrices
A
andB
should be realizations, not sparsity patterns.