Wikipedia / Controllability Gramian
Wikipedia / Controllability Gramian
In control theory is controllable outputs.
One of the many ways one can achieve such goal is by the use of the Controllability Gramian.
Controllability in LTI Systems
are invariant with respect to time.
. Then, we can say that the following statements are equivalent:
is controllable.
.
has rank n.
.
is stable), and the unique solution of the Lyapunov equation
In the following section we are going to take a closer look at the Controllability Gramian.
Controllability Gramian
The controllability Gramian can be found as the solution of the Lyapunov equation
is indeed the solution for the Lyapunov equation under analysis.
Properties
is a symmetric matrix.
has to be unique.
is positive for any t a positive definite matrix.
More properties of controllable systems can be found in is controllable” presented in section Controllability in LTI Systems.
Discrete Time Systems
is controllable” (the equivalences are much alike for the continuous time case).
is controllable. More properties and proofs can be found in .
Linear Time Variant Systems
is controllable or not. This can be done in a very similar way of the preceding cases.
, is nonsingular.
Again, we have a similar method to determine if a system is or is not a controllable system.
Properties of Wc(t0,t1)
More about the Controllability Gramian can be found in .