Controllability Gramian

Controllability Gramian
In control theory, we may need to find out whether or not a system such as
is controllable, where and are, respectively , and matrices for a system with inputs, state variables and outputs.
One of the many ways one can achieve such goal is by the use of the Controllability Gramian.
Controllability in LTI Systems
Linear Time Invariant (LTI) Systems are those systems in which the parameters and are invariant with respect to time.
One can observe if the LTI system is or is not controllable simply by looking at the pair . Then, we can say that the following statements are equivalent:
1. The pair is controllable.
2. The matrix
is nonsingular for any .
3. The controllability matrix
has rank n.
4. The matrix
has full row rank at every eigenvalue of .
If, in addition, all eigenvalues of have negative real parts (is stable), and the unique solution of the Lyapunov equation
is positive definite, the system is controllable. The solution is called the Controllability Gramian and can be expressed as
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 given by
In fact, we can see that if we take
as a solution, we are going to find that:
Where we used the fact that at for stable (all its eigenvalues have negative real part). This shows us that is indeed the solution for the Lyapunov equation under analysis.
Properties
We can see that is a symmetric matrix, therefore, so is .
We can use again the fact that, if is stable (all its eigenvalues have negative real part) to show that is unique. In order to prove so, suppose we have two different solutions for
and they are given by and . Then we have:
Multiplying by by the left and by by the right, would lead us to
Integrating from to :
using the fact that as :
In other words, has to be unique.
Also, we can see that
is positive for any t (assuming the non-degenerate case where is not identically zero). This makes a positive definite matrix.
More properties of controllable systems can be found in, as well as the proof for the other equivalent statements of “The pair is controllable” presented in section Controllability in LTI Systems.
Discrete Time Systems
For discrete time systems as
One can check that there are equivalences for the statement “The pair is controllable” (the equivalences are much alike for the continuous time case).
We are interested in the equivalence that claims that, if “The pair is controllable” and all the eigenvalues of have magnitude less than (is stable), then the unique solution of
is positive definite and given by
That is called the discrete Controllability Gramian. We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that is positive definite, and all eigenvalues of have magnitude less than , the system is controllable. More properties and proofs can be found in.
Linear Time Variant Systems
Linear time variant (LTV) systems are those in the form:
That is, the matrices , and have entries that varies with time. Again, as well as in the continuous time case and in the discrete time case, one may be interested in discovering if the system given by the pair is controllable or not. This can be done in a very similar way of the preceding cases.
The system is controllable at time if and only if there exists a finite such that the matrix, also called the Controllability Gramian, given by
where is the state transition matrix of , 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)
We have that the Controllability Gramian have the following property:
that can easily be seen by the definition of and by the property of the state transition matrix that claims that:
More about the Controllability Gramian can be found in.
See also
Controllability
Observability Gramian
Gramian matrix
Minimum energy control
Hankel singular value
References
External links
Mathematica function to compute the controllability Gramian
Control theory