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:
- The pair
is controllable. - The
matrix
is nonsingular for any
. - The
controllability matrix
has rank n. - 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 Chen (1999, p. 145), 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 Chen (1999, p. 169).
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 Chen (1999, p. 176).
See also
- Controllability
- Observability Gramian
- Gramian matrix
- Minimum energy control
- Hankel singular value
References
- Chen, Chi-Tsong (1999). Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. ISBN 0-19-511777-8.
External links
- Mathematica function to compute the controllability Gramian