The finite controllability grammian at time
t<∞t
is defined as follows.
Pt=∫0tⅇAτBB*ⅇA*τdτ
P
t
τ
t
0
A
τ
B
B*
A*
τ
(1)
This grammian has two important properties. First,
Pt=P*t≥0
P
t
P*
t
0
.
Secondly, the columns of
PtPt
span the controllable space, i.e.
imPt=imCAB
im
P
t
im
C
A
B
It can be shown that the state defined by AA and BB
is controllable if, and only if,
PtPt
is positive definite for some
t>0t0.
Using the controllability grammian, we can determine how to most efficiently take a system from the zero state to a certain state
x―x―.
Given that
x―x―
is in the controllable space, there exists ξξ such that
x―=PT―ξ
x―
P
T―
ξ
(2)
for some
T―>0T―0. In this case, the minimum energy input required to move the system from zero to
x―x―
is
u―=B*ⅇA*T―-tξ―
u―
B*
A*
T―
t
ξ―
If the controllability matrix is invertible, we can use
the relation equation between ξ and certain state to put
u―u―
in terms of
x―x―:
u―
=B*ⅇA*T―-tP-1T―x―
u―
B*
A*
T―
t
P
T―
x―
(3)
In general, this minimal energy is exactly equal to
ξ―*PT―ξ―
ξ―
*
P
T―
ξ―
.
If the system is controllable, then this formula becomes
Energyu―=x―*P-1T―x―
Energy
u―
x―
*
P
T―
x―
(4)
If you don't want to start at the zero state, the formulas above can still be applied for taking a system at state
x1x1
to a state
x2x2.
This holds even if
x1x1
and
x2x2
are not controllable; in this case, all that is necessary is for
x2-x1
x2
x1
to be in the controllable space. (This makes sense if you think of
x1x1
as being the zero state and
x2x2
as being the general state we are trying to reach; it is the exact analog of the previous case. Using
x1x1
and
x2x2
is just like using
00 and
xx
with an appropriate offset.)
The finite observability grammian at time
t<∞t
is defined as
Qt=∫0tⅇA*τC*CⅇAτdτ
Q
t
τ
t
0
A*
τ
C*
C
A
τ
(5)
Parallel to the finite controllability grammian, the kernel of finite observability grammian is equal to the kernel of the observability matrix. (This relationship holds for positive time only.)
kerQt=kerOCA
ker
Q
t
ker
O
C
A
Using this grammian, we can find an expression for
the energy of the output yy at time TT
caused by the system's initial state xx:
Energyy=x*QTx
Energy
y
x*
Q
T
x
(6)
Consider a continuous-time linear system defined,
as per normal, by the matrices AA, BB, CC, andDD.
Assuming that this system is stable (i.e. all of its eigenvalues have negative real parts), both the controllability and observability grammians are defined for
t=∞t.
P=∫0∞ⅇAτBB*ⅇA*τdτ
P
τ
0
A
τ
B
B*
A*
τ
(7)
Q=∫0∞ⅇA*τC*CⅇAτdτ
Q
τ
0
A*
τ
C*
C
A
τ
(8)
These are called the infinite controllability and infinite observability grammians, respectively. These grammians satisfy the linear matrix equations known as the
Lyapunov equations.
AP+PA*+BB*=0
A
P
P
A*
B
B*
0
(9)
A*Q+QA+C*C=0
A*
Q
Q
A
C*
C
0
(10)
In the case of infinite grammians, the equations for minimal energy state transfer and observation energy drop their dependence on time. Assuming stability and complete controllability, the minimal energy required to transfer from zero to state
xcxc
is
xc*P-1xc
xc
*
P
xc
(11)
Similarly, the largest observation energy produced by the state
xoxo
is obtained for an infinite observation interval and is equal to:
xo*Qxo
xo
*
Q
xo
(12)
