Digital Signal Processing Reference
In-Depth Information
problems has been limited to relatively slow systems, i.e., systems for which
is
large enough to insure that the computations will be completed before the next value
of the control signal is needed. As a result, there is a great deal of research at present
on ways to speed up the computations involved in MPC.
The time required to complete the control computations becomes even longer
if it is necessary to account explicitly for noise. Consider the following relatively
simple version of MPC in which the plant is linear and time-invariant except for the
saturation of the actuators. Furthermore, the plant has a white Gaussian noise input
and the output signal contains additive white Gaussian noise as well. The plant is
then modeled by
δ
x
(
i
+
1
)=
Ax
(
i
)+
Bu
(
i
)+
D
ξ
(
i
)
(29)
y
(
i
)=
Cx
(
i
)+
ν
(
i
)
(30)
The two noise signals are zero mean white Gaussian noises with covariance ma-
trices
E
T
T
(
ξ
(
i
)
ξ
(
i
)) =
Ξ
∀
i
and
E
(
ν
(
i
)
ν
(
i
)) =
I
∀
i
where
E
(
·
)
denotes expectation.
The two noise signals are independent of each other.
The performance measure is
i
=
∞
i
=
0
(
y
T
1
2
u
T
J
(
u
(
0
,
∞
]
)=
E
(
(
i
)
Qy
(
i
)+
(
i
)
Ru
(
i
)))
(31)
In the equation above,
Q
and
R
are symmetric real matrices of appropriate
dimensions. To avoid technical difficulties
R
is taken to be positive definite (i.e.,
u
Ru
0) and
Q
positive semidefinite (i.e.,
y
Qy
0forall
y
).
In the absence of saturation, i.e., if the linear model is accurate for all inputs,
It separates into two independent components. One component is the optimal
feedback control
u
o
>
0forall
u
=
≥
F
o
x
, where the superscript “
o
” denotes optimal. This
control uses the actual state vector which is, of course, unavailable. The other
component of the optimal control is a Kalman filter which produces the optimal
estimate of
x
(
i
)=
(
i
)
(
i
)
given the available data at time
i
. Denoting the output of the filter
by
x
(
i
|
i
)
, the control signal becomes
u
o
F
o
x
(
i
)=
(
i
|
i
)
(32)
This is known as the certainty equivalent control because the optimal state
estimate is used in place of the actual state. For this special case, all of the
parameters can be computed in advance. The only computations needed in real
time are the one step ahead Kalman filter /predictor and the matrix multiplication in
If actuator saturation is important, as it is in many applications, then the optimal
control is no longer linear and it is necessary to use MPC. As before, approximate
the performance measure by replacing
∞
by some finite
N
. An exact feedback
