Digital Signal Processing Reference
In-Depth Information
3.2
Demanding Cases
The signal processing necessary for control can be very demanding. One such
control technique that is extensively used is commonly known as Model Predictive
Horizon Control, Dynamic Matrix Control, and numerous variants. The application
of MPC is limited to systems with “slow” dynamics because the computations
involved require too much time to be accomplished in the sampling interval for
“fast” systems. The simplest version of MPC occurs when there is no exogenous
input and the objective of the controller is to drive the system state to zero. This is
the case described below.
The starting point for MPC is a known plant which may have multiple inputs and
multiple outputs (MIMO) and a precise mathematical measure of the performance
of the closed-loop system. It is most convenient to use a state-space description of
the plant
x
(
t
)=
f
(
x
(
t
)
,
u
(
t
))
(20)
y
(
t
)=
g
(
x
(
t
)
,
u
(
t
))
(21)
Note that we have assumed that the system is time-invariant and that the noise can
be adequately handled indirectly by designing a robust controller (a controller that
is relatively insensitive to disturbances) rather than by means of a more elaborate
controller whose design accounts explicitly for the noise. It is assumed that the state
x
(
t
)
is an
n
-vector, the control
u
(
t
)
is an
m
-vector, and the output
y
(
t
)
is a
p
-vector.
The performance measure is generically
∞
J
(
u
[
0
,
∞
)
)=
l
(
x
(
t
)
,
u
(
t
))
,
dt
(22)
0
.
The control objective is to design and build a feedback controller that will
minimize the performance measure. It is possible to solve this problem analytically
in a few special cases. When an analytic solution is not possible one can resort
to the approximate solution known as MPC. First, discretize both the plant and
the performance measure with respect to time. Second, further approximate the
performance measure by a finite sum. Lastly, temporarily simplify the problem by
abandoning the search for a feedback control and ask only for an open-loop control.
The open-loop control will depend on the initial state so assume that
x
The notation
u
[
0
,
∞
)
denotes the entire signal
{
u
(
t
)
:0
≤
t
<
∞
}
(
0
)=
x
0
and
this is known.
The approximate problem is then to find the sequence
[
u
(
0
)
u
(
1
)
···
u
(
N
−
1
]
that
solves the constrained nonlinear programming problem given below
