Digital Signal Processing Reference
In-Depth Information
solution to this new problem would be extremely complicated. Because of the finite
time horizon, it is no longer true that the optimal control is time invariant. Because
of the saturation, it is no longer true that the closed-loop system is linear nor is it
true that the state estimation and control are separate and independent problems.
Even though this separation is not true, it is common to design the MPC controller
by using a Kalman filter to estimate the state vector. Denote this estimate by
x
(
i
|
i
)
and the one step ahead prediction by
x
(
i
|
i
−
1
)
i
N
i
=
0
(
y
T
=
1
2
u
T
y
T
(
u
(
0
,
N
]
)=
(
(
)
(
)+
(
)
(
))+
(
+
)
(
+
))
J
E
i
Qy
i
i
Ru
i
N
1
Qy
N
1
(33)
is then solved open loop with initial condition
x
(
0
)=
x
(
i
|
i
−
1
)
and with the
constraint
⎨
u
max
,
if
u
(
i
)
≥
u
max
;
u
(
i
)=
(34)
u
(
i
)
,
if
|
u
(
i
)
| <
u
max
;
⎩
−
u
max
,
if
u
(
i
)
≤−
u
max
;
This is a convex programming problem which can be quickly and reliably solved.
As is usual in MPC, only the first term of the computed control sequence is actually
implemented. The substitution of the one step prediction for the filtered estimate is
done so as to provide time for the computations.
There is much more to MPC than has been discussed here. An introduction to
important to omit is that of stability. The rationale behind MPC is an heuristic notion
that the performance of such a controller is likely to be very good. While good
performance implicitly requires stability, it certainly does not guarantee it. Thus,
it is comforting to know that stability is theoretically guaranteed under reasonably
3.3
Exemplary Case
ABS brakes are a particularly vivid example of the challenges and benefits
associated with the use of DSP in control. The basic problem is simply stated.
Control the automobile's brakes so as to minimize the distance it takes to stop.
The theoretical solution is also simple. Because the coefficient of sliding friction
is smaller than the coefficient of rolling friction, the vehicle will stop in a shorter
distance if the braking force is the largest it can be without locking the wheels.
All this is well known. The difficulty is the following. The smallest braking force
that locks the wheels depends on how slippery the road surface is. For example, a
very small braking force will lock the wheels if they are rolling on glare ice. A much
