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APPENDIX
Model Predictive Control (MPC)
Following the state space dynamics in (2), consid-
ering the state reference vector r ∈ R
n
, the predic-
tion horizon H
p
∈ N and the input horizon H
u
∈ N,
the MPC standard quadratic cost function can be for-
mulated, (Maciejowski, 2001)
J(k) = kX(k) −Ξ(k)k
2
Q
+ k∆U(k)k
2
R
(14)
for the optimization problem
min
∆U(k)
J(k), (15)
where X(k) ∈ R
H
p
n
is the future state sequence
prediction, Ξ(k) ∈ R
H
p
n
the future reference trajec-
tory, and ∆U(k) ∈ R
(H
u
+1)m
the control input change
sequence. Considering with the cost of con-
trol error given by the positive semi-definite ma-
trix Q ∈ R
H
p
n×H
p
n
and the cost of control effort by
the positive definite matrix R ∈ R
(H
u
+1)m×(H
u
+1)m
.
The solution of (15) is given by the optimal con-
trol input sequence ∆U
?
(k), which is fed to the plant,
as shown in the standard MPC control loop of Fig-
ure 6.
Controller
Optimization
Prediction
Model
Plant
u
x
r
Figure 6: Model Predictive Controller: Classical Scheme.
In order to find the solution ∆U
?
(k) in the uncon-
strained scenario in a numerically well-conditioned
manner, the cost function J(k) can be rearranged as
J(k) =
S
Q
(X(k) −Ξ(k))
S
R
∆U(k)
2
, (16)
with the cost matrices factorization
Q = S
Q
|
S
Q
, (17)
R = S
R
|
S
R
, (18)
which can be efficiently solved in the least-square
sense, e.g. using a QR algorithm, leading to a linear
time invariant system controller (Maciejowski, 2001).
Typically the MPC controller uses the receding
horizon principle to set an update time strategy for the
controlled input signal, where only the first element of
the computed optimal input vector sequence ∆U
?
(k)
is given to the system, then the state values are mea-
sured or observed, resp., (Maciejowski, 2001).
Comparison of Linear State Signal Shaping Model Predictive Control with Classical Concepts for Active Power Filter Design
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