The Development of Neural Network Based System Identification ...

6.8 MODEL PREDICTIVE CONTROL WITH CONSTRAINTS 173
The general statement of MPC optimisation problem with the effect of constraints
involves finding the optimal control solution ∆U that minimises the cost function (6.31),
which is reproduced here as follows:
J = (R s − Γx(k)) T (R s − Γx(k)) − 2∆U T Φ T (R s − Γx(k))
+ ∆U T (Φ T Φ + ¯R)∆U
The minimisation of (6.31) is subject to inequality constraints:
⎡ ⎤ ⎡ ⎤
M 1
N 1
⎢M 2 ⎥
⎣ ⎦ ∆U ≤ ⎢N 2 ⎥
⎣ ⎦
M 3 N 3
(6.35)
where the data matrices are:
⎡ ⎤ ⎡
⎤
⎢
M 1 = ⎣ C 1⎥
⎢
⎦ ; N 1 = ⎣ −U min + C 1 u(k − 1) ⎥
⎦
C 2 U max − C 1 u(k − 1)
⎡ ⎤ ⎡ ⎤
⎢
M 2 = ⎣ −I ⎥ ⎢−∆U min
⎥
⎦ ; N 2 = ⎣ ⎦
I
∆U max
⎡ ⎤ ⎡
⎤
⎢
M 3 = ⎣ −Φ ⎥ ⎢
⎦ ; N 3 = ⎣ −Y min + Γx(k) ⎥
⎦
Φ
Y min − Γx(k)
Considering a MIMO case, U min and U max are defined as column vectors with mN c
elements of u min and u max respectively.
∆U min and ∆U max are also defined as a
column vectors with mN c elements of ∆u min and ∆u max respectively. M 1 and N 1 are
matrices associated with constraints on the amplitude of control variables, M 2 and N 2
are matrices associated with constraints on the increment of control variables, and the
matrices M 3 and N 3 refer to constraints on output variable. In the case of manipulated