Page 57 - Distributed model predictive control for plant-wide systems

Model Predictive Control 31

Then, the optimization problem of MPC in each control period becomes
min J(k)= ||THx(k|k)+ TFu(k − 1|k − 1)+ TVW(k|k)
ΔU(k|k)
2 2
+TGΔU(k|k)− Y (k|k)|| + ||ΔU(k|k)|| (2.26)
r Q R
which is a standard quadratic programming problem. Let
Z(k)= Hx(k|k)+ TFu(k − 1|k − 1)+ TVW(k|k)− Y (k|k) (2.27)
r
The explicit solution of (2.26) can be deduced as
T
−1
T
ΔU(k|k)=((TG) QTG + R) (TG) QZ(k) (2.28)
Thus, in every control period, the manipulated variable should be
T
T
−1
u(k)= D((TG) QTG + R) (TG) QZ(k)+ u(k − 1) (2.29)
[ ]
where D = I 0 ··· 0 .
2.3.5 State Space MPC with Constraint
In the real application of MPC, the constraints on actuator slew rates, actuator ranges, and
constraints on the controlled variables usually exist. We use the following equations to express
those constraints.

1. Output constraint: y min ≤ y(k + l|k) ≤ y max .
At each optimization cycle, the output prediction can be calculated by (2.24). Hence, we
can let the optimization problem satisfy the following constraint:
Y ≤ T(Hx(k|k)+ GΔU(k|k)+ Fu(k − 1|k − 1)+ VW(k|k)) ≤ Y (2.30)
min max
or
Y min ≤ T(HÂ x(k|k − 1)+ Bu(k − 1)+ Ed(k − 1)+ L(̂ y(k)− ̂ y(k|k − 1))
+ GΔU(k|k)+ Fu(k − 1|k − 1)+ VW(k|k)) ≤ Y max (2.31)

where
[ ] T
Y = y T y T ··· y T
min min min min
[ T T T ] T
Y max = y max y max ··· y max
2. Input increment constraint: Δu ≤ u(k + l|k) − u(k + l − 1|k) ≤ u .
min max
We can let the optimization problem satisfy the following constraint:
ΔU ≤ ΔU(k|k) ≤ ΔU (2.32)
min max
where
[ T T T ] T
ΔU min = Δu min Δu min ··· Δu min
[ T T T ] T
ΔU = Δu Δu ··· Δu
max max max max