Model Predictive Control System Design and Implementation Using ...

1.3 Predictive Control within One Optimization Window 11
The coefficients in the F and Φ matrices are calculated as follows:
CA = [ s 1 1 ]
CA 2 = [ s 2 1 ]
CA 3 = [ s 3 1 ]
where s 1 = a, s 2 = a 2 + s 1 , ..., s k = a k + s k−1 ,and
.
CA k = [ s k 1 ] , (1.20)
CB = g 0 = b
CAB = g 1 = ab + g 0
CA 2 B = g 2 = a 2 b + g 1
.
CA k−1 B = g k−1 = a k−1 b + g k−2
CA k B = g k = a k b + g k−1 . (1.21)
With the plant parameters a =0.8 andb =0.1, N p =10andN c =4,we
calculate the quantities
⎡
⎤
1.1541 1.0407 0.9116 0.7726
Φ T Φ = ⎢ 1.0407 0.9549 0.8475 0.7259
⎥
⎣ 0.9116 0.8475 0.7675 0.6674⎦
0.7726 0.7259 0.6674 0.5943
⎡
⎤ ⎡ ⎤
9.2325 3.2147
3.2147
Φ T F = ⎢ 8.3259 2.7684
⎥
⎣ 7.2927 2.3355⎦ ; ΦT ¯Rs = ⎢ 2.7684
⎥
⎣ 2.3355⎦ .
6.1811 1.9194
1.9194
Note that the vector Φ T ¯Rs is identical to the last column in the matrix Φ T F .
This is because the last column of F matrix is identical to ¯R s .
At time k i = 10, the state vector x(k i )=[0.1 0.2] T . In the first case, the
error between predicted Y and R s is reduced without any consideration to
the magnitude of control changes. Namely, r w = 0. Then, the optimal ΔU is
found through the calculation
ΔU =(Φ T Φ) −1 (Φ T R s − Φ T Fx(k i )) = [ 7.2 −6.4 00 ] T
.
We note that without weighting on the incremental control, the last two elements
Δu(k i +2)=0 andΔu(k i + 3) = 0, while the first two elements have a
rather large magnitude. Figure 1.1a shows the changes of the state variables
where we can see that the predicted output y has reached the desired set-point