Page 135 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control 109

According to Equation (6.13), the solution to problem (6.6) can be deduced as
−1
ΔU (k, M|k)= 1∕2 ⋅ H G (k + 1, P|k)
i i i
By noting that only the first element of the optimal sequence is actually applied to the pro-
cess, Theorem 6.1 is obtained.

Theorem 6.1 (Closed-form solution) Under Assumption 6.1, for each controller C ,
i
i = 1, … , m, the closed-form control law applied at time k to subsystem S is given by
i
d
u (k)= u (k − 1)+ K [Y (k + 1, P|k)− Z (k + 1, P|k)] (6.19)
̂
i i i i
where
K = K i
i
i
T
−1
K = H N Q
i
i
i
[ ]
= I n u i n u i ×(M−1)n u i (6.20)
i
Remark 6.2 In C , the complexity to obtain the closed-form solution is mainly incurred by the
i
inversion of H . By using the Gauss–Jordan algorithm for this task and considering that the
i
3
3
size of H equals M ⋅ n , the complexity of the inversion algorithm is O(M ⋅ n ). Therefore,
i
u i
( u i )
3
the total computational complexity of solving C-DMPC is only O M ⋅ ∑ n n 3 , while the
i=1 u i
( )
( ) 3
3
computational complexity of the centralized control strategy equals O M ⋅ ∑ n n .
i=1 u i
6.2.4 Stability and Performance Analysis
6.2.4.1 Stability Analysis
On the basis of the closed-form solution stated by Theorem 6.4, the closed-loop dynamics can
be specified and the stability condition can be verified by analyzing the closed-loop dynamic
matrix. Define
= ··· T ] T
[
T
1 P
= diag{ , … , }
l 1l ml
[ ]
= n x i ×(l−1)n x i I n x i n x i ×(P−l)n x i
il
(i = 1, … , m, l = 1, … , P); (6.21)
= ··· T ] T
[
T
1 M
= diag{ , … , }
l
1l
ml
[ ]
= I
il n u i ×(l−1)n u i n u i n u i ×(M−l)n u i
(i = 1, … , m, l = 1, … , M) ; (6.22)