Page 177 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Information Structure Constraints 151

T
H QH =Φ +Φ nd
d
{
H , j ∈ P i
ij
H(i, j)=
, j ∉ P i
H(i, j) represents the block element matrix at the ith row and jth column of H.
T
−1
At the time instant k, R (k) and Ĝ x(k) are known in advance; hence, ( + R) H Q
P d
[R (k)− Ĝ x(k)] is the constant term irrelevant to the iteration. The convergence of expression
P
(7.55) is equivalent to that of the following equation:
(l+1) −1 (l)
ΔU (k)=−( + R) ΔU (k) (7.57)
d
nd
M M
From the above analysis, the convergent condition for the algorithm in application to net-
worked linear MPC is
−1
| (( + R) )| < 1 (7.58)
d nd
That is, the spectrum radius must be less than 1 to guarantee a convergent computation.
If the convergent condition (7.58) is satisfied, the integral optimal control decision of the
whole system at the time instant k is
∗
∗
−1
ΔU (k)=−( + R) ΔU (k)
M d nd M
−1
T
+( + R) H Q[R (k)− Ĝ x(k)] (7.59)
d
P
which can be rewritten as
[ ]
( ) −1 −1( ) −1 [ ]
T
∗
ΔU (k)= I + + R + R H Q R (k) − Ĝ x (k)
M d nd d P
{ [ ]} −1
( ) ( ) −1 [ ]
T
= + R ⋅ I + + R H Q R (k) − Ĝ x (k)
d d nd P
(7.60)
( ) −1 [ ]
= + R + nd H Q R (k) − Ĝ x (k)
T
P
d
( ) −1
T T [ ]
= H QH + R H Q R (k) − Ĝ x (k)
P
At the sampling time instant k, the output prediction model for the whole system can be
described as
c
Y (k)= Ĝ x(k)+ HΔU (k) (7.61)
̂
P M
c
where ΔU (k) represents control decision derived by centralized MPC,
M
[ T T ] T
c
ΔU (k)= ΔU (k) ··· ΔU (k)
M 1,M m,M
[ ] T
̂ T
̂ T
Y (k)= Y 1,P (k) ··· Y m,P (k)
̂
P