Page 136 - Distributed model predictive control for plant-wide systems

110 Distributed Model Predictive Control for Plant-Wide Systems

A = diag {A }
m a
B = diag{B , … , B }
1
m
C = diag {C }
m a
[ ] T
̃ T
̃ T
̃
B = B ··· B , (6.23)
1 m
T
L = diag {L }
i
P
i
L = diag{L , … , L }
1 m
L = diag{L , … , L }
1
m
[ ′T ′T ] T
′
L = L ··· L m
1
[ ]
′
L = L I (6.24)
̃
n x n x ×(P−1)n x
′
′
′
= diag{ , … , }
1 m
= diag{ , … , }
1
m
S = diag {S}
m
= diag{ K , … , K } (6.25)
1 1 m m
=− CSAL
=− CSAL
̃
′
′
̃
= − CS(B + B ) (6.26)
Then, Theorem 6.2 can be deduced.
Theorem 6.2 (C-DMPC stability) The closed-loop system given by the feedback connection
of plant S with the set of independent controller C ,i = 1, … , m, whose closed-form control
i
laws are given by Equation (6.19), is asymptotically stable if and only if
| {A }| < 1, ∀j = 1, … , n N (6.27)
j
N
where
A B
⎡ ⎤
⎢ LSAL LSAL LSB LSB ⎥
̃
̃
A = ⎢ ⎥
N ̃ ̃
⎢ A + LSAL LSAL B + LSB + LSB ⎥

⎢ I ⎥
⎣ Mn u ⎦
n = Pn + n + 2Mn is the order of the whole closed-loop system.
N x x u
Proof. By Equations (6.7) and (6.13), the stacked prediction of the states S in C at time k
i
i
is expressed as
T
X (k + 1, P|k)= L S[A L x (k)+ B U (k, M|k)
̂
i i a i i i i
′
+ A L ̂ x(k|k − 1)+ B U(k − 1, M|k − 1)] (6.28)
̃
i
i i