Page 191 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Information Structure Constraints 165

( )
∑
T
=ΔU T (k) H Q H + R ΔU (k)
i,M ji j ji i i,M
j∈P i
( ) T
∑ T
− 2 H Q R (k) ΔU (k)+ constant (F.2)
ji j j,P i,M
j∈P i
where
∑
R (k)= R (k)− G ̂ x (k)− H ΔU h,M (k)
jh
j,P
j j
j,P
h∈P j ,h≠i
[ T T ] T
R (k)= r (k + 1) ··· r (k + P)
i,P
i
i
Q = block-diag(Q , … , Q )
i i i
⏟⏞⏞⏟⏞⏞⏟
P
R = block-diag(R , … , R ).
i
i
i
⏟⏞⏞⏟⏞⏞⏟
M
By removing the constant terms, the local optimization problem (7.51) can be written in the
following quadratic form:
1 T T
min J (k) ⇐⇒ min ΔU (k)Π ΔU i,M (k)+ f (k)ΔU i,M (k) (F.3)
i
i
ΔU i,M (k) ΔU i,M (k) 2 i,M i
where
∑ T
= H Q H + R > 0
i ji j ji i
j∈ℕ i
∑ T
f (k)=− H Q R (k)
i ji j j,P
j∈ℕ i
The inequality constraints in (7.51) can be converted into
ΔU i,M (k) ≤ b (k) (F.4)
i
i
where
⎡ U i ⎤
⎢ ⎥
T i ⎢ −U ⎥
⎡ ⎤
⎢ ⎥ i
⎢ ⎥
−T i ⎢ ΔU ⎥
⎢ ⎥
⎢ ⎥ i
⎢ ⎥
I ⎢ −ΔU ⎥
⎢ ⎥
= ⎢ Mn ui ⎥ , b (k)= ⎢ i ⎥
i
i
−I ⎢Y − G ̂ x (k) − H ΔU (k)⎥
⎢ ⎥ ∑
⎢ Mn ui ⎥ i i i ij j,M
⎢ ⎥
H ii ⎢ j≠i ⎥
⎢ ⎥ j∈ℕ i
⎢ ⎥
⎢ ∑ ⎥
−H ⎢ G ̂ x (k)+ H ΔU j,M (k)− Y ⎥
⎢ ⎥
i i
ij
⎣ ii ⎦ i
⎢ j∈ℕ i ⎥
⎣ j≠i ⎦