Page 239 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Inputs and Information Structure Constraints 213

Table 10.1 Notations in this chapter
Notation Explanation

i The subscript denoting all downstream subsystems of S i
+ i The subscript denoting all upstream subsystems of S i
̃ i The subscript denoting all upstream subsystems of S and S , i
−i
excluding S and S themselves
−i
i
p
x (k + s|k) The predicted state sequence of S , calculated by C at time
j,i j i
p
x (k + s|k) The predicted state sequence of S , calculated by C at time k,
i p p i i
x (k + s|k)= x (k + s|k)
i
i,i
p
u (k + s|k) The predicted control sequence of S , calculated by C at time k
i i i
̂ x (k + s|k) The presumed state sequence of S , calculated by C at time k
j,i j i
̂ x (k + s|k) The presumed state sequence of S , calculated by C at time k,
i i i
̂ x (k + s|k)= ̂ x (k + s|k)
i i,i
û (k + s|k) The presumed control sequence of S , defined by C at time k
i
i
i
û (k + s|k) The presumed control sequence of S , defined in C at time k
j
j,i
i
f
x (k + s|k) The feasible state sequence of S , calculated by C at time k
j,i j i
f
x (k + s|k) The feasible state sequence of S , calculated by C at time k,
i i i
f
f
x (k + s|k)= x (k + s|k)
i i,i
f
u (k + s|k) The feasible control sequence of S ,definedin C at time k
j,i j i
f
u (k + s|k) The feasible control sequence of S ,definedby C at time k,
i i i
f
f
u (k + s|k)= u (k + s|k)
i i,i
T
̂
where Q = Q + K R K . Denote
i
i
i
i
i
P = block-diag{P , P , … , P }
1 2 m
Q = block-diag{Q , Q , … , Q }
1 2 m
R = block-diag{R , R , … , R }
1 2 m
A = block-diag{A , A , … , A }
d d1 d2 dm
Then, it follows that
T
A PA − P =−Q
̂
d d
T
where Q = Q + K RK > 0.
̂
p
To get the predicted state sequence x (k + s|k) of subsystem S under the control decision
j,i j
p
sequence of u (k + s|k) in (10.3), the system evolution model should be deduced first. Since
i
every subsystem-based controller is updated synchronously, the state and control sequences
of other subsystems are unknown to subsystem S . Thus, at the time instant k, the presumed
i
and the presumed control sequence
state sequence {̂ x̃ i (k|k), ̂ x̃ i (k + 1|k), … , ̂ x̃ i (k + N |k)} of S̃ i
{̂ u (k | k), ̂ u (k + 1| k), … , ̂ u (k + N | k)} of S are used in the predictive model of the MPC
− i
− i
− i
−i
in S , which is given as
i
[ p ] [ ] s
x (k + s|k) s x (k|k)
i i ∑ s−l p
p = A i + A i B u (k + l − 1|k)
x (k + s|k) x (k|k) i i
i,i i l=1