Page 198 - Distributed model predictive control for plant-wide systems

172 Distributed Model Predictive Control for Plant-Wide Systems

subsystem-based MPC optimizes only for its own open-loop control sequence, given the
current states and the estimated inputs of the whole system.
To proceed, we need the following assumption, and we also define the necessary notation
in Table 8.1.

Assumption 8.1 For every subsystem S , ∀i ∈ P there exists a state feedback u = K x
i i,k i i,k
such that the closed-loop system x(k + 1) = A x(k) is asymptotically stable, where
c
A = A + BK and K = block − diag{K , K , … , K }
c 1 2 m
Remark 8.1 This assumption is usually used in the design of stabilizing DMPC [50]. It pre-
sumes that each subsystem is able to be stabilized by a decentralized control K x ,i ∈ P, and
i i
the decentralized control gain K can be obtained by LMI or LQR.
We also define the necessary notation in Table 8.1.
Each subsystem-based MPC minimizes the cost function of its corresponding subsystem.
More specifically, the performance index is defined as

N−1 ( )
2 ∑ 2
‖ p ‖ ‖ p ‖ 2
J (k)= ‖x (k + N|k)‖ + ‖x (k + s|k)‖ + u (k + s|k) ‖ (8.3)
‖
‖ i
i
‖ i
‖ i
‖R i
‖P i
s=0 ‖Q i
T
T
T
where Q = Q > 0, R = R > 0 and P = P > 0. The matrix P is chosen to satisfy the
i i i i i i i
Lyapunov equation
T
̂
A P A − P =−Q
di i di 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 }
dm
d2
d1
d
Table 8.1 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
p
p
p
x (k + s|k) The predicted state sequence of S , calculated by C at time k, x (k + s|k)= x (k + s|k)
i i i 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, ̂ x (k + s|k)= ̂ x (k + s|k)
i i i i i,i
̂ u (k + s|k) The presumed control sequence of S ,definedby C at time k
i i i
f
f
f
x (k + s|k) The feasible state sequence of S , calculated by C at time k, x (k + s|k)= x (k + s|k)
i i i i i,i
f
f
f
u (k + s|k) The feasible control sequence of S ,definedby C at time k, u (k + s|k)= u (k + s|k)
i i i i i,i