Page 95 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 69

The neighborhood P of subsystem S is the set of all its neighbors:
i
i
P = P ∪ P ∪ S i
i
−i
+i
and defines the state and output interaction vectors w and v :
i
i
n m
∑ ∑
⎧
w (k) ≜
ij j
ij j
⎪ i A x (k)+ B u (k)
⎪ j=1( j≠1) j=1( j≠1)
(5.2)
⎨ m
∑
⎪ v (k) ≜ C x (k)
⎪ i ij j
⎩ j=1( j≠1)
The whole system can be expressed as
{
x (k + 1) = Ax(k)+ Bu(k)
(5.3)
y(k)= Cx(k)
where x ∈ ℝ , u ∈ ℝ , and y ∈ ℝ n y are the state vector, control input vector, and control
n u
n x
output vector, respectively.
The control objective of this system is to minimize a global performance index at time k
under the distributed control framework, and
[ ]
m P M
∑ ∑ d 2 ∑ 2
J(k)= ‖ i i ‖ + ‖Δu (k + l − 1)‖ (5.4)
‖y (k + l) − y (k + l)‖
i
‖ ‖Q i R i
i=1 l=1 l=1
where Q and R are the weight matrices, P and M ∈ ℕ are the predictive horizon
i
i
and control horizon, respectively, and P ≥ M, y d is the set-point of subsystem S ;
i
i
Δu (k) = u (k) −Δu (k − 1) is the input increment vector of subsystem S .
i
i
i
i
5.2.2 DMPC Formulation
In the DMPC, the control framework introduced is based on a set of an independent
subsystem-based controller C ; i = 1, … , n, implementing an MPC algorithm for subsystem
i
S using both local information acquired on S and the estimate of the interactions among S
i i i
and is upstream neighbors P . The resulting optimal sequence and the future prediction of
+i
the state over the prediction horizon have to be exchanged among subsystems through a local
area network.
The simplifying hypothesis of accessible local states x (k) is considered in this chapter.
i
Moreover, the sets of the prediction and control horizons are the same to each MPC controller
C ; i = 1, … , n, and are considered as P and M, respectively. Let all subsystem-based MPCs
i
be synchronous. In addition, the following assumption is required.
Assumption 5.1
1. Control agents communicate only once within a sampling time interval.
2. The communication channel introduces a delay of a single sampling time interval.