Page 152 - Distributed model predictive control for plant-wide systems

126 Distributed Model Predictive Control for Plant-Wide Systems

local optimization problem, the optimality of the iteration-based networked MPC algorithm
is analyzed and the nominal stability is derived for distributed control systems without the
control and output constraints. An illustrative example is provided to verify the optimality of
the networked MPC algorithm.
The contents are organized as follows. Section 7.1 describes the noniterative networked DPC
and gives its closed-form solution, as well as the stability condition of the closed-loop system.
Section 7.2 details the networked DMPC with an iterative algorithm based on neighborhood
optimization.

7.2 Noniterative Networked DMPC
7.2.1 Problem Description
Without losing generality, suppose that the whole system is composed of n linear, discrete-time
subsystems S , i = 1, 2, … , m, and each subsystem interacts with each other by both inputs and
i
states; then the state-space model of subsystem S can be expressed as
i
m m
⎧ ∑ ∑
⎪x (k + 1) = A x (k)+ B u (k)+ A x (k)+ B u (k)
i ii i ii i ij j ij j
⎪ j=1(j≠i) j=1(j≠i)
n (7.1)
⎨
∑
y (k)= C x (k)+ C x (k)
⎪
ii i
i
ij j
⎪
j=1(j≠i)
⎩
where vectors x ∈ ℝ , u ∈ ℝ , and y ∈ ℝ n y are the local state, control input, and output
n x
n u
vectors, respectively. When at least one of the matrices A , B , C is not null, it is said that S j
ij
ij
ij
interacts with S . The whole system can be expressed as
i
{
x (k + 1) = Ax(k)+ Bu(k)
(7.2)
y(k)= Cx(k)
n u
n y
n x
where x ∈ ℝ , u ∈ ℝ , and y ∈ ℝ are the state, control input, and output vectors, respec-
tively. The control objective of this system is to minimize a global performance index J(k) at
time k, and
[ ]
n P M
∑ ∑ d 2 ∑ 2
J(k)= ‖ i i ‖ + ‖ Δu (k + l − 1) ‖ (7.3)
‖y (k + l) − y (k + l)‖
i
‖ ‖Q i ‖ ‖R i
i=1 l=1 l=1
where Q and R are weight matrices, P, M ∈ ℕ are the predictive horizon and control horizon,
i
i
d
respectively, and P ≥ M, y is the set-point of subsystem S , and Δu (k)= u (k)−Δu (k − 1)
i
i
i
i
i
is the input increment vector of subsystem S .
i
Moreover, in many situations, the communication resources are not unlimited for the safety
reason and communication bandwidth limitation, or the global information is unavailable to
every subsystem due to the physical or man-made reasons. Those require a simple structure of
a local controller. Thus, as pointed out in [100], how to improve the performance of the entire
system is still a challenge for this class of system under the distributed control framework with
a simple control structure.