Page 216 - Distributed model predictive control for plant-wide systems

190 Distributed Model Predictive Control for Plant-Wide Systems

[69, 78] strategy is adopted. These consistency constraints and the dual mode strategy
guarantee that the remaining part of the solution at the previous time instant is a feasible solu-
tion if there is a feasible solution at initial time instant. They also guarantee the asymptotical
stability of the closed-loop system.
The remainder of this chapter is organized as follows: Section 9.2 describes the problem to
be solved in this chapter. Section 9.3 presents the design of the stabilizing C-DMPC with com-
municating once a sampling time. The feasibility of the proposed C-DMPC and the stability
of the resulting closed-loop system are analyzed in Section 9.4. Section 9.5 presents the sim-
ulation results to demonstrate the effectiveness of the proposed C-DMPC algorithm. Finally,
a brief conclusion to the chapter is drawn in Section 9.6.

9.2 System Description
Consider a spatially distributed system, with each subsystem-based controller which in turn is
able to exchange information with all other subsystem-based controllers.
Without losing generality, suppose the whole system is composed of m discrete-time linear
subsystems S , i ∈ P, P ={1, … , m}. Let the subsystems interact with each other through
i
their states. Then, subsystem S can be expressed as
i
∑
⎧
x (k + 1) = A x (k)+ B u (k)+ A x (k)
ij j
i
ii i
ii i
⎪
j∈P +i
⎨
⎪y (k)= C x (k)
i ii i
⎩
where x ∈ ℝ xi , u ∈ U ⊂ ℝ ui , and y ∈ ℝ n yi are, respectively, the local state, input and out-
n
n
i
i
i
i
put vectors, and U is the feasible set of the input u , which is used to bound the input according
i i
to the physical constraints on the actuators, the control requirements, or the characteristics of
the plant. A nonzero matrix A indicates that S is affected by S , j ∈ P, and subsystem S is
ij i j j
said to be an upstream system of S .Let P denote the set of the subscripts of the upstream
j +i
systems of S , that is, j ∈ P , and set P be the set of the subscripts of the downstream sys-
i +i −i
tems of S . In addition, set P ={j| j ∈ P, and, j ≠ i}. In the concatenated vector form, the
i i
system dynamics can be written as
{
x (k + 1) = Ax(k)+ Bu(k)
y(k)= Cx(k)
T T
T
T
T T
T
T
T T
T
n
n
T
where x =[x , x , … , x ] ∈ ℝ x , u =[u , u , … , u ] ∈ ℝ u , and y =[y , y , … , y ]
m
m
m
2
1
2
1
1
2
∈ ℝ n y are, respectively, the concatenated state, control input, and output vectors of the
overall system S, and A, B, and C are constant matrices of appropriate dimensions. Also,
u ∈ U = U × U ×···× U and U contain a neighborhood of the origin.
1
m
1
The control objective is to stabilize the overall system S in a DMPC framework with
the limited communication resources. Meanwhile, the achieved performance index of the
overall system should be as close as possible to the performance index achievable under a
centralized MPC.
When the global information is available for each subsystem-based MPC, the coordination
strategy where each subsystem-based MPC optimizes the cost of entire system is very suitable
for the control of this class of system, since it is able to achieve a very good global performance.