Page 238 - Distributed model predictive control for plant-wide systems
P. 238
212 Distributed Model Predictive Control for Plant-Wide Systems
The control objective is to stabilize the overall system S in a DMPC framework. And the
performance of entire closed-loop system should be as close as that under the control of cen-
tralized MPC and the communication cost should not be too high.
The coordination strategy as proposed in Chapter 7 is a preferable method to tradeoff
between the communication burden and the global performance. However, the DMPC under
this coordination strategy does not take the constraints into consideration. A noniterative
stabilizing DMPC that takes constraints into consideration remains to be developed either
under this coordination strategy or some other coordination strategies reviewed earlier. The
objective of this chapter is to develop such a DMPC design.
10.3 Constrained N-DMPC
10.3.1 Formulation
In this section, m separate optimal control problems, one for each subsystem, and the N-DMPC
algorithm are defined. In each of these optimal control problems, the same constant prediction
horizon N, N ≥ 1, is used. The resulting m separate subsystem-based MPC laws are updated
synchronously. At each update, every subsystem-based MPC optimizes only over its own
predicted open-loop control, given its impacted region’s current states, and their upstream
subsystems’ estimated inputs and states.
To proceed, we need the following assumption:
Assumption 10.1 For every subsystem S , ∀i ∈ P, there exists a state feedback u =
i
i,k
K x such that the closed-loop system x(k + 1) = A x(k) is asymptotically stable, where
i i,k c
A = A + BK and K = block-diag{K , K , … , K }.
c 1 2 m
This assumption is usually used in the design of stabilizing DMPC [26, 34]. It presumes
that each subsystem is able to be stabilized by a decentralized control K x , i ∈ P.
i i
We also define the necessary notation in Table 10.1.
As the state evolution of the downstream subsystems of S is affected by the optimal control
j
decision of S , the performance of these downstream subsystems may be affected negatively
i
by the control decision of S . Thus, in the ICO-DMPC, each subsystem-based MPC takes into
i
account the cost functions of its downstream subsystems. More specifically, the performance
index is defined as
∑ ‖ p 2
J (k)= ‖
i ‖x (k + N |k)‖
‖ j,i
‖P j
j∈P i
( )
N−1
∑ ∑ ‖ p 2 ‖ p 2
‖
‖
+ ‖x (k + s|k)‖ + ‖u (k + s|k)‖ (10.3)
‖ j,i ‖Q j ‖ i ‖R i
s=0 j∈P 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 i (10.4)
di i
di
i