Page 196 - Distributed model predictive control for plant-wide systems

170 Distributed Model Predictive Control for Plant-Wide Systems

It is well known that the explicitly handling constraint is an important advantage of MPC.
Designing stabilized DMPC with constraints is definitely an important and difficult task. In
this part, we will develop the stabilized DMPC with constraints. And in this chapter the basic
and most used local optimization based DMPC is designed first.
Control design that takes state and/or input constraints into account, whether or not under
the MPC framework, is an important and challenging problem. Many methods can be found
in the literature [78, 111–113]. Under the MPC framework, closed-loop stability is ensured
by judiciously integrating designs of the terminal cost, the terminal constraint set, and the
local controllers [78]. In DMPC, the future state sequences of upstream neighbors, which are
calculated based on the solution in the previous time instant, may not be equal to the predictive
states calculated by the corresponding subsystem at the current time instant, and the errors
between them are hard to estimate. In addition, in the presence of constraints, the feasibility of
each subsystem-based MPC cannot be guaranteed. The remaining part of the optimal control
sequence calculated at the previous time instant may not be a feasible solution at the current
time instant. It is difficult to construct a feasible solution in the current time instant. All these
make it difficult to design a stabilizing LCO-DMPC that takes constraints into consideration.
In this chapter, a stabilized LCO-DMPC algorithm is developed, which uses a constraint to
limit the error between the future state sequences (or called presumed sequences) of upstream
neighbors, which are calculated based on the solution in the previous time instant, and the
predictive states calculated by the corresponding subsystem in the current time instant. Then
the stability is ensured by judiciously integrating designs of the bound of the error between
presumed state sequence and predictive state sequence, the terminal cost, the constraint set,
and the local controllers.
The remainder of this chapter is organized as follows: Section 8.2 describes the problem to
be solved in this chapter. Section 8.3 presents the design of the stabilizing LCO-DMPC with
communicating once a sampling time. The feasibility of the proposed LCO-DMPC and the
stability of the resulting closed-loop system are analyzed in Section 8.4. Section 8.5 presents
the simulation results to demonstrate the effectiveness of the proposed LCO-DMPC algorithm.
Finally, a brief conclusion to the chapter is drawn in Section 8.6.

8.2 Problem Description

Consider a spatially distributed system, as illustrated in Figure 4.1, which is composed of
many physically partitioned interacted subsystems, and each subsystem is controlled by a
subsystem-based controller which in turn is able to exchange information with its neighboring
subsystem-based controllers.
Suppose the distributed system S is composed of m discrete-time linear subsystems S ,
i
i ∈ P = {1, 2, … , m} and m controllers C , i ∈ P = {1, 2, … , m}. Let the subsystems interact
i
with each other through their states. If subsystem S is affected by S , for any i ∈ P and j ∈ P,
i
j
subsystem S is said to be a downstream subsystem of subsystem S , and subsystem S is
i
j
j
an upstream system of S .Let P +i denote the set of the subscripts of the upstream systems
i
of S and P −i is the set of the subscripts of the downstream systems of S . Then, using the
i
i
state-interacted model introduced in Chapter 4, each 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 (8.1)
⎨
⎪y (k)= C x (k)
i ii i
⎩