Page 236 - Distributed model predictive control for plant-wide systems

210 Distributed Model Predictive Control for Plant-Wide Systems

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 [51], the
terminal cost, the constraint set, and the local controllers [69]. Farina and Scattolini [50] gave
another design for linear system, which uses a fixed reference trajectory with a moving win-
dow to substitute the presumed state/input of upstream neighbors used in Ref. [51]. Both these
methods are design for DMPC in which each subsystem-based MPC optimizes the cost of the
corresponding subsystem itself. As for the DMPC, which uses the global cost function, some
convergence conditions are deduced if using iterative algorithms, then the distributed problems
can be reformulated into a centralized problem, and the stabilizing DMPC can be designed with
similar method of centralized MPC. For the coordination strategy used here, there is no global
model that can be used. And except that there are errors between the presumed state/input
sequences and predictive state sequences of upstream neighbors, the predictive state sequences
of downstream neighbors calculated by current subsystem may not equal to those calculated
by the downstream neighbors themselves; these error are difficult to estimate. In the presence
of constraints, 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. All these make it difficult
to design a stabilizing N-DMPC that takes constraints into consideration.
In this chapter, the coordination strategy that optimizes the impacted-region cost in each
subsystem-based MPC is adopted to achieve a DMPC performance that is close to that of a
centralized MPC. The consistency constraints, which limit the error between the presumed
state and the state predicted at the current time instant within a prescribed bound, are designed
and included in the optimization problem of each subsystem-based MPC. These constraints
can bound the error between the presumed state and the predictive state of upstream neigh-
bors and the error between the predictive state of downstream neighbor calculated by current
subsystem-based MPC and that of the downstream neighbors themselves. And these con-
straints guarantee that the remaining part of the solution at the current time instant is a feasible
solution at the next time instant. In the meantime, stabilization constraints and the dual mode
predictive control strategy are adopted to result in a stabilizing N-DMPC.
The remainder of this chapter is organized as follows: Section 10.2 describes the prob-
lem to be solved in this chapter. Section 10.3 presents the design of the stabilizing N-DMPC.
The feasibility of the proposed N-DMPC and the stability of the resulting closed-loop sys-
tem are analyzed in Section 10.4. Section 10.5 discusses DMPC formulations under other
coordination strategies. Section 10.6 presents the simulation results to demonstrate the effec-
tiveness of the proposed DMPC algorithm. Finally, a brief conclusion to the chapter is drawn in
Section 10.7.

10.2 Problem Description
A distributed system, as illustrated in Figure 4.1, is considered here. Suppose the distributed
system S is composed of m discrete-time linear subsystems S , i ∈ P ={1, 2, … , m} and m
i
controllers C , i ∈ P ={1, 2, … , m}. Let the subsystems interact with each other through their
i
states. If subsystem S is affected by S , for any i ∈ P and j ∈ P subsystem S is said to be a
i
i
j