Page 94 - Distributed model predictive control for plant-wide systems

68 Distributed Model Predictive Control for Plant-Wide Systems

subsystem’s cost and uses the state prediction of the previous time instant to approximate the
state sequence at the current time instant in the procedure of computing an optimal solution.
Another commonly used coordinating strategy is that each subsystem-based MPC optimizes
the cost of the overall system to improve the global performance [89]. This strategy could
achieve a good global performance in some cases, but it reduces the flexibility and increases
the communication load. In an effort to achieve a tradeoff between the global performance
of the entire system and the computational burden, the authors of [54] proposed the idea
that each subsystem-based MPC optimizes the costs of all subsystems that it interacts with.
Recently, an intuitively reasonable strategy was proposed in [19, 47, 56, 57], where each
subsystem-based MPC only considers the cost of the corresponding subsystem and the
subsystems it directly impacts on. Such a design can be referred to as networked distributed
MPC (N-DMPC).
In this chapter, the most used and very practiced strategy local cost optimization-based
DMPC (LCO-DMPC) is introduced; both the unconstrained iterative and noniterative methods
are presented. In Section 5.2, the noniterative LCO-DMPC using the input- and state-interacted
model is introduced; the majority of this part of content comes from [46]. The closed-loop
solution and stable condition are deduced. Section 5.3 gives the iterative LCO-DMPC which
could achieve Nash optimality when the convergence conditions are satisfied. The computa-
tional convergence and nominal stability of the distributed MPC as well as the performance
analysis with single-step horizon control under communication failure are given in detail that
characterizes the Nash optimization-based DMPC.

5.2 Local Cost Optimization-based Distributed Predictive Control
5.2.1 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 A :
j
i
m m
⎧ ∑ ∑
⎪ x (k + 1) = A x (k)+ B u (k)+ A x (k)+ B u (k)
ii i
i
ij j
ij j
ii i
j=1( j≠i) j=1( j≠i)
⎪
(5.1)
⎨ n
∑
⎪ y (k)= C x (k)+ C x (k)
⎪ i ii i ij j
⎩ j=1( j≠i)
where vectors x ∈ ℝ , u ∈ ℝ , and y ∈ ℝ n y i are the local state, control input, and output
n u
n xi
i i i
vectors, respectively. When at least one of the matrices A , B ,or C is not null, it means that
ij ij ij
S interacts with S . In this case, S is called the upstream neighboring subsystem of S and S i
i
j
i
j
is called the downstream neighboring subsystem of S . S and S are called the neighboring
j
i
j
subsystems or neighbors. To proceed, we define that the upstream (downstream) neighbor-
hood P (P ) of subsystem S is the set of the subscripts of all its upstream (downstream)
i
+i
−i
neighbors:
P ={S , S |S is an upstream neighbor of S }
+i i j j i
P −i ={S , S |S is a downstream neighbor of S }
j
i
j
i