Page 140 - Distributed model predictive control for plant-wide systems

114 Distributed Model Predictive Control for Plant-Wide Systems

Decompose S into two SISO subsystems, S and S . The corresponding state-space real-
1
2
izations of S and S have the form (5.1), with matrices {A , B , C } and {A , B , C },
22
11
22
22
11
2
1
11
respectively. The constant parameter is used to study the interactions between S and S .
1 2
Similar to N-DMPC, the computational complexity of C-DMPC depends on the plant
size, in particular on the number of inputs of independent controllers, while the feasibility
depends on the possibility of satisfying Theorem 6.2 for a specific set of parameters P, M,
Q, and R , i = 1, … , m. Different weighting matrices and horizons can be selected. Here,
i
for simplifying the graphical representations of the results, we choose P = M, R = I , and
u
Q = I .
y
The maximum eigenvalues of the corresponding closed-loop systems are computed and
plotted in three-dimensional graphs of Figures 6.1a, 6.2a, and 6.3a for different combina-
tions of and P. (More degrees of freedom can be used if the achieved performances and
stability are not satisfactory.) The Z axis represents the maximum eigenvalues, and the X
and Y axes represent the logarithms of and P, respectively. The control performance of the
closed-loop system is plotted in Figures 6.1b, 6.2b, and 6.3b, where the black dashed lines cor-
respond to the desired outputs, the blue solid lines correspond to the system outputs and inputs
using LCO-DMPC, and the red dashed lines represent the system outputs and inputs using
C-DMPC.
The stability depends on the choice of the tuning parameters and P. For weak interactions
(see Figures 6.1a and 6.2a), the ranges of tuning parameters in C-DMPC are similar to that of
LCO-DMPC. At the same time, a wider range of tuning parameters in C-DMPC, than that in
LCO-DMPC, is available for strong interactions (see Figure 6.3a).
Most importantly, a better global performance of the closed-loop system is observed for the
proposed C-DMPC where the subsystems exhibit interactions (see Figures 6.1b and 6.2b). The
MSEs of outputs with C-DMPC are less than that using LCO-DMPC when = 0.1 and = 1,
which are (0.2086, 0.2034) for C-DMPC and (0.2568, 0.2277) for N-DMPC, respectively. The
closed-loop system is unstable using N-DMPC, while it is stable using C-DMPC when = 10,
= 1, and P = 20 (see Figure 6.3).
In conclusion, for the given example, C-DMPC provides larger regions of tuning parameters
than LCO-DMPC. Usually, the stable regions are associated with the big prediction horizon
P and small weight . The C-DMPC can achieve a satisfactory global performance whether
the interactions among subsystems are stronger or not. Furthermore, the cost of computation
is very small as compared with the centralized.

6.3 Distributed Predictive Control based on Pareto Optimality
In this section, we will introduce the DMPC algorithm which could achieve Pareto optimality,
the main idea of this method is that: each subsystem-based MPC communicates with each other
many times a control period and computes the optimal control law through iteration, and each
subsystem-based MPC optimizes the cost of the entire closed-loop system, then improves the
global performance of the closed-loop system. The detail of the DMPC-based Nash optimality
is detailed in the following context.