Page 107 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 81

Similar to the centralized MPC, the feasibility depends on the possibility of satisfying
Theorem 5.2 for a specific set of parameters of P, M, Q, and R , i = 1, … , m. Here,
i
for simplifying the graphical representations of the results, we choose P = M, R = I ,
u
and Q = I .
y
The three-dimensional graphs of Figures 5.1 and 5.3 show the maximum eigenvalues of the
corresponding closed-loop systems calculated for different combinations of and P, respec-
tively. In Figures 5.1 and 5.3, the Z axis represents the maximum eigenvalues, the X and Y axes
represent the logarithm of and P, respectively. The control performance of the closed-loop
system is plotted in Figures 5.2, and 5.4, where the black dashed lines correspond to the
desired outputs, and the blue solid lines correspond to the system outputs and inputs using
LCO-DMPC.
It can be seen from these figures that the stability depends on the choice of the tuning
parameters and P. For weak interactions, the range of tuning parameters in LCO-DMPC
is acceptable. And a good global performance of the closed-loop system is obtained by the
control of the introduced LCO-DMPC where the subsystems exhibit interactions (see Figures
5.2, and 5.4). The MSEs of outputs with LCO-DMPC are 0.2568 and 0.2277 when = 0.1 and
= 1, respectively.
In conclusion, for the given example, unconstrained LCO-DMPC provides acceptable
regions of tuning parameters. Usually, the stable regions are associated with a big prediction
horizon P and small weight . The LCO-DMPC can achieve a good performance of the
entire closed-loop system when the interactions among subsystems are not very stronger.
Furthermore, the cost of computation is very small as compared with the centralized (see the
analysis at the end of Section 5.2.3).

LCO−DMPC

4
3.5
3
2.5
2
1.5
1
0.5
3
2 30
1 25
0 20
−1 15
log 10 γ −2 10
−3 5 P
Figure 5.1 Maximum closed-loop eigenvalues with LCO-DMPC when = 0.1