Page 211 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 185

Table 8.2 Parameters of the LCO-DMPC
Subsystem K i P i Q i R i Δu max u max
i
i
Δu min u min
i i
S 1 −0.35 5.36 4 0.2 ± 1 ± 2
S 2 −0.25 5.35 4 0.2 ± 1 ± 2
S 3 −0.28 5.36 4 0.2 ± 1 ± 2
S 4 −0.43 5.38 4 0.2 ± 1 ± 2

x 1
1
x 2
x 3
x 4
0.5
x i
0

–0.5
5 10 15 20
Time (s)

Figure 8.2 The evolution of the states under the LCO-DMPC

FMINCON has already been provided in MATLAB and it is able to solve multivariable cost
function with nonlinear constraints.
Some parameters of the controllers are shown in Table 8.2. Among these parameters, P is
i
obtained by solving the Lyapunov function. The eigenvalue of each closed-loop system under
T
T
the feedback control shown in the Table 8.2 is 0.5. The eigenvalues of A PA + A PA +
d
o
o
o
T
A PA − Q∕2are {−2.42, − 2.26, − 1.80, − 1.29}, all of which are negative. Thus Assump-
d o
√
tion 10.2 is satisfied. Set = 0.2. Consequently, if x ≤ ∕ N ≤ 0.1, then ‖K x‖ would
‖ ‖
‖ i‖p i i i2
be less than 0.1, and the constraints on the inputs and the increments of inputs, as shown in
Table 8.2, are satisfied. Set the control horizon of all the controllers to be N = 10. Set the initial
presumed inputs and states, at time k = 0, be the solution calculated by a centralized MPC
0
and the corresponding predictive states.
The state responses and the inputs of the closed-loop system are shown in Figures 8.2
and 8.3, respectively.
The states of all four subsystems converge close to zeroes in about 14 sec. The state of S 4
undershoots by 0.05 before converging to zero.
8.5.2 Performance Comparison with the Centralized MPC
To further demonstrate the performance of the proposed DMPC, a dual model centralized MPC
are applied to the system described by (8.40). In what follows, we discuss the performance
comparison with the centralized MPC.