Page 104 - Distributed model predictive control for plant-wide systems

78 Distributed Model Predictive Control for Plant-Wide Systems

• Apply the first element u (k) of the optimal sequence U (k, M|k) as a control input to the
i
i
physical system S .
i
Step 3. Estimate the future state
• Compute the estimation of the future state trajectory of subsystem S over the horizon P
i
by the following equation:
̂
̃ ̂
X (k + 1, P|k)= S [A ̂x (k|k)+ B U (k, M|k)+ A X(k, P|k − 1)
i
i
i
i
i i
i
̃
+ B U(k − 1, M|k − 1).
i
Step 4. Go to the next time instant
• At time k + 1, let k + 1 → k; then go to Step 1 and repeat the algorithm.
Remark 5.2 According to the fault-tolerant control proposed in [91–93], the LCO-DMPC
control solution is able to manage also eventual subsystem faults. For example, for controller
C , if a fault occurs and leads to a structural or parametric change on the model of sub-
i
system S , by model-based techniques [94], then controllers C detects the occurred fault
i
i
and determines the new configuration of LCO-DMPC and broadcasts the new configura-
tion to its downstreaming neighbors’s controller C ,j ∈ P . The controller C switches to
−i
j
j
a new MPC policy according to the configuration and goes on controlling its correspond-
ing subsystem. Here, the local fault-detection system has to have the functions of detecting
faults, determining the fault types and automatic selecting configuration. Is should also have
the function of informing the downstream neighbor controller’s new configuration. In addi-
tion, the fault-detection algorithm should have the online MPC switching policy which does
not affect the normal operating of the corresponding subsystem-based controller. Finally, the
different MPC policies should be designed primarily, which have to guarantee the stability
of the overall closed-loop system. The different MPC polices for each controller C , j ≠ i,
j
could be obtained by changing the tuning parameters of the subsystem-based MPC in the new
configuration.
5.2.3.6 Computational Complexity
By Theorem 5.1, the resulting computational complexity to obtain the explicit solution for the
local subsystem S is mainly given by the inversion of matrix H . If using a Gauss–Jordan
i
i
algorithm for this task and considering that the size of matrix H is equal to M ⋅ n , the com-
i
u i
3
3
plexity of the inversion algorithm is O(M ⋅ n ). Thus, if the LCO-DMPC is implemented
u i
in a distributed framework, the computational complexity of each subsystem-based MPC in
3
3
each control cycle is O(M ⋅ n ). If the centralized control structure is employed, the compu-
u i
( )
( ) 3
tational complexity of the centralized MPC in each control cycle is O M ⋅ ∑ m n .
3
i=1 u i
Obviously, the computational complexity of the distributed implementation is much less than
the centralized one, especially for the large-scale system where there is a large number of
subsystems.