Page 103 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 77

U (k, M|k) is used only by the controllers S , j ∈ P −i ; therefore, in the real implementation
i
j
of the LCO-DMPC, each subsystem-based controller C can build its U(k, M|k) with nonzero
i
blocks U (k − 1, M|k − 1), j ∈ P and with zero blocks U(k, M|k) in the other parts.
j +i
Consider that only the first element of U (k, M|k), that is, u (k), i = 1, … , m, is applied at
i i
time k to subsystem S , and it is only this element that defines the feedback control law at time
i
k. Therefore, the feedback control law of controller C can also be expressed as
i
u (k)= U (k, M|k), i = 1, … , m (5.38)
i i i
and the complete feedback control law computed by all controllers as
u(k)= U(k, M|k) (5.39)

So far, the stacked control sequence over the control horizon M computed by controller
C and the complete stacked control sequence formed by all these stacked control sequences
i
computed by all controllers C ,i = 1, … , m, are provided in Theorem 5.1, the elements up to
i
time k + M − 1 of which is broadcasted to its downstream neighborhood S , j ∈ P by the
j −i
subsystem-based controller C and is used to evaluate the interactions among the subsystems,
i
and then to get the complete stacked control sequence U(k, M|k) for the next control step.
5.2.3.5 Unconstrained LCO-DMPC Algorithm

d
Algorithm 5.1 Let the desired output Y (k + 1, P|k) is provided; then the LCO-DMPC
i
algorithm for the subsystem-based MPC controller C at each time instant k is as follows.
i
Step 1. Communication and interaction calculation
̂
• Sent U (k − 1, M|k − 1) and X (k, P|k − 1) to its downstream neighbors’s controller C ,
i
j
i
j ∈ P .
−i
̂
• Get the estimation of the future state trajectories X (k, P|k − 1) and control inputs
j
U (k − 1, M|k − 1) from its upstream neighbors’ controller C , j ∈ P through network
j j +i
information exchange.
d
• Set the desired trajectory Y (k + 1, P|k) over the horizon P according to the MPC’s
configuration.
• Get the measurement of x (k) through field instruments or a designed observer.
i
̂
̂
• Build X(k, P|k − 1) and U(k, P|k) by combining the local state trajectory X (k, P|k − 1)
i
and control input U(k, P|k) with the acquired upstream neighbors’ information of
̂
X (k, P|k − 1) U (k − 1, M|k − 1), j ∈ P , and compute the corresponding predictions of
j
+i
j
the interactions:
̃ ̂
̃
̂
W (k, P|k − 1)= A X(k, P|k − 1)+ B U(k − 1, M|k − 1)
i
i
i
̂
̂ ̂
V (k, P|k − 1)= C X(k, P|k − 1)
i
i
Step 2. Compute control law and apply it
• Compute the optimal control sequence according to Equation (5.32), that is,
′
d
U (k, M|k)= u (k − 1)+ K [Y (k + 1, P|k)− Z (k + 1, P|k)]
̂
i 1 i i i i i