Page 244 - Distributed model predictive control for plant-wide systems

218 Distributed Model Predictive Control for Plant-Wide Systems

Comparing with the method proposed in Chapter 8, both the optimization index and the
consistent constraints are different. In Problem 10.1, the constraints in (10.13) are necessary
since the estimation error cannot be expressed by the states sequence. In addition, the terminal
constraint should bound both the final states of corresponding subsystem but also that of the
subsystems it directly impacted on.

10.3.2 Algorithm Design for Resolving Each Subsystem-based Predictive
Control
Before stating the N-DMPC algorithm, we make the following assumption to facilitate the
initialization phase.
p
Assumption 10.3 At initial time k , there exists a feasible control u (k + s)∈ U ,
i
0
0
i
s = 1, 2, … , N − 1, for each i ∈ P, such that the solution to the full system x(s + 1 + k ) =
0
p
p
p
Ax(s + k ) + Bu (s + k ), denoted as x (⋅|k ), satisfies x (N + k )∈Ω ( ∕2) and results in a
0
0
0
i
0
i
i
p
bounded cost J (k ). Moreover, each subsystem has access to u (⋅|k ).
0
i
0
i
Assumption 10.3 bypasses the difficult task of actually constructing an initially feasible
solution in a distributed way. In fact, finding an initially feasible solution for many optimiza-
tion problems is often a primary obstacle, whether or not such problems are used in a control
setting. As such, many centralized implementations of MPC also assume that an initially fea-
sible solution is available.
Algorithm 10.1 (Constraint N-DPC Algorithm) The dual mode N-DPC law for any S is
i
constructed as follows:
Step 1: Initialization
p
• Initialize x(k ), u (k + s|k ), s = 1, 2, … , N, to satisfy Assumption 10.3
0
0
0
i
• At time k ,if x(k ) ∈Ω( ), then apply the terminal controller u (k) = K (x (k)), for all
i
i
i
0
0
k ≥ k ,else
0
• Compute ̂ x (k + s + 1|k + 1) according to (10.5) and transmit ̂ x (k + s + 1|k + 1)=
j,i
0
0
0
0
i
̂ x (k + s + 1|k + 1) to S , a ⊂ P ∪ P and ̂ u (k + s|k + 1) to S , j ∈ P
i,i 0 0 a +−i −i i 0 0 j +i
Step 2: Communicating at time k + 1
• Communication: Measure x (k), transmit x (k)to S , and receive ̂ x (k), from S
i i i i i
• Transmit ̂ x (k + s + 1|k)= ̂ x (k + s + 1|k) to S , a ⊂ P ∪ P and ̂ u (k + s|k)to
i i,i a +−i −i i
j +i i j −i
S , j ∈ P ; receive ̂ x̃ i (k + s + 1|k) and ̂ u (k + s|k)from S , j ∈ P
Step 3: Update of control law at time k + 1
•If x(k) ∈Ω( ), then apply the terminal controller u (k) = K (x (k)), else
i
i
i
p p
• Solve Problem 10.1 for u (k|k) and apply u (k|k)
i i
• Compute ̂ x (k + s + 1|k + 1) according to (10.5) and
j,i
Step 4: Update of control at time k + 1
•Let k + 1 → k, repeat Step 2.
Algorithm 10.1 presumes that all local controllers C , i ∈ P, have access to the full state x(k).
i
This requirement results solely from the use of the dual mode control, in which the switching