Page 203 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 177

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
• Solve Problem 8.1for u (k|k) and apply u (k|k)
i i
• Compute ̂ x (k + s + 1|k + 1) according to (8.4) and transmit ̂ x (k + s + 1|k + 1) to S , j ∈
i i j
P .
−i
Step 4: Update of control at time k + 1
Let k + 1 → k, repeat Step 2.
Algorithm 8.1 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
occurs synchronously only when x(k) ∈Ω( ), with Ω( ) being as defined in Lemma 8.1. In the
next section, it will be shown that the constraint LCO-DMPC policy drives the state x(k + s)to
Ω( ) in a finite number of updates. As a result, if Ω ( ) is chosen sufficiently small, then MPC
i
can be employed for all time without switching to a terminal controller, eliminating the need
of the local controllers to access the full state. Of course, in this case, instead of asymptotic
stability at the origin, we can only drive the state toward the small set Ω( ).
The analysis in the next section shows that the constraint LCO-DMPC algorithm is feasible
at every update and is stabilizing.

8.4 Analysis
8.4.1 Recursive Feasibility of Each Subsystem-based Predictive Control

The main result of this section is that, provided that an initially feasible solution is available and
f
Assumption 8.3 holds true, for any S and at any time k ≥ 1, u (⋅|k)= u (⋅|k) is a feasible con-
i i i
trol solution to Problem 1. This feasibility result refers that, for any S and at any update k ≥ 1,
i
f
f
the control and state pair (u (⋅|k), x (⋅|k)), j ∈ P satisfy the consistency constraints (8.10) and
i i i
(8.11), the control constraint (8.13), and the terminal state constraint (8.14). Theorem 8.2 iden-
′
′
tifies sufficient conditions that ensure ̂ x (k + N|k)∈Ω ( ∕2), where = (1 − ) . Lemma 8.2
i i
‖ f ‖ √
identifies sufficient conditions that ensure ‖x (s + k|k) − ̂ x (s + k|k)‖ ≤ ∕(2 m) for all
i
‖ i ‖P i
i ∈ P. Lemma 8.3 establishes that the control constraint is satisfied. Finally, the results in
Lemma 8.2– 8.4 are combined to arrive at the conclusion that, for any i ∈ P, the control and
f
f
state pair (u (⋅|k), x (⋅|k)) are a feasible solution to Problem 8.1 at any update k ≥ 1.
i i
Lemma 8.2 Suppose Assumptions 8.1–8.3 hold and x(k )∈ X. For any k ≥ 0, if Problem 8.1
0
has a solution at time k − 1, and ̂ x (k + N − 1|k − 1)∈Ω ( ∕2) for any i ∈ P, then
i i
̂ x (k + N − 1|k)∈Ω ( ∕2)
i i
and
′
̂ x (k + N|k)∈Ω ( ∕2)
i i
Provided P and Q satisfy that
̂
i
i
max( ) ≤ 1 − (8.18)
i
i∈P
√
′ −1
−1 T ̂
̂
where, = (1 − ) , = max (Q P ) Q P .
i i
i i