Page 199 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 173

Then, it follows that
T
̂
A PA − P =−Q
d d
T
̂
where Q = Q + K RK > 0.
p
To get the predicted state sequence x (k + s|k) of subsystem S under the control decision
i
i
sequence of u (k + s|k) in (8.3), the system evolution model should be deduced first. Since
i
every subsystem-based controller is updated synchronously, the state and control sequences
of other subsystems are unknown to subsystem S . Thus, at the time instant k, the presumed
i
state sequence {̂ x (k|k), ̂ x (k + 1|k), ··· , ̂ x (k + N|k)} of S is used in the predictive model of
j j j j
the MPC in S , which is given as
i
l
∑
p l p l−h
x (k + l|k)= A x (k|k)+ A B u (k + l|k)
i ii i ii ii i
h=1
l
∑ ∑ l−h
+ A A ̂ x (k + h − 1|k) (8.4)
ii ij j
j∈P +i h=1
p
Given x (k|k)= x (k|k), the presumed control sequence for subsystem S for any i ∈ P is
i i i
given by
{
u (k + s − 1|k − 1), s = 1, 2, … , N − 1
i
̂ u (k + s − 1|k)= (8.5)
i p
K x (k + N − 1|k − 1), s = N
i i
Set each presumed state sequence ̂ x to be the remainder of the sequence predicted at time
i
instant k − 1, concatenated with the closed-loop response under the state feedback control
p
̂ u (k + s − 1|k)= K x (k + N − 1|k − 1), that is
i i i
p
⎧ ̂ x (k + s − 1|k) = x (k + s − 1|k − 1), s = 1, 2, … , N,
i i
⎪
p
⎪ ̂ x (k + N + 1 − 1|k)= A x (k + N − 1|k − 1)
i di i (8.6)
⎨
∑ p
+ A x (k + N − 1|k − 1)
⎪
⎪ ij j
⎩ j∈P +i
It should be noticed that since ̂ x (k + N|k) is only a media viable here, ̂ x (k + N|k) is not equal
i i
to the solution of substituting ̂ u (k + N − 1|k) into (8.4) based on the presumed state sequence
i
obtained at time k − 1.
In MPC systems, it is an important proposition to focus on the systems sequence feasibility
and the stability. The same things happen in the distributed MPC systems. To enlarge the
feasible region, a terminal state constraint is included in each subsystem-based MPC, which
guarantees that the terminal controllers are stabilizing inside a terminal set. To define this
terminal state set, we need to make an assumption and establish a technical lemma.
Assumption 8.2 The block-diagonal matrix
A = block-diag{A , A , … , A }
d d1 d2 dm
and the off-diagonal matrix A = A − A satisfy the following inequality:
o
c
d
T
T
T
A PA + A PA + A PA < Q∕2
̂
o o o d d o
T
̂
where Q = Q + K RK > 0