Page 179 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Information Structure Constraints 153

If the convergent condition (7.58) is satisfied, the optimal instant control law of the whole
system at the time instant k is
∗
∗
Δu (k)= LΔU (k)= W[R (k)− Ĝ x(k)] (7.66)
M P
where
⎛ ⎞
⎜ ⎟
L = block-diag L ··· L 0⎟
⎜ 0
⎜⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟⎟
⎝ m ⎠
[ ]
L = I
0 n ui n ui ×(M−1)n ui
−1
T
T
W = L(H QH + R) H Q
The whole closed-loop system with the networked MPC control and predictive state
observer can be described as
[ ] [ ][ ] [ ]
x(k + 1) A − BWG BWG x(k) BW
= + R (k) (7.67)
P
̃ x(k + 1) (I − VC) A ̃ x(k)
The nominal stability of the whole closed-loop system can be guaranteed if only if
{[ ]}
| |
| A − BWG BWG |
| j | < 1, ∀j = 1, … , 2Nn y (7.68)
| (I − VC) A |
| |
That is, the eigenvalues of the above matrix are all in the unit circle.
Remark 7.3 It has been noticed that the convergence of N-MPC Algorithm 7.1 is local,
that is, whether the distributed computation is convergent is only related to the current sam-
pling time instant. While the stability analysis in this section is global, the convergence of the
distributed computation and stability for distributed control systems are concerned during the
whole receding horizon.

7.3.6 Simulation Study
A simple simulation example is illustrated to verify the optimality of the N-DMPC interative
Algorithm 7.1, and the N-DMPC Algorithm 7.1 is applied to the fuel feed flow control for the
walking beam reheating furnace in this section.

7.3.6.1 Illustrative Example
Consider the following two-input two-output process:
1 2
⎛ ⎞
( ) ( )
y 1 = ⎜ 10 s + 1 11 s + 1 ⎟ u 1
y 2 ⎜ 1 1.5 ⎟ u 2
⎜ − ⎟
⎝ 8 s + 1 9 s + 1 ⎠