Page 101 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 75

5.2.3.3 Quadratic Program Tansformation
The N-DMPC problem stated in Problem 5.1 will now be formulated as a quadratic
program, and then using the minimum principle to obtain the explicit solution of the
unconstrained LCO-DMPC.
To simplify, we give the result directly, and the deducing procedure is detailed in Appendix
A at the end of this chapter. The readers can also refer to [46].

Problem 5.2 (Quadratic program). Under Assumption 5.1, for each subsystem-based
controller C , i = 1, … , m, the MPC optimization Problem 5.1 at time k can be transferred to
i
the following quadratic program problem:
T
T
min [ U (k, M|k)H U (k, M|k)− G (k + 1, P|k)ΔU (k, M|k)] (5.24)
i
i
i
Δu i (k, M|k) i i
where the positive definite matrix H has the form
i
T
H ≜ N Q N + R (5.25)
i i i i i
d
T
G (k + 1, P|k) ≜ 2N Q [Y (k + 1, P|k)− Z (k + 1, P|k)] (5.26)
̂
i i i i i
′
̂
Z (k + 1, P|k) ≜ S [B u (k − 1)+ A ̂ x (k|k)+ W (k, P|k − 1)]
̂
i i i i i i i i
+ T V (k, P|k − 1) (5.27)
̂
i
i
where
S ≜ C S
i
i i
N ≜ S B
i i i
i
(5.28)
Q ≜ diag {Q }
i P i
R ≜ diag {R }
i
i
P
I
⎡ n u i ⎤
′ ≜ ⎢ ⋮ ⎥
i ⎢ ⎥
(M block) ⎢ ⎥
⎣I ⎦
n u i
(5.29)
I ··· 0
⎡ n u i ⎤
≜ ⎢ ⋮ ⋱ ⋮ ⎥
i ⎢ ⎥
(M×M blocks)
⎢ ⎥
⎣I ··· I ⎦
n u i n u i
where Q and R are the weight matrices of the cost function (5.5).
i i
In this way, the subsystem-based MPC Problem 5.1 has been transformed into an equiv-
alent unconstrained QP Problem 5.2 which has to be locally solved online at each sampling
instant.