Page 58 - Distributed model predictive control for plant-wide systems

32 Distributed Model Predictive Control for Plant-Wide Systems

3. Input magnitude constraint: u min ≤ u(k + l|k) ≤ u max .
We can let the optimization problem satisfy the following constraint:

U ≤ SΔU(k|k)+ u(k − 1) ≤ U (2.33)
min max
where

[ T T T ] T
U = u u ··· u
min min min min
[ T T T ] T
U max = u max u max ··· u max
⎡ I 0 ··· 0⎤
⎢I I ⋱ ⋮⎥
S = ∈ ℝ Mn u ×Mn u
⎢ ⋮ ⋱ ⋱ 0 ⎥
⎢ ⎥
⎣I ··· I I⎦
Equations (2.30)–(2.33) can be written in a uniform form as C ΔU(k|k) ≤ b, where C and
u
u
b are the matrix and vector known at time k, respectively. The MPC optimization problem
considering these constraints can be written as follows.
Problem 2.2
2 2
min J(k)= ‖Y(k|k)− Y (k|k)‖ + ‖ΔU(k|k)‖ R
r
Q
ΔU(k|k)
s.t. C ΔU(k|k) ≤ b (2.34)
u
Problem (2.34) is a quadratic optimization problem. The feedback law solution to the
constrained quadratic optimization problem is, in general, nonlinear. In the MATLAB MPC
Toolbox, for an MPC optimization problem for constrained systems, one can adopt “cmpc.”
The above constraint DMC algorithm can be summarized as follows.

Algorithm 2.2 State Space MPC algorithm
Step 0. Obtain system model (2.21). Choose L.
Step 1. At k = 0,
• measure y(0);
• determine Y (0);
r
• solve Problem 2.2 to calculate Δu(0);
• implement u(0) = u(0) +Δu(0).
Step 2. At time k > 0,
• measure y(k);
• determine Y (k);
r
• solve Problem 2.2 to calculate Δu(k);
• implement u(k) = u(k − 1) +Δu(k).
Step 3. Let at time k + 1. Let k + 1 → k,gotoStep2.