Page 52 - Distributed model predictive control for plant-wide systems

26 Distributed Model Predictive Control for Plant-Wide Systems

where C and b are the matrix and vector, respectively, provided at time k.The DMC
optimization problem considering these constraints can be written as follows.

Problem 2.1
2 2
min J(k)= ‖E(k|k)‖ + ‖ΔU(k|k)‖
ΔU(k|k) ̃ W ̃ R
(2.20)
s.t. CΔU(k|k) ≤ b
Problem (2.20) is a quadratic optimization problem. The feedback law solution to the con-
strained quadratic optimization problem is, in general, nonlinear. In MATLAB MPC Toolbox,
for the DMC optimization problem for constrained systems, one can adopt “cmpc.”
The above constraint DMC algorithm can be summarized as follows.

Algorithm 2.1 DMC Algorithm
Step 0. Obtain step response coefficient matrix (2.3). Choose F.
̃
Step 1. At k = 0,
• measure y(0);
• determine ̃ y (0);
r
• choose y (i|0), i ∈ {1, 2, … , P} and construct Y (0|0)
0 0
• solve Problem 2.1 to calculate Δu(0);
• implement u(0) = u(0) +Δu(0).
Step 2. At time k > 0,
• measure y(k);
• determine ̃ y (k);
r
• use (2.11) to calculate Υ(k);
• use (2.15) to calculate Y (k|k);
0
• solve Problem 2.1 to calculate Δu(k);
• implement u(k) = u(k − 1) +Δu(k).
Step 3. At time k + 1, let k + 1 → k, and go to Step 2.

2.3 Predictive Control with the State Space Model
In the above section, we introduced the DMC algorithm, which is based on the step response
model. In this section, we will present the MPC algorithm which is based on the state space
model, since, so far, this kind of MPC is a most discussed and studied MPC method in the
literature works. And this part of content mainly refers to [2, 26, 74].
For the basic formulation of predictive control and making the formulation useful in the real
world, we shall assume that:

• the plant model is linear time invariant,
• there are measurement disturbances in the plant model. Handling feed forwarding measure-
ment disturbances is very important in distributed predictive control,
• the output variables can be measured,
• the real plant is governed by the same equations as the model, although, in most cases, this
is not really true,