Model Predictive Control System Design and Implementation Using ...

1.3 Predictive Control within One Optimization Window 13
Solution. From (1.14), by adding and subtracting the term
(R s − Fx(k i )) T Φ(Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i ))
to the original cost function J, its value remains unchanged. This leads to
J =(R s − Fx(k i )) T (R s − Fx(k i ))
{ }} {
−2ΔU T Φ T (R s − Fx(k i )) + ΔU T (Φ T Φ + ¯R)ΔU
{ }} {
+ (R s − Fx(k i )) T Φ(Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i ))
− (R s − Fx(k i )) T Φ(Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i )), (1.23)
where the quantities under the {}}{ . are the completed ‘squares’:
J 0 = ( ΔU − (Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i )) ) T
× (Φ T Φ + ¯R) ( ΔU − (Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i )) ) . (1.24)
This can be easily verified by opening the squares. Since the first and last
terms in (1.23) are independent of the variable ΔU (sometimes, we call this
a decision variable), and (Φ T Φ + ¯R) is assumed to be positive definite, then
the minimum of the cost function J is achieved if the quantity J 0 equals zero,
i.e.,
ΔU =(Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i )). (1.25)
This is the optimal control solution. By substituting this optimal solution into
the cost function (1.23), we obtain the minimum of the cost as
J min =(R s − Fx(k i )) T (R s − Fx(k i ))
− (R s − Fx(k i )) T Φ(Φ T Φ + ¯R) −1 Φ T (R s − Fx(k i )).
1.3.3 MATLAB Tutorial: Computation of MPC Gains
Tutorial 1.2. The objective of this tutorial is to produce a MATLAB function
for calculating Φ T Φ, Φ T F , Φ T ¯Rs . The key here is to create F and Φ matrices.
Φ matrix is a Toeplitz matrix, which is created by defining its first column,
and the next column is obtained through shifting the previous column.
Step by Step
1. Create a new file called mpcgain.m.
2. The first step is to create the augmented model for MPC design. The
input parameters to the function are the state-space model (A p ,B p ,C p ),
prediction horizon N p and control horizon N c . Enter the following program
into the file: