Control Formulation

To solve the control problem, at each time step an optimization problem is solved for a control move that minimizes the following objective function:

$$\begin{array}{c} \min \\ i=k...k+p \end{array}\vert\vert\Gam... ...rt\vert _{2}+\vert\vert\Gamma _{u}\, \Delta u(i)\vert\vert _{2}$$

where r(i) is the reference at time i for the process measurements y(i). The process inputs are given as u(i) and is the difference between u(i) and u(i-1). The values for the process input beyond a point m in the horizon are assumed constant:

m(i+m)=m(i+m+1)=...=m(i+p)

The values and are matrices that can be used to scale and weight process outputs and changes in process inputs. The values can express preference for control of one measurement over another. The values suppress chatter and extreme moves in the calculated process inputs.

The formulation takes advantage of a prediction of the future process outputs.

y(i)=y(i-1)+Mu(i)+M(i)+f_d(y_m(k)-y(k))

Here, M is the impulse response matrix relating u to y. M_d is a model relating the disturbances d to the process outputs y. The model error, or disturbance term, at the current time step relating the actual measurement, y_m, to the modeled value y(k) can be multiplied by filter f_d in order to reduce noise effects.