# Why the plant in MPC is assumed to be strictly proper?

The model of plant in Model Predictive Control (MPC) is assumed to be strictly proper (Wang 2009: page 4).

\begin{aligned} & x_m(k+1)= A_m x_m(k)+B_mu(k)\\ & y(k)=C_m x_m(k) \end{aligned}

However, no convincing reason has been mentioned for why should $D_m=0$.

..., due to the principle of receding horizon control, where a current information of the plant is required for prediction and control, we have implicitly assumed that the input $u(k)$ cannot affect the output $y(k)$ at the same time. Thus, $D_m = 0$ in the plant model.

So, what happens if $D_m\ne0$? If so, does it mean MPC controller will not be able to optimize?

No, there is no requirement that $D$ is zero in general in MPC. I would assume they have this to simplify when talking about state-estimation etc in combination with MPC, as you then don't have the complication that the measurement-update of the state estimate, which is used for the computation of the input, depends on the input, i.e. a catch-22 situation.
• Thank you Johan. Is there any reference which has handled MPC with $D\ne0$? – Adams May 17 '18 at 6:45
• I would switch around that and say that I've never seen anyone working explicitly with that requirement before. It is not related to the MPC setting really and really makes no difference when setting up the MPC problem, but more an issue with observer implementation. In practice, $D$ is typically 0 anyway. – Johan Löfberg May 17 '18 at 7:42