Converting a nonlinear model predictive problem to parametric optimization problem

Question

It is very well known that a linear model predictive control problem \begin{align} \label{eq:linear-original problem} \begin{aligned} &\text{minimize}_{(u_{t})_{t=0}^{N-1}} && \sum_{t=0}^{N-1}\left(x_t^{\top}Qx_t+u_t^{\top}Ru_t\right) +x_N^{\top}Px_N\\ &\hspace{2mm}\mathrm{subject}\;\mathrm{to} && \begin{cases} x_{t+1}=Ax_t+Bu_t\\ x_{t} \in X,\, t=0,\ldots,N-1,\\ u_t \in U,\,t=0,\ldots,N-1,\\ x_N \in X_F,\\ x_0=\bar{x}, \end{cases} \end{aligned} \end{align} can be converted to a multiparametric optimization problem: \begin{align} \begin{aligned} &\text{minimize}_{u} && J^{*}(\bar{x},u)\\ &\hspace{2mm}\mathrm{subject}\,\mathrm{to} && \begin{cases} G(\bar{x},u)\le 0, \end{cases} \end{aligned} \end{align} where $u=\begin{pmatrix}u_0^{\top}\cdots&u_{N-1}^{\top}\end{pmatrix}^{\top}$, $J^{*}(\bar{x},u)=\frac{1}{2}u^{\top}Eu+ \bar{x}^{\top}Fu$ and $G(x,u)= Gu - w -H\bar{x}\le 0$ (see this). I want to do the same for a Nonlinear MPC problem: \begin{align} \begin{aligned} &\text{minimize}_{(u_{t})_{t=0}^{N-1}} && \sum_{t=0}^{N-1}L(x_t,u_t) +V(x_N)\\ &\hspace{2mm}\mathrm{subject}\,\mathrm{to} && \begin{cases} x_{t+1}=f(x_t,u_t)\\ x_{t} \in X,\, t=0,\ldots,N-1,\\ u_t \in U,\,t=0,\ldots,N-1,\\ x_N \in X_F,\\ x_0=\bar{x}, \end{cases} \end{aligned} \end{align} The problem here is the nonlinear dynamics, which does not permit writing things in matrix form and eliminating (x) variables. Can anybody provide ideas or any sources where this is given?

Answer 1

There is no reason to eliminate the $x$-variables in neither the linear nor the nonlinear case. The problem though is that there is no (realistic) computational method to compute the parametric solution in the general nonlinear case.

The closest you have is the special case of piecewise affine dynamics, where you obtain parametric MILPs which is sort of computationally tractable in low dimensions.