Direct Linear-Quadratic MPC

The standard realization of an MPC controller is such that solves the optimization problem $$\begin{array}{rl}\underset{\{{\mathbf{x}}_k{\}}_{k=1}^{N+1},\{{\mathbf{u}}_k{\}}_{k=0}^N}{\min }& {\mathbf{x}}_k^{\top }\mathbf{Q}{\mathbf{x}}_k+{\mathbf{u}}_k^{\top }\mathbf{R}{\mathbf{u}}_k\\ \text{s.t.}& {\mathbf{x}}_{k+1}=\mathbf{A}{\mathbf{x}}_k+\mathbf{B}{\mathbf{u}}_k\\ & {\mathbf{u}}_{\min }\le {\mathbf{u}}_k\le {\mathbf{u}}_{\max },\end{array}$$ where the initial state ${\mathbf{x}}_0$ is given¹.

Canonical QP form

Optimized variables

$$\begin{array}{c}\mathbf{x}={\left[\begin{array}{cccccccc}{\mathbf{u}}_0^{\top }& \dots & \dots & {\mathbf{u}}_N^{\top }& {\mathbf{x}}_1^{\top }& \dots & \dots & {\mathbf{x}}_{N+1}^{\top }\end{array}\right]}^{\top }\end{array}$$

Objective function's terms

$$\begin{array}{c}\mathbf{P}=\left[\begin{array}{cccccccc}\mathbf{R}& & & & & & & \\ & \ddots & & & & & & \\ & & \ddots & & & & & \\ & & & \mathbf{R}& & & & \\ & & & & \mathbf{Q}& & & \\ & & & & & \ddots & & \\ & & & & & & \ddots & \\ & & & & & & & \mathbf{Q}\end{array}\right],\mathbf{q}=\left[\begin{array}{c}0\\ ⋮\\ ⋮\\ 0\\ 0\\ ⋮\\ ⋮\\ 0\end{array}\right]\end{array}$$

Constraint's terms

$$\begin{array}{c}\mathbf{l}=\left[\begin{array}{c}-\mathbf{A}{\mathbf{x}}_0\\ 0\\ ⋮\\ 0\\ {\mathbf{u}}_{\min }\\ ⋮\\ ⋮\\ {\mathbf{u}}_{\min }\end{array}\right],\mathbf{A}=\left[\begin{array}{cccccccc}\mathbf{B}& & & & -\mathbf{I}& & & \\ & \mathbf{B}& & & \mathbf{A}& -\mathbf{I}& & \\ & & \ddots & & & \ddots & \ddots & \\ & & & \mathbf{B}& & & \mathbf{A}& -\mathbf{I}\\ \mathbf{I}& & & & & & & \\ & \ddots & & & & & & \\ & & \ddots & & & & & \\ & & & \mathbf{I}& & & & \end{array}\right],\mathbf{u}=\left[\begin{array}{c}-\mathbf{A}{\mathbf{x}}_0\\ 0\\ ⋮\\ 0\\ {\mathbf{u}}_{\max }\\ ⋮\\ ⋮\\ {\mathbf{u}}_{\max }\end{array}\right]\end{array}$$