Linear-Quadratic MPC Examples

2-DOF double integrator

Let control a 2-DOF double integrator with an MPC controller with a prediction horizon of one second and quadratic running cost penalizing the state-error and input level.

We will use the first order approximation $${\mathbf{x}}_{k+1}=\underset{\mathbf{A}}{\underbrace{(\mathbf{I}+{\mathbf{A}}_{c}h)}}{\mathbf{x}}_k+\underset{\mathbf{B}}{\underbrace{{\mathbf{B}}_{c}h}}{\mathbf{u}}_k$$ with a timestep of $10^{-2}$s to discretize its continuous dynamics $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{u})=\underset{{\mathbf{A}}_c}{\underbrace{\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}\mathbf{x}+\underset{{\mathbf{B}}_c}{\underbrace{\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]}}\mathbf{u}.$$

We will formulate the running cost as $$l({\mathbf{x}}_k,{\mathbf{u}}_k)={\mathbf{x}}_k^{\top }\mathbf{Q}{\mathbf{x}}_k+{\mathbf{u}}_k^{\top }\mathbf{R}{\mathbf{u}}_k$$ and place additional constraints $$-2\le {\mathbf{u}}_k\le 2$$ on the controllers inputs.

Code

• Direct
  • julia
    • JuMP
    • OSQP
  • python
    • CVXPY
    • OSQP