LinearMPC.jl

Documentation under development

The documentation for LineaerMPC.jl is currently under development

The focus of LinearMPC.jl is Model Predictive Control (MPC) of linear systems. The aim of the package is to produce high-performant and lightweight C-code that can easily be used on embedded systems, while at the same time give a user-friendly and expressive development environment for MPC. The package supports code generation for the Quadratic Programming solver DAQP, and for explicit solutions computed by ParametricDAQP.jl.

Model Predictive Control

In MPC, an optimal control decision is computed at every sampling instance by solving an optimization probelm. On a high-level, the optimization problems solved are of the form

\[\begin{aligned} &\underset{u_0,\dots,u_{N-1}}{\text{minimize}}&& \textcolor{purple}{\frac{1}{2}\sum_{k=0}^{N-1} {\left((Cx_{k}-r)^T Q (C x_{k}-r) + u_{k}^T R u_{k} + \Delta u_{k}^T R_r \Delta u_k\right)}}\\ &\text{subject to} &&\textcolor{blue}{{x_{k+1} = F x_k + G u_k}}, \quad k=0,\dots, N-1\\ &&& \textcolor{red}{x_0 = \hat{x}} \\ &&& \textcolor{green}{\underline{b} \leq A_x x_k + A_u u_k \leq \overline{b}}, \quad k=0, \dots, N-1 \end{aligned}\]

where an $\textcolor{purple}{\text{objective}}$ is minimized , subject to a $\textcolor{blue}{\text{dynamical system}}$ that is simulated over a horizon $N$, $\textcolor{red}{\text{starting from}}$ an estimate $\hat{x}$ of the current state. Additionally, $\textcolor{green}{\text{constraints}}$ like actuator limits and state constraints are accounted for. The objective is comprised by the deviation of an output $y= Cx$ from a reference value $r$, the control effort ($u^T R u$) , and the change of the control action $\Delta u^T R_r \Delta u$.

LinearMPC.jl generates a condensed problem by eliminating the equality constraint. The resuliting optimization problem is a dense Quadratic Program (QP). LinearMPC.jl uses the QP sovler DAQP, a dual active-set solver that has been specialized to solved such problems.

The solution map for the optimization problem is a piecewise affine function over polyhedral regions. LinearMPC.jl supports the computation of such explicit solutions by interfacing the multi-parameteric QP solver ParametricDAQP.jl

Why LinearMPC.jl?

Code generation of high-performant, allocation-free, library-free, and lightweight C-code that can be embedded on any micro controller.
State-of-the-art computation of explicit solutions (~100x faster than other software packages)
Tools to determine real-time certificates of the complexity of the solver, allowing for MPC in with guarantees on the memory and computational requirements before deploying the solver.

Why not LinearMPC.jl?

As its name suggests, the package is specialized for MPC for linear systems. If a linear (or linearized) model does not suffice for your use case, consider the following packages.

ModelPredictiveControl.jl - A high-level MPC packages in Julia. While it does not generate embeddable C-code, it is an excellent package during development of MPC controllers.
acados - provides fast and embedded solvers for nonlinear optimal control, specifically designed for real-time applications and embedded systems. This is the current state-of-the art if you are interested in real-time nonlinear MPC on embedded systems.
Are you more of a Python person? You can still use LinearMPC.jl through its sister package lmpc.