lmpc is a Python package for Model Predictive Control (MPC) of linear systems, brining the functionality of the Julia package LinearMPC.jl to Python. It aims to provide a user-friendly experience, while simultaneously being able to generate high-performant and lightweight C-code that can easily be used on embedded systems. The package supports code generation for the Quadratic Programming solver DAQP, and for explicit solutions computed by ParametricDAQP.jl.

A simplified version (see the, soon to be released, documentation for a more complete formulation) of the solved problem is

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

where $\hat{x}$ is the current state and $r$ is the desired reference value of $Cx$.

Installation

pip install lmpc

Example

The following code show a simple MPC example of controlling an inverted pendulum on a cart, inspired by this example in the Model Predictive Toolbox in MATLAB.

import numpy
from lmpc import MPC,ExplicitMPC

# Continuous time system dx = A x + B u
A = numpy.array([[0, 1, 0, 0], [0, -10, 9.81, 0], [0, 0, 0, 1], [0, -20, 39.24, 0]])
B = 100*numpy.array([0,1.0,0,2.0])
C = numpy.array([[1.0, 0, 0, 0], [0, 0, 1.0, 0]])


# create an MPC control with sample time 0.01, prediction horizon 10 and control horizon 5 
Np,Nc = 10,5
Ts = 0.01
mpc = MPC(A,B,Ts,C=C,Nc=Nc,Np=Np);

# set the objective functions weights
mpc.set_objective(Q=[1.44,1],R=[0.0],Rr=[1.0])

# set actuator limits
mpc.set_bounds(umin=[-2.0],umax=[2.0])

A control, given the state x and reference value r, is computed with

u = mpc.compute_control(x=[0,0,0,0],r=[1,0])

Embeddable C-code for the MPC controller is generated with the command

mpc.codegen(dir="codgen_dir")

which produce allocation-free C-code in the directory codegen_dir for setting up optimization problems and solving them with the Quadratic Programming solver DAQP.

The C-function mpc_compute_control(control, state, reference, disturbance) computes the optimal control, given the current state, reference, and measured disturbances disturbance, which are all floating-point arrays. The optimal control is stored in the floating-point array control.

Citation

If you find the package useful, consider citing one of the following papers, which are the backbones of the package:

@article{arnstrom2022daqp,
  author={Arnström, Daniel and Bemporad, Alberto and Axehill, Daniel},
  journal={IEEE Transactions on Automatic Control},
  title={A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive {LDL}$^{T}$ Updates},
  year={2022},
  volume={67},
  number={8},
  pages={4362-4369},
  doi={10.1109/TAC.2022.3176430}
}

@inproceedings{arnstrom2024pdaqp,
  author={Arnström, Daniel and Axehill, Daniel},
  booktitle={2024 IEEE 63rd Conference on Decision and Control (CDC)}, 
  title={A High-Performant Multi-Parametric Quadratic Programming Solver}, 
  year={2024},
  volume={},
  number={},
  pages={303-308},
}