History

Shunichi09 4b8ff4c0ac Done v1.0		2020-04-02 13:40:42 +09:00
..
README.md	add	2018-12-29 16:00:19 +09:00
animation.py	add ACC.py of mpc	2018-12-29 15:49:04 +09:00
main_ACC.py	add ACC.py of mpc	2018-12-29 15:49:04 +09:00
main_ACC_TEMP.py	modify mpc	2019-02-07 18:08:17 +09:00
main_example.py	Done v1.0	2020-04-02 13:40:42 +09:00
mpc_func_with_cvxopt.py	add ACC.py of mpc	2018-12-29 15:49:04 +09:00
mpc_func_with_scipy.py	add ACC.py of mpc	2018-12-29 15:49:04 +09:00
test_compare_methods.py	add ACC.py of mpc	2018-12-29 15:49:04 +09:00

README.md

Model Predictive Control Basic Tool

This program is about template, generic function of linear model predictive control

Documentation of the MPC function

Linear model predicitive control should have state equation. So if you want to use this function, you should model the plant as state equation. Therefore, the parameters of this class are as following

class MpcController()

Attributes :

A : numpy.ndarray
- system matrix
B : numpy.ndarray
- input matrix
Q : numpy.ndarray
- evaluation function weight for states
Qs : numpy.ndarray
- concatenated evaluation function weight for states
R : numpy.ndarray
- evaluation function weight for inputs
Rs : numpy.ndarray
- concatenated evaluation function weight for inputs
pre_step : int
- prediction step
state_size : int
- state size of the plant
input_size : int
- input size of the plant
dt_input_upper : numpy.ndarray, shape(input_size, ), optional
- constraints of input dt, default is None
dt_input_lower : numpy.ndarray, shape(input_size, ), optional
- constraints of input dt, default is None
input_upper : numpy.ndarray, shape(input_size, ), optional
- constraints of input, default is None
input_lower : numpy.ndarray, shape(input_size, ), optional
- constraints of input, default is None

Methods:

initialize_controller() initialize the controller
calc_input(states, references) calculating optimal input

More details, please look the mpc_func_with_scipy.py and mpc_func_with_cvxopt.py

We have two function, mpc_func_with_cvxopt.py and mpc_func_with_scipy.py Both functions have same variable and member function. However the solver is different. Plese choose the right method for your environment.

example of import

from mpc_func_with_scipy import MpcController as MpcController_scipy
from mpc_func_with_cvxopt import MpcController as MpcController_cvxopt

Examples

Problem Formulation

first order system

$\frac{d}{dt} \boldsymbol{X} = \begin{bmatrix} -1/ \tau & 0 & 0 & 0\\ 0 & -1/ \tau & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} v_x \\ v_y \\ x \\ y \end{bmatrix} + \begin{bmatrix} 1/ \tau & 0 \\ 0 & 1/ \tau \\ 0 & 0 \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} u_x \\ u_y \\ \end{bmatrix} = \boldsymbol{A}\boldsymbol{X} + \boldsymbol{B}\boldsymbol{U}$

ACC (Adaptive cruise control)

The two wheeled model are expressed the following equation.

$\frac{d}{dt} \boldsymbol{X}= \frac{d}{dt} \begin{bmatrix} x \\ y \\ \theta \end{bmatrix} = \begin{bmatrix} \cos(\theta) & 0 \\ \sin(\theta) & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} u_v \\ u_\omega \\ \end{bmatrix} = \boldsymbol{B}\boldsymbol{U}$

However, if we assume the velocity are constant, we can approximate the equation as following,

$\frac{d}{dt} \boldsymbol{X}= \frac{d}{dt} \begin{bmatrix} y \\ \theta \end{bmatrix} = \begin{bmatrix} 0 & V \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} y \\ \theta \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \begin{bmatrix} u_\omega \\ \end{bmatrix} = \boldsymbol{A}\boldsymbol{X} + \boldsymbol{B}\boldsymbol{U}$

then we can apply this model to linear mpc, we should give the model reference V although.

evaluation function

the both examples have same evaluation function form as following equation.

$J = \sum_{i = 0}^{prestep}||\boldsymbol{\hat{X}}(k+i|k)-\boldsymbol{r}(k+i|k) ||^2_{{\boldsymbol{Q}}(i)} + ||\boldsymbol{\Delta {U}}(k+i|k)||^2_{{\boldsymbol{R}}(i)}$

$\boldsymbol{\hat{X}}$ is predicit state by using predict input
$\boldsymbol{{r}}$ is reference state
$\boldsymbol{\Delta \boldsymbol{U}}$ is predict amount of change of input
$\boldsymbol{\boldsymbol{R}}, \boldsymbol{\boldsymbol{Q}}$ are evaluation function weights

Expected Results

first order system
time history

input

ACC (Adaptive cruise control)
time history of states

animation

Usage

for example(first order system)

$ python main_example.py

for example(ACC (Adaptive cruise control))

$ python main_ACC.py

for comparing two methods of optimization solvers

$ python test_compare_methods.py

Requirement

python3.5 or more
numpy
matplotlib
cvxopt
scipy1.2.0 or more
python-control

Reference

I`m sorry that main references are written in Japanese

モデル予測制御―制約のもとでの最適制御著：Jan M. Maciejowski　訳：足立修一　東京電機大学出版局

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