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CGMRES method of Nonlinear Model Predictive Control

This program is about Continuous gmres method for NMPC Although usually we have to calculate the partial differential of optimal matrix, it could be really complicated. By using CGMRES, we can pass the calculating step and get the optimal input quickly.

Problem Formulation

example
model

$\begin{bmatrix} \dot{x_1} \\ \dot{x_2} \\ \end{bmatrix} = \begin{bmatrix} x_2 \\ (1-x_1^2-x_2^2)x_2-x_1+u \\ \end{bmatrix}, |u| \leq 0.5$

evaluation function

$J = \frac{1}{2}(x_1^2(t+T)+x_2^2(t+T))+\int_{t}^{t+T}\frac{1}{2}(x_1^2+x_2^2+u^2)-0.01vd\tau$

two wheeled model
model

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

evaluation function

$J = \boldsymbol{X}(t_0+T)^2 + \int_{t_0}^{t_0 + T} \boldsymbol{U}(t)^2 - 0.01 dummy_{u_v} - dummy_{u_\omega} dt$

if you want to see more detail about this methods, you should go https://qiita.com/MENDY/items/4108190a579395053924. However, it is written in Japanese

Expected Results

example

two wheeled model
trajectory

time history

Usage

for example

$ python main_example.py

for two wheeled

$ python main_two_wheeled.py

Requirement

python3.5 or more
numpy
matplotlib

Reference

I`m sorry that main references are written in Japanese

main (commentary article) (Japanse)　https://qiita.com/MENDY/items/4108190a579395053924
Ohtsuka, T., & Fujii, H. A. (1997). Real-time Optimization Algorithm for Nonlinear Receding-horizon Control. Automatica, 33(6), 1147–1154. https://doi.org/10.1016/S0005-1098(97)00005-8
非線形最適制御入門（コロナ社）
実時間最適化による制御の実応用（コロナ社）

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