65 lines
2.5 KiB
Markdown
65 lines
2.5 KiB
Markdown
# Newton method of Nonlinear Model Predictive Control
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This program is about NMPC with newton method.
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Usually we have to calculate the partial differential of optimal matrix.
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In this program, in stead of using any paticular methods to calculate the partial differential of optimal matrix, I used numerical differentiation.
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Therefore, I believe that it easy to understand and extend your model.
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# Problem Formulation
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- **example**
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- model
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<a href="https://www.codecogs.com/eqnedit.php?latex=\begin{bmatrix}&space;\dot{x_1}&space;\\&space;\dot{x_2}&space;\\&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;x_2&space;\\&space;(1-x_1^2-x_2^2)x_2-x_1+u&space;\\&space;\end{bmatrix},&space;|u|&space;\leq&space;0.5" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\begin{bmatrix}&space;\dot{x_1}&space;\\&space;\dot{x_2}&space;\\&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;x_2&space;\\&space;(1-x_1^2-x_2^2)x_2-x_1+u&space;\\&space;\end{bmatrix},&space;|u|&space;\leq&space;0.5" title="\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" /></a>
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- evaluation function
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To consider the constraints of input u, I introduced dummy input.
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<a href="https://www.codecogs.com/eqnedit.php?latex=J&space;=&space;\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" target="_blank"><img src="https://latex.codecogs.com/gif.latex?J&space;=&space;\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" title="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" /></a>
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- **two wheeled model**
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coming soon !
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# Expected Results
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- example
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![Figure_1.png]()
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you can confirm that the my method could consider the constraints of input.
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- two wheeled model
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coming soon !
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# Usage
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- for example
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```
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$ python main_example.py
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```
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- for two wheeled
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coming soon !
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# Requirement
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- python3.5 or more
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- numpy
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- matplotlib
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# Reference
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I`m sorry that main references are written in Japanese
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- main (commentary article) (Japanse) https://qiita.com/MENDY/items/4108190a579395053924
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- 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
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- 非線形最適制御入門（コロナ社）
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- 実時間最適化による制御の実応用（コロナ社） |