[足式机器人]Part2 Dr. CAN学习笔记-数学基础Ch0-3线性化Linearization

本文仅供学习使用
本文参考：
B站：DR_CAN

---

1. 线性系统 Linear System 与 叠加原理 Superposition

$$\dot{x}=f\left(x\right)$$

1. $x_1,x_2$ 是解
2. $x_3=k_1{x}_1+k_2{x}_2,k_1,k_2\in \mathbb{R}$
3. $x_3$ 是解

eg:
$$\begin{gathered} \ddot{x}+2\dot{x}+\sqrt{2}x=0\surd \\ \ddot{x}+2\dot{x}+\sqrt{2}{x}^2=0\times \\ \ddot{x}+\sin \dot{x}+\sqrt{2}x=0\times \end{gathered}$$

2. 线性化：Taylor Series

$$f\left(x\right)=f\left(x_0\right)+\frac{f^{\prime }\left(x_0\right)}{1!}\left(x-x_0\right)+\frac{{f^{\prime }}^{\prime }\left(x_0\right)}{2!}{\left(x-x_0\right)}^2+\cdots +\frac{f^n\left(x_0\right)}{n!}{\left(x-x_0\right)}^n$$

若$x-x_0\to 0,{\left(x-x_0\right)}^n\to 0$，则有：$\Rightarrow f\left(x\right)=f\left(x_0\right)+f^{\prime }\left(x_0\right)\left(x-x_0\right)\Rightarrow f\left(x\right)=k_1+k_{2}x-k_3{x}_0\Rightarrow f\left(x\right)=k_{2}x+b$

eg1:

eg2:

eg3:

3. Summary

1. $f\left(x\right)=f\left(x_0\right)+\frac{f^{\prime }\left(x_0\right)}{1!}\left(x-x_0\right),x-x_0\to 0$
2. [ x ˙ 1 d x ˙ 2 d ] = [ ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ] ∣ x = x 0 [ x 1 d x 2 d ] \left[ \begin{array}{c} \dot{x}_{1\mathrm{d}}\\ \dot{x}_{2\mathrm{d}}\\ \end{array} \right] =\left. \left[ \begin{matrix} \frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}\\ \end{matrix} \right] \right|_{\mathrm{x}=\mathrm{x}_0}\left[ \begin{array}{c} x_{1\mathrm{d}}\\ x_{2\mathrm{d}}\\ \end{array} \right] [x˙1d​x˙2d​​]=[∂x1​∂f1​​∂x1​∂f2​​​∂x2​∂f1​​∂x2​∂f2​​​] ​x=x0​​[x1d​x2d​​]