跟踪误差

定义跟踪误差$$z_1=y-y_r$$ 其中$y_r$为参考信号（期望信号）。
则对$z_1$求导
$$\begin{array}{rl}{\dot{z}}_1& =\dot{y}-{\dot{y}}_r\\ & =x_2+f_1(x_1)-{\dot{y}}_r\end{array}$$
若把$x_2$看做某种控制输入，并选择
$$x_2=-c_1{z}_1-f_1(x_1)+{\dot{y}}_r(c_1>0)$$
那么这样一来
$${\dot{z}}_1=-c_1{z}_1$$
现考虑李雅普诺夫函数
$$V_1=\frac{1}{2}{z}_1^2$$
求导有
$$\begin{array}{rl}{\dot{V}}_1& =z_1{\dot{z}}_1\\ & =-c_1{z}_1^2\le 0\end{array}$$
然而$x_2$不是控制输入，故$x_2$称为虚拟控制，记为
$${\alpha }_1=-c_1{z}_1-f_1(x_1)+{\dot{y}}_r$$
则${\dot{z}}_1$可表示为
$${\dot{z}}_1=x_2-{\alpha }_1-c_1{z}_1$$
此时${\dot{V}}_1$记为
$$\begin{array}{rl}{z}_1{\dot{z}}_1& =z_1(x_1-{\alpha }_1-c_1{z}_1)\\ & =z_1(x_2-{\alpha }_1)-c_1{z}_1^2\end{array}$$

坐标变换

定义坐标变换
$$z_2=x_2-{\alpha }_1$$
则
$$\begin{array}{rl}{\dot{z}}_2& ={\dot{x}}_2-{\dot{\alpha }}_1\\ & =u+f_2(x_1,x_2)+(c_1{\dot{z}}_1+{\dot{f}}_1(x_1)-{\ddot{y}}_r)\\ & =u+f_2(x_1,x_2)-(\frac{\partial {\alpha }_1}{\partial y_r}\cdot {\dot{y}}_r+\frac{\partial {\alpha }_1}{\partial {\dot{y}}_r}\cdot {\dot{y}}_r+\frac{\partial {\alpha }_1}{\partial x_1}\cdot {\dot{x}}_1)\\ & =u+f_2(x_1,x_2)-\frac{\partial {\alpha }_1}{\partial x_1}[x_1+f_1(x_1)]-\frac{\partial {\alpha }_1}{\partial y_r}{\dot{y}}_r-\frac{\partial {\alpha }_1}{\partial {\dot{y}}_r}{\ddot{y}}_r\end{array}$$
令
$$u=-z_1-c_2{z}_2-f_2(x_1,x_2)+\frac{\partial {\alpha }_1}{\partial x_1}[x_1+f_1(x_1)]+\frac{\partial {\alpha }_1}{\partial y_r}{\dot{y}}_r+\frac{\partial {\alpha }_1}{\partial {\dot{y}}_r}{\ddot{y}}_r$$
则${\dot{z}}_2=-z_1-c_2{z}_2$。
再设
$$\begin{array}{rl}{V}_2& =V_1+\frac{1}{2}{z}_2^2\\ & =\frac{1}{2}{z}_1^2+\frac{1}{2}{z}_2^2\end{array}$$
则
$$\begin{array}{rl}{\dot{V}}_2& =z_1(x_2-{\alpha }_1)-c_1{z}_1^2+z_2(-z_1-c_2{z}_2)\\ & =z_1{z}_2-c_1{z}_1^2-z_1{z}_2-c_2{z}_2^2\\ & =-c_1{z}_1^2-c_2{z}_2^2\le 0\end{array}$$
这样一来即可满足李雅普诺夫稳定性。

在以后的文章中，会把反步法应用到四旋翼的控制中去，并详细解释用simulink进行建模的过程。