﻿ 二阶系统线性自抗扰控制的稳定性条件
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 自动化学报  2018, Vol. 44 Issue (9): 1725-1728 PDF

1. 厦门大学航空航天学院 厦门 361005

On Stability Condition of Linear Active Disturbance Rejection Control for Second-order Systems
JIN Hui-Yu1, LIU Li-Li1, LAN Wei-Yao1
1. School of Aerospace Engineering, Xiamen University, Xiamen 361005
Manuscript received : December 10, 2016, accepted: May 31, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61374035), Natural Science Foundation of Fujian Province (2016J01317), and State Scholarship Fund of China (201606315084)
Corresponding author. JIN Hui-Yu Assistant professor at the School of Aerospace Engineering, Xiamen University.His research interest covers data-driven control, sampled-data control, and nonlinear systems.Corresponding author of this paper.
Abstract: The mechanism for linear active disturbance rejection control (LADRC) to reject internal disturbance is investigated.For the linear second-order systems without external disturbance, a sufficient and necessary stability condition of LADRC is given.With the condition, it is proved that the bandwidth method widely used in practice can overcome plant parametric uncertainty and find a suitable observer bandwidth to guarantee the stability of LADRC.
Key words: Linear active disturbance rejection control (LADRC)     second-order system     stability     internal disturbance     observer bandwidth

 \begin{eqnarray} &\left\{\begin{aligned} &\dot{x}_1 = x_2\\ &\dot{x}_2 = -a_2 x_1 - a_1 x_2 + bu\\ \end{aligned}\right. \end{eqnarray} (1)
 $y=x_1$ (2)

 $H_1(s) = - \frac {s(a_1 s + a_2)} {s^2 + l_1 s + l_2} \\ H_2(s) = \frac {s^2 + (k_1 +l_1)s + k_1 l_1 +k_2 +l_2} {s^3 + k_1 s^2 + k_2 s + k_3}$

 图 2 LADRC的反馈互联结构, 其中子系统用传递函数形式 Figure 2 Feedback interconnection structure of LADRC, in which subsystems are described with transfer functions

 \begin{align}\label{condition1} &s(a_1 s + a_2)(s^2 + (k_1 +l_1)s + k_1 l_1 +k_2 +l_2)+\nonumber\\ &\qquad (s^2 + l_1 s + l_2)(s^3 + k_1 s^2 + k_2 s + k_3) =0 \end{align} (11)

 $$$\label{condition2} 1-H_1(s)H_2(s)=0$$$ (12)

1.3 带宽法有效性的新证明

 $\begin{equation*} |H_1({\rm j}\omega)| < \gamma, ~~~~~\forall \omega \in [0, +\infty) \end{equation*}$

 $$$\nonumber H_2(s)=\frac{s^2+(3\omega_o+l_1)s+3\omega_ol_1+3\omega_o^2+l_2}{(s+\omega_o)^3}$$$

 $\begin{eqnarray*} \xi_1=\frac{3\omega_o+l_1}{2\sqrt{3\omega_ol_1+3\omega_o^2+l_2}}\\ \omega_o^{1*} = \frac{1}{3}l_1-\frac{l_2}{l_1} \end{eqnarray*}$

$\omega_o>\max \{0, \omega_o^{1*}\}$时, 有

 $$$\nonumber \xi_1=\frac{3(\omega_o+\frac{1}{3}l_1)}{2\sqrt{3(\omega_o+\frac{1}{3}l_1)^2+\omega_ol_1-\frac{1}{3}l_1^2+l_2}} <\frac{\sqrt{3}}{2}$$$

 $\begin{equation*}\label{pro01} |H_2(s)|=\left|\frac{s^2+2\xi_1\omega_1s+\omega_1^2}{(s+\omega_o)^3}\right|=\frac{\omega_1^2}{\omega_o^3}\left|\frac{\frac{1}{\omega_1^2}s^2+\frac{2\xi_1}{\omega_1}s+1}{(\frac{1}{\omega_o}s+1)^3}\right| \end{equation*}$

 \begin{align*}\nonumber &\left|\frac{1}{\omega_1^2}({\rm j}\omega)^2+2\xi_1\frac{1}{\omega_1}{\rm j}\omega+1\right|^2=\\ &\left[1-(\frac{\omega}{\omega_1})^2\right]^2+4\xi_1^2(\frac{\omega}{\omega_1})^2\leq\\ &1 + 4 \cdot {\frac 3 4} \frac{\omega^2}{3\omega_o^2}= 1 + \frac{\omega^2}{\omega_o^2} = \\ &\left| 1 + \frac {{\rm j}\omega} {\omega_o}\right|^2\leq \left| 1 + \frac {{\rm j}\omega} {\omega_o}\right|^3 \end{align*}

 $$$\label{H_2Ma} |H_2({\rm j}\omega)| \leq \frac{3\omega_ol_1+3\omega_o^2+l_2}{\omega_o^3}, ~~\forall \omega \in [0, +\infty)$$$ (15)

 $\begin{equation*} \lim\limits_{\omega_o \to +\infty} \frac{3\omega_ol_1+3\omega_o^2+l_2}{\omega_o^3} =0 \end{equation*}$

 $$$\label{H_2M} \frac{3\omega_ol_1+3\omega_o^2+l_2}{\omega_o^3} < \frac {1} {\gamma}$$$ (16)

$\omega_o^* =\max \{\omega_o^{1*}, \omega_o^{2*}\}$.当$\omega_o>\omega_o^{*}$时, 因为(16), 对任意$\omega \in [0, +\infty)$, 都有

 $$$\label{small_gain_7} |H_1({\rm j}\omega)H_2({\rm j}\omega)|\leq |H_1({\rm j}\omega)||H_2({\rm j}\omega)| <1$$$ (17)

2 仿真算例

 \begin{eqnarray*} &\left\{\begin{aligned} &\dot{x}_1 = x_2\\ &\dot{x}_2 = 3x_1-x_2+u\\ \end{aligned}\right.\label{plant1}\\ &y=x_1\label{out1*} \end{eqnarray*}

 $\begin{equation*}\label{simu203} u=-\hat{x}_1-2\hat{x}_2-\hat{x}_3 \end{equation*}$

 $\begin{equation*}\label{simu202} \left\{ \begin{array}{lll} \dot{\hat{x}}_1 = 3\omega_o(y-\hat{x}_1)+\hat{x}_2\\ \dot{\hat{x}}_2 = 3\omega_o^2(y-\hat{x}_1)+\hat{x}_3+u\\ \dot{\hat{x}}_3 = \omega_o^3(y-\hat{x}_1) \end{array}\right. \end{equation*}$

 $\begin{equation*}\label{simu204b} H_1(s)=-\frac{s(s-3)}{s^2+2s+1} \end{equation*}$

 $\begin{equation*}\label{simu006} H_2(s)= \frac {s^2+(3 \omega_o+2)s+3\omega_o^2+6\omega_o+1}{s^3+3\omega_o s^2+3\omega_o^2 s + \omega_o^3} \end{equation*}$

$\omega_o$分别等于$5.0$$7.0$.当$\omega_o=5.0$时, 特征方程(11)的根为

 $\begin{equation*} 0.0516\pm 0.5944{\rm j}, -4.6204\pm 4.2742{\rm j}, -8.8624 \end{equation*}$

 $\begin{equation*} -0.1151\pm 0.6802{\rm j}, -6.1649\pm 4.9994{\rm j}, -11.4401 \end{equation*}$

 图 3 系统状态和控制量 Figure 3 States of the plant and the control signal
 图 4 LESO的误差 Figure 4 Errors of LESO
3 结论

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