﻿ 非线性系统的动态面自抗扰控制器设计及应用
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 哈尔滨工程大学学报  2017, Vol. 38 Issue (8): 1278-1284  DOI: 10.11990/jheu.201606005 0

### 引用本文

LI Juan, QIU Junting, GAO Haitao. Design and application of dynamic surface auto-disturbance rejection control for nonlinear systems[J]. Journal of Harbin Engineering University, 2017, 38(8), 1278-1284. DOI: 10.11990/jheu.201606005.

### 文章历史

Design and application of dynamic surface auto-disturbance rejection control for nonlinear systems
LI Juan, QIU Junting, GAO Haitao
College of Automation, Harbin Engineering University, Harbin 150001, China
Abstract: Focusing on the control problem of a class of nonlinear systems with strict feedback, by combination with the dynamic surface and active disturbance rejection control method, this paper proposes an active disturbance rejection control algorithm. The controller can produce the expected signal and its derivative with a tracking differentiator (TD) and estimate the external disturbance with an extended state observer (ESO), which can be compensated by the ESO. The controller can effectively avoid the phenomenon of differential explosion, which appears in the traditional backstepping control method, and reduce the dependence on accurate mathematical models for the control system. The controller design is based on the Lyapunov stability theory, the stability of the ESO and dynamic surface control are analyzed. A simulation of an unmanned underwater vehicle (UUV) model was conducted, which resulted in the track error in the range of (-4, 4). The experimental results show effectiveness of the proposed approach.
Key words: nonlinear system    dynamic surface control    tracking differentiator    observer    disturbance compensation    active disturbance rejection control    robustness    unmanned underwater vehicle

1 动态面自抗扰控制器设计 1.1 系统描述

 $\left\{ \begin{array}{l} {{\dot x}_i} = {x_{i + 1}},\;\;\;i = 1,2, \cdots ,n - 1\\ {{\dot x}_n} = f\left( {{x_1},{x_2}, \cdots ,{x_n}} \right) + bu\\ y = {x_1} \end{array} \right.$ (1)

1.2 控制器设计

 图 1 动态面自抗扰控制器原理 Fig.1 Principle of dynamic surface autodisturbance rejection controller
1.2.1 跟踪微分器设计

TD的主要作用是安排过渡过程，其设计思路是利用一个惯性环节来尽快(通常选取较小的时间常数即可实现，但不宜过小，需根据系统的实际要求)地跟踪上期望信号，并获取期望信号的微分值，即一边尽快地跟踪上期望，同时给出其近似微分。

 $\left\{ \begin{array}{l} {f_h} = {f_h}\left( {{v_1}\left( k \right) - {y_d}\left( k \right),{v_2}\left( k \right),r,h} \right)\\ {v_1} = \left( {k + 1} \right) = {v_1}\left( k \right) + h{v_2}\left( k \right)\\ {v_2} = \left( {k + 1} \right) = {v_2}\left( k \right) + h{f_h} \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} d = rh\\ {d_0} = hd\\ y = {x_1} + h{x_2}\\ {a_0} = \sqrt {{d^2} + 8r\left| y \right|} \\ a = \left\{ \begin{array}{l} {x_2} + \frac{{\left( {{a_0} - d} \right)}}{2}{\rm{sign}}\left( y \right),\;\;\;\;\left| y \right| > {d_0}\\ {x_2} + \frac{y}{h},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left| y \right| \le {d_0} \end{array} \right.\\ {f_h} = \left\{ \begin{array}{l} - r\;{\rm{sign}}\left( a \right),\;\;\;\;\left| a \right| > d\\ - r\frac{a}{d},\;\;\;\;\;\;\;\;\;\;\;\;\left| a \right| \le d \end{array} \right. \end{array} \right.$ (3)

1.2.2 扩张状态观测器设计

 $\left\{ \begin{array}{l} {{\dot x}_i} = {x_{i + 1}},\;\;i = 1, \cdots ,n - 1\\ {{\dot x}_n} = {x_{n + 1}} + bu\\ {{\dot x}_{n + 1}} = \zeta \left( t \right)\\ y = {x_1} \end{array} \right.$ (4)

 $\left\{ \begin{array}{l} e\left( k \right) = {z_1}\left( k \right) - y\left( k \right)\\ {z_1}\left( {k + 1} \right) = {z_1}\left( k \right) + h\left( {{z_2}\left( k \right) - {\beta _1}e\left( k \right)} \right)\\ {z_2}\left( {k + 1} \right) = {z_2}\left( k \right) + h\left( {{z_3}\left( k \right) - } \right.\\ \;\;\;\;\;\;\;\left. {{\beta _2}{\rm{fal}}\left( {e\left( k \right),\frac{1}{2},\delta } \right)} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ {z_n}\left( {k + 1} \right) = {z_n}\left( k \right) + h\left( {{z_{n + 1}}\left( k \right) - } \right.\\ \;\;\;\;\;\;\;\;\left. {{\beta _n}{\rm{fal}}\left( {e\left( k \right),\frac{1}{{{2^{n - 1}}}},\delta } \right) + bu\left( k \right)} \right)\\ {z_{n + 1}}\left( {k + 1} \right) = {z_{n + 1}}\left( k \right) + \\ \;\;\;\;\;\;\;h\left( { - {\beta _{n + 1}}{\rm{fal}}\left( {e\left( k \right),\frac{1}{{{2^n}}},\delta } \right)} \right) \end{array} \right.$ (5)

 ${\rm{fal}}\left( {e,\alpha ,\delta } \right) = \left\{ \begin{array}{l} \frac{e}{{{\delta ^{1 - \alpha }}}},\;\;\;\;\;\;\;\;\left| e \right| \le \delta \\ {\left| e \right|^\alpha }{\rm{sign}}\left( e \right),\;\;\;\;\left| e \right| > \delta \end{array} \right.$ (6)

1.2.3 动态面控制器设计

 $\left\{ \begin{array}{l} {{\dot x}_i} = {x_{i + 1}},\;\;\;i = 1, \cdots ,n - 1\\ {{\dot x}_n} = {u_0}/{{\dot x}_n} = b{u_0}\\ y = {x_1} \end{array} \right.$ (7)

1) 对于第一个子系统$\dot x$1=x2,

 ${S_1} = {x_1} - {y_d}$ (8)

 ${{\dot S}_1} = {{\dot x}_1} - {{\dot y}_d}$ (9)

 ${{\bar \alpha }_2} = - {k_1}{S_1} + {{\dot y}_d}$ (10)

 ${\tau _2}{{\dot \alpha }_2} + {\alpha _2} = {{\bar \alpha }_2},\;\;{\alpha _2}\left( 0 \right) = {{\bar \alpha }_2}\left( 0 \right)$ (11)

2) 对于第二个子系统$\dot x$2=x3

 ${S_2} = {x_2} - {\alpha _2}$ (12)

 ${{\dot S}_2} = {x_3} - {{\dot \alpha }_2}$ (13)

 ${{\bar \alpha }_3} = - {k_2}{S_2} + {{\dot \alpha }_2}$ (14)

 ${\tau _3}{{\dot \alpha }_3} + {\alpha _3} = {{\bar \alpha }_3},\;\;{\alpha _3}\left( 0 \right) = {{\bar \alpha }_3}\left( 0 \right)$ (15)

 ${S_i} = {x_i} - {\alpha _i}$ (16)

 ${{\dot S}_i} = {x_{i + 1}} - {{\dot \alpha }_i}$ (17)

 ${{\bar \alpha }_{i + 1}} = - {k_i}{S_i} + {{\dot \alpha }_i}$ (18)

 ${\tau _{i + 1}}{{\dot \alpha }_{i + 1}} + {\alpha _{i + 1}} = {{\bar \alpha }_{i + 1}},\;\;{\alpha _{i + 1}}\left( 0 \right) = {{\bar \alpha }_{i + 1}}\left( 0 \right)$ (19)

n)对于最后一个子系统$\dot x$n=u0

 ${S_n} = {x_n} - {\alpha _n}$ (20)

 ${{\dot S}_n} = {u_0} - {{\dot \alpha }_n}$ (21)

 ${u_0} = - {k_n}{S_n} + {{\dot \alpha }_n}$ (22)

1.3 扰动补偿设计

 $u = {u_0} - {z_{n + 1}}/b\;或者\;u = \left( {{u_0} - {z_{n + 1}}} \right)/b$ (23)

2 稳定性分析 2.1 扩张状态观测器稳定性分析

 $\left\{ \begin{array}{l} {{\dot x}_1} = f\left( {{x_1}} \right) + bu\\ y = {x_1} \end{array} \right.$ (24)

 $\left\{ \begin{array}{l} {e_1} = {z_1} - {x_1}\\ {e_2} = {z_2} - {x_2}\\ {{\dot e}_1} = {e_2} - {\beta _1}{e_1}\\ {{\dot e}_2} = \zeta \left( t \right) - {\beta _2}{f_e} \end{array} \right.$ (25)

 图 2 (e1, e2)划分图 Fig.2 (e1, e2)partition area

 $\left\{ \begin{array}{l} {A_0} = \left\{ {\left( {{e_1},{e_2}} \right)\left| {\left| {{e_1}} \right| < {c_0},\frac{{{\beta _1}}}{2}\left( {{e_1} - {c_0}} \right) \le {e_2} \le } \right.} \right.\\ \left. {\frac{{{\beta _1}}}{2}\left( {{e_1} + {c_0}} \right)} \right\}\\ {A_1} = \left\{ {\left( {{e_1},{e_2}} \right)\left| {{e_1} > {c_0},0 \le {e_2} \le {\beta _1}{e_1}} \right.} \right\}\\ {A_2} = \left\{ {\left( {{e_1},{e_2}} \right)\left| {{e_2} > 0,{e_2} \ge \frac{{{\beta _1}}}{2}\left( {{e_1} + {c_0}} \right),} \right.} \right.\\ \left. {{e_2} \ge {\beta _1}{e_1}} \right\}\\ {A_3} = \left\{ {\left( {{e_1},{e_2}} \right)\left| {{e_1} < - {c_0},0 \ge {e_2} \ge {\beta _1}{e_1}} \right.} \right\}\\ {A_4} = \left\{ {\left( {{e_1},{e_2}} \right)\left| {{e_2} < 0,{e_2} \le \frac{{{\beta _1}}}{2}\left( {{e_1} - {c_0}} \right),} \right.} \right.\\ \left. {{e_2} \le {\beta _1}{e_1}} \right\} \end{array} \right.$ (26)

 $\left\{ \begin{array}{l} {V_1}\left( {{e_1},{e_2}} \right) = \frac{{{\beta _1}}}{2}\left( {{e_1} - {c_0}} \right),\;\;\left( {{e_1},{e_2}} \right) \in {A_1}\\ {V_2}\left( {{e_1},{e_2}} \right) = {e_2} - \frac{{{\beta _1}}}{2}\left( {{e_1} + {c_0}} \right),\;\;\left( {{e_1},{e_2}} \right) \in {A_2}\\ {V_3}\left( {{e_1},{e_2}} \right) = - \frac{{{\beta _1}}}{2}\left( {{e_1} + {c_0}} \right),\;\;\left( {{e_1},{e_2}} \right) \in {A_3}\\ {V_4}\left( {{e_1},{e_2}} \right) = - {e_2} + \frac{{{\beta _1}}}{2}\left( {{e_1} - {c_0}} \right),\;\;\left( {{e_1},{e_2}} \right) \in {A_4} \end{array} \right.$ (27)

 $\left\{ \begin{array}{l} \left( {\frac{{\partial {V_1}}}{{\partial {e_1}}},\frac{{\partial {V_1}}}{{\partial {e_2}}}} \right) = \left( {\frac{{{\beta _1}}}{2},0} \right),\\ \left( {\frac{{\partial {V_2}}}{{\partial {e_1}}},\frac{{\partial {V_2}}}{{\partial {e_2}}}} \right) = \left( { - \frac{{{\beta _1}}}{2},1} \right)\\ \left( {\frac{{\partial {V_3}}}{{\partial {e_1}}},\frac{{\partial {V_3}}}{{\partial {e_2}}}} \right) = \left( { - \frac{{{\beta _1}}}{2},0} \right),\\ \left( {\frac{{\partial {V_4}}}{{\partial {e_1}}},\frac{{\partial {V_4}}}{{\partial {e_2}}}} \right) = \left( {\frac{{{\beta _1}}}{2}, - 1} \right) \end{array} \right.$ (28)

 ${{\dot V}_1}\left( {{e_1},{e_2}} \right) = \frac{{\partial {V_1}}}{{\partial {e_1}}}{{\dot e}_1} + \frac{{\partial {V_1}}}{{\partial {e_2}}}{{\dot e}_2} = \frac{{{\beta _1}}}{2}\left( {{e_2} - {\beta _1}{e_1}} \right)$ (29)

 ${{\dot V}_1}\left( {{e_1},{e_2}} \right) < 0$ (30)

 $\begin{array}{*{20}{c}} {{{\dot V}_2}\left( {{e_1},{e_2}} \right) = \frac{{\partial {V_2}}}{{\partial {e_1}}}{{\dot e}_1} + \frac{{\partial {V_2}}}{{\partial {e_2}}}{{\dot e}_2} = }\\ { - \frac{{{\beta _1}}}{2}\left( {{e_2} - {\beta _1}{e_1}} \right) + \left[ {\zeta \left( t \right) - {\beta _2}fe} \right]} \end{array}$ (31)

 $\begin{array}{*{20}{c}} {{{\dot V}_2}\left( {{e_1},{e_2}} \right) = \frac{{{\beta _1}}}{2}\left( {{e_2} - {\beta _1}{e_1}} \right) + \left[ {\zeta \left( t \right) - {\beta _2}fe} \right] = }\\ {\frac{{{\beta _1}}}{4}{e_1} - {\beta _2}{\rm{sign}}\left( {{e_1}} \right){{\left| {{e_1}} \right|}^{0.5}} + \zeta \left( t \right) - \frac{{\beta _1^2}}{4}c} \end{array}$ (32)

 $\frac{{{\beta _1}}}{4}{e_1} - {\beta _2}{\rm{sign}}\left( {{e_1}} \right){\left| {{e_1}} \right|^{0.5}} + {w_0} - \frac{{\beta _1^2}}{4}c < 0$ (33)

 $c > \max \left\{ {\frac{{4{w_0}}}{{\beta _1^2}},{{\left| {\frac{{{w_0}}}{{{\beta _2}}}} \right|}^2},\frac{1}{4}{{\left( {\frac{{4{\beta _2}}}{{\beta _1^2}}} \right)}^2} + \frac{{4{w_0}}}{{\beta _1^2}}} \right\}$ (34)

 $c > \frac{4}{{\beta _1^2}}\left( {{w_0} - {\beta _2}{\delta ^\alpha }} \right) + \delta$ (35)

2.2 动态面稳定性分析

 ${e_2} = - {\tau _2}{{\dot \alpha }_2}$ (36)

 ${{\dot S}_1} = {S_2} + {\alpha _2} - {{\dot y}_d}$ (37)

 ${{\dot S}_1} = {S_2} + {e_2} - {k_1}{S_1}$ (38)

 ${e_i} = - {\tau _i}{{\dot \alpha }_i},i = 1, \cdots ,n - 1$ (39)

 ${{\dot S}_i} = {S_{i + 1}} + {e_{i + 1}} - {k_i}{S_i}$ (40)

i=n时，有

 ${{\dot S}_n} = - {k_n}{S_n}$ (41)

 $\left\{ \begin{array}{l} {{\dot e}_2} = {{\dot \alpha }_2} + {k_1}{{\dot S}_1} - {{\ddot y}_d} = \\ \;\;\;\; - \frac{{{e_2}}}{{{\tau _2}}} + {g_1}\left( {{S_1},{S_2},{e_2},{k_1},{y_d},{{\dot y}_d},{{\ddot y}_d}} \right)\\ {{\dot e}_i} = {{\dot \alpha }_i} + {k_i}{{\dot S}_i} + \frac{{{e_{i - 1}}}}{{{\tau _{i - 1}}}} = \\ \;\;\;\; - \frac{{{e_i}}}{{{\tau _i}}} + {g_i}\left( {{S_1}, \cdots ,{S_{i + 1}},{e_2} \cdots {e_i},{k_1}, \cdots ,{k_i},{\tau _2},} \right.\\ \;\;\;\;\;\;\;\;\;\;\left. { \cdots ,{\tau _i},{y_d},{{\dot y}_d},{{\ddot y}_d}} \right),i = 2,3, \cdots ,n \end{array} \right.$ (42)

 ${V_{is}} = \frac{{S_i^2}}{2},i = 1,2, \cdots ,n$ (43)
 ${V_{ie}} = \frac{{e_i^2}}{2},i = 2,3, \cdots ,n$ (44)

 $\left\{ \begin{array}{l} {{\dot V}_{is}} = {S_i}\left( {{S_{i + 1}} + {e_{i + 1}} - {k_i}{S_i}} \right) = \\ \;\;\;\; - {k_i}S_i^2 + {S_i}{S_{i + 1}} + {S_i}{e_{i + 1}},i = 1, \cdots ,n - 1\\ {{\dot V}_{ns}} = - {k_n}S_n^2 \end{array} \right.$ (45)

 ${{\dot V}_{ie}} = - \frac{{e_i^2}}{{{\tau _i}}} + {e_i}{g_i},i = 2,3, \cdots ,n$ (46)

 $V = \sum\limits_{i = 1}^n {{V_{is}}} + \sum\limits_{j = 2}^n {{V_{je}}}$ (47)

 $S_1^2 + e_2^2 + \cdots + S_{n - 1}^2 + e_n^2 + S_n^2 \le 2p$ (48)
 ${g_i} \le C$ (49)

 $\begin{array}{l} \dot V \le \left. { - \left( {2 + a} \right)\sum\limits_{i = 1}^n {S_i^2} } \right) + \sum\limits_{i = 1}^{n - 1} {\left[ {\frac{{2S_i^2 + S_{i + 1}^2 + e_{i + 1}^2}}{2} + } \right.} \\ \;\;\;\;\;\;\left. {\left( {1 + \frac{{{C^2}}}{{2\varepsilon }} + a} \right)e_{i + 1}^2 + \frac{{{C^2}e_{i + 1}^2}}{{2\varepsilon }}\frac{{g_i^2}}{{{C^2}}}} \right] + \frac{{\left( {n - 1} \right)\varepsilon }}{2} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - 2aV + \frac{{\left( {n - 1} \right)\varepsilon }}{2} \end{array}$ (50)

V=p，且a满足a>((n－1)ε)/2p，则有

 $\dot V < 0$ (51)

3 UUV的特性及其仿真研究 3.1 UUV的数学模型

UUV在海流干扰下的水平面非线性状态方程[8-9]

 $\left\{ \begin{array}{l} {{\dot u}_r} = \left( { - {d_{11}}{u_r} + {\tau _u}} \right)/{m_{11}}\\ {{\dot v}_r} = \left( {A{m_{66}} - B{m_{26}}} \right)/\left( {{m_{22}}{m_{66}} - m_{26}^2} \right)\\ \dot r = \left( {B{m_{22}} - A{m_{26}}} \right)/\left( {{m_{22}}{m_{66}} - m_{26}^2} \right)\\ \dot x = u\cos \psi - v\sin \psi \\ \dot y = u\sin \psi + v\cos \psi \\ \dot \psi = r \end{array} \right.$ (52)

 $\left\{ \begin{array}{l} {m_{11}} = m - {X_{\dot u}}\;\;\;{d_{11}} = {X_{{u_r}}} + {X_{\left| {{u_r}} \right|{u_r}}}\left| {{u_r}} \right|\\ {m_{22}} = m - {Y_{\dot v}}\;\;\;\;{d_{22}} = {Y_{{v_r}}} + {Y_{\left| {{v_r}} \right|{v_r}}}\left| {{v_r}} \right|\\ {m_{26}} = - {Y_{\dot r}}\;\;\;\;\;\;\;\;{d_{26}} = {Y_r}\\ {m_{66}} = {I_z} - {N_{\dot r}}\;\;\;{d_{62}} = {N_{{v_r}}}\\ {c_{26}} = m - {X_{\dot u}}\;\;\;\;{d_{66}} = {N_r} + {N_{\left| r \right|r}}\left| r \right|\\ {c_{62}} = {X_{\dot u}} - {Y_{\dot v}} \end{array} \right.$ (53)

3.2 UUV循迹控制器设计

UUV循迹过程中，由视线导引法[10-11]提供实时期望艏向vψ，纵向速度为定常值。根据动态面自抗扰控制器的设计思路，分别设计该UUV的艏向控制器和纵向速度控制器。

1) 艏向控制器设计：

 $\left\{ \begin{array}{l} e = {v_{11}} - {v_\psi }\\ {f_h} = {f_h}\left( {e,{v_{12}},r,h} \right)\\ {v_{11}} = {v_{11}} + h{v_{12}}\\ {v_{12}} = {v_{12}} + h \cdot fh\\ {e_1} = {z_{11}} - {y_1}\\ fe = {\rm{fal}}\left( {{e_1},0.5,\delta } \right),f{e_1} = {\rm{fal}}\left( {{e_1},0.25,\delta } \right)\\ {z_{11}} = {z_{11}} + h\left( {{z_{12}} - {\beta _{11}}{e_1}} \right)\\ {z_{12}} = {z_{12}} + h\left( {{z_{13}} - {\beta _{12}} \cdot fe + {b_{10}}{u_1}} \right)\\ {z_{13}} = {z_{13}} + h\left( { - {\beta _{13}} \cdot f{e_1}} \right)\\ {s_{11}} = {x_{11}} - {v_{11}}\\ {{\bar \alpha }_{11}} = {v_{12}} - {k_{11}}{s_{11}},{\tau _{11}}{{\dot \alpha }_{11}} + {\alpha _{11}} = {{\bar \alpha }_{11}},\\ {\alpha _{11}}\left( 0 \right) = {{\bar \alpha }_{11}}\left( 0 \right)\\ {s_{12}} = {x_{12}} - {\alpha _{11}}\\ {u_{10}} = {{\dot \alpha }_{11}} - {k_{12}}{s_{12}}\\ {u_1} = \left( {{u_{10}} - {z_{13}}} \right)/{b_{10}} \end{array} \right.$ (54)

2) 纵向速度控制器设计：

 $\left\{ \begin{array}{l} e = {v_{21}} - {v_u}\\ {f_h} = {f_h}\left( {e,{v_{22}},r,h} \right)\\ {v_{21}} = {v_{21}} + h{v_2}\\ {v_2} = {v_2} + h \cdot {f_h}\\ {e_{21}} = {z_{21}} - {y_2}\\ fe = {\rm{fal}}\left( {{e_1},\alpha ,\delta } \right)\\ {z_{21}} = {z_{21}} + h\left( {{z_{22}} - {\beta _{21}}{e_1} + {b_{20}}{u_2}} \right)\\ {z_{22}} = {z_{22}} + h\left( { - {\beta _{22}} \cdot fe} \right)\\ {s_{21}} = {x_{21}} - {v_{21}}\\ {u_{20}} = {v_{22}} - {k_{21}}{s_{21}}\\ {u_2} = \left( {{u_{20}} - {z_{22}}} \right)/{b_{20}} \end{array} \right.$ (55)

3.3 UUV循迹仿真

 图 3 UUV航迹图 Fig.3 UUV track map
 图 4 UUV航迹误差图 Fig.4 UUV track error chart
 图 5 跟踪微分器的输出信号 Fig.5 Output signal of tracking differentiator
 图 6 扩张状态观测器的观测值 Fig.6 Observed value of extended state observer

4 结论

1) 该控制算法成功避免了反步法对虚拟控制量求导的过程中会出现“微分爆炸”的现象，对系统的未建模部分或者未知部分进行实时估计，然后根据估计量的大小对系统的实际控制量进行相应的补偿，使得控制器具有更强的鲁棒性。

2) 该控制算法对UUV航迹跟踪、UUV纵向速度和艏向速度估计以及估计外界扰动具有很好的效果。

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