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CN113093553B - An Adaptive Backstepping Control Method Based on Command Filter Disturbance Estimation - Google Patents

An Adaptive Backstepping Control Method Based on Command Filter Disturbance Estimation Download PDF

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CN113093553B
CN113093553B CN202110396089.8A CN202110396089A CN113093553B CN 113093553 B CN113093553 B CN 113093553B CN 202110396089 A CN202110396089 A CN 202110396089A CN 113093553 B CN113093553 B CN 113093553B
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郑晓龙
杨学博
李湛
高会军
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Harbin Institute of Technology Shenzhen
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Abstract

一种基于指令滤波扰动估计的自适应反步控制方法，属于非线性系统自适应控制方法领域。解决了目前自适应反步控制技术对扰动上界进行估计，导致所设计的控制器过于保守且能量消耗较大问题。本发明根据实际应用的非线性系统的状态变量和系统期望输出信号，建立含有扰动项的非线性二阶系统状态空间模型；根据含有扰动项的非线性二阶系统状态空间模型，建立扩维的非线性三阶系统状态空间模型，设定误差变量，利用误差变量设计李雅普诺夫函数；对李雅普诺夫函数对时间求一阶导数；利用反步法和指令滤波器设计虚拟控制函数以及系统控制输入；实现对期望输出信号的跟踪。本发明适用于非线性系统控制使用。

Figure 202110396089

An adaptive backstepping control method based on command filter disturbance estimation belongs to the field of nonlinear system adaptive control methods. It solves the problem that the current adaptive backstepping control technology estimates the upper bound of the disturbance, which leads to the design of the controller is too conservative and consumes a lot of energy. The present invention establishes a nonlinear second-order system state space model containing disturbance items according to the state variables of the actually applied nonlinear system and system expected output signals; according to the nonlinear second-order system state space model containing disturbance items, an expanded dimension Nonlinear third-order system state-space model, setting error variables, using error variables to design Lyapunov functions; calculating first-order derivatives of Lyapunov functions with respect to time; using backstepping and command filters to design virtual control functions and system control Input; enables tracking of the desired output signal. The invention is suitable for use in nonlinear system control.

Figure 202110396089

Description

一种基于指令滤波扰动估计的自适应反步控制方法An Adaptive Backstepping Control Method Based on Command Filter Disturbance Estimation

技术领域technical field

本发明属于非线性系统自适应控制方法领域。The invention belongs to the field of nonlinear system adaptive control method.

背景技术Background technique

自适应反步控制技术是非线性系统控制中的一种方法，其基本思想是利用自适应参数对系统中的不确定参数进行估计，然后利用该估计参数的负反馈来达到降低参数不确定性对系统性能影响的目的。关于自适应反步控制技术，可以参考中国发明专利CN106329986A、CN106438593A以及CN111679582A。由于自适应反步控制技术主要针对的是含有参数不确定的非线性系统，当系统中含有未知非线性扰动项时，传统的自适应反步控制技术不再适用，此时一般假设扰动有界，然后利用自适应参数对扰动上界进行估计，最后通过负反馈降低扰动对系统性能的影响。由于该策略直接对扰动上界进行估计，导致所设计的控制器过于保守且能量消耗较大。Adaptive backstepping control technology is a method in nonlinear system control. Its basic idea is to use adaptive parameters to estimate the uncertain parameters in the system, and then use the negative feedback of the estimated parameters to reduce the uncertainty of the parameters. The purpose of system performance impact. Regarding the self-adaptive backstepping control technology, reference may be made to Chinese invention patents CN106329986A, CN106438593A and CN111679582A. Since the adaptive backstepping control technology is mainly aimed at nonlinear systems with uncertain parameters, when the system contains unknown nonlinear disturbance items, the traditional adaptive backstepping control technology is no longer applicable. At this time, it is generally assumed that the disturbance is bounded , and then use the adaptive parameters to estimate the upper bound of the disturbance, and finally reduce the influence of the disturbance on the system performance through negative feedback. Since this strategy directly estimates the upper bound of the disturbance, the designed controller is too conservative and consumes a lot of energy.

发明内容Contents of the invention

本发明是为了解决目前自适应反步控制技术对扰动上界进行估计，导致所设计的控制器过于保守且能量消耗较大问题，提出了一种基于指令滤波扰动估计的自适应反步控制方法。The present invention aims to solve the problem that the current self-adaptive backstepping control technology estimates the disturbance upper bound, resulting in the designed controller being too conservative and consuming large energy, and proposes an adaptive backstepping control method based on command filter disturbance estimation .

本发明所述的一种基于指令滤波扰动估计的自适应反步控制方法，该方法包括：An adaptive backstepping control method based on command filter disturbance estimation according to the present invention, the method includes:

步骤一、根据实际应用的非线性系统的状态变量x1、x2和期望输出信号yd，建立含有扰动项的非线性二阶系统状态空间模型；Step 1. According to the state variables x 1 , x 2 and the expected output signal y d of the nonlinear system used in practice, a nonlinear second-order system state space model including disturbance items is established;

步骤二、根据含有扰动项的非线性二阶系统状态空间模型，建立扩维的非线性三阶系统状态空间模型，同时设定误差变量z1＝x1-yd，z2＝x2-α1和z3＝x3-α2，其中，α1和α2表示待设计的虚拟控制函数，状态变量x3＝u，u为待设计的系统控制输入；Step 2. Based on the state-space model of the nonlinear second-order system containing disturbance items, establish an expanded nonlinear third-order system state-space model, and set the error variables z 1 = x 1 -y d , z 2 = x 2 - α 1 and z 3 =x 3 -α 2 , where α 1 and α 2 represent the virtual control function to be designed, state variable x 3 =u, and u is the system control input to be designed;

步骤三、利用步骤二中得到的误差变量z1，z2和z3设计李雅普诺夫函数V；Step 3, using the error variables z 1 , z 2 and z 3 obtained in step 2 to design the Lyapunov function V;

步骤四、对步骤三中的李雅普诺夫函数V对时间求一阶导数得到

Figure BDA0003018617620000011
Step 4. Calculate the first derivative of the Lyapunov function V in step 3 with respect to time to obtain
Figure BDA0003018617620000011

步骤五、根据李雅普诺夫函数的一阶导数

Figure BDA0003018617620000012
利用反步法和指令滤波器设计虚拟控制函数α1和α2以及系统控制输入u；获得基于指令滤波扰动估计的自适应反步控制器，实现对期望输出信号yd的跟踪。Step 5. According to the first derivative of the Lyapunov function
Figure BDA0003018617620000012
The virtual control functions α 1 and α 2 and the system control input u are designed by using the backstepping method and command filter; an adaptive backstepping controller based on command filter disturbance estimation is obtained to realize the tracking of the desired output signal y d .

进一步地，本发明中，步骤一中，建立含有扰动项的非线性二阶系统状态空间模型为：Further, in the present invention, in step 1, a nonlinear second-order system state-space model containing a disturbance term is established as:

Figure BDA0003018617620000021
Figure BDA0003018617620000021

其中，x1，x2代表非线性二阶系统的状态变量，

Figure BDA0003018617620000022
表示x2的一阶导数，b为常数，f(x1,x2)为实际已知非线性函数，代表系统的非线性，d(t)表示非线性二阶系统的扰动项，u表示非线性二阶系统的控制输入信号，y表示非线性二阶系统的输出，控制目的为设计控制输入u使系统输出y跟踪期望输出信号yd。Among them, x 1 , x 2 represent the state variables of the nonlinear second-order system,
Figure BDA0003018617620000022
Indicates the first-order derivative of x 2 , b is a constant, f(x 1 , x 2 ) is an actual known nonlinear function, representing the nonlinearity of the system, d(t) indicates the disturbance term of the nonlinear second-order system, and u indicates The control input signal of the nonlinear second-order system, y represents the output of the nonlinear second-order system, and the purpose of control is to design the control input u to make the system output y track the desired output signal y d .

优选地，本发明中，常数b不为0。Preferably, in the present invention, the constant b is not 0.

优选地，本发明中，非线性函数f(x1,x2)为局部李普希茨连续函数。Preferably, in the present invention, the nonlinear function f(x 1 , x 2 ) is a local Lipschitz continuous function.

进一步地，本发明中，步骤二中，建立扩维的非线性三阶系统状态空间模型为：Further, in the present invention, in step 2, the state-space model of the expanded nonlinear third-order system is established as:

Figure BDA0003018617620000023
Figure BDA0003018617620000023

其中，

Figure BDA0003018617620000024
表示x3的一阶导数，
Figure BDA0003018617620000025
表示u的一阶导数。in,
Figure BDA0003018617620000024
represents the first derivative of x 3 ,
Figure BDA0003018617620000025
Indicates the first derivative of u.

进一步地，本发明中，步骤三中利用步骤二中设定的误差变量z1，z2和z3设计的李雅普诺夫函数V为：Further, in the present invention, the Lyapunov function V designed using the error variables z 1 , z 2 and z 3 set in step 2 in step 3 is:

Figure BDA0003018617620000026
Figure BDA0003018617620000026

进一步地，本发明中，步骤四中，对步骤三中的李雅普诺夫函数V对时间求一阶导数为：Further, in the present invention, in step 4, the first-order derivative of the Lyapunov function V in step 3 to time is:

Figure BDA0003018617620000027
Figure BDA0003018617620000027

其中，

Figure BDA0003018617620000028
表示期望输出信号yd的一阶导数；
Figure BDA0003018617620000029
和
Figure BDA00030186176200000210
分别表示虚拟控制函数α1和α2的一阶导数；
Figure BDA00030186176200000211
表示控制输入u的一阶导数。in,
Figure BDA0003018617620000028
Indicates the first derivative of the desired output signal y d ;
Figure BDA0003018617620000029
and
Figure BDA00030186176200000210
represent the first derivatives of virtual control functions α 1 and α 2 respectively;
Figure BDA00030186176200000211
Indicates the first derivative of the control input u.

进一步地，本发明中，步骤五中，获得的虚拟控制函数α1和α2以及系统控制输入u为：Further, in the present invention, in step five, the obtained virtual control functions α 1 and α 2 and the system control input u are:

Figure BDA0003018617620000031
Figure BDA0003018617620000031

Figure BDA0003018617620000032
Figure BDA0003018617620000032

Figure BDA0003018617620000033
Figure BDA0003018617620000033

其中k1,k2,k3为常数，

Figure BDA0003018617620000034
表示期望输出信号yd的一阶导数；
Figure BDA0003018617620000035
为α1的一阶导数，
Figure BDA0003018617620000036
和
Figure BDA0003018617620000037
分别为指令滤波器的输出：Where k 1 , k 2 , k 3 are constants,
Figure BDA0003018617620000034
Indicates the first derivative of the desired output signal y d ;
Figure BDA0003018617620000035
is the first derivative of α 1 ,
Figure BDA0003018617620000036
and
Figure BDA0003018617620000037
are the output of the instruction filter respectively:

Figure BDA0003018617620000038
Figure BDA0003018617620000038

Figure BDA0003018617620000039
Figure BDA0003018617620000039

其中，λ1，λ2为常数，

Figure BDA00030186176200000310
和
Figure BDA00030186176200000311
分别为指令滤波器的状态变量。Among them, λ 1 , λ 2 are constants,
Figure BDA00030186176200000310
and
Figure BDA00030186176200000311
are the state variables of the instruction filter, respectively.

优选地，本发明中，k1，k2和k3均大于0。Preferably, in the present invention, k 1 , k 2 and k 3 are all greater than 0.

优选地，本发明中，λ1和λ2均大于0。Preferably, in the present invention, both λ 1 and λ 2 are greater than 0.

本发明提出了一种基于指令滤波扰动估计的自适应反步控制方法，利用指令滤波器对系统扰动进行估计，相较于传统自适应反步控制技术对扰动上界进行估计，本发明所述方法能直接对系统未知扰动进行有效估计，同时消耗的控制能量更小，同时有效地实现了系统对期望输出信号yd的跟踪。The present invention proposes an adaptive backstepping control method based on command filter disturbance estimation, which utilizes command filters to estimate system disturbances. The method can directly and effectively estimate the unknown disturbance of the system, consume less control energy, and effectively realize the tracking of the expected output signal y d by the system.

附图说明Description of drawings

图1为本发明方法流程图；Fig. 1 is a flow chart of the method of the present invention;

图2为系统采用本发明方法和传统方法输出的响应曲线对比图；Fig. 2 is that the system adopts the response curve contrast figure of the inventive method and traditional method output;

图3为系统采用本发明方法和传统方法的跟踪误差响应曲线对比图；Fig. 3 is a comparison chart of the tracking error response curves of the system adopting the method of the present invention and the traditional method;

图4为采用本发明方法时未知扰动与扰动估计曲线图；Fig. 4 is unknown disturbance and disturbance estimation graph when adopting the method of the present invention;

图5为采用传统方法时未知扰动与扰动上界估计曲线图；Fig. 5 is a curve diagram of unknown disturbance and disturbance upper bound estimation when adopting the traditional method;

图6为采用本发明方法时系统控制输入信号曲线图；Fig. 6 is a system control input signal curve diagram when adopting the inventive method;

图7为采用传统方法时系统控制输入信号曲线图；Fig. 7 is the curve diagram of system control input signal when adopting traditional method;

图8为系统采用本发明方法和传统方法的能量消耗曲线对比图；Fig. 8 is a comparison chart of energy consumption curves of the system adopting the method of the present invention and the traditional method;

图9为直线电机系统采用本发明方法和传统方法的输出的响应曲线对比图；Fig. 9 is a comparison diagram of the response curves of the output of the linear motor system using the method of the present invention and the traditional method;

图10为直线电机系统采用本发明方法和传统方法的跟踪误差响应曲线对比图；Fig. 10 is a comparison chart of the tracking error response curves of the linear motor system using the method of the present invention and the traditional method;

图11为采用本发明方法时直线电机系统未知扰动与扰动估计曲线图；Fig. 11 is a curve diagram of unknown disturbance and disturbance estimation of the linear motor system when the method of the present invention is adopted;

图12为采用传统方法时直线电机系统未知扰动与扰动上界估计曲线图；Fig. 12 is a linear motor system unknown disturbance and disturbance upper bound estimation curve when using the traditional method;

图13为采用本发明方法使直线电机系统控制输入信号曲线图；Fig. 13 is a curve diagram of the control input signal of the linear motor system by adopting the method of the present invention;

图14为采用传统方法时直线电机系统控制输入信号曲线图；Fig. 14 is a linear motor system control input signal curve diagram when adopting the traditional method;

图15为直线电机系统采用本发明方法和传统方法的能量消耗曲线对比图。Fig. 15 is a graph comparing the energy consumption curves of the linear motor system using the method of the present invention and the traditional method.

具体实施方式detailed description

下面将结合本发明实施例中的附图，对本发明实施例中的技术方案进行清楚、完整地描述，显然，所描述的实施例仅仅是本发明一部分实施例，而不是全部的实施例。基于本发明中的实施例，本领域普通技术人员在没有作出创造性劳动的前提下所获得的所有其他实施例，都属于本发明保护的范围。The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without creative efforts fall within the protection scope of the present invention.

需要说明的是，在不冲突的情况下，本发明中的实施例及实施例中的特征可以相互组合。It should be noted that, in the case of no conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

具体实施方式一：下面结合图1说明本实施方式，本实施方式所述一种基于指令滤波扰动估计的自适应反步控制方法，该方法包括：步骤一、根据实际应用的非线性系统的状态变量x1、x2和期望输出信号yd，建立含有扰动项的非线性二阶系统状态空间模型；Specific Embodiment 1: The present embodiment will be described below in conjunction with FIG. 1 . The adaptive backstepping control method based on command filter disturbance estimation described in this embodiment includes: Step 1. According to the state of the nonlinear system in actual application variables x 1 , x 2 and expected output signal y d , establish a nonlinear second-order system state-space model with disturbance items;

步骤二、根据含有扰动项的非线性二阶系统状态空间模型，建立扩维的非线性三阶系统状态空间模型，同时设定误差变量z1＝x1-yd，z2＝x2-α1和z3＝x3-α2，其中，α1和α2表示待设计的虚拟控制函数，状态变量x3＝u，u为待设计的系统控制输入；Step 2. Based on the state-space model of the nonlinear second-order system containing disturbance items, establish an expanded nonlinear third-order system state-space model, and set the error variables z 1 = x 1 -y d , z 2 = x 2 - α 1 and z 3 =x 3 -α 2 , where α 1 and α 2 represent the virtual control function to be designed, state variable x 3 =u, and u is the system control input to be designed;

步骤三、利用步骤二中得到的误差变量z1，z2和z3设计李雅普诺夫函数V；Step 3, using the error variables z 1 , z 2 and z 3 obtained in step 2 to design the Lyapunov function V;

步骤四、对步骤三中的李雅普诺夫函数V对时间求一阶导数得到

Figure BDA0003018617620000041
Step 4. Calculate the first derivative of the Lyapunov function V in step 3 with respect to time to obtain
Figure BDA0003018617620000041

步骤五、根据李雅普诺夫函数的一阶导数

Figure BDA0003018617620000042
利用反步法和指令滤波器设计虚拟控制函数α1和α2以及系统控制输入u；获得基于指令滤波扰动估计的自适应反步控制器，实现对期望输出信号yd的跟踪。Step 5. According to the first derivative of the Lyapunov function
Figure BDA0003018617620000042
The virtual control functions α 1 and α 2 and the system control input u are designed by using the backstepping method and command filter; an adaptive backstepping controller based on command filter disturbance estimation is obtained to realize the tracking of the desired output signal y d .

本发明采用适用于电机控制系统、仪表控制系统等多中非线性系统，有效实现了系统对期望输出信号yd的跟踪，保证了系统输出的准确性。The invention adopts multiple nonlinear systems suitable for motor control systems, instrument control systems, etc., effectively realizes the tracking of the expected output signal y d by the system, and ensures the accuracy of the system output.

进一步地，本发明中，步骤一中，建立含有扰动项的非线性二阶系统状态空间模型为：Further, in the present invention, in step 1, a nonlinear second-order system state-space model containing a disturbance term is established as:

Figure BDA0003018617620000051
Figure BDA0003018617620000051

其中，x1，x2代表非线性二阶系统的状态变量，

Figure BDA0003018617620000052
表示x2的一阶导数，b为常数，f(x1,x2)为实际已知非线性函数，代表系统的非线性，d(t)表示非线性二阶系统的扰动项，u表示非线性二阶系统的控制输入信号，y表示非线性二阶系统的输出，控制目的为设计控制输入u使系统输出y跟踪期望输出信号yd。Among them, x 1 , x 2 represent the state variables of the nonlinear second-order system,
Figure BDA0003018617620000052
Indicates the first-order derivative of x 2 , b is a constant, f(x 1 , x 2 ) is an actual known nonlinear function, representing the nonlinearity of the system, d(t) indicates the disturbance term of the nonlinear second-order system, and u indicates The control input signal of the nonlinear second-order system, y represents the output of the nonlinear second-order system, and the purpose of control is to design the control input u to make the system output y track the desired output signal y d .

进一步地，本实施方式中，常数b不为0。Further, in this embodiment, the constant b is not 0.

进一步地，本实施方式中，非线性函数f(x1,x2)为局部李普希茨连续函数。Further, in this embodiment, the nonlinear function f(x 1 , x 2 ) is a local Lipschitz continuous function.

进一步地，本实施方式中，步骤二中，建立扩维的非线性三阶系统状态空间模型为：Further, in this embodiment, in step 2, the state-space model of the expanded nonlinear third-order system is established as follows:

Figure BDA0003018617620000053
Figure BDA0003018617620000053

其中，

Figure BDA0003018617620000054
表示x3的一阶导数，
Figure BDA0003018617620000055
表示u的一阶导数。in,
Figure BDA0003018617620000054
represents the first derivative of x 3 ,
Figure BDA0003018617620000055
Indicates the first derivative of u.

进一步地，本实施方式中，步骤三中利用步骤二中设定的误差变量z1，z2和z3设计的李雅普诺夫函数V为：Further, in this embodiment, the Lyapunov function V designed using the error variables z 1 , z 2 and z 3 set in step 2 in step 3 is:

Figure BDA0003018617620000056
Figure BDA0003018617620000056

进一步地，本实施方式中，步骤四中，对步骤三中的李雅普诺夫函数V对时间求一阶导数为：Further, in this embodiment, in step 4, the first order derivative of the Lyapunov function V in step 3 with respect to time is:

Figure BDA0003018617620000057
Figure BDA0003018617620000057

其中，

Figure BDA0003018617620000058
表示期望输出信号yd的一阶导数；
Figure BDA0003018617620000059
和
Figure BDA00030186176200000510
分别表示虚拟控制函数α1和α2的一阶导数；
Figure BDA00030186176200000511
表示控制输入u的一阶导数。in,
Figure BDA0003018617620000058
Indicates the first derivative of the desired output signal y d ;
Figure BDA0003018617620000059
and
Figure BDA00030186176200000510
represent the first derivatives of virtual control functions α 1 and α 2 respectively;
Figure BDA00030186176200000511
Indicates the first derivative of the control input u.

进一步地，本实施方式中，步骤五中，获得的虚拟控制函数α1和α2以及系统控制输入u为：Further, in this embodiment, in step five, the obtained virtual control functions α1 and α2 and the system control input u are:

Figure BDA00030186176200000512
Figure BDA00030186176200000512

Figure BDA00030186176200000513
Figure BDA00030186176200000513

Figure BDA0003018617620000061
Figure BDA0003018617620000061

其中k1,k2,k3为常数，

Figure BDA0003018617620000062
表示期望输出信号yd的一阶导数；
Figure BDA0003018617620000063
为α1的一阶导数，
Figure BDA0003018617620000064
和
Figure BDA0003018617620000065
分别为指令滤波器的输出：Where k 1 , k 2 , k 3 are constants,
Figure BDA0003018617620000062
Indicates the first derivative of the desired output signal y d ;
Figure BDA0003018617620000063
is the first derivative of α 1 ,
Figure BDA0003018617620000064
and
Figure BDA0003018617620000065
are the output of the instruction filter respectively:

Figure BDA0003018617620000066
Figure BDA0003018617620000066

Figure BDA0003018617620000067
Figure BDA0003018617620000067

其中，λ1，λ2为常数，

Figure BDA0003018617620000068
和
Figure BDA0003018617620000069
分别为指令滤波器的状态变量。Among them, λ 1 , λ 2 are constants,
Figure BDA0003018617620000068
and
Figure BDA0003018617620000069
are the state variables of the instruction filter, respectively.

进一步地，本发明中，k1，k2和k3均大于0。Further, in the present invention, k 1 , k 2 and k 3 are all greater than 0.

进一步地，本实施方式中，λ1和λ2均大于0。Further, in this embodiment, both λ 1 and λ 2 are greater than 0.

证明基于指令滤波器设计的控制输入(7)能保证系统跟踪误差收敛到原点附近；证明过程如下：Prove that the control input (7) based on the command filter design can ensure that the system tracking error converges to the vicinity of the origin; the proof process is as follows:

定义误差变量

Figure BDA00030186176200000610
选择李雅普诺夫函数为
Figure BDA00030186176200000611
对V1求一阶导数可得：Define the error variable
Figure BDA00030186176200000610
Choose the Lyapunov function as
Figure BDA00030186176200000611
Taking the first derivative with respect to V 1 gives:

Figure BDA00030186176200000612
Figure BDA00030186176200000612

其中δ1，δ2为正常数，

Figure BDA00030186176200000613
Figure BDA00030186176200000614
Among them, δ 1 and δ 2 are normal numbers,
Figure BDA00030186176200000613
Figure BDA00030186176200000614

由式(10)可得：From formula (10) can get:

Figure BDA00030186176200000615
Figure BDA00030186176200000615

其中，常数c1与c2满足V1(0)≥c2/c1。Wherein, the constants c 1 and c 2 satisfy V 1 (0)≥c 2 /c 1 .

由式(11)可得：From formula (11), we can get:

Figure BDA00030186176200000616
Figure BDA00030186176200000616

Figure BDA00030186176200000617
Figure BDA00030186176200000617

将式(5)-(7)以及式(12)-(13)代入式(4)可得：Substituting formulas (5)-(7) and formulas (12)-(13) into formula (4), we can get:

Figure BDA0003018617620000071
Figure BDA0003018617620000071

其中ξ1，ξ2为正常数，

Figure BDA0003018617620000072
where ξ 1 , ξ 2 are normal numbers,
Figure BDA0003018617620000072

由式(14)可得：From formula (14), we can get:

Figure BDA0003018617620000073
Figure BDA0003018617620000073

由式(15)可得：From formula (15), we can get:

Figure BDA0003018617620000074
Figure BDA0003018617620000074

式(16)表明所设计的控制输入(7)能保证系统跟踪误差收敛到原点附近，实现对本发明所述方法的验证。Equation (16) shows that the designed control input (7) can ensure that the system tracking error converges to the vicinity of the origin, and realizes the verification of the method of the present invention.

具体实施例一Specific embodiment one

取非线性二阶系统状态空间模型(1)的初始值为x1(0)＝0.8，x2(0)＝-0.3，常数b＝2。为了演示仿真，假设已知系统非线性函数为

Figure BDA0003018617620000075
系统扰动项d(t)＝0.5cos0.04t，该扰动项对于设计者是未知的，不能直接用于设计系统控制输入u。系统期望输出信号设为yd(t)＝1.2sin(0.5t)。The initial values of the nonlinear second-order system state-space model (1) are x 1 (0)=0.8, x 2 (0)=-0.3, and the constant b=2. To demonstrate the simulation, assume that the known system nonlinear function is
Figure BDA0003018617620000075
The system disturbance item d(t)=0.5cos0.04t, this disturbance item is unknown to the designer and cannot be directly used to design the system control input u. The expected output signal of the system is set as y d (t)=1.2sin(0.5t).

虚拟控制函数(5)和(6)以及控制输入(7)中的参数取为k1＝2，k2＝2，k3＝2；指令滤波器(8)和(9)中的参数为λ1＝20，λ2＝60；指令滤波器(8)和(9)的状态变量初始值为

Figure BDA0003018617620000076
为了对比结果，采用本发明基于指令滤波扰动估计的自适应反步控制方法(以下简称本发明方法)和传统基于扰动上界估计的自适应反步控制方法(以下简称传统方法)进行对比。The parameters in the virtual control functions (5) and (6) and the control input (7) are taken as k 1 =2, k 2 =2, k 3 =2; the parameters in the command filters (8) and (9) are λ 1 = 20, λ 2 = 60; the initial values of the state variables of command filters (8) and (9) are
Figure BDA0003018617620000076
In order to compare the results, the adaptive backstepping control method based on command filter disturbance estimation (hereinafter referred to as the method of the present invention) of the present invention is used for comparison with the traditional adaptive backstepping control method based on disturbance upper bound estimation (hereinafter referred to as the traditional method).

图2给出了在本发明方法和传统方法作用下，系统输出响应曲线对比图，其中实线为期望输出信号yd，短划线为本发明方法下系统输出y，点划线为传统方法下系统输出y；图3为本发明所述方法和传统方法下，系统跟踪误差响应曲线对比图，其中短划线为本发明方法下系统输出与期望输出信号的差值，点划线为传统方法下系统输出与期望输出信号的差值；图4为本发明方法下，指令滤波扰动估计与系统未知扰动曲线图，其中虚线为系统未知扰动，点划线为本发明方法下的扰动估计值；图5为传统方法下，扰动上界估计与系统未知扰动曲线图，其中虚线为系统未知扰动，短划线为传统方法下的扰动上界估计值；图6为本发明方法下，系统控制输入曲线图，用短划线表示；图7为传统方法下，系统控制输入曲线图，用点划线表示；图8为本发明方法和传统方法下，系统控制输入能量消耗曲线对比图，其中短划线为本发明方法下控制输入能量消耗，点划线为传统方法下控制输入能量消耗。Fig. 2 has provided under the action of the method of the present invention and traditional method, the system output response curve comparative figure, wherein solid line is expected output signal yd, dashed line is system output y under the method of the present invention, dotted line is traditional method Lower system output y; Fig. 3 is under the method of the present invention and the traditional method, the system tracking error response curve comparison chart, wherein the dashed line is the difference between the system output and the expected output signal under the method of the present invention, and the dotted line is the traditional The difference between the system output and the expected output signal under the method; Fig. 4 is under the method of the present invention, the command filter disturbance estimation and the system unknown disturbance curve, wherein the dotted line is the system unknown disturbance, and the dotted line is the disturbance estimation value under the method of the present invention ; Fig. 5 is under the traditional method, the disturbance upper bound estimate and the system unknown disturbance curve diagram, wherein the dotted line is the system unknown disturbance, and the dashed line is the disturbance upper bound estimated value under the traditional method; Fig. 6 is under the method of the present invention, the system control Input graph, represented by a dashed line; Fig. 7 is under the traditional method, the system control input graph, represented by a dotted line; Fig. 8 is under the method of the present invention and the traditional method, the system control input energy consumption curve comparative figure, wherein The dashed line is the control input energy consumption under the method of the present invention, and the dotted line is the control input energy consumption under the traditional method.

具体实施例二Specific embodiment two

系统(1)可以描述直线电机位置控制系统动力学，其状态空间模型如下：System (1) can describe the dynamics of the linear motor position control system, and its state space model is as follows:

Figure BDA0003018617620000081
Figure BDA0003018617620000081

其中x1(单位m)代表电机线圈位置，x2(单位m/s)为电机线圈速度，m(单位kg)代表电机线圈质量，u(单位V)为电机控制电压，σ为粘滞摩擦系数，d(t)为系统扰动项。Where x 1 (unit m) represents the motor coil position, x 2 (unit m/s) is the motor coil speed, m (unit kg) represents the motor coil mass, u (unit V) is the motor control voltage, σ is the viscous friction Coefficient, d(t) is the system disturbance item.

系统的初始值为x1(0)＝0.8m，x2(0)＝0m/s，电机线圈质量m＝1.2kg，粘滞摩擦系数σ＝0.011。为了演示仿真，假设系统扰动项为d(t)＝0.2sin0.25t，该函数对于设计者是未知的，不能直接用于设计电机控制电压u。系统期望输出信号设为yd＝0.6m。The initial values of the system are x 1 (0)=0.8m, x 2 (0)=0m/s, motor coil mass m=1.2kg, viscous friction coefficient σ=0.011. In order to demonstrate the simulation, it is assumed that the system disturbance term is d(t) = 0.2sin0.25t, this function is unknown to the designer and cannot be directly used to design the motor control voltage u. The expected output signal of the system is set to y d =0.6m.

虚拟控制函数(5)和(6)以及控制输入(7)中的参数取b＝0.83，k1＝2，k2＝2，k3＝2；指令滤波器(8)和(9)中的参数为λ1＝20，λ2＝60；指令滤波器(8)和(9)的状态变量初始值为

Figure BDA0003018617620000082
为了对比结果，采用本发明基于指令滤波扰动估计的自适应反步控制方法(以下简称本发明方法)和传统基于扰动上界估计的自适应反步控制方法(以下简称传统方法)进行对比。The parameters in virtual control functions (5) and (6) and control input (7) take b=0.83, k 1 =2, k 2 =2, k 3 =2; in command filter (8) and (9) The parameters of λ 1 = 20, λ 2 = 60; the initial values of the state variables of the instruction filters (8) and (9) are
Figure BDA0003018617620000082
In order to compare the results, the adaptive backstepping control method based on command filter disturbance estimation (hereinafter referred to as the method of the present invention) of the present invention is used for comparison with the traditional adaptive backstepping control method based on disturbance upper bound estimation (hereinafter referred to as the traditional method).

图9给出了在本发明方法和传统方法作用下，系统输出响应曲线对比图，其中实线为期望输出信号yd，短划线为本发明方法下系统输出y，点划线为传统方法下系统输出y；图10为本发明所述方法和传统方法下，系统跟踪误差响应曲线图，其中短划线为本发明方法下系统输出与期望输出信号的差值，点划线为传统方法下系统输出与期望输出信号的差值；图11为本发明方法下，指令滤波扰动估计与系统未知扰动曲线图，其中虚线为系统未知扰动，点划线为本发明方法下的扰动估计值；图12为传统方法下，扰动上界估计与系统未知扰动曲线图，其中虚线为系统未知扰动，短划线为传统方法下的扰动上界估计值；图13为本发明方法下，系统控制输入曲线图，用短划线表示；图14为传统方法下，系统控制输入曲线图，用点划线表示；图15为本发明方法和传统方法下，系统控制输入能量消耗曲线对比图，其中短划线为本发明方法下控制输入能量消耗，点划线为传统方法下控制输入能量消耗。Fig. 9 shows a comparison diagram of system output response curves under the action of the method of the present invention and the traditional method, wherein the solid line is the expected output signal yd, the dashed line is the system output y under the method of the present invention, and the dotted line is the traditional method The lower system output y; Fig. 10 is the system tracking error response curve diagram under the method of the present invention and the traditional method, wherein the dashed line is the difference between the system output and the expected output signal under the method of the present invention, and the dotted line is the traditional method The difference between the lower system output and the expected output signal; Figure 11 is a curve diagram of command filter disturbance estimation and system unknown disturbance under the method of the present invention, wherein the dotted line is the unknown disturbance of the system, and the dotted line is the disturbance estimation value under the method of the present invention; Fig. 12 is a curve diagram of the estimated upper bound of disturbance and the unknown disturbance of the system under the traditional method, wherein the dotted line is the unknown disturbance of the system, and the dashed line is the estimated value of the upper bound of the disturbance under the traditional method; Fig. 13 is the system control input under the method of the present invention The graph is represented by a dashed line; Fig. 14 is a graph of the system control input under the traditional method, represented by a dotted line; Fig. 15 is a comparison diagram of the energy consumption curve of the system control input under the method of the present invention and the traditional method, where the short The dashed line is the energy consumption of the control input under the method of the present invention, and the dotted line is the energy consumption of the control input under the traditional method.

结论一：从图4、图5、图11以及图12可以看出，传统方法只能得到扰动上界的估计值，而本发明方法采用指令滤波器可以直接得到扰动的估计值。Conclusion 1: From Figure 4, Figure 5, Figure 11 and Figure 12, it can be seen that the traditional method can only obtain the estimated value of the upper bound of the disturbance, but the method of the present invention can directly obtain the estimated value of the disturbance by using the instruction filter.

结论二：从图8和图15可以看出，本发明方法下系统控制输入能量消耗要小于传统方法。Conclusion 2: It can be seen from Fig. 8 and Fig. 15 that the system control input energy consumption under the method of the present invention is smaller than that of the traditional method.

本发明的上述算例仅为详细地说明本发明的计算模型和计算流程，而并非是对本发明的实施方式的限定。对于所属领域的普通技术人员来说，在上述说明的基础上还可以做出其它不同形式的变化或变动，这里无法对所有的实施方式予以穷举，凡是属于本发明的技术方案所引伸出的显而易见的变化或变动仍处于本发明的保护范围之列。The above calculation example of the present invention is only to describe the calculation model and calculation process of the present invention in detail, but not to limit the implementation of the present invention. For those of ordinary skill in the art, on the basis of the above description, other changes or changes in different forms can also be made, and all implementation modes cannot be exhaustively listed here. Obvious changes or modifications are still within the protection scope of the present invention.

Claims (6)

1.一种基于指令滤波扰动估计的自适应反步控制方法，其特征在于，该方法包括：1. An adaptive backstepping control method based on command filter disturbance estimation, characterized in that the method comprises: 步骤一、根据实际应用的非线性系统的状态：直线电机位置控制系统的变量电机线圈位置x1、电机线圈速度x2和期望输出信号yd，建立含有扰动项的非线性二阶系统状态空间模型；Step 1. According to the state of the nonlinear system in practical application: the variable motor coil position x 1 , the motor coil speed x 2 and the expected output signal y d of the linear motor position control system, establish a nonlinear second-order system state space with disturbance items Model; 其中，建立含有扰动项的非线性二阶系统状态空间模型为：Among them, the state-space model of nonlinear second-order system with disturbance term is established as:
Figure FDA0003912299010000011
Figure FDA0003912299010000011
 其中，x1，x2代表非线性二阶系统的状态变量，
Figure FDA0003912299010000012
表示x2的一阶导数，b为常数，f(x1,x2)为实际已知非线性函数，代表系统的非线性，d(t)表示非线性二阶系统的扰动项，u表示非线性二阶系统的控制输入信号，y表示非线性二阶系统的输出，控制目的为设计控制输入u使系统输出y跟踪期望输出信号yd；Among them, x 1 , x 2 represent the state variables of the nonlinear second-order system,
Figure FDA0003912299010000012
Indicates the first-order derivative of x 2 , b is a constant, f(x 1 , x 2 ) is an actual known nonlinear function, representing the nonlinearity of the system, d(t) indicates the disturbance term of the nonlinear second-order system, and u indicates The control input signal of the nonlinear second-order system, y represents the output of the nonlinear second-order system, and the control purpose is to design the control input u to make the system output y track the expected output signal y d ; 步骤二、根据含有扰动项的非线性二阶系统状态空间模型，建立扩维的非线性三阶系统状态空间模型，同时设定误差变量z1＝x1-yd，z2＝x2-α1和z3＝x3-α2，其中，α1和α2表示待设计的虚拟控制函数，状态变量x3＝u，u为待设计的系统控制输入；Step 2. Based on the state-space model of the nonlinear second-order system containing disturbance items, establish an expanded nonlinear third-order system state-space model, and set the error variables z 1 = x 1 -y d , z 2 = x 2 - α 1 and z 3 =x 3 -α 2 , where α 1 and α 2 represent the virtual control function to be designed, state variable x 3 =u, and u is the system control input to be designed; 步骤三、利用步骤二中得到的误差变量z1，z2和z3设计李雅普诺夫函数V；Step 3, using the error variables z 1 , z 2 and z 3 obtained in step 2 to design the Lyapunov function V; 其中，设计的李雅普诺夫函数V为：Among them, the designed Lyapunov function V is:
Figure FDA0003912299010000013
Figure FDA0003912299010000013
 步骤四、对步骤三中的李雅普诺夫函数V对时间求一阶导数得到
Figure FDA0003912299010000014
Step 4. Calculate the first derivative of the Lyapunov function V in step 3 with respect to time to obtain
Figure FDA0003912299010000014
 其中，对步骤三中的李雅普诺夫函数V对时间求一阶导数为：Among them, the first-order derivative of the Lyapunov function V in step 3 with respect to time is:
Figure FDA0003912299010000015
Figure FDA0003912299010000015
 其中，
Figure FDA0003912299010000016
表示期望输出信号yd的一阶导数；
Figure FDA0003912299010000017
和
Figure FDA0003912299010000018
分别表示虚拟控制函数α1和α2的一阶导数；
Figure FDA0003912299010000019
表示控制输入u的一阶导数；in,
Figure FDA0003912299010000016
Indicates the first derivative of the desired output signal y d ;
Figure FDA0003912299010000017
and
Figure FDA0003912299010000018
represent the first derivatives of virtual control functions α 1 and α 2 respectively;
Figure FDA0003912299010000019
Indicates the first derivative of the control input u; 步骤五、根据李雅普诺夫函数的一阶导数
Figure FDA00039122990100000110
利用反步法和指令滤波器设计虚拟控制函数α1和α2以及系统控制输入u；获得基于指令滤波扰动估计的自适应反步控制器，实现对期望输出信号yd的跟踪；Step 5. According to the first derivative of the Lyapunov function
Figure FDA00039122990100000110
Design virtual control functions α 1 and α 2 and system control input u by using backstepping method and instruction filter; obtain an adaptive backstepping controller based on instruction filter disturbance estimation, and realize tracking of desired output signal yd; 其中，获得的虚拟控制函数α1和α2以及系统控制输入u为：Among them, the obtained virtual control functions α 1 and α 2 and the system control input u are:
Figure FDA0003912299010000021
Figure FDA0003912299010000021
Figure FDA0003912299010000022
Figure FDA0003912299010000022
Figure FDA0003912299010000023
Figure FDA0003912299010000023
 其中k1,k2,k3为常数，
Figure FDA0003912299010000024
表示期望输出信号yd的一阶导数；
Figure FDA0003912299010000025
为α1的一阶导数，
Figure FDA0003912299010000026
和
Figure FDA0003912299010000027
分别为指令滤波器的输出：Where k 1 , k 2 , k 3 are constants,
Figure FDA0003912299010000024
Indicates the first derivative of the desired output signal y d ;
Figure FDA0003912299010000025
is the first derivative of α 1 ,
Figure FDA0003912299010000026
and
Figure FDA0003912299010000027
are the output of the instruction filter respectively:
Figure FDA0003912299010000028
Figure FDA0003912299010000028
Figure FDA0003912299010000029
Figure FDA0003912299010000029
 其中，λ1，λ2为常数，
Figure FDA00039122990100000210
和
Figure FDA00039122990100000211
分别为指令滤波器的状态变量。Among them, λ 1 and λ 2 are constants,
Figure FDA00039122990100000210
and
Figure FDA00039122990100000211
are the state variables of the instruction filter, respectively. 2.根据权利要求1所述的一种基于指令滤波扰动估计的自适应反步控制方法，其特征在于，常数b不为0。2. An adaptive backstepping control method based on command filter disturbance estimation according to claim 1, characterized in that the constant b is not 0. 3.根据权利要求2所述的一种基于指令滤波扰动估计的自适应反步控制方法，其特征在于，非线性函数f(x1,x2)为局部李普希茨连续函数。3. An adaptive backstepping control method based on command filter disturbance estimation according to claim 2, characterized in that the nonlinear function f(x 1 , x 2 ) is a local Lipschitz continuous function. 4.根据权利要求1所述的一种基于指令滤波扰动估计的自适应反步控制方法，其特征在于，步骤二中，建立扩维的非线性三阶系统状态空间模型为：4. a kind of adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 1, is characterized in that, in step 2, the nonlinear third-order system state-space model of setting up dimension expansion is:
Figure FDA00039122990100000212
Figure FDA00039122990100000212
 其中，
Figure FDA00039122990100000213
表示x3的一阶导数，
Figure FDA00039122990100000214
表示u的一阶导数。in,
Figure FDA00039122990100000213
represents the first derivative of x 3 ,
Figure FDA00039122990100000214
Indicates the first derivative of u. 5.根据权利要求1所述的一种基于指令滤波扰动估计的自适应反步控制方法，其特征在于，k1，k2和k3均大于0。5. An adaptive backstepping control method based on command filter disturbance estimation according to claim 1, characterized in that k 1 , k 2 and k 3 are all greater than 0. 6.根据权利要求1所述的一种基于指令滤波扰动估计的自适应反步控制方法，其特征在于，λ1和λ2均大于0。6. A kind of adaptive backstepping control method based on instruction filter disturbance estimation according to claim 1, is characterized in that, λ 1 and λ 2 are all greater than 0.
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