CN114721274B - A sliding mode controller design method and system based on improved fal function - Google Patents

A sliding mode controller design method and system based on improved fal function Download PDF

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CN114721274B
CN114721274B CN202210434471.8A CN202210434471A CN114721274B CN 114721274 B CN114721274 B CN 114721274B CN 202210434471 A CN202210434471 A CN 202210434471A CN 114721274 B CN114721274 B CN 114721274B
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鲁仁全
陈汉泉
陶杰
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Guangdong University of Technology
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Abstract

本发明的目的在于提出一种基于改进fal函数的滑模控制器设计方法及系统，其中包括以下步骤：步骤S1：对物理系统建模，得出对应的二阶物理系统模型；步骤S2：根据所述二阶物理系统模型的输入信号类型，根据输入信号类型构建通用的误差公式；步骤S3：以所述误差公式的误差值作为滑模控制器的输入，使用sfal函数构建滑模控制器的滑模面；步骤S4：构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，完成滑膜控制器的设计，本发明利用非线性sfal函数使控制系统收敛更快，减少系统相角滞后，提升系统的跟踪性能。另一方面，本发明利用非线性sfal函数使控制系统在误差大时减少增益，提升系统的稳定性能。

Figure 202210434471

The purpose of the present invention is to propose a sliding mode controller design method and system based on the improved fal function, which includes the following steps: step S1: modeling the physical system to obtain a corresponding second-order physical system model; step S2: according to According to the input signal type of the second-order physical system model, a general error formula is constructed according to the input signal type; Step S3: The error value of the error formula is used as the input of the sliding mode controller, and the sfal function is used to construct the sliding mode controller. Sliding mode surface; Step S4: constructing the approach rate equation of the sliding mode surface, and selecting the approach rate equation and substituting it into the derivative of the sliding mode surface to complete the design of the sliding film controller, the present invention utilizes nonlinear The sfal function makes the control system converge faster, reduces the phase angle lag of the system, and improves the tracking performance of the system. On the other hand, the present invention utilizes the nonlinear sfal function to reduce the gain of the control system when the error is large, thereby improving the stability of the system.

Figure 202210434471

Description

一种基于改进fal函数的滑模控制器设计方法及系统A sliding mode controller design method and system based on improved fal function

技术领域technical field

本发明涉及滑模控制器技术领域，特别是一种基于改进fal函数的滑模控制器设计方法及系统。The invention relates to the technical field of sliding mode controllers, in particular to a design method and system for a sliding mode controller based on an improved fal function.

背景技术Background technique

滑模控制是一类特殊的非线性控制，且非线性表现为不连续性。由于滑模函数可以进行设计且与对象参数及扰动无关，这就使得滑模控制具有快速响应、对应参数变化及扰动不灵敏、无需系统在线辨识、物理实现简单等优点。Sliding mode control is a special kind of nonlinear control, and the nonlinearity is discontinuous. Since the sliding mode function can be designed and has nothing to do with the object parameters and disturbances, the sliding mode control has the advantages of fast response, insensitive to corresponding parameter changes and disturbances, no need for online identification of the system, and simple physical implementation.

当在滑模控制中采用fal函数作用于滑模控制的非线性控制时，fal函数在误差少于预设值时为线性函数，在大于预设值时为非线性函数，在线性和非线性函数的切换中，存在非平滑切换，导致了在选用非线性滑模面的滑模控制器中由于非线性函数非平滑，导致系统在切换点附近有一定颤抖。When the fal function is used in the sliding mode control to act on the nonlinear control of the sliding mode control, the fal function is a linear function when the error is less than the preset value, and it is a nonlinear function when the error is greater than the preset value, linear and nonlinear In the switching of functions, there is non-smooth switching, which causes the system to tremble near the switching point due to the non-smooth nonlinear function in the sliding mode controller with nonlinear sliding mode surface.

发明内容Contents of the invention

针对上述缺陷，本发明的目的在于提出一种基于改进fal函数的滑模控制器设计方法及系统，使采用fal函数作用于滑模控制的非线性控制时，能够在系统切换点平滑的过渡。In view of the above defects, the object of the present invention is to propose a sliding mode controller design method and system based on the improved fal function, so that when the fal function is used to act on the nonlinear control of the sliding mode control, a smooth transition can be made at the switching point of the system.

为达此目的，本发明采用以下技术方案：一种基于改进fal函数的滑模控制器设计方法，包括以下步骤：For reaching this purpose, the present invention adopts following technical scheme: a kind of sliding mode controller design method based on improved fal function comprises the following steps:

步骤S1：对物理系统建模，得出对应的二阶物理系统模型；Step S1: Model the physical system to obtain the corresponding second-order physical system model;

步骤S2：根据所述二阶物理系统模型的输入信号类型，根据输入信号类型构建通用的误差公式；Step S2: according to the input signal type of the second-order physical system model, construct a general error formula according to the input signal type;

步骤S3：以所述误差公式的误差值作为滑模控制器的输入，使用sfal函数构建滑模控制器的滑模面；Step S3: using the error value of the error formula as the input of the sliding mode controller, using the sfal function to construct the sliding mode surface of the sliding mode controller;

步骤S4：构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，完成滑膜控制器的设计。Step S4: constructing the reaching rate equation of the sliding mode surface, and substituting the selected reaching rate equation into the derivative of the sliding mode surface to complete the design of the sliding film controller.

优选的，所述步骤S1中所述二阶物理系统模型具体如下：Preferably, the second-order physical system model in the step S1 is specifically as follows:

y＝x1；y = x 1 ;

Figure BDA0003612436660000021
Figure BDA0003612436660000021

其中，y为二阶物理系统模型的输出，f(x1,x2)为非线性函数，x1和x2分别为二阶物理系统模型的控制参数,u为滑模控制器的输出，b为增益参数。Among them, y is the output of the second-order physical system model, f(x 1 , x 2 ) is a nonlinear function, x 1 and x 2 are the control parameters of the second-order physical system model, u is the output of the sliding mode controller, b is the gain parameter.

优选的，所述步骤S2中通用的误差公式为e＝r-y，e为误差值，r为输入信号类型的输入值，y为二阶物理系统模型的输出。Preferably, the common error formula in the step S2 is e=r-y, e is the error value, r is the input value of the input signal type, and y is the output of the second-order physical system model.

优选的，所述步骤S3中构建滑模控制器的滑模面如下所示：Preferably, the sliding mode surface of the sliding mode controller constructed in the step S3 is as follows:

Figure BDA0003612436660000022
其中s为滑膜面，e为误差值，c、a和
Figure BDA0003612436660000023
为调节参数，sfal()为改进的fal函数；
Figure BDA0003612436660000022
Where s is the synovial surface, e is the error value, c, a and
Figure BDA0003612436660000023
To adjust parameters, sfal() is an improved fal function;

Figure BDA0003612436660000024
的具体表达式如下所示：
Figure BDA0003612436660000024
The specific expression of is as follows:

Figure BDA0003612436660000025
Figure BDA0003612436660000025

其中

Figure BDA0003612436660000026
in
Figure BDA0003612436660000026

Figure BDA0003612436660000027
Figure BDA0003612436660000027

Figure BDA0003612436660000028
Figure BDA0003612436660000028

Figure BDA0003612436660000031
Figure BDA0003612436660000031

优选的，所述步骤S4的具体过程如下：Preferably, the specific process of the step S4 is as follows:

构建所述滑模面的趋近率方程为以下方程其中之一：The approach rate equation for constructing the sliding mode surface is one of the following equations:

Figure BDA0003612436660000032
Figure BDA0003612436660000032

其中k为调节参数，ε为有界扰动的估值，s为滑模面，

Figure BDA0003612436660000033
为滑模面的趋近率即滑模面的导数，a为调节参数；Where k is the adjustment parameter, ε is the estimate of the bounded disturbance, s is the sliding surface,
Figure BDA0003612436660000033
is the approach rate of the sliding mode surface, that is, the derivative of the sliding mode surface, and a is the adjustment parameter;

根据所述调节参数k和ε的取值选择对应的滑模面的趋近率方程；Select the approach rate equation of the corresponding sliding mode surface according to the values of the adjustment parameters k and ε;

所述滑模面根据所述调节参数

Figure BDA0003612436660000034
的取值选择对应的滑模面并进行求导，得到滑模面导数；The sliding mode surface according to the adjustment parameters
Figure BDA0003612436660000034
Select the corresponding sliding mode surface for the value of , and perform derivation to obtain the sliding mode surface derivative;

将滑模面的趋近率与滑模面导数进行等比，得到所述滑模控制器输出的表达式；The rate of approach of the sliding mode surface is proportional to the derivative of the sliding mode surface to obtain the output expression of the sliding mode controller;

通过以所述输入信号类型的误差值作为滑模控制器输出表达式的输入值，得到所述滑模控制器的输出值。The output value of the sliding mode controller is obtained by using the error value of the input signal type as the input value of the output expression of the sliding mode controller.

一种基于改进fal函数的滑模控制器设计系统，使用上述一种基于改进fal函数的滑模控制器设计方法，其特征在于，包括：物理模型构建模块、误差获取模块、滑模面构建模块以及滑模控制器输出获取模块；A sliding mode controller design system based on the improved fal function, using the above-mentioned sliding mode controller design method based on the improved fal function, is characterized in that it includes: a physical model building module, an error acquisition module, and a sliding surface building module And the sliding mode controller output acquisition module;

所述物理模型构建模块用于对物理系统建模，得出对应的二阶物理系统模型；The physical model building block is used to model the physical system to obtain a corresponding second-order physical system model;

所述误差获取模块用于根据所述二阶物理系统模型的输入信号类型r，根据输入信号类型构建通用的误差公式；The error acquisition module is used to construct a general error formula according to the input signal type according to the input signal type r of the second-order physical system model;

所述滑模面构建模块用于以所述误差公式的误差值作为滑模控制器的输入，构建滑模控制器的滑模面；The sliding mode surface construction module is used to use the error value of the error formula as the input of the sliding mode controller to construct the sliding mode surface of the sliding mode controller;

所述滑模控制器输出获取模块用于构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，以所述输入信号类型的误差作为滑模控制器的输入值，得到滑模控制器的输出。The sliding mode controller output acquisition module is used to construct the reaching rate equation of the sliding mode surface, and select the reaching rate equation to substitute into the derivative of the sliding mode surface, and use the error of the input signal type as The input value of the sliding mode controller, the output of the sliding mode controller is obtained.

上述技术方案中的一个技术方案具有如下优点或有益效果：本发明利用非线性fal函数使控制系统收敛更快，减少系统相角滞后，提升系统的跟踪性能。另一方面，本发明利用非线性fal函数使控制系统在误差大时减少增益，提升系统的稳定性能。其次，本发明改进了传统fal函数使得非线性和线性之间切换更平滑。One of the above technical solutions has the following advantages or beneficial effects: the present invention uses the nonlinear fal function to make the control system converge faster, reduce the system phase angle lag, and improve the tracking performance of the system. On the other hand, the present invention uses the non-linear fal function to reduce the gain of the control system when the error is large, and improve the stability of the system. Secondly, the present invention improves the traditional fal function to make the switch between nonlinear and linear smoother.

附图说明Description of drawings

图1是本发明方法中一个实施例的流程图。Figure 1 is a flow chart of one embodiment of the method of the present invention.

图2是本发明系统中一个实施例的结构示意图。Fig. 2 is a schematic structural diagram of an embodiment of the system of the present invention.

图3是传统fal函数中不同调节参数a的对比图像。Figure 3 is a comparison image of different adjustment parameters a in the traditional fal function.

图4是传统fal函数中不同调节参数

Figure BDA0003612436660000041
的对比图像。Figure 4 shows different adjustment parameters in the traditional fal function
Figure BDA0003612436660000041
comparison images.

图5是传统fal函数与本发明中sfal函数对比图像。Fig. 5 is a comparison image of the traditional fal function and the sfal function in the present invention.

具体实施方式Detailed ways

下面详细描述本发明的实施例，所述实施例的示例在附图中示出，其中自始至终相同或类似的标号表示相同或类似的元件或具有相同或类似功能的元件。下面通过参考附图描述的实施例是示例性的，仅用于解释本发明，而不能理解为对本发明的限制。Embodiments of the present invention are described in detail below, examples of which are shown in the drawings, wherein the same or similar reference numerals designate the same or similar elements or elements having the same or similar functions throughout. The embodiments described below by referring to the figures are exemplary only for explaining the present invention and should not be construed as limiting the present invention.

在本发明的描述中，需要理解的是，术语“中心”、“纵向”、“横向”、“长度”、“宽度”、“厚度”、“上”、“下”、“左”、“右”、“竖直”、“水平”、“顶”、“底”、“内”、“外”、“轴向”、“径向”、“周向”等指示的方位或位置关系为基于附图所示的方位或位置关系，仅是为了便于描述本发明和简化描述，而不是指示或暗示所指的装置或元件必须具有特定的方位、以特定的方位构造和操作，因此不能理解为对本发明的限制。In describing the present invention, it should be understood that the terms "center", "longitudinal", "transverse", "length", "width", "thickness", "upper", "lower", "left", " The orientation or positional relationship indicated by "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "axial", "radial", "circumferential" is Based on the orientation or positional relationship shown in the drawings, it is only for the convenience of describing the present invention and simplifying the description, and does not indicate or imply that the referred device or element must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood To limit the present invention.

此外，术语“第一”、“第二”仅用于描述目的，而不能理解为指示或暗示相对重要性或者隐含指明所指示的技术特征的数量。由此，限定有“第一”、“第二”的特征可以明示或者隐含地包括一个或者更多个该特征。在本发明的描述中，除非另有说明，“多个”的含义是两个或两个以上。In addition, the terms "first" and "second" are used for descriptive purposes only, and cannot be interpreted as indicating or implying relative importance or implicitly specifying the quantity of indicated technical features. Thus, a feature defined as "first" and "second" may explicitly or implicitly include one or more of these features. In the description of the present invention, unless otherwise specified, "plurality" means two or more.

在本发明的描述中，需要说明的是，除非另有明确的规定和限定，术语“安装”、“相连”、“连接”应做广义理解，例如，可以是固定连接，也可以是可拆卸连接，或一体地连接；可以是直接相连，也可以通过中间媒介间接相连，可以是两个元件内部的连通。对于本领域的普通技术人员而言，可以具体情况理解上述术语在本发明中的具体含义。In the description of the present invention, it should be noted that unless otherwise specified and limited, the terms "installation", "connection" and "connection" should be understood in a broad sense, for example, it can be a fixed connection or a detachable connection. Connected, or integrally connected; it can be directly connected, or indirectly connected through an intermediary, and it can be the internal communication of two elements. Those of ordinary skill in the art can understand the specific meanings of the above terms in the present invention in specific situations.

如图1～5所示，一种基于改进fal函数的滑模控制器设计方法，包括以下步骤：As shown in Figures 1 to 5, a sliding mode controller design method based on the improved fal function includes the following steps:

步骤S1：对物理系统建模，得出对应的二阶物理系统模型；Step S1: Model the physical system to obtain the corresponding second-order physical system model;

步骤S2：根据所述二阶物理系统模型的输入信号类型，根据输入信号类型构建通用的误差公式；Step S2: according to the input signal type of the second-order physical system model, construct a general error formula according to the input signal type;

步骤S3：以所述误差公式的误差值作为滑模控制器的输入，使用sfal函数构建滑模控制器的滑模面；Step S3: using the error value of the error formula as the input of the sliding mode controller, using the sfal function to construct the sliding mode surface of the sliding mode controller;

步骤S4：构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，完成滑膜控制器的设计。Step S4: constructing the reaching rate equation of the sliding mode surface, and substituting the selected reaching rate equation into the derivative of the sliding mode surface to complete the design of the sliding film controller.

所述二阶物理系统模型在运用时可以为实际的操作对象，如电机、风扇、无人机等。在使用滑膜控制器时，滑膜控制器的输出值为二阶物理系统模型的输入值，通过误差值控制二阶物理系统模型使其能跟踪目标值，该目标值为二阶物理系统的输出，以提高二阶物理系统模型的鲁棒性。When the second-order physical system model is used, it can be an actual operation object, such as a motor, a fan, an unmanned aerial vehicle, and the like. When using a sliding film controller, the output value of the sliding film controller is the input value of the second-order physical system model, and the second-order physical system model is controlled by the error value so that it can track the target value, which is the second-order physical system. output to improve the robustness of second-order physical system models.

而本发明与现有技术相比，一方面，本发明利用非线性sfal函数使控制系统收敛更快，减少系统相角滞后，提升系统的跟踪性能。另一方面，本发明利用非线性sfal函数使控制系统在误差大时减少增益，提升系统的稳定性能。其次本发明改进了传统sfal函数使得非线性和线性之间切换更平滑。Compared with the prior art, the present invention, on the one hand, utilizes the nonlinear sfal function to make the control system converge faster, reduce the phase angle lag of the system, and improve the tracking performance of the system. On the other hand, the present invention uses the nonlinear sfal function to reduce the gain of the control system when the error is large, and improve the stability of the system. Secondly, the present invention improves the traditional sfal function to make the switch between nonlinear and linear smoother.

综上所述，本发明在非线性滑模控制中具有系统收敛更快、非线性切换平滑优点。In summary, the present invention has the advantages of faster system convergence and smooth nonlinear switching in nonlinear sliding mode control.

优选的，所述步骤S1中所述二阶物理系统模型具体如下：Preferably, the second-order physical system model in the step S1 is specifically as follows:

y＝x1；y = x 1 ;

Figure BDA0003612436660000061
Figure BDA0003612436660000061

其中，y为二阶物理系统模型的输出，f(x1,x2)为非线性函数，x1和x2分别为二阶物理系统模型的控制参数,u为滑模控制器的输出，b为增益参数。Among them, y is the output of the second-order physical system model, f(x 1 , x 2 ) is a nonlinear function, x 1 and x 2 are the control parameters of the second-order physical system model, u is the output of the sliding mode controller, b is the gain parameter.

x1和x2没有具体的定义，是根据所述二阶物理系统模型实际运用的场景不同，而具有不同的状态参数含义，但是x1和x2之间是存在特定的关系，所以本申请可以使用二阶物理系统模型对不同的工作场景的工作对象进行常规的定义；例如运用在无人机定位的场景下，x1为位置状态参数，则x2对应为速度状态参数；x 1 and x 2 have no specific definition, and have different meanings of state parameters according to the actual application scenarios of the second-order physical system model, but there is a specific relationship between x 1 and x 2 , so this application The second-order physical system model can be used to routinely define the working objects of different working scenarios; for example, in the scenario of UAV positioning, x 1 is the position state parameter, and x 2 corresponds to the speed state parameter;

当运用在电机功率控制的场景下时，x1为转动角度状态参数，则x2对应为转动角速度状态参数。对于x1和x2在不同应用场景的定义。可视具体情况可参考《现代控制理论》和《矩阵分析》。When used in the scene of motor power control, x 1 is the rotation angle state parameter, and x 2 corresponds to the rotation angular velocity state parameter. For the definition of x 1 and x 2 in different application scenarios. Depending on the specific situation, you can refer to "Modern Control Theory" and "Matrix Analysis".

优选的，所述步骤S2中通用的误差公式为e＝r-y，e为误差值，r为输入信号类型的输入值，y为二阶物理系统模型的输出。Preferably, the common error formula in the step S2 is e=r-y, e is the error value, r is the input value of the input signal type, and y is the output of the second-order physical system model.

优选的，所述步骤S3中构建滑模控制器的滑模面如下所示：Preferably, the sliding mode surface of the sliding mode controller constructed in the step S3 is as follows:

Figure BDA0003612436660000071
其中s为滑膜面，e为误差值，c、a和
Figure BDA0003612436660000072
为调节参数，sfal()为改进的fal函数；
Figure BDA0003612436660000071
Where s is the synovial surface, e is the error value, c, a and
Figure BDA0003612436660000072
To adjust parameters, sfal() is an improved fal function;

Figure BDA0003612436660000073
的具体表达式如下所示：
Figure BDA0003612436660000073
The specific expression of is as follows:

Figure BDA0003612436660000074
Figure BDA0003612436660000074

其中

Figure BDA0003612436660000075
in
Figure BDA0003612436660000075

Figure BDA0003612436660000076
Figure BDA0003612436660000076

Figure BDA0003612436660000077
Figure BDA0003612436660000077

Figure BDA0003612436660000078
Figure BDA0003612436660000078

值得一提的是对于非线性滑模面函数的选取一般有一下原则：It is worth mentioning that there are generally the following principles for the selection of nonlinear sliding mode surface functions:

1)非线性函数在原点有较好的收敛性和平滑性；1) The nonlinear function has better convergence and smoothness at the origin;

2)非线性函数在原点处的值恒为0；2) The value of the nonlinear function at the origin is always 0;

3)非线性函数在原点处可导、连续。3) The nonlinear function is derivable and continuous at the origin.

本发明中，对传统滑模控制中滑模面进行改进，引入非线性滑模面，同时在非线性滑模面引入了调节参数a和

Figure BDA0003612436660000081
作为调节，而c，a，
Figure BDA0003612436660000082
是根据实际应用场景取值，针对不同应用场景，这些参数的值不同，以实际应用为准。利用调节参数
Figure BDA0003612436660000083
控制系统稳态边界，让系统在误差小的时候呈线性滑模面，利用参数a控制非线性曲率，在反馈误差较大时，产生较小的反馈增益，误差较小时，产生较大的反馈增益，这样既可以保证系统的稳定性能，又使系统快速的达到稳定。如图3和图4所示，其中图X中为当
Figure BDA0003612436660000084
时，a＝0.25、0.5、0.75时的传统fal函数图像，而图4为当a＝0.25时，
Figure BDA0003612436660000085
0.5、0.75时的传统fal函数图像，图3和图4可以看出a的大小是影响函数曲率，而
Figure BDA0003612436660000086
的大小是控制函数线性区域的大小。所以非线性函数fal函数的参数调节过程中，可以使控制在反馈误差较大时，产生相对较小的反馈增益，不容易导致执行器输出饱和，误差较小时，产生相对较大的反馈增益，减少系统稳态误差，这样既可以保证系统的稳定性能，又使系统快速的达到稳定。滑模控制的研究中使用改进后的sfal函数能具有更快收敛的效果。In the present invention, the sliding mode surface in the traditional sliding mode control is improved, and the nonlinear sliding mode surface is introduced, and the adjustment parameters a and
Figure BDA0003612436660000081
as regulation, while c, a,
Figure BDA0003612436660000082
The values are selected according to the actual application scenario. The values of these parameters are different for different application scenarios, and the actual application shall prevail. Utilize tuning parameters
Figure BDA0003612436660000083
Control the steady-state boundary of the system, so that the system is a linear sliding surface when the error is small, and use the parameter a to control the nonlinear curvature. When the feedback error is large, a small feedback gain is generated, and when the error is small, a large feedback is generated. Gain, which can not only ensure the stable performance of the system, but also make the system reach stability quickly. As shown in Figure 3 and Figure 4, where in Figure X is when
Figure BDA0003612436660000084
, when a=0.25, 0.5, 0.75, the traditional fal function image, and Fig. 4 is when a=0.25,
Figure BDA0003612436660000085
The traditional fal function images at 0.5 and 0.75, as shown in Figure 3 and Figure 4, it can be seen that the size of a affects the curvature of the function, while
Figure BDA0003612436660000086
The size of is the size of the linear region of the control function. Therefore, in the parameter adjustment process of the nonlinear function fal function, the control can generate a relatively small feedback gain when the feedback error is large, and it is not easy to cause the output saturation of the actuator. When the error is small, a relatively large feedback gain is generated. Reduce the steady-state error of the system, which can not only ensure the stable performance of the system, but also make the system reach stability quickly. In the study of sliding mode control, the improved sfal function can have a faster convergence effect.

而图4中也表明fal函数在

Figure BDA0003612436660000087
时由于其函数的形式导致非平滑切换，因此对
Figure BDA0003612436660000088
段函数进行改进，利用函数拟合的方式代替原本的函数，通过利用高阶拟合的方法改进的sfal函数的特性使系统在非线性段和线性段切换时更平滑，减少系统稳态时颤抖。Figure 4 also shows that the fal function is in
Figure BDA0003612436660000087
When the form of its function causes non-smooth switching, so for
Figure BDA0003612436660000088
The segment function is improved, and the original function is replaced by the function fitting method. The characteristics of the sfal function improved by using the high-order fitting method make the system smoother when switching between the nonlinear segment and the linear segment, and reduce the trembling of the system in a steady state .

其中

Figure BDA0003612436660000089
时sfal的函数拟合过程如下：in
Figure BDA0003612436660000089
The function fitting process of sfal is as follows:

其中sfal的函数为sfal＝β1e+β2e2+β2tan(e),

Figure BDA00036124366600000810
在本申请中拟合利用的是泰勒展开原理，在本申请中的第三部分采用正切函数tan(e)，而不使用e3是因为tan(e)在原点附近的收敛更好。The function of sfal is sfal=β 1 e+β 2 e 2 +β 2 tan(e),
Figure BDA00036124366600000810
In this application, the fitting uses the Taylor expansion principle. In the third part of this application, the tangent function tan(e) is used instead of e 3 because tan(e) converges better near the origin.

当

Figure BDA00036124366600000811
时，需要满足如下：when
Figure BDA00036124366600000811
, the following needs to be met:

Figure BDA0003612436660000091
Figure BDA0003612436660000091

而当

Figure BDA0003612436660000092
时，需要满足如下：And when
Figure BDA0003612436660000092
, the following needs to be met:

Figure BDA0003612436660000093
Figure BDA0003612436660000093

通过上述两个公式结合，即可得到所述公式(3)。The formula (3) can be obtained by combining the above two formulas.

优选的，所述步骤S4的具体过程如下：Preferably, the specific process of the step S4 is as follows:

构建所述滑模面的趋近率方程为以下方程其中之一：The approach rate equation for constructing the sliding mode surface is one of the following equations:

Figure BDA0003612436660000094
Figure BDA0003612436660000094

Figure BDA0003612436660000095
Figure BDA0003612436660000095

Figure BDA0003612436660000096
Figure BDA0003612436660000096

其中k为调节参数，ε为有界扰动的估值，s为滑模面，

Figure BDA0003612436660000097
为滑模面的趋近率即滑模面的导数，a为调节参数；Where k is the adjustment parameter, ε is the estimate of the bounded disturbance, s is the sliding surface,
Figure BDA0003612436660000097
is the approach rate of the sliding mode surface, that is, the derivative of the sliding mode surface, and a is the adjustment parameter;

根据所述调节参数k和ε的取值选择对应的滑模面的趋近率方程；Select the approach rate equation of the corresponding sliding mode surface according to the values of the adjustment parameters k and ε;

所述滑模面根据所述调节参数

Figure BDA0003612436660000098
的取值选择对应的滑模面并进行求导，得到滑模面导数；The sliding mode surface according to the adjustment parameters
Figure BDA0003612436660000098
Select the corresponding sliding mode surface for the value of , and perform derivation to obtain the sliding mode surface derivative;

将滑模面的趋近率与滑模面导数进行等比，得到所述滑模控制器输出的表达式；The rate of approach of the sliding mode surface is proportional to the derivative of the sliding mode surface to obtain the output expression of the sliding mode controller;

通过以所述输入信号类型的误差值作为滑模控制器输出表达式的输入值，得到所述滑模控制器的输出值。The output value of the sliding mode controller is obtained by using the error value of the input signal type as the input value of the output expression of the sliding mode controller.

由于所述滑模面的导数的数学含义为滑模面的趋近律，故将将滑模面的趋近率与滑模面导数进行等比后，可以得到所述滑模控制器输出的表达式，下面以一个实施例作为解释：当ε为有界扰动的估值大于0时，所述所述滑模面的趋近率方程选定为

Figure BDA0003612436660000101
Since the mathematical meaning of the derivative of the sliding mode surface is the approach law of the sliding mode surface, after the approach rate of the sliding mode surface is proportional to the sliding mode surface derivative, the output of the sliding mode controller can be obtained expression, an embodiment is used as an explanation below: when ε is an estimate of a bounded disturbance greater than 0, the approach rate equation of the sliding mode surface is selected as
Figure BDA0003612436660000101

此时滑模面的表达式如下：

Figure BDA0003612436660000102
对滑模面进行求导
Figure BDA0003612436660000103
由于滑模面导数的数学物理意义就是滑模面的趋近率，所以将滑模面导数与构建的滑模面的趋近率进行等比得出：At this time, the expression of the sliding mode surface is as follows:
Figure BDA0003612436660000102
Differentiate the sliding surface
Figure BDA0003612436660000103
Since the mathematical and physical meaning of the derivative of the sliding mode surface is the approach rate of the sliding mode surface, the derivative of the sliding mode surface is compared with the approach rate of the constructed sliding mode surface to obtain:

Figure BDA0003612436660000104
Figure BDA0003612436660000104

Figure BDA0003612436660000105
Figure BDA0003612436660000105

此时将公式(1)与公式(2)带出到公式(5)中的e处既可以得出如下表达式：At this time, taking formula (1) and formula (2) to e in formula (5) can give the following expression:

Figure BDA0003612436660000107
Figure BDA0003612436660000107

Figure BDA0003612436660000106
Figure BDA0003612436660000106

此时只需要输入误差值e以及信号类型的输入值r即可获得滑模控制器的输出值u。At this time, it is only necessary to input the error value e and the input value r of the signal type to obtain the output value u of the sliding mode controller.

一种基于改进fal函数的滑模控制器设计系统，使用上述一种基于改进fal函数的滑模控制器设计方法，包括：物理模型构建模块、误差获取模块、滑模面构建模块以及滑模控制器输出获取模块；A sliding mode controller design system based on an improved fal function, using the above-mentioned sliding mode controller design method based on an improved fal function, including: a physical model building block, an error acquisition module, a sliding mode surface building block, and sliding mode control The output acquisition module of the device;

所述物理模型构建模块用于对物理系统建模，得出对应的二阶物理系统模型；The physical model building block is used to model the physical system to obtain a corresponding second-order physical system model;

所述误差获取模块用于根据所述二阶物理系统模型的输入信号类型r，根据输入信号类型构建通用的误差公式；The error acquisition module is used to construct a general error formula according to the input signal type according to the input signal type r of the second-order physical system model;

所述滑模面构建模块用于以所述误差公式的误差值作为滑模控制器的输入，构建滑模控制器的滑模面；The sliding mode surface construction module is used to use the error value of the error formula as the input of the sliding mode controller to construct the sliding mode surface of the sliding mode controller;

所述滑模控制器输出获取模块用于构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，以所述输入信号类型的误差作为滑模控制器的输入值，得到滑模控制器的输出。The sliding mode controller output acquisition module is used to construct the reaching rate equation of the sliding mode surface, and select the reaching rate equation to substitute into the derivative of the sliding mode surface, and use the error of the input signal type as The input value of the sliding mode controller, the output of the sliding mode controller is obtained.

优选的，其中所述物理模型构建模块中的二阶物理系统模型具体如下：Preferably, the second-order physical system model in the physical model building block is specifically as follows:

y＝x1；y = x 1 ;

Figure BDA0003612436660000111
Figure BDA0003612436660000111

其中，y为二阶物理系统模型的输出，f(x1,x2)为非线性函数,u为滑模控制器的输出，b为增益参数。Among them, y is the output of the second-order physical system model, f(x 1 , x 2 ) is the nonlinear function, u is the output of the sliding mode controller, and b is the gain parameter.

优选的，所述误差获取模块中所述中通用的误差公式为e＝r-y，e为误差值，r为输入信号类型的输入值，y为二阶物理系统模型的输出。Preferably, the common error formula in the error acquisition module is e=r-y, where e is the error value, r is the input value of the input signal type, and y is the output of the second-order physical system model.

优选的，所述滑模面构建模块中构建滑模控制器的滑模面如下所示：Preferably, the sliding mode surface of constructing the sliding mode controller in the described sliding mode surface building block is as follows:

Figure BDA0003612436660000112
其中c，a，
Figure BDA0003612436660000113
为可以调节参数；
Figure BDA0003612436660000112
where c, a,
Figure BDA0003612436660000113
for adjustable parameters;

Figure BDA0003612436660000114
的具体表达式如下所示：
Figure BDA0003612436660000114
The specific expression of is as follows:

Figure BDA0003612436660000115
Figure BDA0003612436660000115

其中

Figure BDA0003612436660000116
in
Figure BDA0003612436660000116

Figure BDA0003612436660000121
Figure BDA0003612436660000121

Figure BDA0003612436660000122
Figure BDA0003612436660000122

Figure BDA0003612436660000123
Figure BDA0003612436660000123

优选的，所述滑模控制器输出获取模块包括趋近率方程获取模块、选取模块以及输出模块；Preferably, the sliding mode controller output acquisition module includes a rate of approach acquisition module, a selection module and an output module;

所述趋近率方程获取模块用于构建所述滑模面的趋近率方程，其中趋近率方程为以下方程其中之一：The reaching rate equation acquisition module is used to construct the reaching rate equation of the sliding mode surface, wherein the reaching rate equation is one of the following equations:

Figure BDA0003612436660000124
Figure BDA0003612436660000124

Figure BDA0003612436660000125
Figure BDA0003612436660000125

Figure BDA0003612436660000126
Figure BDA0003612436660000126

选取模块用于根据所述调节参数k和有界扰动的估值ε的取值选择对应的滑模面的趋近率方程；The selection module is used to select the approach rate equation of the corresponding sliding mode surface according to the value of the adjustment parameter k and the estimated value ε of the bounded disturbance;

所述滑模面根据所述调节参数

Figure BDA0003612436660000127
的取值选择对应的滑模面并进行求导，得到滑模面导数；The sliding mode surface according to the adjustment parameters
Figure BDA0003612436660000127
Select the corresponding sliding mode surface for the value of , and perform derivation to obtain the sliding mode surface derivative;

所述输出模块用于将滑模面的趋近率与滑模面导数进行等比，得到所述滑模控制器输出的表达式；The output module is used to proportionalize the rate of approach of the sliding mode surface to the derivative of the sliding mode surface to obtain an expression output by the sliding mode controller;

通过以所述输入信号类型的误差值作为滑模控制器输出表达式的输入值，得到所述滑模控制器的输出值。The output value of the sliding mode controller is obtained by using the error value of the input signal type as the input value of the output expression of the sliding mode controller.

在本说明书的描述中，参考术语“一个实施例”、“一些实施例”、“示意性实施例”、“示例”、“具体示例”、或“一些示例”等的描述意指结合该实施例或示例描述的具体特征、结构、材料或者特点包含于本发明的至少一个实施例或示例中。在本说明书中，对上述术语的示意性表述不一定指的是相同的实施例或示例。而且，描述的具体特征、结构、材料或者特点可以在任何的一个或多个实施例或示例中以合适的方式结合。In the description of this specification, references to the terms "one embodiment," "some embodiments," "exemplary embodiments," "example," "specific examples," or "some examples" are intended to mean that the implementation A specific feature, structure, material, or characteristic described by an embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

尽管已经示出和描述了本发明的实施例，本领域的普通技术人员可以理解：在不脱离本发明的原理和宗旨的情况下可以对这些实施例进行多种变化、修改、替换和变型，本发明的范围由权利要求及其等同物限定。Although the embodiments of the present invention have been shown and described, those skilled in the art can understand that various changes, modifications, substitutions and modifications can be made to these embodiments without departing from the principle and spirit of the present invention. The scope of the invention is defined by the claims and their equivalents.

Claims (3)

1.一种基于改进fal函数的滑模控制器设计方法，其特征在于，包括以下步骤：1. a sliding mode controller design method based on improved fal function, is characterized in that, comprises the following steps: 步骤S1：对物理系统建模，得出对应的二阶物理系统模型；Step S1: Model the physical system to obtain the corresponding second-order physical system model; 步骤S2：根据所述二阶物理系统模型的输入信号类型，根据输入信号类型构建通用的误差公式；Step S2: according to the input signal type of the second-order physical system model, construct a general error formula according to the input signal type; 步骤S3：以所述误差公式的误差值作为滑模控制器的输入，使用sfal函数构建滑模控制器的滑模面；Step S3: using the error value of the error formula as the input of the sliding mode controller, using the sfal function to construct the sliding mode surface of the sliding mode controller; 步骤S4：构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，完成滑膜控制器的设计；Step S4: Construct the approach rate equation of the sliding mode surface, and substitute the selected approach rate equation into the derivative of the sliding mode surface to complete the design of the sliding film controller; 所述步骤S1中所述二阶物理系统模型具体如下：The second-order physical system model described in the step S1 is specifically as follows: y＝x1；y = x 1 ;
Figure FDA0003838727680000011
Figure FDA0003838727680000011
 其中，y为二阶物理系统模型的输出，f(x1，x2)为非线性函数，x1和x2分别为二阶物理系统模型的状态参数，u为滑模控制器的输出，b为增益参数；Among them, y is the output of the second-order physical system model, f(x 1 , x 2 ) is a nonlinear function, x 1 and x 2 are the state parameters of the second-order physical system model, u is the output of the sliding mode controller, b is the gain parameter; 所述步骤S2中通用的误差公式为e＝r-y，e为误差值，r为输入信号类型的输入值，y为二阶物理系统模型的输出；The general error formula in the step S2 is e=r-y, e is an error value, r is an input value of the input signal type, and y is an output of a second-order physical system model; 所述步骤S3中构建滑模控制器的滑模面如下所示：In the step S3, the sliding mode surface of the sliding mode controller is constructed as follows:
Figure FDA0003838727680000012
其中s为滑膜面，e为误差值，c、a和
Figure FDA0003838727680000013
为调节参数，sfal()为改进的fal函数；
Figure FDA0003838727680000012
Where s is the synovial surface, e is the error value, c, a and
Figure FDA0003838727680000013
To adjust parameters, sfal() is an improved fal function;
Figure FDA0003838727680000014
的具体表达式如下所示：
Figure FDA0003838727680000014
The specific expression of is as follows:
Figure FDA0003838727680000015
Figure FDA0003838727680000015
 其中
Figure FDA0003838727680000021
in
Figure FDA0003838727680000021
Figure FDA0003838727680000022
Figure FDA0003838727680000022
Figure FDA0003838727680000023
Figure FDA0003838727680000023
Figure FDA0003838727680000024
Figure FDA0003838727680000024
 2.根据权利要求1所述的一种基于改进fal函数的滑模控制器设计方法，其特征在于，所述步骤S4的具体过程如下：2. a kind of sliding mode controller design method based on improved fal function according to claim 1, is characterized in that, the concrete process of described step S4 is as follows: 构建所述滑模面的趋近率方程为以下方程其中之一：The approach rate equation for constructing the sliding mode surface is one of the following equations:
Figure FDA0003838727680000025
Figure FDA0003838727680000025
 其中k为调节参数，ε为有界扰动的估值，s为滑模面，
Figure FDA0003838727680000026
为滑模面的趋近率即滑模面的导数，a为调节参数；Where k is the adjustment parameter, ε is the estimate of the bounded disturbance, s is the sliding surface,
Figure FDA0003838727680000026
is the approach rate of the sliding mode surface, that is, the derivative of the sliding mode surface, and a is the adjustment parameter; 根据所述调节参数k和ε的取值选择对应的滑模面的趋近率方程；Select the approach rate equation of the corresponding sliding mode surface according to the values of the adjustment parameters k and ε; 所述滑模面根据所述调节参数
Figure FDA0003838727680000027
的取值选择对应的滑模面并进行求导，得到滑模面导数；The sliding mode surface according to the adjustment parameters
Figure FDA0003838727680000027
Select the corresponding sliding mode surface for the value of , and perform derivation to obtain the sliding mode surface derivative; 将滑模面的趋近率与滑模面导数进行等比，得到所述滑模控制器输出的表达式；The rate of approach of the sliding mode surface is proportional to the derivative of the sliding mode surface to obtain the output expression of the sliding mode controller; 通过以所述输入信号类型的误差值作为滑模控制器输出表达式的输入值，得到所述滑模控制器的输出值。The output value of the sliding mode controller is obtained by using the error value of the input signal type as the input value of the output expression of the sliding mode controller. 3.一种基于改进fal函数的滑模控制器设计系统，使用权利要求1～2任一项所述一种基于改进fal函数的滑模控制器设计方法，其特征在于，包括：物理模型构建模块、误差获取模块、滑模面构建模块以及滑模控制器输出获取模块；3. A sliding mode controller design system based on the improved fal function, using a kind of sliding mode controller design method based on the improved fal function described in any one of claims 1 to 2, characterized in that, comprising: physical model construction module, error acquisition module, sliding mode surface construction module and sliding mode controller output acquisition module; 所述物理模型构建模块用于对物理系统建模，得出对应的二阶物理系统模型；The physical model building block is used to model the physical system to obtain a corresponding second-order physical system model; 所述误差获取模块用于根据所述二阶物理系统模型的输入信号类型r，根据输入信号类型构建通用的误差公式；The error acquisition module is used to construct a general error formula according to the input signal type according to the input signal type r of the second-order physical system model; 其中所述二阶物理系统模型具体如下：Wherein the second-order physical system model is specifically as follows: y＝x1；y = x 1 ;
Figure FDA0003838727680000031
Figure FDA0003838727680000031
 其中，y为二阶物理系统模型的输出，f(x1，x2)为非线性函数，x1和x2分别为二阶物理系统模型的状态参数，u为滑模控制器的输出，b为增益参数；Among them, y is the output of the second-order physical system model, f(x 1 , x 2 ) is a nonlinear function, x 1 and x 2 are the state parameters of the second-order physical system model, u is the output of the sliding mode controller, b is the gain parameter; 其中误差公式为e＝r-y，e为误差值，r为输入信号类型的输入值，y为二阶物理系统模型的输出；Wherein the error formula is e=r-y, e is the error value, r is the input value of the input signal type, and y is the output of the second-order physical system model; 所述滑模面构建模块用于以所述误差公式的误差值作为滑模控制器的输入，构建滑模控制器的滑模面；The sliding mode surface construction module is used to use the error value of the error formula as the input of the sliding mode controller to construct the sliding mode surface of the sliding mode controller; 其中构建滑模控制器的滑模面如下所示：The sliding mode surface for constructing the sliding mode controller is as follows:
Figure FDA0003838727680000032
其中s为滑膜面，e为误差值，c、a和
Figure FDA0003838727680000033
为调节参数，sfal()为改进的fal函数；
Figure FDA0003838727680000032
Where s is the synovial surface, e is the error value, c, a and
Figure FDA0003838727680000033
To adjust parameters, sfal() is an improved fal function;
Figure FDA0003838727680000034
的具体表达式如下所示：
Figure FDA0003838727680000034
The specific expression of is as follows:
Figure FDA0003838727680000041
Figure FDA0003838727680000041
 其中
Figure FDA0003838727680000042
in
Figure FDA0003838727680000042
Figure FDA0003838727680000043
Figure FDA0003838727680000043
Figure FDA0003838727680000044
Figure FDA0003838727680000044
Figure FDA0003838727680000045
Figure FDA0003838727680000045
 所述滑模控制器输出获取模块用于构建所述滑模面的趋近率方程，并选定趋近率方程代入到所述滑模面的导数中，以所述输入信号类型的误差作为滑模控制器的输入值，得到滑模控制器的输出。The sliding mode controller output acquisition module is used to construct the reaching rate equation of the sliding mode surface, and select the reaching rate equation to substitute into the derivative of the sliding mode surface, and use the error of the input signal type as The input value of the sliding mode controller, the output of the sliding mode controller is obtained.
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