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

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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CN114721274A (en
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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

The invention aims to provide a sliding mode controller design method and a system based on an improved fal function, wherein the sliding mode controller design method comprises the following steps: step S1: modeling a physical system to obtain a corresponding second-order physical system model; step S2: constructing a general error formula according to the input signal type of the second-order physical system model; and step S3: taking the error value of the error formula as the input of the sliding mode controller, and constructing a sliding mode surface of the sliding mode controller by using an sfal function; and step S4: and constructing an approach rate equation of the sliding mode surface, and selecting the approach rate equation to substitute into a derivative of the sliding mode surface to complete the design of the sliding mode controller. On the other hand, the 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.

Description

Sliding mode controller design method and system based on improved fal function
Technical Field
The invention relates to the technical field of sliding mode controllers, in particular to a sliding mode controller design method and a sliding mode controller design system based on an improved fal function.
Background
Sliding mode control is a special type of non-linear control, and non-linearity appears as a discontinuity. The sliding mode function can be designed and is irrelevant to object parameters and disturbance, so that the sliding mode control has the advantages of quick response, insensitive corresponding parameter change and disturbance, no need of system online identification, simple physical realization and the like.
When a fal function is adopted in sliding mode control to act on nonlinear control of sliding mode control, the fal function is a linear function when an error is less than a preset value, and is a nonlinear function when the error is greater than the preset value, and in the switching of the linear function and the nonlinear function, non-smooth switching exists, so that a system has certain flutter near a switching point due to the fact that the nonlinear function is non-smooth in a sliding mode controller with a selected nonlinear sliding mode surface.
Disclosure of Invention
In view of the above-mentioned drawbacks, the present invention provides a sliding mode controller design method and system based on an improved fal function, so that when the fal function is applied to the nonlinear control of the sliding mode control, smooth transition can be performed at the system switching point.
In order to achieve the purpose, the invention adopts the following technical scheme: a design method of a sliding mode controller based on an improved fal function comprises the following steps:
step S1: modeling a physical system to obtain a corresponding second-order physical system model;
step S2: constructing a general error formula according to the input signal type of the second-order physical system model;
and step S3: taking the error value of the error formula as the input of the sliding mode controller, and constructing a sliding mode surface of the sliding mode controller by using an sfal function;
and step S4: and constructing an approach rate equation of the sliding mode surface, and selecting the approach rate equation to substitute into the derivative of the sliding mode surface to complete the design of the sliding mode controller.
Preferably, the second-order physical system model in step S1 is specifically as follows:
y＝x1；
Figure BDA0003612436660000021
where y is the output of the second order physical system model, f (x)1,x2) Is a non-linear function, x1And x2And the control parameters are respectively control parameters of a second-order physical system model, u is the output of the sliding mode controller, and b is a gain parameter.
Preferably, the general error formula in step S2 is e = r-y, e is an error value, r is an input value of an input signal type, and y is an output of the second-order physical system model.
Preferably, the sliding mode surface for constructing the sliding mode controller in step S3 is as follows:
Figure BDA0003612436660000022
wherein s is the slide film surface, e is the error value, c, a and
Figure BDA0003612436660000023
for adjusting parameters, sfal () is a modified fal function;
Figure BDA0003612436660000024
the specific expression of (a) is as follows:
Figure BDA0003612436660000025
wherein
Figure BDA0003612436660000026
Figure BDA0003612436660000027
Figure BDA0003612436660000028
Figure BDA0003612436660000031
Preferably, the specific process of step S4 is as follows:
the approximation rate equation for constructing the sliding mode surface is one of the following equations:
Figure BDA0003612436660000032
where k is the tuning parameter,. Epsilon.is the estimate of the bounded perturbation, s is the sliding mode surface,
Figure BDA0003612436660000033
the approach rate of the slip form surface is the derivative of the slip form surface, and a is an adjusting parameter;
selecting an approach rate equation of a corresponding sliding mode surface according to the values of the adjusting parameters k and epsilon;
the sliding form surface is adjusted according to the adjusting parameters
Figure BDA0003612436660000034
Selecting a corresponding sliding mode surface and carrying out derivation to obtain a sliding mode surface derivative;
carrying out equal ratio on the approach rate of the sliding mode surface and the derivative of the sliding mode surface to obtain an expression output by the sliding mode controller;
and obtaining an output value of the sliding mode controller by taking the error value of the input signal type as an input value of an output expression of the sliding mode controller.
A sliding mode controller design system based on an improved fal function uses the sliding mode controller design method based on the improved fal function, and is characterized by comprising the following steps: the system comprises a physical model construction module, an error acquisition module, a sliding mode surface construction module and a sliding mode controller output acquisition module;
the physical model building module is used for modeling a physical system to obtain a corresponding second-order physical system model;
the error acquisition module is used for constructing a general error formula according to the input signal type r of the second-order physical system model and the input signal type;
the sliding mode surface construction module is used for constructing a sliding mode surface of the sliding mode controller by taking an error value of the error formula as the input of the sliding mode controller;
and the output acquisition module of the sliding mode controller is used for constructing an approach rate equation of the sliding mode surface, selecting the approach rate equation to be substituted into a derivative of the sliding mode surface, and taking the error of the input signal type as an input value of the sliding mode controller to obtain the output of the sliding mode controller.
One of the above technical solutions has the following advantages or beneficial effects: the invention utilizes the nonlinear fal function to lead the convergence of the control system to be faster, reduce the phase angle lag of the system and improve the tracking performance of the system. On the other hand, the invention utilizes the nonlinear fal function to reduce the gain of the control system when the error is large, thereby improving the stability of the system. Second, the present invention improves the conventional fal function to make the switch between non-linear and linear smoother.
Drawings
FIG. 1 is a flow chart of one embodiment of the method of the present invention.
Fig. 2 is a schematic structural diagram of an embodiment of the system of the present invention.
Fig. 3 is a comparison image of different adjustment parameters a in the conventional fal function.
FIG. 4 is a graph of different tuning parameters in a conventional fal function
Figure BDA0003612436660000041
The contrast image of (1).
FIG. 5 is a comparison of the conventional fal function and the sfal function of the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
In the description of the present invention, it is to be understood that the terms "center", "longitudinal", "lateral", "length", "width", "thickness", "upper", "lower", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "axial", "radial", "circumferential", and the like, indicate orientations and positional relationships based on those shown in the drawings, and are used merely for convenience of description and for simplicity of description, and do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus, should not be considered as limiting the present invention.
Furthermore, the terms "first", "second" and "first" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of the present invention, "a plurality" means two or more unless otherwise specified.
In the description of the present invention, it should be noted that, unless otherwise explicitly specified or limited, the terms "mounted," "connected," and "connected" are to be construed broadly, e.g., as meaning either a fixed connection, a removable connection, or an integral connection; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood in specific cases to those skilled in the art.
As shown in fig. 1 to 5, a sliding mode controller design method based on an improved fal function includes the following steps:
step S1: modeling a physical system to obtain a corresponding second-order physical system model;
step S2: constructing a general error formula according to the input signal type of the second-order physical system model;
and step S3: taking the error value of the error formula as the input of the sliding mode controller, and constructing a sliding mode surface of the sliding mode controller by using an sfal function;
and step S4: and constructing an approach rate equation of the sliding mode surface, and selecting the approach rate equation to substitute into the derivative of the sliding mode surface to complete the design of the sliding mode controller.
The second order physical system model can be an actual operation object when in use, such as a motor, a fan, an unmanned aerial vehicle and the like. When the sliding film controller is used, the output value of the sliding film controller is the input value of the second-order physical system model, the second-order physical system model is controlled through the error value to enable the second-order physical system model to track the target value, and the target value is the output of the second-order physical system, so that the robustness of the second-order physical system model is improved.
Compared with the prior art, on one hand, the nonlinear sfal function is utilized to enable the control system to be converged more quickly, reduce the phase angle lag of the system and improve the tracking performance of the system. On the other hand, the 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. The invention improves the traditional sfal function to make the switch between non-linearity and linearity smoother.
In summary, the present invention has the advantages of faster system convergence and smooth nonlinear switching in the nonlinear sliding mode control.
Preferably, the second-order physical system model in step S1 is specifically as follows:
y＝x1；
Figure BDA0003612436660000061
where y is the output of the second order physical system model, f (x)1,x2) As a non-linear function, x1And x2And the control parameters are respectively control parameters of a second-order physical system model, u is the output of the sliding mode controller, and b is a gain parameter.
x1And x2There is no specific definition, but x has different meaning of state parameter according to different actual operation scenes of the second order physical system model1And x2The two-order physical system model can be used for carrying out conventional definition on the working objects of different working scenes; for example, in the positioning of unmanned aerial vehicles, x1Is a position state parameter, then x2Corresponding to a speed state parameter;
when applied in the context of motor power control, x1Is a rotation angle state parameter, then x2Corresponding to the state parameter of the rotational angular velocity. For x1And x2And (4) definition of different application scenes. Reference may be made to modern control theory and matrix analysis, depending on the specific situation.
Preferably, the general error formula in step S2 is e = r-y, e is an error value, r is an input value of an input signal type, and y is an output of the second-order physical system model.
Preferably, the sliding mode surface of the sliding mode controller constructed in step S3 is as follows:
Figure BDA0003612436660000071
wherein s is the slide film surface, e is the error value, c, a and
Figure BDA0003612436660000072
to adjust the parameters, sfal () is a modified fal function;
Figure BDA0003612436660000073
the specific expression of (a) is as follows:
Figure BDA0003612436660000074
wherein
Figure BDA0003612436660000075
Figure BDA0003612436660000076
Figure BDA0003612436660000077
Figure BDA0003612436660000078
It is worth mentioning that there is generally the following principle for the selection of the nonlinear sliding mode surface function:
1) The nonlinear function has better convergence and smoothness at the origin;
2) The value of the nonlinear function at the origin is constant 0;
3) The non-linear function is conductive and continuous at the origin.
In the invention, the sliding mode surface in the traditional sliding mode control is improved, the nonlinear sliding mode surface is introduced, and simultaneously, the adjusting parameter a and the adjusting parameter a are introduced into the nonlinear sliding mode surface
Figure BDA0003612436660000081
As a regulation, the pressure of c, a,
Figure BDA0003612436660000082
the values of the parameters are different according to the values of the actual application scenes and aiming at different application scenes, and the actual application is taken as the standard. Using regulating parameters
Figure BDA0003612436660000083
The steady-state boundary of the system is controlled to enable the system to be in a linear sliding mode surface when the error is small, the nonlinear curvature is controlled by using the parameter a, when the feedback error is large, small feedback gain is generated, and when the error is small, large feedback gain is generated, so that the stability of the system can be ensured, and the system can be quickly stabilized. As shown in fig. 3 and 4, wherein in the figure X is
Figure BDA0003612436660000084
The conventional fal function image when a =0.25, 0.5, 0.75, while fig. 4 is the image when a =0.25,
Figure BDA0003612436660000085
0.5 and 0.75, the sizes of a are the influence function curvature, and the sizes of a are shown in figures 3 and 4
Figure BDA0003612436660000086
Is the size of the linear region of the control function. Therefore, in the parameter adjusting process of the fal function of the nonlinear function, when the feedback error is large, the control can generate relatively small feedback gain, the output saturation of the actuator is not easy to cause, when the error is small, the relatively large feedback gain is generated, and the steady-state error of the system is reduced, so that the stability of the system can be ensured, and the system can be quickly stabilized. The improved sfal function used in the sliding mode control research can have the effect of faster convergence.
While FIG. 4 also shows the fal function in
Figure BDA0003612436660000087
When it is due to the form of its function causing non-smooth switching, therefore
Figure BDA0003612436660000088
The segment function is improved, the original function is replaced by a function fitting method, and the characteristic of the sfal function improved by a high-order fitting method enables the system to be non-linearThe linear segment and the linear segment are smoother when switched, and the judder of the system in a steady state is reduced.
Wherein
Figure BDA0003612436660000089
The function fitting procedure for sfal is as follows:
wherein the function of sfal is sfal = β1e+β2e2+β2tan(e),
Figure BDA00036124366600000810
The fitting in this application uses the Taylor expansion principle, and the third part in this application uses the tangent function tan (e) instead of e3Because the convergence of tan (e) is better near the origin.
When in use
Figure BDA00036124366600000811
Then, the following needs to be satisfied:
Figure BDA0003612436660000091
when in
Figure BDA0003612436660000092
Then, the following needs are satisfied:
Figure BDA0003612436660000093
the formula (3) can be obtained by combining the two formulas.
Preferably, the specific process of step S4 is as follows:
and constructing an approach rate equation of the sliding mode surface as one of the following equations:
Figure BDA0003612436660000094
Figure BDA0003612436660000095
Figure BDA0003612436660000096
where k is the tuning parameter,. Epsilon.is the estimate of the bounded perturbation, s is the sliding mode surface,
Figure BDA0003612436660000097
the approach rate of the slip form surface is the derivative of the slip form surface, and a is an adjusting parameter;
selecting an approach rate equation of a corresponding sliding mode surface according to the values of the adjusting parameters k and epsilon;
the sliding mode surface is adjusted according to the adjusting parameter
Figure BDA0003612436660000098
Selecting a corresponding sliding mode surface and carrying out derivation to obtain a sliding mode surface derivative;
carrying out equal ratio on the approach rate of the sliding mode surface and the derivative of the sliding mode surface to obtain an expression output by the sliding mode controller;
and obtaining an output value of the sliding mode controller by taking the error value of the input signal type as an input value of an output expression of the sliding mode controller.
Since the mathematical meaning of the derivative of the sliding mode surface is the approximation rule of the sliding mode surface, after the approximation rate of the sliding mode surface is equal to the derivative of the sliding mode surface, the expression output by the sliding mode controller can be obtained, and the following explanation is made by taking an embodiment: when the estimated value of epsilon as bounded disturbance is more than 0, the approach rate equation of the sliding mode surface is selected as
Figure BDA0003612436660000101
The expression of the slip form surface at this time is as follows:
Figure BDA0003612436660000102
face-to-face slip formLine derivation
Figure BDA0003612436660000103
Because the mathematical and physical meaning of the slip form surface derivative is the approach rate of the slip form surface, the slip form surface derivative and the approach rate of the constructed slip form surface are subjected to equal ratio to obtain:
Figure BDA0003612436660000104
Figure BDA0003612436660000105
taking equations (1) and (2) out to e in equation (5) at this time may result in the following expression:
Figure BDA0003612436660000107
Figure BDA0003612436660000106
at this time, the output value u of the sliding mode controller can be obtained only by inputting the error value e and the input value r of the signal type.
A sliding mode controller design system based on an improved fal function uses the sliding mode controller design method based on the improved fal function, and comprises the following steps: the system comprises a physical model construction module, an error acquisition module, a sliding mode surface construction module and a sliding mode controller output acquisition module;
the physical model building module is used for modeling a physical system to obtain a corresponding second-order physical system model;
the error acquisition module is used for constructing a general error formula according to the input signal type r of the second-order physical system model and the input signal type;
the sliding mode surface construction module is used for constructing a sliding mode surface of the sliding mode controller by taking an error value of the error formula as the input of the sliding mode controller;
and the sliding mode controller output acquisition module is used for constructing an approach rate equation of the sliding mode surface, selecting the approach rate equation to be substituted into a derivative of the sliding mode surface, and taking the error of the input signal type as an input value of the sliding mode controller to obtain the output of the sliding mode controller.
Preferably, the second-order physical system model in the physical model building module is specifically as follows:
y＝x1；
Figure BDA0003612436660000111
where y is the output of the second order physical system model, f (x)1,x2) Is a nonlinear function, u is the output of the sliding mode controller, and b is the gain parameter.
Preferably, the general error formula in the error obtaining module is e = r-y, e is an error value, r is an input value of an input signal type, and y is an output of a second-order physical system model.
Preferably, the sliding mode surface of the sliding mode controller built in the sliding mode surface building module is as follows:
Figure BDA0003612436660000112
wherein the ratio of c, a,
Figure BDA0003612436660000113
to allow adjustment of the parameters;
Figure BDA0003612436660000114
the specific expression of (a) is as follows:
Figure BDA0003612436660000115
wherein
Figure BDA0003612436660000116
Figure BDA0003612436660000121
Figure BDA0003612436660000122
Figure BDA0003612436660000123
Preferably, the sliding mode controller output acquisition module comprises an approach rate equation acquisition module, a selection module and an output module;
the approach rate equation obtaining module is used for constructing an approach rate equation of the sliding mode surface, wherein the approach rate equation is one of the following equations:
Figure BDA0003612436660000124
Figure BDA0003612436660000125
Figure BDA0003612436660000126
the selection module is used for selecting an approach rate equation of a corresponding sliding mode surface according to the adjusting parameter k and the value of the estimation value epsilon of the bounded disturbance;
the sliding mode surface is adjusted according to the adjusting parameter
Figure BDA0003612436660000127
Selecting a corresponding sliding mode surface and carrying out derivation to obtain a sliding mode surface derivative;
the output module is used for carrying out equal ratio on the approach rate of the sliding mode surface and the derivative of the sliding mode surface to obtain an expression output by the sliding mode controller;
and obtaining an output value of the sliding mode controller by taking the error value of the input signal type as an input value of an output expression of the sliding mode controller.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims (3)

1. A design method of a sliding mode controller based on an improved fal function is characterized by comprising the following steps:
step S1: modeling a physical system to obtain a corresponding second-order physical system model;
step S2: constructing a general error formula according to the input signal type of the second-order physical system model;
and step S3: taking the error value of the error formula as the input of the sliding mode controller, and constructing a sliding mode surface of the sliding mode controller by using an sfal function;
and step S4: constructing an approach rate equation of the sliding mode surface, and selecting the approach rate equation to substitute into a derivative of the sliding mode surface to complete the design of the sliding mode controller;
the second-order physical system model in step S1 is specifically as follows:
y＝x1；
Figure FDA0003838727680000011
where y is the output of the second order physical system model, f (x)1，x2) Is a non-linear function, x1And x2Respectively are state parameters of a second-order physical system model, u is the output of the sliding mode controller, and b is a gain parameter;
the general error formula in the step S2 is e = r-y, e is an error value, r is an input value of an input signal type, and y is an output of a second-order physical system model;
the sliding mode surface for constructing the sliding mode controller in the step S3 is as follows:
Figure FDA0003838727680000012
wherein s is the slide film surface, e is the error value, c, a and
Figure FDA0003838727680000013
to adjust the parameters, sfal () is a modified fal function;
Figure FDA0003838727680000014
the specific expression of (a) is as follows:
Figure FDA0003838727680000015
wherein
Figure FDA0003838727680000021
Figure FDA0003838727680000022
Figure FDA0003838727680000023
Figure FDA0003838727680000024
2. The design method of the sliding-mode controller based on the improved fal function according to claim 1, wherein the specific process of step S4 is as follows:
the approximation rate equation for constructing the sliding mode surface is one of the following equations:
Figure FDA0003838727680000025
where k is the tuning parameter,. Epsilon.is the estimate of the bounded perturbation, s is the sliding mode surface,
Figure FDA0003838727680000026
the approach rate of the slip form surface is the derivative of the slip form surface, and a is an adjusting parameter;
selecting an approach rate equation of a corresponding sliding mode surface according to the values of the adjusting parameters k and epsilon;
the sliding form surface is adjusted according to the adjusting parameters
Figure FDA0003838727680000027
Selecting a corresponding sliding mode surface and carrying out derivation to obtain a sliding mode surface derivative;
carrying out equal ratio on the approach rate of the sliding mode surface and the derivative of the sliding mode surface to obtain an expression output by the sliding mode controller;
and obtaining an output value of the sliding mode controller by taking the error value of the input signal type as an input value of an output expression of the sliding mode controller.
3. A sliding-mode controller design system based on an improved fal function, which uses the sliding-mode controller design method based on the improved fal function as claimed in any one of claims 1-2, and is characterized by comprising: the system comprises a physical model construction module, an error acquisition module, a sliding mode surface construction module and a sliding mode controller output acquisition module;
the physical model building module is used for modeling a physical system to obtain a corresponding second-order physical system model;
the error acquisition module is used for constructing a general error formula according to the input signal type r of the second-order physical system model and the input signal type;
the second-order physical system model specifically comprises the following steps:
y＝x1；
Figure FDA0003838727680000031
where y is the output of the second order physical system model, f (x)1，x2) Is a non-linear function, x1And x2Respectively are state parameters of a second-order physical system model, u is the output of the sliding mode controller, and b is a gain parameter;
wherein the error formula is e = r-y, e is an error value, r is an input value of an input signal type, and y is an output of a second-order physical system model;
the sliding mode surface construction module is used for constructing a sliding mode surface of the sliding mode controller by taking an error value of the error formula as the input of the sliding mode controller;
the sliding mode surfaces in which the sliding mode controller is constructed are as follows:
Figure FDA0003838727680000032
wherein s is the slide film surface, e is the error value, c, a and
Figure FDA0003838727680000033
to adjust the parameters, sfal () is a modified fal function;
Figure FDA0003838727680000034
the specific expression of (a) is as follows:
Figure FDA0003838727680000041
wherein
Figure FDA0003838727680000042
Figure FDA0003838727680000043
Figure FDA0003838727680000044
Figure FDA0003838727680000045
And the output acquisition module of the sliding mode controller is used for constructing an approach rate equation of the sliding mode surface, selecting the approach rate equation to be substituted into a derivative of the sliding mode surface, and taking the error of the input signal type as an input value of the sliding mode controller to obtain the output of the sliding mode controller.
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