CN114035436A - Backstepping control method based on saturation adaptive law, storage medium and equipment - Google Patents

Backstepping control method based on saturation adaptive law, storage medium and equipment Download PDF

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CN114035436A
CN114035436A CN202111405311.2A CN202111405311A CN114035436A CN 114035436 A CN114035436 A CN 114035436A CN 202111405311 A CN202111405311 A CN 202111405311A CN 114035436 A CN114035436 A CN 114035436A
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郑晓龙
杨学博
李湛
高会军
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Harbin Institute of Technology
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Abstract

A backstepping control method, a storage medium and equipment based on a saturation adaptive law belong to the technical field of nonlinear system control. The method aims to solve the problems that the existing self-adaptive backstepping control method cannot process unknown nonlinear functions in a system and the existing saturation control is unsmooth. Aiming at a controlled object, a two-dimensional nonlinear system state space model is established, and two state variables exist in the two-dimensional nonlinear system; then according to the state variable x of the system1And constructing an error variable z from the target signal1According to the state variable x2And a virtual control function alpha to be designed1Construction of the error variable z2Designing a Lyapunov function, solving a first derivative of the Lyapunov function with respect to time, and designing a virtual control function alpha based on the first derivative of the Lyapunov function1And controlling the input u, and finally designing to obtain a saturation adaptive law. The method is mainly used for controlling the nonlinear system.

Description

Backstepping control method based on saturation adaptive law, storage medium and equipment
Technical Field
The invention belongs to the technical field of nonlinear system control, and particularly relates to a backstepping control method based on a saturation adaptive law, a storage medium and equipment.
Background
The basic idea of the method is to estimate the unknown constant in the system by using the adaptive parameter, and then to complete the design of the control law by using the adaptive parameter and the backstepping control idea. At present, the adaptive backstepping control method has been widely applied to industrial automation systems, automobile control systems, robot control systems, and the like, and for the design details of the adaptive backstepping control method, reference may be made to chinese patent CN108869420B, chinese patent CN106787940B, and chinese patent CN 105573125A.
It should be noted that in the conventional adaptive back-stepping control method, the adaptive parameters are designed to only handle the uncertainty of the system parameters, i.e.: the adaptive parameters can only estimate the unknown constants in the system. When the system contains unknown nonlinear functions, the traditional self-adaptive backstepping control method is not applicable any more. For an actual nonlinear system, the system uncertainty is often caused by some unknown nonlinear functions, such as viscous friction nonlinearity, actuator dead zone/saturation nonlinearity, and the like. The traditional self-adaptive backstepping control method cannot process unknown nonlinear functions of the system, so that the method is greatly limited in practical application. Therefore, how to design an adaptive backstepping control strategy capable of handling the unknown nonlinear function of the system is a key problem.
Disclosure of Invention
The invention aims to solve the problems that the existing self-adaptive backstepping control method cannot process unknown nonlinear functions in a system and the existing saturation control is unsmooth.
A backstepping control method based on a saturation adaptive law is characterized in that a two-dimensional nonlinear system state space model is established for a controlled object, and the controlled object is controlled by the saturation adaptive law;
the design process of the saturation adaptive law comprises the following steps:
the method comprises the following steps of firstly, establishing a two-dimensional nonlinear system state space model aiming at a controlled object, wherein the established state space model of the two-dimensional nonlinear system is in the following specific form:
Figure BDA0003372022880000011
Figure BDA0003372022880000012
y＝x1
wherein x is1,x2Represents the state variable of the system and is,
Figure BDA0003372022880000013
representing the system state variable x1,x2The first derivative of (a), y is system output, f is a nonlinear smooth function unknown to the system, d is unknown disturbance outside the system, u represents a system control input signal, and the control aim is to design the system control input u so that the system output y tracks a given target signal;
step two, according to the state variable x of the system1And a target signal ydAnd the state variable x of the system2And a virtual control function alpha to be designed1Separately constructing the error variable z1And z2；
Step three, utilizing the error variable z obtained in the step two1And z2Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Figure BDA0003372022880000021
Step five, according to the first derivative of the Lyapunov function
Figure BDA0003372022880000022
Designing a virtual control function alpha1And based on a virtual control function alpha1Design control input u:
Figure BDA0003372022880000023
wherein the content of the first and second substances,
Figure BDA0003372022880000024
representing a virtual control function alpha1The first derivative of (a) is,
Figure BDA0003372022880000025
for estimation of an unknown non-linear function f, k2Is a constant greater than zero;
Figure BDA0003372022880000026
the function is a saturation function;
Figure BDA0003372022880000027
the function is a saturation function and is defined as follows:
Figure BDA0003372022880000028
wherein the constant c satisfies
Figure BDA0003372022880000029
Step six, based on the first derivative of the Lyapunov function
Figure BDA00033720228800000210
And a virtual control function alpha1And finally designing a control input u to obtain a saturation adaptive law.
Furthermore, the controlled object is a motor system, and the corresponding system state variable x1,x2The rotational angle and the rotational speed of the motor system,
Figure BDA00033720228800000211
the rotating speed and the angular acceleration of the motor system are shown, and the system output y is the rotating angle of the motor system.
Further, a solution of a state space model (1) of the two-dimensional nonlinear system exists and is unique; unknown non-linear smoothing function f andits first order second derivative satisfies
Figure BDA00033720228800000212
Wherein
Figure BDA00033720228800000213
Is a constant greater than zero; target signal ydAnd its first and second derivatives are bounded.
Further, the error variable z in step two1And z2The following were used:
Figure BDA00033720228800000214
wherein alpha is1Representing the virtual control function to be designed.
Further, the error variable z set in the step two is used1And z2Designed Lyapunov function
Figure BDA00033720228800000215
Further, the first derivative of the lyapunov function V with respect to time in step three is:
Figure BDA0003372022880000031
wherein the content of the first and second substances,
Figure BDA0003372022880000032
representing a virtual control function alpha1The first derivative of (a).
Further, the virtual control function α1As follows
Figure BDA0003372022880000033
Wherein k is1Is a constant greater than zero.
Further, the saturation adaptive law in the step six is designed as follows:
Figure BDA0003372022880000034
output of filter
Figure BDA0003372022880000035
Figure BDA0003372022880000036
Wherein, mu1,μ2,μ3Is a constant greater than zero, η is a constant greater than zero, and s is a filter state variable.
A storage medium having stored therein at least one instruction that is loaded and executed by a processor to implement a method of backstepping control based on a saturated adaptive law.
An apparatus comprising a processor and a memory, the memory having stored therein at least one instruction that is loaded and executed by the processor to implement the method for saturation adaptive law based backstepping control.
The invention has the beneficial effects that:
the invention provides a backstepping control method based on a saturation self-adaptive law. Compared with the traditional self-adaptive backstepping control method which cannot process the unknown nonlinear function of the system, the method can directly estimate and process the unknown nonlinear function of the system; meanwhile, the saturation function designed by the invention can be conducted in a second order, so that the control effect of the system is smoother.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a system tracking performance curve under the method of the present invention;
FIG. 3 is a system tracking error curve under the method of the present invention;
FIG. 4 shows a system state x according to the method of the present invention2A response curve;
FIG. 5 is a diagram illustrating a system adaptive estimation performance curve according to the present invention;
FIG. 6 is a response curve of the system control input u according to the method of the present invention.
Detailed Description
The first embodiment is as follows: the present embodiment is described in connection with figure 1,
the embodiment is a backstepping control method based on a saturation adaptive law, aiming at a controlled object, establishing a two-dimensional nonlinear system state space model, and controlling the controlled object by using the saturation adaptive law;
the design process of the saturation adaptive law comprises the following steps:
step one, aiming at a controlled object, establishing a two-dimensional nonlinear system state space model, wherein two state variables x exist in the two-dimensional nonlinear system1And x2A control input and a given target signal yd；
Step two, according to the state variable x of the system1And a target signal ydAnd the state variable x of the system2And a virtual control function alpha to be designed1Separately constructing the error variable z1And z2；
Step three, utilizing the error variable z obtained in the step two1And z2Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Figure BDA0003372022880000041
Step five, according to the first derivative of the Lyapunov function
Figure BDA0003372022880000042
Designing a virtual control function alpha1And a control input u;
step sixFirst derivative based on Lyapunov function
Figure BDA0003372022880000043
And a virtual control function alpha1And finally designing a control input u to obtain a saturation adaptive law.
The second embodiment is as follows:
the embodiment is a backstepping control method based on a saturation adaptive law, and in the step one, the specific form of the established state space model of the two-dimensional nonlinear system is as follows:
Figure BDA0003372022880000044
wherein x is1,x2Represents the state variable of the system and is,
Figure BDA0003372022880000045
representing the system state variable x1,x2Is the first derivative of (a), y is the system output, f is a nonlinear smooth function unknown to the system, d is an unknown disturbance external to the system, u represents the system control input signal, the control objective is to design the system control input u so that the system output y tracks a given target signal yd。
The controlled object of the invention can be a motor system, and can also be other systems or objects, and when the controlled object is the motor system, the corresponding system state variable x1,x2The rotational angle and the rotational speed of the motor system,
Figure BDA0003372022880000046
the rotating speed and the angular acceleration of the motor system are shown, and the system output y is the rotating angle of the motor system.
Other steps and parameters are the same as in the first embodiment.
The third concrete implementation mode:
the embodiment is a backstepping control method based on a saturation self-adaption law, and the solution of a state space model (1) of the two-dimensional nonlinear system exists and is unique; unknown non-linear smoothnessFunction f and its first and second derivatives satisfy
Figure BDA0003372022880000047
Wherein
Figure BDA0003372022880000048
Is a constant greater than zero; target signal ydAnd its first and second derivatives are bounded.
Other steps and parameters are the same as in the second embodiment.
The fourth concrete implementation mode:
the embodiment is a backstepping control method based on a saturation adaptive law, and an error variable z in the step two1And z2The following were used:
Figure BDA0003372022880000051
wherein alpha is1Representing the virtual control function to be designed.
Other steps and parameters are the same as in one of the first to third embodiments.
The fifth concrete implementation mode:
the present embodiment is a back-stepping control method based on the saturation adaptive law, which uses the error variable z set in the second step1And z2The designed Lyapunov function V is as follows:
Figure BDA0003372022880000052
other steps and parameters are the same as in embodiment four.
The sixth specific implementation mode:
the embodiment is a reverse step control method based on a saturation adaptive law, and the first derivative of the lyapunov function V in the third step to the time is:
Figure BDA0003372022880000053
wherein the content of the first and second substances,
Figure BDA0003372022880000054
representing a virtual control function alpha1The first derivative of (a).
Other steps and parameters are the same as those in the fifth embodiment.
The seventh embodiment:
the embodiment is a backstepping control method based on a saturation adaptive law, and a virtual control function alpha1And the control input u is:
Figure BDA0003372022880000055
Figure BDA0003372022880000056
wherein the content of the first and second substances,
Figure BDA0003372022880000057
representing a virtual control function alpha1The first derivative of (a) is,
Figure BDA0003372022880000058
for estimation of an unknown non-linear function f, k1、k2Respectively being a constant greater than zero
Figure BDA0003372022880000059
The function is a saturation function and is defined as follows:
Figure BDA0003372022880000061
wherein the constant c satisfies
Figure BDA0003372022880000062
The saturation function of the invention is different from the traditional saturation function, the traditional saturation function is continuous but can not be conducted, and the saturation function designed by the invention can be conducted in a second order, so that the system control effect is smoother.
Other steps and parameters are the same as in one of the first to sixth embodiments.
The specific implementation mode is eight:
the embodiment is a backstepping control method based on a saturation adaptive law, and the saturation adaptive law in the sixth step is designed as follows:
Figure BDA0003372022880000063
wherein, mu1,μ2,μ3Is a constant number greater than zero and is,
Figure BDA0003372022880000064
the following filter outputs:
Figure BDA0003372022880000065
where η is a constant greater than zero and s is a filter state variable.
The traditional adaptive rate only estimates the system unknown constant, and the adaptive law designed by the invention can directly estimate the system unknown function.
The other steps and parameters are the same as in the seventh embodiment.
It will be demonstrated below that the back-stepping controller (6) based on the saturated adaptation law enables the system tracking error to converge to a smaller neighborhood near the origin. The demonstration process is as follows:
substituting the formula (6) into the formula (4) to obtain the final product
Figure BDA0003372022880000066
Therein, ζ1Is a constant number greater than zero and is,
Figure BDA0003372022880000067
is obtainable from the formula (10)
Figure BDA0003372022880000068
The formula indicates the error variable z1And z2Are bounded.
To z2Derived by derivation
Figure BDA0003372022880000071
From f, z1And z2Are all bounded and available
Figure BDA0003372022880000072
Is bounded, so there is a constant for the filter (9)
Figure BDA0003372022880000073
So that
Figure BDA0003372022880000074
Where ξ is the filter estimation error.
Substituting the formulas (12) and (13) into the formula (8) to obtain the final product
Figure BDA0003372022880000075
If it is
Figure BDA0003372022880000076
Selecting Lyapunov function
Figure BDA0003372022880000077
Can obtain the product
Figure BDA0003372022880000078
The first derivative with respect to time is
Figure BDA0003372022880000079
Therein, ζ2Is a normal number, and is,
Figure BDA00033720228800000710
further can obtain
Figure BDA00033720228800000711
Wherein, the proper mu is selected2And mu3So that
Figure BDA00033720228800000712
Order to
Figure BDA00033720228800000713
Can obtain the product
Figure BDA00033720228800000714
Wherein b is 2c4/c3. From the formula (17), t is not less than t1At a time there is
Figure BDA00033720228800000715
If it is
Figure BDA00033720228800000716
Then there is
Figure BDA00033720228800000717
Wherein
Figure BDA00033720228800000718
Selecting Lyapunov function
Figure BDA00033720228800000719
Can obtain the product
Figure BDA00033720228800000720
Therein, ζ3Is a normal number, and is,
Figure BDA00033720228800000721
further can obtain
Figure BDA00033720228800000722
Wherein a suitable c is selected5So that
Figure BDA00033720228800000723
Can obtain the product
Figure BDA00033720228800000724
The constant c is chosen such that
Figure BDA0003372022880000081
It can be obtained that t is more than or equal to all t1，
Figure BDA0003372022880000082
This is true.
Thus, it is possible to obtain t ≧ t for all t ≧ t1Is obtained by
Figure BDA0003372022880000083
By substituting formula (19) for formula (20)
Figure BDA0003372022880000084
Wherein
Figure BDA0003372022880000085
From the formula (21)
Figure BDA0003372022880000086
From the formulae (19) and (22)
Figure BDA0003372022880000087
Equation (23) shows that as time goes to infinity, the system tracking error converges to a small neighborhood around the origin, and the result is verified.
The specific implementation method nine:
the embodiment is a storage medium, wherein at least one instruction is stored in the storage medium, and the at least one instruction is loaded and executed by a processor to realize a backstepping control method based on a saturation adaptive law.
The detailed implementation mode is ten:
the embodiment is an apparatus, which includes a processor and a memory, where the memory stores at least one instruction, and the at least one instruction is loaded and executed by the processor to implement a method for controlling backstepping based on a saturation adaptive law.
Example one
For the system (1), the initial value is taken as x1(0)＝0，x2(0) 0, function f 1.25sin (x)1x2) +0.25cos (0.5t), system target signal ydSint. Taking the controller parameter as k1＝3，k2＝0.5，c＝100，η＝50，μ1＝100，μ2＝80，μ 35. The system sampling interval time was 0.002 seconds.
FIG. 2 is a system tracking performance curve under the method of the present invention; FIG. 3 is a system tracking error curve under the method of the present invention; FIG. 4 shows a system state x according to the method of the present invention2A response curve; FIG. 5 shows a system for implementing the method of the present inventionSelf-adaptively estimating a performance curve; FIG. 6 is a response curve of the system control input u according to the method of the present invention;
and (4) conclusion: from fig. 2, the back-stepping controller based on the saturation adaptive law designed by equation (6) can make the system tracking error converge to a smaller neighborhood near the origin.
The above-described calculation examples of the present invention are merely to explain the calculation model and the calculation flow of the present invention in detail, and are not intended to limit the embodiments of the present invention. It will be apparent to those skilled in the art that other variations and modifications of the present invention can be made based on the above description, and it is not intended to be exhaustive or to limit the invention to the precise form disclosed, and all such modifications and variations are possible and contemplated as falling within the scope of the invention.

Claims (10)

1. A backstepping control method based on a saturation adaptive law is characterized in that a two-dimensional nonlinear system state space model is established for a controlled object, and the controlled object is controlled by the saturation adaptive law;
the design process of the saturation adaptive law comprises the following steps:
the method comprises the following steps of firstly, establishing a two-dimensional nonlinear system state space model aiming at a controlled object, wherein the established state space model of the two-dimensional nonlinear system is in the following specific form:
Figure FDA0003372022870000011
Figure FDA0003372022870000012
y＝x1
wherein x is1,x2Represents the state variable of the system and is,
Figure FDA0003372022870000013
representsSystem state variable x1,x2The first derivative of (a), y is system output, f is a nonlinear smooth function unknown to the system, d is unknown disturbance outside the system, u represents a system control input signal, and the control aim is to design the system control input u so that the system output y tracks a given target signal;
step two, according to the state variable x of the system1And a target signal ydAnd the state variable x of the system2And a virtual control function alpha to be designed1Separately constructing the error variable z1And z2；
Step three, utilizing the error variable z obtained in the step two1And z2Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Figure FDA0003372022870000014
Step five, according to the first derivative of the Lyapunov function
Figure FDA0003372022870000015
Designing a virtual control function alpha1And based on a virtual control function alpha1Design control input u:
Figure FDA0003372022870000016
wherein the content of the first and second substances,
Figure FDA0003372022870000017
representing a virtual control function alpha1The first derivative of (a) is,
Figure FDA0003372022870000018
for estimation of an unknown non-linear function f, k2Is a constant greater than zero;
Figure FDA0003372022870000019
the function is a saturation function;
Figure FDA00033720228700000110
the function is a saturation function and is defined as follows:
Figure FDA00033720228700000111
wherein the constant c satisfies
Figure FDA00033720228700000112
Step six, based on the first derivative of the Lyapunov function
Figure FDA00033720228700000113
And a virtual control function alpha1And finally designing a control input u to obtain a saturation adaptive law.
2. The saturation adaptive law-based backstepping control method according to claim 1, wherein the controlled object is a motor system, and the corresponding system state variable x1,x2The rotational angle and the rotational speed of the motor system,
Figure FDA00033720228700000114
the rotating speed and the angular acceleration of the motor system are shown, and the system output y is the rotating angle of the motor system.
3. A method for backstepping control based on saturation adaptive law according to claim 1 or 2, characterized in that the solution of the state space model (1) of the two-dimensional non-linear system exists and is unique; the unknown nonlinear smooth function f and the first-order second-order derivative thereof satisfy
Figure FDA0003372022870000021
Wherein
Figure FDA0003372022870000022
Is a constant greater than zero; target signal ydAnd its first and second derivatives are bounded.
4. The method according to claim 3, wherein the error variable z in step two is1And z2The following were used:
Figure FDA0003372022870000023
wherein alpha is1Representing the virtual control function to be designed.
5. The method according to claim 4, wherein the error variable z set in step two is used1And z2Designed Lyapunov function
Figure FDA0003372022870000024
6. The method of claim 5, wherein the first derivative of the Lyapunov function V in step III with respect to time is:
Figure FDA0003372022870000025
wherein the content of the first and second substances,
Figure FDA0003372022870000026
representing a virtual control function alpha1The first derivative of (a).
7. A substrate according to claim 6Method for backstepping control on the saturated adaptive law, characterized in that the virtual control function α1As follows
Figure FDA0003372022870000027
Wherein k is1Is a constant greater than zero.
8. The method according to claim 7, wherein the saturation adaptive law in the sixth step is designed as follows:
Figure FDA0003372022870000028
output of filter
Figure FDA0003372022870000029
Figure FDA00033720228700000210
Wherein, mu1,μ2,μ3Is a constant greater than zero, η is a constant greater than zero, and s is a filter state variable.
9. A storage medium having stored therein at least one instruction that is loaded and executed by a processor to implement a method of saturation adaptive law based backstepping control as claimed in any one of claims 1 to 8.
10. An apparatus comprising a processor and a memory, the memory having stored therein at least one instruction that is loaded and executed by the processor to implement a method of saturation adaptive law based backstepping control as claimed in any one of claims 1 to 8.
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