CN114035436B - Backstepping control method, storage medium and device based on saturation adaptive law - Google Patents
Backstepping control method, storage medium and device based on saturation adaptive law Download PDFInfo
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Abstract
A backstepping control method, a storage medium and a device based on a saturation self-adaptive law belong to the technical field of nonlinear system control. The method aims to solve the problems that the current self-adaptive backstepping control method cannot process an unknown nonlinear function in a system and the existing saturation control has 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 system 1 And a target signal construction error variable z 1 According to state variable x 2 And a virtual control function alpha to be designed 1 Build error variable z 2 The Lyapunov function is designed and the first derivative is calculated over time, and then the virtual control function alpha is designed based on the first derivative of the Lyapunov function 1 And controlling the input u, and finally designing to obtain a saturation adaptive law. The method is mainly used for controlling the nonlinear system.
Description
Technical Field
The invention belongs to the technical field of nonlinear system control, and particularly relates to a backstepping control method, a storage medium and equipment based on a saturation self-adaption law.
Background
The basic idea of the method is to estimate the unknown constant in the system by using the adaptive parameters, and then to complete the design of the control law by using the adaptive parameters and the backstepping control idea. Currently, the adaptive back-step control method has been widely used in industrial automation systems, automobile control systems, robot control systems, etc., and reference may be made to chinese patent No. CN108869420B, chinese patent No. CN106787940B, and chinese patent No. CN105573125a for design details of the adaptive back-step control method.
It should be noted that in the conventional adaptive back-off control method, the designed adaptive parameters can only deal with the uncertainty of the system parameters, namely: the adaptive parameters can only estimate the unknown constants in the system. When the system contains an unknown nonlinear function, the traditional self-adaptive backstepping control method is not applicable any more. For a real nonlinear system, the system uncertainty is often caused by some unknown nonlinear function, such as viscous friction nonlinearity, actuator dead zone/saturation nonlinearity, etc. The inability of conventional adaptive backstepping control methods to handle unknown nonlinear functions of the system results in significant limitations in practical applications. Therefore, how to design an adaptive back-off control strategy that can handle the unknown nonlinear function of the system is a critical issue.
Disclosure of Invention
The invention aims to solve the problems that the current self-adaptive backstepping control method cannot process an unknown nonlinear function in a system and the existing saturation control has unsmooth.
A backstepping control method based on a saturation adaptive law aims at a controlled object, a two-dimensional nonlinear system state space model is established, and the controlled object is controlled by the saturation adaptive law;
the design process of the saturation adaptive law comprises the following steps:
step one, aiming at a controlled object, a state space model of a two-dimensional nonlinear system is established, and the established state space model of the two-dimensional nonlinear system is in a specific form that:
y＝x 1
wherein x is 1 ,x 2 Representing a state variable of the system,representing a system state variable x 1 ,x 2 Y is the system output, f is the nonlinear smoothness unknown to the systemD is an external unknown disturbance of the system, u represents a system control input signal, and the control purpose is to design the system control input u to enable the system output y to track a given target signal;
step two, according to the state variable x of the system 1 And a target signal y d And state variable x of the system 2 And a virtual control function alpha to be designed 1 Respectively constructing error variables z 1 Z 2 ；
Step three, utilizing the error variable z obtained in the step two 1 And z 2 Designing a Lyapunov function V;
step four, obtaining the first derivative of the Lyapunov function V in the step three with respect to time
Step five, according to the first derivative of Lyapunov functionDesigning virtual control function alpha 1 And based on virtual control function alpha 1 Design control input u:
wherein,representing a virtual control function alpha 1 First derivative of>For estimation of unknown nonlinear function f, k 2 A constant greater than zero; />The function is a saturation function;
the function is a saturation function, defined as follows:
wherein the constant c satisfies
Step six, first derivative based on Lyapunov functionVirtual control function alpha 1 And the control input u is finally designed to get a saturation adaptive law.
Further, the controlled object is a motor system, and the corresponding system state variable x is 1 ,x 2 For the rotational angle and rotational speed of the motor system,the system output y is the rotation angle of the motor system for the rotation speed and the angular acceleration of the motor system.
Further, a solution of a state space model (1) of the two-dimensional nonlinear system exists and is unique; the unknown nonlinear smooth function f and the first-order second-order derivative thereof satisfyWherein->A constant greater than zero; target signal y d And its first and second derivatives are bounded.
Further, the error variable z in step two 1 And z 2 The following are provided:
wherein alpha is 1 Representing the virtual control function to be designed.
Further, the error variable z set in the second step is used 1 And z 2 Designed Lyapunov function
Further, the first derivative of the lyapunov function V in the third step with respect to time is:
wherein,representing a virtual control function alpha 1 Is a first derivative of (a).
Further, virtual control function α 1 The following are listed below
Wherein k is 1 Is a constant greater than zero.
Further, the saturation adaptive law in the sixth step is designed as follows:
filter output
Wherein mu 1 ,μ 2 ,μ 3 Is a constant that is greater than zero and,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 saturation adaptation law.
An apparatus comprising a processor and a memory having stored therein at least one instruction that is loaded and executed by the processor to implement the method of controlling backstepping based on a saturation adaptation law.
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 second order of the saturation function designed by the invention is conductive, so that the system control effect is smoother.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a graph of system tracking performance under the method of the present invention;
FIG. 3 is a graph of the tracking error of the system under the method of the present invention;
FIG. 4 shows the system state x in the method of the present invention 2 A response curve;
FIG. 5 is a system adaptive estimation performance curve under the method of the present invention;
FIG. 6 is a response curve of the system control input u under the method of the present invention.
Detailed Description
The first embodiment is as follows: the present embodiment will be described with reference to figure 1,
the embodiment is a backstepping control method based on a saturation adaptive law, which aims at a controlled object, establishes a two-dimensional nonlinear system state space model, and controls 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, a two-dimensional nonlinear system state space model is established, wherein two state variables x exist in a two-dimensional nonlinear system 1 And x 2 A control input and a given target signal y d ；
Step two, according to the state variable x of the system 1 And a target signal y d And state variable x of the system 2 And a virtual control function alpha to be designed 1 Respectively constructing error variables z 1 Z 2 ；
Step three, utilizing the error variable z obtained in the step two 1 And z 2 Designing a Lyapunov function V;
step four, obtaining the first derivative of the Lyapunov function V in the step three with respect to time
Step five, according to the first derivative of Lyapunov functionDesigning virtual control function alpha 1 A control input u;
step six, first derivative based on Lyapunov functionVirtual control function alpha 1 And the control input u is finally designed to get a saturation adaptive law.
The second embodiment is as follows:
the embodiment is a backstepping control method based on a saturation adaptive law, in the first step, a state space model of a two-dimensional nonlinear system is established in a specific form as follows:
wherein x is 1 ,x 2 Representing a state variable of the system,representing a system state variable x 1 ,x 2 Is the first derivative of 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, and the control objective is to design the system control input u such that the system output y tracks a given target signal y d 。
The controlled object of the invention can be a motor system, or can be other systems or objects, and when the controlled object is a motor system, the corresponding system state variable x is used 1 ,x 2 For the rotational angle and rotational speed of the motor system,the system output y is the rotation angle of the motor system for the rotation speed and the angular acceleration of the motor system.
Other steps and parameters are the same as in the first embodiment.
And a third specific embodiment:
the embodiment is a backstepping control method based on a saturation adaptive law, wherein the solution of a state space model (1) of the two-dimensional nonlinear system exists and is unique; the unknown nonlinear smooth function f and the first-order second-order derivative thereof satisfyWherein->A constant greater than zero; target signal y d And its first and second derivatives are bounded.
Other steps and parameters are the same as in the second embodiment.
The specific embodiment IV is as follows:
the present embodiment is a backstepping control method based on a saturation adaptive law, and the error variable z in the second step 1 And z 2 The following are provided:
wherein alpha is 1 Representing the virtual control function to be designed.
Other steps and parameters are the same as in one to three embodiments.
Fifth embodiment:
the present embodiment is a method for controlling a backstepping based on a saturation adaptive law, which uses the error variable z set in the second step 1 And z 2 The designed Lyapunov function V is:
other steps and parameters are the same as in the fourth embodiment.
Specific embodiment six:
the embodiment is a backstepping control method based on a saturation adaptive law, and the first derivative of the Lyapunov function V in the third step with respect to time is as follows:
wherein,representing a virtual control function alpha 1 Is a first derivative of (a).
Other steps and parameters are the same as in embodiment five.
Seventh embodiment:
the embodiment is a backstepping control method based on a saturation adaptive law, and the virtual control function alpha 1 The control input u is:
wherein,representing a virtual control function alpha 1 First derivative of>For estimation of unknown nonlinear function f, k 1 、k 2 Constants respectively greater than zero
The function is a saturation function, defined as follows:
wherein the constant c satisfies
The saturation function of the invention is different from the traditional saturation function, the traditional saturation function is continuous but not conductive, and the saturation function designed by the invention is second-order conductive, so that the system control effect is smoother.
Other steps and parameters are the same as in one to six of the embodiments.
Eighth embodiment:
the present embodiment is a backstepping control method based on a saturation adaptive law, and the saturation adaptive law in the step six is designed as follows:
wherein mu 1 ,μ 2 ,μ 3 Is a constant that is greater than zero and,the following filter outputs:
where η is a constant greater than zero and s is a filter state variable.
The traditional self-adaptive rate only estimates the unknown constant of the system, and the self-adaptive law designed by the invention can directly estimate the unknown function of the system.
Other steps and parameters are the same as in embodiment seven.
The backstepping controller (6) based on the saturation adaptation law will be demonstrated to be able to converge the system tracking error into a small neighborhood around the origin. The proving process is as follows:
substituting the formula (6) into the formula (4) to obtain the final product
Wherein ζ 1 Is a constant that is greater than zero and,
is obtained from (10)
The indication of the error variable z 1 And z 2 Are all bounded.
For z 2 Can obtain the derivation
From f, z 1 And z 2 Are all available in a bounded mannerIs bounded and therefore has a constant for the filter (9)>So that
Where ζ is the filter estimation error.
Substituting the formulas (12) and (13) into the formula (8), and finishing to obtain
If it isSelecting Lyapunov function ++>Available->The first derivative with respect to time is
Wherein ζ 2 Is a normal number of times, and the number of times is equal to the normal number,further can obtain
Wherein, a proper mu is selected 2 Sum mu 3 So thatLet->Is available in the form of
Wherein b=2c 4 /c 3 . The proper t is more than or equal to t by the formula (17) 1 Sometimes have
If it isThere is->Wherein->Selecting Lyapunov functionIs available in the form of
Wherein ζ 3 Is a normal number of times, and the number of times is equal to the normal number,further can obtain
Wherein, selecting proper c 5 So thatAvailable->The constant c is chosen such that +.>Can be obtained for all t.gtoreq.t 1 ，/>This is true.
Thus, it is possible to obtain for all t.gtoreq.t 1 Can be obtained
Substitution of formula (19) into formula (20) can be achieved
Wherein the method comprises the steps of
Obtainable from (21)
Obtainable by formulae (19) and (22)
Equation (23) shows that the time tends to infinity, and the system tracking error converges to a smaller neighborhood near the origin, indicating the pich.
Detailed description nine:
the present embodiment is a storage medium having at least one instruction stored therein, the at least one instruction being loaded and executed by a processor to implement a method of backstepping control based on a saturation adaptation law.
Detailed description ten:
the embodiment is an apparatus comprising a processor and a memory, wherein at least one instruction is stored in the memory, 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 1
For the system (1), the initial value is taken as x 1 (0)＝0，x 2 (0) =0, function f=1.25 sin (x 1 x 2 ) +0.25cos (0.5 t), system target signal y d =sint. Taking the controller parameter as k 1 ＝3，k 2 ＝0.5，c＝100，η＝50，μ 1 ＝100，μ 2 ＝80，μ 3 =5. The system sampling interval time is 0.002 seconds.
FIG. 2 is a graph of system tracking performance under the method of the present invention; FIG. 3 is a graph of the tracking error of the system under the method of the present invention; FIG. 4 shows the system state x in the method of the present invention 2 A response curve; FIG. 5 is a system adaptive estimation performance curve under the method of the present invention; FIG. 6 is a response curve of the system control input u under the method of the present invention;
conclusion: the saturated-adaptive law-based backstepping controller designed from equation (6) of fig. 2 can enable the system tracking error to converge into a smaller neighborhood around the origin.
The above examples of the present invention are only for describing the calculation model and calculation flow of the present invention in detail, and are not limiting of the embodiments of the present invention. Other variations and modifications of the above description will be apparent to those of ordinary skill in the art, and it is not intended to be exhaustive of all embodiments, all of which are within the scope of the invention.
Claims (3)
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 aiming at a controlled object, and the controlled object is controlled by the saturation adaptive law; the controlled object is a motor system, and the corresponding system state variable x 1 ,x 2 For the rotational angle and rotational speed of the motor system,the system output y is the rotation angle of the motor system for the rotation speed and the angular acceleration of the motor system;
the design process of the saturation adaptive law comprises the following steps:
step one, aiming at a controlled object, a state space model of a two-dimensional nonlinear system is established, and the established state space model of the two-dimensional nonlinear system is in a specific form that:
wherein x is 1 ,x 2 Representing a state variable of the system,representing a system state variable x 1 ,x 2 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, and the control purpose is to design the system control input u so that the system output y tracks a given target signal;
the solution of the state space model (1) of the two-dimensional nonlinear system exists and is unique; the unknown nonlinear smooth function f and the first-order second-order derivative thereof satisfyWherein->A constant greater than zero; target signal y d And its first and second derivatives are bounded;
step two, according to the state variable x of the system 1 And a target signal y d And state variable x of the system 2 And a virtual control function alpha to be designed 1 Respectively constructing error variables z 1 Z 2 The method comprises the steps of carrying out a first treatment on the surface of the Error variable z 1 And z 2 The following are provided:
wherein alpha is 1 Representing a virtual control function to be designed; virtual control function alpha 1 The following are listed below
Wherein k is 1 A constant greater than zero;
step three, utilizing the error variable z obtained in the step two 1 And z 2 Design of Lyapunov function V, lyapunov function
Step four, obtaining the first derivative of the Lyapunov function V in the step three with respect to timeThe first derivative of Lyapunov function V over time is:
wherein,representing a virtual control function alpha 1 Is the first derivative of (a);
step five, according to the first derivative of Lyapunov functionDesigning virtual control function alpha 1 And based on virtual control function alpha 1 Design control input u:
wherein,representing a virtual control function alpha 1 First derivative of>For estimation of unknown nonlinear function f, k 2 A constant greater than zero; />The function is a saturation function;
the function is a saturation function, defined as follows:
wherein the constant c satisfies
Step six, first derivative based on Lyapunov functionVirtual control function alpha 1 The control input u is finally designed to obtain a saturation self-adaptive law; the saturation adaptive law is designed as follows:
filter output
Wherein mu 1 ,μ 2 ,μ 3 Is a constant greater than zero, η is a constant greater than zero, s is a filter state variable.
2. A storage medium having stored therein at least one instruction that is loaded and executed by a processor to implement a method of controlling backstepping based on a saturation adaptation law as claimed in claim 1.
3. An apparatus comprising a processor and a memory having stored therein at least one instruction that is loaded and executed by the processor to implement a method of controlling backstepping based on a saturation adaptation law as claimed in claim 1.
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| GB201911738D0 (en) * | 2019-07-12 | 2019-10-02 | Huaiyin Inst Technology | Adaptive backstepping optimal control method of fractional-order chaotic electromechanical transducer system |
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| CN113093553A (en) * | 2021-04-13 | 2021-07-09 | 哈尔滨工业大学 | Adaptive backstepping control method based on instruction filtering disturbance estimation |
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| JP2009228605A (en) * | 2008-03-24 | 2009-10-08 | Daihatsu Motor Co Ltd | Method for controlling engine speed and device for controlling engine speed |
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