CN114035436B - A backstepping control method, storage medium and device based on saturation adaptive law - Google Patents

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

A back-stepping control method, storage medium and device based on saturation adaptive law

Technical Field

The invention belongs to the technical field of nonlinear system control, and specifically relates to a back-stepping control method, storage medium and equipment based on a saturation adaptive law.

Background technique

Adaptive backstepping control is a very popular control method in the field of nonlinear system control. The basic idea of this method is to use adaptive parameters to estimate the unknown constants in the system, and then use adaptive parameters and backstepping control ideas to complete the design of the control law. At present, the adaptive backstepping control method has been widely used in industrial automation systems, automotive control systems, robot control systems, etc. For the design details of the adaptive backstepping control method, please refer to Chinese invention patents CN108869420B, Chinese invention patents CN106787940B and Chinese invention patents CN105573125A.

It should be noted that in the traditional adaptive backstepping control method, the designed adaptive parameters can only handle the uncertainty of system parameters, that is, the adaptive parameters can only estimate the unknown constants in the system. When the system contains unknown nonlinear functions, the traditional adaptive backstepping control method will no longer be applicable. For actual nonlinear systems, the system uncertainty is often caused by some unknown nonlinear functions, such as viscous friction nonlinearity, actuator dead zone/saturation nonlinearity, etc. The traditional adaptive backstepping control method cannot handle the unknown nonlinear functions of the system, which leads to great limitations in its practical application. Therefore, how to design an adaptive backstepping control strategy that can handle the unknown nonlinear functions of the system is a key issue.

Contents of the invention

The purpose of the present invention is to solve the problem that the current adaptive backstepping control method cannot handle the unknown nonlinear function in the system and the problem that the existing saturation control is not smooth.

A back-stepping control method based on the saturation adaptive law, which establishes a two-dimensional nonlinear system state space model for the controlled object, and uses the saturation adaptive law to control the controlled object;

The design process of the saturation adaptive law includes the following steps:

Step 1: Establish a two-dimensional nonlinear system state space model for the controlled object. The specific form of the established two-dimensional nonlinear system state space model is:

y＝ _x1

Among them, x ₁ and x ₂ represent the state variables of the system, represents the first-order derivative of the system state variables x ₁ and x ₂ , y is the system output, f is the unknown nonlinear smooth function of the system, d is the unknown external disturbance of the system, u represents the system control input signal, and the control purpose is to design the system control input u causes the system output y to track the given target signal;

Step 2: Construct error variables z ₁ and z ₂ respectively according to the system state variable x ₁ and target signal y _d , as well as the system state variable x ₂ and the virtual control function α ₁ to be designed;

Step 3: Use the error variables z ₁ and z ₂ obtained in step 2 to design the Lyapunov function V;

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

Step 5. According to the first derivative of Lyapunov function Design the virtual control function α ₁ and design the control input u based on the virtual control function α ₁ :

in, Represents the first derivative of the virtual control function α ₁ ,/> is the estimate of the unknown nonlinear function f, and k ₂ is a constant greater than zero;/> The function is a saturated function;

The function is a saturation function, defined as follows:

Among them, the constant c satisfies

Step 6. First derivative based on Lyapunov function And the virtual control function α ₁ and the control input u are finally designed to obtain the saturated adaptive law.

Further, the controlled object is a motor system, and the corresponding system state variables x ₁ and x ₂ are the rotation angle and speed of the motor system, are the rotation speed and angular acceleration of the motor system, and the system output y is the rotation angle of the motor system.

Furthermore, the solution to the state space model (1) of the two-dimensional nonlinear system exists and is unique; the unknown nonlinear smooth function f and its first-order and second-order derivatives satisfy Among them/> is a constant greater than zero; the target signal y _d and its first-order and second-order derivatives are bounded.

Further, the error variables z ₁ and z ₂ in step 2 are as follows:

Among them, α ₁ represents the virtual control function to be designed.

Furthermore, the Lyapunov function designed using the error variables z ₁ and z ₂ set in step 2

Furthermore, the first derivative of the Lyapunov function V in step 3 with respect to time is:

in, Represents the first derivative of the virtual control function α ₁ .

Further, the virtual control function α ₁ is as follows

Among them, k ₁ is a constant greater than zero.

Furthermore, the saturation adaptive law in step six is designed as follows:

filter output

Among them, μ ₁ , μ ₂ , μ ₃ are constants greater than zero, eta is a constant greater than zero, and s is the filter state variable.

A storage medium stores at least one instruction in the storage medium, and the at least one instruction is loaded and executed by a processor to implement a back-stepping control method based on a saturation adaptive law.

A device, the device includes a processor and a memory, at least one instruction is stored in the memory, and the at least one instruction is loaded and executed by the processor to implement the back-stepping control based on the saturation adaptive law method.

Beneficial effects of the present invention:

The present invention proposes a back-stepping control method based on the saturation adaptive law. The present invention uses a saturation function and a back-stepping control method to design a back-stepping control method based on the saturation adaptive law. Compared with the traditional adaptive back-stepping control method that cannot process unknown nonlinear functions of the system, the present invention can directly estimate and process the unknown nonlinear functions of the system; at the same time, the saturation function designed by the present invention is second-order differentiable, which can make System control effects are smoother.

Description of drawings

Figure 1 is a flow chart of the method of the present invention;

Figure 2 shows the system tracking performance curve under the method of the present invention;

Figure 3 shows the system tracking error curve under the method of the present invention;

Figure 4 is the system state x ₂ response curve under the method of the present invention;

FIG5 is a performance curve of system adaptive estimation 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 ways

Specific implementation mode one: This implementation mode will be described with reference to Figure 1.

This implementation mode is a back-stepping control method based on the saturation adaptive law. For the controlled object, a two-dimensional nonlinear system state space model is established, and the saturated adaptive law is used to control the controlled object;

The design process of the saturation adaptive law includes the following steps:

Step 1: Establish a two-dimensional nonlinear system state space model for the controlled object. There are two state variables x ₁ and x ₂ in the two-dimensional nonlinear system, a control input and a given target signal y _d ;

Step 2: Construct error variables z ₁ and z ₂ respectively according to the system state variable x ₁ and target signal y _d , as well as the system state variable x ₂ and the virtual control function α ₁ to be designed;

Step 3: Use the error variables z ₁ and z ₂ obtained in step 2 to design the Lyapunov function V;

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

Step 5: Based on the first-order derivative of the Lyapunov function Design virtual control function α ₁ and control input u;

Step 6. First derivative based on Lyapunov function And the virtual control function α ₁ and the control input u are finally designed to obtain the saturated adaptive law.

Specific implementation method two:

This implementation mode is a back-stepping control method based on the saturation adaptive law. In step one, the specific form of the state space model of the two-dimensional nonlinear system established is:

Among them, x ₁ and x ₂ represent the state variables of the system, represents the first-order derivative of the system state variables x ₁ and x ₂ , y is the system output, f is the unknown nonlinear smooth function of the system, d is the unknown external disturbance of the system, u represents the system control input signal, and the control purpose is to design the system control input u causes the system output y to track the given target signal y _d .

The controlled object of the present invention can be a motor system or other systems or objects. When the controlled object is a motor system, the corresponding system state variables x ₁ and x ₂ are the rotation angle and rotation speed of the motor system, are the rotation speed and angular acceleration of the motor system, and the system output y is the rotation angle of the motor system.

Other steps and parameters are the same as in the first embodiment.

Specific implementation method three:

This implementation mode is a back-stepping control method based on the saturation adaptive law. The solution to the state space model (1) of the two-dimensional nonlinear system exists and is unique; the unknown nonlinear smooth function f and its first-order and second-order derivatives satisfy Among them/> is a constant greater than zero; the target signal y _d and its first-order and second-order derivatives are bounded.

Other steps and parameters are the same as the second embodiment.

Specific implementation method four:

This implementation is a back-stepping control method based on the saturation adaptive law. The error variables z ₁ and z ₂ in step 2 are as follows:

Among them, α ₁ represents the virtual control function to be designed.

The other steps and parameters are the same as those in the first to third embodiments.

Specific implementation method five:

This implementation is a back-stepping control method based on the saturation adaptive law. The Lyapunov function V designed using the error variables z ₁ and z ₂ set in step 2 is:

Other steps and parameters are the same as the fourth embodiment.

Specific implementation method six:

This implementation is a back-stepping control method based on the saturation adaptive law. The first-order derivative of the Lyapunov function V in step 3 with respect to time is:

in, Represents the first derivative of the virtual control function α ₁ .

The other steps and parameters are the same as those in the fifth embodiment.

Specific implementation method seven:

This implementation is a back-stepping control method based on the saturation adaptive law. The virtual control function α ₁ and the control input u are:

in, Represents the first derivative of the virtual control function α ₁ ,/> is the estimate of the unknown nonlinear function f, k ₁ and k ₂ are constants greater than zero respectively.

The function is a saturation function, defined as follows:

Among them, the constant c satisfies

The saturation function of the present invention is different from the traditional saturation function. The traditional saturation function is continuous but not differentiable. The saturation function designed by the present invention is second-order differentiable, which can make the system control effect smoother.

Other steps and parameters are the same as in one of the first to sixth embodiments.

Specific implementation method eight:

This implementation is a back-stepping control method based on the saturation adaptive law. The saturation adaptive law in step 6 is designed as follows:

Among them, μ ₁ , μ ₂ , μ ₃ are constants greater than zero, The output is the following filter:

Among them, eta is a constant greater than zero, and s is the filter state variable.

The traditional adaptive rate only estimates the unknown constants of the system, but the adaptive law designed in the present invention can directly estimate the unknown functions of the system.

Other steps and parameters are the same as the seventh embodiment.

It will be proved below that the backstepping controller (6) based on the saturation adaptive law can make the system tracking error converge to a smaller neighborhood near the origin. The proof process is as follows:

Substituting equation (6) into equation (4), we can get

Where ζ ₁ is a constant greater than zero,

From formula (10), we can get

This formula shows that the error variables z ₁ and z ₂ are both bounded.

Taking the derivative with respect to z _2, we get

From f, z ₁ and z ₂ are all bounded, we can get is bounded, so there are constants/> for filter (9) make

Among them, ξ is the filter estimation error.

Substituting (12) and (13) into (8), we can get

like Select Lyapunov function /> Available/> The first-order derivative with respect to time is

Among them, ζ ₂ is a positive constant, further can be obtained

Among them, select appropriate μ ₂ and μ ₃ such that Order/> Available

Where b=2c ₄ /c ₃ . From formula (17), we can get that when t≥t ₁

like Then there/> Among them/> Select Lyapunov function function Available

Among them, ζ ₃ is a positive constant, Further we can get

Among them, choosing a suitable c ₅ makes Available/> Choose the constant c such that/> It can be obtained that for all t≥t ₁ ,/> established.

Therefore, it can be obtained that for all t≥t ₁ , we can obtain

Substituting equation (19) into equation (20), we can get

in

From formula (21) we can get

From equations (19) and (22), we can get

Equation (23) shows that when time tends to infinity, the system tracking error converges to a smaller neighborhood near the origin, and the proof is completed.

Specific implementation method nine:

This embodiment is a storage medium in which at least one instruction is stored, and the at least one instruction is loaded and executed by a processor to implement a back-stepping control method based on a saturation adaptive law.

Specific implementation method ten:

This embodiment is a device. The device includes a processor and a memory. At least one instruction is stored in the memory. The at least one instruction is loaded and executed by the processor to implement a backstepping method based on the saturation adaptive law. Control Method.

Embodiment 1

For system (1), take the initial values as x ₁ (0) = 0, x ₂ (0) = 0, function f = 1.25sin (x ₁ x ₂ ) + 0.25cos (0.5t), and the system target signal is y _d =sint. The controller parameters are k ₁ =3, k ₂ =0.5, c =100, eta =50, μ ₁ =100, μ ₂ =80, μ ₃ =5. The system sampling interval is 0.002 seconds.

Figure 2 is the system tracking performance curve under the method of the present invention; Figure 3 is the system tracking error curve under the method of the present invention; Figure 4 is the system state x ₂ response curve under the method of the present invention; Figure 5 is the system state x 2 response curve under the method of the present invention. Adaptive estimation performance curve; Figure 6 shows the system control input u response curve under the method of the present invention;

Conclusion: It can be seen from Figure 2 that the back-stepping controller based on the saturation adaptive law designed by formula (6) can make the system tracking error converge to a smaller neighborhood near the origin.

The above-mentioned calculation examples of the present invention are only to explain the calculation model and calculation process of the present invention in detail, but are not intended to limit the implementation of the present invention. For those of ordinary skill in the art, other changes or modifications in different forms can be made on the basis of the above description. It is impossible to exhaustively enumerate all the implementation modes here. All the technical solutions that belong to the present invention are derived. Obvious changes or modifications are still within the protection scope of the present 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.