阅读说明：本技术 一种基于反步迭代学习的欧拉-伯努利梁的振动控制方法 (Euler-Bernoulli beam vibration control method based on backstepping iterative learning ) 是由 刘屿 郑小惠 湛文康 于 2019-07-23 设计创作，主要内容包括：本发明公开了一种基于反步迭代学习的欧拉-伯努利梁的振动控制方法,该方法过程如下：根据欧拉-伯努利梁的动力学特征构建欧拉-伯努利梁系统；根据欧拉-伯努利梁系统,结合李雅普诺夫方法,构建基于反步迭代学习的振动控制方法,包括虚拟控制量设计、反步项设计及迭代项设计；验证上述欧拉-伯努利梁系统在振动控制方法下的稳定性；利用MATLAB仿真软件对欧拉-伯努利梁系统进行数字仿真,验证控制效果是否符合预期；若不符合,则根据仿真结果调节控制器的增益参数,使之具有较好的控制效果。本发明所提出的基于反步迭代学习的振动控制方法能够有效抑制欧拉-伯努利梁系统的振动,使得欧拉-伯努利梁系统工作更加稳定。(The invention discloses a vibration control method of an Euler-Bernoulli beam based on backstep iterative learning, which comprises the following steps: constructing an Euler-Bernoulli beam system according to the dynamic characteristics of the Euler-Bernoulli beam; according to an Euler-Bernoulli beam system, a vibration control method based on backstepping iterative learning is constructed by combining a Lyapunov method, and the vibration control method comprises virtual control quantity design, backstepping item design and iterative item design; verifying the stability of the Euler-Bernoulli beam system under a vibration control method; performing digital simulation on the Euler-Bernoulli beam system by using MATLAB simulation software, and verifying whether the control effect is in accordance with expectation; if not, the gain parameter of the controller is adjusted according to the simulation result, so that the controller has a better control effect. The vibration control method based on the backstepping iterative learning can effectively inhibit the vibration of the Euler-Bernoulli beam system, so that the Euler-Bernoulli beam system works more stably.)

1. A vibration control method of an Euler-Bernoulli beam based on backstepping iterative learning is characterized by comprising the following steps of:

according to the dynamic characteristics of the Euler-Bernoulli beam, a dynamic model of the Euler-Bernoulli beam system is constructed;

constructing a boundary controller based on backstepping iterative learning based on the Euler-Bernoulli beam system, wherein the boundary controller comprises a virtual control quantity, a backstepping item and an iteration item;

constructing a Lyapunov function of the Euler-Bernoulli beam system based on the Euler-Bernoulli beam system and the boundary controller;

verifying the stability of the Euler-Bernoulli beam system according to the Lyapunov function;

when the Euler-Bernoulli beam system can meet the preset stability requirement under the action of a boundary controller based on backstepping iterative learning, carrying out digital simulation on the Euler-Bernoulli beam system by using simulation software to obtain a simulation result;

if the control effect obtained by the simulation result is in line with the expectation, the gain parameters of the constructed boundary controller based on the backstepping iterative learning are reserved, and the operation is ended;

and if the simulation result is not in accordance with the expectation, correcting the gain parameters of the constructed boundary controller based on the backstepping iterative learning, and carrying out digital simulation again.

2. The method of claim 1, wherein the dynamic characteristics include kinetic energy, potential energy and virtual work done by non-conservative forces on the Euler-Bernoulli beam system, and the Euler-Bernoulli beam system is obtained by substituting the kinetic energy, potential energy and virtual work into Hamilton's principle:

wherein the content of the first and second substances,

the boundary conditions are as follows:

wherein the content of the first and second substances,

3. the method of claim 2, wherein the virtual control quantity is Euler-Bernoulli beam vibration control method

α(t)＝-k₁w'(L,t)+k₂w”'(L,t)；

The backstepping item is

Wherein the content of the first and second substances,

the iterative learning item is

Wherein, beta, gamma, eta, k₁、k₂、k₃Is a gain parameter of the controllerNumber, beta, gamma, eta, k₁、k₂、k₃Are all greater than 0, and the error variable is z₂(t)＝y₂(t)-α(t)，

4. The method for controlling vibration of an Euler-Bernoulli beam based on backstepping iterative learning according to claim 3, wherein the Lyapunov function of the Euler-Bernoulli beam system is constructed based on the Euler-Bernoulli beam system and the virtual control quantity, the backstepping term and the iteration term, and specifically comprises the following steps:

(1) according to the virtual control quantity, constructing a Lyapunov function as follows:

V₁(t)＝V_a(t)+V_b(t)；

wherein the content of the first and second substances,

(2) and constructing a Lyapunov function according to the backstepping term as follows:

(3) according to the iterative learning term, a Lyapunov function of the closed-loop Euler-Bernoulli beam system is constructed as follows:

5. the method for controlling the vibration of the Euler-Bernoulli beam based on the backstepping iterative learning as claimed in claim 1, wherein the stability of the Euler-Bernoulli beam system model is verified according to the Lyapunov function, specifically as follows:

by verifying the positive nature of the Lyapunov function, the stability of the La-Bernoulli beam system in accordance with the Lyapunov meaning is obtained;

and (3) verifying the negative nature of the first-order derivative of the Lyapunov function to obtain the consistency and gradual stability of the Euler-Bernoulli beam system.

6. The method for controlling the vibration of the euler-bernoulli beam based on the back-stepping iterative learning of claim 1, wherein if the simulation result is not in expectation, the gain parameter of the controller is corrected, and the digital simulation is performed again, specifically:

correcting the gain parameter of the controller, verifying the positive qualitative of the Lyapunov function and the negative qualitative of the first derivative of the Lyapunov function according to the gain parameter, and performing digital simulation on the Euler-Bernoulli beam system by using MATLAB simulation software.

7. The method of controlling vibration of an Euler-Bernoulli beam based on inverse iterative learning according to any one of claims 1 to 6,

the simulation result comprises the vibration amplitude of the Euler-Bernoulli beam without control action, the vibration amplitude of different iteration times with control action, and the relationship between the maximum error of the boundary vibration amplitude of the Euler-Bernoulli beam and the iteration times.

Technical Field

The invention relates to the technical field of vibration control, in particular to a vibration control method of an Euler-Bernoulli beam based on backstepping iterative learning.

Background

The flexible structure is widely applied to the engineering fields of mechanical arms, mechanical engineering, spacecrafts and the like because of the advantages of light weight, low energy consumption and the like. Euler-bernoulli beams are often used as the fundamental model for these flexible structural systems in the research of flexible robotic arm, flexible riser, flexible satellite, etc. systems. However, due to the action of external disturbance, the euler-bernoulli beam can generate elastic deformation, and further generate long-time continuous elastic vibration, which can affect the normal operation of the system and also become an obstacle to the application of the flexible structure in the engineering field. Therefore, how to reduce or eliminate the elastic deformation and vibration of the euler-bernoulli beam from the aspect of control is a problem to be solved. The euler-bernoulli beam is a typical distributed parameter system, i.e. the model parameters and the working characteristics are functions of time and space coordinates, so the dynamic response of the euler-bernoulli beam in elastic vibration is complex. The vibration control of the Euler-Bernoulli beam is researched, so that a flexible structure system based on the model, such as a flexible mechanical arm, a flexible riser, a flexible satellite and the like, can obtain higher precision in practical engineering.

At present, most vibration control researches on Euler-Bernoulli beams adopt methods such as PID control and robust control, and few reports are reported about a backstepping iterative learning control method. Therefore, the research of the invention provides theoretical reference for vibration control of a system with an Euler-Bernoulli beam structure in the fields of aerospace, mechanical engineering and the like.

Disclosure of Invention

The invention aims to solve the defects in the prior art and provides a vibration control method of an Euler-Bernoulli beam based on backstepping iterative learning.

The purpose of the invention can be achieved by adopting the following technical scheme:

a vibration control method of an Euler-Bernoulli beam based on backstepping iterative learning, the vibration control method comprising the steps of:

according to the dynamic characteristics of the Euler-Bernoulli beam, a dynamic model of the Euler-Bernoulli beam system is constructed;

constructing a boundary controller based on backstepping iterative learning based on the Euler-Bernoulli beam system, wherein the boundary controller comprises a virtual control quantity, a backstepping item and an iteration item;

constructing a Lyapunov function of the Euler-Bernoulli beam system based on the Euler-Bernoulli beam system and the boundary controller;

verifying the stability of the Euler-Bernoulli beam system according to the Lyapunov function;

when the Euler-Bernoulli beam system can meet the preset stability requirement under the action of a boundary controller based on backstepping iterative learning, carrying out digital simulation on the Euler-Bernoulli beam system by using simulation software to obtain a simulation result;

if the control effect obtained by the simulation result is in line with the expectation, the gain parameters of the constructed boundary controller based on the backstepping iterative learning are reserved, and the operation is ended;

and if the simulation result is not in accordance with the expectation, correcting the gain parameters of the constructed boundary controller based on the backstepping iterative learning, and carrying out digital simulation again.

Further, the kinetic characteristics include kinetic energy, potential energy and virtual work of non-conservative force of the euler-bernoulli beam system, and the kinetic energy, potential energy and virtual work are substituted into the hamilton principle to obtain the euler-bernoulli beam system as follows:

wherein the content of the first and second substances,respectively, the first derivative and the second derivative of w (x, t) with respect to time, and the first derivative, the second derivative, the third derivative and the fourth derivative of w (x, t) with respect to x, respectively;

the boundary conditions are as follows:

wherein the content of the first and second substances,l is the length of the Euler-Bernoulli beam, ρ is the uniform mass per unit length of the Euler-Bernoulli beam, EI is the bending stiffness of the Euler-Bernoulli beam, T is the tension of the Euler-Bernoulli beam, M is the mass per unit length of the Euler-Bernoulli beam_sW (x, t) is the elastic deformation of the euler-bernoulli beam at time t position x in the xoy coordinate system,

further, the virtual control quantity is

α(t)＝-k₁w′(L，t)+k₂w″′(L，t)；

The backstepping item is

Wherein the content of the first and second substances,

the first derivatives of w '(L, t), w' "(L, t) with respect to time, respectively.

Is an iterative learning term;

the iterative learning item is

Wherein, beta, gamma, eta, k₁、k₂、k₃As gain parameters of the controller, beta, gamma, eta, k₁、 k₂、k₃Are all greater than 0, and the error variable is z₂(t)＝y₂(t)-α(t)，

Respectively representing the kth iteration term and its last iteration term, z_2，k(t) denotes the value of the k-th error variable.

Further, the Lyapunov function of the euler-bernoulli beam system is constructed based on the euler-bernoulli beam system, the virtual control quantity, the backstepping term and the iteration term, and specifically the following steps are carried out:

(1) according to the virtual control quantity, constructing a Lyapunov function as follows:

V₁(t)＝V_a(t)+V_b(t)；

wherein the content of the first and second substances,

(2) and constructing a Lyapunov function according to the backstepping term as follows:

(3) according to the iterative learning term, a Lyapunov function of the closed-loop Euler-Bernoulli beam system is constructed as follows:

further, according to the Lyapunov function, the stability of the Euler-Bernoulli beam system model is verified, and the specific steps are as follows:

by verifying the positive nature of the Lyapunov function, the stability of the La-Bernoulli beam system in accordance with the Lyapunov meaning is obtained;

and (3) verifying the negative nature of the first-order derivative of the Lyapunov function to obtain the consistency and gradual stability of the Euler-Bernoulli beam system.

Further, if the simulation result is not in accordance with the expectation, the gain parameter of the controller is corrected, and the digital simulation is performed again, specifically:

correcting the gain parameter of the controller, verifying the positive qualitative of the Lyapunov function and the negative qualitative of the first derivative of the Lyapunov function according to the gain parameter, and performing digital simulation on the Euler-Bernoulli beam system by using MATLAB simulation software.

Further, the simulation result includes the vibration amplitude of the euler-bernoulli beam without control action, the vibration amplitude of different iteration times with control action, and the relationship between the maximum error of the boundary vibration amplitude of the euler-bernoulli beam and the iteration times.

Compared with the prior art, the invention has the following advantages and effects:

compared with the traditional control method, the control method based on the backstepping iterative learning only needs less prior knowledge and calculated amount, has strong adaptability and is easy to realize. The controller designed by the invention comprises a virtual control quantity, a backstepping item and an iteration item, and the control quality of the Euler-Bernoulli beam is effectively improved by repeatedly applying information obtained by previous experiments to generate control input of expected output. And with the increase of iteration times, the elastic deformation of the Euler-Bernoulli beam is obviously reduced and is continuously close to zero, which shows that the designed controller has good control effect and is beneficial to improving the control precision of the controller in the industry.

Drawings

FIG. 1 is a schematic flow chart of a vibration control method based on backstepping iterative learning for an Euler-Bernoulli beam according to an embodiment of the present invention;

FIG. 2 is another schematic flow diagram of an embodiment of the present invention;

FIG. 3 is a schematic structural diagram of an Euler-Bernoulli beam system in an embodiment of the present invention;

FIG. 4 is a diagram showing the results of a simulation of the elastic deformation w (x, t) of an Euler-Bernoulli beam without control applied in an embodiment of the present invention;

FIG. 5 shows the elastic deformation w of an Euler-Bernoulli beam after applying control according to an embodiment of the present invention, with the number of iterations k being 15_kA simulation result diagram of (x, t);

FIG. 6 shows the elastic deformation w of an Euler-Bernoulli beam after applying control according to an embodiment of the present invention, with the number of iterations k being 35_kA simulation result diagram of (x, t);

FIG. 7 illustrates the elastic deformation w of the boundary of an Euler-Bernoulli beam after control has been applied in an embodiment of the present invention_kAnd (L, t) the maximum error is shown in the relation of the iteration number k.

Detailed Description

In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.