A boundary cooperative control method for Timoshenko beams based on event-triggered mechanism

Abstract

This invention discloses a boundary cooperative control method for Timoshenko beams based on an event-triggered mechanism. The method involves the following steps: constructing a dynamic model of n Timoshenko beams that takes into account boundary disturbances and parameter uncertainty; using the n Timoshenko beams as followers to track a leader, and designing auxiliary variables for the followers; constructing a Lyapunov function based on the dynamic model and the auxiliary variables; designing a boundary cooperative controller and an adaptive parameter estimation update law based on the event-triggered mechanism; and applying boundary cooperative control to the n Timoshenko beams. This method can effectively suppress the vibration of the Timoshenko beams, mitigate the effects of boundary disturbances and parameter uncertainty on control performance, and reduce the communication burden between the actuator and the controller. Furthermore, the joint angles of the Timoshenko beams can track the leader, achieving joint angle coordination.

Description

Technical Field

[0001] The present invention relates to the technical fields of vibration control, adaptive control, event-triggered control and multi-agent collaborative control, and more particularly to a boundary collaborative control method for a Timoshenko beam based on an event-triggered mechanism. Background Art

[0002] The Timoshenko beam is one of the basic research models for flexible manipulators. Studying the complex control methods of the Timoshenko beam is of great significance for the application of flexible manipulators. Due to their flexible properties, flexible manipulators have the characteristics of light weight, low energy consumption, and high speed, and can be applied in industrial production, aerospace missions and other fields. In the study of flexible manipulators, when the ratio of the manipulator length to the cross-sectional diameter is small, the Timoshenko beam is used as the basic model. However, in engineering, the Timoshenko beam will inevitably produce shear deformation and elastic deformation due to influences such as boundary perturbations, which in turn leads to long-term vibration. This unwanted vibration can significantly affect the joint angle control accuracy of the Timoshenko beam, causing mechanical fatigue damage and even production safety accidents. Therefore, vibration control of the Timoshenko beam is a problem that needs to be solved when applying the Timoshenko beam.

[0003] Traditional Timoshenko beams are often used in applications with a boundary control strategy based on a time-triggered mechanism, where signals are acquired and actuators are updated at fixed sampling intervals. These intervals are generally short, which inevitably increases the communication burden between the actuator and the controller, leading to continuous energy loss.

[0004] Currently, most research on Timoshenko beam control is based on single-beam control methods such as PID and robust control. However, there are few methods for collaborative control of multiple Timoshenko beams, and single-beam control methods cannot achieve complex and efficient tasks, such as synchronized assembly and coordinated transport in the production process. Summary of the Invention

[0005] The purpose of the present invention is to address the above-mentioned defects in the prior art and to provide a boundary cooperative control method for Timoshenko beams based on an event-triggered mechanism, so as to provide a theoretical reference for vibration control, adaptive control, event-triggered control and multi-agent cooperative control of Timoshenko beams in the fields of industrial production, mechanical engineering, etc.

[0006] The purpose of the present invention can be achieved by taking the following technical solutions:

[0007] A boundary collaborative control method for a Timoshenko beam based on an event triggering mechanism, the boundary collaborative control method comprising the following steps:

[0008] S1. Based on the dynamic characteristics of n Timoshenko beams, a dynamic model considering boundary disturbances and parameter uncertainties is constructed;

[0009] S2, select an external reference signal as a leader, use n Timoshenko beams as followers to track the leader, and design auxiliary variables for the n Timoshenko beams as followers;

[0010] S3. constructing a Lyapunov function based on the dynamic model and auxiliary variables;

[0011] S4. Based on the Lyapunov function, an adaptive parameter estimation technique is used to process boundary disturbances and parameter uncertainties, a continuous-time controller is designed, an event-triggered technique is used to process the continuous-time controller, and a boundary cooperative controller and an adaptive parameter estimation update law based on an event-triggered mechanism are designed;

[0012] S5. Apply boundary cooperative control to n Timoshenko beams based on the boundary cooperative controller.

[0013] Furthermore, the dynamic characteristics of the n Timoshenko beams in step S1 include the kinetic energy, potential energy, and virtual work done by the non-conservative force on the Timoshenko beam. Substituting the above kinetic energy, potential energy, and virtual work into the Hamiltonian principle, the dynamic model of the Timoshenko beam is finally obtained as follows:

[0014] Wherein, the subscript i∈{1,2,...,n} represents the number of the i-th Timoshenko beam, the variable with the subscript i represents that the variable belongs to the i-th Timoshenko beam, and n is the total number of Timoshenko beams; a is the spatial position variable, t is the time variable; y i (a,t), represent the elastic deformation and shear deformation of the ith Timoshenko beam at time t and position a, respectively; z i (a, t) represents the joint angle of the i-th Timoshenko beam. The displacement at time t, and the elastic deformation y i (a,t) has the following relationship: L i represents the arm length of the i-th Timoshenko beam, ρ irepresents the mass per unit length of the beam arm of the i-th Timoshenko beam, E i represents the bending stiffness of the beam arm of the i-th Timoshenko beam, I 1i represents the moment of inertia per unit length of the beam arm of the i-th Timoshenko beam, K i is a parameter related to the cross section; the variables of the time partial derivative and space partial derivative in the dynamic model are marked with the following definitions:

[0015] The boundary conditions of the i-th Timoshenko beam are:

[0016]

[0017]

[0018]

[0019]

[0020] in, (*)(L i ,t) represents the variable (*) at position a=L i The value at I 2i is the hub moment of inertia of the ith Timoshenko beam, J i 、m i are the moment of inertia and mass of the end load of the i-th Timoshenko beam respectively; F i1 (t) is the control force output by the end force actuator of the i-th Timoshenko beam, F i2 (t), F i3 (t) are the torques output by the motors at the end of the i-th Timoshenko beam and the hub respectively; d i1 (t), d i2 (t) are the time-varying boundary force and torque perturbations acting on the end of the i-th Timoshenko beam, d i3 (t) is the time-varying boundary torque disturbance acting on the hub of the i-th Timoshenko beam, and satisfies in, are the boundary perturbations d i1 (t)~d i3 (t) is the upper bound; in the boundary conditions, the variables superscripted on the time partial derivative and the space partial derivative are defined as follows:

[0021] Note that the above parameter ρ i 、E i , I1i , K i , I 2i 、J i 、m i and boundary perturbation d i1 (t), d i2 (t), d i3 (t) is assumed to be unknown in the control method. The above dynamic model constructed based on the Hamiltonian principle is a set of high-order partial differential equations related to both time and spatial position. The boundary conditions are a set of ordinary differential equations related only to time. The state variables in the dynamic model and boundary conditions of the Timoshenko beam can reflect the energy contained in it and are the basis for the subsequent construction of the Lyapunov function. The control force and torque are generated by the force actuator and the motor respectively. The force actuator and the motor can both be called actuators. The actuator is updated by the controller and installed at the boundary of the Timoshenko beam, that is, the hub (a=0) or the end (a=L of the Timoshenko beam). i ).

[0022] Furthermore, in step S2, the external reference signal is selected as a leader, n Timoshenko beams are used as followers to track the leader, and the process of designing auxiliary variables for the n Timoshenko beams as followers is as follows:

[0023] Select the external reference signal as a leader and number the leader as 0. Define the leader's joint angle trajectory as in, represents the external reference signal, The first derivative of with respect to time t and the second-order derivative All are 0;

[0024] Each Timoshenko beam is regarded as a follower. The n Timoshenko beams as followers are connected according to an undirected graph, and the following auxiliary variables are designed for the n followers:

[0025]

[0026]

[0027]

[0028]

[0029] Among them, ψ i (t) is the first auxiliary variable, φ i (t) is the second auxiliary variable, is the third auxiliary variable, si (t) is the fourth auxiliary variable; represents the joint angle of the Timoshenko beam numbered j, where j is defined as j∈{1,2,...,n}; a ij is the element with subscript (i, j) in the adjacency matrix A. The values ​​of subscript i and j are consistent with the numbers of the n Timoshenko beams. The adjacency matrix A={a ij}∈R n×n is a non-negative matrix describing the connection between n Timoshenko beams as followers, defined as if there is signal communication between Timoshenko beams numbered i and j, then a ij =1, otherwise a ij =0;a i0 The diagonal matrix M0 is the element on the diagonal line that is in the i-th row and i-th column at the same time. The diagonal matrix M0 = diag{a i0}∈R n×n is a non-negative matrix describing the connection between the Timoshenko beams as n followers and the leader, defined as if the Timoshenko beam numbered i can obtain the signal of the leader, then a i0 =1, otherwise a i0 =0;c i is a positive constant. In the control method, it is assumed that at least one Timoshenko beam among the n followers can obtain the leader's signal. Among the auxiliary variables mentioned above, the variables related to the time partial derivative and the space partial derivative are marked with the following definitions:

[0030] The auxiliary variables mentioned above include error signals for achieving coordinated control of joint angles and vibration suppression. They also imply the connection relationships between the Timoshenko beams acting as followers and between the Timoshenko beams and the leader. They are an important component for the subsequent construction of Lyapunov functions and the design of boundary collaborative controllers.

[0031] Furthermore, the process of constructing the Lyapunov function based on the dynamic model and auxiliary variables in step S3 is as follows:

[0032] Based on the dynamic model and auxiliary variables, the Lyapunov function is constructed as:

[0033] V(t)=V1(t)+V2(t)+V3(t),

[0034] in,

[0035]

[0036]

[0037]

[0038] Where, β i , γ i are the first energy parameter and the second energy coefficient for realizing the overall energy constraint of the i-th Timoshenko beam, and are defined as follows: i >0,γ i > 0; the diagonal matrix Γ is defined as Γ=diag{k i4 +k i5}∈R n×n , k i4 、k i5 are the fourth and fifth positive control parameters for realizing boundary cooperative control of the i-th Timoshenko beam, respectively; is the error vector, defined as in, 1 n =[1,1,...,1]∈R n×1 ; is the Laplace matrix, defined as Among them, l ij yes For elements with subscripts (i, j), when i≠j, there is l ij =-a ij ,otherwise, is the boundary perturbation d i1 (t)~d i3 The upper bound of (t) estimated value of; They are Θ 1i 、Θ 2i , I 2i The estimation error is defined as Among them, Θ 1i 、Θ 2i are the first parameter vector and the second parameter vector of the i-th Timoshenko beam, respectively, defined as Θ 1i =[K i ,m i ] T ,Θ 2i =[E i ,J i ] T , are Θ 1i 、Θ 2i The estimated value of is the moment of inertia of the end load I 2i The estimated value of Λ 1i , Λ 2iThe adjustment estimate of the ith Timoshenko beam is The first adjustment matrix and the second adjustment matrix of the update law, and Λ 1i , Λ 2i are both defined as positive definite diagonal matrices; γ i1 , γ i2 , γ i3 ,λ i The adjustment estimate of the ith Timoshenko beam is The first positive adjustment parameter, the second positive adjustment parameter, the third positive adjustment parameter, and the fourth positive adjustment parameter of the update law.

[0039] The Lyapunov function above includes the dynamic model of n Timoshenko beams and the state variables in the boundary conditions. It can reflect the overall energy of the n Timoshenko beams. The overall energy of the n Timoshenko beams is closely related to the design of the subsequent boundary cooperative controller, because the design of the boundary cooperative controller is based on the fact that the boundary cooperative controller can attenuate the overall energy of the n Timoshenko beams.

[0040] Furthermore, in step S4, based on the Lyapunov function, an adaptive parameter estimation technique is used to process boundary disturbances and parameter uncertainties, a continuous-time controller is designed, an event-triggered technique is used to process the continuous-time controller, and a boundary cooperative controller and an adaptive parameter estimation update law based on an event-triggered mechanism are designed as follows:

[0041] Take the first-order derivative of the Lyapunov function V(t) with respect to time t According to the Lyapunov stability theory, the adaptive parameter estimation technology is used to deal with boundary disturbances and parameter uncertainties. Then, the following continuous-time controller F for the ith Timoshenko beam can be designed: im (t),m=1,2,3:

[0042]

[0043]

[0044]

[0045] Among them, k i1 、k i2 、k i3 are the first, second and third positive control parameters for implementing boundary cooperative control of the i-th Timoshenko beam respectively; ζ i1 ,ζ i2 ,ζ i3The perturbation upper bound estimate for the i-th Timoshenko beam is The first attenuation factor, second attenuation factor, and third attenuation factor of attenuation are defined as μ im are the first attenuation factor adjustment parameter and the second attenuation factor adjustment parameter of the mth attenuation factor of the i-th Timoshenko beam respectively; Ξ1 and Ξ2 are the first state vector and the second state vector respectively, which are defined as Among them, the superscript It means to find the first-order partial derivative of (*) with respect to time t and position a at the same time;

[0046] Accordingly, the adaptive parameter estimation update law is designed as follows:

[0047]

[0048]

[0049] Define an incremental sequence As a set of event triggering moments, where represents the time when the kth event of the mth controller of the i-th Timoshenko beam is triggered; let the initial time Then we can finally design the following mth boundary cooperative controller based on event triggering mechanism:

[0050]

[0051]

[0052]

[0053] Accordingly, the following event trigger function f is designed m (e im (t),t) and event triggering conditions:

[0054]

[0055]

[0056]

[0057] Among them, ε im ,θ im 、ω im are the triggering functions f of the mth event of the i-th Timoshenko beam respectively. m (e im (t), t) first event trigger parameter, second event trigger parameter, third event trigger parameter, eim (t) is the mth continuous time controller F im (t) and the mth controller based on event triggering mechanism The error between them is defined as

[0058] In the above boundary cooperative controller, the unknown parameter ρ i 、E i , I 1i , K i , I 2i 、J i 、m i and boundary perturbation d i1 (t), d i2 (t), d i3 The adverse effects of (t) on the control performance of the i-th Timoshenko beam are addressed by the adaptive parameter estimation update law designed using the adaptive parameter estimation technique. This involves directly estimating the unknown parameters and the upper bound of the boundary disturbance in the boundary cooperative controller. Furthermore, the state signals in both the boundary cooperative controller and the adaptive parameter estimation update law can be obtained using sensor sampling or finite difference calculations.

[0059] Furthermore, the process of designing the boundary cooperative controller and the adaptive parameter estimation update law based on the event-triggered mechanism also includes the step of verifying the asymptotic stability of n Timoshenko beams under the action of the boundary cooperative controller, and the process is as follows:

[0060] By constraining the first energy parameter coefficient β in the Lyapunov function i and the second energy coefficient γ i , ensuring the positive definiteness of the Lyapunov function;

[0061] Calculating a first-order derivative of the Lyapunov function with respect to time t to verify the semi-negative definiteness of the first-order derivative of the Lyapunov function;

[0062] By applying Lyapunov stability theory and Barbalat's lemma, it is concluded that n Timoshenko beams are asymptotically stable under the boundary cooperative controller.

[0063] The above-mentioned step of verifying the asymptotic stability of the n Timoshenko beams under the action of the boundary cooperative controller is actually to verify the rationality of the boundary cooperative controller, because a reasonable boundary cooperative controller should be able to make the total energy of the n Timoshenko beams decay over time, and this step is indispensable; according to the Lyapunov stability theory, when the n Timoshenko beams are asymptotically stable under the action of the boundary cooperative controller, it can be said that the total energy of the n Timoshenko beams decays over time, and further, it can be concluded that the boundary cooperative controller is reasonable.

[0064] Furthermore, the process of applying boundary cooperative control to n Timoshenko beams based on the boundary cooperative controller based on the event trigger mechanism in step S5 is as follows:

[0065] When the i-th Timoshenko beam is in the time interval The continuous-time controller F can be calculated in the calculation unit of the mth boundary controller. im (t), Boundary collaborative controller based on event triggering mechanism Event trigger function f m (e im (t),t), adaptive parameter estimation update law Calculate, if the event triggers the condition f m (e im (t),t)≥0 is not satisfied, then the mth actuator of the i-th Timoshenko beam will always keep outputting the same The same value, once the event triggers the condition f m (e im (t),t)≥0 is satisfied, the trigger time is assigned to And with time The m-th actuator of the i-th Timoshenko beam is updated with the numerical value of . When the event triggering conditions are continuously satisfied and the actuators are continuously updated, the boundary cooperative control effect of the n Timoshenko beams can be achieved, that is, vibration suppression and joint angle cooperative control can be achieved.

[0066] The above-mentioned process of applying boundary coordinated control to n Timoshenko beams is also the specific implementation process of the control method of the present invention in practical applications. The actual control effect of the control method on the Timoshenko beam is reflected in the calculation controller, the updating of the actuator according to the event triggering conditions, and the output of the control force or torque by the actuator at the boundary to directly adjust the elastic deformation, shear deformation, and joint angle.

[0067] Furthermore, the boundary cooperative controller can achieve vibration suppression of the Timoshenko beam under the influence of boundary disturbances and parameter uncertainties, and can also enable the joint angle of the Timoshenko beam as a follower to track the leader. In addition, the controller based on the event trigger mechanism can also reduce the communication burden between the actuator and the controller, and ultimately achieve the boundary cooperative control effect of the Timoshenko beam based on the event trigger mechanism.

[0068] The present invention has the following advantages and effects compared to the prior art:

[0069] (1) The present invention proposes a boundary cooperative control method for a Timoshenko beam. Compared with the traditional single Timoshenko beam control method, the boundary cooperative control method can not only achieve vibration suppression of the Timoshenko beam, but also achieve cooperative control of the joint angles of multiple Timoshenko beams. In addition, since parameter uncertainty and boundary disturbances are estimated by adaptive parameter estimation technology, the present invention can also alleviate the impact of parameter uncertainty and boundary disturbances on control performance.

[0070] (2) The present invention proposes a boundary cooperative control method for Timoshenko beams based on an event trigger mechanism. Compared with the traditional control method based on a time trigger mechanism, the boundary cooperative control method based on the event trigger mechanism only updates the actuator at the event trigger moment. The event trigger moment is uneven and non-periodic. Therefore, the present invention can alleviate the communication burden between the actuator and the controller and reduce energy loss.

[0071] (3) The present invention proposes a boundary cooperative control method for Timoshenko beams based on an event trigger mechanism. Each Timoshenko beam mainly includes three types of boundary controllers: 1) One type of boundary controller mainly realizes joint angle cooperative control (follower tracks leader). The controller signal mainly comes from the difference between its own sampling signal and the sampling signal of the adjacent Timoshenko beam. In addition, the follower connected to the leader can also obtain the leader signal. In particular, only when the event trigger condition is met will the controller communicate with the actuator and update the actuator with the sampling signal at the current trigger moment. Otherwise, the actuator will always maintain the actuator output synthesized from the sampling signal at the last event trigger moment. 2) One type of boundary controller mainly realizes the suppression of the elastic deformation of the Timoshenko beam. The control signal comes from its own signal. The actuator output is also updated only when the corresponding event trigger condition is met. 3) One type of boundary controller mainly realizes the suppression of the shear deformation of the Timoshenko beam. The control signal is also obtained from its own signal. The actuator update method is also updated only when the event trigger is met. By selecting appropriate control parameters and adjustment parameters, the joint angle of the Timoshenko beam can quickly track the upper leader and achieve vibration suppression, and can also effectively alleviate the impact of boundary disturbances and parameter uncertainties on control performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0073] Figure 1 This is a flow chart of a boundary cooperative control method of a Timoshenko beam based on an event triggering mechanism disclosed in the present invention;

[0074] Figure 2 This is a schematic structural diagram of a typical Timoshenko beam disclosed in the present invention;

[0075] Figure 3 This is an example diagram of the connection topology of the Timoshenko beam in Example 1 of the present invention;

[0076] Figure 4 2 is a schematic diagram of the elastic deformation simulation results of the Timoshenko beam in Example 2 of the present invention;

[0077] Figure 5 2 is a schematic diagram of the shear deformation simulation results of the Timoshenko beam in Example 2 of the present invention;

[0078] Figure 6 1 is a schematic diagram of the joint angle simulation results of the Timoshenko beam in Example 2 of the present invention;

[0079] Figure 7 Schematic diagram of the simulation results of the event triggering moment of the Timoshenko beam in Example 2 of the present invention. DETAILED DESCRIPTION

[0080] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0081] Example 1

[0082] Figure 1 The flowchart of the boundary cooperative control method of Timoshenko beam based on event trigger mechanism disclosed by the present invention specifically includes the following steps:

[0083] S1. According to the dynamic characteristics of n Timoshenko beams, a dynamic model considering boundary disturbances and parameter uncertainties is constructed.

[0084] like Figure 2 The figure shows a typical Timoshenko beam structure diagram. The subscript i represents the Timoshenko beam numbered i, i∈{1,2,...,n}. The variable with the subscript i in the figure represents that the variable belongs to the i-th Timoshenko beam, and n is the total number of Timoshenko beams. Figure 2 The circular rotatable pattern on the left represents the installation of a device that can generate torque F i3 The hub of the motor (t) may be subjected to a torque disturbance d in the same direction i3 (t); the strip image consisting of two solid lines and one dotted line in the middle represents the length L i The beam arm of the Timoshenko beam; the small ball on the right connected to the beam arm represents the mass m i The end load is subjected to the end motor output torque F i2 (t) and the end force actuator output force F i1 (t), and are also subject to the same-direction disturbance d i2 (t), d i1 The influence of (t). The coordinate system AoY represents the inertial coordinate system with the hub center o as the origin and the horizontal direction as the A axis. The coordinate system aoy represents the self-coordinate system with the hub center o as the origin and the tangent direction of the beam arm starting point as the a axis. Therefore, y i(a,t) is the position of the ith Timoshenko beam in its own coordinate system aoy a∈[0,L i ] and the elastic deformation at time t∈[0,∞), is the shear deformation of the ith Timoshenko beam at position a and time t, is the joint angle of the hub rotation of the i-th Timoshenko beam in the AoY coordinate system, It is the displacement of position a on the beam arm in the AoY coordinate system at time t.

[0085] The dynamic model of the i-th Timoshenko beam is:

[0086]

[0087]

[0088] in, a is the spatial position variable, t is the time variable; L i represents the arm length of the i-th Timoshenko beam, ρ i represents the mass per unit length of the beam arm of the i-th Timoshenko beam, E i represents the bending stiffness of the beam arm of the i-th Timoshenko beam, I 1i represents the moment of inertia per unit length of the beam arm of the i-th Timoshenko beam, K i is a parameter related to the cross section. The variables of the time partial derivative and space partial derivative in the dynamic model are marked with the following definitions:

[0089]

[0090] The boundary conditions of the i-th Timoshenko beam are:

[0091]

[0092]

[0093]

[0094]

[0095] in, (*)(L i ,t) represents the variable (*) at position a=L i The value at I 2i is the hub moment of inertia of the ith Timoshenko beam, J i 、m iare the moment of inertia and mass of the end load of the i-th Timoshenko beam respectively; F i1 (t) is the control force output by the end force actuator of the i-th Timoshenko beam, F i2 (t), F i3 (t) are the torques output by the motors at the end of the i-th Timoshenko beam and the hub respectively; d i1 (t), d i2 (t) are the time-varying boundary force and torque perturbations acting on the end of the i-th Timoshenko beam, d i3 (t) is the time-varying boundary torque disturbance acting on the hub of the i-th Timoshenko beam, and satisfies in, are the boundary perturbations d i1 (t)~d i3 In the boundary conditions, the variables of time partial derivative and space partial derivative are defined as follows:

[0096] Note that the above parameter ρ i 、E i , I 1i , K i , I 2i 、J i 、m i and boundary perturbation d i1 (t), d i2 (t), d i3 (t) is assumed to be unknown in the control method. The dynamic model constructed according to the Hamiltonian principle is a set of high-order partial differential equations related to both time and spatial position, and the boundary conditions are a set of ordinary differential equations related only to time. The control force and torque are generated by the force actuator and the motor respectively. The force actuator and the motor can both be called actuators. The actuator is updated by the controller and installed at the boundary of the Timoshenko beam, that is, the hub (a=0) or the end (a=L of the Timoshenko beam. i ).

[0097] S2. Select an external reference signal as a leader, use n Timoshenko beams as followers to track the leader, and design auxiliary variables for the n Timoshenko beams as followers.

[0098] like Figure 3The figure shows an example of a topology that connects six Timoshenko beams and one leader. In the figure, a one-way arrow indicates that the Timoshenko beam at the tip of the arrow can obtain signals from the Timoshenko beam at the tail of the arrow. A two-way arrow indicates that two Timoshenko beams can exchange signals with each other. As you can see, Figure 3 In the method, six Timoshenko beams are used as followers and are numbered 1 to 6, corresponding to n=6 in the control method. The leader is numbered 0, and only the Timoshenko beams numbered 1 and 2 can obtain signals from the leader 0.

[0099] Select the external reference signal as a leader and number the leader as 0. Define the leader's joint angle trajectory as in, represents the external reference signal, The first derivative of with respect to time t and the second-order derivative All are 0;

[0100] Each Timoshenko beam is regarded as a follower. The n Timoshenko beams as followers are connected according to an undirected graph, and the following auxiliary variables are designed for the n followers:

[0101]

[0102]

[0103]

[0104]

[0105] Among them, ψ i (t) is the first auxiliary variable, φ i (t) is the second auxiliary variable, is the third auxiliary variable, s i (t) is the fourth auxiliary variable; represents the joint angle of the Timoshenko beam numbered j, where j is defined as j∈{1,2,...,n}; a ij is the element with subscript (i, j) in the adjacency matrix A. The values ​​of subscript i and j are consistent with the numbers of the n Timoshenko beams. The adjacency matrix A={a ij}∈R n×n is a non-negative matrix describing the connection between n Timoshenko beams as followers, defined as if there is signal communication between Timoshenko beams numbered i and j, then a ij =1, otherwise aij =0;a i0 The diagonal matrix M0 is the element on the diagonal line that is in the i-th row and i-th column at the same time. The diagonal matrix M0 = diag{a i0}∈R n×n is a non-negative matrix describing the connection between the Timoshenko beam as a follower and the leader, defined as if the Timoshenko beam numbered i can obtain the signal of the leader, then a i0 =1, otherwise a i0 =0;c i is a positive constant; among the auxiliary variables mentioned above, the variables of time partial derivative and space partial derivative are marked with the following definitions: In addition, according to the connection relationship between n Timoshenko beams, the Laplace matrix can be defined as Among them, l ij yes For elements with subscripts (i, j), when i≠j, there is l ij =-a ij ,otherwise, In the control method, it is necessary to assume that there is at least one Timoshenko beam among the n followers that can obtain the leader's signal.

[0106] The auxiliary variables mentioned above include error signals for achieving coordinated control of joint angles and vibration suppression. They also imply the connection relationships between the Timoshenko beams acting as followers and between the Timoshenko beams and the leader. They are an important component for the subsequent construction of Lyapunov functions and the design of boundary collaborative controllers.

[0107] S3. Constructing a Lyapunov function based on the dynamic model and auxiliary variables.

[0108] Based on the dynamic model and auxiliary variables, the Lyapunov function is constructed as:

[0109] V(t)=V1(t)+V2(t)+V3(t),

[0110] in

[0111]

[0112]

[0113]

[0114] Among them, β i , γ iare the first energy parameter and the second energy coefficient for realizing the overall energy constraint of the i-th Timoshenko beam, and are defined as follows: i >0,γ i > 0; the diagonal matrix Γ is defined as Γ=diag{k i4 +k i5}∈R n×n , k i4 、k i5 are the fourth and fifth positive control parameters for realizing boundary cooperative control of the i-th Timoshenko beam, respectively; is the error vector, defined as in,

[0115] 1 n =[1,1,...,1]∈R n×1 ; is the Laplace matrix, defined as Among them, l ij yes For elements with subscripts (i, j), when i≠j, there is l ij =-a ij ,otherwise, is the boundary perturbation d i1 (t)~d i3 The upper bound of (t) estimated value of; They are Θ 1i 、Θ 2i , I 2i The estimation error is defined as Among them, Θ 1i 、Θ 2i are the first parameter vector and the second parameter vector of the i-th Timoshenko beam, respectively, defined as Θ 1i =[K i ,m i ] T ,Θ 2i =[E i ,J i ] T , They are Θ 1i 、Θ 2i The estimated value of is the moment of inertia of the end load I 2i The estimated value of Λ 1i , Λ 2i The adjustment estimate of the ith Timoshenko beam is The first adjustment matrix and the second adjustment matrix of the update law, and Λ 1i , Λ 2i are defined as positive definite diagonal matrices; γ i1 , γ i2 , γ i3 ,λ i The adjustment estimate of the ith Timoshenko beam is The first positive adjustment parameter, the second positive adjustment parameter, the third positive adjustment parameter, and the fourth positive adjustment parameter of the update law.

[0116] The above Lyapunov function contains the dynamic model of n Timoshenko beams and the state variables in the boundary conditions, which can reflect the overall energy situation of n Timoshenko beams. The subsequent design of the boundary collaborative controller needs to be based on the overall energy attenuation reflected by the above Lyapunov function.

[0117] S4. Based on the Lyapunov function, adaptive parameter estimation technology is used to deal with boundary disturbances and parameter uncertainties, a continuous-time controller is designed, event triggering technology is used to deal with the continuous-time controller, and a boundary cooperative controller and adaptive parameter estimation update law based on the event triggering mechanism are designed.

[0118] Find the first derivative of the Lyapunov function V(t) with respect to time t According to the Lyapunov stability theory, the adaptive parameter estimation technology is used to deal with boundary disturbances and parameter uncertainties. Then, the following continuous-time controller F for the ith Timoshenko beam can be designed: im (t),m=1,2,3:

[0119]

[0120]

[0121]

[0122] Among them, k i1 、k i2 、k i3 are the first, second and third positive control parameters for implementing boundary cooperative control of the i-th Timoshenko beam respectively; ζ i1 ,ζ i2 ,ζ i3 The perturbation upper bound estimate for the i-th Timoshenko beam is The first attenuation factor, second attenuation factor, and third attenuation factor of attenuation are defined as im 、μim are the first attenuation factor adjustment parameter and the second attenuation factor adjustment parameter of the mth attenuation factor of the i-th Timoshenko beam respectively; Ξ1 and Ξ2 are the first state vector and the second state vector respectively, which are defined as Among them, the superscript It means to find the first-order partial derivative of (*) with respect to time t and position a at the same time;

[0123] Accordingly, the adaptive parameter estimation update law is designed as follows:

[0124]

[0125]

[0126] Define an incremental sequence As a set of event triggering moments, where represents the time when the kth event of the mth controller of the i-th Timoshenko beam is triggered; let the initial time Then we can finally design the following mth boundary cooperative controller based on event triggering mechanism:

[0127]

[0128]

[0129]

[0130] Accordingly, the following event trigger function f is designed m (e im (t),t) and event triggering conditions:

[0131]

[0132]

[0133]

[0134] Among them, ε im ,θ im 、ω im are the triggering functions f of the mth event of the i-th Timoshenko beam respectively. m (e im (t), t) first event trigger parameter, second event trigger parameter, third event trigger parameter, e im (t) is the mth continuous time controller F im (t) and the mth controller based on event triggering mechanism The error between

[0135] In the above boundary cooperative controller, the unknown parameter ρ i 、E i , I 1i , K i , I 2i 、J i 、m i and boundary perturbation d i1 (t), d i2 (t), d i3 The adverse effects of (t) on the control performance of the i-th Timoshenko beam are all addressed by the adaptive parameter estimation update law designed using the adaptive parameter estimation technique. Furthermore, the state signals in the boundary cooperative controller and the adaptive parameter estimation update law can be obtained using sensor sampling or finite difference calculation.

[0136] Note that the design of the event-triggered boundary cooperative controller and the adaptive parameter estimation update law is based on the assumption that the boundary cooperative controller can attenuate the total energy reflected by the Lyapunov function. Therefore, the asymptotic stability of n Timoshenko beams under the boundary cooperative controller needs to be verified to demonstrate the rationality of the boundary cooperative controller. The specific process is as follows:

[0137] (1) By constraining the first energy parameter coefficient β in the Lyapunov function i and the second energy coefficient γ i , to ensure the positive definiteness of the Lyapunov function, as follows: First, introduce the following inequality lemma:

[0138] Lemma 1: For n-dimensional vectors x1,x2∈R n×1 ,have

[0139] Lemma 2: For z(a,t), if z(0,t)=z a (0,t)=0,a∈[0,L],t∈[0,∞), then

[0140] Define a positive definite function as follows:

[0141]

[0142] in, σ1=min{ρ i ,I 1i ,E i ,K i},σ2=max{ρi ,I 1i ,E i ,K i}, min{*,...,*} means taking the minimum value of the elements in the set {*,...,*}, and max{*,...,*} means taking the maximum value of the elements in the set {*,...,*}.

[0143] Using inequality Lemmas 1 and 2 to scale V2(t) in the Lyapunov function V(t), we obtain:

[0144]

[0145] in,

[0146] is an inequality Adding V1(t) in the Lyapunov function V(t) to both sides of , we can get: Constrained β i , γ i By making σ1-σ3>0, the positive definiteness of the Lyapunov function V(t) can be guaranteed.

[0147] (2) Take the first derivative of the Lyapunov function with respect to time t and verify the semi-negative definiteness of the first derivative of the Lyapunov function, as follows:

[0148] Find the first derivative of the Lyapunov function V(t) with respect to time And by scaling the inequality Lemmas 1 and 2, we can get:

[0149]

[0150] Where s=[s i (t)] T ∈R n×1 , z(L)=[z i (L i ,t)] T ∈R n×1 , y(L)=[y i (L i ,t)] T ∈R n×1 , φ(t)=[φ i (t)] T ∈R n×1 , ψ(t)=[ψ i (t)] T ∈R n×1 , K=diag{K i}∈Rn×n ,E=diag{E i}∈R n×n , βρL=diag{β i ρ i L i}∈R n×n , βI1L=diag{β i I 1i L i}∈R n×n , βEL=diag{β i E i L i}∈R n×n , κ g =diag{k ig}∈R n×n (g=1,2...,5).

[0151] 2=min{(β i -2γ i )I 1i}, δ1, δ2, δ2, δ4, and δ5 are the first scaling parameter, the second scaling parameter, the third scaling parameter, the fourth scaling parameter, and the fifth scaling parameter, respectively.

[0152] From the above event triggering conditions, we can see that at two specific triggering moments and Between, that is m=1,2,3, the following conditions are met:

[0153] Therefore, when The following inequality holds:

[0154]

[0155] in,

[0156] To verify the semi-negative definiteness of the first-order derivative of the Lyapunov function with respect to time t, the following inequality must also be satisfied:

[0157] σ1-σ3>0, ε1~ε4>0, K-βρl≥0, K-(1+δ2)βKL≥0,

[0158] E-βI1L≥0,E-βEL-δ3γE≥0,

[0159]

[0160]

[0161] If the above inequality conditions are met, it can be concluded that when t→∞, the first-order derivative of the Lyapunov function is is semi-negative definite, that is Therefore, the semi-negative definiteness of the first-order derivative of the Lyapunov function with respect to time t is verified.

[0162] (3) Applying Lyapunov stability theory and Barbalat’s lemma, it is concluded that n Timoshenko beams are asymptotically stable under the boundary cooperative controller, as follows:

[0163] First, we introduce the Barbalat lemma based on Lyapunov stability theory: if the differentiable scalar function p(t) has a lower bound and its first-order derivative with respect to time is is semi-negative definite and uniformly continuous (semi-negative definite is equivalent to Uniformly continuous is equivalent to the second-order derivative with respect to time t is bounded), then when t→∞,

[0164] Trigger interval for the above events Inside, the first-order derivative of the Lyapunov function Integrating both sides of the inequality, we can get:

[0165]

[0166] in, Symbol λ min (*) represents the smallest eigenvalue of the matrix *, and the symbol ||*||2 represents the 2-norm of the vector *.

[0167] Regarding the above The inequality from Iterate to At this moment, we can get:

[0168]

[0169] From the above inequality, we can know that: V(t), It is bounded, as we know from the limited energy during modeling. is also bounded, so applying Lyapunov stability theory and Barbalat's lemma, we can obtain: when t→∞, s i (t)→0,φ i (t)→0,ψ i(t)→0 holds, and the final result is as follows:

[0170]

[0171] The above results show that the asymptotic stability of n Timoshenko beams under the above-mentioned boundary cooperative controller is verified by applying Lyapunov stability theory and Barbalat's lemma.

[0172] The above asymptotic stability verification results show that the total energy of the n Timoshenko beams reflected by the Lyapunov function is attenuated under the action of the above boundary cooperative controller. Therefore, the boundary cooperative controller designed in this step is reasonable.

[0173] S5. Based on the boundary cooperative controller, boundary cooperative control is applied to n Timoshenko beams.

[0174] When the i-th Timoshenko beam is in the time interval The continuous-time controller F can be calculated in the calculation unit of the mth boundary controller. im (t), Boundary collaborative controller based on event triggering mechanism Event trigger function f m (e im (t),t), adaptive parameter estimation update law Calculate, if the event triggers the condition f m (e im (t),t)≥0 is not satisfied, then the mth actuator of the i-th Timoshenko beam will always keep outputting the same The same value, once the event triggers the condition f m (e im (t),t)≥0 is satisfied, the trigger time is assigned to And with time The m-th actuator of the i-th Timoshenko beam is updated with the numerical value of . When the event triggering conditions are continuously satisfied and the actuators are continuously updated, the boundary cooperative control effect of the n Timoshenko beams can be achieved, that is, vibration suppression and joint angle cooperative control can be achieved.

[0175] In summary, this embodiment provides a boundary cooperative control method for Timoshenko beams based on an event trigger mechanism, including: constructing a dynamic model of n Timoshenko beams that takes into account boundary disturbances and parameter uncertainties; using n Timoshenko beams as followers to track one leader, and designing auxiliary variables for the followers; constructing a Lyapunov function based on the dynamic model and the auxiliary variables; designing a boundary cooperative controller and an adaptive parameter estimation update law based on the event trigger mechanism based on the Lyapunov function, combined with event triggering technology and adaptive parameter estimation technology; and applying boundary cooperative control to the n Timoshenko beams based on the boundary cooperative controller. The control method of the present invention can effectively suppress the vibration of the Timoshenko beam, alleviate the impact of boundary disturbances and parameter uncertainty on control performance, reduce the communication burden between the actuator and the controller, and the joint angle of the Timoshenko beam can track the leader, thereby achieving the boundary cooperative control effect of the Timoshenko beam based on the event trigger mechanism.

[0176] Example 2

[0177] For the event-triggered boundary collaborative control method in Example 1, the MATLAB simulation software can be used to digitally simulate the Timoshenko beam to further verify the effectiveness of the event-triggered boundary collaborative control method for the Timoshenko beam proposed in this invention. Therefore, this embodiment is based on the event-triggered boundary collaborative control method in Example 1, combined with the simulation parameters in Tables 1, 2, and 3, to perform a digital simulation of the Timoshenko beam.

[0178] Table 1. System simulation parameters of the i-th Timoshenko beam (i = 1, 2, ..., 6)

[0179]

[0180] In Table 1, It represents the leader’s joint angle trajectory and is also the target of the joint angle coordinated control.

[0181] Table 2. Simulation parameters of the boundary cooperative controller, parameter estimation update law, and event triggering conditions for the i-th Timoshenko beam

[0182]

[0183]

[0184] Table 2 6×6 Represents the identity matrix of dimension 6×6.

[0185] Table 3. Simulated values ​​of boundary perturbations for the ith Timoshenko beam

[0186] parameter Numerical <![CDATA[d i1 (t)]]> 0.001×i×[sin(3πt)+sin(10πt)]N <![CDATA[d i2 (t)]]> 0.001×i×[0.1+sin(3πt)+sin(10πt)]Nm <![CDATA[d i3 (t)]]> 0.001×i×cos(3πt)Nm

[0187] The π in Table 3 refers to pi.

[0188] This embodiment uses Figure 3 As the connection topology of the 6 Timoshenko beams in the simulation, the matrix describing the connection relationship between Timoshenko beams and between Timoshenko beams and leaders is for:

[0189]

[0190] Figures 4 to 7 These are all simulation result diagrams in this embodiment. Figure 4 The figure shows the elastic deformation y of the six Timoshenko beam arms after applying the boundary cooperative control method based on the event trigger mechanism. i (a, t)(i=1,2,...,6)Simulation result diagram. Figure 5 The figure shows the shear deformation of the six Timoshenko beam arms after applying the boundary cooperative control method based on the event trigger mechanism. Schematic diagram of simulation results. Figure 4 and Figure 5 It can be seen that under the action of the boundary cooperative controller based on the event-triggered mechanism, even in the presence of parameter uncertainty and boundary disturbances, the elastic deformation and shear deformation of the Timoshenko beam arm are suppressed to 0, thus achieving vibration suppression of the Timoshenko beam. Figure 6 As shown, in the boundary cooperative controller based on event trigger mechanism and Figure 3 The joint angles of the six Timoshenko beams under the connection topology shown Can track the joint angle trajectory of the leader That is, to achieve coordinated control of joint angles. Figure 7 Shown are the event triggering moments of the boundary cooperative controllers of the six Timoshenko beams. The horizontal axis represents time, and the vertical axis represents the number i of the Timoshenko beam. Figure 7 It can be seen that the event triggering time is uneven and non-periodic. The boundary cooperative controller based on the event triggering mechanism is only updated at the event triggering time, which greatly reduces the communication burden between the actuator and the controller and reduces energy consumption.

[0191] It can be seen from the simulation results that the control method designed in the present invention can effectively suppress the vibration of the Timoshenko beam, alleviate the impact of boundary disturbances and parameter uncertainty on the control performance, and reduce the communication burden between the actuator and the controller. In addition, the joint angle of the Timoshenko beam can track the leader, realizing the Timoshenko beam boundary collaborative control effect based on the event trigger mechanism.

[0192] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A boundary cooperative control method for Timoshenko beam based on event triggering mechanism, characterized in that: The boundary collaborative control method comprises the following steps: S1. Based on the dynamic characteristics of n Timoshenko beams, a dynamic model considering boundary disturbances and parameter uncertainties is constructed; S2, select an external reference signal as a leader, use n Timoshenko beams as followers to track the leader, and design auxiliary variables for the n Timoshenko beams as followers; S3. constructing a Lyapunov function based on the dynamic model and auxiliary variables; S4, based on the Lyapunov function, using adaptive parameter estimation technology to deal with boundary disturbances and parameter uncertainties, designing a continuous time controller, using event triggering technology to deal with the continuous time controller, designing a boundary cooperative controller based on the event triggering mechanism and an adaptive parameter estimation update law, in step S4, defining an incremental sequence As a set of event triggering moments, where represents the time when the kth event of the mth controller of the i-th Timoshenko beam is triggered; let the initial time Then we can finally design the following mth boundary cooperative controller based on event triggering mechanism: Accordingly, the following event trigger function f is designed m (e im (t),t) and event triggering conditions: Among them, ε im ,θ im 、ω im are the triggering functions f of the mth event of the i-th Timoshenko beam respectively. m (e im (t), t) first event trigger parameter, second event trigger parameter, third event trigger parameter, e im (t) is the mth continuous time controller F im (t) Collaborative controller with the mth boundary based on event triggering mechanism The error between S5. Apply boundary cooperative control to n Timoshenko beams based on the boundary cooperative controller.

2. The method for boundary cooperative control of Timoshenko beams based on event triggering mechanism according to claim 1 is characterized in that: The dynamic characteristics of the n Timoshenko beams in step S1 include the kinetic energy, potential energy, and virtual work done by the non-conservative force on the Timoshenko beam. Substituting the above kinetic energy, potential energy, and virtual work into the Hamiltonian principle, the dynamic model of the Timoshenko beam is finally obtained as follows: Wherein, the subscript i∈{1,2,...,n} represents the number of the i-th Timoshenko beam, the variable with the subscript i represents that the variable belongs to the i-th Timoshenko beam, and n is the total number of Timoshenko beams; a is the spatial position variable, t is the time variable; y i (a,t), represent the elastic deformation and shear deformation of the ith Timoshenko beam at time t and position a, respectively; z i (a, t) represents the joint angle of the i-th Timoshenko beam. The displacement at time t, and the elastic deformation y i (a,t) has the following relationship: L i represents the arm length of the i-th Timoshenko beam, ρ i represents the mass per unit length of the beam arm of the i-th Timoshenko beam, E i represents the bending stiffness of the beam arm of the i-th Timoshenko beam, I 1i represents the moment of inertia per unit length of the beam arm of the i-th Timoshenko beam, K i is a parameter related to the cross section; the variables of the time partial derivative and space partial derivative in the dynamic model are marked with the following definitions: The boundary conditions of the i-th Timoshenko beam are: in (*)(L i ,t) represents the variable (*) at position a=L i The value at I 2i is the hub moment of inertia of the ith Timoshenko beam, J i 、m i are the moment of inertia and mass of the end load of the i-th Timoshenko beam respectively; F i1 (t) is the control force output by the end force actuator of the i-th Timoshenko beam, F i2 (t), F i3 (t) are the torques output by the motors at the end of the i-th Timoshenko beam and the hub respectively; d i1 (t), d i2 (t) are the time-varying boundary force disturbance and time-varying boundary torque disturbance acting on the end of the i-th Timoshenko beam, respectively, and d i3 (t) is the time-varying boundary torque disturbance acting on the hub of the i-th Timoshenko beam, and satisfies in, are the boundary perturbations d i1 (t)~d i3 (t) is the upper bound; in the boundary conditions, the variables superscripted on the time partial derivative and the space partial derivative are defined as follows: Note that the above parameter ρ i 、E i , I 1i , K i , I 2i 、J i 、m i and boundary perturbation d i1 (t), d i2 (t), d i3 (t) is assumed to be unknown in the control method.

3. The method for boundary cooperative control of Timoshenko beams based on event triggering mechanism according to claim 2, characterized in that: In step S2, the external reference signal is selected as a leader, n Timoshenko beams are used as followers to track the leader, and the process of designing auxiliary variables for the n Timoshenko beams as followers is as follows: Select the external reference signal as a leader and number the leader as 0. Define the leader's joint angle trajectory as in, represents the external reference signal, The first derivative of with respect to time t and the second-order derivative All are 0; Each Timoshenko beam is regarded as a follower. The n Timoshenko beams as followers are connected according to an undirected graph, and the following auxiliary variables are designed for the n followers: Among them, ψ i (t) is the first auxiliary variable, φ i (t) is the second auxiliary variable, is the third auxiliary variable, s i (t) is the fourth auxiliary variable; represents the joint angle of the Timoshenko beam numbered j, where j is defined as j∈{1,2,...,n}; a ij is the element with subscript (i, j) in the adjacency matrix A. The values ​​of subscript i and j are consistent with the numbers of the n Timoshenko beams. The adjacency matrix A = {a ij }∈R n×n is a non-negative matrix describing the connection between n Timoshenko beams as followers, defined as if there is signal communication between Timoshenko beams numbered i and j, then a ij =1, otherwise a ij =0;a i0 The diagonal matrix M0 is the element on the diagonal line that is in the i-th row and i-th column at the same time. The diagonal matrix M0 = diag{a i0 }∈R n×n is a non-negative matrix describing the connection between the Timoshenko beams as n followers and the leader, defined as if the Timoshenko beam numbered i can obtain the signal of the leader, then a i0 =1, otherwise a i0 =0;c i is a positive constant. In the control method, it is assumed that at least one Timoshenko beam among the n followers can obtain the leader's signal. Among the auxiliary variables mentioned above, the variables related to the time partial derivative and the space partial derivative are marked with the following definitions:

4. The method for boundary cooperative control of Timoshenko beams based on event triggering mechanism according to claim 3 is characterized in that: In step S3, the process of constructing the Lyapunov function based on the dynamic model and auxiliary variables is as follows: Based on the dynamic model and auxiliary variables, the Lyapunov function is constructed as: V(t)=V1(t)+V2(t)+V3(t), in Where, β i , γ i are the first energy parameter and the second energy coefficient for realizing the overall energy constraint of the i-th Timoshenko beam, and are defined as follows: i >0,γ i > 0; the diagonal matrix Γ is defined as Γ=diag{k i4 +k i5 }∈R n×n , k i4 、k i5 are the fourth and fifth positive control parameters for realizing boundary cooperative control of the i-th Timoshenko beam, respectively; is the error vector, defined as in, 1 n =[1,1,...,1]∈R n×1 ; is the Laplace matrix, defined as Among them, l ij yes For elements with subscripts (i, j), when i≠j, there is l ij =-a ij ,otherwise, is the boundary perturbation d i1 (t)~d i3 The upper bound of (t) estimated value of; They are Θ 1i 、Θ 2i , I 2i The estimation error is defined as Among them, Θ 1i 、Θ 2i are the first parameter vector and the second parameter vector of the i-th Timoshenko beam, respectively, defined as Θ 1i =[K i ,m i ] T ,Θ 2i =[E i ,J i ] T , are Θ 1i 、Θ 2i The estimated value of is the moment of inertia of the end load I 2i The estimated value of Λ 1i , Λ 2i The adjustment estimate of the ith Timoshenko beam is The first adjustment matrix and the second adjustment matrix of the update law, and Λ 1i , Λ 2i are defined as positive definite diagonal matrices; γ i1 , γ i2 , γ i3 ,λ i The adjustment estimate of the ith Timoshenko beam is The first positive adjustment parameter, the second positive adjustment parameter, the third positive adjustment parameter, and the fourth positive adjustment parameter of the update law.

5. The method for boundary cooperative control of Timoshenko beams based on event triggering mechanism according to claim 4 is characterized in that: In step S4, based on the Lyapunov function, an adaptive parameter estimation technique is used to process boundary disturbances and parameter uncertainties, a continuous-time controller is designed, an event-triggered technique is used to process the continuous-time controller, and a boundary cooperative controller and an adaptive parameter estimation update law based on an event-triggered mechanism are designed as follows: The first-order derivative V(t) of the Lyapunov function V(t) with respect to time t is obtained. Based on the Lyapunov stability theory, the adaptive parameter estimation technology is used to deal with the boundary disturbance and parameter uncertainty. Then, the following continuous-time controller F for the i-th Timoshenko beam can be designed: im (t),m=1,2,3: Among them, k i1 、k i2 、k i3 are the first, second and third positive control parameters for implementing boundary cooperative control of the i-th Timoshenko beam respectively; ζ i1 ,ζ i2 ,ζ i3 The perturbation upper bound estimate for the i-th Timoshenko beam is The first attenuation factor, second attenuation factor, and third attenuation factor of attenuation are defined as μ im are the first attenuation factor adjustment parameter and the second attenuation factor adjustment parameter of the mth attenuation factor of the i-th Timoshenko beam respectively; Ξ1 and Ξ2 are the first state vector and the second state vector respectively, which are defined as Among them, the superscript It means to find the first-order partial derivative of (*) with respect to time t and position a at the same time; Accordingly, the adaptive parameter estimation update law is designed as follows:

6. The method for boundary cooperative control of Timoshenko beams based on event triggering mechanism according to claim 5, characterized in that: The process of designing a boundary cooperative controller based on an event-triggered mechanism and an adaptive parameter estimation update law also includes the step of verifying the asymptotic stability of n Timoshenko beams under the action of the boundary cooperative controller, and the process is as follows: By constraining the first energy parameter coefficient β in the Lyapunov function i and the second energy coefficient γ i , ensuring the positive definiteness of the Lyapunov function; Calculating a first-order derivative of the Lyapunov function with respect to time t to verify the semi-negative definiteness of the first-order derivative of the Lyapunov function; By applying Lyapunov stability theory and Barbalat's lemma, it is concluded that n Timoshenko beams are asymptotically stable under the boundary cooperative controller.

7. The method for boundary cooperative control of Timoshenko beams based on event triggering mechanism according to claim 6, characterized in that: The process of applying boundary cooperative control to n Timoshenko beams based on the boundary cooperative controller based on the event trigger mechanism in step S5 is as follows: When the i-th Timoshenko beam is in the time interval The continuous-time controller F can be calculated in the calculation unit of the mth boundary controller. im (t), Boundary collaborative controller based on event triggering mechanism Event trigger function f m (e im (t),t), adaptive parameter estimation update law Calculate, if the event triggers the condition f m (e im (t),t)≥0 is not satisfied, then the mth actuator of the i-th Timoshenko beam will always keep outputting the same The same value, once the event triggers the condition f m (e im (t),t)≥0 is satisfied, the trigger time is assigned to And with time The m-th actuator of the i-th Timoshenko beam is updated with the numerical value of . When the event triggering conditions are continuously satisfied and the actuators are continuously updated, the boundary cooperative control effect of the n Timoshenko beams can be achieved, that is, vibration suppression and joint angle cooperative control can be achieved.

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