A mechanical system robust vibration control method based on event triggering mechanism

Abstract

The application relates to a mechanical system robust vibration control method based on an event triggering mechanism, which comprises the following steps: S1, establishing a mechanical system dynamic model; S2, introducing a switching event triggering mechanism; S3, designing an output feedback controller; S4, integrating the output feedback controller in S3 and the switching event triggering mechanism in S2 into the mechanical system dynamic model in S1, forming a mechanical system dynamic model containing the switching event triggering mechanism; S5, constructing a functional, calculating a sufficient condition for the stability of the mechanical system dynamic model in S4 through the functional, and realizing robust vibration control of the mechanical system dynamic model by meeting the sufficient condition for the stability of the mechanical system dynamic model in S4. Compared with the prior art, the application has the advantages of reducing unnecessary data transmission, saving limited network resources, obviously improving and upgrading the communication efficiency and control efficiency of the control system, and the like while guaranteeing that the system obtains desired stable control performance.

Description

Technical Field

[0001] This invention relates to the field of mechanical system control technology, and in particular to a robust vibration control method for mechanical systems based on an event-triggered mechanism. Background Technology

[0002] With the rapid development of modern industry, mechanical systems are widely used in fields such as robotics, industrial control, and aerospace. Stability is a fundamental requirement for any system, and stabilization requires designing an appropriate controller to stabilize the closed-loop system. Engineers often design proportional-integral-derivative (PID) controls based on the system signal to achieve system stabilization. However, due to the special internal structure of some systems, traditional PID controllers (Proportional-Integral-Derivative controllers) with any constant gain may not be able to stabilize the system.

[0003] Robust control is a control method capable of handling uncertainties and disturbances. It enables a system to remain stable and robust even under conditions of uncertainty and disturbances. Commonly used robust control methods include H∞ control and μ synthesis.

[0004] In recent years, a design method for high-gain stabilized controllers relying on square wave jitter has emerged. This method provides appropriate model transformation techniques to eliminate the high-gain terms introduced by the high-gain stabilized controller and accurately defines a small parameter tuning range to ensure system stability. However, this controller relies on continuous measurement signals, generating a large amount of sampling data that is ineffective for system performance during information transmission and consuming significant network resources. Ensuring mechanical system stability and implementing a collaboratively designed, rational control strategy to guarantee the desired control performance while minimizing limited network resources remains a major technical challenge. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the existing technology by providing a robust vibration control method for mechanical systems based on an event-triggered mechanism. This method is a method in the field of networked control, which aims to reduce network load and energy consumption, reduce the number of data transmissions, stabilize the system, and enhance the robustness of the system.

[0006] The objective of this invention can be achieved through the following technical solutions:

[0007] This invention discloses a robust vibration control method for mechanical systems based on an event-triggered mechanism, comprising the following steps:

[0008] S1. Based on the actual uncertain mechanical system, establish a dynamic model of the mechanical system;

[0009] S2. Introduce a switching event triggering mechanism into the dynamic model of the mechanical system;

[0010] S3. Based on the switching event triggering mechanism, design an output feedback controller;

[0011] S4. Integrate the output feedback controller described in S3 and the switching event triggering mechanism described in S2 into the mechanical system dynamic model in S1 to form a mechanical system dynamic model that includes the switching event triggering mechanism.

[0012] S5. Construct a functional, and use the functional to calculate the sufficient conditions for the stability of the dynamic model of the mechanical system described in S4. By satisfying the sufficient conditions for the stability of the dynamic model of the mechanical system described in S4, robust vibration control of the dynamic model of the mechanical system can be achieved.

[0013] Furthermore, the dynamic model of the mechanical system described in S1 includes a second-order linear mechanical system dynamic model.

[0014] Furthermore, the dynamic model of the second-order linear mechanical system is as follows:

[0015]

[0016] in: Let u(t)∈R be the system state, y(t)∈R be the system input, t be the system output, and t be the time. C = [c1 c2], where A, B, and C are all matrices, and b, c1, and c2 are constants, with c2 ≠ 0. For the uncertain term, The inequality ‖ΔA(t)‖≤σ0 is satisfied, where σ0 is a very small positive constant.

[0017] Furthermore, the switching event triggering mechanism described in S2 is as follows:

[0018]

[0019] Where: t l+1 For the next triggering time that satisfies the triggering condition, l∈N0, t0=0, h>0, h and h are the event triggering threshold parameters to be designed, ε>0, and ε is a parameter that is inversely proportional to the jitter frequency.

[0020] Furthermore, the output feedback controller described in S3 is:

[0021]

[0022] Where: k is the control gain. For a square wave signal, it can be represented by the following formula:

[0023]

[0024] Where: j∈N0, ε is a parameter inversely proportional to the jitter frequency.

[0025] Furthermore, the dynamic model of the mechanical system described in S4 is as follows:

[0026]

[0027]

[0028] in: l∈N0, e(t)=y(t)-y(t) l ).

[0029] Furthermore, the switching event triggering mechanism receives the measured y(t) l After obtaining the value, the calculation is performed using the formula for switching the event triggering mechanism, and the value of y(t) that meets the triggering condition is obtained. l The value is sent to the output feedback controller.

[0030] Furthermore, the switching event triggering mechanism has a minimum event interval ε. 2 h.

[0031] Furthermore, the functional is the Lyapunov-Krasovskii functional.

[0032] Furthermore, the Lyapunov-Krasovskii functional is:

[0033]

[0034] in:

[0035] V P (t)=z T (t)Pz(t),0 <P∈R n×n z(t) = ζ(t) - G(t)

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] (1) The key of the present invention lies in the introduction of a switching event triggering mechanism, which reduces unnecessary data transmission and saves limited network resources while ensuring that the system obtains expected stable control performance, so that the communication efficiency and control efficiency of the control system are significantly improved and enhanced.

[0045] (2) Aiming at the time-varying characteristic of the mechanical system, the present invention designs a novel output feedback controller dependent on square wave jitter and small parameters to stabilize the mechanical system, and provides an appropriate model transformation method to eliminate the high-gain term caused by the small parameter stabilizing output feedback controller, which can adjust the system state in a more flexible manner.

[0046] (3) The present invention constructs a functional, and obtains the sufficient condition for system stability through solving, and provides an effective upper bound of the square wave frequency that ensures system stability. Brief Description of the Drawings

[0047] Figure 1 is a flowchart of the robust vibration control method for a mechanical system of the present invention;

[0048] Figure 2 is a schematic diagram of a Kapitza pendulum of the present embodiment;

[0049] Figure 3 is the system state stability curve under the event triggering mechanism when c2≠0 in the present embodiment;

[0050] Reference Numerals in the Drawings:

[0051] Figure 2 : 1. Motor; 2. Crank; 3. Lever arm; 4. Pendulum bob. Detailed Description of the Embodiments

[0052] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. The present embodiment is implemented on the premise of the technical solution of the present invention, and provides detailed implementation modes and specific operation processes, but the protection scope of the present invention is not limited to the following embodiments.

[0053] Embodiment

[0054] This embodiment proposes a robust vibration control method for mechanical systems based on an event-triggered mechanism. First, a second-order mechanical system model is established based on a rigid pendulum—the Kappa pendulum. A switching event-triggered mechanism is introduced into the mechanical system model. To address the time-varying nature of the mechanical system model, a novel output feedback controller dependent on square wave jitter and small parameters is designed to stabilize the model. An appropriate model transformation method is provided to eliminate the high-gain term introduced by the small-parameter stabilizing output feedback controller, resulting in a neutral system model that guarantees the exponential stability of the original system. The effective upper bound of the square wave frequency for ensuring the stability of the mechanical system is given by solving for sufficient conditions.

[0055] like Figure 1 As shown, the specific steps include:

[0056] Step (1): In actual mechanical systems, there are many uncertain factors or nonlinearities in system parameters. Mechanical systems are complex nonlinear dynamic systems. When the mechanical system is operating normally, the load changes little. A linearized model can be used near its operating point to describe the dynamic characteristics of the system.

[0057] like Figure 2 The image shows a second-order uncertain mechanical system model of a Kapitza pendulum. A Kapitza pendulum is a rigid pendulum, its basic construction being: a motor drives a crank to rotate at high speed, the crank causes a lever arm to vibrate up and down, and a pendulum is pivotally connected to the lever arm. The unique feature of the Kapitza pendulum is that the vibrating suspension allows it to maintain stable equilibrium in an inverted position, with the pendulum positioned above the suspension point. Therefore, its linearized model at the equilibrium point can be expressed as:

[0058]

[0059] Where: θ(t) is the angle between the Kapichakra swing and the vertical direction, and β>0 is the amplitude of the suspension oscillation. is the natural frequency of the mathematical pendulum, g is the acceleration due to free fall, and l is the length of the pendulum.

[0060] Based on the Kapichacha mechanical system, this embodiment constructs an uncertain second-order mechanical system dynamic model, which is as follows:

[0061]

[0062] in: Let u(t)∈R be the system state, y(t)∈R be the system input, t be the system output, and t be the time. C = [c1 c2], where A, B, and C are all matrices, and b, c1, and c2 are constants, with c2 ≠ 0. For the uncertain term, The inequality ‖ΔA(t)‖≤σ0 is satisfied, where σ0 is a very small positive constant.

[0063] Step (2): Introduce a switching event triggering mechanism into the dynamic model of the mechanical system. The switching event triggering mechanism satisfies the following formula:

[0064]

[0065] Where: t l+1 For the next triggering time that satisfies the triggering condition, l∈N0, t0=0, h>0 is the trigger threshold parameter for the event to be designed, and ε>0 is a small parameter inversely proportional to the jitter frequency. The measured output y(t) is only measured at discrete time t. l (t l Available at (∈N0), where t l The timing is determined by the aforementioned switching trigger mechanism.

[0066] The event triggering mechanism is switched to receive y(t) l The value is calculated based on the formula of the switching event triggering mechanism, and the value that satisfies the triggering condition y(t) of the switching event triggering mechanism is determined. l The value is sent to the feedback controller in output step (3). The switching event triggering mechanism introduced in this embodiment has a minimum event interval ε. 2 h can prevent the Zeno phenomenon from occurring.

[0067] Step (3): Design an output feedback controller based on the switching event triggering mechanism. The output feedback controller satisfies the following formula:

[0068]

[0069] Where: k is the control gain. j∈N0.

[0070] Step (4): Integrate the output feedback controller described in step (3) and the switching event triggering mechanism described in step (2) into the mechanical system dynamic model in step (1) to form a mechanical system dynamic model containing the switching event triggering mechanism.

[0071] The dynamic model of a mechanical system that includes a switching event triggering mechanism satisfies the following formula:

[0072]

[0073] in: l∈N0, e(t)=y(t)-y(t) l );

[0074] The output measurement point y(t) in step (3) l Based on its different time intervals, it can be defined in the following form:

[0075]

[0076] Step (5): By constructing an appropriate Lyapunov-Krasovskii functional, the linear matrix inequality (LMI) is calculated to find the effective upper bound of the square wave frequency that guarantees the stability of the dynamic model of the mechanical system, and sufficient conditions for the stability of the dynamic model of the mechanical system are obtained. The sufficient conditions for the system to be exponentially stable are given by using the switching event triggering mechanism. An appropriate coordinate transformation is constructed to cancel the high gain, and the system is periodically averaged by applying the delay method under the new coordinates.

[0077] The corresponding calculation process is shown below:

[0078] The Lyapunov-Krasovskii functional is constructed as follows:

[0079]

[0080] in:

[0081] V P (t)=z T (t)Pz(t),0 <P∈R n×n

[0082] z(t) = ζ(t) - G(t)

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] t∈[t l ,t l +ε 2 h), l∈N0,W i ∈R + i = 1, 2

[0089]

[0090] Here, α is the decay rate, and ζ(t) satisfies coordinate transformation, where for:

[0091]

[0092] Applying Jensen's inequality to V(t), we can make it satisfy the following inequality:

[0093] Since V(t) is positive definite, applying the S-procedure to calculate the time derivatives of each term in the functional yields the following inequality:

[0094]

[0095]

[0096] in:

[0097] When χ(t) = 1,

[0098]

[0099]

[0100]

[0101]

[0102]

[0103] When χ(t) = 0,

[0104]

[0105]

[0106]

[0107] The results of the above functional calculations are expressed in the form of linear matrix inequalities (LMIs), and relevant system parameters are set: that is, the matrix is ​​set. σ>0, α>0, ε * >0, h>0 and ψ i >0 (i=1,2). If it is uncertain whether the system has an n×n dimension matrix P>0, R>0, and scalar Q>0, W i For any ε > 0 (i = 1, 2) and λ > 0, satisfying the following linear matrix inequalities (LMIs), then for all ε ∈ (0, ε * c2≠0 and trigger time t lWhen determined by the given switching event triggering mechanism, the dynamic model of the mechanical system with the added switching event triggering mechanism is exponentially stable, and its decay rate is α.

[0108] This is a sufficient condition for system stability, through which an effective upper bound can be obtained for the stable square wave frequency of the dynamic model of the mechanical system. When the Kapitza pendulum mechanical system satisfies the following linear matrix inequalities (LMIs), robust vibration control of the second-order uncertain Kapitza pendulum mechanical system model can be achieved, thus realizing the stability of the uncertain mechanical system.

[0109] Linear matrix inequalities (LMIs) are shown below:

[0110]

[0111] Where i = 1, ..., N: Here It is a symmetric matrix composed of the following parts:

[0112]

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119] The rest are zero matrices.

[0120] As shown in Table 1, this embodiment calculates some relevant parameters of the mechanical system:

[0121]

[0122] Table 1 shows the maximum effective upper bound ε for ensuring system stability when taking different values ​​of α, σ, h, and ρ. * .

[0123] Different values ​​of c2 will affect the system's behavior during the calculation process. Among them, the system state where c2 = 0 is a special case of c2 not equal to 0. The case of c2 = 0 is relatively simple and will not be considered in this embodiment. Here, we mainly consider the system state analysis of c2 not equal to 0.

[0124] When c2 ≠ 0, under the initial condition x(0) = [-0.2, 0.2]T Numerical simulations show that when the parameter ε = 2 and h = 1.10 -3 αρ=0.5·10 -3 Even with large values, the vibration control method based on the switching event triggering mechanism designed in this embodiment can still stabilize the mechanical system, as shown in the simulation image. Figure 3 As shown, the horizontal axis represents the simulation time (t = 50s), and the vertical axis is θ and... Let and represent the angle and rate of change of the angle of the Kapichakra mechanical system, respectively. Compared to the traditional periodic sampling method, the measurement quantity sent under the switching event triggering mechanism is less than that under periodic sampling. Under the control method of this embodiment, the measurement quantity sent when the mechanical system tends to stabilize is significantly reduced compared to the traditional control method. This also reduces the amount of data in the information transmission process and greatly saves network communication resources. It can be seen that the robust vibration control method for mechanical systems based on the switching event triggering mechanism proposed in this embodiment can ensure stable system operation while saving limited network bandwidth. Using the design method of this embodiment, the effective upper bound of the stable square wave frequency of the mechanical system can also be obtained, and the curve of the system state finally tending to stabilize can be obtained.

[0125] This embodiment addresses the challenge of stabilizing a type of mechanical system through static time-invariant output feedback. It proposes a high-gain stabilizing controller design method relying on square wave jitter. By introducing periodic square wave jitter, the system can adjust its state more flexibly. The key to this method lies in introducing a switching event triggering mechanism, which reduces unnecessary data transmission and conserves limited network resources while ensuring the system achieves the desired control performance. This embodiment significantly improves the communication and control efficiency of the control system.

[0126] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A robust vibration control method for a mechanical system based on an event-triggered mechanism, characterized in that, Includes the following steps: S1. Based on the actual uncertain mechanical system, establish a dynamic model of the mechanical system; S2. Introduce a switching event triggering mechanism into the dynamic model of the mechanical system; S3. Based on the switching event triggering mechanism, design an output feedback controller; S4. Integrate the output feedback controller described in S3 and the switching event triggering mechanism described in S2 into the mechanical system dynamic model in S1 to form a mechanical system dynamic model that includes the switching event triggering mechanism. S5. Construct a functional, calculate the sufficient conditions for the stability of the dynamic model of the mechanical system described in S4 using the functional, and achieve robust vibration control of the dynamic model of the mechanical system by satisfying the sufficient conditions for the stability of the dynamic model of the mechanical system described in S4. The dynamic model of the mechanical system described in S1 includes a second-order linear mechanical system dynamic model; The dynamic model of the second-order linear mechanical system is as follows: in: For system status, For system input, For system output, For time, and , A, B, C All are matrices. , as well as Both are constants and , ,for Satisfying the inequality , It is a very small positive number; The switching event triggering mechanism described in S2 is as follows: in: To meet the triggering conditions at the next triggering time, , , and The threshold parameter for triggering the event to be designed. , It is the reciprocal of the mechanical vibration frequency; The dynamic model of the mechanical system described in S4 is: ， in: , , It is a square wave signal.

2. The robust vibration control method for a mechanical system based on an event-triggered mechanism according to claim 1, characterized in that, The output feedback controller described in S3 is: in: To control the gain, the following formula is used: ； in: , This parameter is inversely proportional to the jitter frequency.

3. The robust vibration control method for a mechanical system based on an event-triggered mechanism according to claim 1, characterized in that, The switching event triggering mechanism receives measurements. After setting the value, the formula is used to perform calculations by switching the event triggering mechanism, and the values ​​that meet the triggering conditions are then calculated. The value is sent to the output feedback controller.

4. The robust vibration control method for a mechanical system based on an event-triggered mechanism according to claim 3, characterized in that, The switching event triggering mechanism has a minimum event interval. .

5. A robust vibration control method for a mechanical system based on an event-triggered mechanism according to claim 4, characterized in that, The functional mentioned is the Lyapunov-Krasovskii functional.

6. A robust vibration control method for a mechanical system based on an event-triggered mechanism according to claim 5, characterized in that, The Lyapunov-Krasovskii functional is: ， in: ， ； ； ； ； ； 。

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