Cantilever beam vibration control method based on rapid nonsingular terminal composite sliding mode control

Abstract

The invention discloses a cantilever beam vibration control method based on rapid nonsingular terminal composite sliding mode control, and belongs to the field of vibration control. The method comprises the following steps: firstly, constructing a kinetic model of a piezoelectric cantilever beam, and establishing a state-space equation; then, according to a state-space equation of the piezoelectric cantilever beam, constructing an extended state observer, and obtaining estimated values of internal and external total disturbance, a displacement tracking error and a speed tracking error of the vibration control system; and finally, based on the internal and external total disturbance, displacement tracking error and speed tracking error of the vibration control system estimated by the extended state observer, constructing a rapid nonsingular terminal composite sliding mode controller, and using the controller to control a piezoelectric actuator on the piezoelectric cantilever beam. A convergence factor of a high-order exponential term is introduced into a nonsingular terminal sliding mode surface, so that a displacement tracking error and a speed tracking error of a vibration control system can be converged to zero within finite time, the problem of low convergence speed is solved, and meanwhile, the chattering phenomenon in the sliding mode control process is reduced.

Description

Technical Field

[0001] The present invention belongs to the technical field of vibration control, and in particular relates to a cantilever beam vibration control method based on fast non-singular terminal composite sliding mode control. Background Art

[0002] Vibration is a widespread phenomenon in industrial equipment. Structural vibrations, particularly those caused by environmental disturbances or system uncertainties, can accelerate material fatigue, reduce equipment accuracy, shorten service life, and even cause structural damage. As modern industry's demand for precision manufacturing continues to increase, vibration control has become a key technology for ensuring equipment reliability and safety.

[0003] To improve the vibration suppression capabilities of beam structures, traditional feedback-based active control techniques (such as PID control and optimal control) have been widely used. However, due to the complexity of the boundary conditions and system uncertainties of piezoelectric cantilever beams, it is often difficult to establish an accurate mathematical model. As a result, active control techniques are unable to directly and effectively suppress internal and external disturbances in the system. Consequently, these active control methods struggle to achieve satisfactory vibration suppression performance.

[0004] Sliding mode control (SMC) offers strong anti-interference capabilities, but traditional sliding mode control based on exponential reaching laws not only suffers from significant chattering but also fails to guarantee a rapid steady-state state. Therefore, improving the convergence speed to the sliding mode surface and mitigating chattering are key to applying sliding mode control to beam structure vibration control.

[0005] The Extended State Observer (ESO) is a core component of active disturbance rejection control technology. Its core concept is to dynamically estimate the total system disturbance as a new state. The ESO can simultaneously observe the system state and disturbances and compensate for the disturbance observations through a feedforward channel, effectively improving system performance. ESO-based control methods, through feedforward compensation, can significantly reduce the impact of internal and external disturbances and model uncertainty. Therefore, introducing the ESO into sliding mode control significantly reduces the switching gain of the sliding mode controller by sharing the disturbance rejection burden of sliding mode control. This not only enhances the system's anti-interference capability, but also effectively suppresses the chattering phenomenon inherent in traditional sliding mode control. Summary of the Invention

[0006] In view of the deficiencies in the prior art, the technical problem to be solved by the present invention is to provide a cantilever beam vibration control method based on fast non-singular terminal composite sliding mode control.

[0007] The present invention solves the technical problem by adopting the following technical solutions:

[0008] A cantilever beam vibration control method based on fast non-singular terminal composite sliding mode control is characterized by comprising the following steps:

[0009] Step 1: Construct the dynamic model of the piezoelectric cantilever beam and establish the state space equation;

[0010] Step 2: Based on the state space equation of the piezoelectric cantilever beam, an extended state observer is constructed to obtain the estimated values of the total internal and external disturbances, displacement tracking error, and velocity tracking error of the vibration control system;

[0011] Step 3: Based on the total internal and external disturbances, displacement tracking error, and velocity tracking error of the vibration control system estimated by the extended state observer, a fast non-singular terminal composite sliding mode controller is constructed and used to control the piezoelectric actuator on the piezoelectric cantilever beam.

[0012] The fast non-singular terminal sliding surface is:

[0013]

[0014] Where s represents the fast non-singular terminal sliding surface, λ and β are the design constants of the sliding surface; p, q, g, and h are odd numbers, and 1 < p / q < 2, g / h > p / q; e1 is the displacement tracking error of the vibration control system, and e2 is the velocity tracking error of the vibration control system;

[0015] The sliding mode reaching law is:

[0016]

[0017] Where, represents the first-order differential of the fast non-singular terminal sliding mode surface, n is the rate at which the tracking error reaches the fast non-singular terminal sliding mode surface, η is the sign function, and softsign(·) is the switching gain;

[0018] The fast non-singular terminal composite sliding mode control law is:

[0019]

[0020] Where u is the control voltage, b0 is the estimated value of the controller gain, z1 is the estimated value of the displacement tracking error of the vibration control system, z2 is the estimated value of the velocity tracking error of the vibration control system, and z3 is the estimated value of the total internal and external disturbance of the vibration control system.

[0021] Compared with the prior art, the present invention has the following beneficial effects:

[0022] 1. This application introduces a convergence factor of a high-order exponential term into the non-singular terminal sliding surface. This effectively increases the absolute value of the state derivative when moving away from the terminal sliding surface, allowing the displacement tracking error and velocity tracking error of the vibration control system to converge to zero along the non-singular terminal sliding surface within a finite time. This solves the slow convergence problem of existing non-singular terminal sliding mode control when moving away from the terminal sliding surface. The fast non-singular terminal composite sliding mode controller features fast response and excellent transient performance, and can effectively address system uncertainty, hysteresis, and nonlinearity issues caused by the use of piezoelectric actuators.

[0023] 2. The extended state observer is used to observe the various state quantities of the vibration system. There is no need to calculate the velocity tracking error by differentiating the displacement tracking error in traditional sliding mode control, thus avoiding the noise problem caused by large differential gain. At the same time, the total interference of the vibration control system can be observed and corresponding compensation can be performed through the feedforward channel to reduce the anti-interference burden of the sliding mode controller, the size of the sliding mode switching gain, and the chattering phenomenon of the sliding mode control process. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 is a control flow chart of the present invention;

[0025] Figure 2 is a voltage diagram of a sinusoidal excitation signal of the present invention;

[0026] Figure 3 is the oscillation displacement diagram of the piezoelectric cantilever beam of this embodiment without control;

[0027] Figure 4 : The oscillation displacement diagram of the piezoelectric cantilever beam in this embodiment under the sliding mode controller (SMC) and without control;

[0028] Figure 5 The oscillation displacement diagram of the piezoelectric cantilever beam in this embodiment under the fast non-singular terminal composite sliding mode controller (ESO-FNTSMC) and without control. DETAILED DESCRIPTION

[0029] Specific embodiments are given below in conjunction with the accompanying drawings. The specific embodiments are only used to introduce the technical solutions of the present invention in detail and are not intended to limit the scope of protection of the present application.

[0030] The present invention provides a cantilever beam vibration control method based on fast non-singular terminal composite sliding mode control, comprising the following steps:

[0031] Step 1: Construct the dynamic model of the piezoelectric cantilever beam and establish the state space equation;

[0032] The dynamic model of the piezoelectric cantilever beam is:

[0033]

[0034] Where r is the node displacement vector, is the first and second order differential of r, M uu is the mass matrix, C uu is the damping matrix, K uu is the stiffness matrix, is the piezoelectric coupling stiffness matrix, is the voltage vector applied to the piezoelectric actuator, F ue is the external force vector applied to the piezoelectric actuator;

[0035] Based on the dynamic model, the state space variables are introduced and the state space equation of the piezoelectric cantilever beam is established as follows:

[0036]

[0037] Where x(t) represents the state quantity, A represents the state matrix, and B represents the input matrix; u(t) represents the system control quantity, that is, the control voltage of the piezoelectric actuator; y(t) represents the output quantity, and C represents the output matrix;

[0038] The basic second-order form of the state-space equation is:

[0039]

[0040] Where x1 represents the tracking state quantity of the input signal, x2 represents the first-order differential of the tracking state quantity, y represents the actual output displacement signal, m represents the control torque, b represents the actual gain of the controller, f(x1, x2, T, t) represents the total internal and external disturbance of the vibration control system, and T represents the external disturbance of the vibration control system.

[0041] Step 2: Based on the state space equation of the piezoelectric cantilever beam, an extended state observer is constructed to estimate the total internal and external disturbances, displacement tracking error, and velocity tracking error of the vibration control system. The specific equation of the extended state observer is:

[0042]

[0043] Where e is the error between the estimated value and the actual value of the input signal, β1 = 3ω, β2 = 3ω 2 β3=3ω 3is the gain of the extended state observer, ω>0 is the bandwidth of the extended state observer, z1(·) is the estimated value of the displacement tracking error of the vibration control system, z2(·) is the estimated value of the velocity tracking error of the vibration control system, z3(·) is the estimated value of the total internal and external disturbance of the vibration control system, h is the sampling step size, fal(·) is the nonlinear function, α is the nonlinear factor, δ is the length of the linear segment, b0 is the estimated value of the controller gain, and k represents the time;

[0044] The nonlinear function is:

[0045]

[0046] Step 3: Based on the displacement and velocity tracking errors and combined with the finite time convergence mechanism, a fast non-singular terminal composite sliding mode controller is constructed, and the controller is used to control the vibration of the piezoelectric cantilever beam.

[0047] The traditional non-singular terminal sliding mode control will have a slow convergence problem when it is far away from the terminal sliding surface area. To address this phenomenon, this application introduces a convergence factor of high-order exponential terms. In order to make the displacement tracking error and velocity tracking error of the vibration control system converge to zero along the non-singular terminal sliding mode surface in a finite time, the fast non-singular terminal sliding mode surface is designed as:

[0048]

[0049] Where s represents the fast non-singular terminal sliding surface, λ∈R + and β∈R + are sliding mode surface design constants; p, q, g, h∈N are odd numbers satisfying 1<p / q<2, g / h>p / q; e1 is the displacement tracking error of the vibration control system, and e2 is the velocity tracking error of the vibration control system. In traditional sliding mode controller design, the displacement tracking error of the vibration control system is generally obtained by direct measurement, and the velocity tracking error of the vibration control system is obtained by differential calculation. In this application, the displacement tracking error and velocity tracking error of the vibration control system are estimated using an extended state observer, which can effectively reduce chattering.

[0050] Design the sliding mode reaching law with attractor as follows:

[0051]

[0052] Where, represents the first-order differential of the fast non-singular terminal sliding mode surface, η>0 is the sign function, n is the rate at which the tracking error reaches the fast non-singular terminal sliding mode surface, and softsign(·) is the switching gain;

[0053] Through the extended state observer, z1, z2, and z3 are estimated, where z1 is the estimated value of the displacement tracking error e1 of the vibration control system, z2 is the estimated value of the velocity tracking error e2 of the vibration control system, and z3 is the estimated value of the total internal and external disturbance f of the vibration control system. Substituting them into the fast non-singular terminal composite sliding mode controller, the fast non-singular terminal composite sliding mode control law is obtained:

[0054]

[0055] Where, u is the control voltage;

[0056] The piezoelectric actuator is controlled according to the fast non-singular terminal composite sliding mode control law to realize the vibration control of the piezoelectric cantilever beam.

[0057] Example

[0058] This embodiment uses Matlab / Simulink simulation research and Labview experiment to verify the first-order modal vibration control effect of the piezoelectric cantilever beam.

[0059] The key parameters of the cantilever beam and piezoelectric actuator were selected, including the hysteresis, creep, and nonlinear characteristics exhibited by the piezoelectric actuator during use, as well as the external disturbances to which the cantilever beam is subjected. Based on these key parameters, the response curve of the cantilever beam when actuated by the piezoelectric actuator, as well as the natural frequency and mode shape of the cantilever beam, were obtained. Based on the response curve, natural frequency, and mode shape, the optimal installation location of the piezoelectric actuator (at the root of the cantilever beam) was determined. The key parameters of the piezoelectric actuator and cantilever beam are shown in Table 1.

[0060] Table 1 Main parameters of piezoelectric actuator and cantilever beam

[0061]

[0062] The fast non-singular terminal composite sliding mode control is simulated using Matlab / Simulink. Under the same excitation conditions, the free oscillation curves of the piezoelectric cantilever beam under no control, sliding mode controller and fast non-singular terminal composite sliding mode controller are obtained respectively. A sinusoidal excitation signal with a first-order modal frequency and an amplitude of 10V is applied to the piezoelectric cantilever beam for 80 seconds, sampled every 0.001 seconds, and the free oscillation curve of the piezoelectric cantilever beam without control is observed. Under the same excitation conditions, the piezoelectric cantilever beam is excited for 80 seconds with a sliding mode controller (SMC) in effect. A laser displacement sensor is used to collect the vibration displacement signal of the free end of the piezoelectric cantilever beam in real time, and the data is transmitted to a data acquisition card. The data acquisition card transmits the vibration displacement signal collected by the laser displacement sensor to a sliding mode controller for processing. The sliding mode controller outputs a control voltage, which is amplified by a power amplifier to a certain multiple to reach the effective working voltage of the piezoelectric actuator. The amplified control voltage is applied to the piezoelectric actuator to achieve control of the piezoelectric cantilever beam and obtain the free oscillation curve of the free end of the piezoelectric cantilever beam under the sliding mode controller. See [1]. Figure 4 Similarly, obtain the free oscillation curve of the piezoelectric cantilever beam under the fast non-singular terminal composite sliding mode controller, see Figure 5 .

[0063] from Figure 4 、 5 It can be seen that under no control, the vibration response takes about 10 seconds to approach stability, and under the sliding mode controller, the vibration response takes about 7 seconds to approach stability. Under the fast non-singular terminal composite sliding mode controller, the vibration response quickly approaches the stable value in a short time (about 2 seconds), indicating that the system can quickly reach a stable state. This is because the application introduces a fast non-singular terminal sliding mode surface, which significantly shortens the time required for the system to reach stability, which is better than the traditional sliding mode control method. The experimental results show that the vibration amplitude of the piezoelectric cantilever beam controlled by the method of the application is attenuated by 88.13%, indicating that the combination of the extended state observer (ESO) for state estimation has shown excellent control effects in suppressing system chattering. In summary, the fast non-singular terminal composite sliding mode controller of the present application can effectively suppress the first-order modal vibration of the piezoelectric cantilever beam.

[0064] Any matters not described in the present invention are applicable to the prior art.

Claims

1. A cantilever beam vibration control method based on fast non-singular terminal composite sliding mode control, characterized in that: The following steps are involved: Step 1: Construct the dynamic model of the piezoelectric cantilever beam and establish the state space equation; Step 2: Based on the state space equation of the piezoelectric cantilever beam, an extended state observer is constructed to obtain the estimated values of the total internal and external disturbances, displacement tracking error, and velocity tracking error of the vibration control system; Step 3: Based on the total internal and external disturbances, displacement tracking error, and velocity tracking error of the vibration control system estimated by the extended state observer, a fast non-singular terminal composite sliding mode controller is constructed and used to control the piezoelectric actuator on the piezoelectric cantilever beam. The fast non-singular terminal sliding surface is: Where s represents the fast non-singular terminal sliding surface, λ and β are the design constants of the sliding surface; p, q, g, and h are odd numbers, and 1 < p / q < 2, g / h > p / q; e1 is the displacement tracking error of the vibration control system, and e2 is the velocity tracking error of the vibration control system; The sliding mode reaching law is: Where, represents the first-order differential of the fast non-singular terminal sliding mode surface, n is the rate at which the tracking error reaches the fast non-singular terminal sliding mode surface, η is the sign function, and softsign(·) is the switching gain; The fast non-singular terminal composite sliding mode control law is: Where u is the control voltage, b0 is the estimated value of the controller gain, z1 is the estimated value of the displacement tracking error of the vibration control system, z2 is the estimated value of the velocity tracking error of the vibration control system, and z3 is the estimated value of the total internal and external disturbance of the vibration control system.

2. The cantilever beam vibration control method based on fast non-singular terminal composite sliding mode control according to claim 1 is characterized in that: The extended state observer is expressed as: Where e is the error between the estimated value and the actual value of the input signal, y represents the actual output displacement signal; β1 = 3ω, β2 = 3ω 2 β3=3ω 3 is the gain of the extended state observer, ω>0 is the bandwidth of the extended state observer; h is the sampling step, fal(·) is the nonlinear function, α is the nonlinear factor, δ is the length of the linear segment, k represents the time, and m represents the control torque; The nonlinear function is:

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