Fractional order modeling and fractional order composite sliding mode control method for four-side fixed support plate vibration system based on inertial actuator

Abstract

The invention discloses a fractional order modeling and fractional order composite sliding mode control method for a four-side fixed support plate vibration system based on an inertial actuator in the technical field of vibration active control. The method comprises the following steps: S1, establishing a state-space equation of a fractional order electromechanical coupling system; s2, building a fractional order integral sliding mode control algorithm based on a fractional order expansion state observer; s3, estimating the system state quantity and the total disturbance of the vibration system in real time to obtain estimation information; s4, in combination with feedforward compensation of total disturbance, driving an inertial actuator to realize vibration suppression; the fractional order expansion state observer and fractional order integral sliding mode control are introduced, and feed-forward compensation and memory characteristics are utilized, so that the switching gain is reduced, the vibration suppression effect is improved, and the service life of equipment is prolonged; the NI hardware and the code automatic generation technology are combined, the efficient development process from graphical modeling to real-time monitoring is achieved, manual programming errors are avoided, and state monitoring is facilitated.

Description

Technical Field

[0001] This invention relates to the field of active vibration control technology, and in particular to a fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed plate vibration system based on an inertial actuator. Background Technology

[0002] In aerospace, automotive manufacturing, and shipbuilding engineering, lightweight and highly rigid thin-plate structures are widely used due to their superior mechanical properties and structural efficiency. However, under actual operating conditions, these thin-plate structures are inevitably subjected to continuous influence from complex external environmental loads (such as fluid excitation, mechanical impact, and acoustic loads). Due to the inherent structural characteristics of thin plates, they are highly susceptible to inducing severe structural vibrations under external excitation, accompanied by problems such as high-order harmonics. This excessive vibration not only leads to fatigue damage, fastener loosening, and acoustic noise pollution, but also seriously affects the operational accuracy and system stability of the corresponding equipment. Therefore, designing efficient vibration suppression schemes for typical structures such as four-sided fixed plates has significant scientific research value and engineering practical significance.

[0003] With the advent of smart structure theory, active vibration control technology based on a "sensor-actuator" closed-loop architecture has gradually replaced traditional passive vibration reduction methods. Among numerous actuators, electromagnetic inertial actuators have become the preferred choice for vibration suppression of thin-plate structures due to their low control voltage, large output force, wide bandwidth, and excellent control performance in the low-frequency range. However, in practical engineering applications, the electromechanical coupling system formed by the inertial actuator and the four-sided fixed plate has complex characteristics. Traditional integer-order modeling assumes that the system response is instantaneous and the damping is constant, neglecting the inductance effect inside the actuator and the widespread nonlocal memory effect and frequency dependence in the damping characteristics. In contrast, fractional-order operators, due to their infinite memory capacity and nonlocal characteristics, can accurately characterize the historical dependence of the physical system's evolution over time through non-integer orders, thus more realistically restoring the system's dynamic behavior and providing a theoretical basis for the design of subsequent control methods.

[0004] While traditional sliding mode control (SMC) offers strong robustness, it often requires significant switching gain to offset internal and external system disturbances, leading to severe chattering and compromising actuator lifespan. Introducing an extended state observer (ESO) can mitigate chattering through feedforward compensation, but traditional ESOs and SMCs, based on integer-order frameworks, suffer from observation lag and accuracy bottlenecks when handling electromechanical coupling systems with fractional-order characteristics. The introduction of fractional-order operators in control adds two adjustable degrees of freedom—differential and integral orders—making the controller's dynamic adjustment in the frequency domain more flexible. Fractional-order observers can capture total system disturbances more quickly, while fractional-order integral sliding surfaces can smooth the control output using their order and integral characteristics, fundamentally reducing chattering while maintaining robustness and achieving refined suppression of vibration signals.

[0005] Furthermore, the traditional "algorithm simulation - manual programming - chip burning" development model suffers from problems such as relatively independent stages, long development cycles, difficulty in writing complex algorithms, and difficulty in troubleshooting errors. This invention employs a model-based design approach, using MATLAB / Simulink as the core, combined with the NI PCIe-6343 acquisition card and Desktop Real-Time environment, to achieve a complete development process from control algorithm construction to automatic real-time code generation. This hardware-in-the-loop system not only improves the verification efficiency of advanced composite control algorithms but also ensures the real-time performance and reliability of fractional-order operations on actual physical platforms. In summary, this invention proposes a fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on inertial actuators, achieving a deep integration of simulation design and practical development, and possessing significant theoretical value and application prospects. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-support plate vibration system based on an inertial actuator.

[0007] The objective of this invention is achieved as follows: a fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator, comprising the following steps:

[0008] S1. A four-sided fixed plate vibration system based on an inertial actuator is modeled using the Caputo fractional operator. The fractional characteristics caused by damping and inductance are used to establish the state-space equation of the fractional electromechanical coupling system.

[0009] S2. Build a control algorithm implementation platform in a real-time simulation environment. Under the condition of disturbance of a four-sided fixed plate vibration system based on inertial actuator, build a fractional integral sliding mode control algorithm based on a fractional extended state observer.

[0010] S3. For vibration systems, parameter perturbation, higher-order terms, external excitation and interaction force are uniformly defined as total disturbance. A fractional-order extended state observer is designed to estimate the system state variables and total disturbance of the vibration system in real time and obtain estimation information.

[0011] S4. The obtained estimation information is used to construct a fractional integral sliding surface. The fractional integral operator is used to accumulate and compensate for the residual vibration error. Combined with the feedforward compensation of the total disturbance, the inertial actuator is driven to achieve vibration suppression.

[0012] Furthermore, the state-space equations of the fractional-order electromechanical coupling system in S1 are derived from the four-sided fixed plate electromechanical coupling model based on the inertial actuator, and the construction of the four-sided fixed plate electromechanical coupling model based on the inertial actuator is as follows:

[0013] ;

[0014] In the formula, , , , , , , and These represent the mass, displacement, damping, and stiffness coefficients of the fixed plate and the inertial actuator, respectively. , and These are the resistance, inductance, and force constants inside the inertial actuator; and To control voltage and current, As an external incentive, The vibration velocity of the inertial actuator. The vibration acceleration of the inertial actuator. The vibration velocity of the fixed plate, The vibration acceleration of the fixed plate, The vibration jerk of the inertial actuator. The rate of change of the external stimulus.

[0015] Furthermore, the state-space equations of the fractional-order electromechanical coupling system introduce fractional-order operators into the four-sided fixed-plate electromechanical coupling model based on inertial actuators. , ,in, , Fractional order:

[0016] ;

[0017] In the formula, It represents the sum of higher-order terms, external stimuli, and interaction forces;

[0018] Define the tracking error as ,in, To control the vibration of the fixed plate, a fractional-order operator is introduced, and state variables are defined. , The system state-space equations can be established as follows:

[0019] ;

[0020] in, The output value is the measured displacement value. For the displacement of the system, Defined as the fractional-order velocity of the system.

[0021] Furthermore, in S3, the sum of parameter perturbations, higher-order terms, external excitations, and interaction forces is defined as the total perturbation, and the total perturbation is defined by the following formula for the extended state:

[0022] ,in, To expand the state variables, the system state-space equations are expanded to:

[0023]

[0024] In the formula, For the system gain, the designed fractional-order extended state observer is as follows:

[0025]

[0026] In the formula, These are the observed values ​​of the system state and the total disturbance; The observer gain is adjusted according to the bandwidth method. , , , The observer bandwidth is used to adjust the gain. The tuning parameters, This is for tracking error.

[0027] Furthermore, the fractional integral sliding surface in S4 is constructed as follows:

[0028] ;

[0029] In the formula, , All of these are controller gain coefficients. Using Lyapunov functions to design auxiliary control laws and combining them with equivalent control laws, the overall system control law is obtained as follows:

[0030] ;

[0031] In the formula, Indicates switching gain. For switching functions, For equivalent control, To switch control, To control the gain, it represents the efficiency coefficient of converting the input voltage into the system's driving force.

[0032] Furthermore, the real-time simulation environment in S2 is specifically built on the NI PCIe-6343 data acquisition card in the Desktop Real-Time environment of the software MATLAB / Simulink.

[0033] Furthermore, the convergence of the fractional-order extended state observer is defined by the tracking error. The dynamic equation for the tracking error of the fractional-order extended state observer is as follows:

[0034] ;

[0035] In the formula, , , , ;

[0036] right Performing the Laplace transform, we get:

[0037] ,in, The observation error state vector is, i.e. , The error system matrix is ​​formed by adjusting the gain. The coefficient matrix formed determines the stability and speed of error convergence. Let be the disturbance input vector, representing the rate of change of the disturbance. The channel vector entering the error system has the following specific form: This is because of the expansion of state variables. The derivative is This directly affects the error dynamics of the third state. For the Laplace operator, corresponding to complex variables in the frequency domain, For the fractional differential operator in the corresponding time domain In the complex frequency domain, For the Laplace transform of the tracking error, The rate of change of the disturbance The Laplace transform of .

[0038] Furthermore, the calculation formula for the fractional-order extended state observer is improved as follows:

[0039] ;

[0040] ;

[0041] In the formula, It is a 3×3 identity matrix. Let be the inverse of the characteristic matrix of the error system. The characteristic polynomial of the error state-space model is represented by the bandwidth method. To further express:

[0042] ;

[0043] In the formula, Given the observer bandwidth, according to the final value theorem, we can obtain:

[0044] ;

[0045] when At that time, the error is bounded:

[0046] ;

[0047] Complete the convergence proof of the fractional-order extended state observer, where the tracking error converges to the neighborhood of the desired point. That is, when the disturbance is constant, the tracking error asymptotically converges to zero. Let i be the steady-state error of the tracking error for sequence i. This represents the maximum rate of change of the disturbance. The constant coefficient is greater than zero.

[0048] Furthermore, the equivalent control The determination is as follows:

[0049] right Taking the derivative, we get:

[0050] ;

[0051] When the sliding surface Then, the equivalent control law is:

[0052] .

[0053] Furthermore, the switching control The formula for determining is as follows:

[0054] The Lyapunov function is as follows:

[0055] ;

[0056] To ensure system stability, the following conditions must be met:

[0057] ;in, To estimate the sliding surface, For the estimated Lyapunov function;

[0058] ;

[0059] Switching control for:

[0060] .

[0061] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0062] This invention uses the MATLAB / Simulink environment to perform fractional-order modeling of a four-sided fixed plate electromechanical coupling vibration system based on an inertial actuator. The Caputo fractional-order calculus operator is used for model building, and the Oustaloup filter approximation algorithm is used to compare and verify the order of the fractional-order model for model fitting, so as to more realistically restore the dynamic process of the electromechanical coupling system.

[0063] This invention utilizes a fractional-order extended state observer to perform real-time feedforward compensation of the total system disturbance, significantly reducing the dependence of sliding mode control on switching gain. Combined with fractional-order integral sliding surface design, it leverages the memory characteristics and smoothing filtering effect of fractional-order integral operators to not only eliminate the lack of robustness in the approach phase of the system, but also fundamentally weaken the high-frequency chattering phenomenon inherent in traditional sliding mode control. While improving vibration suppression, it also reduces the power consumption and wear of inertial actuators, extending the service life of the equipment.

[0064] This invention utilizes MATLAB / Simulink and the NI PCIe-6343 acquisition card, combined with automatic code generation technology, to construct a complete algorithm design and development process. Through graphical modeling and real-time environment compilation, it avoids logical errors that are easily generated when manually writing complex algorithms. It also utilizes a host computer to observe system state changes in real time, making it convenient for relevant technical personnel to monitor the system status. Attached Figure Description

[0065] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0066] Figure 1 This is a schematic diagram of signal transmission in the vibration system of this invention.

[0067] Figure 2 This is a schematic diagram of the fractional-order composite sliding mode controller structure in this invention.

[0068] Figure 3 This is a diagram showing the model comparison experiment in this invention.

[0069] Figure 4 This is a time-domain comparison experiment of different algorithms of this invention.

[0070] Figure 5 This is a frequency domain comparison experiment diagram of different algorithms of this invention.

[0071] Figure 6 This is a comparison diagram of voltage control experiments using different algorithms of the present invention.

[0072] Figure 7 This is a flowchart illustrating the development process of the model-based design of this invention. Detailed Implementation

[0073] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0074] like Figures 1 to 7 The method for fractional-order modeling and fractional-order composite sliding mode control of a four-sided fixed plate vibration system based on an inertial actuator, as shown, includes the following steps:

[0075] S1. A four-sided fixed plate vibration system based on an inertial actuator is modeled using the Caputo fractional operator. The fractional characteristics caused by damping and inductance are used to establish the state-space equation of the fractional electromechanical coupling system.

[0076] S2. Build a control algorithm implementation platform in a real-time simulation environment. Under the condition of disturbance of a four-sided fixed plate vibration system based on inertial actuator, build a fractional integral sliding mode control algorithm based on a fractional extended state observer.

[0077] S3. For vibration systems, parameter perturbation, higher-order terms, external excitation and interaction force are uniformly defined as total disturbance. A fractional-order extended state observer is designed to estimate the system state variables and total disturbance of the vibration system in real time and obtain estimation information.

[0078] S4. The obtained estimation information is used to construct a fractional integral sliding surface. The fractional integral operator is used to accumulate and compensate for the residual vibration error. Combined with the feedforward compensation of the total disturbance, the inertial actuator is driven to achieve vibration suppression.

[0079] This invention addresses the problems of strong coupling, internal inductance effect, and damping characteristics in a four-sided fixed plate vibration system based on inertial actuators. It also addresses the limitations of traditional development methods by implementing a model-based design using the NI PCIe-6343 as the core. This involves a fractional-order modeling and fractional-order integral sliding mode control composite control algorithm based on a fractional-order extended state observer to suppress plate vibration. The invention combines MATLAB / Simulink simulation with hardware, improving development efficiency and control effectiveness.

[0080] Fractional-order modeling of a four-sided fixed plate based on an inertial actuator

[0081] The first-order mode of the fixed plate is represented by a single-degree-of-freedom system. Therefore, the system model of the four-sided fixed plate system based on the inertial actuator is as follows:

[0082] (1)

[0083] (2)

[0084] (3)

[0085] (4)

[0086] By combining equations (1) to (4), the system equations are obtained as follows:

[0087] (5)

[0088] Where, in the formula, , , , , , , and These represent the mass, displacement, damping, and stiffness coefficients of the fixed plate and the inertial actuator, respectively. , and These are the resistance, inductance, and force constants inside the inertial actuator; and To control voltage and current, As an external incentive, The vibration velocity of the inertial actuator. The vibration acceleration of the inertial actuator. The vibration velocity of the fixed plate, The vibration acceleration of the fixed plate, The vibration jerk of the inertial actuator. The rate of change of the external stimulus;

[0089] As can be seen from equation (5), the complex dynamics are caused by the internal damping and inductance of the inertial actuator, especially the higher-order terms. In addition, there are interaction forces and external excitations.

[0090] Traditional integer derivatives assume instantaneous response and constant damping coefficients, neglecting the inherent nonlocal memory effect and frequency-dependent energy dissipation within the actuator. Therefore, to improve the theoretical and identification accuracy of the model, the Caputo fractional operator is introduced. and To replace the integer derivatives in the model, where, , Given a fractional order, we obtain the equation:

[0091] (6)

[0092] In the formula, , It is a fractional order; It represents the sum of higher-order terms, external stimuli, and interaction forces.

[0093] To verify the feasibility of the model, the following experiment was designed:

[0094] ① Set the sampling value to 0.0001, use a sinusoidal voltage with a frequency of 48.5Hz and an amplitude of 5V as the control input to the inertial actuator, and collect the output displacement by a combination of accelerometer and integrator circuit. Use this input and output data as the basis for model fitting.

[0095] ② Using the System Identification Toolbox in MATLAB, the Levenberg-Marquardt algorithm was employed with a maximum of 20 iterations to identify the integer-order model, which achieved a goodness of fit of 97.67%.

[0096] ③ The parameters of the fitted integer-order model are given to the fractional-order model, the fractional-order order is adjusted, and the two models are compared and verified in Simulink. The same control signal as in the experiment is applied, and the results are compared with the actual system output.

[0097] Through the above experimental steps, the fractional order was finally determined to be... The model comparison results are as follows Figure 3 As shown, specific error quantization is performed using the integral of absolute error (IAE), and the results are obtained. Therefore, the feasibility and superiority of the fractional-order model were confirmed, and the experimental verification of the fractional-order model was completed.

[0098] Design of a fractional-order extended state observer:

[0099] For equation (6), a fractional-order extended state observer is introduced to estimate the fractional-order characteristics and unmeasurable state variables in the system, while simultaneously estimating the total disturbance and compensating for it in real time. The tracking error is defined as: ,in To control the vibration of the fixed plate, a fractional-order operator is introduced, and state variables are defined. , The system state-space equations can be established as follows:

[0100] (7)

[0101] The parameter perturbation, higher-order terms, external excitation, and interaction forces are defined as the total perturbation. Equation (7) is further derived as follows:

[0102] (8)

[0103] In the formula, Let be the system gain, where The output value is the measured displacement value. For the displacement of the system, Defined as the fractional-order velocity of the system; according to equation (8), the fractional-order extended state observer is designed as follows:

[0104] (9)

[0105] In the formula, These are the observed values ​​of the system state and the total disturbance; The observer gain can be adjusted using the bandwidth method. , , , The observer bandwidth is used to adjust the gain. The tuning parameters, This is for tracking error.

[0106] Prove the convergence of fractional integral sliding surfaces and define observation error. The dynamic equation for the observation error of the fractional integral sliding surface can be obtained as follows:

[0107] (10)

[0108] In the formula,

[0109] , , , ;

[0110] Assuming zero initial conditions, performing a Laplace transform on equation (10) yields:

[0111] (11)

[0112] in, The observation error state vector is, i.e. , The error system matrix is ​​formed by adjusting the gain. The coefficient matrix formed determines the stability and speed of error convergence. Let be the disturbance input vector, representing the rate of change of the disturbance. The channel vector entering the error system has the following specific form: This is because of the expansion of state variables. The derivative is This directly affects the error dynamics of the third state. For the Laplace operator, corresponding to complex variables in the frequency domain, For the fractional differential operator in the corresponding time domain In the complex frequency domain, For the Laplace transform of the tracking error, The rate of change of the disturbance The Laplace transform of;

[0113] Further rewriting yields:

[0114] (12)

[0115] (13)

[0116] In the formula, It is a 3×3 identity matrix. Let be the inverse of the characteristic matrix of the error system. The characteristic polynomial of the error state-space model is represented by the bandwidth method. To further express:

[0117] (14)

[0118] In the formula, Given the observer bandwidth, according to the final value theorem, we can obtain:

[0119] (15)

[0120] when At that time, the error is bounded:

[0121] (16)

[0122] This completes the convergence proof of the fractional-order extended state observer; the observation error can converge to the neighborhood of the desired point. That is, when the perturbation is constant, the observation error will asymptotically converge to zero. Let i be the steady-state error of the tracking error for sequence i. This represents the maximum rate of change of the disturbance. The constant coefficient is greater than zero.

[0123] Design of fractional integral sliding mode controller:

[0124] By utilizing a fractional-order extended state observer for real-time estimation of the system state, a fractional-order integral sliding mode controller is designed. Compared with the traditional integer-order integral sliding mode surface, it retains the advantages of eliminating arrival phase and suppressing steady-state error. Furthermore, it optimizes the dynamic response through fractional-order operators and avoids overshoot problems caused by traditional integral saturation. The fractional-order integral sliding mode surface is constructed as follows:

[0125] (17)

[0126] In the formula, , For the controller gain; taking the derivative of equation (17) yields:

[0127] (18)

[0128] When the sliding surface When this happens, the equivalent control law can be obtained as follows:

[0129] (19)

[0130] Because the fractional-order extended state observer has observation errors in disturbance estimation, it cannot move along the ideal sliding surface, resulting in a decrease in observation performance. To eliminate the influence of observation errors, an auxiliary control law is designed. The Lyapunov function is designed as follows:

[0131] (20)

[0132] To ensure system stability, the following condition must be met as shown in equation (21):

[0133] ;(twenty one)

[0134] The complete control law is defined as follows:

[0135] ;(twenty two)

[0136] Equation (21) can be further rewritten as:

[0137] ;(twenty three)

[0138] In the formula, To account for observation error, substituting equation (22) into equation (23) yields:

[0139] ;(twenty four)

[0140] Substituting equation (19) into equation (24), we get:

[0141] (25)

[0142] Further simplifying equation (25) yields:

[0143] (26)

[0144] To ensure system stability, i.e., equation (26) is less than zero, we obtain the control law. for:

[0145] (27)

[0146] At this point, the system is stable, and the complete control law is expressed as follows:

[0147] (28)

[0148] In the formula, in the formula, Indicates switching gain. For switching functions, For equivalent control, To switch control, To control the gain, it represents the efficiency coefficient of converting the input voltage into the system's driving force.

[0149] In the designed control law, the gain is switched. Only the perturbation estimation error of FOOSO needs to be compensated, not the initial total perturbation. Therefore, only a relatively small switching gain is required. This is sufficient to satisfy the Lyapunov stability condition.

[0150] Based on the above theoretical derivation, the effectiveness of the fractional-order modeling of the designed four-sided fixed-plate vibration system based on inertial actuators, and the stability of the designed fractional-order integral sliding mode control based on a fractional-order extended state observer, can be obtained. The above composite control algorithm is built in Simulink and runs using NI PCIe-6343 as the core through automatically generated C code. The entire development process is as follows: Figure 7 As shown.

[0151] To verify its effectiveness, the following experiment was designed:

[0152] ① First, set the external excitation frequency to 48.5Hz. After passing through the power amplifier, the signal is applied to the exciter to excite the board surface.

[0153] ② Secondly, the fractional order of the system is obtained through modeling experiments, and the fractional order of the designed composite controller is determined accordingly. A model of a fractional integral sliding mode controller based on a fractional extended state observer is established, and vibration control experiments are carried out.

[0154] ③ Input the desired target vibration displacement as 0 into the host computer and observe the voltage signal of the plate surface vibration displacement. Experiments were conducted using three different control algorithms: an integral sliding mode controller based on an extended state observer (ISMC+ESO), a fractional-order integral sliding mode controller based on an extended state observer (FOISMC+ESO), and a fractional-order integral sliding mode controller based on a fractional-order extended state observer (FOISMC+FOESO). Their vibration displacement responses are as follows: Figure 4 As shown.

[0155] ④ Next, the experimental results were further verified and analyzed by performing a Fast Fourier Analysis on the time-domain displacement voltage signal to examine its amplitude response in the frequency domain. Figure 5 As shown. Also check the control voltage level as follows: Figure 6 As shown.

[0156] from Figure 4 It can be observed that when the board is running without any control applied, the vibration displacement voltage in the time domain is approximately 9.85V. Firstly, adding the ISMC+ESO controller reduces the vibration displacement voltage from 9.85V to 2.72V, a decrease of approximately 7.13V, achieving a suppression performance of 72.3%. When using the FOISMC+ESO controller, the voltage further decreases to 2.43V, a reduction of approximately 7.42V, with a suppression level of 75.3%. Finally, the FOISMC+FOESO controller achieves the optimal vibration control performance, reducing the vibration displacement voltage to 2.08V, a total reduction of 7.73V, with a suppression level of 78.9%.

[0157] from Figure 5 In the study, the frequency domain responses of the first-order mode and other higher harmonic modes were obtained. Compared with the FOISMC+ESO and ISMC+ESO controllers, the FOISMC+FOESO controller exhibits the best suppression performance. For the first-order mode, the FOISMC+FOESO controller demonstrates better vibration suppression performance, reducing the amplitude from 19.11 dB to 5.36 dB, a total reduction of 13.75 dB.

[0158] from Figure 6By examining the control voltages of the three different controllers, it can be found that FOISMC+FOESO only needs to generate a voltage of 0.06V, demonstrating excellent energy efficiency.

[0159] The above description of the embodiments is only for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make several improvements and modifications to the present invention without departing from the principles of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-support plate vibration system based on an inertial actuator, characterized in that, Includes the following steps: S1. A four-sided fixed plate vibration system based on an inertial actuator is modeled using the Caputo fractional operator. The fractional characteristics caused by damping and inductance are used to establish the state-space equation of the fractional electromechanical coupling system. S2. Build a control algorithm implementation platform in a real-time simulation environment. Under the condition of disturbance of a four-sided fixed plate vibration system based on inertial actuator, build a fractional integral sliding mode control algorithm based on a fractional extended state observer. S3. For vibration systems, parameter perturbation, higher-order terms, external excitation and interaction force are uniformly defined as total disturbance. A fractional-order extended state observer is designed to estimate the system state variables and total disturbance of the vibration system in real time and obtain estimation information. S4. The obtained estimation information is used to construct a fractional integral sliding surface. The fractional integral operator is used to accumulate and compensate for the residual vibration error. Combined with the feedforward compensation of the total disturbance, the inertial actuator is driven to achieve vibration suppression.

2. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 1, characterized in that: The state-space equations of the fractional-order electromechanical coupling system in S1 are presented in a four-sided fixed plate electromechanical coupling model based on an inertial actuator. The construction of the four-sided fixed plate electromechanical coupling model based on an inertial actuator is as follows: ； In the formula, , , , , , , and These represent the mass, displacement, damping, and stiffness coefficients of the fixed plate and the inertial actuator, respectively. , and These are the resistance, inductance, and force constants inside the inertial actuator; and To control voltage and current, As an external incentive, The vibration velocity of the inertial actuator. The vibration acceleration of the inertial actuator. The vibration velocity of the fixed plate, The vibration acceleration of the fixed plate, The vibration jerk of the inertial actuator. The rate of change of the external stimulus.

3. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 2, characterized in that: The state-space equation of the fractional electromechanical coupling system introduces fractional operators into the four-sided fixed plate electromechanical coupling model based on inertial actuators. , ,in, , Fractional order: ； In the formula, It represents the sum of higher-order terms, external stimuli, and interaction forces; The tracking error is defined as ,in, To control the vibration of the fixed plate, a fractional-order operator is introduced, and state variables are defined. , The system state-space equations can be established as follows: ； in, The output value is the measured displacement value. For the displacement of the system, Defined as the fractional-order velocity of the system.

4. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 3, characterized in that: In S3, the sum of parameter perturbations, higher-order terms, external excitations, and interaction forces is defined as the total perturbation, and the total perturbation is defined as the extended state using the following formula: ,in, To expand the state variables, the system state-space equations are expanded to: In the formula, For the system gain, the designed fractional-order extended state observer is as follows: In the formula, These are the observed values ​​of the system state and the total disturbance; The observer gain is adjusted according to the bandwidth method. , , , The observer bandwidth is used to adjust the gain. The tuning parameters, This is for tracking error.

5. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 4, characterized in that: The fractional integral sliding surface in S4 is constructed as follows: ； In the formula, , All of these are controller gain coefficients. Using Lyapunov functions to design auxiliary control laws and combining them with equivalent control laws, the overall system control law is obtained as follows: ； In the formula, Indicates switching gain. For switching functions, For equivalent control, To switch control, To control the gain, it represents the efficiency coefficient of converting the input voltage into the system's driving force.

6. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 5, characterized in that: The real-time simulation environment in S2 is specifically built on the NI PCIe-6343 data acquisition card and in the Desktop Real-Time environment of the software MATLAB / Simulink.

7. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 6, characterized in that: The convergence of the fractional-order extended state observer is defined by the tracking error. The dynamic equation for the tracking error of the fractional-order extended state observer is as follows: ； In the formula, , , , ; right Performing the Laplace transform, we get: ,in, The observation error state vector is, i.e. , The error system matrix is ​​formed by adjusting the gain. The coefficient matrix formed determines the stability and speed of error convergence. Let be the disturbance input vector, representing the rate of change of the disturbance. The channel vector entering the error system has the following specific form: This is because of the expansion of state variables. The derivative is This directly affects the error dynamics of the third state. For the Laplace operator, corresponding to complex variables in the frequency domain, For the fractional differential operator in the corresponding time domain In the complex frequency domain, For the Laplace transform of the tracking error, The rate of change of the disturbance The Laplace transform of .

8. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 7, characterized in that: The calculation formula for the fractional-order extended state observer is improved as follows: ； ； In the formula, It is a 3×3 identity matrix. Let be the inverse of the characteristic matrix of the error system. The characteristic polynomial of the error state-space model is represented by the bandwidth method. To further express: ； In the formula, Given the observer bandwidth, according to the final value theorem, we can obtain: ； when At that time, the error is bounded: ； Complete the convergence proof of the fractional-order extended state observer, where the tracking error converges to the neighborhood of the desired point. That is, when the disturbance is constant, the tracking error asymptotically converges to zero. Let i be the steady-state error of the tracking error for sequence i. This represents the maximum rate of change of the disturbance. The constant coefficient is greater than zero.

9. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 8, characterized in that: The equivalent control The determination is as follows: right Perform differentiation; We can obtain: ； When the sliding surface Then, the equivalent control law is: 。 10. The fractional-order modeling and fractional-order composite sliding mode control method for a four-sided fixed-plate vibration system based on an inertial actuator according to claim 9, characterized in that: The switching control The formula for determining is as follows: The Lyapunov function is as follows: ； To ensure system stability, the following conditions must be met: ;in, To estimate the sliding surface, For the estimated Lyapunov function; ； Switching control for: 。

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