Air floating bearing quasi-zero stiffness active vibration reduction method based on model predictive control

Abstract

The invention relates to the technical field of vibration isolation control, discloses an air floating support quasi-zero stiffness active vibration reduction method based on model predictive control, and aims to solve the problems of high vibration isolation requirements of precision equipment in a low-frequency vibration environment, response lag of a traditional vibration isolation technology and insufficient control precision under uncertainty. The method comprises the following steps: step 1, carrying out system modeling on the vibration damping device to obtain a state space model of the vibration damping device, and introducing a Kalman filter into the state space model to predict state information in real time; step 2, constructing an optimization problem and a constraint condition of the MPC controller; step 3, carrying out robustness and stability analysis on the MPC controller; and step 4, fusing sensor data through the MPC controller after robustness analysis, solving an optimization problem through an optimization algorithm, calculating an optimal control input # imgabs0 # in real time, driving a voice coil motor, and suppressing low-frequency vibration.

Description

Technical Field

[0001] The invention relates to a quasi-zero stiffness active vibration reduction method of an air floating support based on model predictive control, and belongs to the technical field of vibration isolation control. Background Art

[0002] In order to improve the ability of passive vibration reduction devices to isolate low-frequency vibrations, it is necessary to reduce the natural frequency of the vibration reduction device. Reducing the stiffness of the vibration reduction device can directly reduce the natural frequency of the system, but the lower the system stiffness, the lower the load capacity. Therefore, the passive vibration reduction device has limited vibration reduction ability in the low-frequency band of 0.5-70Hz, which is difficult to meet the needs of precision equipment. We need to improve the low-frequency vibration reduction efficiency of the system through active control.

[0003] In recent years, model predictive control (MPC), as an advanced control method, aims to provide better isolation effect and wider working range, and has attracted widespread attention due to its advantages in dealing with uncertainty and external disturbances. However, there is still room for optimization of traditional MPC in ultra-low frequency micro-vibration control. To solve this problem, the MPC algorithm provides an effective prediction and optimization method. By designing a controller to predict future vibration trends in real time and make dynamic adjustments, the vibration impact caused by external disturbances can be minimized, and the system can maintain robustness under the condition of large external disturbance uncertainty. Summary of the invention

[0004] In order to solve the problems of high vibration isolation demand for precision equipment in low-frequency vibration environments, delayed response of traditional vibration isolation technologies, and insufficient control accuracy under uncertainty, the present invention proposes a quasi-zero stiffness active vibration reduction method of air floating support based on model predictive control.

[0005] The technical solution adopted by the present invention to solve the above-mentioned problem is: the steps of the present invention include:

[0006] Step 1: System modeling of the vibration reduction device is carried out to obtain a state space model of the vibration reduction device, and a Kalman filter is introduced into the state space model to predict state information in real time;

[0007] Step 2: Construct the optimization problem and constraints of the MPC controller;

[0008] Step 3: Perform robustness and stability analysis on the MPC controller;

[0009] Step 4: The MPC controller after robustness analysis fuses the sensor data, solves the optimization problem through the optimization algorithm, and calculates the optimal control input in real time And drives the voice coil motor to suppress low-frequency vibrations.

[0010] Further, in step 1, the state variables of the state space model are the displacement and moving speed of the vibration isolation platform, the disturbance input is the external disturbance, and the control input is one of the current and voltage of the voice coil motor;

[0011] The state space expression is:

[0012] (1);

[0013] (2);

[0014] In formulas (1) and (2), is the state vector of the system, including the displacement and velocity of the platform, is the control input, i.e., the current or voltage of the voice coil motor; is the disturbance input, A, B, C, D, E, F are the system matrices, and the system matrix is ​​determined by system identification. is the expected output value, is the time derivative of the state vector of the system. Further, the step of real-time prediction of state information in step 1 includes:

[0015] Step 1.1: Use the least square method to identify the system parameters using the actual parameters of the vibration reduction device, minimize the error between the system modeling and the actual parameters of the vibration reduction device, and obtain the transfer function of the vibration device from the vibration input to the current / voltage output of the voice coil motor;

[0016] Step 1.2: Introduce the Kalman filter to predict and update the measurement state of the vibration device in real time, and obtain the state information of the state space model of the vibration reduction device;

[0017] The expression of the least squares method is:

[0018] (3);

[0019] In formula (3), is the actual parameter of the vibration reduction device, is the system parameter, is the number of parameters;

[0020] The expression of state prediction is:

[0021] (4);

[0022] (5);

[0023] In formulas (4) and (5), is the control input, i.e. the current or voltage of the voice coil motor, is the state information of the state space model of the vibration reduction device at time t, is the state information of the state space model of the vibration reduction device at time t+1, is the system estimation error covariance matrix at time t, is the system estimation error covariance matrix at time t+1, A is the system state transfer matrix, B is the control input matrix, is the process noise covariance matrix, is the transposed matrix of A;

[0024] The expression for state update is:

[0025] (6);

[0026] (7);

[0027] (8);

[0028] In formulas (6)-(8), is the expected output value, R is the measurement noise covariance, is the transpose of the observation matrix, is the identity matrix, is the state information of the state space model of the vibration reduction device at time t-1, is the state information of the state space model of the vibration reduction device at time t, is the observation matrix, is the Kalman gain matrix, is the system estimation error covariance matrix at time t-1, is the system estimation error covariance matrix at time t.

[0029] Furthermore, step 2 specifically includes:

[0030] Step 2.1: Define the prediction time domain and control time domain, minimize the cost function at each time t, and complete the construction of the MPC controller optimization problem;

[0031] Step 2.2: Obtain the constraints of the MPC controller optimization problem based on the control input u and the state information x of the vibration reduction device;

[0032] The expression for minimizing the cost function is:

[0033] (9);

[0034] In formula (9), is the cost function to be minimized, is the expected output value at time t+k, is the weight matrix of the output error, is the weight matrix of the control input, is the prediction of the system output value at time t+k, is the prediction of the control input at time t+k, is the number of time steps for future output, The forecast range for the control input;

[0035] The constraint expression of the MPC controller optimization problem is:

[0036] (10);

[0037] (11);

[0038] In formulas (10) and (11), is the minimum value of the state information of the vibration reduction device, is the maximum value of the status information of the vibration reduction device, is the minimum value of the control input, is the maximum value of the control input, is the state information of the state space model of the vibration reduction device at time t+k, is the control input of the vibration reduction device at time t+k.

[0039] Furthermore, step 3 specifically includes:

[0040] Step 3.1: Construct the Lyapunov function to determine the stability of the state space model of the vibration reduction device;

[0041] Step 3.2: Set the robustness condition of the state space model of the vibration reduction device based on the constructed Lyapunov function to complete the robustness analysis of the MPC controller;

[0042] The robustness condition of the state space model of the vibration reduction device is expressed as:

[0043] (12);

[0044] In formula (12), is the system stability factor, >0, is the Lyapunov function value of the system at time t+1, is the Lyapunov function value of the system at time t, is the state vector of the system at time t.

[0045] Furthermore, in step 4, the optimal control input is calculated in real time The steps include:

[0046] Step 4.1: The MPC controller after robustness analysis receives sensor data, wherein the sensor includes a velocity sensor and a displacement sensor;

[0047] Step 4.2: Combine the received sensor data and use the Kalman filter to perform data fusion and provide real-time feedback on the vibration state of the vibration isolation platform;

[0048] Step 4.2: Based on the vibration state of the feedback vibration isolation platform, the optimization problem is solved through the optimization algorithm, and the optimal control input is calculated in real time. ;

[0049] The expression of data fusion is:

[0050] (13);

[0051] In formula (13), is the Kalman filter, is the expected output value, is the control input, is the estimated state vector of the system at time t;

[0052] Optimal control input The expression is:

[0053] (14).

[0054] The beneficial effects of the present invention are as follows: the present invention establishes a state space model of the platform through modeling and experimental identification, and uses Kalman filtering to estimate unmeasured states to enhance system robustness and accuracy; the MPC controller combines multi-sensor data to predict vibration trends in real time, dynamically adjusts the output of the voice coil motor, and realizes active vibration reduction; in order to cope with periodic interference, the present invention introduces a repetitive control (RC) strategy, and adjusts controller parameters through an adaptive algorithm to adapt to changes; the method proposed in the present invention achieves significant vibration reduction effects in the frequency band of 0.5~70Hz, is suitable for precision instruments in complex environments, and provides stable vibration isolation support for precision instruments. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 It is a schematic diagram of the process of the present invention;

[0056] Figure 2 Schematic diagram of the structure of the quasi-zero stiffness air floating spring active vibration isolator provided by the present invention. Figure 2 In the figure, 1-horizontal voice coil motor, 2-vertical voice coil motor, 3-sensor bracket, 4-motor bracket, 5-water cooling splitter, 6-limiting mechanism, 7-isolator upper plate, 8-limiting pile, 9-isolator air cavity, 10-isolator lower plate;

[0057] Figure 3 is the system amplitude-frequency response curve diagram under the KF-DMPC controller, passive control and PID controller provided by the present invention, Figure 3 In the figure, the green curve is the amplitude-frequency response of the system under passive control, the blue curve is the amplitude-frequency response of the system under PID controller, and the red curve is the amplitude-frequency response of the system under KF-DMPC controller;

[0058] Figure 4 It is the system phase-frequency response curve diagram under the KF-DMPC controller, passive control and PID controller provided by the present invention, Figure 4 In the figure, the green curve is the phase-frequency response of the system under passive control, the blue curve is the phase-frequency response of the system under PID controller, and the red curve is the phase-frequency response of the system under KF-DMPC controller;

[0059] Figure 5 It is the system speed response curve diagram under the KF-DMPC controller, passive control and PID controller provided by the present invention, Figure 5 In the figure, the green curve is the system speed response under passive control, the blue curve is the system speed response under PID controller, and the red curve is the system speed response under KF-DMPC controller;

[0060] Figure 6 It is a comparative curve diagram of the braking force output under the KF-DMPC controller and the PID controller provided by the present invention. Figure 6 In the figure, the blue curve is the braking force output under the PID controller, and the red curve is the braking force output under the KF-DMPC controller;

[0061] Figure 7 It is the state estimation of the Kalman filter, the output of the KF-DMPC controller and the observed residual curve diagram provided by the present invention, Figure 7 In the figure, the red curve is the state estimation of the Kalman filter, the blue curve is the output of the KF-DMPC controller, and the green curve is the observation residual. DETAILED DESCRIPTION

[0062] Specific implementation method 1: Figure 2 As shown in the figure, the vibration isolation platform adopts a quasi-zero stiffness structure with air-floating support, and uses the static pressure air-floating surface of the air spring to achieve extremely low stiffness in the vertical and horizontal directions, thereby enhancing the passive vibration isolation performance of low-frequency micro-vibration. The support structure consists of a top plate, a piston, an air cavity and other aerodynamic components. The high-precision suspension of the platform is achieved through air cavity pressure control, providing an ideal low-stiffness environment for the MPC controller.

[0063] like Figure 1 As shown, the steps of the air floating support quasi-zero stiffness active vibration reduction method based on model predictive control described in this embodiment include:

[0064] S1: System Modeling and Identification;

[0065] S101: System modeling of the vibration reduction device is performed to obtain a state space model of the vibration reduction device, wherein the state variables of the state space model are the displacement and movement speed of the vibration isolation platform, the disturbance input is external interference, wherein the external interference includes but is not limited to ground vibration, and the control input is one of the current and voltage of the voice coil motor;

[0066] The state space expression is:

[0067] (1);

[0068] (2);

[0069] In formulas (1) and (2), is the state vector of the system, including the displacement and velocity of the platform, is the control input, i.e., the current or voltage of the voice coil motor; is the disturbance input, A, B, C, D, E, F are the system matrices, and the system matrix is ​​determined by system identification. is the expected output value, is the time derivative of the system's state vector.

[0070] S102: Using the actual parameters of the vibration reduction device obtained from the experiment to identify the system parameters, the transfer function from the vibration input of the vibration reduction device to the current / voltage output of the voice coil motor is calculated. In this embodiment, the identification method adopts the least square method to minimize the error between the output of the state space model of the vibration reduction device and the experimental measurement value:

[0071] The expression of the least squares method is:

[0072] (3);

[0073] In formula (3), is the actual parameter of the vibration reduction device, is the system parameter, is the number of parameters;

[0074] S103: In order to improve the accuracy of state estimation, this implementation method introduces a Kalman filter to estimate the unmeasured state (such as torque or unmeasured acceleration) in the system. The prediction and update equations of the Kalman filter are as follows:

[0075] The expression of state prediction is:

[0076] (4);

[0077] (5);

[0078] In formulas (4) and (5), is the control input, i.e. the current or voltage of the voice coil motor, is the state information of the state space model of the vibration reduction device at time t, is the state information of the state space model of the vibration reduction device at time t+1, is the system estimation error covariance matrix at time t, is the system estimation error covariance matrix at time t+1, A is the system state transfer matrix, B is the control input matrix, is the process noise covariance matrix, is the transposed matrix of A;

[0079] The expression for state update is:

[0080] (6);

[0081] (7);

[0082] (8);

[0083] In formulas (6)-(8), is the expected output value, R is the measurement noise covariance, is the transpose of the observation matrix, is the identity matrix, is the state information of the state space model of the vibration reduction device at time t-1, is the state information of the state space model of the vibration reduction device at time t, is the observation matrix, is the Kalman gain matrix, is the system estimation error covariance matrix at time t-1, is the system estimation error covariance matrix at time t.

[0084] S2: MPC controller design;

[0085] S201: MPC optimization problem construction: define the prediction time domain and the control time domain, minimize the cost function at each time t, and complete the construction of the MPC controller optimization problem;

[0086] The expression for minimizing the cost function is:

[0087] (9);

[0088] In formula (9), is the cost function to be minimized, is the expected output value at time t+k, is the weight matrix of the output error, is the weight matrix of the control input, is the prediction of the system output value at time t+k, is the prediction of the control input at time t+k, is the number of time steps for future output, The forecast range for the control input;

[0089] S202: Obtaining the constraint conditions of the MPC controller optimization problem according to the control input u and the state information x of the vibration reduction device;

[0090] The constraint expression of the MPC controller optimization problem is:

[0091] (10);

[0092] (11);

[0093] In formulas (10) and (11), is the minimum value of the state information of the vibration reduction device, is the maximum value of the status information of the vibration reduction device, is the minimum value of the control input, is the maximum value of the control input, is the state information of the state space model of the vibration reduction device at time t+k, is the control input of the vibration reduction device at time t+k.

[0094] S3: Robustness and stability analysis of MPC controller;

[0095] The MPC controller should ensure the robustness of the system when considering model errors and external disturbances. , to ensure that the closed-loop system output by the controller has anti-disturbance capability. The robustness and stability analysis are as follows:

[0096] S301: constructing a Lyapunov function to determine the stability of the state space model of the vibration reduction device;

[0097] S302: Setting the robustness condition of the state space model of the vibration reduction device based on the constructed Lyapunov function, and completing the robustness analysis of the MPC controller;

[0098] The robustness condition of the state space model of the vibration reduction device is expressed as:

[0099] (12);

[0100] In formula (12), is the system stability factor, >0, is the Lyapunov function value of the system at time t+1, is the Lyapunov function value of the system at time t, is the state vector of the system at time t.

[0101] S4: The MPC controller combines real-time feedback from multi-function sensor data to dynamically generate precise control signals to drive the voice coil motor to achieve active vibration isolation.

[0102] S401: The MPC controller after robustness and stability analysis combines multi-sensor data, including displacement sensors and velocity sensors, and uses Kalman filters for signal fusion to ensure the real-time and accuracy of system state estimation;

[0103] The expression of data fusion is:

[0104] (13);

[0105] In formula (13), is the Kalman filter, is the expected output value, is the control input, is the estimated state vector of the system at time t;

[0106] S402: Based on the vibration state of the feedback vibration isolation platform, the optimization problem is solved through the optimization algorithm, and the optimal control input is calculated in real time. ;

[0107] Optimal control input The expression is:

[0108] (14).

[0109] In summary, the present invention establishes a state space model of the platform through modeling and experimental identification, and uses Kalman filtering to estimate unmeasured states to enhance the robustness and accuracy of the system. The MPC controller combines multi-sensor data to predict vibration trends in real time and dynamically adjust the output of the voice coil motor to achieve active vibration reduction. In order to cope with periodic interference, the present invention introduces a repetitive control (RC) strategy and adjusts the controller parameters through an adaptive algorithm to adapt to changes.

[0110] Specific implementation method 2: Combination Figure 3-7This embodiment is described. To verify the technical effect of the present invention, this embodiment conducts a simulation experiment on a quasi-zero stiffness active vibration reduction method of an air floating support based on model predictive control proposed in the specific embodiment 1 in a simulation environment such as MATLAB / Simulink. The simulation experiment results are analyzed as follows:

[0111] like Figure 3 As shown in the figure, according to the amplitude-frequency response curves of the system under the KF-DMPC controller, passive control and PID controller, the passive control (FRF: Passive): the gain is close to 20dB in the low frequency band (about 0.1Hz), and reaches a peak value of about 40dB near the natural frequency of the system (about 10Hz). This shows that the system has a significant amplification effect on vibration under passive control; KF-DMPC controller (FRF: KF-DMPC): In most frequency bands, the gain is significantly lower than that of passive control, especially between 1Hz and 100Hz, the gain drops to about -40dB. It can be seen that in this frequency band, the KF-DMPC controller greatly reduces the vibration response of the system; PID controller (FRF: PID): The gain of the PID controller is about -10dB in the low frequency band (about 0.1Hz), which is improved compared with the passive control, but in the high frequency band (greater than 100Hz), the gain increases significantly to about 0dB, showing a weaker vibration suppression ability than the KF-DMPC.

[0112] like Figure 4 As shown in the figure, according to the system phase-frequency response curves under KF-DMPC controller, passive control and PID controller, KF-DMPC and PID controller: In the low frequency band (about 0.1Hz), the phase of KF-DMPC and PID controller is close to 0°, indicating that the low-frequency response of the two is similar. However, in the high frequency band (100Hz to above 1000Hz), the phase of the PID controller changes rapidly, even reaching a phase drift of ±300° at certain frequencies, while the phase change of the KF-DMPC controller is relatively stable, within ±100°, indicating that the stability in the high frequency band is better.

[0113] like Figure 5As shown in the figure, according to the system speed response curves under the KF-DMPC controller, passive control and PID controller, the initial speed response peak of the passive control is about 0.06m / s. The decay is very slow, there is still obvious residual vibration after 60 seconds, and it stabilizes after 200 seconds. This shows that the vibration attenuation effect of passive control is very limited, and the vibration of the system lasts for a long time; the initial response of the KF-DMPC controller is about 0.06m / s, but it decays rapidly to close to 0m / s within 20 seconds. This shows that the KF-DMPC controller can effectively suppress vibration in a very short time; the initial response of the PID controller is also 0.06m / s, but the decay rate is slightly slower than that of the KF-DMPC controller, and it is basically stable after about 40 seconds. Although the PID controller can suppress vibration, its decay rate is slightly slower than that of the KF-DMPC, and there is still a small residual vibration after 60 seconds.

[0114] like Figure 6 As shown in the figure, according to the comparison curve of the braking force output under the KF-DMPC controller and the PID controller, the initial braking force output of the KF-DMPC controller reaches about -12N, which is used to quickly suppress the vibration of the system. After that, the braking force quickly approaches 0N and remains stable in the following time. It shows that the KF-DMPC controller can effectively control the vibration in a short time; the initial braking force of the PID controller is about -10N, which is slightly smaller than that of the KF-DMPC. Although the PID controller shows a certain stability in the initial stage, its braking force fluctuates continuously in the subsequent time, ranging from about ±5N. This continuous fluctuation shows that the control signal of the PID controller is not stable enough, which may lead to increased energy loss.

[0115] like Figure 7 As shown in Figure 2, according to the state estimation of the Kalman filter, the output of the KF-DMPC controller and the observed residual curve, the KF-DMPC controller output: after the initial response, the output of the KF-DMPC controller quickly stabilizes, and the output stable value is close to 0m / s, indicating that the controller can quickly restore the system to a stable state. The Kalman filter state estimation: in the initial stage, the state estimation closely follows the actual output, and the deviation is very small, less than 10 -5 m / s, indicating that the Kalman filter can track the system state with high accuracy. This accurate state estimation further verifies the effectiveness of the Kalman filter. The observation residual is basically maintained near 0 (within 10 -6 m / s), indicating that the accuracy of system state estimation is very high, and the filter can effectively eliminate noise interference, ensuring the accuracy of state estimation.

[0116] In summary, according to the simulation results of this embodiment, it can be seen that the method proposed in the present invention can achieve a significant vibration suppression effect in the frequency band of 0.5~70Hz, providing stable vibration isolation support for precision instruments.

[0117] The above is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement made to the above embodiments without departing from the content of the technical solution of the present invention, based on the technical essence of the present invention, within the spirit and principles of the present invention, still fall within the protection scope of the technical solution of the present invention.

Claims

1. A quasi-zero stiffness active vibration reduction method for air-floating support based on model predictive control, characterized in that: The steps of the air floating support quasi-zero stiffness active vibration reduction method based on model predictive control include: Step 1: System modeling of the vibration reduction device is carried out to obtain a state space model of the vibration reduction device, and a Kalman filter is introduced into the state space model to predict state information in real time; Step 2: Construct the optimization problem and constraints of the MPC controller; Step 3: Perform robustness and stability analysis on the MPC controller; Step 4: The MPC controller after robustness analysis fuses the sensor data, solves the optimization problem through the optimization algorithm, and calculates the optimal control input in real time And drives the voice coil motor to suppress low-frequency vibrations.

2. According to the model predictive control-based air floating support quasi-zero stiffness active vibration reduction method of claim 1, it is characterized in that: In step 1, the state variables of the state space model are the displacement and moving speed of the vibration isolation platform, the disturbance input is the external disturbance, and the control input is one of the current and voltage of the voice coil motor; The state space expression is: （1）； （2）； In formulas (1) and (2), is the state vector of the system, including the displacement and velocity of the platform, is the control input, i.e., the current or voltage of the voice coil motor; is the disturbance input, A, B, C, D, E, F are the system matrices, and the system matrix is ​​determined by system identification. is the expected output value, is the time derivative of the system's state vector.

3. The method for quasi-zero stiffness active vibration reduction of air-floating support based on model predictive control according to claim 1 is characterized in that: The steps of real-time prediction of status information in step 1 include: Step 1.1: Use the least square method to identify the system parameters using the actual parameters of the vibration reduction device, minimize the error between the system modeling and the actual parameters of the vibration reduction device, and obtain the transfer function of the vibration device from the vibration input to the current / voltage output of the voice coil motor; Step 1.2: Introduce the Kalman filter to predict and update the measurement state of the vibration device in real time, and obtain the state information of the state space model of the vibration reduction device; The expression of the least squares method is: （3）； In formula (3), is the actual parameter of the vibration reduction device, is the system parameter, is the number of parameters; The expression of state prediction is: （4）； （5）； In formulas (4) and (5), is the control input, i.e. the current or voltage of the voice coil motor, is the state information of the state space model of the vibration reduction device at time t, is the state information of the state space model of the vibration reduction device at time t+1, is the system estimation error covariance matrix at time t, is the system estimation error covariance matrix at time t+1, A is the system state transfer matrix, B is the control input matrix, is the process noise covariance matrix, is the transposed matrix of A; The expression for state update is: （6）； （7）； （8）； In formulas (6)-(8), is the expected output value, R is the measurement noise covariance, is the transpose of the observation matrix, is the identity matrix, is the state information of the state space model of the vibration reduction device at time t-1, is the state information of the state space model of the vibration reduction device at time t, is the observation matrix, is the Kalman gain matrix, is the system estimation error covariance matrix at time t-1, is the system estimation error covariance matrix at time t.

4. The method for quasi-zero stiffness active vibration reduction of air-floating support based on model predictive control according to claim 1 is characterized in that: Step 2 specifically includes: Step 2.1: Define the prediction time domain and control time domain, minimize the cost function at each time t, and complete the construction of the MPC controller optimization problem; Step 2.2: Obtain the constraints of the MPC controller optimization problem based on the control input u and the state information x of the vibration reduction device; The expression for minimizing the cost function is: （9）； In formula (9), is the cost function to be minimized, is the expected output value at time t+k, is the weight matrix of the output error, is the weight matrix of the control input, is the prediction of the system output value at time t+k, is the prediction of the control input at time t+k, is the number of time steps for future output, The forecast range for control inputs; The constraint expression of the MPC controller optimization problem is: （10）； （11）； In formulas (10) and (11), is the minimum value of the state information of the vibration reduction device, is the maximum value of the status information of the vibration reduction device, is the minimum value of the control input, is the maximum value of the control input, is the state information of the state space model of the vibration reduction device at time t+k, is the control input of the vibration reduction device at time t+k.

5. The method for quasi-zero stiffness active vibration reduction of air bearing based on model predictive control according to claim 1 is characterized in that: Step 3 specifically includes: Step 3.1: Construct the Lyapunov function to determine the stability of the state space model of the vibration reduction device; Step 3.2: Set the robustness condition of the state space model of the vibration reduction device based on the constructed Lyapunov function to complete the robustness analysis of the MPC controller; The robustness condition of the state space model of the vibration reduction device is expressed as: （12）； In formula (12), is the system stability factor, >0, is the Lyapunov function value of the system at time t+1, is the Lyapunov function value of the system at time t, is the state vector of the system at time t.

6. The method for quasi-zero stiffness active vibration reduction of air-floating support based on model predictive control according to claim 1, characterized in that: In step 4, the optimal control input is calculated in real time The steps include: Step 4.1: The MPC controller after robustness analysis receives sensor data, wherein the sensor includes a velocity sensor and a displacement sensor; Step 4.2: Combine the received sensor data and use the Kalman filter to perform data fusion and provide real-time feedback on the vibration state of the vibration isolation platform; Step 4.2: Based on the vibration state of the feedback vibration isolation platform, the optimization problem is solved through the optimization algorithm, and the optimal control input is calculated in real time. ; The expression of data fusion is: （13）； In formula (13), is the Kalman filter, is the expected output value, is the control input, is the estimated state vector of the system at time t; Optimal control input The expression is: （14）。

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