Feedback linearization sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuator and air spring

Abstract

The invention discloses a feedback linearization sliding mode control method for hybrid vibration isolation of a giant magnetostrictive actuator and an air spring, and belongs to the technical field of electronic design models. In order to ensure the accurate control of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, the influence of eddy current loss, temperature effect and prestress is considered, the relationship between the expansion and contraction rate of a giant magnetostrictive material and the magnetic field intensity is established, and a constitutive equation of the giant magnetostrictive actuator is constructed; establishing a dynamic balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring; controlling the dynamic balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring by using a feedback linearization sliding mode control strategy to obtain a feedback linearization sliding mode control equation; the control method of the giant magnetostrictive actuator and air spring mixed vibration isolation system is constructed based on a feedback linearization sliding mode control equation and the relation between the expansion rate and the magnetic field intensity of the giant magnetostrictive material. The control method is accurate in control.

Description

Technical Field

[0001] The present invention belongs to the technical field of electronic design models and their vibration control, and particularly relates to a feedback linearization sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuators and air springs. Background Technique

[0002] With the improvement of the requirements for the working environment of precision equipment, traditional vibration isolation technologies can no longer meet the growing demands. Due to advantages such as fast response speed and large output force, the giant magnetostrictive material GMM shows great potential in the field of active vibration isolation. A significant drawback of the giant magnetostrictive actuator is that its deformation amount is relatively small, usually in the micron level. This means that there are limitations in application scenarios that require large displacements or handle large vibration amplitudes, such as vibration control in rail transit and situations where large displacement compensation is needed for suppressing ground vibrations. Combining the giant magnetostrictive actuator GMA with an air spring can effectively solve the problem of the small output displacement of the giant magnetostrictive actuator and improve the vibration isolation effect. At the same time, as a passive vibration isolation device, the air spring has high stability and reliability, improving the robustness of the hybrid vibration isolation system.

[0003] There is a complex nonlinear hysteresis phenomenon in the input-output relationship of the giant magnetostrictive actuator. Traditional linear models are difficult to reflect the mechanical behavior of the giant magnetostrictive actuator. Under high-frequency excitation, eddy current effects, magnetic domain response delays, etc. lead to the attenuation of the output force and displacement amplitude. Solving the hysteresis problem of the mechanical model of the giant magnetostrictive actuator is a hot issue in the control of giant magnetostrictive actuators.

[0004] Feedback linearization linearizes the state equation of the system. Through the nonlinear transformation of the state and nonlinear state feedback, the original nonlinear system is transformed into a linearly controllable and observable system, and then a linear control method is used to construct the controller. The feedback linearization control method can provide precise control for complex nonlinear systems, such as characteristics like dead zones and saturation, compensating for the problem that traditional linear control is difficult to effectively control complex nonlinear behaviors. Feedback linearization has extremely high requirements for the accurate modeling of the system. When there are influencing factors and parameter uncertainties that are not considered in the actual system, it will lead to poor control effects of feedback linearization or even loss of stability.

[0005] Sliding mode control is a special nonlinear control method. By establishing a sliding surface, the system moves along the sliding surface. Once the system reaches the sliding surface, the control action will ensure that the system reaches the system origin along the sliding surface. Sliding mode control is highly robust to parameter changes, model uncertainty and external disturbances of the controlled object. When the controlled object reaches the sliding surface, the dynamic behavior of the controlled object only depends on the sliding surface and has nothing to do with the specific parameters of the controlled object. Sliding mode control also has the ability to respond quickly. Due to its discontinuous control characteristics, sliding mode control can respond quickly to make the state of the controlled object converge quickly to the equilibrium point. However, there is an obvious chattering phenomenon in sliding mode control. After the system state reaches the sliding surface, the ideal sliding mode is difficult to strictly achieve, and the system state often moves back and forth on both sides of the sliding surface. This phenomenon is very unfavorable for the acceleration control of the control target. Summary of the invention

[0006] The problem to be solved by the present invention is to ensure the precise control of a hybrid vibration isolation system of a giant magnetostrictive actuator and an air spring, and a feedback linearized sliding mode control method for hybrid vibration isolation of a giant magnetostrictive actuator and an air spring is proposed.

[0007] To achieve the above object, the present invention is implemented through the following technical solutions:

[0008] A feedback linearized sliding mode control method for hybrid vibration isolation of a giant magnetostrictive actuator and an air spring comprises the following steps:

[0009] S1. Considering the influence of eddy current loss, temperature effect and prestress, the relationship between the expansion and contraction rate of giant magnetostrictive material and magnetic field intensity is established, and the constitutive equation of giant magnetostrictive actuator is constructed;

[0010] S2. Based on the expansion and contraction rate of the giant magnetostrictive material obtained in step S1, a dynamic balance equation of a hybrid vibration isolation system of a giant magnetostrictive actuator and an air spring is established;

[0011] S3. The dynamic balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S2 is controlled by using a feedback linearized sliding mode control strategy to obtain a feedback linearized sliding mode control equation;

[0012] S4. Based on the feedback linearized sliding mode control equation obtained in step S3 and the relationship between the expansion rate λ of the giant magnetostrictive material and the magnetic field intensity H obtained in step S1, a control method for a hybrid vibration isolation system of a giant magnetostrictive actuator and an air spring is constructed.

[0013] Furthermore, the specific implementation method of step S1 includes the following steps:

[0014] S1.1. According to the Bertotti core loss separation theory, set the core loss W of the giant magnetostrictive material in one cycle as:

[0015] W = W hys + W eddy + W anom (1)

[0016] Among them, W hys is the hysteresis loss in one cycle, W eddy is the eddy current loss generated in one cycle, W anom is the abnormal loss in one cycle;

[0017]

[0018] Among them, ρ is the resistivity of the GMM material, β is the shape factor of the GMM material, d is the diameter of the GMM material, dB is the change in magnetic induction intensity, and dt is the change in time;

[0019]

[0020] Among them, G is a dimensionless constant, S is the cross-sectional area of the GMM material, V 0 is a parameter characterizing the statistical distribution of the local coercive field;

[0021] S1.2. Based on the law of secondary domain rotation, determine the magnetostriction rate λ of the giant magnetostrictive material as:

[0022]

[0023] Among them, λ s is the saturation magnetostriction rate, M s is the saturation magnetization intensity, and M is the magnetization intensity;

[0024] S1.3. For the magnetization intensity M of the giant magnetostrictive material, it is divided into the reversible magnetization intensity M revh and the irreversible magnetization intensity M irrh , and the calculation formula is:

[0025] M = M revh + M irrh (5)

[0026] M revh = c(M anh - M irrh ) (6)

[0027] Among them, c is the reversible magnetization coefficient, and M anh is the non-hysteresis magnetization intensity;

[0028] M anh The calculation formula of is:

[0029]

[0030] Among them, a is the effective magnetic domain density; H e is the effective magnetic field strength;

[0031] H e The calculation formula of is:

[0032]

[0033] Among them, α is the molecular field constant; μ 0 is the vacuum permeability, H eddy is the eddy current loss magnetic field strength, H anom is the anomalous loss magnetic field strength, H is the internal magnetic field strength of the giant magnetostrictive material;

[0034] S1.4. Obtain the mathematical relationship expression between each magnetization parameter according to the energy conservation equation, and construct the constitutive equation of the giant magnetostrictive actuator. The mathematical relationship expression between each magnetization parameter is:

[0035]

[0036] Among them, δ M is the coefficient to prevent non-physical solutions, δ H is the direction coefficient, and k is the pinning coefficient;

[0037] Derive Equation (9) with respect to B, and the obtained expression is:

[0038]

[0039] Based on the fact that temperature has an impact on the saturation magnetization intensity and effective magnetic domain density of the material, Equations (11) and (12) are obtained, and the expressions are:

[0040]

[0041] Among them, t c is the Curie temperature, and Δy is the temperature change;

[0042]

[0043] Among them, N is the magnetic domain volume domain density, k B is the Boltzmann constant;

[0044] Combine Equation (12), (11), (10) with Equation (4) to form the constitutive equation of the giant magnetostrictive actuator.

[0045] Furthermore, the specific implementation method of step S2 includes the following steps:

[0046] S2.1. Mathematically model the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring. Simplify the air spring into a vibration isolation component containing nonlinear stiffness and linear damping, and simplify the giant magnetostrictive actuator part into a vibration isolation component containing linear stiffness, linear damping, and magnetostrictive force, obtaining the following expressions:

[0047] Set the damping c of the giant magnetostrictive actuator GMA 1 as:

[0048]

[0049] where c g is the damping coefficient, A is the cross-sectional area of the GMM material rod; L is the length of the GMM material rod;

[0050] The stiffness k of GMA 1 is:

[0051]

[0052] where E is the elastic modulus of the GMM material rod;

[0053] Set c 2 as the damping coefficient of the air spring, and k 2 as the stiffness of the air spring, and the expression is:

[0054]

[0055] where P 0 is the internal pressure of the air spring in the initial state, A 0 is the initial effective area of the air spring, V 0 is the initial volume of the air spring, x is the compression of the air spring, h is the initial height of the air spring, r is the stiffness state coefficient, when it is dynamic stiffness, r takes 1.3 - 1.4, and when it is static stiffness, r takes 1;

[0056] S2.2. Construct the motion balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring as:

[0057]

[0058] where F is the applied external excitation, x 1 is the position of the air spring, x 2 is the output position of GMA, m 1 is the mass of the air spring, m 2 is the mass of GMA and the controlled object.

[0059] Furthermore, the specific implementation method of step S3 includes the following steps:

[0060] S3.1. Rewrite the motion balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S2 based on the feedback linearization sliding mode control strategy, and the obtained expression is:

[0061]

[0062] Wherein, is the generalized state vector of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, f(x) is the internal dynamic response matrix of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, g(x) is the control input vector of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, u is the control force of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, h(x) is the observation coefficient matrix of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, and y is the observation output vector of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring;

[0063]

[0064] h(x) = x 1 (21);

[0065] S3.2. Based on the differential geometry principle, transform the motion balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S3.1 by using the Lie derivative, and the obtained expression is:

[0066]

[0067] Based on Derive Equation (22) to obtain:

[0068]

[0069] At this time, take:

[0070]

[0071] Wherein, EAλ is the output force of the giant magnetostrictive actuator, μ 1 is the first feedback gain coefficient, μ 2 is the second feedback gain coefficient;

[0072] Substitute Equation (24) into Equation (23) to obtain:

[0073]

[0074] When μ 1 > 0, μ 2 > 0, Equation (23) satisfies Lyapunov stability;

[0075] S3.3. Define the sliding mode surface s as:

[0076]

[0077] where n 1 is the sliding mode control attenuation gain;

[0078] Take the exponential reaching law as:

[0079]

[0080] where n 2 is the sliding mode control attenuation gain;

[0081] Define the Lyapunov function V(s) as:

[0082]

[0083] When s = 0, V(s) = 0; when s ≠ 0, V(s)>0, so V(s) is a positive definite function;

[0084] Take the derivative of the Lyapunov function to get:

[0085]

[0086] From equation (28), using the feedback linearization sliding mode control strategy for control satisfies Lyapunov stability, so the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring is asymptotically stable.

[0087] Furthermore, the specific implementation method of step S4 includes the following steps:

[0088] S4.1. It is known that the ground vibration is transmitted to the giant magnetostrictive actuator through the air spring and then to the controlled object; first, specify the desired displacement of the controlled object. According to the displacement of the controlled object measured by the displacement sensor, the difference from the desired displacement is input into equations (24) and (27) of the feedback linearization sliding mode control constructed in step S3. The feedback linearization sliding mode control gives the output strain of the giant magnetostrictive actuator according to the displacement difference;

[0089] S4.2 Input the output strain of the giant magnetostrictive actuator into the constitutive equation of the giant magnetostrictive actuator constructed in step S1. Substitute the output strain into equation (4) to calculate the magnetization M, and combine equations (5) - (10) to calculate the magnetic field strength H;

[0090] S4.3. Calculate the magnitude of the input current based on the magnetic field strength H and Biot-Savart's law, and transmit the calculated input current into the giant magnetostrictive actuator. The giant magnetostrictive actuator outputs displacement and transfers it to the controlled object. When the difference between the displacement sensor and the desired displacement is 0, the control process ends, and the giant magnetostrictive actuator gives the output displacement to complete the vibration isolation of the controlled object.

[0091] Advantages of the present invention:

[0092] For the feedback linearization sliding mode control method for hybrid vibration isolation of a giant magnetostrictive actuator and an air spring described in the present invention, the constitutive model of the giant magnetostrictive actuator can simultaneously consider the effects of eddy current loss, temperature effect, and prestress. The model of the present invention is established based on the magnetic domain theory and can better reflect the physical processes inside the material. By modifying the key parameters of the constitutive equation of the giant magnetostrictive actuator, the model can be adjusted according to different working conditions to adapt to various application scenarios.

[0093] For the feedback linearization sliding mode control method for hybrid vibration isolation of a giant magnetostrictive actuator and an air spring described in the present invention, feedback linearization uses the differential geometry method to map the original nonlinear system dynamics into a completely linear form, eliminating the nonlinear coupling terms and simplifying the controller design. The sliding mode control can ensure the stability of the system and achieve good dynamic performance even in the face of parameter changes or external disturbances. Description of the drawings

[0094] Figure 1 It is a flowchart of the feedback linearization sliding mode control method for hybrid vibration isolation of a giant magnetostrictive actuator and an air spring described in the present invention;

[0095] Figure 2 It is a mechanical simplified diagram of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring of the present invention;

[0096] Figure 3 It is a control block diagram of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring of the present invention;

[0097] Figure 4 It is a hybrid vibration isolation tracking control diagram of the present invention, where (a) is low-frequency tracking, (b) is medium-frequency tracking, and (c) is composite frequency tracking;

[0098] Figure 5 It is a comparison diagram of the vibration isolation effect under seismic excitation of the present invention, where (a) is displacement comparison, (b) is velocity comparison, and (c) is acceleration comparison. Detailed implementation manners

[0099] In order to make the objectives, technical solutions and advantages of the present invention more clear and understandable, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention, that is, the specific embodiments described are only a part of the embodiments of the present invention, rather than all of the specific embodiments. The components of the specific embodiments of the present invention usually described and shown in the drawings here can be arranged and designed in various different configurations, and the present invention can also have other embodiments.

[0100] Therefore, the following detailed description of the specific embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents the selected specific embodiments of the present invention. All other specific embodiments obtained by those skilled in the art based on the specific embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0101] To further understand the content, features and effects of the present invention, the following specific embodiments are exemplified and combined with the attached Figure 1 - attached Figure 5 The details are as follows:

[0102] Embodiment 1:

[0103] A feedback linearization sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuators and air springs, comprising the following steps:

[0104] S1. Considering the influence of eddy current loss, temperature effect and prestress, establish the relationship between the elongation rate and magnetic field strength of giant magnetostrictive materials, and construct the constitutive equation of giant magnetostrictive actuators;

[0105] Further, the specific implementation method of step S1 includes the following steps:

[0106] S1.1. According to Bertotti core loss separation theory, set the core loss W of giant magnetostrictive materials in one cycle as:

[0108] W = W hys + W eddy + W anom (1)

[0109] Wherein, W hys is the hysteresis loss in one cycle, W eddy is the eddy current loss generated in one cycle, and W anom is the abnormal loss in one cycle;

[0110]

[0111] Among them, ρ is the resistivity of the GMM material, β is the shape factor of the GMM material, d is the diameter of the GMM material, dB is the change in magnetic induction intensity, and dt is the change in time;

[0112]

[0113] Among them, G is a dimensionless constant, S is the cross-sectional area of the GMM material, V 0 is a parameter characterizing the statistical distribution of the local coercive field;

[0114] S1.2. Determine the elongation rate λ of the giant magnetostrictive material based on the law of secondary domain rotation as:

[0115]

[0116] Among them, λ s is the saturation elongation rate, M s is the saturation magnetization intensity, and M is the magnetization intensity;

[0117] S1.3. For the magnetization intensity M of the giant magnetostrictive material, it is divided into reversible magnetization intensity M revh and irreversible magnetization intensity M irrh , and the calculation formula is:

[0118] M = M revh + M irrh (5)

[0119] M revh = c(M anh - M irrh ) 6)

[0120] Among them, c is the reversible magnetization coefficient, and M anh is the non-hysteretic magnetization intensity;

[0121] M anh The calculation formula of is:

[0122]

[0123] Among them, a is the effective magnetic domain density; H e is the effective magnetic field intensity;

[0124] H e The calculation formula of is:

[0125]

[0126] Among them, α is the molecular field constant; μ 0 is the vacuum permeability, H eddy is the eddy current loss magnetic field intensity, H anom is the anomalous loss magnetic field intensity, and H is the internal magnetic field intensity of the giant magnetostrictive material;

[0127] S1.4. Obtain the mathematical relationship expression between each magnetization parameter according to the energy conservation equation, and construct the constitutive equation of the giant magnetostrictive actuator. The mathematical relationship expression between each magnetization parameter is as follows:

[0128]

[0129] Among them, δ M To prevent the occurrence of non - physical solution coefficients, δ H is the direction coefficient, k is the pinning coefficient, a physical quantity describing the degree of obstruction when microscopic structures such as magnetic domain walls or superconducting vortices inside the material move;

[0130] Furthermore, it has the following expression:

[0131]

[0132] Take the derivative of Equation (9) with respect to B, and the obtained expression is:

[0133]

[0134] Based on the fact that temperature has an impact on the saturation magnetization intensity and the effective magnetic domain density of the material, Equations (11) and (12) are obtained, and the expressions are:

[0135]

[0136] Among them, t c is the Curie temperature, and Δt is the temperature change;

[0137]

[0138] Among them, N is the magnetic domain volume density, k B is the Boltzmann constant;

[0139] Combine Equation (12), (11), (10) with Equation (4) to form the constitutive equation of the giant magnetostrictive actuator.

[0140] S2. Based on the elongation rate of the giant magnetostrictive material obtained in step S1, establish the dynamic equilibrium equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring;

[0141] Furthermore, the specific implementation method of step S2 includes the following steps:

[0142] S2.1. Conduct mathematical modeling on the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring. Simplify the air spring into a vibration isolation component containing nonlinear stiffness and linear damping, and simplify the giant magnetostrictive actuator part into a vibration isolation component containing linear stiffness, linear damping, and magnetostrictive force, and obtain the following expression:

[0143] Set the damping c of the giant magnetostrictive actuator GMA 1 as:

[0144]

[0145] where c g is the damping coefficient, A is the cross-sectional area of the GMM material rod; L is the length of the GMM material rod;

[0146] The stiffness k of the GMA 1 is:

[0147]

[0148] where E is the elastic modulus of the GMM material rod;

[0149] Set c 2 as the damping coefficient of the air spring, k 2 as the stiffness of the air spring, and the expression is:

[0150]

[0151] where P 0 is the internal pressure of the air spring in the initial state, A 0 is the initial effective area of the air spring, V 0 is the initial volume of the air spring, x is the compression of the air spring, h is the initial height of the air spring, r is the stiffness state coefficient, when it is the dynamic stiffness, r takes 1.3 - 1.4, and when it is the static stiffness, r takes 1;

[0152] S2.2. The motion balance equation for constructing the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring is:

[0153]

[0154] where F is the applied external excitation, x 1 is the position of the air spring, x 2 is the output position of the GMA, m 1 is the mass of the air spring, m 2 is the mass of the GMA and the controlled object.

[0155] S3. For the dynamic balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S2, use the feedback linearization sliding mode control strategy for control to obtain the feedback linearization sliding mode control equation;

[0156] Furthermore, the specific implementation method of step S3 includes the following steps:

[0157] S3.1. Rewrite the motion balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S2 based on the feedback linearization sliding mode control strategy, and the obtained expression is:

[0158]

[0159] Where, is the generalized state vector of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, f(x) is the internal dynamic response matrix of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, g(x) is the control input vector of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, u is the control force of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, h(x) is the observation coefficient matrix of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring, and y is the observation output vector of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring;

[0160]

[0161]

[0162] h(x) = x 1 (21);

[0163] S3.2. Based on the differential geometry principle, transform the motion balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S3.1 using the Lie derivative, and the obtained expression is:

[0164]

[0165] Based on Derive Equation (22) to obtain:

[0166]

[0167] At this time, take:

[0168]

[0169] Where, EAλ is the output force of the giant magnetostrictive actuator, μ 1 is the first feedback gain coefficient, μ 2 is the second feedback gain coefficient;

[0170] Substitute Equation (24) into Equation (23) to obtain:

[0171]

[0172] When μ 1 > 0, μ 2> 0, equation (23) satisfies Lyapunov stability;

[0173] The goal of sliding mode control is to drive the system state to a pre-designed sliding mode surface and maintain the motion on the sliding mode surface, which has strong robustness to system parameter uncertainties and external disturbances.

[0174] S3.3. Define the sliding mode surface s as:

[0175]

[0176] where n 1 is the sliding mode control attenuation gain;

[0177] Take the exponential reaching law as:

[0178]

[0179] where n 2 is the sliding mode control attenuation gain;

[0180] When using sliding mode control in practice, the reaching law is designed using the sign function. The discontinuity of the sign function at zero will cause the control input u to switch frequently when s approaches 0, resulting in chattering. In this paper, the tanh function is used instead of the sign function. The tanh function is a smooth non-linear function, which is close to linear when s approaches zero and does not produce discontinuous jumps at s = 0. It is close to the sign function when s is large. This smooth transition characteristic can significantly reduce the high-frequency components of the control input, thus reducing the chattering phenomenon.

[0181] Define the Lyapunov function V(s) as:

[0182]

[0183] When s = 0, V(s) = 0; when s ≠ 0, V(s)> 0, so V(s) is a positive definite function;

[0184] Differentiate the Lyapunov function to get:

[0185]

[0186] From equation (28), using the feedback linearization sliding mode control strategy for control satisfies Lyapunov stability, so the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring is asymptotically stable.

[0187] S4. Based on the feedback linearization sliding mode control equation obtained in step S3 and the relationship between the elongation rate λ and the magnetic field strength H of the giant magnetostrictive material obtained in step S1, construct a control method for the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring.

[0188] Furthermore, the specific implementation method of step S4 includes the following steps:

[0189] S4.1. It is known that ground vibration is transmitted to the giant magnetostrictive actuator through the air spring and then to the controlled object through the giant magnetostrictive actuator. First, specify the desired displacement of the controlled object. The difference between the displacement of the controlled object measured by the displacement sensor and the desired displacement is input into equations (24) and (27) of the feedback linearization sliding mode control constructed in step S3. The feedback linearization sliding mode control gives the output strain of the giant magnetostrictive actuator according to the displacement difference.

[0190] S4.2 Input the output strain of the giant magnetostrictive actuator into the constitutive equation of the giant magnetostrictive actuator constructed in step S1. Substitute the output strain into equation (4) to calculate the magnetization intensity M, and calculate the magnetic field intensity H by combining equations (5) - (10).

[0191] S4.3. Calculate the magnitude of the input current according to the magnetic field intensity H and the Biot - Savart law, and input the calculated input current into the giant magnetostrictive actuator. The giant magnetostrictive actuator outputs displacement and transmits it to the controlled object. When the difference between the displacement sensor and the desired displacement is 0, end the control process. The giant magnetostrictive actuator gives the output displacement to complete the vibration isolation of the controlled object.

[0192] Using the feedback linearization sliding mode control method for the hybrid vibration isolation of the giant magnetostrictive actuator and the air spring described in this embodiment, the control strategy of the feedback linearization sliding mode control is used for the tracking control of the sine input signal. Tracking effect test: The tracking signal is a sine signal, including two single frequencies of 3 Hz and 10 Hz and a composite frequency of 10 / 30 / 50. The tracking results are as Figure 3 shown. Figure 3 In it, TD is the tracking target curve, LCR is the tracking curve of the giant magnetostrictive actuator using the linear piezomagnetic constitutive equation, and the MCR curve is the tracking curve of the giant magnetostrictive actuator using the modified constitutive model described in this embodiment. The tracking effect using the modified constitutive is much better than that using the linear constitutive, especially for the tracking of the composite frequency signal. Using the linear constitutive can hardly track the target signal, while the tracking curve using the modified constitutive is basically consistent with the target signal. It can be seen from the tracking results that the actual output of the hybrid vibration isolation system can effectively track the target signal using the control strategy of the feedback linearization sliding mode control, and the tracking accuracy is satisfactory, meeting the requirements of actual engineering.

[0193] For the feedback linearization sliding mode control method for the hybrid vibration isolation of the giant magnetostrictive actuator and the air spring described in this embodiment, the vibration isolation effect of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring using the feedback linearization sliding mode control under seismic excitation is as Figure 4as shown Figure 4 In Figure 4 , "Uncontrolled" means that no vibration isolation measures are applied to the controlled object, "PVI" means that the passive control is used for the controlled object, and "HVI" means that the giant magnetostrictive actuator and the air spring hybrid vibration isolation system are used for the controlled object. Since the dominant frequency of ground motion is generally between 1 - 10 Hz, it is very difficult for the natural vibration frequency of the passive vibration isolation device to be less than 1 Hz. Therefore, it is difficult for the passive vibration isolation device to effectively isolate ground motion. However, the giant magnetostrictive actuator and the air spring hybrid vibration isolation system using feedback linearization sliding mode control has a good vibration isolation effect on the responses of the displacement, velocity, and acceleration of the controlled object under ground motion.

[0194] It should be noted that relational terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprising", "including" or any other variant thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device comprising a series of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article or device. Without further limitation, an element defined by the statement "comprising an..." does not exclude the presence of additional identical elements in the process, method, article or device comprising the element.

[0195] Although the present application has been described above with reference to specific embodiments, various improvements can be made to it and components thereof can be replaced with equivalents without departing from the scope of the present application. In particular, as long as there is no structural conflict, the various features in the specific embodiments disclosed in the present application can be combined with each other in any way. The exhaustive description of these combinations is not given in this specification only for the sake of saving space and resources. Therefore, the present application is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims

1. A feedback linearized sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuator and air spring, characterized in that: The steps include: S1. Considering the influence of eddy current loss, temperature effect and prestress, the relationship between the expansion and contraction rate of giant magnetostrictive material and magnetic field intensity is established, and the constitutive equation of giant magnetostrictive actuator is constructed; S2. Based on the expansion and contraction rate of the giant magnetostrictive material obtained in step S1, a dynamic balance equation of a hybrid vibration isolation system of a giant magnetostrictive actuator and an air spring is established; S3. The dynamic balance equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S2 is controlled by using a feedback linearized sliding mode control strategy to obtain a feedback linearized sliding mode control equation; S4. Based on the feedback linearized sliding mode control equation obtained in step S3 and the relationship between the expansion rate λ of the giant magnetostrictive material and the magnetic field intensity H obtained in step S1, a control method for a hybrid vibration isolation system of a giant magnetostrictive actuator and an air spring is constructed.

2. A feedback linearized sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuator and air spring according to claim 1, characterized in that: The specific implementation method of step S1 includes the following steps: S1.

1. According to Bertotti core loss separation theory, the core loss W of the giant magnetostrictive material in one cycle is set to: W=W hys +W eddy +W anom (1) Among them, W hys is the hysteresis loss in one cycle, W eddy is the eddy current loss generated in one cycle, W anom Abnormal loss within a cycle; Wherein, ρ is the resistivity of the GMM material, β is the shape factor of the GMM material, d is the diameter of the GMM material, dB is the change in magnetic induction intensity, and dt is the change in time; Among them, G is a dimensionless constant, S is the cross-sectional area of ​​the GMM material, and V0 is a parameter that describes the statistical distribution of the local coercive field; S1.

2. Based on the secondary domain rotation law, the expansion rate λ of the giant magnetostrictive material is determined as: Among them, λ s is the saturation expansion ratio, M s is the saturation magnetization, M is the magnetization intensity; S1.

3. For the magnetization intensity M of giant magnetostrictive materials, it is divided into reversible magnetization intensity M revh and irreversible magnetization M irrh , the calculation formula is: M=M revh +M irrh (5) M revh =c(M anh -M irrh ) (6) Where c is the reversible magnetic susceptibility, M anh is the hysteresis-free magnetization; M anh The calculation formula is: Where a is the effective magnetic domain density; H e is the effective magnetic field strength; H e The calculation formula is: Among them, α is the molecular field constant; μ0 is the vacuum magnetic permeability, H eddy is the eddy current loss magnetic field intensity, H anom is the abnormal loss magnetic field intensity, H is the magnetic field intensity inside the giant magnetostrictive material; S1.

4. According to the energy conservation equation, the mathematical relationship between the magnetization parameters is obtained, and the constitutive equation of the giant magnetostrictive actuator is constructed. The mathematical relationship between the magnetization parameters is as follows: Among them, δ M To prevent the occurrence of nonphysical solutions, δ H is the directional coefficient, k is the pinning coefficient; Derivative (9) with respect to B, we get the following expression: Based on the influence of temperature on the saturation magnetization and effective magnetic domain density of the material, equations (11) and (12) are obtained, and the expression is: Among them, t c is the Curie temperature, Δt is the temperature change; Where N is the volume density of magnetic domains, k B is the Boltzmann constant; Combining equations (12), (11), (10) and (4) together constitutes the constitutive equation of the giant magnetostrictive actuator.

3. The feedback linearized sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuator and air spring according to claim 2, characterized in that: The specific implementation method of step S2 includes the following steps: S2.

1. Mathematically model the hybrid vibration isolation system of giant magnetostrictive actuator and air spring. The air spring is simplified into a vibration isolation component containing nonlinear stiffness and linear damping, and the giant magnetostrictive actuator is simplified into a vibration isolation component containing linear stiffness, linear damping and magnetostrictive force. The following expression is obtained: The damping c1 of the giant magnetostrictive actuator GMA is set to: Among them, c g is the damping coefficient, A is the cross-sectional area of ​​the GMM material rod; L is the length of the GMM material rod; The stiffness k1 of GMA is: Where, E is the elastic modulus of the GMM material rod; Let c2 be the damping coefficient of the air spring, and k2 be the stiffness of the air spring. The expression is: Among them, P0 is the internal pressure of the air spring in the initial state, A0 is the initial effective area of ​​the air spring, V0 is the initial volume of the air spring, x is the compression of the air spring, h is the initial height of the air spring, r is the stiffness state coefficient, when it is dynamic stiffness, r is 1.3~1.4, when it is static stiffness, r is 1; S2.

2. The motion equilibrium equation of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring is: Where F is the applied external excitation, x1 is the position of the air spring, x2 is the GMA output position, m1 is the mass of the air spring, and m2 is the mass of the GMA and the controlled object.

4. The feedback linearized sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuator and air spring according to claim 3, characterized in that: The specific implementation method of step S3 includes the following steps: S3.

1. Based on the feedback linearization sliding mode control strategy, the motion equilibrium equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S2 is rewritten to obtain the expression: in, is the generalized state vector of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring, f(x) is the internal dynamic response matrix of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring, g(x) is the control input vector of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring, u is the control force of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring, h(x) is the observation coefficient matrix of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring, y is the observation output vector of the hybrid vibration isolation system of giant magnetostrictive actuator and air spring; h(x)=x1 (21); S3.

2. Based on the principle of differential geometry, the motion equilibrium equation of the hybrid vibration isolation system of the giant magnetostrictive actuator and the air spring obtained in step S3.1 is transformed using Lie derivative to obtain the expression: based on By taking the derivative of formula (22), we can obtain: At this time, take: Wherein, EAλ is the output force of the giant magnetostrictive actuator, μ1 is the first feedback gain coefficient, and μ2 is the second feedback gain coefficient; Substituting formula (24) into formula (23), we get: When μ1>0,μ2>0, equation (23) satisfies Lyapunov stability; S3.

3. Define the sliding surface s as: Where n1 is the sliding mode control attenuation gain; Take the exponential reaching law as: Where n2 is the sliding mode control attenuation gain; Define the Lyapunov function V(s) as: When s=0, V(s)=0; when s≠0, V(s)>0, so V(s) is a positive definite function; Taking the derivative of the Lyapunov function, we get: According to formula (28), the feedback linearized sliding mode control strategy satisfies Lyapunov stability, so the hybrid vibration isolation system of giant magnetostrictive actuator and air spring is asymptotically stable.

5. The feedback linearized sliding mode control method for hybrid vibration isolation of giant magnetostrictive actuator and air spring according to claim 4, characterized in that: The specific implementation method of step S4 includes the following steps: S4.

1. It is known that the ground vibration is transmitted to the giant magnetostrictive actuator through the air spring, and then to the controlled object through the giant magnetostrictive actuator; first, the desired displacement of the controlled object is specified, and the difference between the displacement of the controlled object measured by the displacement sensor and the desired displacement is input into equations (24) and (27) of the feedback linearized sliding mode control constructed in step S3, and the feedback linearized sliding mode control gives the output strain of the giant magnetostrictive actuator according to the displacement difference; S4.2 The output strain of the giant magnetostrictive actuator is input into the constitutive equation of the giant magnetostrictive actuator constructed in step S1, the output strain is substituted into equation (4) to calculate the magnetization intensity M, and the magnetic field intensity H is calculated by combining equations (5) to (10); S4.

3. The input current is calculated based on the magnetic field strength H and the Biot-Sadell law, and the calculated input current is transmitted to the giant magnetostrictive actuator. The output displacement of the giant magnetostrictive actuator is transmitted to the controlled object. When the difference between the displacement sensor and the expected displacement is 0, the control process ends, the giant magnetostrictive actuator gives the output displacement, and the vibration isolation of the controlled object is completed.

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