A nonlinear magnetic suspension ball system active disturbance rejection control method

Abstract

The present application relates to the technical field of magnetic levitation ball system, in particular to a nonlinear magnetic levitation ball system active disturbance rejection control method, in order to solve the problems of the existing magnetic levitation ball system in the off-equilibrium point working condition, such as nonlinear dynamic instability, insufficient coupling inhibition of multi-source interference, and hardware damage caused by limited control input, a new nonlinear magnetic levitation ball system active disturbance rejection control method is provided, including the following steps: 1) establishing the mechanism model of the magnetic levitation ball system under the action of voltage; 2) constructing the third-order nonlinear dynamic differential equation of the magnetic levitation ball system; 3) designing an extended state observer; 4) limiting the control input. The present application defines a new control input to reduce the complexity of the controller, and designs an extended state observer to utilize the state feedback information to design the control input, so as to realize the adjustment of the system and effectively suppress the external disturbance.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of magnetic suspension systems, in particular to a self-disturbance control method for a nonlinear magnetic suspension system. BACKGROUND

[0002] A magnetic suspension small ball system balances gravity by electromagnetic force to stably suspend the magnetic suspension small ball. When deviating from the equilibrium point, the traditional PID or linear control method cannot achieve high-precision robust control due to strong nonlinearity, parameter uncertainty and external disturbances (such as mechanical vibration and air flow disturbance). In the prior art, the disturbance observer or the sliding mode control can suppress the disturbance, but there are problems of chattering, strong modeling dependence or slow dynamic response. The self-disturbance control estimates the system "total disturbance" (including model uncertainty and external disturbance) in real time through the extended state observer, and can achieve robust control without accurate modeling. At present, in the research of the time-varying gain extended state observer, a method of dynamically adjusting the gain is proposed, that is, the disturbance estimation speed is improved in the transient stage, and the influence of sensor noise is reduced in the steady state. Compared with the traditional linear extended state observer, the time-varying extended state observer improves the high-frequency noise attenuation ability while ensuring the transient performance, but the time-varying parameter depends on the experience adjustment and lacks the self-adaptive mechanism. Another research introduces the particle swarm algorithm to set the self-disturbance control parameters, reduces the difficulty of artificial parameter adjustment through global optimization, and improves the dynamic response and steady accuracy of the magnetic suspension system under a specific disturbance mode. However, the optimization process of the method does not strictly analyze the calculation time and convergence stability of the algorithm, and the parameter optimization depends on the specific disturbance mode, so the anti-disturbance performance may be insufficient when facing complex and coupled multi-element disturbance (such as mechanical vibration and air flow fluctuation), and the robustness is limited.

[0003] In addition, the current researchers model the magnetic suspension small ball system by linear expansion according to the Taylor formula at the equilibrium point, which can only control the small ball in the local equilibrium position, and the control accuracy is low. When the small ball deviates far from the equilibrium point, the strong nonlinearity of the system is highlighted, and the performance of the controller designed based on the linear model may decrease sharply or even fail, resulting in the small ball being unable to stably suspend, which seriously limits the working range and application scenarios of the system. In addition, the existing control scheme does not fully consider the physical limitations of the electromagnetic coil (such as maximum current and power), and when a larger disturbance signal is encountered, the controller may generate an excessive current. If the control signal is not constrained, the electromagnetic coil may overheat and cause thermal breakdown. At the same time, the change of the out-of-control electromagnetic force may also cause air gap oscillation, and then cause the small ball to collide, causing hardware damage. SUMMARY

[0004] The application provides a new nonlinear magnetic suspension ball system active disturbance rejection control method to solve the problems of nonlinear dynamic instability, insufficient multi-source interference coupling suppression, and control input limited hardware damage of the existing magnetic suspension ball system under the working condition deviating from the equilibrium point.

[0005] The application is implemented by adopting the following technical scheme:

[0006] A nonlinear magnetic suspension ball system active disturbance rejection control method comprises the following steps:

[0007] 1) According to the working principle of the magnetic suspension ball system, a mechanism model of the magnetic suspension ball system under the action of voltage is established as follows:

[0008]

[0009] wherein, is the mass of the ball, is the gravity acceleration, is the actual position of the ball, is the actual position current of the ball, is the equivalent resistance of the electromagnetic coil, is the self-inductance of the electromagnetic coil, is the vacuum permeability, is the coil turns of the electromagnetic coil, is the magnetic conductive cross-sectional area, is the equilibrium position of the ball, is the equilibrium position current of the ball, is the input voltage of the power amplifier, is the electromagnetic force received by the ball at the actual position, is the electromagnetic force received by the ball at the equilibrium position, is time, is the mutual inductance coefficient of the electromagnetic coil;

[0010] 2) A third-order nonlinear dynamic differential equation of the magnetic suspension ball system is constructed:

[0011] Let the actual velocity of the ball be the mechanism acceleration of the ball be the system input be the system output be and the mechanism model of the magnetic suspension ball system be derived as follows:

[0012]

[0013] Let the actual acceleration of the ball be According to the mechanism model of the magnetic suspension ball system and considering the existence of uncertainty factors and external disturbances of the system , obtain , nonlinear time-varying gain , obtain the whole as a control input of the controller , i.e. , is the actual disturbance of the magnetic levitation ball system containing external disturbance, and , obtain , thus the differential equation of the magnetic levitation ball system can be obtained as follows:

[0014]

[0015] 3) Design an extended state observer

[0016] Let the extended state of the magnetic levitation ball system be , define the rate of change of is the time derivative of , i.e. , the extended state model of the magnetic levitation ball system is as follows:

[0017]

[0018] Design the extended state observer as follows:

[0019]

[0020] wherein, is the estimated position error of the ball and , is the estimated position of the ball observed by the extended state observer, is the estimated velocity of the ball observed by the extended state observer, is the estimated acceleration of the ball observed by the extended state observer, is the estimated disturbance of the ball, , , and are adjustable gains of the extended state observer, and , , , wherein is the bandwidth of the extended state observer;

[0021] The system state and disturbance are estimated in real time by using the extended state observer, and the control input is designed by using the estimated system state and disturbance , so as to realize the control of the magnetic levitation ball system.

[0022] Further, the active disturbance rejection control method further comprises step 4), i.e. clipping the control input

[0023] saturating function design for the control input , i.e.

[0024] u ' = Sat [ u min , u max ] k p ( y r - z 1 )- k d z 2 - k f z 3 - z 4

[0025] controlling the control input to be within [ u min , u max ] , i.e.

[0026]

[0027] wherein, is a target position of the ball, is a tracking error of the ball, , , , respectively, are feedback gains of the tracking error, the estimated velocity and the estimated acceleration of the ball, is a bandwidth of the controller, and , , , , respectively, are maximum and minimum values of the control input .

[0028] The present application has the following beneficial effects: first, the present application proposes a third-order nonlinear dynamic model and expresses it in the form of a differential equation, and defines a new control input , which effectively reduces the complexity of the controller; second, the present application considers the factor of external disturbance, designs an extended state observer to estimate the system state and external disturbance in real time, and then designs the control input using the state feedback information, so as to realize the adjustment of the system and effectively suppress the external disturbance. BRIEF DESCRIPTION OF DRAWINGS

[0029] The accompanying drawings, which are incorporated herein and form part of the specification, illustrate embodiments consistent with the present application and, together with the description, further serve to explain the principles of the application.

[0030] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or prior art description will be briefly introduced. Obviously, those skilled in the art can obtain other drawings according to these drawings without any creative effort.

[0031] Fig. 1 A flowchart of the self-disturbance control method of the present application;

[0032] Fig. 2 A structural diagram of the magnetic levitation ball system;

[0033] Fig. 3 A diagram of the target position and actual position of the ball in the present application;

[0034] Fig. 4 A diagram of the actual position and estimated position of the ball in the present application;

[0035] Fig. 5 A diagram of the actual speed and estimated speed of the ball in the present application;

[0036] Fig. 6 A diagram of the actual acceleration and estimated acceleration of the ball in the present application;

[0037] Fig. 7 A diagram of the actual disturbance and estimated disturbance of the ball in the present application.

[0038] In the figure: 1-electromagnet, 2-electromagnetic coil, 3-power amplifier, 4-controller, 5-laser sensor, 6-ball. DETAILED DESCRIPTION

[0039] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or prior art description will be briefly introduced. Obviously, those skilled in the art can obtain other drawings according to these drawings without any creative effort.

[0040] In the following description, many specific details are set forth in order to provide a thorough understanding of the present application, but the present application can also be implemented in other ways different from those described herein; obviously, the embodiments in the specification are only some embodiments of the present application, not all embodiments.

[0041] The specific embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0042] A self-disturbance control method of a nonlinear magnetic levitation ball system, as shown in Fig. 1 , includes the following steps:

[0043] 1) as shown in Fig. 2As shown, the magnetic levitation ball system is composed of ball 6, laser sensor 5, power amplifier 3, electromagnet 1 and electromagnetic coil 2, controller 4. The working principle of the system is as follows: the laser sensor 5 monitors the vertical position of the ball 6 in real time, transmits the position signal to the controller 4, the controller 4 analyzes the deviation between the target position and the actual position, calculates the required electromagnetic force, and outputs the control signal, the power amplifier 3 converts the voltage signal into current signal, drives the electromagnetic coil 2 to generate electromagnetic force, when the electromagnetic force and the gravity of the ball 6 are equal, the ball 6 reaches equilibrium.

[0044] According to the working principle of the magnetic levitation ball system, the mechanism model of the magnetic levitation ball system under voltage is established as follows:

[0045]

[0046] Wherein, is the mass of the ball, is the acceleration of gravity, is the actual position of the ball, is the actual position current of the ball, is the equivalent resistance of the electromagnetic coil, is the self-inductance of the electromagnetic coil, is the vacuum permeability, is the number of turns of the electromagnetic coil, is the magnetic cross-sectional area, is the equilibrium position of the ball, is the equilibrium position current of the ball, is the input voltage of the power amplifier, is the electromagnetic force received by the ball at the actual position, is the electromagnetic force received by the ball at the equilibrium position, is the time, is the mutual inductance coefficient of the electromagnetic coil;

[0047] 2) Construct the third-order nonlinear dynamic differential equation of the magnetic levitation ball system:

[0048] Let the actual velocity of the ball 6 be , the mechanism acceleration of the ball 6 be , the system input be , the system output be , and be derived from the mechanism model of the magnetic levitation ball system:

[0049]

[0050] Let the actual acceleration of the ball 6 be , according to the mechanism model of the magnetic levitation ball system and considering the existence of uncertainty factors and external disturbances , get nonlinear time-varying gain , the overall as a control input of the controller 4 i.e. , is the actual disturbance including external disturbance suffered by the magnetic levitation ball system, and , get , thus the differential equation of the magnetic levitation ball system can be obtained as follows:

[0051]

[0052] wherein the overall is regarded as a new control input of the controller 4 , without online estimation of the nonlinear time-varying gain , thereby reducing the complexity of the controller 4;

[0053] 3) design an extended state observer

[0054] Let the extended state of the magnetic levitation ball system be , define the rate of change of as the time derivative of , that is , the extended state model of the magnetic levitation ball system is as follows:

[0055]

[0056] Design the extended state observer as follows:

[0057]

[0058] wherein is the estimated position error of the ball 6 and , is the estimated position of the ball 6 observed by the extended state observer, is the estimated velocity of the ball 6 observed by the extended state observer, is the estimated acceleration of the ball 6 observed by the extended state observer, is the estimated disturbance suffered by the ball 6, , , and are adjustable gains of the extended state observer, and , , , wherein is the bandwidth of the extended state observer;

[0059] ​The system state and disturbance are estimated in real time by using an extended state observer, and the estimated system state and disturbance are used to design the control input , so as to realize the control of the magnetic levitation ball system.

[0060] 4) The control input is limited:

[0061] The control input is designed by using a saturation function, i.e.

[0062] u ' = Sat [ u min , u max ] k p ( y r - z 1 )- k d z 2 - k f z 3 - z 4

[0063] The control input is controlled within [ u min , u max ] , i.e.

[0064]

[0065] wherein, is the target position of the ball 6, is the tracking error of the ball 6, , , are respectively the feedback gains of the tracking error, the estimated velocity and the estimated acceleration of the ball 6, is the bandwidth of the controller 4, and , , , are respectively the maximum value and the minimum value of the control input .

[0066] According to Joule's law, when the control voltage exceeds the rated range, the square growth of the coil current will cause the winding temperature to rise exponentially. By designing a saturation function for the control input , the control voltage is strictly limited within [ u min , u max ] , which ensures that the current does not exceed the current-carrying capacity of the electromagnetic coil 2, and effectively prevents thermal breakdown and permanent magnet demagnetization caused by overloading of the electromagnetic coil 2.

[0067] The above control method is verified by simulation experiments.

[0068] This embodiment uses MATLAB / Simulink to build a model of a magnetically levitated ball system and verifies the effectiveness of the active disturbance rejection control method for the magnetically levitated ball system described in this invention.

[0069] To improve engineering applicability, simulation reliability, and the efficiency and computational performance of modeling in Simulink, the Euler method is used to discretize the continuous-time system model in Simulink. An appropriate sampling period is selected based on the system characteristics to ensure that the discretized continuous-time system model accurately preserves the dynamic characteristics of the original continuous-time system model. The discretized continuous-time system model is as follows:

[0070] & x 1 [k+1]= x 1 [k] + τ · x 2 [k] & x 2 |k+1|= x 2 [k] + τ · x 3 [k] & x 3 [k+1]= x 3 [k] + τ · (f[k] + u ' )    &&

[0071] & z 1 [k+1]= z 1 [k] + τ · z 2 [k]+ ρ 1 e[k]) & z 2 [k+1]= z 2 [k] + τ · z 3 [k]+ ρ 2 e[k]) & z 3 [k+1]= z 3 [k] + τ · z 4 [k]+ ρ 3 e[k] + u ' ) & z 4 [k+1]= z 4 [k] + τ · ρ 4 e [k]      &&

[0072] in, Indicates the discrete time point index. The sampling step size, e[k] Indicates the 6th ball The estimated position error of the step, and e[ k ]=y[ k ]- z 1 [ k ] , x 1 [k] For the 6th ball The actual position of the step x 2 [k] For the 6th ball The actual speed of the step x 3 [k] For the 6th ball The actual acceleration of the step, y[ k ] For the 6th ball The actual output of the step, x 1 [k+1] For the 6th ball The actual position of the step x 2 [k+1] For the 6th ball The actual speed of the step x 3 [k+1] For the 6th ball The actual acceleration of the step, For the 6th ball The estimated position of the step, For the 6th ball The estimated speed of the step, For the 6th ball The estimated acceleration of the step, For the 6th ball The estimated perturbation experienced by the step. For the 6th ball The estimated position of the step, For the 6th ball The estimated speed of the step, For the 6th ball The estimated acceleration of the step, For the 6th ball The estimated disturbance experienced by the step.

[0073] A magnetically levitated ball system model controlled by the active disturbance rejection control method described in this invention was built in Simulink using the discretized continuous-time system model, and the active disturbance rejection control method described in this invention was verified by simulation.

[0074] The results obtained from the above simulation experiments are as follows: Fig. 3 - Fig. 7 As shown: by Fig. 3 It can be seen that the actual position of ball 6 basically matches the target position of ball 6; from Fig. 4 It can be seen that the actual position of ball 6 and the estimated position of ball 6 basically match; from Fig. 5 It can be seen that the actual velocity of ball 6 and the estimated velocity of ball 6 are basically consistent; from Fig. 6 It can be seen that the actual acceleration of ball 6 and the estimated acceleration of ball 6 are basically consistent; from Fig. 7 It can be seen that the actual perturbation experienced by ball 6 matches the estimated perturbation experienced by ball 6. In summary... Fig. 3 - 7 The results demonstrate the effectiveness of the active disturbance rejection control method for the magnetic levitation ball system described in this invention.

[0075] The above description is merely a specific embodiment of the present invention, enabling those skilled in the art to understand or implement the present invention. Although detailed descriptions have been provided with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments, and they should all be covered within the protection scope of the claims.

Claims

1. A nonlinear magnetic levitation ball system active disturbance rejection control method, characterized in that, Comprising the following steps: 1) According to the working principle of the magnetic suspension ball system, the mechanism model of the magnetic suspension ball system under the action of voltage is established as follows: ， wherein, is the mass of the ball (6), is the acceleration of gravity, is the actual position of the ball (6), is the actual position current of the ball (6), is the equivalent resistance of the electromagnetic coil (2), is the self-inductance of the electromagnetic coil (2), is the permeability of vacuum, is the number of turns of the electromagnetic coil (2), is the cross-sectional area of the magnetic flux, is the equilibrium position of the ball (6), is the equilibrium position current of the ball (6), is the input voltage of the power amplifier (3), is the electromagnetic force experienced by the ball (6) at the actual position, is the electromagnetic force experienced by the ball (6) at the equilibrium position, is the time, is the mutual inductance of the electromagnetic coil (2); 2) Construct the third-order nonlinear dynamic differential equation of the magnetic suspension ball system: Let the actual velocity of the ball (6) , the mechanism acceleration of the ball (6) , the system input quantity is , the system output quantity , and is derived from the mechanism model of the magnetic levitation ball system: ， Let the actual acceleration of the small ball (6) be Based on the mechanism model of the magnetically levitated ball system and considering the uncertainties and external disturbances in the system, ,get Nonlinear time-varying gain ,Will The whole is used as the control input of the controller (4) ,Right now , The actual disturbances experienced by the magnetically levitated sphere system, including external interference, and ,get Therefore, the differential equation of the magnetically levitated ball system can be obtained as follows: ； 3) Design the extended state observer Let the expansion state of the magnetic levitation ball system be defined as the rate of change of , the time derivative of , that is The expansion state model of the magnetic levitation ball system is as follows: ， The extended state observer is designed as follows: ， wherein is an estimated position error of the ball (6) and , is an estimated position of the ball (6) as observed by the extended state observer, is an estimated velocity of the ball (6) as observed by the extended state observer, is an estimated acceleration of the ball (6) as observed by the extended state observer, is an estimated disturbance experienced by the ball (6), , , and is an adjustable gain of the extended state observer, and , , , wherein is a bandwidth of the extended state observer; Real-time estimation of system states and disturbances using an extended state observer, and design of control inputs using the estimated system states and disturbances Thus, the control of the magnetic levitation ball system is realized.

2. The active disturbance rejection control method of a nonlinear magnetic levitation sphere system according to claim 1, wherein, Also included is step 4), i.e. clipping the control input to the limit For control input Design a saturation function, i.e. ， The control input Control is Within the interval, i.e. ， wherein is a target position of the ball (6), is a tracking error of the ball (6), , , are feedback gains for the tracking error, the estimated velocity and the estimated acceleration of the ball (6), respectively, is a bandwidth of the controller (4), and , , , are maximum and minimum values of the control input , respectively.

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