A fast mirror resonance control method based on extended state observer

Abstract

The present invention provides a fast mirror resonance control method based on an extended state observer, comprising the following steps: S1: establishing a dynamic mathematical model of a fast mirror driven by a voice coil motor; S2: designing inner and outer loop controllers based on existing model information, and providing stability conditions to achieve the improvement of damping and bandwidth; S3: designing an interference observer for the inner loop controller, and combining the existing model to derive the conditions that need to be met when the interference observer is stable. The present invention suppresses the influence of multi-source interference and load changes, and improves the robustness of the system.

Description

Technical Field

[0001] The invention belongs to the field of precision tracking and precision motion control, and in particular relates to a fast mirror resonance control method based on an extended state observer. Background Art

[0002] A fast reflector is a precision tracking component that controls the light beam between the receiver and the light source through a mirror, and is composed of a support frame, a mirror body, and a driver. Fast reflectors have the characteristics of fast response and small lag, which can make up for the shortcomings of the coarse tracking system in the composite axis system. Therefore, they are widely used in optical detection equipment and play an important role in laser communication, situational awareness, and reconnaissance and confrontation. The existing fast reflector drives are generally divided into two types: voice coil motors and piezoelectric ceramics. Voice coil motors have small lag and low driving voltage, and have been widely used. The present invention mainly focuses on fast reflectors driven by voice coil motors.

[0003] Fast reflectors must have ultra-high pointing accuracy and high control bandwidth to meet increasingly complex application scenarios. On the one hand, fast reflectors are mainly used in space environments. Fast reflectors will be affected by multiple sources of interference such as space environment torque, self-modeling errors, and flexible structure vibrations, which will reduce the pointing accuracy of the line of sight. On the other hand, the fast reflector driven by the voice coil motor faces the problem of too low resonant frequency, which inhibits the improvement of the closed-loop control bandwidth and limits the improvement of the overall system dynamic performance. As the key to connecting the fast reflector drive, sensor and mechanical parts, the control system directly affects the pointing accuracy and dynamic characteristics of the fast reflector. However, the traditional PID control method has difficulty in improving the closed-loop bandwidth of the system when facing a weakly damped controlled object, and it is difficult to ensure the control accuracy of the closed-loop system when facing multiple sources of interference. Summary of the invention

[0004] Due to the weak damping problem of the controlled object caused by the driver and the mechanical structure, the closed-loop bandwidth of the control system is difficult to improve; in addition, due to the existence of multi-source interference in the space environment, it is difficult for the fast reflector to achieve precise beam pointing. In order to solve the above technical problems, the present invention proposes a fast reflector resonance control method based on an extended state observer. First, the desired system characteristic polynomial is designed in combination with the existing model information of the actual system and the desired controlled object model characteristics; thereafter, the damping controller in the inner loop feedback path is derived according to the desired characteristic polynomial; then, the extended state observer is designed according to the desired characteristic polynomial, and the desired dynamic characteristics of the controlled object are embedded into the extended state observer, thereby suppressing the influence of multi-source interference and load changes and improving the robustness of the system.

[0005] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A fast mirror resonance control method based on an extended state observer comprises the following steps:

[0007] S1: Establish the dynamic mathematical model of the fast reflector driven by voice coil motor;

[0008] S2: Design inner and outer loop controllers based on the existing model information, and give stability conditions to improve damping and bandwidth;

[0009] S3: Design a disturbance observer for the inner loop controller and derive the conditions that must be met for the disturbance observer to be stable based on the existing model.

[0010] Specifically, the specific modeling process of step S1 is as follows:

[0011] The mechanical balance equation of a fast reflector using a voice coil motor is:

[0012]

[0013] Where J is the moment of inertia of the load and the flexible support rotating structure, T is the torque applied to the shaft, θ is the deflection angle, c m is the equivalent damping of the support and voice coil motor, m c is the equivalent mass of the support and the voice coil motor, l is the distance from the driver to the support center, k is the axial stiffness, is the deflection angular velocity, is the deflection angular acceleration.

[0014] The voltage balance equation for the voice coil motor is:

[0015]

[0016] Among them, L is the armature inductance, R is the resistance, U is the input voltage, E is the back electromotive force, and i is the current.

[0017] And the torque output equation of the voice coil motor is:

[0018] T=k t i

[0019]

[0020] Among them, k t is the voice coil motor torque coefficient, k e is the back electromotive force coefficient.

[0021] Combining the above equations and performing Laplace transform, we get the dynamic mathematical model of the fast reflector system:

[0022]

[0023] Where s is the Laplace operator, θ(s) is the Laplace transform of the deflection angle, and U(s) is the Laplace transform of the input voltage. Ignoring the inductance and back electromotive force coefficient of the fast reflector, the dynamic mathematical model is simplified to:

[0024]

[0025] Among them, nd is the object damping ratio, ω nd is the undamped natural frequency, and σ is the transfer function gain.

[0026] Furthermore, the inner and outer loop controllers of step S2 are specifically designed as follows:

[0027] The design of the inner loop controller is as follows:

[0028]

[0029] Among them, c is the controller damping ratio, ω c is the natural frequency of the controller, Γ1 and Γ2 are the feedback gains of the velocity signal and displacement signal respectively.

[0030] The inner loop controller is embedded in the overall system in the form of positive feedback, so the outer loop controller controls the output u t The transfer function to the output y of the controlled object is expressed as:

[0031]

[0032] The above G ty After substituting each term in (s) into the formula, we can get its characteristic polynomial in the following form:

[0033]

[0034] According to the relationship between the roots and coefficients of the equation, the characteristic polynomial that satisfies the desired dynamic characteristics is derived using the desired pole position as follows:

[0035] Q(s)=s 4 +K1s 3 +K2s 2 +K3s+K4

[0036] Among them, K1, K2, K3, K4 represent the coefficients of the cubic term, quadratic term, linear term and constant term in the desired characteristic polynomial Q(s) respectively. The controller damping ratio ζ can be solved by using the corresponding relationship of the following coefficients: c and the controller natural frequency ω c The parameter 2ζ c ω c ,ω c 2As well as the feedback gains Γ1 and Γ2 of the velocity signal and displacement signal, the inner loop controller is then determined.

[0037] 2ζ nd ω nd +2ζ c ω c =K1

[0038]

[0039]

[0040]

[0041] After that, the outer loop controller is designed, which can be expressed as:

[0042]

[0043] Among them, k p , k i Represent the proportional gain and integral gain of the outer loop tracking controller respectively. Combined with the outer loop controller, the characteristic root of the entire closed-loop system is derived as the zero point of the following expression:

[0044]

[0045] Using the Routh–Hurwitz stability criterion to determine k p With k i Based on the stability constraint, the controller gain with the largest bandwidth is obtained through optimization.

[0046] Furthermore, the interference observer of step S3 is specifically designed as follows:

[0047] The observer that embeds the model information is expressed as:

[0048]

[0049] Among them, u is the actual control quantity, is the observation value of the disturbance observer on the system state, is the observed value of the interference information, l1, l2, l3 are gain parameters;

[0050] After Laplace transform, we get:

[0051]

[0052] Based on the above formula, the characteristic equation of the disturbance observer is derived as follows:

[0053]

[0054] The method for determining the observer gain parameters l1, l2, and l3 is selected according to the following formula:

[0055] 2ζ nd ω nd +l1=3ω od

[0056]

[0057]

[0058] Among them, ω od is the observer bandwidth;

[0059] The observation of interference is expressed as:

[0060]

[0061] Among them, Q F (s) is a low-pass filter, is the inverse of the system identification model, which can be specifically expressed as follows:

[0062]

[0063]

[0064] The actual control quantity u is further expressed as:

[0065]

[0066] Based on the small gain theorem, the stability condition of the disturbance observer is derived to ensure that the disturbance observer is effective. Through the above observation of the disturbance, the following formula is derived based on the small gain theorem:

[0067]

[0068] Where j is an imaginary unit, ω is the frequency, and G(s) represents the model information of the actual controlled object, which is consistent with the identification model G N The relationship between (s) is expressed as follows:

[0069] G(s)=G N (s)(1+Δ(s))

[0070] Where Δ(s) represents the multiplicative uncertainty between the actual controlled object model and the identified model;

[0071] The stability condition of the disturbance observer is obtained as follows:

[0072] ||Δ(jω)||||Q F (jω)||<1

[0073] By solving the maximum observer bandwidth ω that satisfies the above equation od , ensure the stability of the observer, and obtain the values ​​of the gain parameters l1, l2, and l3 according to the expressions of the observer.

[0074] The beneficial effects of the present invention are:

[0075] 1. The method of the present invention can realize ultra-precision servo control of a type of weak damping system represented by a voice coil motor driving a fast reflector, and further suppress the system oscillation caused by weak damping through a damping controller, thereby improving the control performance of the system and the service life of the controlled object.

[0076] 2. Compared with traditional PID control, the present invention introduces an interference observer embedded with the existing model information of the controlled object, which can not only deal with the mechanical resonance problem caused by weak damping, but also realize the observation of system status, external interference and changes in the controlled object model, and can effectively suppress the degradation of control performance caused by changes in the controlled object model and external interference.

[0077] 3. The present invention adopts a method of combining resonance control with a modified disturbance observer, which is more suitable for the actual application of a type of weak damping system represented by a fast reflector driven by a voice coil motor, and has stronger anti-interference ability and robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 This is a flow chart of the fast mirror resonance control based on the extended state observer of the present invention.

[0079] Figure 2 It is a schematic diagram of the structure of the fast reflection mirror of the present invention.

[0080] Figure 3 This is the overall control block diagram of the present invention.

[0081] Figure 4 Schematic diagram of the relationship between disturbance compensation, control quantity and system output.

[0082] Figure 5 is the expected position signal θ r Tracking effect diagram with the system angular position output θ.

[0083] Figure 6 is the expected position signal θ after the fast mirror parameter perturbation occurs r Tracking effect diagram with the system angular displacement output θ2.

[0084] Figure 7 This is a diagram showing the tracking effect of the angular displacement output θ of the original system and the angular displacement output θ2 of the system after parameter perturbation on the same expected position signal. DETAILED DESCRIPTION

[0085] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0086] like Figure 1 As shown, the present invention proposes a fast mirror resonance control method based on an extended state observer, comprising the following steps:

[0087] Step S1: Establishing a mathematical model of the dynamics of a fast reflector driven by a voice coil motor.

[0088] like Figure 2 As shown, the mechanical balance equation can be obtained according to the schematic diagram of the fast reflector structure of the present invention:

[0089]

[0090] Where J is the moment of inertia of the load and the flexible support rotating structure, T is the torque applied to the shaft, θ is the deflection angle, c m is the equivalent damping of the support and voice coil motor, m c is the equivalent mass of the support and the voice coil motor, l is the distance from the driver to the support center, k is the axial stiffness, is the deflection angular velocity, is the deflection angular acceleration.

[0091] At the same time, the voltage balance equation of the voice coil motor is:

[0092]

[0093] Among them, L is the armature inductance, R is the resistance, U is the power supply voltage, E is the back electromotive force, and i is the current.

[0094] And the torque output equation of the voice coil motor is:

[0095] T=k t i

[0096]

[0097] Among them, k t is the voice coil motor torque coefficient, k e is the back electromotive force coefficient.

[0098] Combining the above three equations and performing Laplace transform, we can get the dynamic model of the fast mirror system:

[0099]

[0100] Among them, s is the Laplace operator, θ(s) is the Laplace transform of the deflection angle, and U(s) is the Laplace transform of the input voltage. Since L is very small, the inductance and back electromotive force coefficient of the fast reflector are often very small and often ignored. Therefore, the mathematical model of the system can be simplified to the following mathematical model, that is, the model information is obtained through system identification.

[0101]

[0102] Among them, nd is the object damping ratio, ω nd is the undamped natural frequency, and σ is the transfer function gain.

[0103] Step S2: Based on the dynamic mathematical model established in step S1, design the inner and outer loop controllers according to the existing model information, and give stability conditions to achieve the improvement of damping and bandwidth. The specific design steps are as follows:

[0104] The design of the inner loop controller is as follows:

[0105]

[0106] Among them, c is the controller damping ratio, ω c is the natural frequency of the controller, Γ1 and Γ2 are the feedback gains of the velocity signal and displacement signal respectively.

[0107] The inner loop controller is embedded in the overall system in the form of positive feedback, so the outer loop controller controls the output u t The transfer function to the controlled object output y can be expressed as:

[0108]

[0109] For parameter selection, we can use the object model G N The position of the pole of (s) is confirmed. In the present invention, its complex pole is translated 2000 units toward the negative real axis to ensure that the system has sufficient damping to suppress oscillation. ty The characteristic polynomial of (s) can be written in the following form:

[0110]

[0111] According to the relationship between the polynomial roots and coefficients, combined with the expected pole positions, the characteristic polynomial can be derived as follows:

[0112] Q(s)=s 4+K1s 3 +K2s 2 +K3s+K4

[0113] Among them, K1, K2, K3, K4 represent the coefficients of the cubic term, quadratic term, linear term and constant term in the desired characteristic polynomial Q(s) respectively. The controller damping ratio ζ can be solved by using the corresponding relationship of the following coefficients: c and the controller natural frequency ω c The parameter 2ζ c ω c ,ω c 2 And the feedback gains Γ1 and Γ2 of the velocity signal and the displacement signal, and then determine the inner loop controller. According to the pole selection in the present invention, the transfer function of the inner loop controller can be solved as follows:

[0114] 2ζ nd ω nd +2ζ c ω c =K1

[0115]

[0116]

[0117]

[0118] After that, the outer loop controller is designed, using the form of proportional-integral control to reduce the steady-state error. The outer loop controller can be expressed as:

[0119]

[0120] Among them, k p , k i Represent the proportional gain and integral gain of the outer loop tracking controller respectively;

[0121] Combined with the outer loop controller, the characteristic root of the entire closed-loop system can be derived as the zero of the following expression:

[0122]

[0123] The Routh–Hurwitz stability criterion can be used to determine k p With k i The stability constraint of k can be obtained by optimizing the controller gain with the largest bandwidth. p =10, k p =600.

[0124] Step S3: Design a disturbance observer for the inner loop controller in step S2, and derive the conditions that must be met when the disturbance observer is stable by combining the existing model. The specific design steps are as follows:

[0125] The disturbance observer embedded with model information can be expressed as:

[0126]

[0127] Among them, u is the actual control quantity, is the observation value of the disturbance observer on the system state, is the observed value of the interference information, l1, l2, l3 are gain parameters.

[0128] After Laplace transform, we can get:

[0129]

[0130] Based on the above formula, the characteristic equation of the observer can be derived as:

[0131]

[0132] The method for determining observer parameters such as l1, l2, and l3 can be selected according to the following formula:

[0133] 2ζ nd ω nd +l1=3ω od

[0134]

[0135]

[0136] Among them, ω od is the observer bandwidth, which needs to be further determined. Figure 3 The overall control block diagram and Figure 4 The relationship between the disturbance compensation and the control quantity and the system output is shown in the schematic diagram. The relationship between the disturbance observation value and the control quantity and the system output can be expressed as:

[0137]

[0138] Among them, Q F (s) is a low-pass filter, is the inverse of the system identification model, which can be specifically expressed as follows:

[0139]

[0140]

[0141] The actual control quantity u can be further expressed as:

[0142]

[0143] Based on the small gain theorem, the stability condition of the disturbance observer is derived to ensure that the disturbance observer is effective. Through the above observation of the disturbance, the following formula can be derived based on the small gain theorem:

[0144]

[0145] Where j is an imaginary unit, ω is the frequency, and G(s) represents the model information of the actual controlled object, which often changes due to changes in its own structure or load. N The relationship between (s) can be expressed as follows:

[0146] G(s)=G N (s)(1+Δ(s))

[0147] Where Δ(s) represents the multiplicative uncertainty between the actual controlled object model and the identified model. The stability condition of the disturbance observer can be obtained as:

[0148] ||Δ(jω)||||Q F (jω)||<1

[0149] Therefore, the maximum observer bandwidth ω that satisfies the above equation can be obtained by solving od , to ensure the stability of the observer. And obtain the values ​​of observer gains such as l1, l2, and l3 according to their expressions.

[0150] The present invention mainly aims at the problem of parameter perturbation of the controlled object caused by load change in actual use. In view of the influence of load change, ω is selected od =948, according to the corresponding relationship of observer parameters, we can get l1=2795.37, l2=-7859826.84, l3=851971392.

[0151] The simulation experiment of the present invention comprises the following steps:

[0152] (1) Simulation settings:

[0153] In this example, the simulation step is set to 0.0001s (i.e., the sampling frequency is 10kHz), and the total simulation time is 0.12s; the fast mirror system identification model is The controller damping ratio ζ in the inner loop controller c and the controller natural frequency ω c The parameters of the composition are: 2ζ c ω c =8048.63,ωc 2 =34614520, the feedback gains of the velocity signal and the displacement signal are Γ1=-1580.25, Γ2=7404072.12, then the transfer function of the inner loop controller is: On this basis, the outer loop controller parameters are selected as k p =10, k i =600; then the observer bandwidth ω is selected according to the possible parameter perturbation range of the controlled object. od =948, the corresponding gain parameters of the observer are calculated as follows: l1 = 2795.37, l2 = -7859826.84, l3 = 851971392. When the parameter perturbation occurs later, the actual controlled object is

[0154] (2) Triangle wave reference signal tracking:

[0155] First, a triangle wave signal with an amplitude of 2 μrad and a frequency of 20 Hz is input as the simulated tracking trajectory signal. Simulation is carried out to obtain the tracking effect of the fast reflector under this condition. Figure 5 is the expected position signal θ r Tracking effect diagram with the system angular position output θ. Figure 6 is the expected position signal θ after the fast mirror parameter perturbation occurs r The tracking effect diagram of the system angular displacement output θ2 shows that the relative error does not exceed 1.8%. Figure 7 This is a diagram showing the tracking effect of the angular displacement output θ of the original system and the angular displacement output θ2 of the system after parameter perturbation on the same expected position signal.

[0156] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A fast mirror resonance control method based on an extended state observer, characterized in that: The following steps are involved: S1: Establish the dynamic mathematical model of the fast reflector driven by voice coil motor; S2: Design inner and outer loop controllers based on the existing model information, and give stability conditions to improve damping and bandwidth; The inner and outer loop controllers are specifically designed as follows: Design inner loop controller The form is: ； in, is the controller damping ratio, is the natural frequency of the controller, , are the feedback gains of velocity signal and displacement signal respectively; s is the Laplace operator; The inner loop controller is embedded in the overall system in the form of positive feedback, so the outer loop controller controls the output u t Transfer function to the controlled object output y It is expressed as: ； in, To identify the model; The above G ty After substituting each term in (s) into the formula, we can obtain its characteristic polynomial In the following form: ； in, is the object damping ratio, is the undamped natural frequency, is the transfer function gain; According to the relationship between the equation roots and coefficients, the desired characteristic polynomial that satisfies the desired dynamic characteristics is derived using the desired pole position. The form is as follows: ； Among them, K1, K2, K3, K4 represent the coefficients of the cubic term, quadratic term, linear term and constant term in the desired characteristic polynomial Q(s) respectively; the corresponding relationship of the coefficients is used to obtain the following linear equations, and the controller damping ratio ζ is obtained by solving c and the controller natural frequency ω c The parameter 2ζ c ω c ,ω c 2 And the feedback gains Γ1 and Γ2 of the velocity signal and the displacement signal, and then determine the inner loop controller; ； Then design the outer loop controller. It is expressed as: ； Among them, k p , k i Represent the proportional gain and integral gain of the outer loop tracking controller respectively. Combined with the outer loop controller, the characteristic root of the entire closed-loop system is derived as the zero point of the following expression: ； Using the Routh–Hurwitz stability criterion to determine k p With k i The stability constraint is conditioned on this basis, and the controller gain with the largest bandwidth is obtained through optimization. S3: Design a disturbance observer for the inner loop controller and derive the conditions that must be met for the disturbance observer to be stable based on the existing model; Based on the small gain theorem, the stability condition of the disturbance observer is derived to ensure that the disturbance observer is effective. Through the above observation of the disturbance, the following formula is derived based on the small gain theorem: ； Where j is an imaginary unit, ω is the frequency, and G(s) represents the model information of the actual controlled object, which is consistent with the identification model G N The relationship between (s) is expressed as follows: ； Where Δ(s) represents the multiplicative uncertainty between the actual controlled object model and the identified model; The stability condition of the disturbance observer is obtained as follows: ； By solving the maximum observer bandwidth ω that satisfies the above equation od , ensure the stability of the observer, and obtain the values ​​of the gain parameters l1, l2, and l3 according to the expressions of the observer.

2. The fast mirror resonance control method based on extended state observer according to claim 1, characterized in that: The specific modeling process of the S1 step is as follows: The mechanical balance equation of a fast reflector using a voice coil motor is: ； in, is the moment of inertia of the load and the flexibly supported rotating structure, is the torque applied to the shaft, is the deflection angle, To support the equivalent damping of the voice coil motor, is the equivalent mass of the support and voice coil motor, is the distance from the drive to the support center, is the stiffness in the axial direction; is the deflection angular velocity, is the deflection angular acceleration; The voltage balance equation for the voice coil motor is: ； in, is the armature inductance, is the resistance, is the input voltage, is the back electromotive force, is the current; And the torque output equation of the voice coil motor is: ； in, is the voice coil motor torque coefficient, is the back electromotive force coefficient; Combining the above equations and performing Laplace transform, we can get the dynamic mathematical model of the fast reflector: ； Where s is the Laplace operator, is the Laplace transform of the deflection angle, is the Laplace transform of the input voltage; ignoring the inductance and back electromotive force coefficient of the fast reflector, the dynamic mathematical model is simplified to: ； in, is the object damping ratio, is the undamped natural frequency, is the transfer function gain.

3. The fast mirror resonance control method based on extended state observer according to claim 2, characterized in that: The interference observer in step S3 is specifically designed as follows: The observer that embeds the model information is expressed as: ； , is the observation value of the disturbance observer on the system state, is the observed value of the interference information, , , is the gain parameter; After Laplace transform, we get: ； Based on the above formula, the characteristic equation of the disturbance observer is derived as follows: ； The method for determining the observer gain parameters l1, l2, and l3 is selected according to the following formula: ； Among them, ω od is the observer bandwidth; The observation of interference is expressed as: ； Among them, Q F (s) is a low-pass filter, is the inverse of the system identification model, which is specifically expressed in the following form: ； ； The actual control quantity u is further expressed as: 。

Images