CN115453874A - Rapid reflector resonance control method based on extended state observer - Google Patents
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Abstract
The invention provides a rapid reflector resonance control method based on an extended state observer, which comprises the following steps: s1: establishing a dynamic mathematical model of the fast reflector based on the driving of the voice coil motor; s2: designing an inner-outer ring controller according to the existing model information, and giving stability conditions to realize the improvement of damping and bandwidth; s3: and designing an interference observer aiming at the inner loop controller, and deducing the conditions required to be met when the interference observer is stable by combining the existing model. The invention inhibits the influence of multi-source interference and load change and improves the robustness of the system.
Description
Technical Field
The invention belongs to the field of precision tracking and precision motion control, and particularly relates to a fast reflector resonance control method based on an extended state observer.
Background
The fast reflector is a precise tracking part for controlling light beam between the receiver and the light source via the mirror surface and consists of support, mirror body, driver and other structures. The fast reflector has the characteristics of fast response and small lag, can make up the defects of a coarse tracking system in a composite axis system, is widely applied to optical-mechanical detection equipment, and plays an important role in the fields of laser communication, situation perception, reconnaissance and confrontation and the like. The existing fast reflecting mirror driving is generally divided into a voice coil motor and piezoelectric ceramics, the voice coil motor has small hysteresis and low driving voltage, and the fast reflecting mirror driving device is widely applied.
Fast mirrors must have ultra-high pointing accuracy and high control bandwidth to meet increasingly complex application scenarios. On one hand, the fast reflector is mainly applied to a space environment, and the fast reflector is influenced by multi-source interference such as space environment moment, self modeling error, flexible structure vibration and the like, so that the pointing accuracy of a visual axis is reduced; on the other hand, the fast reflector driven by the voice coil motor faces the problem of too low resonant frequency, the improvement of the closed-loop control bandwidth is inhibited, and the improvement of the dynamic performance of the whole system is limited. The control system is used as a key for connecting the driving part, the sensing part and the mechanical part of the quick reflector, and the design of the control system directly influences the pointing accuracy and the dynamic characteristic of the quick reflector. The traditional PID control method is difficult to improve the closed-loop bandwidth of the system when facing to a controlled object with weak damping, and is difficult to ensure the control accuracy of the closed-loop system when facing to multi-source interference.
Disclosure of Invention
Due to the problem of weak damping of a controlled object caused by a driver and a mechanical structure, the closed loop bandwidth of a control system is difficult to promote; in addition, due to the multi-source interference in the space environment, the fast reflector is difficult to realize precise beam pointing. In order to solve the technical problem, the invention provides a fast reflector resonance control method based on an extended state observer, which comprises the steps of firstly, designing an expected system characteristic polynomial by combining the existing model information of an actual system and the expected characteristics of a controlled object model; thereafter, deriving a damping controller in the inner loop feedback path according to the desired characteristic polynomial; and then, designing the extended state observer according to the expected characteristic polynomial, and embedding the expected dynamic characteristics of the controlled object into the extended state observer, so that the influence of multi-source interference and load change is inhibited, and the robustness of the system is improved.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a fast reflector resonance control method based on an extended state observer comprises the following steps:
s1: establishing a dynamic mathematical model of the fast reflector based on the driving of the voice coil motor;
s2: designing an inner-outer ring controller according to the existing model information, and giving stability conditions to realize the improvement of damping and bandwidth;
s3: and designing a disturbance observer aiming at the inner loop controller, and deducing the condition required to be met when the disturbance observer is stable by combining the existing model.
Specifically, 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:
wherein J is the moment of inertia of the load and flexible support rotating structure, T is the moment applied to the shaft, θ is the deflection angle, c m To support equivalent damping with the voice coil motor, m c For supporting the equivalent mass of the voice coil motor, l is the distance from the driver to the center of the support, k is the stiffness in the axial direction,
as to the yaw angular velocity,
is the yaw angular acceleration.
The voltage balance equation of the voice coil motor is as follows:
wherein, L is armature inductance, R is resistance, U is input voltage, E is counter electromotive force, and i is current.
And the torque output equation of the voice coil motor is as follows:
T＝k t i
wherein k is t Is the torque coefficient, k, of the voice coil motor e Is the back electromotive force coefficient.
And (3) simultaneously establishing the above formula and performing Laplace transform to obtain a dynamic mathematical model of the fast reflector system:
where s is a laplacian operator, θ(s) is a laplacian transform of the deflection angle, and U(s) is a laplacian transform of the input voltage. Neglecting the inductance and back electromotive force coefficient of the fast reflector, the kinetic mathematical model is simplified as follows:
therein, ζ nd Is the target damping ratio, ω nd For undamped natural frequencies, σ is the transfer function gain.
Further, the inner and outer ring controller of step S2 is specifically designed as follows:
the inner ring controller is designed in the form that:
therein, ζ c To control the damping ratio, omega c For the natural frequency of the controller, gamma 1 、Γ 2 Respectively velocity signal and displacementThe feedback gain of the signal.
The inner loop controller is embedded in the whole system in a positive feedback mode, and the outer loop controller controls the output u t The transfer function to the controlled object output y is expressed as:
mixing the above G ty And(s) substituting and sorting the various items in the(s) to obtain a characteristic polynomial in the following form:
according to the relation between the equation root and the coefficient, the form of a characteristic polynomial which meets the expected dynamic characteristic is derived by using the expected pole position is as follows:
Q(s)＝s 4 +K 1 s 3 +K 2 s 2 +K 3 s+K 4
wherein, K 1 ,K 2 ,K 3 ,K 4 Respectively representing coefficients of a cubic term, a quadratic term, a primary term and a constant term in the desired characteristic polynomial Q(s). The controller damping ratio ζ can be solved by using the correspondence relationship of the following coefficients c And the natural frequency omega of the controller c Parameter 2 ζ of composition c ω c 、ω c 2 And feedback gain gamma of the velocity signal and the displacement signal 1 、Γ 2 And then determines the inner loop controller.
2ζ nd ω nd +2ζ c ω c ＝K 1
Hereafter, an outer loop controller is designed, which can be expressed as:
wherein k is p 、k i Respectively representing the proportional gain and the integral gain of the outer loop tracking controller, and combining the outer loop controller to deduce the characteristic root of the whole closed loop system to be the zero point of the following expression:
judging k by utilizing Route-Hurwitz stability criterion p And k is i The controller gain with the largest bandwidth is obtained through optimization on the basis of the stability constraint.
Further, the disturbance observer of step S3 is specifically designed as follows:
the observer embedding the model information is represented as:
wherein u is an actual control amount,
in order to observe the state of the system by the disturbance observer,
as an observation of interference information,/ 1 、l 2 、l 3 Is a gain parameter;
after laplace transformation, we obtain:
the characteristic equation of the disturbance observer is derived based on the formula as follows:
gain parameter l of observer 1 、l 2 、l 3 The determination method of (2) is selected according to the following formula:
2ζ nd ω nd +l 1 ＝3ω od
wherein, ω is od Is the observer bandwidth;
the observations of interference are expressed as:
wherein Q is F (s) is a low-pass filter,
the inverse of the system identification model can be expressed in the following form:
the actual control amount u is further expressed as:
deducing the stability condition of the disturbance observer based on the small gain theorem to ensure that the disturbance observer can be effective, and deducing the following formula based on the small gain theorem through the observation of the disturbance:
wherein j is an imaginary unit, ω is a frequency, and G(s) represents model information of an actual controlled object, and the model information and the identification model G N The relationship of(s) is represented by the following formula:
G(s)＝G N (s)(1+Δ(s))
wherein, Δ(s) represents multiplicative uncertainty between the actual controlled object model and the identified model;
the stability condition of the disturbance observer is obtained as follows:
||Δ(jω)||||Q F (jω)||＜1
by solving for the maximum observer bandwidth ω satisfying the above equation od Ensuring the stability of the observer and according to the gain parameter l of the observer 1 、l 2 、l 3 The expression of (2) obtains its value.
The invention has the beneficial effects that:
1. the method can realize the ultra-precise servo control of a weak damping system represented by a voice coil motor-driven quick reflector, further inhibit the system oscillation caused by weak damping through the damping controller, and improve the control performance of the system and the service life of a controlled object.
2. Compared with the traditional PID control, the method can not only solve the problem of mechanical resonance caused by weak damping, but also realize the observation of the system state, external interference and the change of the controlled object model by introducing the interference observer embedded with the existing model information of the controlled object, and can effectively inhibit the reduction of the control performance caused by the change of the controlled object model and the external interference.
3. The invention adopts a mode of combining resonance control and a correction disturbance observer, is more suitable for the practical application condition of a weak damping system represented by a fast reflector driven by a voice coil motor, and has stronger anti-jamming capability and robustness.
Drawings
FIG. 1 is a flow chart of the resonance control of the fast reflector based on the extended state observer of the invention.
Fig. 2 is a schematic diagram of the fast reflector structure of the present invention.
Fig. 3 is a block diagram of the overall control of the present invention.
Fig. 4 is a schematic diagram of the relationship between the disturbance compensation and the control quantity and the system output.
FIG. 5 is a diagram of the desired position signal θ r A tracking effect plot with the system angular position output theta.
FIG. 6 shows the desired position signal θ after the perturbation of the generation parameters of the fast reflector r And system angular displacement output theta 2 The tracking effect map of (1).
FIG. 7 is the angular displacement output θ of the original system and the angular displacement output θ of the system after the parameter perturbation 2 A plot of the effect of tracking on the same desired position signal.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
As shown in fig. 1, the present invention provides a fast mirror resonance control method based on an extended state observer, which includes the following steps:
step S1: and establishing a dynamic mathematical model of the fast reflector based on the driving of the voice coil motor.
As shown in fig. 2, the schematic diagram of the fast reflector structure according to the present invention can obtain the mechanical balance equation as follows:
wherein J is the moment of inertia of the load and flexible support rotating structure, T is the moment applied to the shaft, θ is the deflection angle, c m To support equivalent damping with the voice coil motor, m c For supporting the equivalent mass of the voice coil motor, l is the distance from the driver to the center of the support, k is the stiffness in the axial direction,
as to the yaw angular velocity,
is the yaw angular acceleration.
Meanwhile, the voltage balance equation of the voice coil motor is as follows:
wherein, L is armature inductance, R is resistance, U is power voltage, E is counter electromotive force, and i is current.
And the torque output equation of the voice coil motor is as follows:
T＝k t i
wherein k is t Is the torque coefficient, k, of the voice coil motor e Is the back electromotive force coefficient.
And (3) combining the three formulas and performing Laplace transform to obtain a dynamic model of the fast reflector system:
where s is a laplacian operator, θ(s) is a laplacian transform of the deflection angle, and U(s) is a laplacian transform of the input voltage. And because L is very small, the inductance and the back electromotive force coefficient of the quick reflector are very small and are often ignored, so the mathematical model of the system can be simplified into the following mathematical model, namely, model information is obtained through system identification.
Therein, ζ nd Is the target damping ratio, ω nd For undamped natural frequencies, σ is the transfer function gain. In the invention, selection is carried out
Step S2: and on the basis of the dynamic mathematical model established in the step S1, designing an inner-outer ring controller according to the existing model information, and giving a stability condition to realize the improvement of damping and bandwidth. The specific design steps are as follows:
the inner ring controller is designed in the form that:
therein, ζ c To control the damping ratio, omega c For controlling the natural frequency of the device, F 1 、Γ 2 The feedback gains of the velocity signal and the displacement signal, respectively.
The inner loop controller is embedded in the whole system in a positive feedback mode, and the outer loop controller controls the output u t The transfer function to controlled object output y can be expressed as:
for parameter selection, the object model G can be used N And(s) confirming the position of the pole, and shifting the compound pole of the damping device to the negative real axis direction by 2000 units in the invention to ensure that the system has enough damping so as to inhibit oscillation. And the above G ty The characteristic polynomial of(s) can be written in the form:
from the polynomial root and coefficient relationships, in combination with the desired pole positions, a characteristic polynomial of the form:
Q(s)＝s 4 +K 1 s 3 +K 2 s 2 +K 3 s+K 4
wherein, K 1 ,K 2 ,K 3 ,K 4 Respectively representing coefficients of a cubic term, a quadratic term, a primary term and a constant term in the desired characteristic polynomial Q(s). The controller damping ratio ζ can be solved by using the correspondence relationship of the following coefficients c And the natural frequency omega of the controller c Parameter 2 ζ of composition c ω c 、ω c 2 And feedback gain gamma of the velocity signal and the displacement signal 1 、Γ 2 And thus the inner loop controller. According to the pole selection condition in the invention, the transfer function of the inner ring controller can be solved as follows:
2ζ nd ω nd +2ζ c ω c ＝K 1
and designing an outer loop controller to reduce the steady-state error in a proportional-integral control mode. The outer loop controller can be represented as:
wherein k is p 、k i Respectively representing the proportional gain and the integral gain of the outer loop tracking controller;
the characteristic root of the whole closed-loop system can be deduced to be the zero point of the following expression by combining the outer-loop controller:
k can be judged by utilizing the Router-Hurwitz stability criterion p And k is i The controller gain with the largest bandwidth can be obtained through optimization on the basis of the stability constraint. In the invention, k is selected p ＝10，k p ＝600。
And step S3: and (3) designing a disturbance observer aiming at the inner loop controller in the step (S2), and deducing a condition to be met when the disturbance observer is stable by combining an existing model. The specific design steps are as follows:
the disturbance observer embedded with the model information can be expressed as:
wherein u is an actual control amount,
in order to observe the state of the system by the disturbance observer,
as an observation of interference information,/ 1 、l 2 、l 3 Is a gain parameter.
After laplace transformation, we can get:
the characteristic equation of the observer can be derived based on the above equation:
l 1 、l 2 、l 3 the determination method of the iso-observer parameters can be selected according to the following formula:
2ζ nd ω nd +l 1 ＝3ω od
wherein, ω is od To observer bandwidth, a further determination is needed. As shown in the overall control block diagram of fig. 3 and the relationship between the disturbance compensation and the control quantity and the system output of fig. 4, the relationship between the disturbance observation value and the control quantity and the system output can be expressed as follows:
wherein Q F (s) is a low-pass filter,
is the inverse number of the system identification model, and can be specifically expressed asThe following forms:
the actual control amount u can be further expressed as:
the stability condition of the disturbance observer is deduced based on the small gain theorem, the disturbance observer can be ensured to be effective, and the following formula can be deduced based on the small gain theorem through the observation of the disturbance:
where j is an imaginary unit, ω is a frequency, and G(s) represents model information of an actual controlled object, which is often changed due to a change in its structure or a change in load, and the identification model G N The relationship of(s) can be represented by the following formula:
G(s)＝G N (s)(1+Δ(s))
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 found as follows:
||Δ(jω)||||Q F (jω)||＜1
therefore, the maximum observer bandwidth ω satisfying the above equation can be solved od And the stability of the observer is ensured. And according to l 1 、l 2 、l 3 The expression of the observer gain is equal to obtain its value.
The invention mainly aims at solving the problem of parameter perturbation of a controlled object caused by load change in actual use and aims at solving the problem caused by load changeIs selected from ω od =948, solve according to observer parameter corresponding relation 1 ＝2795.37,l 2 ＝-7859826.84,l 3 ＝851971392.
The simulation experiment of the invention comprises the following steps:
(1) Simulation setting:
in this example, the simulation step size is set to 0.0001s (i.e. the sampling frequency is 10 kHz), and the total simulation duration is 0.12s; the fast reflector system is identified by
Damping ratio zeta of controller in inner ring controller c And the natural frequency omega of the controller c The parameters of the composition are as follows: 2 ζ c ω c ＝8048.63、ω c 2 =34614520, and the feedback gain of the velocity signal and the displacement signal is gamma 1 ＝-1580.25、Γ 2 If =7404072.12, the transfer function of the inner loop controller is
On the basis, the outer ring controller parameter is selected to be k p ＝10，k i =600; then selecting the bandwidth omega of the observer according to the possible parameter perturbation range of the controlled object od =948, the calculated gain parameters corresponding to the observer are respectively: l 1 ＝2795.37,l 2 ＝-7859826.84,l 3 =851971392, and the actual controlled object is
(2) Triangular wave reference signal tracking:
first, a triangular wave signal having an amplitude of 2 μ rad and a frequency of 20Hz was inputted as a track signal to be tracked. And carrying out simulation to obtain the tracking effect of the fast reflector under the condition. FIG. 5 is a diagram of the desired position signal θ r A tracking effect plot with the system angular position output theta. FIG. 6 is a diagram of a parameter perturbation post-expectation position signal theta of the fast reflector r Output theta of angular displacement with system 2 Is trackedEffect diagram, relative error is not more than 1.8%. FIG. 7 shows the angular displacement output θ of the original system and the angular displacement output θ of the system after the perturbation of the parameters 2 A plot of the effect of tracking on the same desired position signal.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (4)
1. A fast reflector resonance control method based on an extended state observer is characterized by comprising the following steps:
s1: establishing a dynamic mathematical model of the fast reflector based on the driving of the voice coil motor;
s2: designing an inner-outer ring controller according to the existing model information, and giving stability conditions to realize the improvement of damping and bandwidth;
s3: and designing a disturbance observer aiming at the inner loop controller, and deducing the condition required to be met when the disturbance observer is stable by combining the existing model.
2. The extended state observer-based fast mirror resonance control method according to claim 1, wherein the specific modeling process of the step S1 is as follows:
the mechanical balance equation for a fast mirror using a voice coil motor is:
wherein J is the moment of inertia of the load and flexible support rotating structure, T is the moment applied to the shaft, θ is the deflection angle, c m To support equivalent damping with the voice coil motor, m c In order to support the equivalent mass of the voice coil motor, l is the distance from the driver to the center of the support, and k is the rigidity in the axial direction;
as to the yaw angular velocity,
is the yaw angular acceleration;
the voltage balance equation of the voice coil motor is as follows:
wherein, L is armature inductance, R is resistance, U is input voltage, E is counter electromotive force, and i is current;
and the torque output equation of the voice coil motor is as follows:
T＝k t i
wherein k is t Is the torque coefficient, k, of the voice coil motor e Is the back electromotive force coefficient;
and (3) establishing the above formula and performing Laplace transformation to obtain a dynamic mathematical model of the fast reflector:
wherein s is a Laplace operator, theta(s) is Laplace transformation of a deflection angle, and U(s) is Laplace transformation of input voltage; neglecting the inductance and back emf coefficients of the fast mirror, the mathematical model of dynamics reduces to:
therein, ζ nd Is the target damping ratio, ω nd For undamped natural frequencies, σ is the transfer function gain.
3. The extended state observer-based fast mirror resonance control method according to claim 2, wherein the inner and outer loop controllers of step S2 are specifically designed as follows:
the inner ring controller is designed in the form that:
therein, ζ c To control the damping ratio, omega c For the natural frequency of the controller, gamma 1 、Γ 2 Feedback gains for the velocity signal and the displacement signal, respectively;
the inner-loop controller is embedded in the whole system in a positive feedback mode, and the outer-loop controller controls the output u t The transfer function to the controlled object output y is expressed as:
mixing the above G ty And(s) substituting and sorting the various items in the(s) to obtain a characteristic polynomial of the(s) in the following form:
according to the relation between the equation root and the coefficient, the form of a characteristic polynomial which meets the expected dynamic characteristic is derived by using the expected pole position is as follows:
Q(s)＝s 4 +K 1 s 3 +K 2 s 2 +K 3 s+K 4
wherein, K 1 ,K 2 ,K 3 ,K 4 Coefficients respectively representing a cubic term, a quadratic term, a primary term and a constant term in the desired characteristic polynomial Q(s); the following linear equation set is obtained by utilizing the corresponding relation of the coefficients, and the linear equation set is obtained by solvingDamping ratio ζ of controller c And the natural frequency omega of the controller c Parameter 2 ζ of composition c ω c 、ω c 2 And feedback gain gamma of the velocity signal and the displacement signal 1 、Γ 2 And then determines the inner loop controller.
2ζ nd ω nd +2ζ c ω c ＝K 1
Hereafter, an outer ring controller is designed, and the outer ring controller is expressed as:
wherein k is p 、k i Respectively representing the proportional gain and the integral gain of the outer loop tracking controller, and combining the outer loop controller to deduce the characteristic root of the whole closed loop system to be the zero point of the following expression:
k is judged by utilizing a Router-Hurwitz stability criterion p And k is i The controller gain with the largest bandwidth is obtained through optimization on the basis of the stability constraint of the controller.
4. The extended state observer-based fast mirror resonance control method according to claim 3, wherein the disturbance observer of step S3 is specifically designed as follows:
the observer embedding the model information is represented as:
after laplace transformation, we obtain:
the characteristic equation of the disturbance observer is derived based on the formula as follows:
s 3 +(2ζ nd ω nd +l 1 )s 2 +(2ζ nd ω nd l 1 +ω n 2 d +l 2 )s+l 3 ＝0
gain parameter l of observer 1 、l 2 、l 3 The determination method of (2) is selected according to the following formula:
2ζ nd ω nd +l 1 ＝3ω od
wherein, ω is od Is the observer bandwidth;
the observations on interference are expressed as:
wherein Q is F (s) is a low-pass filter,
is the reciprocal of the system identification model, and is specifically expressed in the following form:
the actual control amount u is further expressed as:
deducing the stability condition of the disturbance observer based on the small gain theorem to ensure that the disturbance observer can be effective, and deducing the following formula based on the small gain theorem through the observation of the disturbance:
wherein j is an imaginary unit, ω is a frequency, and G(s) represents model information of an actual controlled object, which is associated with the identification model G N The relationship of(s) is represented by the following formula:
G(s)＝G N (s)(1+Δ(s))
wherein, Δ(s) represents multiplicative uncertainty between the actual controlled object model and the identified model;
the stability condition of the disturbance observer is obtained as follows:
||Δ(jω)|| ||Q F (jω)||＜1
by solving for the maximum observer bandwidth ω satisfying the above equation od Ensuring the stability of the observer and according to the gain parameter l of the observer 1 、l 2 、l 3 The expression of (c) obtains its value.
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