阅读说明：本技术 基于扰动观测器的车载光电跟踪平台有限时间控制方法 (Finite time control method of vehicle-mounted photoelectric tracking platform based on disturbance observer ) 是由 朱海荣 吴瑜 张先进 李俊红 茅靖峰 陈益飞 张英聪 马文静 张慧 张怀才 于 2019-08-23 设计创作，主要内容包括：本发明公开了一种基于扰动观测器的车载光电跟踪平台有限时间控制方法。该方法将扰动观测器和有限时间控制器有机结合,两者互补,用于对光电跟踪平台进行控制。在光电跟踪平台数学模型的基础上,设计扰动观测器来观测光电跟踪平台所受到的扰动,将其作为前馈量进行补偿。随后将给定信号与实际信号进行比较,得到包含等效扰动项的一阶误差模型,在该模型的基础上运用有限时间控制的方法设计反馈控制率。通过数学分析,给出在等效干扰情况下闭环系统稳态误差的界与控制器参数之间的关系。该复合控制方法使光电跟踪平台具有更好的抗扰动性能和更优越的收敛性能。(The invention discloses a finite time control method of a vehicle-mounted photoelectric tracking platform based on a disturbance observer. The method organically combines a disturbance observer and a finite time controller, and the disturbance observer and the finite time controller are complementary and are used for controlling the photoelectric tracking platform. On the basis of a mathematical model of the photoelectric tracking platform, a disturbance observer is designed to observe the disturbance on the photoelectric tracking platform and compensate the disturbance as a feedforward quantity. And then comparing the given signal with the actual signal to obtain a first-order error model containing an equivalent disturbance term, and designing a feedback control rate by using a finite time control method on the basis of the model. Through mathematical analysis, the relation between the boundary of the steady state error of the closed loop system and the controller parameter under the condition of equivalent interference is given. The composite control method enables the photoelectric tracking platform to have better disturbance resistance and superior convergence performance.)

1. A limited time control method of a vehicle-mounted photoelectric tracking platform based on a disturbance observer is characterized by comprising the following steps: (1) establishing a mathematical model of the photoelectric tracking platform, designing a disturbance observer on the basis of the mathematical model, observing various disturbances suffered by the photoelectric tracking platform, and compensating a feed forward quantity; (2) comparing the given signal with the actual signal to obtain a first-order error model containing an equivalent disturbance term; (3) a finite time control method is used for designing a feedback control rate, and the relation between the boundary of the steady-state error of the closed-loop system and the parameters of the controller under the equivalent interference condition is given through mathematical analysis, so that the photoelectric tracking platform can be effectively controlled under the influence of various interferences.

2. The method for controlling the limited time of the vehicle-mounted photoelectric tracking platform based on the disturbance observer according to claim 1, wherein the disturbance observer is used for observing the coupling influence caused by the nonlinear friction torque, the mass unbalance torque, the cable constraint, the model mismatch and the moving locomotive on the photoelectric tracking platform, and the influence of the interference on the system is reduced by compensating in a feed-forward mode.

3. the disturbance observer-based limited time control method for the vehicle-mounted photoelectric tracking platform according to claim 1, wherein a feedback control rate is designed for the photoelectric tracking platform by using the limited time control method, so that the anti-interference capability of the system is further improved, and a tracking error is converged to a smaller area.

4. The disturbance observer-based limited time control method for the vehicle-mounted photoelectric tracking platform according to claim 2, wherein in the design of the disturbance observer, a low-pass filter Q(s) is connected in series at the front end of a disturbance observation signal to solve the inverse of a system reference modelThe problem of being unable to realize;

Wherein Q(s) is a low pass filter,Is the inverse of the nominal model of the photoelectric tracking platform.

5. The disturbance observer-based finite time control method for the vehicle-mounted photoelectric tracking platform according to claim 4, wherein the disturbance observer satisfies the following expression:

wherein G is_p(s) is a real model of the photoelectric tracking platform, G_n(s) isNominal model G of photoelectric tracking platform_uy(s) is the transfer function from input signal to output signal, G_dy(s) is the transfer function of the disturbance signal to the output signal, G_ξy(s) is the transfer function of the noise signal to the output signal.

6. The disturbance observer-based vehicle-mounted photoelectric tracking platform finite time control method according to claim 5, wherein the filter Q(s) is configured in a form of Q(s) ═ g/(s + g), and when the output of the disturbance observer is:When the parameter g → ∞ of the filter is satisfied, thenNamely, the observation values of various disturbances of the photoelectric tracking platform by the disturbance observer approach the actual disturbance values, and the disturbance can be compensated through feedforward control;

Wherein the content of the first and second substances,Is the output of the disturbance observer; b is₀＝J_n/K_m，J_nIs an estimate of the inertia J of the photoelectric tracking platform, K_mIs the torque coefficient of the drive motor; s is the laplace operator; omega_l(t) is the actual angular velocity of the drive motor; i.e. i_r(t) is the output of the finite time controller; g is a parameter of the filter; d (t) is the sum of all disturbances including various external disturbances, system model mismatch and current tracking errors suffered by the photoelectric tracking platform.

7. the method for controlling the finite time of the vehicle-mounted photoelectric tracking platform based on the disturbance observer according to claim 3, wherein a finite time controller is adopted in the finite time control method, and the control quantity i of the given current is_rThe expression of (t) is:

Wherein i_r(t) is a current control amount of the driving motor; b is₀＝J_n/K_m，J_nIs an estimate of the inertia J of the photoelectric tracking platform, K_mIs the torque coefficient of the drive motor; k is a proportionality coefficient; sgn is a sign function; e (t) is tracking error, e (t) ═ ω_r(t)-ω_l(t)，ω_r(t) is a given angular velocity of the drive motor, ω_l(t) is the actual angular velocity of the drive motor; alpha is a constant, and alpha is more than 0 and less than 1;Is the output of the disturbance observer.

Technical Field

The invention relates to a finite time control method of a vehicle-mounted photoelectric tracking platform based on a disturbance observer, and belongs to the field of precise servo control systems.

background

Install the photoelectric tracking platform on the locomotive, can receive the influence of multiple interference: when the locomotive and the target position move relatively, the influence of interference such as nonlinear friction torque, mass unbalance torque, cable constraint and the like needs to be overcome, so that the visual axis of the tracking platform keeps accurate tracking on the target; when the locomotive runs on a concave-convex road surface to cause posture change, the photoelectric tracking platform needs to overcome the coupling influence caused by the motion of the locomotive additionally. A gyro is used as a sensitive element to provide a space inertial coordinate system for a photoelectric tracking platform on a locomotive. When the photoelectric tracking platform is detected not to move along a preset track, the motor drives the platform to rapidly move towards the opposite direction so as to accurately keep the attitude reference of the tracking platform, thereby effectively isolating locomotive motion coupling, overcoming the influence of various interference factors and realizing the identification and tracking of the tracking platform on a target.

compared with a land-based tracking platform, the vehicle-mounted photoelectric tracking platform has more interference factors and greater control difficulty, and how to better inhibit the adverse effect of external interference on the performance of the tracking platform system becomes a research hotspot of broad scholars. A disturbance compensation control scheme based on the combination of a LuGre friction model and a disturbance observer is provided in documents (Jinchao, Zhang Bao, Lixiantao, and the like, a photoelectric stabilized platform friction compensation strategy [ J ] based on the disturbance observer, university of Jilin, academic edition, 2017,47(6): 1876-1885), and the anti-interference capability of the aviation photoelectric stabilized platform is further improved. However, in practical application, it is generally difficult to obtain an accurate LuGre friction model of the photoelectric tracking platform, so that the compensation effect is limited. According to the research on a low-speed and stable performance control method of a servo mechanism of a photoelectric stable platform in the literature (Qiqiao, Fangshi, Xixin, and the like) [ J ] in the war industry, 2018,39(10): 1873-1882.), a state expansion Kalman filtering algorithm is provided aiming at the inevitable torque disturbance in the servo mechanism of the photoelectric stable platform, the servo bandwidth is improved by combining with feedforward control, and the optimization of the low-speed stability and stable precision of the servo mechanism is realized. In the literature (from Zhoutao, Juzege, airborne photoelectric tracking platform servo system active disturbance rejection control [ J ] photoelectric engineering, 2011,38(4):31-36.), aiming at low-speed instability of the airborne photoelectric tracking platform servo system caused by friction nonlinearity, a second-order discrete system slowest control function is adopted to design an active disturbance rejection controller which is used as a position ring controller of the airborne photoelectric tracking platform servo system, so that the friction nonlinearity is effectively compensated, and the position and speed tracking precision of the servo system is improved. However, this control method does not guarantee that the optical tracking platform converges to a specified error region within a limited time.

disclosure of Invention

Aiming at the defects of the prior art, the invention provides a vehicle-mounted photoelectric tracking platform finite time control method based on a disturbance observer; the method organically combines a disturbance observer and a finite time controller, and the disturbance observer and the finite time controller are complementary and are used for controlling the vehicle-mounted photoelectric tracking platform. On one hand, a disturbance observer is used for observing the disturbance on the vehicle-mounted photoelectric tracking platform, and the disturbance is used as a feed-forward quantity for compensation; meanwhile, a given signal is compared with an actual signal to obtain a first-order error model containing an equivalent disturbance term, a feedback control rate is designed by using a finite time control method on the basis of the model, and the vehicle-mounted photoelectric tracking platform has better disturbance resistance and better convergence performance by using the composite control method.

In order to achieve the technical purpose, the invention adopts the following technical scheme: a limited time control method of a vehicle-mounted photoelectric tracking platform based on a disturbance observer is characterized by comprising the following steps: (1) establishing a mathematical model of the photoelectric tracking platform, designing a disturbance observer on the basis of the mathematical model, observing various disturbances suffered by the photoelectric tracking platform, and compensating a feed forward quantity; (2) comparing the given signal with the actual signal to obtain a first-order error model containing an equivalent disturbance term; (3) a finite time control method is used for designing a feedback control rate, and the relation between the boundary of the steady-state error of the closed-loop system and the parameters of the controller under the equivalent interference condition is given through mathematical analysis, so that the photoelectric tracking platform can be effectively controlled under the influence of various interferences.

The vehicle-mounted photoelectric tracking platform is directly driven by a direct current torque motor, namely a rotating shaft of the motor is directly connected with a photoelectric detector, a shell of the motor is connected with a base, and the base is arranged on a locomotive.

the mathematical model of the vehicle-mounted photoelectric tracking platform needs to be obtained by combining a kinetic equation of the vehicle-mounted photoelectric tracking platform, a voltage balance equation of the direct-current torque motor and a torque equation of the direct-current torque motor.

the dynamic equation of the vehicle-mounted photoelectric tracking platform is as follows:

wherein J is the sum of the inertia moment (including the fiber-optic gyroscope, the photoelectric detector, the optical lens and the motor shaft) equivalent to the motor shaft of the whole photoelectric detector and the supporting device, and theta_lIs the rotation angle (pitch direction) of the axis of the photodetector in the inertial space, T_MIs the output torque, T, of the torque motor_DVarious disturbance torques (including shafting friction torque, disturbance torque caused by bending of a lead, mass unbalance torque, coupling caused by locomotive motion and the like) are adopted. The dynamic equation of the vehicle-mounted photoelectric tracking platform is subjected to Laplace transform to obtain the dynamic equation,

The torque equation of the torque motor is as follows: t is_M＝K_mi。

wherein, K_mIs electricityThe moment coefficient of the machine; i is the armature current of the motor.

the voltage balance equation of the direct current torque motor is

Wherein, K_eIs the electromagnetic coefficient of the motor; theta is the angular displacement of the motor, theta is equal to theta_l-θ_b(θ_lIs the angular displacement, theta, of the photoelectric detector in the inertial space_bIs the angular displacement of the base in the inertial space, theta is the difference of the angular displacement of the base and the angular displacement of the base); r is the total resistance of an armature loop of the motor; i is the armature current of the motor; and L is the total inductance of the armature loop of the motor.

The torque equation of the direct-current torque motor is as follows:Wherein J is the sum of the rotational inertia of the whole photoelectric detector and the supporting device equivalent to the rotating shaft of the motor; b is a viscous damping coefficient, b ═ b_m+b_L，b_mIs the viscous damping coefficient of the motor itself, b_La viscous damping coefficient for the load; theta is the angular displacement of the motor; t is_∑Is the sum of various disturbance torques applied to the motor rotating shaft; k_mIs the torque coefficient of the motor.

The disturbance observer satisfies the following expression:

Wherein G is_p(s) is a real model of the photoelectric tracking platform, G_n(s) is a nominal model of the on-board opto-electronic tracking platform, G_uy(s) is input signal to output signalTransfer function of number, G_dy(s) is the transfer function of the disturbance signal to the output signal, G_ξy(s) is the transfer function of the noise signal to the output signal; q(s) is a low pass filter.

The low-pass filter Q(s) is of the form Q(s) ═ g/(s + g), where g is a parameter of the filter; s is the laplacian operator.

The output of the disturbance observer is:When the parameter g → ∞ of the filter is satisfied, thenNamely, the observed values of various disturbances of the photoelectric tracking platform by the disturbance observer approach the actual disturbance values, and the disturbance can be compensated through feedforward control.

Wherein the content of the first and second substances,Is the output of the disturbance observer; b is₀＝J_n/K_m，J_nIs the estimation of the inertia J of the vehicle-mounted photoelectric tracking platform, K_mIs the torque coefficient of the drive motor; s is the laplace operator; omega_l(t) is the actual angular velocity of the drive motor; i.e. i_r(t) is the output of the finite time controller; g is a parameter of the filter; d (t) is the sum of all disturbances including various external disturbances, system model mismatch and current tracking errors suffered by the vehicle-mounted photoelectric tracking platform.

The current control quantity i given by the finite time controller_rThe expression of (t) is:

Wherein i_r(t) is a current control amount of the driving motor; b is₀＝J_n/K_m，J_nIs an estimate of J, which is the sum of the moments of inertia on the motor shaft, K_mIs the torque coefficient of the motor; k is a proportionality coefficient; sgn is a symbolA function; e (t) is tracking error, e (t) ═ ω_r(t)-ω_l(t)，ω_r(t) is a given angular velocity of the drive motor, ω_l(t) is the actual angular velocity of the drive motor; alpha is a constant, and alpha is more than 0 and less than 1;Is the output of the disturbance observer.

according to the technical scheme, the following effects can be achieved:

1) the invention adopts a method for observing disturbance by the disturbance observer, realizes compensation of various disturbances of the vehicle-mounted photoelectric tracking platform, such as shafting friction moment, disturbance moment caused by bending of a lead, mass unbalance moment, coupling caused by locomotive motion and the like, and overcomes the influence of the disturbances on the control precision of the vehicle-mounted photoelectric tracking platform. The disturbance observer does not depend on an accurate mathematical model of a controlled object, the realization is simple, the disturbance observer is used for observing the uncertainty caused by external interference and system modeling errors of the vehicle-mounted photoelectric tracking platform, the uncertainty is used as a feedforward quantity for compensation, and the influence of disturbance on the vehicle-mounted photoelectric tracking platform is reduced.

2) The invention combines the finite time control and the disturbance observer, designs the feedback control rate by using a finite time control method on the basis of a first-order error model containing an equivalent disturbance term, and deduces the relation between the boundary of the speed error and the parameters of the controller by using a mathematical method.

3) the disturbance observer does not need an accurate mathematical model of an interference signal, has a simple structure, avoids a large amount of mathematical calculations when observing the interference signal, and can meet the real-time requirement; the finite time stabilization system has better convergence near the balance point and better anti-jamming property.

Drawings

FIG. 1 is a schematic diagram of the operation of the electro-optical tracking platform of the present invention;

FIG. 2 is a system block diagram of the electro-optical tracking stage of the present invention;

FIG. 3 is a diagram of the structure of the disturbance observer of the photoelectric tracking platform of the present invention;

FIG. 4 is a block diagram of the speed loop control of the opto-electronic tracking platform of the present invention;

FIG. 5 is a block diagram of an improved speed loop control for the opto-electronic tracking platform of the present invention;

Fig. 6 is a composite control block diagram of the photoelectric tracking platform of the present invention.

Description of reference numerals: 1. a locomotive; 2. a photoelectric tracking platform; 21. a base; 22. a motor; 23. a photodetector; 24. a gyroscope.

Detailed Description

The accompanying drawings disclose a schematic structural diagram of a preferred embodiment of the present invention without limitation, and the technical solution of the present invention will be described in detail with reference to the accompanying drawings.

As shown in fig. 1, the vehicle-mounted photoelectric tracking platform of the present invention includes a locomotive 1 and a photoelectric tracking platform 2, wherein the photoelectric tracking platform 2 includes a base 21 disposed on the locomotive 1, a motor 22 is mounted on the base 21, a rotating shaft of the motor 22 is connected to a photoelectric detector 23, the photoelectric detector 23 is further provided with a gyroscope 24, the motor 22 is directly driven by a direct current torque motor, that is, the rotating shaft of the motor 22 is directly connected to the photoelectric detector 23, a housing of the motor 22 is connected to the base 21, and the base 21 is mounted on the locomotive 1.

Since the locomotive 1 would pitch during travel, without a stability control system, this pitching motion would couple to the detector, causing the axis of the detector to deviate from the tracking target, resulting in a tracking failure. In order to overcome the coupling influence caused by the movement of the locomotive 1, a gyroscope is used for measuring the angular velocity of the photoelectric detector 23 in an inertial space in the system design, and when the photoelectric detector 23 is detected to deviate from a preset track, the photoelectric detector 23 is driven to move in the opposite direction through the direct current torque motor 22, so that the coupling influence caused by the movement of the locomotive is compensated, the attitude reference of the photoelectric detector is maintained, and the target tracking function is realized.

The invention discloses a vehicle-mounted photoelectric tracking platform finite time control method based on a disturbance observer, which comprises the following steps of: (1) establishing a mathematical model of the photoelectric tracking platform, designing a disturbance observer on the basis of the mathematical model, observing various disturbances suffered by the photoelectric tracking platform, and compensating a feed forward quantity; (2) comparing the given signal with the actual signal to obtain a first-order error model containing an equivalent disturbance term; (3) a finite time control method is used for designing a feedback control rate, and the relation between the boundary of the steady-state error of the closed-loop system and the parameters of the controller under the equivalent interference condition is given through mathematical analysis, so that the photoelectric tracking platform can be effectively controlled under the influence of various interferences.

according to the vehicle-mounted photoelectric tracking platform finite time control method based on the disturbance observer, the disturbance observer is used for observing nonlinear friction torque, mass unbalance torque, cable constraint, model mismatch and coupling influence caused by a moving locomotive on the photoelectric tracking platform, compensation is carried out in a feed-forward mode, and the influence of the interference on a system is reduced.

According to the vehicle-mounted photoelectric tracking platform finite time control method based on the disturbance observer, the feedback control rate is designed for the photoelectric tracking platform by using the finite time control method, the anti-interference capability of the system is further improved, and the tracking error is converged to a smaller area.

the mathematical model of the vehicle-mounted photoelectric tracking platform needs to be obtained by combining a kinetic equation of the vehicle-mounted photoelectric tracking platform, a voltage balance equation of the direct-current torque motor and a torque equation of the direct-current torque motor.

the dynamic equation of the vehicle-mounted photoelectric tracking platform obtained according to Newton's theorem is as follows:

Wherein J is the total sum of the rotational inertia (including fiber optic gyroscope, photodetector, optical lens and motor shaft) equivalent to the motor shaft of the whole photodetector and supporting device, and theta_lIs the axis of the photoelectric detector is in the inertiarotation angle (pitch direction) of sexual space, T_MIs the output torque, T, of the torque motor_Dvarious disturbance torques (including shafting friction torque, disturbance torque caused by bending of a lead, mass unbalance torque, coupling caused by locomotive motion and the like) are adopted. Equation of dynamicsThe laplace transform is performed to obtain,A block diagram of the control system of the electro-optical tracking stage shown in FIG. 2 can thus be obtained, in whichIs a system input speed command, G_gis a transfer function of a fiber optic gyroscope, G_CIs the transfer function of the dc torque motor regulator.

The vehicle-mounted photoelectric tracking platform is driven by a direct-current torque motor, and the torque equation of the torque motor is T_M＝K_mi, wherein K_mThe torque coefficient of the motor; i is the armature current of the motor; the voltage balance equation of the direct current torque motor isWherein K_eis the electromagnetic coefficient of the motor; theta is the angular displacement of the motor, theta is equal to theta_l-θ_b(θ_lIs the angular displacement, theta, of the photoelectric detector in the inertial space_bIs the angular displacement of the base in the inertial space, theta is the difference of the angular displacement of the base and the angular displacement of the base); r is the total resistance of an armature loop of the motor; i is the armature current of the motor; and L is the total inductance of the armature loop of the motor. The torque equation of the direct-current torque motor is as follows:wherein J is the sum of the rotational inertia of the whole photoelectric detector and the supporting device equivalent to the rotating shaft of the motor; b is a viscous damping coefficient, b ═ b_m+b_L，b_mThe viscous damping coefficient of the motor,b_La viscous damping coefficient for the load; theta is the angular displacement of the motor; t is_∑Is the sum of various disturbance torques applied to the motor rotating shaft; k_mIs the torque coefficient of the motor.

In the design of the disturbance observer, the inverse of the system reference model is requiredin the usual case of the use of a magnetic tape,Is not realizable; furthermore, measurement noise of the system can degrade performance of disturbance observations. In practical design, a low-pass filter Q(s) can be connected in series in front of the disturbance observation signal to solve the above problem. The improved disturbance observer structure is shown in fig. 3.

In FIG. 3, G_p(s) is an actual model of the photoelectric tracking platform; g_n(s) is a reference model of the photoelectric tracking platform; g_n ^-1(s) is the inverse of the reference model of the photoelectric tracking platform; q(s) is a low-pass filter connected in series, and is required to have a relative order (difference between the order of the denominator and the numerator order of Q (s)) of G or more in view of realizability_nRelative order of(s); u. of_o、u、d_ex、And y and xi are respectively control input, control output, system interference, an interference observation value, system output and system measurement noise. The expression for the system output y can be derived from fig. 3 as: y is G_uy(s)u+G_dy(s)d_ex+G_ξy(s) ξ in which G_uy(s) is the transfer function of the input to output signal, G_dy(s) is the transfer function of the disturbance to the output signal, G_ξy(s) is the transfer function of the noise to the output signal, their expressions are respectively as follows:

Let f be the frequency band of the PTP system, f_qIs the frequency band of the low-pass filter Q(s), when f is less than or equal to f_qThen, Q(s) ≈ 1, which is deduced to obtain: 1) g_uy(s)≈G_n(s), i.e. the actual model of the system is similar to the reference model; 2) g_dy(s) is approximately equal to 0, namely the system has better anti-interference performance; 3) g_ξy(s) ≈ -1, indicating that measurement noise ξ is introduced at this time. When f > f_qThen, Q(s) ≈ 0, which is deduced to obtain: 1) g_uy(s)≈G_p(s)， G_dy(s)≈G_p(s), the disturbance observer is equivalent to an open loop at the moment, namely the feedforward compensation action is lost; 2) g_ξy(s) ≈ 0, indicating that no measurement noise is introduced at this time. Considering that the system noise is generally high frequency, the influence of the noise introduced by the disturbance observer on the system control performance is not particularly severe.

Due to the velocity signal omega_l(t) and position signal θ_l(t) there is a first derivative relationship, i.e.So that the kinetic equation of the vehicle-mounted photoelectric tracking platform can be changed intoWherein T is_Σ(T) is a group comprising T_Dand (t) the photoelectric tracking platform is subjected to the sum of various interferences. The structure diagram of the speed loop control of the photoelectric tracking platform shown in fig. 5 can be obtained. Wherein G is_i(s) is the current loop transfer function (since the current changes faster, the current loop can be processed in the design in proportion), K_gIs a transfer function of a fiber optic gyroscope, omega_rAnd ω_lThe difference is the reference angular velocity and the actual output angular velocity of the motor, and the ASR is a speed loop regulator.

Analyzing the velocity loop control structure diagram 4 can obtain the external interference T suffered by the system_DWithin the speed loop of the system, but outside the current loop of the system, and therefore can disturb the outside T_Dcompensating corresponding to a current form, and further converting a kinetic equation of the vehicle-mounted photoelectric tracking platform into the following equation for convenient analysis:

Wherein B ═ J/K_m，I_d(t)＝T_Σ/K_m，I_dAnd (t) interference in the form of current, thereby converting the structure diagram of the speed loop control of the photoelectric tracking platform from the graph of FIG. 4 to the graph of FIG. 5.

In the design of the disturbance observer, a reference model G is taken_n(s)＝K_mn/(J_ns) in which J_nAnd K_mnThe inertia J and the moment coefficient K are respectively_mIs estimated. Since the current-to-speed output in the system is a first-order link, a filter of the form Q(s) ═ g/(s + g) is selected. For the convenience of analysis, note B₀＝J_n/K_mThe formula (2) is rewritten as:

Wherein the content of the first and second substances,The system can be considered to be the sum of all disturbances including various external disturbances, system model mismatch and current tracking error. Rewrite formula (3) to:The disturbance observer output can thus be found as:

Get G_p(s)＝1/(Bs)，G_n(s)＝1/(B₀s)，y＝ω_l，u＝i_rThese variables are substituted into equation (4), and the output of the disturbance observer can be obtained by combining equation (3):

As can be seen from the formula (5), when the filter parameter g → ∞ in DOB, thenNamely, the observed value of the disturbance by the DOB approaches the actual disturbance value, and the disturbance can be compensated through feedforward control.

The following is a finite time controller design.

Let the tracking error e (t) be ω_r(t)-ω_l(t), the two-sided derivation, and substituting equation (3) can be obtained:

Control quantity i_rThe expression of (t) is:

wherein, B₀＝J_n/K_m，J_nIs an estimate of J, which is the sum of the moments of inertia on the motor shaft, K_mIs the torque coefficient of the motor; k is a proportionality coefficient; sgn is a sign function; e (t) is tracking error, e (t) ═ ω_r(t)-ω_l(t)，ω_r(t) is a given angular velocity of the drive motor, ω_l(t) is the actual angular velocity of the drive motor; alpha is a constant, and alpha is more than 0 and less than 1;Is the output of the disturbance observer. By substituting equation (7) for equation (6), the tracking error can be obtained:

Wherein the content of the first and second substances,Can be seen as a total disturbance of the system.

The following can be obtained through mathematical derivation: the error e (t) can converge to a certain region,The larger k is, the smaller the error convergence area is, and the value of k cannot be too large in consideration of the system stability and other factors. In the design process of the finite time controller, the convergence region can be made arbitrarily small by reducing α.

Finally, a finite time control block diagram of the photoelectric tracking platform based on the disturbance observer is obtained and is shown in fig. 6.