A Command Filter Control Load Trajectory Tracking Method for a Two-Mass System Based on an Extended State Observer

Abstract

This invention belongs to the field of servo system load trajectory tracking technology, and relates to a command-filtered tracking control based on an extended state observer. The method includes the following steps: establishing a dynamic model and state equations for a two-mass servo system, initializing the system state and system parameters; estimating the load-side information, motor-side information, and total disturbance using an extended state observer; introducing a command-filtered control algorithm to approximate the derivative of the virtual control law, suppressing the differential explosion problem present in the backstepping method, and designing a virtual control function and a load tracking controller. This method achieves effective estimation of the system state and uncertain models, giving the system speed, stability, and accuracy, strong anti-interference capability, and excellent load tracking performance.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of load trajectory tracking of servo systems, and particularly relates to a load trajectory tracking method for a two-mass system command filter control based on an extended state observer. BACKGROUND

[0002] In the process of continuous development of modern industry, two-mass servo systems are widely used in radars, paper machines and medical devices, etc. In actual operation, two-mass servo systems often have the problems of difficult measurement of states, model uncertainty, parameter perturbation and susceptibility to external unknown disturbances, etc. In actual production, the response speed and accuracy of two-mass servo systems are increasingly high, and therefore, it is particularly important to design a reasonable load tracking controller to make the system have rapidity, stability and accuracy. For example, the Chinese patent CN104698844A proposes a sliding mode control method for uncertainty compensation of a hydraulic position servo system, which includes the following steps: first, a mathematical model of the hydraulic position servo system is established; then, a mismatched and matched disturbance observer is designed; second, a sliding mode controller based on the mismatched and matched disturbance observer is designed; and finally, the system is proved to be globally asymptotically stable according to the Lyapunov stability principle. In actual production, the sliding mode control is prone to persistent chattering, which has an impact on the service life of the system. The Chinese patent CN104345640A provides a gradual tracking control method and control system for a motor servo system under input constraints, which includes a first module and a second module. The first module is used to establish a model of the motor servo system, and the second module is used to design an input-constrained gradual tracking controller. Finally, reasonable design parameters are selected to realize gradual tracking control of the system under input constraints.

[0003] The backstepping design method is a recursive design method. The basic idea of the backstepping design method is to divide a complex nonlinear system into subsystems not exceeding the order of the system, and then design a partial Lyapunov function and an intermediate virtual control for each subsystem, and "retreat" to the whole system, and integrate them to complete the design of the whole control law. The Chinese patent CN111781829A proposes a neural network control method for gear gap compensation of a turntable servo system. In view of the problem that the speed information of the load of an industrial robot servo system is difficult to measure, a speed observer is designed to estimate the speed information. The RBF neural network is used to approximate the nonlinear part in the system, and a virtual control law, an adaptive law and an RBF neural network backstepping controller are designed by combining the backstepping control technology. The number of layers of the neural network needs to be considered in the control method, and there are many controller parameters.

[0004] The existing technology also has some problems, such as the problems of external unknown disturbance and model uncertainty in the two-mass system. SUMMARY

[0005] In order to overcome the above-mentioned defects of the prior art, the application provides a load trajectory tracking method for a two-mass system command filter control based on an extended state observer, which combines an extended state observer and a command filter based on backstepping control technology, effectively estimates the state of the system, and can compensate for the model and external disturbance in feedback, effectively realizes load tracking, and improves the anti-interference ability.

[0006] In order to solve the above technical problems, the application provides a load trajectory tracking method for a two-mass system command filter control based on an extended state observer, which includes the following steps:

[0007] S1, a dynamic model and a state equation of a two-mass servo system are established, system states and system parameters are initialized, and the process is as follows:

[0008] S1.1, the expected position of the load is x d ;

[0009] S1.2, the dynamic equation of the two-mass servo system is

[0010]

[0011] In the formula, T s is the flexible shaft transmission torque, k s is the torque torsion coefficient, T dm , T dl are unknown external disturbances at the motor end and the load end respectively, J m , J l are the rotational inertia of the motor and the load respectively, b m , b l are the viscous friction coefficients of the motor and the load respectively, θ m , θ l are the angular displacements of the motor and the load respectively, are the angular velocities of the motor and the load respectively, are the angular accelerations of the motor and the load respectively, and u is the system input torque, that is, the tracking controller to be designed in the application;

[0012] S1.3, the system state variables are selected as x1=θ l , x3=θ m , Let , then the state equation of the system (1) is

[0013]

[0014] where x5=1 / J l (T s -b l x2-Tdl ) x3 is the total disturbance at the motor side, x6 = 1 / J m (-b m x4-T s -T dm ) total disturbance at the motor side, x5, x6 are considered as extended state variables, t is the system running time;

[0015] Assumption 1: x5 and x6 are both unknown and continuously bounded;

[0016] S2, design the extended state observer

[0017] Define the observation error as The extended state observer of the system is designed as

[0018]

[0019] Where β1, β2, β3, β4, β5, β6 are the observer gains, δ1, δ2 are constants that affect the filtering effect of the Fal function, satisfying and 5T≤δ j ≤10T, j = 1, 2, T is the sampling time; two Fal functions are as follows

[0020]

[0021]

[0022] According to formula (2) and (3), the estimation error equation of the observer is

[0023]

[0024] Select appropriate observer gains β1, β2, β3, β4, β5, β6, and the observation error can converge in finite time, for example: l i is a known constant greater than 0;

[0025] S3, based on the two-mass servo system containing unknown external disturbance, a load tracking controller based on extended state observer is designed, a first-order command filter is selected to approximate the derivative of the virtual control law, three virtual control laws and load tracking control laws are recursively set out, and the load trajectory tracking is completed; The specific steps are as follows:

[0026] S3.1 Command filter description

[0027] Consider a first-order command filter as

[0028]

[0029] Where ω is the design parameter of the filter, is the input of the filter, is the state of the filter, is the output of the filter, is used to estimate

[0030] S3.2 Design the load tracking controller u

[0031] Define the tracking error as

[0032] e1 = x1 - x d (8)

[0033] e2 = x2 - a1 (9)

[0034] e3 = x3 - a2 (10)

[0035] e4 = x4 - a3 (11)

[0036] where a1, a2, a3 are virtual control laws;

[0037] According to S3.1, introduce a command filter to estimate the derivative of the virtual control law, that is, there is

[0038]

[0039] where ω j is a design parameter, and the filter error is ε j is a known constant greater than 0, j = 1, 2, 3;

[0040] Step 1: Differentiate equation (8) to get

[0041]

[0042] Design the virtual control law as

[0043]

[0044] Step 2: Differentiate equation (9) to get

[0045]

[0046] Design the virtual control law as

[0047]

[0048] Step 3: Differentiate equation (10) to get

[0049]

[0050] Design the virtual control law as

[0051]

[0052] Step4: Derivation of formula (11) gives

[0053]

[0054] The tracking controller is designed as

[0055]

[0056] u is the input torque in formula (1); k in formulas (14), (16), (18) and (20) i is a design parameter, i = 1, …, 4;

[0057] S4, select Lyapunov function, prove system stability

[0058] The Lyapunov function is selected as

[0059]

[0060] Derivation of (21) gives

[0061]

[0062] Put formulas (14), (16), (18) and (20) into formula (22) to get

[0063]

[0064] From Young's inequality, we get

[0065]

[0066] Put formula (24) into formula (23) to get

[0067]

[0068] where

[0069]

[0070] a and b are both bounded;

[0071] Solving formula (26) gives

[0072]

[0073] From formula (27), it can be seen that all signals in the closed-loop system are semi-globally uniformly bounded, so the system of the invention is stable.

[0074] Compared with the prior art, the present application proposes a command filter tracking control based on an extended state observer for a two-mass system with uncertain system state and model; the extended state observer is used to estimate the load end information, the motor end information and the total disturbance; the command filter control algorithm is introduced to approximately obtain the derivative of the virtual control law, the differential explosion problem existing in the backstepping method is suppressed, and the virtual control function and the load tracking controller are designed. BRIEF DESCRIPTION OF DRAWINGS

[0075] Figure 1 The control flow diagram of the command filter control load trajectory tracking method of the two-mass system based on the extended state observer according to the present application.

[0076] Figure 2 The total disturbance estimation effect diagram of the extended state observer when u=sin(t) according to the present application, wherein represents the estimation of the total disturbance x5 at the load end, represents the estimation of the total disturbance x6 at the motor end.

[0077] Figure 3 The total disturbance estimation effect diagram of the extended state observer when u=cos(t) according to the present application, wherein represents the estimation of the total disturbance x5 at the load end, represents the estimation of the total disturbance x6 at the motor end.

[0078] Figure 4 The system load tracking effect diagram when there is no external disturbance according to the present application, wherein (a) is the curve diagram when the expected load position is x d =cos(0.5t)-1, and (b) is the curve diagram when the expected load position is x d =1.

[0079] Figure 5 The system input torque effect diagram when there is no external disturbance according to the present application, wherein (a) is the curve diagram when the expected load position is x d =cos(0.5t)-1, and (b) is the curve diagram when the expected load position is x d =1.

[0080] Figure 6 The load tracking comparison diagram when there is no external disturbance and the external disturbance T dl =0.3cos(t) and T dm =0.1sin(t) is added when t≥5s according to the present application, wherein (a) is the curve diagram when the expected load position is xd Fig. 3 is a graph of the curve when x d = 1.

[0081] Detailed implementation formula:

[0082] The application will be further described below by specific examples and in conjunction with the drawings.

[0083] Example 1:

[0084] The two-mass system command filter control load trajectory tracking method based on the extended state observer involved in this embodiment includes the following steps:

[0085] S1, establish the dynamic model and state equation of the two-mass servo system, initialize the system state and system parameters, the process is as follows:

[0086] S1.1 Set the expected position of the load as x d ;

[0087] S1.2 The dynamic equation of the two-mass servo system is

[0088]

[0089] In the formula, T s is the flexible shaft transmission torque, k s is the torque torsion coefficient, T dm , T dl are unknown external disturbances at the motor end and the load end respectively, J m , J l are the moments of inertia of the motor and the load respectively, b m , b l are the viscous friction coefficients of the motor and the load respectively, θ m , θ l are the angular displacements of the motor and the load respectively, are the angular velocities of the motor and the load respectively, are the angular accelerations of the motor and the load respectively, and u is the system input torque, that is, the tracking controller to be designed in the application;

[0090] S1.3 Select the system state variables as x1 = θ l , x3 = θ m , Let Then the state equation of the system (1) is

[0091]

[0092] Where x5 = 1 / J l(T s -b l x2-T dl )-x3 is the total disturbance at the load end, x6 = 1 / J m (-b m x4-T s -T dm )total disturbance at the motor end, x5, x6 are regarded as extended state variables, t is the system running time;

[0093] Assumption 1: x5 and x6 are both unknown and continuously bounded;

[0094] S2, design an extended state observer

[0095] Define the observation error as Design the extended state observer of the system as

[0096]

[0097] Where β1, β2, β3, β4, β5, β6 are the observer gains, δ1, δ2 are constants affecting the filtering effect of the Fal function, satisfying and 5T≤δ j ≤10T, j = 1, 2, T is the sampling time; two Fal functions are as follows

[0098]

[0099]

[0100] According to formula (2) and (3), the estimation error equation of the observer is

[0101]

[0102] Select appropriate observer gains β1, β2, β3, β4, β5, β6, and the observation error can converge in finite time, for example: l i is a known constant greater than 0;

[0103] S3, based on the two-mass servo system containing unknown external disturbance, design a load tracking controller based on the extended state observer, select a first-order command filter to approximate the derivative of the virtual control law, recursively set up three virtual control laws and load tracking control law, complete the load trajectory tracking; The specific steps are as follows:

[0104] S3.1 Command filter description

[0105] Consider a first-order command filter as

[0106]

[0107] where ω is a design parameter of the filter, is the input of the filter, is the state of the filter, is the output of the filter, is used to estimate

[0108] S3.2 Design the load tracking controller u

[0109] Define the tracking error as

[0110] e1= x1- x d (8)

[0111] e2= x2- a1 (9)

[0112] e3= x3- a2 (10)

[0113] e4= x4- a3 (11)

[0114] where a1, a2, a3 are the virtual control laws;

[0115] According to S3.1, introduce the command filter to estimate the derivative of the virtual control law, that is, there is

[0116]

[0117] where ω j is a design parameter, and the filter error is ε j is a known constant greater than 0, j = 1, 2, 3;

[0118] Step 1: Differentiate equation (8) to get

[0119]

[0120] The virtual control law is designed as

[0121]

[0122] Step 2: Differentiate equation (9) to get

[0123]

[0124] The virtual control law is designed as

[0125]

[0126] Step 3: Differentiate equation (10) to get

[0127]

[0128] The virtual control law is designed as

[0129]

[0130] Step 4: Derivation of equation (11) gives

[0131]

[0132] The tracking controller is designed as

[0133]

[0134] u is the input torque in equation (1); k in equations (14), (16), (18) and (20) i is a design parameter, i = 1, …, 4.

[0135] S4, select Lyapunov function, prove system stability

[0136] The Lyapunov function is selected as

[0137]

[0138] Derivation of equation (21) gives

[0139]

[0140] Substitute equations (14), (16), (18) and (20) into equation (22) to get

[0141]

[0142] From Young's inequality, we get

[0143]

[0144] Substitute equation (24) into equation (23) to get

[0145]

[0146] where

[0147]

[0148] a and b are both bounded.

[0149] Solving equation (26) gives

[0150]

[0151] From formula (27), all signals in the closed-loop system are semi-globally uniformly bounded, so the system under the application is stable. In order to verify the feasibility of the above method, the control method in the control simulation experiment of a two-mass servo system is given, and the simulation verification is carried out on Simulink, and the specific parameter settings are as follows:

[0152] The dynamic model of the two-mass system is established as

[0153]

[0154] The parameters of the system are shown in Table 1.

[0155] Table 1 Parameter data of two-mass servo system

[0156]

[0157] In order to verify the effectiveness of the observer and the control method, the input torque is given as u=sin(t) and u=cos(t), and the sampling time T=0.01; then two kinds of expected signals are given; expected 1: x d =cos(0.5t)-1; expected 2: x d =1.

[0158] The parameters of the observer are selected as β1=100, β2=200, β3=4000, β4=100, β5=200, β6=4000, δ1=0.05, δ2=0.05;

[0159] The parameters of the command filter controller are: k1=1.6, k2=11, k3=0.1, k4=0.1, w1=w2=w3=0.01.

[0160] The above parameters are substituted into the control law and the simulation model of the application to obtain the simulation results: when u=sin(t), the total disturbance estimation of the extended state observer is as shown in Figure 2 , wherein represents the estimation of the total disturbance x5 at the load end, represents the estimation of the total disturbance x6 at the motor end; when u=cos(t), the total disturbance estimation of the extended state observer is as shown in Figure 3 , wherein represents the estimation of the total disturbance x5 at the load end, represents the estimation of the total disturbance x6 at the motor end; the load position tracking response curve and the control input response curve without external disturbance are as shown in Figure 4 and Figure 5 , (a) represents the expected load position x dThe curve when =cos(0.5t)-1, (b) represents the expected load location x. d The curve when t=1; no external disturbance and external disturbance T added at t≥5s. dl =0.3cos(t), T dm The load tracking comparison diagram when =0.1sin(t) is shown below. Figure 6 As shown, (a) represents the desired load location x. d The curve when =cos(0.5t)-1, (b) is the curve when the desired load position is x. d The graph when = 1.

[0161] from Figure 2 and 3 It can be seen that the extended state observer has a good estimation effect on the total disturbance at the load end and the total disturbance at the motor end.

[0162] from Figure 4 As can be seen, when using the control method of this invention, the load can quickly and stably track the desired signal. Figure 4 In (a), the tracking error range is -0.034 to 0.035. Figure 4 (b) The steady-state error range is 0 to 0.017.

[0163] from Figure 5 It can be seen that the system's control input is relatively stable.

[0164] from Figure 6 As can be seen, even when external disturbances are introduced, the present invention can still guarantee the tracking accuracy of the system.

[0165] In summary, the command filtering control method based on the extended state observer can effectively achieve load tracking and has strong anti-interference ability.

Claims

1. An extended state observer-based command filter control load trajectory tracking method for a two-mass system, characterized by, The method comprises the following steps: S1, establishing a dynamic model and a state equation of a two-mass servo system, initializing system states and system parameters; S2, designing an extended state observer for a load end and a motor end; S3, based on a two-mass servo system containing unknown external disturbances, designing a load tracking controller based on the extended state observer, selecting a first-order command filter to approximate derivatives of a virtual control law, recursively setting three virtual control laws and a load tracking control law, and completing load trajectory tracking; The specific process of step S1 is as follows: S1.1 Set desired position of load to x d ; S1.2 The dynamic equation of the two-mass servo system is: In the formula, T s is the flexible shaft transmission torque, k s is the torque torsion coefficient, T dm ,T dl are unknown external disturbances at the motor end and the load end respectively, J m ,J l are the moments of inertia of the motor and the load respectively, b m ,b l are the viscous friction coefficients of the motor and the load respectively, θ m ,θ l are the angular displacements of the motor and the load respectively, are the angular velocities of the motor and the load respectively, The angular acceleration of the motor and the load respectively, u is the system input torque, that is, the tracking controller to be designed in the application; S1.3 Select the system state variable as x1= 0 l , x3= 0 m , Let The state equation of system (1) is where x5 = 1 / J l (T s -b l x2-T dl ) is the total disturbance at the load side, x6 = 1 / J m (-b m x4-T s -T dm ) is the total disturbance at the motor side, x5, x6 are considered as extended state variables, and t is the system running time. Assumption 1: x5 and x6 are unknown and continuously bounded; The specific process of step S2 is as follows: The observation error is defined as The extended state observer of the system is designed as where β1, β2, β3, β4, β5, β6are observer gains, δ1, δ2are constants affecting the filtering effect of Fal function, satisfying and 5T≤δ j ≤10T, j = 1, 2, T is the sampling time; two Fal functions are as follows According to equations (2) and (3), the estimation error equation of the observer is With proper selection of the observer gains β1, β2, β3, β4, β5, β6, the observation error can converge in finite time, for example: l i is a known constant greater than 0; The specific process of step S3 is as follows: S3.1 Command filter description Consider a first-order command filter where ω is a design parameter of the filter, is an input to the filter, is a state of the filter, is an output of the filter, to estimate S3.2 Design a load tracking controller u Define the tracking error as e1 = x1 - x d (8) e2=x2-α1 (9) e3=x3-α2 (10) e4=x4-α3 (11) Wherein alpha1, alpha2, alpha3 are virtual control laws; According to S3.1, a command filter is introduced to estimate the derivative of the virtual control law, that is, where ω j is a design parameter, and the filtering error is ε j is a known constant greater than 0, j = 1, 2, 3. Step 1: Derive equation (8) Design a virtual control law Step 2: Derive equation (9) Design a virtual control law Step 3: Derive equation (10) Design a virtual control law Step 4: Derive equation (11) Design a tracking controller u is the input torque in equation (1); k in equations (14), (16), (18) and (20) i are design parameters, i = 1,..., 4.

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