CN113934138A - Friction compensation feedforward controller for servo system - Google Patents

Friction compensation feedforward controller for servo system Download PDF

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CN113934138A
CN113934138A CN202111237227.4A CN202111237227A CN113934138A CN 113934138 A CN113934138 A CN 113934138A CN 202111237227 A CN202111237227 A CN 202111237227A CN 113934138 A CN113934138 A CN 113934138A
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feedforward
friction
speed
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servo system
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CN113934138B (en
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朱其新
王嘉祺
金建锋
谢鸥
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Suzhou Jiesen Electric Appliance Co ltd
Suzhou University of Science and Technology
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    • G05B11/01—Automatic controllers electric
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Abstract

The invention relates to a friction compensation feedforward controller for a servo system, which comprises a three-ring control system of a PID (proportion integration differentiation) controller, wherein the three-ring control system sequentially comprises a current ring, a position ring and a speed ring from inside to outside; the system also comprises a feedforward control link and a feedback control link which are added in the three-loop control system, wherein the feedforward control link comprises speed feedforward, acceleration feedforward and friction compensation obtained through a LuGre friction model, the speed feedforward is a first-order differential link input to a position loop, and the acceleration feedback is a second-order differential link input to the speed loop; the feedback control link comprises differential negative feedback; setting three parameters in the feedforward control link by a particle swarm optimization algorithm: velocity feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain Kf. The invention solves the problem of tracking lag of the traditional PID control, and also solves the problems of flat top phenomenon and dead zone phenomenon of speed when the speed passes through zero caused by friction.

Description

Friction compensation feedforward controller for servo system
Technical Field
The invention relates to the technical field of motor control, in particular to a friction compensation feedforward controller for a servo system.
Background
The permanent magnet synchronous motor and the servo system thereof are important components of modern industry and are widely applied to the fields of numerical control machines, military industry production, industrial robots and the like. With the progress of science and technology, the requirements on the servo control technology are increasingly improved, and not only is the response speed block required, but also the motion control is required to have high precision, no overshoot and no error so as to improve the overall efficiency.
The traditional servo system mainly adopts a PID control strategy, and has the advantages of reliability, good use, simple structure and easy realization. However, PID control has some disadvantages such as slow response speed, large overshoot, and dynamic tracking lag. Therefore, it is necessary to optimize the conventional PID control for high-performance servo control.
Since friction is inevitable in the operation of the motor in practical engineering, achieving friction compensation is one of the important points of optimization. In order to solve the influence of friction on the system and achieve a better control effect, a large number of documents propose various friction models for optimizing the effect of friction compensation. However, most of the existing methods directly compensate the output of the friction model into the system when adopting friction compensation feedforward, and the tracking lag problem of the traditional PID control and the position flat top phenomenon and speed dead zone phenomenon at the zero-crossing speed caused by friction can not be solved, so that the optimal control performance can not be achieved.
Disclosure of Invention
The present invention provides a friction compensation feedforward controller for a servo system to solve the above technical problems.
In order to solve the technical problem, the invention provides a friction compensation feedforward controller for a servo system, which comprises a three-ring control system of a PID controller, wherein the three-ring control system sequentially comprises a current ring, a position ring and a speed ring from inside to outside;
the three-loop control system further comprises a feedforward control link and a feedback control link which are added in the three-loop control system, wherein the feedforward control link comprises speed feedforward, acceleration feedforward and friction compensation obtained through a LuGre friction model, the speed feedforward is a first-order differential link input to a position loop, and the acceleration feedback is a second-order differential link input to a speed loop; the feedback control link comprises differential negative feedback;
setting three parameters in the feedforward control link by a particle swarm optimization algorithm: velocity feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain Kf。
Preferably, the transfer functions of the velocity feedforward and the acceleration feedforward are respectively:
Figure BDA0003313733890000021
wherein, KtIs the torque coefficient and J is the moment of inertia.
Preferably, the differential negative feedback coefficient tau in the differential negative feedback is adjustedDAnd adjusting the damping ratio xi of the servo system.
Preferably, the damping ratio ξ of the servo system is: xi is more than 0.8 and less than 0.9.
Preferably, the mathematical model of the LuGre friction model is:
Figure BDA0003313733890000022
wherein z is the average deformation of the bristles, ω is the rotational speed, σ0G (omega) is a nonlinear function for representing the friction effect under different conditions, FfIs the total friction moment, σ1As damping coefficient, σ2Is viscosity coefficient, FcIs coulomb friction torque, omegasIs the Stribeck speed.
Preferably, in the particle swarm optimization algorithm, each feasible solution is a particle, and each particle includes two parameters: the position x and velocity v, continuously approximate the global optimal solution in the solution space according to the following iterative formula:
v＝ω·v+c1r1(pBest-x)+c2r2(gBest-x)
x＝x+v
wherein, ω is an inertia weight for representing the influence degree of the past speed on the present speed; c. C1And c2Is the acceleration coefficient; r is1And r2Is at [0,1 ]]Random numbers uniformly distributed thereon; p is a radical ofBestThe current optimal position is obtained; gBestIs a global optimum position.
Preferably, in each iteration, whether a particle is the optimal solution is judged through a fitness function.
Preferably, the fitness function is:
Figure BDA0003313733890000031
the sampling time ts is 0.001s, the input signal is a sine signal r (k) is 0.1 × sin (0.2 pi · k · ts), k is 1,2,3, …,2500, and the output signal is y (k).
Compared with the prior art, the friction compensation feedforward controller for the servo system has the following advantages:
1. on the basis of a three-loop control system of the PID controller, a feedforward control link and a feedback control link are added, so that the response speed of the system can be improved through feedforward control, and differential negative feedback can play a role in reducing overshoot;
2. the invention adopts a particle swarm optimization algorithm to carry out speed feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain KfParameter optimization is carried out, and the servo system is further enabled to obtain better performance;
3. the invention obtains friction compensation through the LuGre friction model, and solves the problems of flat top phenomenon and dead zone phenomenon of speed at the zero-crossing position caused by friction.
Drawings
FIG. 1 is a block diagram of a feed forward control element according to an embodiment of the present invention;
FIG. 2 is a block diagram of the differential negative feedback principle according to an embodiment of the present invention;
FIG. 3 is a partial enlarged view of the position plateau;
FIG. 4 is a partial enlarged view of a velocity dead band phenomenon;
FIG. 5 is a schematic diagram of a LuGre friction model in accordance with an embodiment of the present invention;
FIG. 6 is a Stribeck curve according to an embodiment of the present invention;
FIG. 7 is a block diagram of a LuGre friction model according to an embodiment of the present invention;
FIG. 8 is a block diagram of friction compensation in an embodiment of the present invention;
FIG. 9 shows velocity feedforward gain K in accordance with an embodiment of the present inventionaOptimizing a curve;
FIG. 10 shows a feed forward gain K for acceleration in accordance with an embodiment of the present inventionbOptimizing a curve;
FIG. 11 shows a friction compensation gain K according to an embodiment of the present inventionfOptimizing a curve;
FIG. 12 is a fitness function optimization curve in accordance with an embodiment of the present invention;
FIG. 13 is a schematic diagram of the overall architecture of a friction compensating feedforward controller in accordance with one embodiment of the present invention;
FIG. 14 is a Simulink simulation block diagram of a friction compensated feedforward controller according to an embodiment of the present invention;
FIG. 15 is a graph of a Simulink simulation for a classical PID control;
FIG. 16 is a simulation graph of Simulink in a friction compensated feedforward controller according to an embodiment of the present invention;
FIG. 17 is an enlarged, comparative plot of the top of the curve of a friction compensating feedforward controller according to the invention versus a comparative example;
FIG. 18 is a velocity response graph of a friction compensating feedforward controller in accordance with an embodiment of the invention.
Detailed Description
In order to more thoroughly express the technical scheme of the invention, the following specific examples are listed to demonstrate the technical effect; it is emphasized that these examples are intended to illustrate the invention and are not to be construed as limiting the scope of the invention.
The friction compensation feedforward controller for the servo system comprises a three-loop control system of a PID (proportion integration differentiation) controller, as shown in FIG. 13, wherein the three-loop control system sequentially comprises a current loop, a position loop and a speed loop from inside to outside; the three-loop control system further comprises a feedforward control link and a feedback control link which are added in the three-loop control system, wherein the feedforward control link comprises speed feedforward, acceleration feedforward and friction compensation obtained through a LuGre friction model, the speed feedforward is a first-order differential link input to a position loop, and the acceleration feedback is a second-order differential link input to a speed loop; the feedback control link comprises differential negative feedback; setting three parameters in the feedforward control link by a particle swarm optimization algorithm: velocity feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain Kf。
Specifically, in order to eliminate the tracking error of position output and ensure that the closed loop transfer function is always 1, two feedforward control links, namely speed feedforward and acceleration feedforward, are designed in front of a speed loop and a current loop according to a feedforward control theory. The control system is shown in fig. 1.
In FIG. 1, KppProportional gain, K, of a position loop proportional controllerspProportional gain, K, for a speed loop PI controllersiIs the integral gain, K, of a speed loop PI controllercCurrent loop gain, K, after simplification of the current loop into an inertial elementtIs the torque coefficient, J is the moment of inertia, GaFor the velocity feedforward loop, GbAn acceleration feedforward link. As can be seen from fig. 1, the closed loop transfer function of the servo system is:
Figure BDA0003313733890000051
therefore, when h(s) is 1, the following results are obtained:
Figure BDA0003313733890000052
wherein the content of the first and second substances,
Figure BDA0003313733890000053
the effect of (2) is very small, and the third-order differential link is considered to be relatively more sensitive to high-frequency disturbance, which is not beneficial to the stability of the system. Therefore, the third-order differential link is omitted, so that the transfer function of the feedforward control is obtained as follows:
Figure BDA0003313733890000054
from the formula (3), GaThe method is a first-order differential link for position loop input, so the method is called speed feedforward; gbIt is a second order differential element to the velocity loop input and is called acceleration feedforward. By the compensation effect of the two feedforward links, the closed-loop transfer function h(s) ═ 1 can be realized, so that no error exists between the actual output and the given input.
When designing differential negative feedback, the position loop simplified model of fig. 2 is first used to construct the closed loop transfer function into a second order model without considering feedforward control.
K in FIG. 2sSpeed loop gain, τ, after simplifying the speed loop into an inertial elementDIs a differential negative feedback coefficient. The closed loop transfer function of the position loop from fig. 2 is:
Figure BDA0003313733890000055
as can be seen from equation (4), the differential negative feedback can be performed without changing the oscillation frequency ωnUnder the condition, the value of the damping ratio xi is changed, and quick response without overshoot is realized. Formula (II)As shown in formula (5):
Figure BDA0003313733890000061
as can be seen from equation (5), the value of the damping ratio ξ follows the differential negative feedback coefficient τDIs increased, and thus, the present application can adjust the differential negative feedback coefficient τ in the differential negative feedbackDAnd adjusting the damping ratio xi of the servo system. When the second-order system is in an underdamping state, the larger the damping ratio is, the smaller the overshoot of the system is. When xi is larger than 1, the second-order system is in an over-damping state, and the response speed is too slow. Therefore, in this embodiment, the damping ratio ξ of the servo system is: xi is more than 0.8 and less than 0.9.
In practical engineering, the motor is inevitably subjected to friction during operation. When the speed crosses zero, a "flat top" of the position curve and a "dead zone" of the speed curve occur, as shown in fig. 3 and 4.
In order to overcome the above problems caused by friction, the present application uses friction compensation obtained by the LuGre friction model, whose principle is modeled as a bristle model approximating the "spring-damper system", as shown in fig. 5, the Stribeck curve is shown in fig. 6, and the mathematical model of the LuGre friction model is shown in equation (6):
Figure BDA0003313733890000062
wherein Z is the average deformation of the bristles, ω is the rotational speed, σ0G (omega) is a nonlinear function representing the friction effect under different conditions, FfIs the total friction moment, σ1As damping coefficient, σ2Is viscosity coefficient, FcIs coulomb friction torque, omegasIs the Stribeck speed.
Preferably, in the particle swarm optimization algorithm, each feasible solution is a particle, and each particle includes two parameters: position x and velocity v. In the process of emptyingThe intermediate is continuously approximated to a globally optimal solution according to the following iterative formula. In each iteration, whether one solution is the optimal solution is judged through a fitness function, and then other particles are enabled to track the two best particles. One of which is the current optimum position, denoted as "pBest"; the other is a global optimum position, denoted as "gBest". Thus, the position and velocity of each particle are iterated according to equations (7), (8):
v＝ω·v+c1r1(pBest-x)+c2r2(gBest-x) (7)
x＝x+v (8)
wherein, ω is an inertia weight for representing the influence degree of the past speed on the present speed; c. C1And c2Is the acceleration coefficient; r is1And r2Is at [0,1 ]]Uniformly distributed random numbers. In the velocity update formula (7), the first part ω · v represents the inertia of the previous behavior of the particle; second part c1r1(pBest-x) is a self-cognizant part of the particle; third part c2r2(gBest-x) is a social part of inter-particle information sharing and collaboration.
The control effect of the friction compensation feedforward controller provided by the present application is described as an embodiment.
The parameter data of the permanent magnet synchronous motor adopted in the embodiment is shown in table 1:
TABLE 1 Servo System parameter Table
Figure BDA0003313733890000071
Through the pole allocation, pole-zero cancellation and other methods, the PID controller parameters configured in the present application and the simplified model parameters are shown in table 2:
TABLE 2 PID controller parameters and simplified model parameters
Kpp Ksp Ksi Kc Ks
358 27 0 150 75
Therefore, according to the parameters and the specific simulation effect, ξ is 0.85, namely:
Figure BDA0003313733890000072
the parameter data used in this patent is shown in table 3:
TABLE 3 LuGre Friction model parameter Table
Fc/(N·m) Fs/(N·m) ωs/(m·s-1) σ0/(N·m·s-1) σ1/(N·s·m-1) σ2/(N·s·m-1)
4.483 5.274 0.0879 682000 1076 0.0065
Therefore, the friction model simulation module built in Simulink is shown in fig. 7 according to the parameters in table 3.
Wherein, Divide is a multiplication and division module, Math Function is a Function module, Gain is a Gain module, Add is an addition and subtraction module, Product is a multiplication module, Integrator is an integration module, In1 is an input end of a rotating speed omega, Out1 is a total friction moment FfTo the output terminal of (a). After the friction model module is built, the friction model module is created into a subsystem, and then the subsystem can be introduced into the whole servo closed-loop system. During the introduction of the overall system, the application is in terms of the total friction torque FfIs followed by a friction compensation gain KfAs shown in fig. 8. The application optimizes K in the method through a particle swarm optimization algorithmfSo as to reach an ideal value.
The particle swarm optimization algorithm is adopted to optimize three parameters: velocity feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain KfDefining the fitness function as:
Figure BDA0003313733890000081
the sampling time ts is 0.001s, the input signal is a sine signal r (k) is 0.1 × sin (0.2 pi · k · ts), k is 1,2,3, …, 2500. Setting the inertia weight omega in the program to be 0.9, wherein the larger the value is, the stronger the global search capability is, and the smaller the value is, the stronger the local search capability is; coefficient of acceleration c1＝c21.4, dimension Dim of solution vector is 3, population size SwarmSize is 100, and maximum number of iterations maxter is 100. After the initialization of the parameters is completed, the operation results are shown in fig. 9-12.
So, as can be seen from FIGS. 9-12, when KaAbout 10.2, KbAbout 3.48, KfAbout 0.57, the total error of the whole system is minimized, which is an ideal condition.
Fig. 13 is a schematic diagram of the overall structure of a feed-forward control system with LuGre friction compensation according to the present invention, a structural block diagram of Simulink simulation after data substitution is shown in fig. 14, and fig. 15-18 are graphs comparing effects after a simulation program is run. As can be seen from fig. 15-17, by introducing the feedforward control element, the tracking error of the response curve is smaller and the control accuracy is higher than that of the conventional PID control. It can also be seen from fig. 17 that the "flat top" of the position curve is corrected by introducing the LuGre friction model. And because the particle swarm optimization algorithm is adopted, the degree of friction compensation is ideal, insufficient compensation and overcompensation are avoided, and the position curve is ideal to be attached to the sine input. It can also be seen from fig. 18 that the "dead band" at speed zero-crossing is improved and the speed-responsive oscillations are reduced after introducing the friction compensation.
In summary, the friction compensation feedforward controller for the servo system provided by the invention comprises a three-loop control system of the PID controller, wherein the three-loop control system sequentially comprises a current loop, a position loop and a speed loop from inside to outside; the three-loop control system also comprises a feedforward control link and a feedback control link which are added in the three-loop control system, wherein the feedforward control link comprises speed feedforward, acceleration feedforward and speed feedforwardFriction compensation is obtained by a LuGre friction model, the speed feedforward is a differential link input to a position loop, and the acceleration feedback is a second-order differential link input to a speed loop; the feedback control link comprises differential negative feedback; setting three parameters in the feedforward control link by a particle swarm optimization algorithm: velocity feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain Kf. The invention simultaneously solves the problem of tracking lag of the traditional PID control, and the position flat-top phenomenon and the speed dead zone phenomenon of zero-crossing caused by friction; and two feedforward gains and one friction compensation gain are optimized simultaneously through a particle swarm optimization algorithm, so that the system performance is further optimized, and an ideal control effect is achieved.
It will be apparent to those skilled in the art that various changes and modifications may be made in the invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

Claims (8)

1. A friction compensation feedforward controller for a servo system is characterized by comprising a three-loop control system of a PID controller, wherein the three-loop control system sequentially comprises a current loop, a position loop and a speed loop from inside to outside;
the three-loop control system further comprises a feedforward control link and a feedback control link which are added in the three-loop control system, wherein the feedforward control link comprises speed feedforward, acceleration feedforward and friction compensation obtained through a LuGre friction model, the speed feedforward is a first-order differential link input to a position loop, and the acceleration feedback is a second-order differential link input to a speed loop; the feedback control link comprises differential negative feedback;
setting three parameters in the feedforward control link by a particle swarm optimization algorithm: velocity feedforward gain KaAcceleration feedforward gain KbAnd a friction compensation gain Kf。
2. A friction compensating feedforward controller for a servo system as in claim 1, wherein the transfer functions of the velocity feedforward and acceleration feedforward are:
Figure FDA0003313733880000011
wherein, KtIs the torque coefficient and J is the moment of inertia.
3. A friction compensating feedforward controller for a servo system as claimed in claim 1, wherein the feedback is controlled by adjusting a differential feedback coefficient τ in the differential feedbackDAnd adjusting the damping ratio xi of the servo system.
4. A friction compensating feedforward controller for a servo system as claimed in claim 3, wherein the damping ratio ξ for the servo system is: xi is more than 0.8 and less than 0.9.
5. A friction compensating feedforward controller for a servo system as claimed in claim 1, wherein the mathematical model of the LuGre friction model is:
Figure FDA0003313733880000012
wherein Z is the average deformation of the bristles, ω is the rotational speed, σ0G (omega) is a nonlinear function for representing the friction effect under different conditions, FfIs the total friction moment, σ1As damping coefficient, σ2Is viscosity coefficient, FcIs coulomb friction torque, omegasIs the Stribeck speed.
6. A friction compensating feedforward controller for a servo system as in claim 1, wherein in the particle swarm optimization algorithm, each feasible solution is a particle, and each particle includes two parameters: the position x and velocity v, continuously approximate the global optimal solution in the solution space according to the following iterative formula:
v＝ω·v+c1r1(PBest-x)+c2r2(gBest-x)
x＝x+v
wherein, ω is an inertia weight for representing the influence degree of the past speed on the present speed; c. C1And c2Is the acceleration coefficient; r is1And r2Is at [0,1 ]]Random numbers uniformly distributed thereon; p is a radical ofBestThe current optimal position is obtained; gBestIs a global optimum position.
7. A friction compensating feedforward controller for a servo system as claimed in claim 6, wherein in each iteration, a fitness function is used to determine whether a particle is the optimal solution.
8. A friction compensating feedforward controller for a servo system as recited in claim 7, wherein the fitness function is:
Figure FDA0003313733880000021
the sampling time ts is 0.00ls, the input signal is a sine signal r (k) is 0.1 x sin (0.2 pi · k · ts), k is 1,2,3, …,2500, and the output signal is y (k).
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CN115167144A (en) * 2022-08-04 2022-10-11 北京航空航天大学 Airplane actuating system based on particle swarm algorithm
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