CN117013897A - Electromagnetic linear actuator high-precision position control method considering nonlinear electromagnetic force compensation - Google Patents
Electromagnetic linear actuator high-precision position control method considering nonlinear electromagnetic force compensation Download PDFInfo
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- H—ELECTRICITY
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Abstract
The invention relates to a high-precision position control method of an electromagnetic linear actuator considering nonlinear electromagnetic force compensation, belonging to the field of motor control. The method comprises the following steps: approximating the relation between the electromagnetic force constant and the position of the rotor as a quadratic function, and establishing a dynamics model of the electromagnetic linear actuator by combining with the LuGre friction model; establishing a dual observer based on a mapping function for facilitating observation of an internal friction state in a dynamic friction model; designing an adaptive compensation term and a parameter adaptive law to compensate nonlinear electromagnetic force and eliminate the influence caused by uncertainty of system parameters; in addition, the unknown nonlinear factors and interference of the system are considered, and a proportional feedback term and an integral robust term for stabilizing the system are designed. The invention has stronger robustness and good tracking performance, and has better control effect in the aspects of control precision, step response time and the like under high load compared with the traditional self-adaptive integral robust control without electromagnetic force compensation under the action of electromagnetic force compensation.
Description
The invention relates to the technical field of electromagnetic linear actuator control.
Background
At present, the driving field of the linear motion mechanism mainly adopts the technical form of a rotary driver and a motion conversion mechanism, but along with the development of scientific technology, the response time, the working efficiency, the structural simplicity and the like of the linear motion mechanism are more required, so that the electromagnetic linear actuator is focused, and the electromagnetic linear actuator becomes one of the products with the market potential in the fields of medicine, robots, machine tools and the like.
In order to pursue better performance of the electromagnetic linear actuator, a great deal of research is carried out by technicians aiming at materials, structures, control algorithms and the like, wherein the control technology becomes a hotspot for the performance research of the electromagnetic linear actuator, and the common control algorithms comprise traditional PID control, decoupling control and the like, and adaptive control, robust control, sliding mode variable structure control and the like. To further improve the control accuracy of the electromagnetic linear actuator, the internal nonlinear state of the electromagnetic linear actuator starts to be compensated, such as nonlinear friction force, temperature compensation and the like. However, it is still assumed that electromagnetic force is directly proportional to current, and the lorentz force actually output has serious nonlinearity due to the electromagnetic property of the magnetic conductive material and the influence of the actuator structure, and needs to be considered in the design of the controller.
The internal state of the electromagnetic linear actuator is inconvenient to observe, unknown uncertainty errors exist, the two problems can be well solved by self-adaptive integral robust control, and the influence caused by parameter uncertainty can be eliminated by the self-adaptive control; robust control with integral robust terms can mitigate the effects of model uncertainty. According to the self-adaptive integral robust control method for the electromagnetic linear actuator with the nonlinear electromagnetic force compensation, the nonlinear electromagnetic force of the electromagnetic linear actuator is effectively compensated, and the uncertain influence of a model is weakened, so that the control algorithm has better tracking performance than the self-adaptive integral robust algorithm without the electromagnetic force compensation under the high-load operation.
Disclosure of Invention
The high-precision position control method of the electromagnetic linear actuator considering nonlinear electromagnetic force compensation is designed to compensate nonlinear Lorentz force caused by electromagnetic properties of magnetic conductive materials and actuator structures, electromagnetic force constants and rotor positions are approximately in a quadratic function relation, unknown parameters of the electromagnetic linear actuator are estimated through a projection adaptive law, good compensation effect is achieved, and control effects in aspects of system stability, control precision and the like are improved.
A high-precision position control method of an electromagnetic linear actuator considering nonlinear electromagnetic force compensation comprises the following steps:
step 1, approximating electromagnetic force constant and rotor position as a quadratic function relation;
step 2, establishing an electromagnetic linear actuator system dynamics model considering the LuGre friction model;
step 3, establishing a dual observer for observing an internal friction state in the dynamic friction model, designing an adaptive compensation term and a parameter adaptive law, compensating nonlinear electromagnetic force and eliminating the influence caused by uncertainty of system parameters;
step 4, designing a proportional feedback term for stabilizing the system, and designing an integral robust term for weakening the influence caused by unknown nonlinear factors of the system;
and step 5, proving the system stability by applying the Lyapunov theory.
The step 1 is specifically as follows:
analyzing the electromagnetic property of the magnetic conductive material and the influence of the structure, wherein the actually output electromagnetic force has serious nonlinearity, and the electromagnetic force constant of the rotor at two ends is smaller than that of the rotor at the middle position, so that the electromagnetic force constant and the rotor position are approximately in a quadratic function relation:
(1)
in the middle ofA、B、CFor three unknown coefficients, the number of coefficients is,x isAnd (5) displacement of the rotor.A、B、CObtained by electromagnetic force test or data fitting of simulation results.
The step 2 is specifically as follows:
the electromagnetic linear actuator system is a mechanical-electromagnetic mutual coupling mathematical model, and expressions of a magnetic circuit, a mechanical subsystem and a circuit subsystem are respectively as follows according to kirchhoff's voltage law, newton's second law and lorentz's law:
(2)
(3)
(4)
in the method, in the process of the invention,F m electromagnetic force generated in a magnetic field by an electrified coil;Nthe total number of turns of the coil;B e is the magnetic field strength;l e is the length of one turn of the coil;iis coil current;K m is the electromagnetic force coefficient;Mis the dynamic mass of the actuator;F dis uncertainty errors and disturbances;uis a phase voltage;RandLcoil resistance and inductance;K e is the counter potential coefficient;vthe mover moving speed. Establishing friction based on LuGre modelF f The nonlinear expression of (2) is as follows:
(5)
in the middle of，，Respectively the rigidity coefficient and the resistanceA coefficient of nylon and a coefficient of viscosity;zis the internal friction state of the system;v s is the Stribeck speed; positive definite functiong(v) To describe the Stribeck phenomenon;F c is coulomb friction;F s is static friction force.
Combining the obtained electromagnetic constant nonlinear formula (1) with an electromagnetic linear actuator system mathematical model, and establishing a system dynamics model of the electromagnetic linear actuator:
(6)
in the method, in the process of the invention,is an uncertain item.
The step 3 is specifically as follows:
(3.1) designing adaptive compensation terms and adaptive approach laws:
(7)
in the method, in the process of the invention,θ 1 =MR/C；θ 2 =K e +(σ 1 +σ 2 )R/C；θ 3 =AR/C；θ 4 =BR/C；θ 5 =σ 0 R/C；θ 6 =σ 1 R/C；θ 7 andrespectively isF dis Constant and time-varying values of/C;
the unknown parameters are estimated using the following projection-modified adaptive law:
(8)
(9)
(10)
in the method, in the process of the invention,and->Respectively estimating a parameter estimation value and an estimation error value; r is a positive diagonal matrix; psi is a row vector formed by variables corresponding to unknown parameters;
(3.2) establishing a dual observer for observing the internal friction state in the dynamic friction model:
(11)
(12)
in the middle ofAnd->The estimates of z in two different terms of equation (7) are made separately,γ 1 andγ 2 the expression form of the projection mapping function is formula (10) respectively the compensation quantity of the state observer.
The step 4 is specifically as follows:
(4.1) design position tracking erroreSliding mode-like variablepAuxiliary variabler：
(13)
In the method, in the process of the invention,x d for the displacement of the object to be achieved,k 1 andk r is the feedback gain;
(4.2) designing a linear feedback term and an integral robust term in the controller:
(14)
in the method, in the process of the invention,kis the gain that is being adjusted in a positive direction,u s2 the specific expression form of the integral robust term is as follows:
(15)
in the method, in the process of the invention,βintegrating the robust feedback gain; sgn () is a sign function.
The step 5 is specifically as follows: system stability is proved by using Lyapunov theory, and auxiliary functions are defined
(16)
(17)
Definition of the lyapunov function:
(18)
stability analysis is performed by using the Lyapunov stability theory, and the control system is proved to be bounded and stable.
The invention relates to a high-precision position control method of an electromagnetic linear actuator considering nonlinear electromagnetic force compensation, which adopts electromagnetic force constant to approximate to a quadratic function relation with a rotor position to compensate nonlinear Lorentz force caused by electromagnetic property of a magnetic conductive material and actuator structure, unknown parameters are estimated through projection mapping law, and the influence of the unknown parameters is eliminated by matching with a self-adaptive compensation term in a controller. The uncertain disturbances of the electromagnetic linear actuator are eliminated by linear feedback terms and integral robustness terms in the controller. The control effects in the aspects of system stability, control precision and the like are improved, and meanwhile, the robustness of the electromagnetic linear actuator system is also improved. Through verification, the system has good tracking performance, the control precision of the sine response signal and the triangular wave response signal is improved compared with uncompensated self-adaptive integral robust control, and huge economic benefits are brought after the system is put into industrial application.
Drawings
FIG. 1 is a general block diagram of a high-precision position control system for an electromagnetic linear actuator that takes into account nonlinear electromagnetic force compensation in accordance with the present invention;
FIG. 2 is a performance comparison of high precision position control of an electromagnetic linear actuator with respect to nonlinear electromagnetic force compensation versus step response of other control methods in accordance with the present invention;
FIG. 3 is a graph showing the displacement error contrast of triangular wave signals under high load between high-precision position control and uncompensated adaptive integral robust control of an electromagnetic linear actuator taking nonlinear electromagnetic force compensation into consideration;
FIG. 4 is a graph showing the displacement error contrast of sine wave signals under high load between high-precision position control and uncompensated adaptive integral robust control of an electromagnetic linear actuator taking nonlinear electromagnetic force compensation into consideration;
Detailed Description
The invention is described in further detail below with reference to the drawings and the detailed description.
In an internal magnetic field of an actual electromagnetic linear actuator, due to the electromagnetic property of a magnetic conductive material and the influence of an actuator structure, the actually output lorentz force has serious nonlinearity, and the electromagnetic force constant of the mover at two ends is smaller than that of the middle position. When the model of the electromagnetic linear actuator is built, nonlinearity of the magnetic field effect is ignored under the condition of low load, and the magnetic field intensity of the electromagnetic linear actuator is assumed to be a fixed value, so that the electromagnetic force is in direct proportion to the coil current. However, this assumption does not allow high-precision control of the actuator under high load conditions, and may even lead to instability of the system. Therefore, it is necessary to eliminate the influence of the nonlinear magnetic field effect, the electromagnetic force constantK m With the position of the moverxCan be approximated as a quadratic function;
(1)
in the middle ofA，B，CIs three unknowns, whereinA<0, the symmetry line of the function is between 0mm and 10mm, soB>0，C>0, the three unknowns belong to unpredictable internal state parameters, so online learning through parameter estimation is required to facilitate accurate control.xThe range of the value of (2) is 0-10mm. The advantage of such an approximated electromagnetic constant model is as follows:
（1）AandBis much smaller thanCTherefore, the driving force is not greatly different under the condition of low load, and the electromagnetic force constant can be approximated to be a fixed valueK m =C；
(2) This non-linear phenomenon of the magnetic field effect can be effectively represented by the quadratic and the first term of the formula (1), and the error is small in both the high load and the low load states.
The electromagnetic linear actuator system is a mechanical-electromagnetic mutual coupling mathematical model, and expressions of a magnetic circuit, a mechanical subsystem and a circuit subsystem are respectively as follows according to kirchhoff's voltage law, newton's second law and lorentz's law:
(2)
(3)
(4)
in the method, in the process of the invention,F m electromagnetic force generated in a magnetic field by an electrified coil;Nthe total number of turns of the coil;B e is the magnetic field strength;l e is the length of one turn of the coil;Iis coil current;K m is the electromagnetic force coefficient;Mis the dynamic mass of the actuator;F dis uncertainty errors and disturbances;uis a phase voltage;RandLcoil resistance and inductance;K e is the counter potential coefficient;vthe mover moving speed. The non-linear expression of the friction force is established based on the LuGre model, and is specifically as follows:
(5)
in the middle of，，Respectively stiffness coefficient, damping coefficient and viscosity coefficient;zis the internal friction state of the system;v s is the Stribeck speed; positive definite functiong(v) To describe the Stribeck phenomenon;F c is coulomb friction;F s is static friction force. The mathematical model constructed based on the above formula is shown in the electromagnetic linear actuator section of fig. 1.
Combining the obtained electromagnetic constant nonlinear formula (1) with an electromagnetic linear actuator system mathematical model, and establishing an overall system dynamics model of required control:
(6)
in the method, in the process of the invention,for the uncertainty term, since the weight coefficient in front of the variable is unknown, the motion of the electromagnetic linear actuator cannot be effectively controlled, so the parameter set θ= [ is definedθ 1, θ 2 ,θ 3 ,θ 4 ,θ 5 ,θ 6 ,θ 7 ] T These parameters are represented by the following formulasθ 1 =MR/C；θ 2 =K e +(σ 1 +σ 2 )R/C；θ 3 =AR/C；θ 4 =BR/C；θ 5 =σ 0 R/C；θ 6 =σ 1 R/C；θ 7 And->Respectively isF dis Constant and time-varying values of/C. The dynamics model of the system is redefined as:
(7)
although it isθAndis unknown, but often the range of variation and the range of uncertainty nonlinearities can be determined, so the following practical assumptions are given:
the parameter uncertainty and uncertainty nonlinear range are known, and the specific equations are as follows:
(8)
(9)
in the method, in the process of the invention,θ imin ，θ imax is thatθ i The value range is the most value, which is a known constant;is a known morphological equation.
Definition of auxiliary variablesr、Sliding mode-like variablepAnd position tracking erroreTo meet the estimation of parameters and facilitate the setting of the controller, the specific expression is:
(10)
in the method, in the process of the invention,x d for the displacement of the object to be achieved,k 1 andk r for feedback gain, the following can be obtained by combining equations (6) and (10):
(11)
the parameters are all within a known bounded set using a discontinuous projection mapping function, so the unknown parameters are estimated using the following adaptive law of projection modification:
(12)
(13)
(14)
in the method, in the process of the invention,and->Respectively estimating a parameter estimation value and an estimation error value; r is a positive diagonal matrix; psi is the row vector formed by the variables corresponding to the unknown parameters, i.e. +.>. There are the following ideal properties according to the adaptation law:
(15)
(16)
since the internal friction state z in the dynamic friction model is not measurable, the dynamic friction model is required to be measured for z 1 And z 2 And performing adaptive estimation. The dual observer based on the mapping function is therefore designed to observe the state quantity z and to make the system somewhat robust to modeling errors. The observer expression is:
(3.2) establishing a dual observer for observing the internal friction state in the dynamic friction model:
(17)
(18)
in the middle ofAnd->The estimated value of z in two different terms of equation (11) respectively,γ 1 andγ 2 the expression of the projection mapping function is shown as formula (14) as the compensation amount of the state observer.
By the characteristics, the influence caused by parameter uncertainty can be eliminated by using an adaptive algorithm, so that the accuracy of the system is improved. The specific design expression of the controller is as follows:
(19)
in the method, in the process of the invention,u a is an adaptive compensation term;u s is a robust feedback term;u s1 is a proportional feedback term for stabilizing the system;u s2 is an integral robust term.
Because the system has non-linear factors of uncertainty, the control effect cannot be well achieved only by a linear feedback term, and therefore the uncertainty influence of a model is weakened by adopting an integral robust term, and the expression is as follows:
(20)
in the method, in the process of the invention,integrating the robust feedback gain; sgn () is a sign function. A block diagram of the controller system is shown in fig. 1.
And (3) performing stability demonstration on the self-adaptive integral robust control system by using the Lyapunov function to obtain a progressive stable result of the system, and firstly giving the following quotients.
Defining auxiliary functionsIf the control gain satisfiesβ >δ 1 +δ 2 /k 2 ThenP(t) Is always more than 0 and is always more than 0,
(21)
in the method, in the process of the invention,π(0) Andp(0) Respectively representπAndpis set to be a constant value.
According to the above quotients, the Lyapunov function is defined as follows:
(22)
deriving formula (22):
(23)
substituting formulas (10) and (11) into (23):
(24)
from the nature of the adaptive law, it is possible to obtain:
(25)
in the method, in the process of the invention,c 1 ，c 2 ，k 3 the expression of (2) is as follows:
(26)
definition of the definitionE=[e,p,r] T ，
(28)
By adjusting parametersk,k 1 ,k r So that the symmetry matrix Λ is positive, there are:
(29)
the method is characterized by comprising the following steps: by adjusting parametersk,k 1 ,k r Can enableNegative setting andVpositive, therefore, the tracking error of the system tends to be 0 under the condition that the time tends to be infinite.
The electromagnetic linear actuator has the following specific parameters: coil resistance r=1.4Ω; coil equivalent inductance l=0.91 mH; actuator moving mass m=0.2 kg; the working range of the rotor is 0mm-10mm.
And (3) selecting parameters of a controller:
robust term parameters: k=400; k (k) 1 =700；k r =30; =50. Adaptive term parameters: θ min =[0 , 10, 0, 0, 0, 0, 0] T ，θ max =[1 , 50, 5, 0.5, 0.5, 0.05, 1] T Adaptive gain f=diag [0.8, 6500, 600, 1300, 0.03, 0.7, 5.3]。
Fig. 2 is a performance comparison of a step response of an electromagnetic linear actuator high-precision position control taking nonlinear electromagnetic force compensation into consideration with other control methods according to the present invention, fig. 3 is a comparison of a triangular wave signal displacement error under high load of an electromagnetic linear actuator high-precision position control taking nonlinear electromagnetic force compensation into consideration with uncompensated adaptive integral robust control according to the present invention, and fig. 4 is a comparison of a sine wave signal displacement error under high load of an electromagnetic linear actuator high-precision position control taking nonlinear electromagnetic force compensation into consideration with uncompensated adaptive integral robust control according to the present invention. In order to check the tracking performance and control accuracy of the control algorithm, the control algorithm is compared with a common PID algorithm and a common sliding mode algorithm by inputting a simulation curve of an 8mm step signal, and the simulation curve is shown in figure 2. The response time of the sliding mode control algorithm and the PID algorithm under the step target is 14.6ms and 18.2ms respectively, and the response time of the control algorithm is 11.4ms, compared with the PID control and the sliding mode control, the adaptive integral robust control taking nonlinear electromagnetic force compensation into consideration has better tracking performance. The stability errors of the PID algorithm and the sliding mode control algorithm are smaller than that of the adaptive integral robust control algorithm considering nonlinear electromagnetic force compensation, which shows that the control algorithm has higher control precision.
To check the effectiveness of nonlinear electromagnetic force compensation, triangular wave signals and sinusoidal signals are input, and the control algorithm is compared with the simulation of the common adaptive integral robust control under the condition of 450N load, and displacement error comparison is shown in figures 3 and 4. As can be seen by comparing the error results of the adaptive integral robust control algorithm with or without electromagnetic force compensation in fig. 3, to further verify it, a sinusoidal signal is set asx=sin(40πt) +5mm, the load was likewise set to 450N. As shown in FIG. 4, the displacement error is compared to determine that there is electromagnetic force compensationThe stability error of the compensated control algorithm is 0.011mm, the error of the control algorithm without electromagnetic force compensation is 0.007mm, and the stability error with electromagnetic force compensation is reduced by 27% for sinusoidal signals. The self-adaptive integral robust control algorithm with electromagnetic force compensation is obviously superior to the self-adaptive integral robust control algorithm without compensation, and the effectiveness of electromagnetic force compensation is further proved.
The foregoing is only a preferred embodiment of the invention, it being noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the present invention, and such modifications and adaptations are intended to be comprehended within the scope of the invention.
Claims (6)
1. A high-precision position control method of an electromagnetic linear actuator considering nonlinear electromagnetic force compensation comprises the following steps:
step 1, approximating electromagnetic force constant and rotor position as a quadratic function relation;
step 2, establishing an electromagnetic linear actuator system dynamics model considering the LuGre friction model;
step 3, establishing a dual observer for observing an internal friction state in the dynamic friction model, designing an adaptive compensation term and a parameter adaptive law, compensating nonlinear electromagnetic force and eliminating the influence caused by uncertainty of system parameters;
step 4, designing a proportional feedback term for stabilizing the system, and designing an integral robust term for weakening the influence caused by unknown nonlinear factors of the system;
and step 5, proving the system stability by applying the Lyapunov theory.
2. The method for controlling the position of the electromagnetic linear actuator with high precision taking into account nonlinear electromagnetic force compensation as recited in claim 1, wherein the step 1 is specifically as follows:
analyzing the electromagnetic property of the magnetic conductive material and the influence of the structure, wherein the actually output electromagnetic force has serious nonlinearity, and the electromagnetic force constant of the rotor at two ends is smaller than that of the rotor at the middle position, so that the electromagnetic force constant and the rotor position are approximately in a quadratic function relation:
(1)
in the middle ofA、B、CFor three unknown coefficients, the number of coefficients is,xis the displacement of the rotor.A、B、CObtained by electromagnetic force test or data fitting of simulation results.
3. The method for controlling the position of the electromagnetic linear actuator with high precision taking into account nonlinear electromagnetic force compensation according to claim 1, wherein the step 2 is specifically as follows:
the electromagnetic linear actuator system is a mechanical-electromagnetic mutual coupling mathematical model, and expressions of a magnetic circuit, a mechanical subsystem and a circuit subsystem are respectively as follows according to kirchhoff's voltage law, newton's second law and lorentz's law:
(2)
(3)
(4)
in the method, in the process of the invention,F m electromagnetic force generated in a magnetic field by an electrified coil;Nthe total number of turns of the coil;B e is the magnetic field strength;l e is the length of one turn of the coil;iis coil current;K m is the electromagnetic force coefficient;Mis the dynamic mass of the actuator; F dis uncertainty errors and disturbances;uis a phase voltage;RandLcoil resistance and inductance;K e is the counter potential coefficient;vis movingSub-movement speed. Establishing friction based on LuGre modelF f The nonlinear expression of (2) is as follows:
(5)
in the middle of，/>，/>Respectively stiffness coefficient, damping coefficient and viscosity coefficient;zis the internal friction state of the system;v s is the Stribeck speed; positive definite functiong(v) To describe the Stribeck phenomenon;F c is coulomb friction;F s is static friction force;
combining the obtained electromagnetic constant nonlinear formula (1) with an electromagnetic linear actuator system mathematical model, and establishing a system dynamics model of the electromagnetic linear actuator:
(6)
in the method, in the process of the invention,is an uncertain item.
4. The method for controlling the position of the electromagnetic linear actuator with high precision taking into account nonlinear electromagnetic force compensation according to claim 1, wherein the step 3 is specifically as follows:
(3.1) designing adaptive compensation terms and adaptive approach laws:
(7)
in the method, in the process of the invention,θ 1 =MR/C；θ 2 =K e +(σ 1 +σ 2 )R/C；θ 3 =AR/C；θ 4 =BR/C；θ 5 =σ 0 R/C；θ 6 =σ 1 R/C；θ 7 andrespectively isF dis Constant and time-varying values of/C;
the unknown parameters are estimated using the following projection-modified adaptive law:
(8)
(9)
(10)
in the method, in the process of the invention,and->Respectively estimating a parameter estimation value and an estimation error value; r is a positive diagonal matrix; psi is a row vector formed by variables corresponding to unknown parameters;
(3.2) establishing a dual observer for observing the internal friction state in the dynamic friction model:
(11)
(12)
in the middle ofAnd->The estimates of z in two different terms of equation (7) are made separately,γ 1 andγ 2 the expression form of the projection mapping function is formula (10) respectively the compensation quantity of the state observer.
5. The method for controlling the position of the electromagnetic linear actuator with high precision taking into account nonlinear electromagnetic force compensation according to claim 1, wherein the step 4 is specifically as follows:
(4.1) design position tracking erroreSliding mode-like variablepAuxiliary variablerThe following are provided:
(13)
in the method, in the process of the invention,x d for the displacement of the object to be achieved,k 1 andk r is the feedback gain;
(4.2) designing a linear feedback term and an integral robust term in the controller:
(14)
in the method, in the process of the invention,kis the gain that is being adjusted in a positive direction,u s2 the specific expression form of the integral robust term is as follows:
(15)
in the method, in the process of the invention,βto integrate the robust feedback gain, sgn () is a sign function.
6. The method for controlling the position of the electromagnetic linear actuator with high precision taking into account nonlinear electromagnetic force compensation according to claim 1, wherein the step 5 is specifically as follows:
system stability is proved by using Lyapunov theory, and auxiliary functions are defined
(16)
(17)
Definition of the lyapunov function:
(18)
stability analysis is performed by using the Lyapunov stability theory, and the control system is proved to be bounded and stable.
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