CN111368400B - Modeling identification method for piezoelectric micro-drive variable-frequency positioning platform based on PSO algorithm - Google Patents

Modeling identification method for piezoelectric micro-drive variable-frequency positioning platform based on PSO algorithm Download PDF

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CN111368400B
CN111368400B CN202010096421.4A CN202010096421A CN111368400B CN 111368400 B CN111368400 B CN 111368400B CN 202010096421 A CN202010096421 A CN 202010096421A CN 111368400 B CN111368400 B CN 111368400B
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positioning platform
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piezoelectric micro
drive variable
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冯颖
李颖
怀斯·艾哈迈德
甘胜利
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South China University of Technology SCUT
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Abstract

The invention discloses a PSO algorithm-based piezoelectric micro-drive variable frequency positioning platform modeling identification method, which aims at a piezoelectric micro-drive variable frequency positioning platform based on a piezoelectric ceramic driver, and measures the displacement of the piezoelectric ceramic driver in the vertical direction under the excitation of voltage signals with different frequencies through a laser displacement sensor; modeling the frequency-dependent symmetric hysteresis characteristic of the piezoelectric ceramic driver by adopting a rate-dependent Prandtl-Ishlinskii hysteresis model to describe strong nonlinearity related to the coupling effect between internal hysteresis and frequency, and establishing an internal dynamic model of a 'hysteresis + linearity' piezoelectric micro-drive variable frequency positioning system based on a Hammerstein cascade structure; the optimization PSO algorithm is adopted to identify the piezoelectric micro-drive variable frequency positioning platform system model, the identification algorithm overcomes the defects that the classic PSO algorithm is easy to fall into local optimization, the convergence is poor and the like, and the identification accuracy and the model effectiveness are guaranteed.

Description

Modeling identification method for piezoelectric micro-drive variable-frequency positioning platform based on PSO algorithm
Technical Field
The invention relates to the technical field of micro-nano positioning and driving, in particular to a piezoelectric micro-driving variable frequency positioning platform modeling identification method based on a PSO algorithm.
Background
In the last decade, micro-drive systems have become an important component of mechanical systems, showing unique features in the relevant fields of robotics, precision manufacturing, medicine, etc. As an actuating mechanism of the micro-driving system, the performance of the micro-driving component based on the intelligent material has great influence on the precision and controllability of the micro-driving system. The piezoelectric actuator shows superiority in the aspect of high-precision micro-driving by the characteristics of micro-nano operation precision, high response speed and the like, and can meet the special precision requirements of some micro-driving systems. Although the piezoelectric actuator has good performance, external disturbance and internal uncertainty of the piezoelectric actuator become major limiting factors for improving the driving accuracy, such as hysteresis effect, creep and rate-related characteristics are major nonlinear factors affecting the improvement of the driving performance. In order to overcome the adverse effects of piezoelectric actuator nonlinearity on the precision, reliability and stability of the micro-drive system, it is necessary to consider the internal nonlinearity of the piezoelectric actuator and accurately model its input-output characteristics.
At present, a great deal of researchers have conducted a great deal of research on the characteristics of the piezoelectric ceramic driver aiming at internal nonlinearity, most importantly hysteresis nonlinearity. At present, a commonly used modeling method for hysteresis nonlinearity is to describe hysteresis characteristics by using a hysteresis model, and simultaneously model the hysteresis nonlinearity by combining a linear dynamic system. In order to establish a model closer to the actual input and output characteristics of the piezoelectric ceramic driver, many scholars research a series of intelligent algorithms, such as genetic algorithm, neural network algorithm, ant colony algorithm, simulated annealing algorithm, particle swarm algorithm and the like. The particle swarm algorithm, namely the PSO algorithm, starts from a random value, finds an optimal value through iteration, is simple in rule, and adopts a real number encoding mode, so that the calculation is simpler and quicker. The algorithm has fast convergence and high precision, so the algorithm is widely applied to the application fields of optimization, identification and the like.
Disclosure of Invention
The invention aims to provide a modeling and identifying method of a piezoelectric micro-drive variable-frequency positioning platform based on a PSO algorithm, aiming at the piezoelectric micro-drive variable-frequency positioning platform with internal nonlinear factors. In the method, an optimized PSO algorithm is adopted to accurately model the input and output characteristics of the piezoelectric micro-drive variable-frequency positioning platform. The algorithm can overcome the defects that the classical particle swarm algorithm is easy to fall into local optimization, poor convergence and the like, so that a model capable of accurately describing the actual input and output characteristics of the piezoelectric ceramic driver can be established, and the operation precision and the stability of the piezoelectric micro-drive variable-frequency positioning platform are improved.
The purpose of the invention can be achieved by adopting the following technical scheme:
a modeling identification method of a piezoelectric micro-drive variable-frequency positioning platform based on a PSO algorithm comprises the following steps:
s1, building a piezoelectric micro-drive frequency conversion positioning platform based on a piezoelectric ceramic driver, and measuring the displacement of the piezoelectric ceramic driver in the vertical direction under the excitation of a frequency conversion voltage signal through a laser displacement measuring device;
s2, establishing an internal dynamic model of the piezoelectric micro-drive variable-frequency positioning platform, wherein the dynamic model of the piezoelectric ceramic driver is as follows:
Figure BDA0002385394170000021
the formula (1) is generalized as:
Figure BDA0002385394170000022
the linear dynamics is defined in part as
Figure BDA0002385394170000023
Wherein upsilon (t) is the driving voltage of the piezoelectric micro-driving frequency conversion positioning platform, and H upsilon](t) is the internal hysteresis characteristic of the piezoceramic driver, yoFor the displacement of the piezo-ceramic actuator in the vertical direction,
Figure BDA0002385394170000031
and
Figure BDA0002385394170000032
are each yoFirst and second order derivatives of (a)1＝Da/M，α0＝K/M，β＝Γυothe/M is related parameters of the internal electromechanical characteristics of the piezoelectric ceramic driver, and the internal hysteresis characteristics of the piezoelectric ceramic driver are described by adopting an RDPI hysteresis model, wherein the RDPI hysteresis model is short for rate-dependent Prandtl-Ishlinskii hysteresis model,
Figure BDA0002385394170000033
at the same time, the user can select the desired position,
Figure BDA0002385394170000034
where ρ is0Is a normal number, and is,
Figure BDA0002385394170000035
for the play operator, ρiWhere i is 1,2, …, n is the weight of the play operator and satisfies ρiNot less than 0, n is the number of play operators,
Figure BDA0002385394170000036
in order to be a function of the dynamic threshold,
Figure BDA0002385394170000037
is the first derivative of the drive voltage upsilon (t), sigma and mu are normal numbers, so that
Figure BDA0002385394170000038
Then there is
H[υ](t)＝ρ0υ(t)+d(t) (6)
I.e. the hysteresis characteristic may be decomposed into linear components p0V (t) and a bounded nonlinear perturbation portion d (t);
s3, identifying the internal dynamic model of the piezoelectric micro-drive variable-frequency positioning platform by adopting an optimized PSO algorithm, wherein the process is as follows:
s31, setting the motion range of the parameters, and randomly initializing the speed of the particles in the population
Figure BDA0002385394170000039
And position
Figure BDA00023853941700000310
The formula is as follows:
Figure BDA00023853941700000311
wherein epsilon is 1, 2.. L, epsilon represents the epsilon-th particle in the population, L is the size of the population,
Figure BDA00023853941700000312
Figure BDA00023853941700000313
for each particle dimension, i.e. the number of parameters to be identified,% represents the χ -th parameter, xminχ、xmaxχAre the chi th respectivelyMaximum and minimum values of the parameter positions, and, similarly, vminχ、vmaxχThe maximum value and the minimum value of the chi-th parameter during speed updating,
Figure BDA0002385394170000041
respectively the initial position and velocity, r, of the epsilon-th particle1、r2Is [0,1 ]]A random value within a range;
s32, calculating the fitness function value of each particle in the population to evaluate the particle, wherein the fitness value of the epsilon-th particle is,
Figure BDA0002385394170000042
wherein
Figure BDA0002385394170000043
Wherein eta is 1,2, …, T is the number of samples selected when identifying parameters,
Δid＝[Δid(1),Δid(2),…,Δid(η),…,Δid(T)]in order to output the model, the model is output,
Δo＝[Δo(1),Δo(2),…,Δo(η),…,Δo(T)]outputting for experimental measurement;
s33, updating the speed and the position of the particles, and calculating the formula as follows:
Figure BDA0002385394170000044
xε(γ+1)＝xε(γ)+vε(γ+1) (11)
wherein γ is 1,2,. xi, γ is the current iteration number, xi is the maximum iteration number, xεIs the position of the epsilon particle, vεIs the velocity of the epsilon particle, pεIs the historical optimum, p, of the epsilon-th particlebIs made as a wholeGlobal optimum of the population, paFor a randomly selected particle in the population, zeta, and kappa are [0,1 ]]Random values in the range, W (γ) being the inertial weight iterated to the γ th, C1(. gamma.) and C2(γ) is the learning factor iterated to the γ -th time,
at the same time, the mutation operation is added,
r5when the carbon content is more than 0.95,
Figure BDA0002385394170000051
and has xεχ＝xminχ+|pmχ-xminχ|*r7，
Wherein r is5、r6And r7Is in the range of [0,1]A random value within a range;
s34, updating the fitness function value of the particle, updating the individual historical optimal value and the global optimal value by using the following rules,
Figure BDA0002385394170000052
and S35, if a termination condition is met, namely gamma xi, the algorithm is terminated, otherwise, the iteration number is added with 1, and the step S33 is returned.
Further, the piezoelectric micro-driving frequency conversion positioning platform comprises: a piezoelectric ceramic driver, a data acquisition and signal transmission device, a voltage amplifier and a laser displacement measuring device,
the data acquisition and signal transmission device adopts a dSPACE real-time simulation system of an A/D, D/A converter with the configuration precision reaching 16 bits, the dSPACE real-time simulation system is connected with an upper computer and receives a control signal generated by the upper computer, the control signal is subjected to D/A conversion to form an analog voltage signal, the signal is amplified by a voltage amplifier connected with the data acquisition and signal transmission device and then drives a piezoelectric ceramic driver to generate displacement in the vertical direction, and a laser displacement measurement device converts the measured displacement into a digital signal through an A/D conversion module of the dSPACE real-time simulation system in the data acquisition and signal transmission device and then transmits the digital signal back to the upper computer.
Further, the air conditioner is provided with a fan,let us (t) at any time be in the time domain [0, gamma ]]All-up-continuous, play operator
Figure BDA0002385394170000053
The definition is as follows,
Figure BDA0002385394170000054
wherein t ∈ (t)r-1,tr]And 0 ═ t0＜t1＜…＜tr-1＜tr＜…tNγ, i.e. the time domain [0, γ ]]Divided into N sub-intervals, [ t ]r-1,tr]Is one of the subintervals, and upsilon (t) is monotonous in each subinterval and t is0At the moment of time, the time of day,
Figure BDA0002385394170000061
further, the random particles p selected at the time of the velocity updateaThe definition is as follows,
when r is3Is greater than 0.8, and
Figure BDA0002385394170000062
when there is pa＝xl；
Otherwise pa＝xεWherein x islIs the first particle in the population.
Further, the inertial weight employed in the velocity update is defined as follows,
Figure BDA0002385394170000063
Winiand WendAn initial value and an end value of W (γ), respectively.
Further, the learning factor employed in the speed update is defined as follows:
Figure BDA0002385394170000064
C1iniand C1endAre respectively C1Initial and terminal values of (. gamma.), C2iniAnd C2endAre respectively C2Initial and terminal values of (γ).
Further, when mutation operation is performed, pm＝[pm1,pm2,pm3,…pmχ]The definition is as follows,
Figure BDA0002385394170000065
i.e. pmIs the average of the historical optima of the particles in the population, pmχIs pmThe χ component of (a).
Compared with the prior art, the invention has the following advantages and effects:
the modeling identification method of the piezoelectric micro-drive variable-frequency positioning platform provided by the invention adopts an optimized PSO algorithm to accurately model the input and output characteristics of the piezoelectric micro-drive variable-frequency positioning platform. The algorithm overcomes the defects that the classical particle swarm algorithm is easy to fall into local optimization, and has poor convergence and the like. The established model can accurately describe the input and output characteristics of the piezoelectric micro-drive platform under different frequencies, and has important significance for improving the operation precision and stability of the piezoelectric micro-drive variable-frequency positioning platform.
Drawings
FIG. 1 is a schematic structural diagram of a piezoelectric micro-drive frequency-conversion positioning platform according to the present invention;
FIG. 2 is a schematic diagram of an internal dynamic model of a piezoelectric micro-drive variable frequency positioning platform according to the present invention;
FIG. 3 is a flow chart of the identification of the optimized PSO algorithm of the present invention;
fig. 4 is a comparison graph of the actual input/output characteristics of the piezoelectric micro-drive frequency-variable positioning platform of the present invention and the internal dynamic model input/output characteristics of the piezoelectric micro-drive frequency-variable positioning platform, wherein,
FIG. 4(a) is a graph comparing the system model output with the measured actual output of the piezoelectric microactuator stage for an input signal frequency of 1 Hz;
FIG. 4(b) is a graph comparing the system model output with the measured actual output of the piezoelectric microactuator platform for an input signal frequency of 20 Hz;
FIG. 4(c) is a graph comparing the system model output with the measured actual output of the piezoelectric microactuator platform for an input signal frequency of 40 Hz;
FIG. 4(d) is a graph comparing the system model output with the measured actual output of the piezoelectric microactuator stage for an input signal frequency of 60 Hz.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Examples
The modeling and identifying method for the piezoelectric micro-drive variable-frequency positioning platform based on the PSO algorithm comprises the following steps:
s1, building a piezoelectric micro-driving platform
In order to realize accurate modeling of a piezoelectric micro-drive positioning platform under a variable frequency condition, a piezoelectric micro-drive platform is built based on a P-844 type piezoelectric ceramic driver of a PI company, and the displacement of the piezoelectric ceramic driver in the vertical direction under the excitation of voltage signals with different frequencies is measured by a laser displacement measurement system. As shown in fig. 1, the experimental apparatus comprises the following specific components:
data acquisition and signal transmission system: the dSPACE real-time simulation system with the A/D, D/A converter with the precision as high as 16 bits is connected with an upper computer to realize data acquisition and signal transmission work and carry out real-time analysis and processing on the acquired data.
Piezoelectric ceramic driver: the maximum displacement of the piezoelectric ceramic actuator can reach 90 μm by using a P-844 piezoelectric ceramic actuator manufactured by PI company.
A voltage amplifier: the output voltage range of the adopted E-617 type voltage amplifier is-30V to +120V, and the voltage signal output by the D/A converter of the dSPACE is amplified to drive the piezoelectric ceramic driver.
Laser displacement measuring device: the laser displacement measuring device is composed of an LK-H008 type laser displacement sensing head of the Kinzhi company and an LK-G5001 type displacement signal amplifier of the Kinzhi company.
The dSPACE real-time simulation system converts a control signal generated by an upper computer through D/A (digital/analog) conversion to form an analog voltage signal, and the signal is amplified by a voltage amplifier and then drives a piezoelectric ceramic driver to generate displacement in the vertical direction. The displacement measured by the laser displacement measuring device is converted into a digital signal through an A/D conversion module of the dSPACE real-time simulation system and then transmitted back to the upper computer.
S2, establishing an internal dynamic model of the piezoelectric micro-drive variable-frequency positioning platform, wherein the dynamic model of the piezoelectric ceramic driver is as follows:
Figure BDA0002385394170000081
the formula (1) is generalized as:
Figure BDA0002385394170000082
the linear dynamics is defined in part as
Figure BDA0002385394170000083
Wherein upsilon (t) is the driving voltage of the piezoelectric micro-driving platform, and H upsilon](t) is the internal hysteresis characteristic of the piezoceramic driver, yoFor the displacement of the piezo-ceramic actuator in the vertical direction,
Figure BDA0002385394170000091
and
Figure BDA0002385394170000092
are each yoFirst and second order derivatives of (a)1＝Da/M，α0＝K/M，β＝Γυothe/M is related parameters of the internal electromechanical characteristics of the piezoelectric ceramic driver, and the internal hysteresis characteristics of the piezoelectric ceramic driver are described by adopting an RDPI hysteresis model, wherein the RDPI hysteresis model is short for rate-dependent Prandtl-Ishlinskii hysteresis model,
Figure BDA0002385394170000093
at the same time, the user can select the desired position,
Figure BDA0002385394170000094
where ρ is0Is a normal number, and is,
Figure BDA0002385394170000095
for the play operator, ρi(i is 1,2, …, n) is a weight of the play operator, and ρ is satisfiediNot less than 0, n is the number of play operators,
Figure BDA0002385394170000096
in order to be a function of the dynamic threshold,
Figure BDA00023853941700000913
is the first derivative of the drive voltage upsilon (t), sigma and mu are normal numbers, so that
Figure BDA0002385394170000098
Then there is
H[υ](t)＝ρ0υ(t)+d(t) (6)
I.e. the hysteresis characteristic may be decomposed into linear components p0V (t) and a bounded nonlinear perturbation portion d (t);
s3, identifying the internal dynamic model of the piezoelectric micro-drive variable-frequency positioning platform by adopting an optimized PSO algorithm, wherein the process is as follows:
s31, setting the motion range of the parameters, and randomly initializing the speed of the particles in the population
Figure BDA0002385394170000099
And position
Figure BDA00023853941700000910
The formula is as follows:
Figure BDA00023853941700000911
wherein epsilon is 1, 2.. L, epsilon represents the epsilon-th particle in the population, L is the size of the population,
Figure BDA00023853941700000912
for each particle dimension, i.e. the number of parameters to be identified,% represents the χ -th parameter, xminχ、xmaxχRespectively the maximum and minimum of the chi-th parameter position, and, similarly, vminχ、vmaxχThe maximum value and the minimum value of the chi-th parameter during speed updating,
Figure BDA0002385394170000101
respectively the initial position and velocity, r, of the epsilon-th particle1、r2Is [0,1 ]]A random value within a range;
s32, calculating the fitness function value of each particle in the population to evaluate the particle, wherein the fitness value of the epsilon-th particle is,
Figure BDA0002385394170000102
wherein
Figure BDA0002385394170000103
Where η is 1,2, …, T is the number of samples when identifying the parameter,
Δid＝[Δid(1),Δid(2),…,Δid(η),…,Δid(T)]in order to output the model, the model is output,
Δo＝[Δo(1),Δo(2),…,Δo(η),…,Δo(T)]outputting for experimental measurement;
s33, updating the speed and the position of the particles, and calculating the formula as follows:
Figure BDA0002385394170000104
xε(γ+1)＝xε(γ)+vε(γ+1) (11)
wherein γ is 1,2,. xi, γ is the current iteration number, xi is the maximum iteration number, xεIs the position of the epsilon particle, vεIs the velocity of the epsilon particle, pεIs the historical optimum, p, of the epsilon-th particlebFor a global optimum of the entire population, paFor a randomly selected particle in the population, zeta, and kappa are [0,1 ]]Random values in the range, W (γ) being the inertial weight iterated to the γ th, C1(. gamma.) and C2(γ) is the learning factor iterated to the γ -th time,
at the same time, the mutation operation is added,
r5when the carbon content is more than 0.95,
Figure BDA0002385394170000111
and has xεχ＝xminχ+|pmχ-xminχ|*r7，
Wherein r is5、r6And r7Is in the range of [0,1]A random value within a range;
s34, updating the fitness function value of the particle, updating the individual historical optimal value and the global optimal value by using the following rules,
Figure BDA0002385394170000112
and S35, if a termination condition is met, namely gamma xi, the algorithm is terminated, otherwise, the iteration number is added with 1, and the step S33 is returned.
Wherein, let us (t) at any time be in time domain [0, gamma ]]All-up-continuous, play operator
Figure BDA0002385394170000113
The definition is as follows,
Figure BDA0002385394170000114
t∈(tr-1,tr]and 0 ═ t0＜t1＜…＜tr-1＜tr＜…tNγ, i.e. the time domain [0, γ ]]Divided into N sub-intervals, [ t ]r-1,tr]Is one of the subintervals, and upsilon (t) is monotonic over each subinterval. At t0At the moment of time, the time of day,
Figure BDA0002385394170000115
wherein the random particles p are selected during the velocity update in the algorithmaThe definition is as follows,
when r is3Is greater than 0.8, and
Figure BDA0002385394170000116
when there is pa＝xl；
Otherwise pa＝xεWherein x islIs the first particle in the population.
Wherein, the inertia weight adopted in the speed updating in the algorithm is defined as follows,
Figure BDA0002385394170000117
Winiand WendAn initial value and an end value of W (γ), respectively.
The learning factor adopted in the speed updating in the algorithm is defined as follows:
Figure BDA0002385394170000121
C1iniand C1endAre respectively C1Initial and terminal values of (. gamma.), C2iniAnd C2endAre respectively C2Initial and terminal values of (γ).
Wherein, when mutation operation is performed in the algorithm, pm＝[pm1,pm2,pm3,…pmχ]The definition is as follows,
Figure BDA0002385394170000122
i.e. pmIs the average of the historical optima of the particles in the population, pmχIs pmThe χ component of (a).
Setting the population size to be L-150; the number of iterations of the algorithm is 500; if the number of the play operators is n-12, the number of the parameters to be identified of the system model is
Figure BDA0002385394170000123
The initial value and the end value of the inertia weight W (gamma) are set to W respectivelyini＝0.8，Wend0.1; learning factor C1(γ)，C2The initial value and the end value of (gamma) are set to C1ini＝2.4，C1end＝1.6，C2ini＝1.6，C2end2.4. FIG. 4 is a comparison of system model output and measured actual output of a piezoelectric microactuator stage for different frequency input signals. The good fitting effect in the experimental result chart shows that the system model obtained by the identification can accurately describe the input and output characteristics of the piezoelectric micro-drive positioning platform under the variable frequency condition, and the effectiveness of the method provided by the invention is proved.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims (7)

1. A modeling identification method for a piezoelectric micro-drive variable-frequency positioning platform based on a PSO algorithm is characterized by comprising the following steps:
s1, building a piezoelectric micro-drive frequency conversion positioning platform based on a piezoelectric ceramic driver, and measuring the displacement of the piezoelectric ceramic driver in the vertical direction under the excitation of a frequency conversion voltage signal through a laser displacement measuring device;
s2, establishing an internal dynamic model of the piezoelectric micro-drive variable-frequency positioning platform, wherein the dynamic model of the piezoelectric ceramic driver is as follows:
Figure FDA0002385394160000011
the formula (1) is generalized as:
Figure FDA0002385394160000012
the linear dynamics is defined in part as
Figure FDA0002385394160000013
Wherein upsilon (t) is the driving voltage of the piezoelectric micro-driving frequency conversion positioning platform, and H upsilon](t) is the internal hysteresis characteristic of the piezoelectric ceramic driver, yo is the displacement of the piezoelectric ceramic driver in the vertical direction,
Figure FDA0002385394160000014
and
Figure FDA0002385394160000015
first and second derivatives, alpha, of yo1＝Da/M，α0＝K/M，β＝Γυothe/M is related parameters of the internal electromechanical characteristics of the piezoelectric ceramic driver, and the internal hysteresis characteristics of the piezoelectric ceramic driver are described by adopting an RDPI hysteresis model, wherein the RDPI hysteresis model is short for rate-dependent Prandtl-Ishlinskii hysteresis model,
Figure FDA0002385394160000016
at the same time, the user can select the desired position,
Figure FDA0002385394160000021
where ρ is0Is a normal number, and is,
Figure FDA0002385394160000022
for the play operator, ρiWhere i is 1,2, …, n is the weight of the play operator and satisfies ρiNot less than 0, n is the number of play operators,
Figure FDA0002385394160000023
in order to be a function of the dynamic threshold,
Figure FDA0002385394160000024
is the first derivative of the drive voltage upsilon (t), sigma and mu are normal numbers, so that
Figure FDA0002385394160000025
Then there is
H[υ](t)＝ρ0υ(t)+d(t) (6)
I.e. the hysteresis characteristic may be decomposed into linear components p0V (t) and a bounded nonlinear perturbation portion d (t);
s3, identifying the internal dynamic model of the piezoelectric micro-drive variable-frequency positioning platform by adopting an optimized PSO algorithm, wherein the process is as follows:
s31, setting the motion range of the parameters, and randomly initializing the speed of the particles in the population
Figure FDA0002385394160000026
And position
Figure FDA0002385394160000027
The formula is as follows:
Figure FDA0002385394160000028
wherein epsilon is 1, 2.. L, epsilon represents the epsilon-th particle in the population, L is the size of the population,
Figure FDA0002385394160000029
Figure FDA00023853941600000210
for each particle dimension, i.e. the number of parameters to be identified,% represents the χ -th parameter, xminχ、xmaxχRespectively the maximum and minimum of the chi-th parameter position, and, similarly, vminχ、vmaxχThe maximum value and the minimum value of the chi-th parameter during speed updating,
Figure FDA00023853941600000211
respectively the initial position and velocity, r, of the epsilon-th particle1、r2Is [0,1 ]]A random value within a range;
s32, calculating the fitness function value of each particle in the population to evaluate the particle, wherein the fitness value of the epsilon-th particle is,
Figure FDA00023853941600000212
wherein
Figure FDA0002385394160000031
Wherein eta is 1,2, …, T is the number of samples selected when identifying parameters,
Δid＝[Δid(1),Δid(2),…,Δid(η),…,Δid(T)]in order to output the model, the model is output,
Δo＝[Δo(1),Δo(2),…,Δo(η),…,Δo(T)]outputting for experimental measurement;
s33, updating the speed and the position of the particles, and calculating the formula as follows:
Figure FDA0002385394160000032
xε(γ+1)＝xε(γ)+vε(γ+1) (11)
wherein γ is 1,2,. xi, γ is the current iteration number, xi is the maximum iteration number, xεIs the position of the epsilon particle, vεIs the velocity of the epsilon particle, pεIs the historical optimum, p, of the epsilon-th particlebFor a global optimum of the entire population, paFor a randomly selected particle in the population, zeta, and kappa are [0,1 ]]Random values in the range, W (γ) being the inertial weight iterated to the γ th, C1(. gamma.) and C2(γ) is the learning factor iterated to the γ -th time,
at the same time, the mutation operation is added,
r5when the carbon content is more than 0.95,
Figure FDA0002385394160000033
and has xεχ＝xminχ+|pmχ-xminχ|*r7，
Wherein r is5、r6And r7Is in the range of [0,1]Within range ofA machine value;
s34, updating the fitness function value of the particle, updating the individual historical optimal value and the global optimal value by using the following rules,
Figure FDA0002385394160000034
and S35, if a termination condition is met, namely gamma xi, the algorithm is terminated, otherwise, the iteration number is added with 1, and the step S33 is returned.
2. The PSO algorithm-based piezoelectric micro-drive variable-frequency positioning platform modeling and identifying method as claimed in claim 1, wherein the piezoelectric micro-drive variable-frequency positioning platform comprises: a piezoelectric ceramic driver, a data acquisition and signal transmission device, a voltage amplifier and a laser displacement measuring device,
the data acquisition and signal transmission device adopts a dSPACE real-time simulation system of an A/D, D/A converter with the configuration precision reaching 16 bits, the dSPACE real-time simulation system is connected with an upper computer and receives a control signal generated by the upper computer, the control signal is subjected to D/A conversion to form an analog voltage signal, the signal is amplified by a voltage amplifier connected with the data acquisition and signal transmission device and then drives a piezoelectric ceramic driver to generate displacement in the vertical direction, and a laser displacement measurement device converts the measured displacement into a digital signal through an A/D conversion module of the dSPACE real-time simulation system in the data acquisition and signal transmission device and then transmits the digital signal back to the upper computer.
3. The PSO algorithm-based piezoelectric micro-drive frequency-conversion positioning platform modeling and identifying method as claimed in claim 1, wherein upsilon (t) at any moment is assumed to be in time domain [0, upsilon ™]All-up-continuous, play operator
Figure FDA0002385394160000041
The definition is as follows,
Figure FDA0002385394160000042
wherein t ∈ (t)r-1,tr]And 0 ═ t0＜t1＜…＜tr-1＜tr＜…tNγ, i.e. time domain [0, γ]Divided into N sub-intervals, [ t ]r-1,tr]Is one of the subintervals, and upsilon (t) is monotonous in each subinterval and t is0At the moment of time, the time of day,
Figure FDA0002385394160000043
4. the PSO algorithm-based piezoelectric micro-drive variable-frequency positioning platform modeling identification method as claimed in claim 1, wherein random particles p selected during speed updateaThe definition is as follows,
when r is3Is greater than 0.8, and
Figure FDA0002385394160000044
when there is pa＝xl；
Otherwise pa＝xεWherein x islIs the first particle in the population.
5. The PSO algorithm-based piezoelectric micro-drive variable-frequency positioning platform modeling identification method according to claim 1, wherein the inertia weight adopted in the speed update is defined as follows,
Figure FDA0002385394160000051
Winiand WendAn initial value and an end value of W (γ), respectively.
6. The PSO algorithm-based piezoelectric micro-drive variable-frequency positioning platform modeling and identifying method as claimed in claim 4, wherein the learning factor adopted in the speed updating is defined as follows:
Figure FDA0002385394160000052
C1iniand C1endAre respectively C1Initial and terminal values of (. gamma.), C2iniAnd C2endAre respectively C2Initial and terminal values of (γ).
7. The PSO algorithm-based piezoelectric micro-drive variable-frequency positioning platform modeling and identifying method as claimed in claim 1, wherein p is p during a mutation operationm＝[pm1,pm2,pm3,…pmχ]The definition is as follows,
Figure FDA0002385394160000053
i.e. pmIs the average of the historical optima of the particles in the population, pmχIs pmThe χ component of (a).
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