CN114296349B - Hysteresis control method and device for nanometer positioning platform - Google Patents
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Abstract
The invention discloses a hysteresis control method and a hysteresis control device for a nanometer positioning platform, which belong to the field of nanometer positioning control, and comprise the following steps: actual track data x of previous moment of piezoelectric positioning platform t‑1 The difference value between the reference track data and the reference track data is input into a predictive controller based on an iterative learning model to obtain a displacement control rate u at the current moment k (t)；u k (t) predicting the sum of the control rate and the iterative learning rate for the optimal model at the current moment; will u k (t) input NARX feedforward compensator to obtain drive voltage v at current moment t The method comprises the steps of carrying out a first treatment on the surface of the By v t Driving the piezoelectric positioning platform to displace so as to obtain actual track data x at the current moment t Will x t The displacement control rate u at the next moment is obtained by feeding back to the predictive controller k (t+1); the NARX feedforward compensator is connected in series with the piezoelectric positioning platform to form an approximately linear system. The invention can improve the control precision and enhance the anti-interference capability when the piezoelectric positioning platform works at different frequencies.
Description
Technical Field
The invention belongs to the technical field of nanometer positioning control, and particularly relates to a hysteresis control method and device of a nanometer positioning platform.
Background
In recent twenty years, piezoelectric driven nano positioning platforms are rapidly developed, and are widely applied to the current high-precision measurement and production and manufacture. The piezoelectric ceramic driven positioning platform plays a vital role in the fields of micro-nano level positioning systems such as atomic force microscopes, photoetching machines and the like. However, the piezoelectric driven positioning platform has strong nonlinear characteristics in the operation process, so that the positioning error is increased, and the problem of how to inhibit the huge error generated by the nonlinearity of the piezoelectric driven positioning platform so as to improve the positioning accuracy is to be solved.
The nonlinear phenomenon existing in the piezoelectric positioning platform mainly comprises characteristics of hysteresis, creep and the like, wherein the hysteresis is more widely focused. For hysteresis, scholars propose a plurality of effective modeling methods, such as Duhem models, bouc-Wen models, preisach models and the like, based on which an open-loop feedforward inverse compensation control algorithm is often adopted to control a piezoelectric positioning platform to improve precision, however, the control strategy is characterized by extremely high sensitivity to model precision, extremely high model precision is required for realizing accurate positioning, but in practice, large errors exist between the identified model parameters and a real system, and the piezoelectric positioning platform also has complex nonlinear characteristics such as creep.
Therefore, high-precision positioning is difficult to realize by feedforward inverse compensation control only, and the anti-jamming capability is poor, but the control rate is high. Closed loop feedback control often has strong immunity to interference and stability, but the controller bandwidth is limited. Therefore, the traditional control method is difficult to effectively realize high-precision control of the piezoelectric positioning platform.
Disclosure of Invention
Aiming at the defects or improvement demands of the prior art, the invention provides a hysteresis control method and a hysteresis control device for a nano positioning platform, which aim to effectively combine the advantages of high feedforward control bandwidth and high feedback control immunity to ensure that the piezoelectric positioning platform provides effective control precision at higher frequency, thereby solving the technical problem of low control precision of the piezoelectric positioning platform.
To achieve the above object, according to one aspect of the present invention, there is provided a hysteresis control method of a nano-positioning stage, comprising:
s1: actual track data x of previous moment of piezoelectric positioning platform t-1 And reference trajectory data x r t-1 Is input to a predictive controller based on an iterative learning model so as to obtain a displacement control rate u at the current moment of its output k (t); the saidCurrent displacement control rate u k (t) superposition of optimal model predictive control rate and iterative learning rate equal to the current moment;
s2: controlling the displacement control rate u k (t) inputting a nonlinear autoregressive model NARX feedforward compensator to cause the NARX feedforward compensator to output a drive voltage v at a current time t ；
S3: by means of the driving voltage v t Driving the piezoelectric positioning platform to displace so as to obtain actual track data x at the current moment t The actual track data x at the current moment t For feedback to the predictive controller to obtain the displacement control rate u at the next moment k (t+1)；
Wherein, the NARX feedforward compensator is connected in series with the piezoelectric positioning platform to form an approximate linear system.
In one embodiment, before S2, the method further includes:
s01: obtaining a mapping relation between driving voltage and displacement data corresponding to the piezoelectric positioning platform;
s02: and fitting according to the mapping relation to obtain an NARX model, and solving the inverse of the NARX model to obtain the NARX feedforward compensator.
In one embodiment, the S02 includes:
using the formulaAcquiring the NARX model, wherein the inverse of the NARX model is the NARX feedforward compensator: />
Wherein n is y And n v Is the maximum hysteresis term of the NARX model output and input, y k And v k Is the input and output of the NARX model at time t, e t To be an uncertain item, F l In the form of the NARX model; g l A feed-forward compensator characterizing the piezoelectric positioning stage.
In one embodiment, the S1 includes:
using the formulaCalculating the displacement control rate u at the time t k (t)；
Wherein,predicting a control rate for an optimal model at time t, < >>The method is characterized in that the method is a D-type differential iterative learning rate at the t moment, k is the iteration number, m is a control time domain mark, and l is an iterative learning rate mark.
In one embodiment, before S1, the method further includes:
using the formulaCalculating the optimal model prediction control rate at the time t;
wherein,the method is characterized in that the method is a coefficient matrix, Q and R are weight matrices respectively, the weight matrices are symmetric positive definite matrices, p is a prediction time domain, and T is matrix transposition; />And predicting the error of the kth iteration of the time domain for the time t+1.
In one embodiment, before S1, the method further includes:
using the formulaCalculating the D-type differential iteration learning rate of the kth iteration at the t moment;
wherein,d-type differential iteration learning rate of kth-1 time iteration at t moment Γδu k-1 ＝u k-1 (t)-u k-1 (t-1)，u k-1 (t) is the predictive control rate of the kth-1 th iteration at time t, u k-1 And (t-1) is the predictive control rate of the kth-1 iteration at the time t-1.
According to another aspect of the present invention, there is provided a hysteresis control apparatus of a nano-positioning stage, comprising:
a subtracter for integrating the actual track data x of the piezoelectric positioning platform at the previous moment t-1 And reference trajectory data x r t-1 Making a difference;
a predictive controller based on an iterative learning model, connected with the subtracter, for inputting the actual track data x of the previous moment t-1 And the reference trajectory data x r t-1 Outputs the displacement control rate u at the current moment k (t); the current displacement control rate u k (t) superposition of optimal model predictive control rate and iterative learning rate equal to the current moment;
NARX feedforward compensator connected to the predictive controller for inputting the displacement control rate u k (t) and outputting the drive voltage v at the current time t ；
A piezoelectric positioning platform connected with the NARX feedforward compensator and the subtracter and used for driving the voltage v t Executing actual track data x at the current time under action t And the actual track data x of the current moment t Feedback to the subtracter to make the subtracter obtain the actual track data x t And reference trajectory data x r t The difference value is input into the predictive controller, so that the displacement control rate u at the next moment is obtained k (t+1)。
In one embodiment, the NARX feedforward compensator obtains a NARX model by acquiring a mapping relation between driving voltage and displacement data corresponding to the piezoelectric positioning platform, and then fitting according to the mapping relation, and solves the inverse of the NARX model.
In general, the above technical solutions conceived by the present invention, compared with the prior art, enable the following beneficial effects to be obtained:
the invention adopts a control structure of feedforward compensation combined with feedback control, approximates the piezoelectric positioning platform to a linear system by a feedforward compensator, and improves the anti-interference capability and the positioning accuracy of the system by combining with feedback control; the feedback control strategy combines iterative learning control and model prediction control, so that the controller has good performance in both iterative domain and time domain, and has better effect than a single control algorithm.
Drawings
FIG. 1 is a schematic diagram of a hysteresis control device of a nano-positioning stage according to an embodiment of the present invention;
FIG. 2 is a hysteresis model effect diagram of the positioning data of the piezoelectric positioning platform and NARX model fitting according to an embodiment of the present invention;
FIG. 3 is a graph showing the tracking control effect of sinusoidal voltage with a tracking stroke of 0-10um and a frequency of 200Hz in an embodiment of the present invention;
FIG. 4 is a graph of absolute error of tracking control of sinusoidal voltage with a frequency of 200Hz for tracking strokes of 0-10um in an embodiment of the present invention;
FIG. 5 is a graph showing the tracking control effect of a triangular wave voltage with a tracking stroke of 0-10um and a frequency of 1000Hz in an embodiment of the present invention;
FIG. 6 is a graph showing the absolute error of a tracking control of a triangular wave voltage with a tracking stroke of 0-10um and a frequency of 1000Hz in an embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
As shown in fig. 1, the present invention provides a hysteresis control device of a nano positioning platform, comprising:
a subtracter for integrating the actual track data x of the piezoelectric positioning platform at the previous moment t-1 And reference trajectory data x r t-1 Making a difference;
the predictive controller based on the iterative learning model is connected with the subtracter and is used for inputting the actual track data x at the previous moment t-1 And reference trajectory data x r t-1 Outputs the displacement control rate u at the current moment k (t); current displacement control rate u k (t) is equal to the superposition of the optimal model predictive control rate and the iterative learning rate at the current moment.
The predictive controller based on the iterative learning model is used for realizing closed-loop feedback control, has relatively strong anti-interference capability in the iterative domain and the time domain, and can realize rapid and stable control.
NARX feedforward compensator connected to the predictive controller for inputting the displacement control rate u k (t) and outputting the drive voltage v at the current time t 。
Wherein, the NARX feedforward compensator is used for compensating hysteresis nonlinearity of the piezoelectric platform to make the controlled object more approximate to a linear system, wherein a nonlinear autoregressive model (NARX) is used as a polynomial model to model nonlinearity, which is expressed as:
n y and n v Is the maximum hysteresis term of system output and input, y k And v k Is the input and output of the system at time t, e t Represents an uncertainty item, F l Is a polynomial function representing its model form. The NARX feedforward compensator is an inverse of the model built with the NARX model, expressed as:
G l as nonlinear function, as feedforward compensator of piezoelectric platform, input reference output displacement y t Can output a reference voltage v t After being connected with the piezoelectric platform in series, the controlled object is theoretically approximate to a linear model.
A piezoelectric positioning platform connected with the NARX feedforward compensator and the subtracter and used for driving the voltage v t Executing actual track data x at the current time under action t And the actual track data x at the current moment t Feedback to the subtracter to make the subtracter obtain the actual track data x t And reference trajectory data x r t The difference value is input into a predictive controller, so that the displacement control rate u at the next moment is obtained k (t+1)。
The piezoelectric positioning platform is a complex nonlinear system as a controlled object, and can generate displacement to realize high-precision positioning by inputting voltage signals.
In one embodiment, the NARX feedforward compensator obtains the NARX model by acquiring the mapping relation between the driving voltage and the displacement data corresponding to the piezoelectric positioning platform, and then fitting according to the mapping relation, and solves the inverse of the NARX model.
The invention provides a hysteresis control method of a nanometer positioning platform, which comprises the following steps:
s1: actual track data x of previous moment of piezoelectric positioning platform t-1 And reference trajectory data x r t-1 Is input to a predictive controller based on an iterative learning model so as to obtain a displacement control rate u at the current moment of its output k (t); current displacement control rate u k (t) superposition of optimal model predictive control rate and iterative learning rate equal to the current moment;
s2: control rate u of displacement k (t) inputting a nonlinear autoregressive model NARX feedforward compensator to make the NARX feedforward compensator output the driving voltage v at the current moment t ；
S3: by means of a driving voltage v t Driving the piezoelectric positioning platform to displace so as to obtain actual track data x at the current moment t Actual track data x at the present moment t For feedback to the predictive controller to obtain the displacement control rate u at the next moment k (t+1)；
Wherein, NARX feedforward compensator and piezoelectric positioning platform connect in series to form approximate linear system.
In one embodiment, prior to S2, the method further comprises:
s01: obtaining a mapping relation between driving voltage and displacement data corresponding to the piezoelectric positioning platform;
specifically, the stroke of the piezoelectric positioning platform is 0-10um, sine wave input voltage and output displacement track data under 200Hz are respectively acquired, and triangular wave input voltage and output displacement track data under 1000Hz are processed in a smooth and denoising way.
S02: and fitting according to the mapping relation to obtain an NARX model, and solving the inverse of the NARX model to obtain the NARX feedforward compensator.
Specifically, collecting open-loop data of a piezoelectric positioning platform, applying driving voltage to the piezoelectric positioning platform, and collecting displacement data; utilizing collected open-loop data of the piezoelectric positioning platform, and solving NARX model parameters according to NARX model fitting; and calculating NARX feedforward compensator by solving NARX model parameters.
Wherein y is k ＝ay k-1 +bv 3k-1 +cv 2k-1 v 1k-1 +dv 3k-1 v 2k-1 y k-1 ，y k Is the output displacement of the piezoelectric positioning platform, v 1k Is the input voltage, v 2k ＝sign(v 1k -v 1k-1 ) Is about v 1k V of a multi-valued function of (v) 3k ＝sign(y k -y k-1 ) Is about y k Is a multi-valued function of (a). According to the input data, a least square linear regression method is adopted to fit model solving parameters a, b, c and d, and an example of model fitting results is shown in fig. 2.
According to the parameters obtained by fitting, calculating an NARX inverse model as follows:
the feedforward compensator is used as a piezoelectric positioning platform to approximate a controlled object to a linear system.
In one embodiment, S02 includes:
using the formulaAn NARX model is obtained, which is inverted to an NARX feedforward compensator: />
Wherein n is y And n v Is the maximum hysteresis term of NARX model output and input, y k And v k Is the input and output of NARX model at time t, e t To be an uncertain item, F l In the form of a NARX model; g l A feed-forward compensator characterizing a piezoelectric positioning stage.
Specifically, setting a control rate and an initial value of a predictive controller based on an iterative learning model, and applying a driving voltage to a piezoelectric control platform; collecting output displacement data of the piezoelectric control platform, and calculating interpolation, namely error, of the output displacement data and the reference displacement dataSolving the iterative learning rate in the iterative domain, summing the control rates at the next moment based on the prediction control rate of the iterative learning model, and repeating the steps until the system is stable.
In one embodiment, S1 comprises:
using the formulaCalculating the displacement control rate u at the time t k (t)；
Wherein,predicting a control rate for an optimal model at time t, < >>The method is characterized in that the method is a D-type differential iterative learning rate at the t moment, k is the iteration number, m is a control time domain mark, and l is an iterative learning rate mark.
Specifically, the predictive controller based on the iterative learning model is used for realizing closed-loop feedback control, has relatively strong anti-interference capability in the iterative domain and the time domain, and can realize rapid and stable control. The concrete structure is as follows:
state space model for linear systemRewritten as delta:
wherein the method comprises the steps ofδu k (t)＝u k (t)-u k (t-1), which is reduced to +.>And making differences between adjacent batches And->Is->And->Is a combination of (a); wherein δu k (t)＝u k (t)-u k (t-1)，/>Will e k ＝y r -y k Carrying out expansion along a prediction time domain and a control time domain to obtain a prediction equation of the iterative learning model, wherein the prediction equation is as follows:
k is the iteration number, the solving form of the control rate of the predictive controller based on the iterative learning model can be expressed as the following secondary performance index, m is the control time domain, p is the predictive time domain, and Q and R are symmetric positive definite matrixes respectively;
when meeting the requirementsWhen using optimality conditions->The prediction control rate of the optimal model is obtained,
the expression is as follows:wherein k is iteration times, m is control time domain identification, p is prediction time domain identification, D-type differential iteration learning rate is adopted on an iteration axis, and the expected track is ensured to be tracked rapidly and accurately, which is expressed as follows
Wherein Γ identifies the derivative Γδu k-1 ＝u k-1 (t)-u k-1 (t-1)。
Combining the model predictive control rate with the iterative learning rate to obtain the iterative learning model predictive control rate at the time t of the kth batch as follows,
specifically, an iterative learning model prediction control initial parameter is set, a prediction time domain and a control time domain p=m=10, and a weight matrix q=i 10×10 ，R＝0.03I 10×10 Setting reference expected tracks as a sine voltage signal with the amplitude of 0-10um and the frequency of 200Hz and a triangular wave voltage signal with the frequency of 1000Hz respectively;
according to the piezoelectric positioning platform, displacement data are output in real time, and the actual control quantity is calculated in real time by combining the error between the piezoelectric positioning platform and the reference expected track through the iterative learning rate and the model prediction control rate, so that the control quantity is continuously regulated, and the tracking positioning control with high precision is realized.
In the embodiment, the tracking effect on the 200Hz sine wave voltage signal is shown in fig. 3, the tracking error result is shown in fig. 4, the basic tracking error is kept within +/-0.1 um under the positioning range of 0-10um, the maximum tracking error is not more than 2%, the positioning accuracy is higher, the tracking error is rapidly reduced in the first period, and the characteristics of rapid convergence and stability are realized; the tracking effect of the 1000Hz triangular wave voltage signal is shown in fig. 5, the tracking error result is shown in fig. 6, and the basic tracking error is still kept within +/-0.1 um, so that the hysteresis tracking control method of the model inverse combination predictive controller based on the iterative learning model has good performance at higher frequency, and can be used for realizing a high-precision positioner.
In one embodiment, prior to S1, the method further comprises:
using the formulaCalculating the optimal model prediction control rate at the time t;
wherein,the method is characterized in that the method is a coefficient matrix, Q and R are weight matrices respectively, the weight matrices are symmetric positive definite matrices, p is a prediction time domain, and T is matrix transposition; />Predicting the error of the kth iteration of the time domain for time t+1And (3) difference.
In one embodiment, prior to S1, the method further comprises:
using the formulaCalculating the D-type differential iteration learning rate of the kth iteration at the t moment;
wherein,d-type differential iteration learning rate of kth-1 time iteration at t moment Γδu k-1 ＝u k-1 (t)-u k-1 (t-1)，u k-1 (t) is the predictive control rate of the kth-1 th iteration at time t, u k-1 And (t-1) is the predictive control rate of the kth-1 iteration at the time t-1.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (2)
1. The hysteresis control method of the nanometer positioning platform is characterized by comprising the following steps of:
s1: actual track data x of previous moment of piezoelectric positioning platform t-1 And reference trajectory data x r t-1 Is input to a predictive controller based on an iterative learning model so as to obtain a displacement control rate u at the current moment of its output k (t); the displacement control rate u at the current moment k (t) superposition of optimal model predictive control rate and iterative learning rate equal to the current moment;
s2: controlling the displacement control rate u k (t) inputting a nonlinear autoregressive model NARX feedforward compensator to cause the NARX feedforward compensator to output a drive voltage v at a current time t ；
S3: by means of the driving voltage v t Driving the piezoelectric positioning platform to displace so as to obtain actual track data x at the current moment t The current timeActual track data of inscription x t For feedback to the predictive controller to obtain the displacement control rate u at the next moment k (t+1); the NARX feedforward compensator is connected in series with the piezoelectric positioning platform to form an approximate linear system;
the method further comprises the steps of: s01: obtaining a mapping relation between driving voltage and displacement data corresponding to the piezoelectric positioning platform; s02: fitting according to the mapping relation to obtain an NARX model, and solving the inverse of the NARX model to obtain the NARX feedforward compensator; the S02 includes: using the formulaObtaining the NARX model, which is inverted to the NARX feedforward compensator: />n y And n v Is the maximum hysteresis term of the NARX model output and input, y k And v k Is the input and output of the NARX model at time t, e t To be an uncertain item, F l In the form of the NARX model; g l A feedforward compensator characterizing the piezoelectric positioning stage;
the S1 comprises the following steps: using the formulaCalculating the displacement control rate u at the time t k (t)；/>Predicting a control rate for an optimal model at time t, < >>The method is characterized in that the method is a D-type differential iterative learning rate at the t moment, k is the iteration number, m is a control time domain mark, and l is an iterative learning rate mark;
prior to S1, the method further comprises: using the formulaCalculating the optimal model prediction control rate at the time t; />The method is characterized in that the method is used as a coefficient matrix, Q and R are respectively used as weight matrices of a prediction time domain and a control time domain, the weight matrices are symmetrical positive definite matrices, p is the prediction time domain, and T is matrix transposition; />Predicting an error of a kth iteration of the time domain for a time t+1;
the method further comprises the steps of: using the formulaCalculating the D-type differential iteration learning rate of the kth iteration at the t moment; />D-type differential iteration learning rate of kth-1 time iteration at t moment, sigma delta u k-1 ＝u k-1 (t)-u k-1 (t-1)，u k-1 (t) is the predictive control rate of the kth-1 th iteration at time t, u k-1 And (t-1) is the predictive control rate of the kth-1 iteration at the time t-1.
2. A hysteresis control apparatus of a nano-positioning stage, for performing the hysteresis control method of a nano-positioning stage according to claim 1, comprising:
a subtracter for integrating the actual track data x of the piezoelectric positioning platform at the previous moment t-1 And reference trajectory data x r t-1 Making a difference;
a predictive controller based on an iterative learning model, connected with the subtracter, for inputting the actual track data x of the previous moment t-1 And the reference trajectory data x r t-1 Outputs the displacement control rate u at the current moment k (t); the current timeDisplacement control rate xk (t) superposition of optimal model predictive control rate and iterative learning rate equal to the current moment;
NARX feedforward compensator connected to the predictive controller for inputting the displacement control rate u k (t) and outputting the drive voltage v at the current time t ；
A piezoelectric positioning platform connected with the NARX feedforward compensator and the subtracter and used for driving the voltage v t Executing actual track data x at the current time under action t And the actual track data x of the current moment t Feedback to the subtracter to make the subtracter obtain the actual track data x t And reference trajectory data x r t The difference value is input into the predictive controller, so that the displacement control rate u at the next moment is obtained k (t+1)。
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