CN107479379A - Piezoelectricity pottery driver feedforward and closed loop composite control method, system based on genetic algorithm - Google Patents
Piezoelectricity pottery driver feedforward and closed loop composite control method, system based on genetic algorithm Download PDFInfo
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Abstract
The present invention relates to a kind of piezoelectricity pottery driver feedforward based on genetic algorithm and closed loop composite control method and system, feedforward and closed loop complex controll are carried out to described piezoelectricity pottery driver with PID closed loops using feed forward models and the feedforward of PID closed-loop control systems composition, improve the control accuracy of piezoelectric ceramics controller.Piezoelectricity pottery driver feedforward and closed loop composite control method and system of the present invention based on genetic algorithm, with the feedforward PID complex control algorithms of present main flow, the closed-loop control for driver of being made pottery to piezoelectricity is realized with genetic algorithm searching optimum PID parameter.Genetic algorithm of the present invention can find suitable functions within the limits prescribed, and adjusting for tri- coefficients of PID is carried out using genetic algorithm, realize the more preferable closed-loop control to piezoelectricity pottery driver.
Description
Technical Field
The invention relates to a piezoelectric ceramic driver feedforward and closed loop composite control method and system based on a genetic algorithm.
Background
The hysteresis model feedforward compensation of the piezoelectric ceramics has errors, the open-loop control has poor anti-interference performance, and the closed-loop control has a feedback link and can effectively resist interference. The PID controller obtains a control deviation e (t) according to a given value r (t) and an actual output value f (t), namely e (t) = r (t) -f (t), and then forms a control variable by linear combination of proportion, integral and differential of the deviation value to control a controlled object, wherein the PID controller is a linear controller, and the control law of the PID controller is as follows:
in the formula, k p -a scaling factor; t is a unit of I -an integration time constant; t is D -a differential time constant; e (t)
-a deviation signal.
Briefly, the functions of the calibration elements of the PID controller are as follows.
And a proportion link P: the deviation signal e (t) of the controller system is proportionally reflected and as soon as the deviation occurs, the controller immediately generates a control action to reduce the deviation.
And an integration link I: the method is mainly used for eliminating the static error and improving the non-difference of the system. The strength of the integration depends on the integration time constant T I ，T I The larger the integral, the weaker and vice versa the stronger.
And a differentiation link D: reflects the variation trend (change rate) of the deviation signal and can introduce an effective early correction signal into the system before the deviation signal becomes too large, thereby accelerating the action speed of the system and reducing the regulation time.
The control effect of the PID controller depends mainly on k p ，k i ，k d The adjustment of the three parameters, and therefore how to obtain the most efficient corresponding parameter, is a key to the design of the PID controller. Optimization of PID controller parameters becomes a concern, and directly influences the quality of control effect, and has an inseparable relationship with safe and economic operation of a system. At present, the optimization methods of PID parameters are many, such as indirect optimization method, gradient method, climbing method and the like, and the simplex method and the expert setting method in a thermal engineering system are widely applied. Although the methods have good optimization characteristics, the methods have some disadvantages, and the simplex method is sensitive to initial values and is easy to fall into a local optimal solution, so that the optimization fails. The expert setting rule needs too much experience, different objective functions correspond to different experiences, and the arrangement of the knowledge base is a long-time project.
In addition, the inherent hysteresis nonlinearity of piezoelectric ceramic actuators is a major factor that limits the control accuracy of precision positioning systems. The hysteresis nonlinear characteristic of the piezoelectric ceramic is macroscopically represented as: for an input voltage, the displacement output of a positive stroke (a voltage rising process) is not overlapped with the displacement output of a reverse stroke (a voltage falling process), a displacement difference exists, the displacement difference is expressed as a multi-value mapping relation, and a hysteresis static curve of the input voltage output displacement is expressed by the following remarkable characteristics: (1) Multivalue, i.e. its output is not unique under the same input condition; (2) The output of the actuator at the next moment is not only dependent on the input and the output at the current moment, but also related to the previous input state; (3) Rate-dependent, i.e. as the input frequency increases, the output hysteresis behavior also increases, and the displacement output generated by the input voltages with the same amplitude and different frequencies also varies, which generally means that the output displacement amount is smaller as the frequency is higher. The hysteresis characteristic makes the output of the driver unpredictable, reduces the performance of the driver and seriously affects the stability of the high-precision motion positioning system.
To limit the positioning error caused by the hysteresis nonlinearity of the piezoelectric ceramic, the hysteresis of the piezoelectric ceramic is modeled. At present, the main hysteresis models include a Preisach model, a PI model, a polynomial model, a Maxwell model and the like. The Preisach model has double integration and more parameters which are difficult to identify, the polynomial model can describe large ring nonlinearity more accurately, but has lower precision for describing small ring hysteresis nonlinearity, and the Maxwell model has physical significance but can only describe a symmetrical hysteresis process. The PI model is used as a model for comparing classical description hysteresis characteristics and is obtained by superposing a plurality of hysteresis operators with different weights. The classical PI model can describe the hysteresis nonlinear process with fewer parameters and no error accumulation, but can only describe the symmetric hysteresis process, but the hysteresis curve of the piezoelectric ceramic is asymmetric.
In view of the above-mentioned drawbacks, the present designer is actively making research and innovation to create a feedforward and closed-loop composite control method for piezoelectric ceramic actuator based on genetic algorithm, so that the method has industrial value.
Disclosure of Invention
In order to solve the technical problems, the invention aims to provide a piezoelectric ceramic driver feedforward and closed loop composite control method and system based on a genetic algorithm with good control effect.
In order to achieve the aim, the piezoelectric ceramic driver feedforward and closed loop composite control method based on the genetic algorithm inputs the expected displacement r (t) of the piezoelectric ceramic driver into a feedforward model, and the feedforward model calculates to obtain the initial driving voltage u of the piezoelectric ceramic driver according to the expected displacement r (t) ε (t); the initial driving voltage u is fed back by the PID controller by taking the displacement of the piezoelectric ceramic or the displacement of the piezoelectric ceramic driver as a feedback signal of the PID controller ε (t) eliminating errors to obtain the driving voltage of the piezoelectric ceramic driver, and utilizing a genetic algorithm to adjust a proportion adjustment parameter k of the PID controller p Integral adjustment parameter k i Differential adjustment parameter k d And (6) optimizing.
Further, the feedforward model is an improved PI hysteresis inversion model, and the improved PI hysteresis inversion model is obtained by inverting the improved PI hysteresis model;
wherein the improved PI lag model is expressed as: the improved PI hysteresis model is divided into a rising part and a falling part, one group of weights is adopted when the input voltage is increased, and the other group of weights is adopted when the input voltage is decreased; the improved PI hysteresis model is expressed by the following equation,
wherein z (t) is the displacement of the piezoelectric ceramic actuator, and w pj Representing the weight of the rise of the input voltage, w qj Representing the weight, r, of the input voltage as it falls j A threshold value that is a hysteresis operator; f j () Is the current lag operator displacement function; x (t) is applied to the piezoceramic actuatorA voltage; i represents a time; j =1, 2 \8230, where \8230, n is the number of hysteresis operators; y is j (t i-1 ) The output value of the j-order hysteresis operator at the previous moment is obtained;
the identification method of the rising weight and the threshold is the same as that of the classic PI hysteresis model; when the input voltage signal x (t) is decreased from the maximum value to zero, the relation between the output angle and the input voltage is the large ring descending part of the piezoelectric ceramic driver hysteresis loop, and the descending weight of the PI hysteresis model can be obtained according to the sectional slope of the large ring descending part of the piezoelectric ceramic hysteresis loop; the size of the threshold is obtained according to the following formula:
the magnitude of the descending weight value is estimated according to the slope value of the descending part of the large loop of the hysteresis curve, and the estimation formula is as follows:
and (3) inverting the improved PI hysteresis model to obtain an improved PI hysteresis inversion model, wherein the improved PI hysteresis inversion model is expressed by the following formula:
and solving an initial driving voltage value of the piezoelectric ceramic driver based on the improved PI hysteresis inverse model.
Further, the proportion regulation parameter k based on the genetic algorithm optimization PID controller p Integral adjustment parameter k i Differential adjustment parameter k d The method specifically comprises the following steps:
s1, respectively determining a proportion adjusting parameter k p Integral adjustment parameter k i Differential adjustment parameter k d Respectively coding each parameter, selecting a binary character string to represent each parameter, and establishing a relation with the parameter;
s2, randomly generating n individual structures to form an initial population p (0);
s3, decoding each population into corresponding parameter values, substituting the parameter values into a target function, and after the target function is determined, carrying out proportion adjustment on a parameter k by taking the target function as an adaptive function p Integral of the regulating parameter k i Differential adjustment parameter k d Optimizing;
s4, operating the population P (t) by using replication, crossover and mutation operators to generate a next generation population P (t + 1);
s5 repeating the steps S3 and S4 until the parameter k p ，k i ，k d Converge or reach a predetermined criterion.
Furthermore, the piezoelectric ceramic driver comprises a platform frame, the piezoelectric ceramic is arranged in the accommodating part of the platform frame, flexible hinges with double parallel plate structures are arranged in the extension direction of the piezoelectric ceramic, and the flexible hinges are divided into two groups and are symmetrically arranged; and (3) performing dynamic analysis on the piezoelectric ceramic driver to obtain a uniform dynamic transfer function of the piezoelectric ceramic and the flexible hinge:
in the formula, U(s) is the driving voltage of the piezoelectric ceramic driver; z(s) is the displacement of the piezoelectric ceramic driver; m is eff Being a piezoelectric ceramicEquivalent mass, c p Is a damping coefficient of piezoelectric ceramics, K p The rigidity of the piezoelectric ceramic; m is the equivalent mass of the hinge, c s For equivalent damping of hinges, K s Is the equivalent stiffness of the hinge;
sampling in a preset sampling period, inputting a step signal as an instruction, and adopting an error absolute value time integral performance index as a parameter k p ，k i ，k d A selected minimum objective function; adding a square term of a control input into the objective function; the following objective function formula is selected as parameter k p ，k i ，k d The selected optimal indexes are as follows:
in the formula, e (t) is the error of the piezoelectric ceramic closed-loop control system, u (t) is the output of the PID controller, and t u For step signal rise time, w 1 ,w 2 ,w 3 Is the weight; once the overshoot is generated, the overshoot is used as one item of the optimal index, and the optimal index is as follows:
wherein ey (t) is an overshoot, w 4 Is an overshoot weight, and w 4 ＞＞w 1 Ey (t) = y (t) -y (t-1), y (t) is a pulled object output.
Further, the number of populations used in the genetic algorithm is 30, and the cross probability and the mutation probability are respectively as follows: p is c ＝0.9，P m ＝0.033；
Parameter k p ，k i ，k d Value range of [0,1 ]]，w 1 ＝0.999，w 2 ＝0.001，w 4 ＝100，w 3 ＝2.0。
In order to achieve the above object, the present invention provides a piezoelectric ceramic driver feedforward and closed loop composite control system based on genetic algorithm, comprising: a feedforward model unit, a PID controller, a PID parameter optimization unit, a sensor measurement unit and a piezoelectric ceramic driver, wherein,
the feedforward model unit is used for obtaining the initial driving voltage u of the piezoelectric ceramic driver according to the input operation of the expected displacement r (t) of the piezoelectric ceramic driver ε (t)；
The sensor measuring unit is used for outputting the displacement of the piezoelectric ceramic driver, acquiring the displacement of the piezoelectric ceramic in the piezoelectric ceramic driver and using the displacement as a feedback signal for the closed-loop control of the PID controller;
the PID controller is used for controlling the initial driving voltage u according to the displacement of the piezoelectric ceramic driver and the displacement of the piezoelectric ceramic in the piezoelectric ceramic driver ε (t) eliminating errors and outputting the driving voltage of the piezoelectric ceramic driver;
a PID parameter optimization unit for optimizing the proportional control parameter k of the PID controller based on the genetic algorithm p Integral adjustment parameter k i Differential adjustment parameter k d 。
Further, the feedforward model unit obtains the initial driving voltage by using an improved PI hysteresis inverse model, where the improved PI hysteresis inverse model is expressed by the following formula:
further, the process of optimizing the PID parameters by the PID parameter optimizing unit specifically includes:
s1, respectively determining a proportion adjusting parameter k p Integral adjustment parameter k i Differential adjustment parameter k d Respectively coding each parameter, selecting a binary character string to represent each parameter, and establishing a relation with the parameter;
s2, randomly generating n individual structures to form an initial population p (0);
s3, decoding each population into corresponding parameter values, determining a target function, and performing proportion adjustment on the parameter k by taking the target function as an adaptive function p Integral adjustment parameter k i Differential adjustment parameter k d Optimizing;
s4, operating the population P (t) by using replication, crossover and mutation operators to generate a next generation population P (t + 1);
s5 repeating the steps S3 and S4 until the parameter k p ，k i ，k d Convergence or reaching a predetermined index;
the piezoelectric ceramic driver comprises a platform frame, wherein a containing part for containing piezoelectric ceramic is arranged in the platform frame, the piezoelectric ceramic is arranged in the piezoelectric ceramic containing part, at least 4 flexible hinges with double parallel plate structures are arranged in the extension direction of the piezoelectric ceramic, the flexible hinges are divided into two groups, and the two groups of flexible hinges are opposite and symmetrically arranged;
and (3) performing dynamic analysis on the piezoelectric ceramic driver to obtain a uniform dynamic transfer function of the piezoelectric ceramic and the flexible hinge:
in the formula, U(s) is the driving voltage of the piezoelectric ceramic driver; z(s) is the displacement of the piezoelectric ceramic driver; m is eff Is the equivalent mass of the piezoelectric ceramic, c p Is a damping coefficient of piezoelectric ceramics, K p The rigidity of the piezoelectric ceramic; m is the equivalent mass of the hinge, c s For equivalent damping of hinges, K s Being hinges or the likeEffective stiffness;
sampling in a preset sampling period, inputting a step signal as an instruction, and adopting an error absolute value time integral performance index as a parameter k p ，k i ，k d A selected minimum objective function; adding a square term of a control input into the objective function; the following objective function formula is selected as parameter k p ，k i ，k d The selected optimal indexes are as follows:
in the formula, e (t) is the error of the piezoelectric ceramic closed-loop control system, u (t) is the output of the PID controller, and t u For step signal rise time, w 1 ,w 2 ,w 3 Is the weight; once the overshoot is generated, the overshoot is used as one item of the optimal index, and the optimal index is as follows:
wherein ey (t) is an overshoot, w 4 Is an overshoot weight, and w 4 ＞＞w 1 Ey (t) = y (t) -y (t-1), y (t) is a pulled object output.
Further, the sensor measuring unit comprises a piezoelectric ceramic displacement measuring circuit and a piezoelectric ceramic driver displacement sensor, wherein the piezoelectric ceramic displacement measuring circuit comprises four groups of resistance strain gauges attached to piezoelectric ceramic, the resistance strain gauges form a full-bridge circuit for measuring the displacement of the piezoelectric ceramic, and each bridge arm of the full-bridge circuit is provided with a group of resistance strain gauges respectively.
By means of the scheme, the piezoelectric ceramic driver feedforward and closed loop composite control method and system based on the genetic algorithm at least have the following advantages:
(1) Compared with the simplex method, the genetic algorithm has the same optimizing characteristic, and overcomes the sensitivity of the initial value of the simplex method parameter. Under the condition that the initial condition is not properly selected, the genetic algorithm can still find a proper function under the condition that initial parameters of the regulator are not required to be given, so that the control target meets the requirements. Meanwhile, the simplex method is difficult to solve the problem of multi-valued function and easily causes optimization failure or overlong time in multi-parameter optimization (such as a cascade system), and the characteristics of the genetic algorithm determine that the method can well overcome the problems.
(2) Compared with the expert setting method, the method has the advantages of convenient operation and high speed, does not need complex rules, and can achieve optimization by simply copying, crossing and mutating the character strings. And a large amount of preliminary knowledge base arrangement work and a large amount of simulation experiments in the expert setting method are avoided.
(3) The genetic algorithm starts to operate in parallel from a plurality of points, and carries out high-efficiency heuristic search in a solution space, so that the defects from a single point and the blindness of search are overcome, the optimization speed is higher, and the phenomenon that the genetic algorithm is trapped in a local optimal solution too early is avoided.
(4) The genetic algorithm is not only suitable for single-target optimization, but also suitable for multi-target optimization. Depending on the control system, the genetic algorithm can find the appropriate function within a specified range for one or more objectives. Genetic algorithm is used as a global optimization algorithm and is more and more widely applied. In recent years, genetic algorithms have been increasingly used for control.
The foregoing description is only an overview of the technical solutions of the present invention, and in order to make the technical solutions of the present invention more clearly understood and to implement them in accordance with the contents of the description, the following detailed description is given with reference to the preferred embodiments of the present invention and the accompanying drawings.
Drawings
FIG. 1 is a flow chart of a genetic algorithm optimizing PID parameters;
FIG. 2 is a target function simulation curve of a genetic algorithm tuning PID;
FIG. 3 is a step response simulation curve after the setting of the genetic algorithm setting PID;
FIG. 4 is an optimized simulation curve of population PID parameters of a genetic algorithm tuning PID;
FIG. 5 is a schematic diagram of a feed forward PID complex control;
FIG. 6 is the output of n Play operators;
FIG. 7 is a graph of the initial load drop and the large loop drop portion of the hysteresis curve;
FIG. 8 is a classic PI lag model fitting error;
FIG. 9 is a graph of improved asymmetric PI hysteresis model fitting error.
Detailed Description
The following detailed description of embodiments of the present invention is provided in connection with the accompanying drawings and examples. The following examples are intended to illustrate the invention but are not intended to limit the scope of the invention.
The invention is based on the PID setting principle of the genetic algorithm:
determination and representation of parameters: the parameter ranges are first determined, which are generally given by the user, and then encoded by the requirements of precision. A binary string is selected to represent each parameter and a relationship is established between the parameters. The binary strings are connected to form a binary string of a factory, and the string is an object which can be operated by a genetic algorithm.
Selecting an initial population: because programming is required to implement the process, the initial population is randomly generated using a computer. For binary coding, random numbers uniformly distributed between 0-1 are generated first, and then it is specified that the generated random numbers 0-0.5 represent 0, 0.5-1 represent 1. In addition, the size of the population is specified in consideration of the complexity of the computer.
Determination of the adaptation function: the general optimization method can obtain a group of parameters meeting the condition under the constraint condition, and a best parameter is searched from the group of parameters in the design. There are three aspects to measure the index of a control system, namely stability, accuracy and rapidity. The rising time reflects the rapidity of the system, and the shorter the rising time is, the faster the control is performed, and the better the system quality is.
If the dynamic characteristic of the system is simply pursued, the obtained parameters are likely to cause the control signal to be overlarge, the system is unstable due to the inherent saturation characteristic of the system in practical application in order to prevent the control signal from being overlarge, and the control quantity is added into the objective function in order to prevent the control quantity from being overlarge. Therefore, in order to make the control effect better, the invention gives control quantity, error and rising time as constraint conditions. Since the adaptation function is related to the objective function, the objective function is determined and then directly used as the adaptation function for parameter optimization. The optimal control parameter is the PID controller parameter corresponding to x when the adaptation function f (x) is maximized under the constraint condition.
Example 1
In the feedforward and closed-loop composite control method of the piezoelectric ceramic driver based on the genetic algorithm, the expected displacement r (t) of the piezoelectric ceramic driver is input into a feedforward model, and the feedforward model calculates to obtain the initial driving voltage u of the piezoelectric ceramic driver according to the expected displacement value r (t) ε (t); the initial driving voltage u is fed back by the PID controller by taking the displacement of the piezoelectric ceramic or the displacement of the piezoelectric ceramic driver as a feedback signal of the PID controller ε (t) eliminating errors to obtain the driving voltage of the piezoelectric ceramic driver, and utilizing a genetic algorithm to regulate a proportional control parameter k of the PID controller p Integral of the regulating parameter k i Differential adjustment parameter k d And (6) optimizing.
In this embodiment, the feedforward model is an improved PI hysteresis inverse model, and the improved PI hysteresis inverse model is obtained by inverting the improved PI hysteresis model;
wherein the improved PI lag model is expressed as: the improved PI hysteresis model is divided into a rising part and a falling part, one group of weights is adopted when the input voltage is increased, and the other group of weights is adopted when the input voltage is decreased; the improved PI hysteresis model is expressed by the following equation,
wherein z (t) is the displacement of the piezoelectric ceramic actuator, w pj Representing the weight of the rise of the input voltage, w qj Representing the weight, r, of the input voltage as it falls j A threshold value that is a hysteresis operator; f j () A displacement function is taken as the current hysteresis operator; x (t) is the voltage applied to the piezoceramic driver; i represents a time; j =1, 2 \8230n, where \8230nis the number of the hysteresis operators; y is j (t i-1 ) The output value of the j-order hysteresis operator at the previous moment is obtained;
the identification method of the rising weight and the threshold is the same as that of the classic PI hysteresis model; when the input voltage signal x (t) is decreased from the maximum value to zero, the relation between the output angle and the input voltage is the large ring descending part of the piezoelectric ceramic driver hysteresis loop, and the descending weight of the PI hysteresis model can be obtained according to the sectional slope of the large ring descending part of the piezoelectric ceramic hysteresis loop; the size of the threshold is obtained according to the following formula:
the magnitude of the descending weight value is estimated according to the slope value of the descending part of the large loop of the hysteresis curve, and the estimation formula is as follows:
inverting the improved PI hysteresis model to obtain an improved PI hysteresis inversion model
The model formula is expressed as follows:
and solving an initial driving voltage value of the piezoelectric ceramic driver based on the improved PI hysteresis inverse model.
Classical PI hysteresis model
The classical PI model considers the lag nonlinearity as a linear weighted superposition of a series of Play operators,
the linear weighted superposition of a plurality of play operators with different threshold values to obtain the output of the hysteresis model is shown in formula (1), w j And r j Satisfies 0= r for the weight and the threshold of the play operator 1 <······<r n &And n is the number of the plan operators.
Parameter identification of classic PI hysteresis model
When the input voltage signal x (t) is a signal with an initial value of zero and monotonically increases to a maximum value, then the output displacement is the large loop rising part of the hysteresis loop in relation to the input voltage. When x (t) is increased from zero to the maximum value, the PI hysteresis model of the piezoelectric ceramic driver can be obtained after the n Play operators are subjected to weighted superposition, and the weight of the PI hysteresis model can be obtained according to the sectional slope of the large ring rising part of the hysteresis loop of the piezoelectric ceramic driver.
When the input voltage increases from zero to the maximum value, the input-output curve of the PI hysteresis model is called an initial loading rising curve, and the initial loading rising curve can be used for fitting the large ring rising part of the hysteresis loop of the piezoelectric ceramic driver.
The size of the threshold value can be equally divided between the minimum value and the maximum value of the input voltage, and the threshold value can be determined by the formula (2):
the slope of the initial loading ascending curve changes after each threshold value is passed, and the corresponding w can be estimated according to the slope value of the large ring ascending part of the hysteresis curve in each interval j The value, estimation formula is as follows:
improved PI hysteresis model
In order to describe an asymmetric hysteresis curve, the PI hysteresis model is improved and is divided into a rising part and a falling part. One set of weights is used when the input voltage increases, and the other set of weights is used when the input voltage decreases. The improved PI hysteresis model can be expressed by equation (4), where w pj Representing the weight of the rise of the input voltage, w qj Representing the weight when the input voltage drops.
Parameter identification for improved PI lag model
The parameters for improving the PI hysteresis model mainly comprise a rising weight value w Pj And a falling weight w qj And a threshold value r. The identification method of the rising weight and the threshold is the same as that of the classic PI hysteresis model, and is not repeated here. When the input voltage signal x (t) decreases from the maximum value to zero, the relationship between the output angle and the input voltage is the large loop decreasing part of the hysteresis loop of the piezoelectric ceramic driver, and the outputs of the n Play operators can be represented by fig. 6, and the output curve is the solid line part in the graph.
After the n playoperators in fig. 6 are weighted and superimposed, the PI hysteresis model of the piezoelectric ceramic can be obtained, and the weight changes the slope of the oblique line in fig. 6, so that the falling weight of the PI hysteresis model can be obtained according to the sectional slope of the large ring falling part of the piezoelectric ceramic hysteresis loop.
When the input voltage is decreased from the maximum value to zero, the input-output curve of the PI hysteresis model is called as an initial loading descending curve, and the initial loading descending curve can be used for fitting a large loop of the piezoelectric ceramic driver hysteresis loop
A descending section. The initial load drop profile and the large loop drop portion of the hysteresis curve are shown in FIG. 7.
The size of the threshold value can be equally divided between the minimum value and the maximum value of the input voltage, and the threshold value can be determined by the formula (5):
the magnitude of the descending weight value can be estimated according to the slope value of the descending part of the large loop of the hysteresis curve, and the estimation formula is as follows:
inverse model of PI hysteresis inverse model
The feedforward control of the piezoelectric ceramic driver is mainly to predict the driving voltage value required by the driving platform before deviation occurs according to expected displacement, so as to improve the control accuracy. The output displacement of the piezoelectric ceramic driver can be estimated according to the improved PI hysteresis model, the feedforward control algorithm needs to obtain a driving voltage value required by a driving platform according to the inverse of the improved PI hysteresis model, the PI model has the greatest advantage that the inverse is easy to solve, the inverse model is still a PI model, and only the threshold value and the weight value need to be correspondingly transformed, and the inverse model of the formula (4) can be recorded as shown in a formula 8.
The hysteresis phenomenon seriously influences the precision of the piezoelectric ceramic as a high-precision positioning platform, the highest tracking error generated by the hysteresis nonlinearity of the piezoelectric ceramic can reach 15% under the condition of an uncontrolled open loop, and in order to reduce the influence of the hysteresis characteristic on the positioning precision, a classic PI hysteresis model is established based on the hysteresis phenomenon of the piezoelectric ceramic. The invention provides an asymmetric PI hysteresis model, the rising weight and the falling weight of the hysteresis are solved by utilizing the slopes of a rising curve and a falling curve, fitting model simulation is carried out by using matlab, the improved PI hysteresis model fitting obtains better model fitting accuracy, and the inverse model of the PI hysteresis model is solved and is used for feedforward compensation of piezoelectric ceramic composite control.
Comparison of fitting accuracy of classic PI hysteresis model and improved PI hysteresis model
Parameter solving
From the above description of the PI lag model, it can be seen that the more the Play operators in the PI lag model are taken, the better the fitting degree of the PI lag model to the lag curve is, but in consideration of the complexity of calculation in practical application, 25 Play operators are selected in the present design. In order to simplify the identification process and the programming process on the computer, the input voltage and the output angle of the piezoelectric ceramic driver are normalized, and the formula is shown as (7).
In equation 7, x (t) is the voltage applied to the piezo-ceramic actuator, x (t) min Is the voltage minimum, x (t) max Is the voltage maximum, z (t) is the corresponding displacement of the piezoceramic actuator, z (t) min Is the minimum value of the displacement, z (t) max Is the maximum value of the displacement, x g (t) and z g And (t) is the corresponding voltage and displacement after normalization. X after normalization g The maximum value of (t) is 1, and the sizes of 25 thresholds can be obtained according to the formula (2), as shown in table 1.
TABLE 1PI hysteresis model thresholds
The variable voltage of 0-150-0V is input to the piezoelectric ceramic driver, and the weight of the classic PI hysteresis model and the rising weight and the falling weight of the improved PI hysteresis model can be identified according to the output displacement value. The weight of the classical PI hysteresis model is equal to the rising weight of the improved PI hysteresis model, and the rising weight can be obtained according to the formula (3), and is shown in table 2. The falling weight can be obtained according to equation (6), and is shown in table 3.
TABLE 2 classical PI lag model weight and improved PI lag model rise weight
TABLE 3 improved PI hysteresis model descent weight
Comparison of fitting accuracy
The fitting error is shown in fig. 8 and 9 by fitting the output angle value to the input voltage value. The fitted data is compared with the actually measured data, data processing is performed on errors, and the processed results are shown in table 4.
TABLE 4 error value data processing
Analysis of the data in fig. 8, 9 and table 4 shows that the fitting accuracy of the improved PI hysteresis model is significantly improved. It can be seen from fig. 8 that the fitting angle value of the classical PI hysteresis model has a smaller error in the voltage rising stage and a larger error in the voltage falling stage, and it can be seen from observing fig. 9 that the fitting error of the fitting value of the improved PI hysteresis model is smaller in the falling stage because the weight of the improved PI hysteresis model is divided into a rising weight and a falling weight, the fitting curve is asymmetric, and the hysteresis curve can be better described, while the weight of the classical PI hysteresis model only includes the rising weight, the fitting curve is symmetric, and the fitting error is larger in the falling stage of the hysteresis curve.
This example, based on the improved PI inverse modelThe introduction of the feedforward control of the type can improve the hysteresis nonlinear characteristic and improve the dynamic characteristic thereof. The PID control method has the characteristics of better robustness and no need of accurate modeling, and is widely applied to a precise positioning control system. According to the classical control theory, a proper feedforward control link can improve the response capability of the system, but the control precision of the system is not influenced. And adopting a control strategy based on improved PI inverse model feedforward combined PID feedback control. The control schematic diagram is shown in fig. 5, wherein the system firstly obtains the corresponding expected voltage u through a feedforward model according to the expected displacement value r (t) ε (t) which is then error eliminated by PID feedback control.
Example 2
In the embodiment of the piezoelectric ceramic driver feedforward and closed loop composite control method based on the genetic algorithm, on the basis of the embodiment 1, the proportional control parameter k of the PID controller is optimized based on the genetic algorithm p Integral of the regulating parameter k i Differential adjustment parameter k d The method specifically comprises the following steps:
s1, respectively determining a proportion adjusting parameter k p Integral adjustment parameter k i Differential adjustment parameter k d Respectively coding each parameter, selecting a binary character string to represent each parameter, and establishing a relation with the parameter;
s2, randomly generating n individual structures to form an initial population p (0);
s3, decoding each population into corresponding parameter values, substituting the parameter values into a target function, and after the target function is determined, carrying out proportion adjustment on a parameter k by taking the target function as an adaptive function p Integral adjustment parameter k i Differential adjustment parameter k d Optimizing;
s4, operating the population P (t) by using replication, crossover and mutation operators to generate a next generation population P (t + 1);
s5 repeating the steps S3 and S4 until the parameter k p ，k i ，k d Convergence or reaching a predetermined level。
In this embodiment, the piezoelectric ceramic driver includes a platform frame, the piezoelectric ceramic is disposed in the accommodating portion of the platform frame, flexible hinges of a double-parallel-plate structure are disposed in the extension direction of the piezoelectric ceramic, and the flexible hinges are divided into two groups and symmetrically disposed; and (3) performing dynamic analysis on the piezoelectric ceramic driver to obtain a uniform dynamic transfer function of the piezoelectric ceramic and the flexible hinge:
in the formula, U(s) is the driving voltage of the piezoelectric ceramic driver; z(s) is the displacement of the piezoelectric ceramic driver; m is a unit of eff Is the equivalent mass of the piezoelectric ceramic, c p Is a damping coefficient of piezoelectric ceramics, K p The rigidity of the piezoelectric ceramic; m is the equivalent mass of the hinge, c s For equivalent damping of the hinge, K s Is the equivalent stiffness of the hinge;
sampling is carried out in a preset sampling period, an input instruction is a step signal, and an absolute value of error time integral performance index is adopted as a parameter k p ，k i ，k d A selected minimum objective function; adding a square term of a control input into the objective function; the following objective function formula is selected as the parameter k p ，k i ，k d The selected optimal indexes are as follows:
in the formula, e (t) is the error of the piezoelectric ceramic closed-loop control system, u (t) is the output of the PID controller, and t u For step signal rise time, w 1 ,w 2 ,w 3 Is the weight; once overshoot is generated, the overshoot is used as one item of the optimal index, and the optimal index is as follows:
wherein ey (t) is the overshoot, w 4 Is an overshoot weight, and w 4 ＞＞w 1 Ey (t) = y (t) -y (t-1), y (t) is a pulled object output.
In this embodiment, the parameter k to be selected is p ，k i ，k d Substituting into PID controller to operate closed-loop control system to obtain actual displacement of piezoelectric ceramic driver, comparing the actual displacement with ideal displacement, and taking parameter k with error value between actual displacement and ideal displacement within error threshold range p ，k i ，k d Are parameters of the PID controller.
In this embodiment, the number of populations used in the genetic algorithm is 30, and the cross probability and the mutation probability are respectively: pc =0.9,P m =0.033. Parameter k p ，k i ，k d Value range of [0,1 ]]，w 1 ＝0.999，w 2 ＝0.001，w 4 ＝100，w 3 =2.0. By adopting a real number encoding mode and 100 generations of evolution, the genetic algorithm setting PID simulation curve shown in figures 2 to 4 is obtained, and the optimal PID parameters are kp =0.9264, ki =0.2492 and kd =0.4416 as shown in figure 4. Performance index =24.2170, optimization of the objective function J and control of the step response with the positive PID are shown in fig. 2 and 3.
In this embodiment, first, a fitness scaling method is used to perform replication, that is, an adaptation value is obtained through an adaptation function, and then a replication probability corresponding to each string is obtained. The product of the copy probability and the number of strings of each generation is the number of strings to be copied in the next generation. The next generation with high replication probability will have more offspring, and will be eliminated. And secondly, carrying out single-point crossing with the crossing probability of Pc. Selecting the character strings from the copied members according to the probability of Pc to form a matching pool, then randomly matching the members in the matching pool, and randomly determining the crossed position. And finally, carrying out mutation by using the probability Pm. If there are 15 strings in each generation, each string has 12 bits, then there are 15x12=180 strings, and the desired number of variant strings is 180x0.01=2 bits, i.e. two strings in each generation are to be changed from 1 to 0 or from 0 to 1. And obtaining a new generation of population by copying, crossing and varying the initial population, carrying the new generation of population into an adaptation function after decoding, observing whether an end condition is met, and if the end condition is not met, repeating the operation until the end condition is met. The termination condition is determined by a specific problem, and the calculation is terminated as long as each target parameter is within a prescribed range. The above operation is shown in fig. 1.
Example 3
The embodiment of the piezoelectric ceramic driver feedforward and closed-loop composite control system based on the genetic algorithm comprises the following components: a feedforward model unit, a PID controller, a PID parameter optimization unit, a sensor measurement unit and a piezoelectric ceramic driver, wherein,
the feedforward model unit is used for obtaining the initial driving voltage u of the piezoelectric ceramic driver according to the input operation of the expected displacement r (t) of the piezoelectric ceramic driver ε (t)；
The sensor measuring unit is used for outputting the displacement of the piezoelectric ceramic driver, acquiring the displacement of the piezoelectric ceramic in the piezoelectric ceramic driver and using the displacement as a feedback signal for the closed-loop control of the PID controller;
the PID controller is used for controlling the initial driving voltage u according to the displacement of the piezoelectric ceramic driver and the displacement of the piezoelectric ceramic in the piezoelectric ceramic driver ε (t) eliminating errors and outputting the driving voltage of the piezoelectric ceramic driver;
a PID parameter optimization unit for optimizing the proportional control parameter k of the PID controller based on the genetic algorithm p Integral of the regulating parameter k i Differential adjustment parameter k d 。
In this embodiment, the feedforward model unit obtains the initial driving voltage by using an improved PI hysteresis inverse model, where the improved PI hysteresis inverse model is expressed by the following formula:
the process of optimizing the PID parameters by the PID parameter optimizing unit specifically includes:
s1, respectively determining a proportion adjusting parameter k p Integral adjustment parameter k i Differential adjustment parameter k d Respectively coding each parameter, selecting a binary character string to represent each parameter, and establishing a relation with the parameter;
s2, randomly generating n individual structures to form an initial population p (0);
s3, decoding each population into corresponding parameter values, determining a target function, and performing proportion adjustment on the parameter k by taking the target function as an adaptive function p Integral adjustment parameter k i Differential adjustment parameter k d Optimizing;
s4, operating the population P (t) by using replication, crossover and mutation operators to generate a next generation population P (t + 1);
s5 repeating the steps S3 and S4 until the parameter k p ，k i ，k d Convergence or reaching a predetermined index;
the piezoelectric ceramic driver comprises a platform frame, wherein a containing part for containing piezoelectric ceramic is arranged in the platform frame, the piezoelectric ceramic is arranged in the piezoelectric ceramic containing part, at least 4 flexible hinges with double parallel plate structures are arranged in the extension direction of the piezoelectric ceramic, the flexible hinges are divided into two groups, and the two groups of flexible hinges are opposite and symmetrically arranged;
and (3) performing dynamic analysis on the piezoelectric ceramic driver to obtain a uniform dynamic transfer function of the piezoelectric ceramic and the flexible hinge:
in the formula, U(s) is the driving voltage of the piezoelectric ceramic driver; z(s) is the displacement of the piezoelectric ceramic driver; m is eff Is the equivalent mass of the piezoelectric ceramic, c p Is a damping coefficient of piezoelectric ceramics, K p The rigidity of the piezoelectric ceramic; m is the equivalent mass of the hinge, c s For equivalent damping of hinges, K s Is the equivalent stiffness of the hinge;
sampling in a preset sampling period, inputting a step signal as an instruction, and adopting an error absolute value time integral performance index as a parameter k p ，k i ，k d Selecting a minimum objective function; adding a square term of a control input into the objective function; the following objective function formula is selected as the parameter k p ，k i ，k d The selected optimal indexes are as follows:
in the formula, e (t) is the error of the piezoelectric ceramic closed-loop control system, u (t) is the output of the PID controller, and t u For step signal rise time, w 1 ,w 2 ,w 3 Is the weight; once overshoot is generated, the overshoot is used as one item of the optimal index, and the optimal index is as follows:
wherein ey (t) is the overshoot, w 4 Is an overshoot weight, and w 4 ＞＞w 1 Ey (t) = y (t) -y (t-1), y (t) is the pulled object output.
The specific control method of the system of this embodiment is the same as that of embodiment 2, and is not described herein again.
In the above embodiments, the sensor measuring unit includes a piezoelectric ceramic displacement measuring circuit and a piezoelectric ceramic driver displacement sensor, wherein the piezoelectric ceramic displacement measuring circuit includes four sets of resistance strain gauges attached to piezoelectric ceramic, the resistance strain gauges form a full bridge circuit for measuring the displacement of the piezoelectric ceramic, and each bridge arm of the full bridge circuit is provided with a set of resistance strain gauge.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, it should be noted that, for those skilled in the art, many modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.
Claims (9)
1. A piezoelectric ceramic driver feedforward and closed loop composite control method based on a genetic algorithm is characterized in that expected displacement r (t) of a piezoelectric ceramic driver is input into a feedforward model, and the feedforward model calculates and obtains initial driving voltage u of the piezoelectric ceramic driver according to the expected displacement r (t) ε (t); the initial driving voltage u is fed back by the PID controller by taking the displacement of the piezoelectric ceramic or the displacement of the piezoelectric ceramic driver as a feedback signal of the PID controller ε (t) eliminating errors to obtain the driving voltage of the piezoelectric ceramic driver, and utilizing a genetic algorithm to regulate a proportional control parameter k of the PID controller p Integral of the regulating parameter k i Differential adjustment parameter k d And (6) optimizing.
2. The piezoelectric ceramic driver feedforward and closed-loop composite control method based on genetic algorithm as claimed in claim 1, wherein the feedforward model is an improved PI hysteresis inverse model obtained by inverting the improved PI hysteresis model;
wherein the improved PI lag model is expressed as: the improved PI hysteresis model is divided into a rising part and a falling part, one group of weights is adopted when the input voltage is increased, and the other group of weights is adopted when the input voltage is decreased; the improved PI hysteresis model is expressed by the following equation,
wherein z (t) is the displacement of the piezoelectric ceramic actuator, and w pj Representing the weight, w, of the rise in input voltage qj Weight, r, representing the drop in input voltage j A threshold value that is a hysteresis operator; f j () A displacement function is taken as the current hysteresis operator; x (t) is the voltage applied to the piezoceramic driver; i represents a time; j =1, 2 \8230n, where \8230nis the number of the hysteresis operators; y is j (t i-1 ) The output value of the j-order hysteresis operator at the previous moment is obtained;
the identification method of the rising weight and the threshold is the same as that of the classic PI hysteresis model; when the input voltage signal x (t) is decreased from the maximum value to zero, the relation between the output angle and the input voltage is the large ring descending part of the piezoelectric ceramic driver hysteresis loop, and the descending weight of the PI hysteresis model can be obtained according to the sectional slope of the large ring descending part of the piezoelectric ceramic hysteresis loop; the size of the threshold is obtained according to the following formula:
the magnitude of the descending weight value is estimated according to the slope value of the descending part of the large loop of the hysteresis curve, and the estimation formula is as follows:
and inverting the improved PI hysteresis model to obtain an improved PI hysteresis inverse model, wherein the improved PI hysteresis inverse model is expressed by the following formula:
and obtaining an initial driving voltage value of the piezoelectric ceramic driver based on the improved PI hysteresis inverse model.
3. The piezoelectric ceramic driver feed-forward and closed-loop compound control method based on genetic algorithm as claimed in claim 1, wherein the proportional control parameter k of the PID controller optimized based on the genetic algorithm p Integral adjustment parameter k i Differential adjustment parameter k d The method specifically comprises the following steps:
s1, respectively determining a proportion adjusting parameter k p Integral adjustment parameter k i Differential adjustment parameter k d Respectively coding each parameter, selecting a binary character string to represent each parameter, and establishing a relation with the parameter;
s2, randomly generating n individual structures to form an initial population p (0);
s3, decoding each population into corresponding parameter values, substituting the parameter values into a target function, and after the target function is determined, carrying out proportion adjustment on a parameter k by taking the target function as an adaptive function p Integral adjustment parameter k i Differential adjustment parameter k d Optimizing;
s4, operating the population P (t) by using replication, crossover and mutation operators to generate a next generation population P (t + 1);
s5 repeating steps S3 and S4 untilParameter k p ，k i ，k d Converge or reach a predetermined target.
4. A piezoelectric ceramic driver feedforward and closed-loop composite control method based on genetic algorithm according to claim 3, wherein the piezoelectric ceramic driver includes a platform frame, the piezoelectric ceramic is disposed in the accommodating portion of the platform frame, flexible hinges of a double parallel plate structure are disposed in the extension direction of the piezoelectric ceramic, and the flexible hinges are divided into two groups and symmetrically arranged; and (3) performing kinetic analysis on the piezoelectric ceramic driver to obtain a uniform kinetic transfer function of the piezoelectric ceramic and the flexible hinge:
in the formula, U(s) is the driving voltage of the piezoelectric ceramic driver; z(s) is the displacement of the piezoelectric ceramic driver; m is eff Is the equivalent mass of the piezoelectric ceramic, c p Is a damping coefficient of piezoelectric ceramics, K p The rigidity of the piezoelectric ceramic; m is the equivalent mass of the hinge, c s For equivalent damping of hinges, K s Is the equivalent stiffness of the hinge;
sampling in a preset sampling period, inputting a step signal as an instruction, and adopting an error absolute value time integral performance index as a parameter k p ，k i ，k d A selected minimum objective function; adding a square term of a control input into the objective function; the following objective function formula is selected as the parameter k p ，k i ，k d The selected optimal indexes are as follows:
in the formula, e (t) is the error of the piezoelectric ceramic closed-loop control system, u (t) is the output of the PID controller, and t u For step signal rise time, w 1 ,w 2 ,w 3 Is the weight; wherein, once the overshoot is generated, the overshoot is used as one item of the optimal indexAt this time, the optimal indexes are as follows:
wherein ey (t) is the overshoot, w 4 Is an overshoot weight, and w 4 ＞＞w 1 Ey (t) = y (t) -y (t-1), y (t) is
And outputting the pulled object.
5. The piezoelectric ceramic driver feedforward and closed-loop composite control method based on genetic algorithm as claimed in claim 4, wherein the population number used in the genetic algorithm is 30, and the cross probability and the variation probability are respectively: p c ＝0.9，P m ＝0.033；
Parameter k p ，k i ，k d Value range of [0,1 ]]，w 1 ＝0.999，w 2 ＝0.001，w 4 ＝100，w 3 ＝2.0。
6. A piezoelectric ceramic driver feedforward and closed loop compound control system based on a genetic algorithm is characterized by comprising the following components: a feedforward model unit, a PID controller, a PID parameter optimization unit, a sensor measurement unit and a piezoelectric ceramic driver, wherein,
the feedforward model unit is used for obtaining the initial driving voltage u of the piezoelectric ceramic driver according to the input operation of the expected displacement r (t) of the piezoelectric ceramic driver ε (t)；
The sensor measuring unit is used for outputting the displacement of the piezoelectric ceramic driver, acquiring the displacement of the piezoelectric ceramic in the piezoelectric ceramic driver and using the displacement as a feedback signal for the closed-loop control of the PID controller;
the PID controller is used for controlling the initial driving voltage u according to the displacement of the piezoelectric ceramic driver and the displacement of the piezoelectric ceramic in the piezoelectric ceramic driver ε (t) eliminating errors and outputting the driving voltage of the piezoelectric ceramic driver;
PID parameter optimization unit for optimizing PID controller based on genetic algorithmProportional adjustment parameter k p Integral of the regulating parameter k i Differential adjustment parameter k d 。
7. A piezo-ceramic actuator feed-forward and closed-loop compound control system based on genetic algorithm as claimed in claim 6,
the feedforward model unit obtains the initial driving voltage by adopting an improved PI hysteresis inverse model, wherein the improved PI hysteresis inverse model is expressed by the following formula:
8. the piezoelectric ceramic driver feedforward and closed-loop composite control system based on genetic algorithm as claimed in claim 7, wherein the process of optimizing the PID parameters by the PID parameter optimizing unit specifically includes:
s1, respectively determining a proportion adjusting parameter k p Integral of the regulating parameter k i Differential adjustment parameter k d Respectively coding each parameter, selecting a binary character string to represent each parameter, and establishing a relation with the parameter;
s2, randomly generating n individual structures to form an initial population p (0);
s3, decoding each population into corresponding parameter values, and determining an objective functionScaling parameter k using the objective function as an adaptation function p Integral adjustment parameter k i Differential adjustment parameter k d Optimizing;
s4, operating the population P (t) by using replication, crossover and mutation operators to generate a next generation population P (t + 1);
s5 repeating the steps S3 and S4 until the parameter k p ，k i ，k d Convergence or reaching a predetermined index;
the piezoelectric ceramic driver comprises a platform frame, wherein a containing part for containing piezoelectric ceramic is arranged in the platform frame, the piezoelectric ceramic is arranged in the piezoelectric ceramic containing part, at least 4 flexible hinges with double parallel plate structures are arranged in the extension direction of the piezoelectric ceramic, the flexible hinges are divided into two groups, and the two groups of flexible hinges are opposite and symmetrically arranged;
and (3) performing kinetic analysis on the piezoelectric ceramic driver to obtain a uniform kinetic transfer function of the piezoelectric ceramic and the flexible hinge:
in the formula, U(s) is the driving voltage of the piezoelectric ceramic driver; z(s) is the displacement of the piezoelectric ceramic driver; m is eff Is the equivalent mass of the piezoelectric ceramic, c p Damping coefficient of piezoelectric ceramic is K p The rigidity of the piezoelectric ceramic; m is the equivalent mass of the hinge, c s For equivalent damping of hinges, K s Is the equivalent stiffness of the hinge;
sampling in a preset sampling period, inputting a step signal as an instruction, and adopting an error absolute value time integral performance index as a parameter k p ，k i ，k d A selected minimum objective function; adding a square term of a control input into the objective function; the following objective function formula is selected as the parameter k p ，k i ，k d The selected optimal indexes are as follows:
in the formula, e (t) is the error of the piezoelectric ceramic closed-loop control system, u (t) is the output of the PID controller, and t u For step signal rise time, w 1 ,w 2 ,w 3 Is the weight; once the overshoot is generated, the overshoot is used as one item of the optimal index, and the optimal index is as follows:
wherein ey (t) is the overshoot, w 4 Is an overshoot weight, and w 4 ＞＞w 1 Ey (t) = y (t) -y (t-1), y (t) is the pulled object output.
9. The piezoelectric ceramic driver feedforward and closed-loop composite control system based on genetic algorithm as claimed in claim 7, wherein the sensor measuring unit includes a piezoelectric ceramic displacement measuring circuit and a piezoelectric ceramic driver displacement sensor, wherein the piezoelectric ceramic displacement measuring circuit includes four sets of resistance strain gauges attached to piezoelectric ceramic, the resistance strain gauges form a full bridge circuit for measuring the displacement of piezoelectric ceramic, and each bridge arm of the full bridge circuit is provided with a set of resistance strain gauge.
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