Hysteresis modeling and loop forming structured control method of piezoelectric actuator

Abstract

The invention relates to a hysteresis modeling and loop forming structured control method for a piezoelectric actuator. The hysteresis modeling and loop forming structured control method comprises the following steps: constructing a piezoelectric actuator model by adopting a Hammerstein structure; performing uncertainty description on the piezoelectric actuator model based on v-gap measurement, and setting open-loop amplitude constraint and a structure and parameter vector search domain of a controller based on the uncertainty description; setting an initial weight function, a precision coefficient and an initial controller, and constructing inequality constraints of a loop shaping system under open-loop amplitude constraints; iteratively solving the first optimization problem and the second optimization problem under the constraint of an inequality until a preset iteration condition is met, and obtaining an optimal weight function and an expected controller; and if a preset iteration condition is met and the solution of the optimization problem is not obtained, resetting the open-loop amplitude constraint and the parameter vector search domain of the controller, and carrying out the next round of iteration solution until the solution is obtained. According to the invention, the dynamic response, the control precision and the robust performance are obviously improved.

Description

Technical Field

[0001] The present invention relates to the field of control technology, and in particular to a hysteresis modeling and loop shaping structured control method for a piezoelectric actuator. Background Art

[0002] Piezoelectric actuators (PEA) have been widely used in ultra-precision positioning systems with nanometer precision due to their excellent performance such as high resolution, high natural frequency and fast response. These PEA-driven positioning systems play a vital role in high-end technology fields such as biomedicine, materials science, and aerospace. However, PEA has a significant defect: the severe hysteresis nonlinearity between input voltage and output displacement, which not only affects the positioning accuracy of the ultra-precision positioning system, but also may cause control system instability. Therefore, for PEA systems with hysteresis nonlinear characteristics, achieving nanometer-level tracking and positioning accuracy is a very challenging task.

[0003] The realization of high-precision control depends on accurate mathematical models. The modeling of hysteresis nonlinear systems can be divided into three categories: physical models, phenomenological models and intelligent models. Physical models are built based on the physical mechanism of hysteresis phenomena, such as JA model, Duhem model and Bouc-Wen model. Phenomenological models are modeled by input and output data, without involving physical structure, and their parameters lack physical interpretation. Typical models include Preisach model and Prandtl-Ishlinskii (PI) model. Intelligent models are modeled using technologies such as artificial neural networks and support vector machines. These models are usually based on the assumption of rate independence, that is, the so-called static hysteresis nonlinear characteristics, whose input and output are only related to the current and historical inputs, and are independent of the input change rate. In recent years, some scholars have proposed a Hammerstein model, which effectively describes rate-dependent hysteresis nonlinearity by cascading a static hysteresis nonlinear module and a linear dynamic module.

[0004] The tracking control technology of PEA can be divided into three categories: control strategy using a charge device, tracking control strategy without inverse compensation, and tracking control strategy based on inverse compensation. The control strategy using a charge device is designed based on the approximately linear relationship between the output displacement and the input charge of the PEA. Although this strategy can effectively reduce the hysteresis nonlinear characteristics of the PEA, the implementation cost of the control system is high, and the charge measurement may be affected by zero drift. The control strategy without inverse compensation directly designs a nonlinear controller to drive the PEA. The controller obtained by this strategy is relatively complex, and it is difficult to ensure the stability of the system. The feedback tracking control strategy based on inverse compensation is the most widely used control strategy at present. This strategy first models the hysteresis phenomenon of the PEA, then constructs the inverse model of the model, and customizes the compensator through the inverse hysteresis model to eliminate the hysteresis nonlinearity. Finally, the feedback control method is used to realize the closed-loop control of the system. Compared with the first two methods, this strategy has more advantages, but due to the possible errors in the hysteresis inverse model modeling and the influence of external interference, it is necessary to adopt advanced feedback control algorithms to ensure the tracking accuracy and robustness of the system.

[0005] In summary, in order to solve the error problem caused by modeling and approximate linearization, the present invention adopts the idea of ​​robust control to design a high-precision controller. The loop-shaping design process is proposed by McFarlane and Glover. The dynamic performance and robust stability of the controller are compromised by reasonably selecting the loop compensator, and the standard left coprime factor decomposition is used to ensure the stability of the closed-loop system. However, the controller has two disadvantages that prevent it from being widely used in industrial production. The first disadvantage is that it is difficult for designers to obtain a suitable open-loop weight function in the design, and repeated experiments are required to obtain a weight function that can meet the desired dynamic performance requirements and maximize the generalized robust stability margin. The second disadvantage is that the optimal controller finally obtained is a high-order controller without structure, which is difficult to implement and maintain in industrial production. Therefore, how to find a weight function that can take into account both performance and robustness and a low-order robust controller that can ensure the closed-loop stability of the system and has good performance is a problem worth studying. Summary of the invention

[0006] The purpose of the present invention is to provide a hysteresis modeling and loop forming structured control method for a piezoelectric actuator for a piezoelectric positioning system with severe hysteresis nonlinear characteristics, so as to obtain a loop forming fixed structure controller with good performance characteristics and robust stability, thereby achieving high-precision positioning and tracking control of PEA and effective suppression of modeling uncertainty and external interference.

[0007] To achieve the above object, the present invention provides the following solutions:

[0008] A hysteresis modeling and loop shaping structured control method for a piezoelectric actuator, comprising:

[0009] Step 1: Use Hammerstein structure to build a piezoelectric actuator model;

[0010] Step 2, describing the uncertainty of the piezoelectric actuator model based on the v-gap metric;

[0011] Step 3: setting an open-loop amplitude constraint, a controller structure, and a parameter vector search domain based on the uncertainty description;

[0012] Step 4: Set the initial weight function, accuracy coefficient and initial controller;

[0013] Step 5: constructing an inequality constraint of the loop shaping system under the open-loop amplitude constraint;

[0014] Step 6: Under the inequality constraint, iteratively solve the first optimization problem and the second optimization problem until a preset iteration condition is reached to obtain an optimal weight function and an expected controller, wherein the first optimization problem is a generalized eigenvalue minimization problem for solving the optimal weight function; and the second optimization problem is a non-convex optimization problem for solving the expected controller;

[0015] Step 7: If the solutions of the first optimization problem and the second optimization problem are not obtained until the preset iteration condition is reached, return to step 3 to reset the open-loop amplitude constraint and the parameter vector search domain of the controller, and perform steps 4 to 6 again until the optimal weight function and the desired controller are obtained.

[0016] Optionally, constructing the piezoelectric actuator model using a Hammerstein structure includes:

[0017] Hammerstein structure is used to model the piezoelectric actuator, and the Bouc-Wen model is used to describe the static hysteresis nonlinear module in the Hammerstein structure, and the Hankel matrix method is used to identify the dynamic linear module in the piezoelectric actuator.

[0018] The hysteresis nonlinear characteristics of the piezoelectric actuator are obtained based on the static hysteresis nonlinear module and the dynamic linear module, and the Bouc-Wen inverse model is used to eliminate the hysteresis nonlinear characteristics to complete the construction of the piezoelectric actuator model.

[0019] Optionally, the Bouc-Wen model is used to describe the static hysteresis nonlinear module in the Hammerstein structure as follows:

[0020]

[0021] Where u is the input voltage of the piezoelectric actuator, y is the output displacement of the piezoelectric actuator, z is the hysteresis component, k, d, n, ρ, σ are the unknown parameters that control the shape of the hysteresis loop, is the derivative of the hysteresis component, is the derivative of the input voltage.

[0022] Optionally, the dynamic linear module in the piezoelectric actuator is identified using the Hankel matrix method as follows:

[0023]

[0024] Among them, P 0 is the dynamic linear module transfer function, and s is a complex domain operator.

[0025] Optionally, a method for describing the uncertainty of the piezoelectric actuator model based on the v-gap metric is:

[0026]

[0027] Among them, δ ν is the uncertainty described by v-gap, P 1 is the piezoelectric actuator model after hysteresis inverse compensation, P 2 is the dynamic linear module identified by the Hankel matrix method, j represents the imaginary unit, and ω represents the angular frequency.

[0028] Optionally, the inequality constraint for constructing the loop shaping system under the open-loop amplitude constraint is:

[0029] P 0 * (jω i ) 0 (jω i )<|s U (jω i )| 2 Λ(ω i ).

[0030] |s L (jω i )| 2 Λ(ω i )>P 0 * (jω i ) 0 (jω i )

[0031] Among them, P 0 * (jω i) is the complex conjugate transpose of the frequency response of the dynamic linear system, P 0 (jω i ) is the frequency response of the dynamic linear system, s U (jω i ) is the frequency response of the upper bound of the open-loop amplitude constraint, s L (jω i ) is the frequency response of the lower bound of the open-loop amplitude constraint, Λ(ω i ) is W -* (jω i )W -1 (jω i ), W -* (jω i ) is the inverse of the complex conjugate transpose of the frequency response of the weight function, W -1 (jω i ) is the inverse of the frequency response of the weight function.

[0032] Optionally, the first optimization problem is:

[0033]

[0034] Among them, P 0 (jω) is the frequency response of the dynamic linear system, Λ(ω) is W j -* (jω)W j -1 (jω), W j -* (jω) is the inverse of the complex conjugate transpose of the frequency response of the weight function obtained at the jth iteration, W j -1 (jω) is the inverse of the frequency response of the weight function obtained at the jth iteration, γ is the inverse of the robust stability margin, K j-1 (jω) is the optimal structured controller obtained in the j-1th iteration.

[0035] Optionally, the second optimization problem is:

[0036] minimizeγ

[0037] subject to||T zw (K j )|| ∞ =||T zw (q)|| ∞ ≤γ

[0038]

[0039] max Re(λ i (A cl (q)))<0

[0040] q∈D

[0041] Among them, T zw (K j ) is the closed-loop system composed of the controller obtained by the jth iteration, T zw (q) is the closed-loop system composed of the controller parameters obtained in the jth iteration, is the uncertainty described by v-gap, Re(λ i (A cl (q))) is the real part of the eigenvalue in the state matrix of the closed-loop system, q is the parameter vector in the controller, and D is the search domain of the controller parameters.

[0042] The beneficial effects of the present invention are:

[0043] The method proposed in the present invention can effectively compensate for the model uncertainty caused by modeling and inverse compensation, and has achieved significant improvements in dynamic response, control accuracy, and robust performance compared to other robust control methods.

[0044] The present invention quantifies the PEA modeling error under the Hammerstein structure through the v-gap metric theory, which can effectively reflect the uncertainty of the hysteresis nonlinear module and the dynamic module. Compared with the previous uncertainty description, the v-gap metric is simple to calculate, the description between systems is more accurate, and it has the advantages of clear frequency domain concept. The uncertainty described by the v-gap metric is equivalent to the uncertainty description of the coprime factor, and the stability condition described by the small gain theorem is still applicable.

[0045] The structured controller (H ∞ Loop Shaping Structured Control,H ∞ _LSSC) can effectively solve the standard H ∞ The shortcomings of the _LS controller design are eliminated, and finally a structured controller and weight function are obtained that can meet the predetermined control performance requirements while maximizing the generalized robust stability margin, achieving high-precision positioning and tracking control of PEA and effective suppression of modeling uncertainties and external disturbances. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0047] Figure 1It is a Hammerstein structure block diagram of an embodiment of the present invention;

[0048] Figure 2 The Bouc-Wen model and the hysteresis characteristic curve of the piezoelectric actuator according to the embodiment of the present invention;

[0049] Figure 3 is the Hankel singular value of the dynamic linear module of the embodiment of the present invention;

[0050] Figure 4 A frequency response comparison diagram of an input-output method of an identification system according to an embodiment of the present invention;

[0051] Figure 5 is a block diagram describing uncertainty in an embodiment of the present invention;

[0052] Figure 6 are the v-gap measurement values ​​of the nominal system and the identification system in the frequency domain of the embodiment of the present invention;

[0053] Figure 7 is the standard H of the embodiment of the present invention ∞ Loop shaping control block diagram;

[0054] Figure 8 are the open-loop singular values ​​of the optimized formed system and the unformed system of the embodiment of the present invention;

[0055] Fig. 9 It is the v-gap metric and the generalized robust stability margin in the frequency domain of the embodiment of the present invention;

[0056] Fig.10 : It is a single frequency tracking control diagram of an embodiment of the present invention, wherein (a) is a tracking control experimental curve comparison diagram of different control schemes under a single frequency sinusoidal signal with a reference input amplitude of 1 μm and a frequency of 20 Hz; (b) is a tracking control experimental curve comparison diagram of different control schemes under a single frequency sinusoidal signal with a reference input amplitude of 1 μm and a frequency of 40 Hz; (c) is a tracking control experimental curve comparison diagram of different control schemes under a single frequency sinusoidal signal with a reference input amplitude of 1 μm and a frequency of 80 Hz; (d) is a tracking control experimental curve comparison diagram of different control schemes under a single frequency sinusoidal signal with a reference input amplitude of 1 μm and a frequency of 100 Hz;

[0057] Fig.11 It is a composite frequency tracking control diagram of an embodiment of the present invention, wherein (a) is a comparison diagram of tracking control experimental curves of different control schemes under a composite frequency sinusoidal signal with a reference input amplitude of 1 μm and a frequency of 20 / 40 / 80 Hz; (b) is a comparison diagram of tracking control experimental curves of different control schemes under a composite frequency sinusoidal signal with a reference input amplitude of 1 μm and a frequency of 40 / 80 / 100 Hz. DETAILED DESCRIPTION

[0058] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0059] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0060] This embodiment provides a hysteresis modeling and loop shaping structured control method for a piezoelectric actuator, including:

[0061] Step 1: Use Hammerstein structure to build a piezoelectric actuator model;

[0062] Step 2, describing the uncertainty of the piezoelectric actuator model based on the v-gap metric;

[0063] Step 3: setting an open-loop amplitude constraint, a controller structure, and a parameter vector search domain based on the uncertainty description;

[0064] Step 4: Set the initial weight function, accuracy coefficient and initial controller;

[0065] Step 5, constructing the inequality constraint of the loop shaping system under the open-loop amplitude constraint;

[0066] Step 6: Under the inequality constraint, iteratively solve the first optimization problem and the second optimization problem until the preset iteration condition is reached to obtain the optimal weight function and the desired controller, wherein the first optimization problem is a generalized eigenvalue minimization problem, which is used to solve the optimal weight function; the second optimization problem is a non-convex optimization problem, which is used to solve the desired controller;

[0067] Step 7: If the solutions to the first optimization problem and the second optimization problem are not obtained until the preset iteration conditions are reached, return to step 3 to reset the open-loop amplitude constraint and the controller parameter vector search domain, and perform steps 4 to 6 again until the optimal weight function and the desired controller are obtained.

[0068] Specifically, the method proposed in this embodiment can effectively compensate for the model uncertainty caused by modeling and inverse compensation, and has achieved significant improvements in dynamic response, control accuracy, and robust performance compared to other robust control methods.

[0069] Combine the following Figure 1-Figure 9The method proposed in this embodiment is described in detail, which specifically includes the following steps:

[0070] Step 1: Use Hammerstein structure to model PEA. Its structural diagram is as follows: Figure 1 shown.

[0071] Furthermore, the modeling of PEA using the Hammerstein structure specifically includes:

[0072] Hammerstein structure is used to model the piezoelectric actuator, and the Bouc-Wen model is used to describe the static hysteresis nonlinear module in the Hammerstein structure, and the Hankel matrix method is used to identify the dynamic linear module in the piezoelectric actuator.

[0073] The hysteresis nonlinear characteristics of the piezoelectric actuator are obtained based on the static hysteresis nonlinear module and the dynamic linear module, and the Bouc-Wen inverse model is used to eliminate the hysteresis nonlinear characteristics to complete the construction of the piezoelectric actuator model.

[0074] Specifically, the improved Bouc-Wen model is used to describe the static hysteresis curve in the Hammerstein structure that is not limited to odd symmetry, and its expression is:

[0075]

[0076] Among them, u is the input voltage of the actuator, y is the output displacement of the actuator, and z is the hysteresis component. The shape of the hysteresis loop can be controlled by reasonably selecting the unknown parameters k, d, n, ρ, σ. is the derivative of the hysteresis component, is the derivative of the input voltage.

[0077] Since the piezoelectric actuator can be approximated as a static hysteresis system in the low frequency band, the relative error RE between the actual output of the piezoelectric actuator and the model output is set as the performance index according to the input / output experimental data in the low frequency band, and the Bouc-Wen model parameters are obtained by minimizing RE using the optimization algorithm. The Bouc-Wen model is modeled using the input and output data measured by the actuator with a 1Hz signal excitation, and the Bouc-Wen model parameters obtained by GA optimization are as follows:

[0078] k=2.683, d=3.386, n=3.310.ρ=1.067, σ=1.991

[0079] The relative error RE between the actual output of PEA and the model output is 0.0243. The input and output of the identified Bouc-Wen model and the actual input and output of PEA are shown in the attached figure. Figure 2 shown.

[0080] Specifically, the Hankel matrix identification method is used to identify the dynamic linear module that compensates the nonlinear module.

[0081] A: Assume that the dynamic linear model of piezoelectric actuator identification is as follows:

[0082]

[0083] in, is the state vector, η(k) is the output vector, u(k) is the input vector, n d are the identification parameters to be sought, D∈R m×q are constant matrices. These four constant matrices can be obtained through the impulse response of the system, and the following equations hold:

[0084] g(0)=D, g(1)=CB, g(2)=CAB,...,

[0085] g(k)=CA k-1 B,....

[0086] Among them, g(kT), k=0,1,2,..., is the impulse response of the discrete system (the initial state is zero).

[0087] B: Use a pseudo-random signal as the input signal to input into the system to be identified, collect the input and output signals of the system, and define the autocorrelation and cross-correlation of the input and output as:

[0088]

[0089] Among them, T is the period, i is the sampling point, N is the sequence length of one period of the input signal, u is the input of the system, and y is the output of the system.

[0090] C: According to the relationship between impulse response and correlation function:

[0091]

[0092] For a sufficiently large N>>n, it can be considered that g(NT+lT)≈0,l≥0. In order to determine the system parameters A, B, C, D, the Hankel matrix H is constructed. At the same time, according to the relationship between the impulse response and the system matrix, the following Hankel matrix is ​​constructed:

[0093]

[0094] D: The order n of the system to be determined dIt is determined by the singular value decomposition of the Hankel matrix. The Hankel matrix constructed by experimental data is usually a non-singular matrix. The singular values ​​of the Hankel matrix are arranged from small to large, and the order of the system is selected according to the position where the singular value changes downward. Perform singular value decomposition on the Hankel matrix H:

[0095]

[0096] Among them, U 1 , U 2 、V 1 、V 2 is an orthogonal matrix and U 1 =[u 1 …u r ],V 1 =[v 1 …v r ],Σ 1 and Σ 2 is a diagonal matrix with decreasing diagonal elements, and here σ (Σ 1 ), Respectively represent Σ 1 The smallest singular value of 2 The largest singular value of 1 The dimension of the state matrix A is equal to the dimension n d .

[0097] The construction dimension is n d The Hankel matrix H 1 :

[0098]

[0099] E: Obtain the parameter matrix A, B, C of the discrete space:

[0100]

[0101] D=g(0)≈0.

[0102] A pseudo-random signal is used as the input signal to the dynamic linear module to be identified, the input and output signals of the system are collected, and the Hankel matrix identification method is used to identify the system. The order of the system is obtained as shown in the attached figure. Figure 3 As shown, the final transfer function P with the order of 2 is obtained 0 :

[0103]

[0104] Where s is a complex domain operator.

[0105] The obtained model frequency response comparison is shown in the attached figure. Figure 4 shown.

[0106] Step 2: Describe the uncertainty of the piezoelectric actuator model based on the v-gap metric. The structural diagram is shown in the attached figure. Figure 5 shown.

[0107] The v-gap value between two systems is calculated using the concept of v-gap metric in the frequency domain:

[0108]

[0109] Among them, δ ν is the uncertainty described by v-gap, P 1 is the PEA system after hysteresis inverse compensation, P 2 is the linear system identified by the Hankel matrix method, j represents the imaginary unit, and ω represents the angular frequency.

[0110] The v-gap values ​​obtained after multiple experiments are shown in the attached figure. Figure 6 shown.

[0111] Step 3: Set the open-loop amplitude constraint, controller structure and parameter vector search domain based on the uncertainty description.

[0112] Set the upper bound S of the open-loop amplitude constraint according to the dynamic performance requirements of PEA U (s) and the lower bound S L (s), and determine the controller structure and set the search domain of the controller parameter vector. Set the desired open-loop amplitude upper bound S for the performance characteristics required by PEA U (s) and the lower bound S L (s) is:

[0113]

[0114] Where s is a complex domain operator.

[0115] Set the controller structure to a standard PID controller, whose structure is given by:

[0116]

[0117] Among them, K P is the proportional gain, K I is the integral gain, K D is the differential gain, τ is the time constant of the filter applied to the differential action, A K is the state matrix, B K is the input matrix, C K is the output matrix, D K is a direct transfer matrix.

[0118] The parameter vector in the controller is: q = vect(K) = vect(K P ,K I .K D ,τ), set the upper bound of the controller’s search domain and the lower bound q =[-5 -100 -10 -5].

[0119] Step 4: Set the initial weight function, accuracy coefficient and initial controller.

[0120] Set the initial weight function W 0 (For example, W 0 =1) and the coefficient of accuracy E r , let j = 1, where j represents the number of iterations, and the unformed system P 0 The optimal H of the design criteria ∞ _LS controller is set as the initial controller K 0 , and calculate the corresponding robust stability margin b 0 (P 0 ,K 0 ).

[0121]

[0122] Where I is the identity matrix.

[0123] Step 5: Construct the inequality constraint of the loop shaping system under the open-loop amplitude constraint.

[0124] For the selected frequency point ω k , k=1,2,...,N, the amplitude constraint of the forming system is transformed into the following two inequality constraints:

[0125] The open loop amplitude of the forming system is located at the upper limit S U (s) is given by the following inequality:

[0126] P 0 * (jω i ) 0 (jω i )<|s U (jω i )| 2 Λ(ω i ).

[0127] The open loop amplitude of the forming system is located at the lower limit S L (s) above is given by the following inequality:

[0128] |s L (jωi )| 2 Λ(ω i )>P 0 * (jω i ) 0 (jω i )

[0129] Among them, P 0 * (jω i ) is the complex conjugate transpose of the frequency response of the dynamic linear system, P 0 (jω i ) is the frequency response of the dynamic linear system, s U (jω i ) is the frequency response of the upper bound of the open-loop amplitude constraint, s L (jω i ) is the frequency response of the lower bound of the open-loop amplitude constraint, Λ(ω i ) is W -* (jω i )W -1 (jω i ), W -* (jω i ) is the inverse of the complex conjugate transpose of the frequency response of the weight function, W -1 (jω i ) is the inverse of the frequency response of the weight function.

[0130] The optimization problem of maximizing b(P,K) is transformed into minimizing γ under the above two inequality constraints: 2 Minimizing the generalized eigenvalue (GEVP) problem:

[0131]

[0132] Among them, P 0 (jω) is the frequency response of the dynamic linear system, Λ(ω) is W j -* (jω)W j -1 (jω), W j -* (jω) is the inverse of the complex conjugate transpose of the frequency response of the weight function obtained at the jth iteration, W j -1 (jω) is the inverse of the frequency response of the weight function obtained at the jth iteration, γ is the inverse of the robust stability margin, K j-1 (jω) is the optimal structured controller obtained in the j-1th iteration.

[0133] By calculating the generalized eigenvalue minimization problem described by this set of LMI at each frequency point, the amplitude curve after shaping can be obtained at S U (s) and S L (s) and can maximize the robust stability margin b(P j ,K j ) is then calculated by fitting a stable minimum phase transfer function to (Λ(ω)) 1 / 2 , we can get the weight function W expected in the design j (s).

[0134] Setting P j (s) = W j (s)P 0 (s), where P j is the forming system obtained in the jth iteration.

[0135] Step 6: Under the inequality constraint, the first optimization problem and the second optimization problem are iteratively solved until the preset iteration condition is reached to obtain the optimal weight function and the expected controller.

[0136] The first optimization problem is a generalized eigenvalue minimization problem, which is used to solve the optimal weight function; the second optimization problem is a non-convex optimization problem, which is used to solve the desired controller;

[0137] The block diagram of the structured controller is shown in the attached Figure 7 As shown, the closed-loop input-output relationship of the system is:

[0138]

[0139] The corresponding structured controller is the solution to the following nonconvex nonsmooth optimization problem:

[0140] minimizeγ

[0141] subject to||T zw (K j )|| ∞ =||T zw (q)|| ∞ ≤γ

[0142]

[0143] max Re(λ i (A cl (q)))<0

[0144] q∈D

[0145] Among them, T zw (K j) is the closed-loop system composed of the controller obtained by the jth iteration, T zw (q) is the closed-loop system composed of the controller parameters obtained in the jth iteration, is the uncertainty described by v-gap, and in this embodiment, the δ calculated above is substituted into v (P 1 ,P 2 ), Re(λ i (A)) represents the real part of the eigenvalue of matrix A. In this embodiment, Re(λ i (A cl (q))) is the real part of the eigenvalue in the state matrix of the closed-loop system, q is the parameter vector in the controller, and D is the search domain of the controller parameters.

[0146] The structured controller K that minimizes γ is solved using a genetic algorithm through the obtained weight function j , get the corresponding minimum value of γ, record b j (P j ,K j )=γ -1 .

[0147] Step 7: If the solutions to the first optimization problem and the second optimization problem are not obtained until the preset iteration conditions are reached, return to step 3 to reset the open-loop amplitude constraint and the controller parameter vector search domain, and perform steps 4 to 6 again until the optimal weight function and the desired controller are obtained.

[0148] If |b j (P j ,K j )-b j-1 (P j-1 ,K j-1 )|<E r Then exit the iteration, otherwise, j=j+1, and return to step 5.

[0149] By continuously iterating the above two optimization problems, we can finally get j (P j ,K j ) maximizes the weight function W(s) and controller K(s), and the final b j (P j ,K j ) may not be optimal, but it can guarantee b j (P j ,K j ) increases monotonically. If no solution to the optimization problem is obtained in steps 5 and 6, the designer should return to step 3 to reselect the upper and lower bounds of the amplitude limit and the search domain of the controller parameter vector.

[0150] Finally, through the above weight function and controller iterative optimization algorithm, the weight function transfer function can be obtained as:

[0151]

[0152] Open loop amplitude upper limit S U (s) and the lower bound S L (s), the open-loop singular value curves of the optimized formed system and the unformed system are shown in Figure 8 shown.

[0153] The optimized controller K PID The parameters are:

[0154] K P =1.3049,K I =1.5765,K D =-0.5466,τ=3.1480

[0155] The final optimized b(P,K PID )=0.663, which satisfies the robust stabilization condition. The generalized robust stability margin and the uncertainty described by v-gap in the frequency domain are described as follows: Fig. 9 shown.

[0156] In the experiment, the tracking trajectory r(t) is selected as a sinusoidal signal of different frequencies, with an amplitude of 1 μm. The inputs are sinusoidal tracking signals of 20 Hz, 40 Hz, 80 Hz, and 100 Hz, respectively. The tracking effects are shown in Figure 2. Fig.10 As shown, (a) is a comparison diagram of the tracking control experimental curves of different control schemes under the reference input amplitude of 1 μm and a single frequency sinusoidal signal of 20 Hz; (b) is a comparison diagram of the tracking control experimental curves of different control schemes under the reference input amplitude of 1 μm and a single frequency sinusoidal signal of 40 Hz; (c) is a comparison diagram of the tracking control experimental curves of different control schemes under the reference input amplitude of 1 μm and a single frequency sinusoidal signal of 80 Hz; (d) is a comparison diagram of the tracking control experimental curves of different control schemes under the reference input amplitude of 1 μm and a single frequency sinusoidal signal of 100 Hz.

[0157] Use a composite frequency sinusoidal signal as the input signal, and input sinusoidal tracking signals with frequencies of 20 / 40 / 80Hz and 40 / 80 / 100Hz respectively. The tracking effect is as follows Fig.11 As shown, (a) is a comparison diagram of tracking control experimental curves of different control schemes under the reference input amplitude of 1 μm and the composite frequency sinusoidal signal of 20 / 40 / 80 Hz; (b) is a comparison diagram of tracking control experimental curves of different control schemes under the reference input amplitude of 1 μm and the composite frequency sinusoidal signal of 40 / 80 / 100 Hz.

[0158] Table 1 shows the loop shaping structured control method (H) under open-loop amplitude constraints at different frequency inputs. ∞ _LSSC)、H ∞ _LS control method and H ∞ Relative error and RMS error corresponding to the control method.

[0159] Table 1

[0160]

[0161] Depend on Fig.10 and Fig.11 It can be seen that the loop shaping structured control method (H) under the open-loop amplitude constraint designed in this embodiment ∞ _LSSC)、H ∞ _LS control method and H ∞ All control methods can achieve real-time tracking control, but the control strategy proposed in this embodiment has more advantages. From the comparison of tracking errors in Table 1, it can be obtained that H ∞ _LSSC method, H ∞ The control effect of LS control method is better than that of H ∞ Control. ∞ _LSSC and H ∞ The reason why the LS control effect is similar is because H ∞ _LS controller adopts H ∞ _The same weight function after optimization of the LSSC control method. The weight function after iterative optimization is in the frequency range of 10 -2 ≤ω≤10 5 rad / s to meet the preset upper and lower limits of the open-loop amplitude. ∞ The final optimized γ value of the LSSC control method is 1.5099, H ∞ The γ value of the optimal controller obtained by the _LS method is 1.2147. Since the γ value is less than 5, it will not change the closed-loop shape too much, so the performance characteristics of the two mapped by the open-loop amplitude are also very similar, but H ∞ The structured controller of LSSC adopts the standard PID controller structure, and its order is smaller than H ∞ The high-order controller obtained by the _LS method is easier to implement and maintain in engineering. ∞ _The controller generalized robust stability margin b(P,K) obtained by iterative optimization of LSSC control method PID )=0.663. It is generally believed that when b(P,K)>0.2~0.3, the controller has good robustness. Therefore, the designed structured controller can fully suppress the error caused by modeling and external disturbances, which inherits the standard H∞ Robustness and performance characteristics under the LS framework. According to the above experimental results, the loop shaping structured control method under the open-loop amplitude constraint proposed in the present invention can track multi-frequency sinusoidal signals in real time and efficiently. The obtained low-order controller is easy to implement and adjust in engineering applications, and its comprehensive control quality is better than the other two control methods used for comparison.

[0162] The method proposed in this embodiment quantifies the PEA modeling error under the Hammerstein structure through the v-gap metric theory, which can effectively reflect the uncertainty of the hysteresis nonlinear module and the dynamic module. Compared with the previous uncertainty description, the v-gap metric is simple to calculate, the description between systems is more accurate, and it has the advantages of clear frequency domain concept. The uncertainty described by the v-gap metric is equivalent to the coprime factor uncertainty description, and the stability condition described by the small gain theorem is still applicable.

[0163] The structured controller (H) designed in this embodiment ∞ Loop Shaping Structured Control,H ∞ _LSSC) can effectively solve the standard H ∞ _LS controller design points, and finally obtain a structured controller and weight function that can meet the predetermined control performance requirements while maximizing the generalized robust stability margin, achieving high-precision positioning and tracking control of PEA and effective suppression of modeling uncertainties and external disturbances.

[0164] The embodiments described above are only descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A hysteresis modeling and loop shaping structured control method for a piezoelectric actuator, characterized in that: include: Step 1: Use Hammerstein structure to build a piezoelectric actuator model; Step 2, describing the uncertainty of the piezoelectric actuator model based on the v-gap metric; Step 3: setting an open-loop amplitude constraint, a controller structure, and a parameter vector search domain based on the uncertainty description; Step 4: Set the initial weight function, precision coefficient and initial controller; Step 5: constructing an inequality constraint of the loop shaping system under the open-loop amplitude constraint; Step 6: Under the inequality constraint, iteratively solve the first optimization problem and the second optimization problem until a preset iteration condition is reached to obtain an optimal weight function and an expected controller, wherein the first optimization problem is a generalized eigenvalue minimization problem for solving the optimal weight function; and the second optimization problem is a non-convex optimization problem for solving the expected controller; Step 7: If the solutions of the first optimization problem and the second optimization problem are not obtained until the preset iteration condition is reached, return to step 3 to reset the open-loop amplitude constraint and the parameter vector search domain of the controller, and perform steps 4 to 6 again until the optimal weight function and the desired controller are obtained.

2. The hysteresis modeling and loop shaping structured control method of a piezoelectric actuator according to claim 1, characterized in that: The piezoelectric actuator model constructed using the Hammerstein structure includes: Hammerstein structure is used to model the piezoelectric actuator, and the Bouc-Wen model is used to describe the static hysteresis nonlinear module in the Hammerstein structure, and the Hankel matrix method is used to identify the dynamic linear module in the piezoelectric actuator. The hysteresis nonlinear characteristics of the piezoelectric actuator are obtained based on the static hysteresis nonlinear module and the dynamic linear module, and the Bouc-Wen inverse model is used to eliminate the hysteresis nonlinear characteristics to complete the construction of the piezoelectric actuator model.

3. The hysteresis modeling and loop shaping structured control method of a piezoelectric actuator according to claim 2, characterized in that: The Bouc-Wen model is used to describe the static hysteresis nonlinear module in the Hammerstein structure: Where u is the input voltage of the piezoelectric actuator, y is the output displacement of the piezoelectric actuator, z is the hysteresis component, k, d, n, ρ, σ are the unknown parameters that control the shape of the hysteresis loop, is the derivative of the hysteresis component, is the derivative of the input voltage.

4. The hysteresis modeling and loop shaping structured control method of a piezoelectric actuator according to claim 2, characterized in that: The dynamic linear module in the piezoelectric actuator is identified using the Hankel matrix method as follows: Among them, P0 is the dynamic linear module transfer function, and s is the complex domain operator.

5. The hysteresis modeling and loop shaping structured control method of a piezoelectric actuator according to claim 1, characterized in that: The method for describing the uncertainty of the piezoelectric actuator model based on the v-gap metric is: Among them, δ ν is the uncertainty described by v-gap, P1 is the piezoelectric actuator model after hysteresis inverse compensation, P2 is the dynamic linear module identified by the Hankel matrix method, j is represented by the imaginary unit, and ω is represented by the angular frequency.

6. The hysteresis modeling and loop shaping structured control method of a piezoelectric actuator according to claim 1, characterized in that: The inequality constraint for constructing the loop shaping system under the open-loop amplitude constraint is: P0 * (y) i )P0(jω i )<|s U (y) i )| 2 L(w i ). |s L (y) i )| 2 L(w i )>P0 * (y) i )P0(jω i ) Among them, P0 * (jω i ) is the complex conjugate transpose of the frequency response of the dynamic linear system, P0(jω i ) is the frequency response of the dynamic linear system, s U (jω i ) is the frequency response of the upper bound of the open-loop amplitude constraint, s L (jω i ) is the frequency response of the lower bound of the open-loop amplitude constraint, Λ(ω i ) is W -* (jω i )W -1 (jω i ), W -* (jω i ) is the inverse of the complex conjugate transpose of the frequency response of the weight function, W -1 (jω i ) is the inverse of the frequency response of the weight function.

7. The hysteresis modeling and loop shaping structured control method of a piezoelectric actuator according to claim 1, characterized in that: The first optimization problem is: Where P0(jω) is the frequency response of the dynamic linear system, Λ(ω) is W j -* (jω)W j -1 (jω), W j -* (jω) is the inverse of the complex conjugate transpose of the frequency response of the weight function obtained at the jth iteration, W j -1 (jω) is the inverse of the frequency response of the weight function obtained at the jth iteration, γ is the inverse of the robust stability margin, K j-1 (jω) is the optimal structured controller obtained in the j-1th iteration.

8. The hysteresis modeling and loop shaping structured control method for a piezoelectric actuator according to claim 7, characterized in that: The second optimization problem is: minimizeγ subject to||T zw (K j )|| ∞ =||T zw (q)|| ∞ ≤γ maxRe(λ i (A cl (q)))<0 q∈D Among them, T zw (K j ) is the closed-loop system composed of the controller obtained by the jth iteration, T zw (q) is the closed-loop system composed of the controller parameters obtained in the jth iteration, is the uncertainty described by v-gap, Re(λ i (A cl (q))) is the real part of the eigenvalue in the state matrix of the closed-loop system, q is the parameter vector in the controller, and D is the search domain of the controller parameters.

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