A method for hysteresis modeling and loop shaping structured control of piezoelectric actuators

Abstract

This invention relates to a hysteresis modeling and loop-forming structured control method for piezoelectric actuators, comprising: constructing a piezoelectric actuator model using a Hammerstein structure; describing the uncertainty of the piezoelectric actuator model based on the v-gap metric, and setting open-loop amplitude constraints and the structure and parameter vector search domain of the controller based on the uncertainty description; setting initial weight functions, accuracy coefficients, and an initial controller, and constructing inequality constraints for the loop-forming system under the open-loop amplitude constraints; iteratively solving a first optimization problem and a second optimization problem under the inequality constraints until a preset iteration condition is met, obtaining the optimal weight function and the desired controller; if no solution to the optimization problem is obtained until the preset iteration condition is met, the open-loop amplitude constraints and the parameter vector search domain of the controller are reset, and the next round of iterative solution is performed until a solution is obtained. This invention achieves significant improvements in dynamic response, control accuracy, and robustness.

Description

Technical Field

[0001] This invention relates to the field of control technology, and in particular to a hysteresis modeling and loop shaping structured control method for piezoelectric actuators. Background Technology

[0002] Piezoelectric actuators (PEAs), with their superior performance characterized by high resolution, high natural frequency, and fast response, have been widely used in ultra-precision positioning systems requiring nanometer-level accuracy. These PEA-driven positioning systems play a crucial role in high-end technology fields such as biomedicine, materials science, and aerospace. However, PEAs suffer from a significant drawback: severe hysteresis nonlinearity between the input voltage and output displacement. This not only affects the positioning accuracy of ultra-precision positioning systems but can also lead to instability in the control system. Therefore, achieving nanometer-level tracking and positioning accuracy for PEA systems with hysteresis nonlinearity is an extremely challenging task.

[0003] The realization of high-precision control relies on accurate mathematical models. Modeling of hysteretic nonlinear systems can be divided into three main categories: physical models, phenomenological models, and intelligent models. Physical models are built upon the physical mechanisms of hysteresis, such as the JA model, Duhem model, and Bouc-Wen model. Phenomenological models model through input-output data, without involving physical structure; their parameters lack physical interpretation. Typical models include the Preisach model and the Prandtl-Ishlinskii (PI) model. Intelligent models utilize techniques such as artificial neural networks and support vector machines. These models are typically based on the assumption of rate independence, i.e., the so-called static hysteresis nonlinearity, where the input and output depend only on the current and historical inputs, and are independent of the rate of change of the input. In recent years, 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] Tracking control techniques for PEAs (Power Emitting Acres) are mainly divided into three categories: control strategies using charge generators, tracking control strategies without inverse compensation, and tracking control strategies based on inverse compensation. The charge generator-based control strategy is designed based on the approximately linear relationship between the output displacement and input charge of the PEA. While this strategy effectively reduces the hysteresis nonlinearity of the PEA, the implementation cost of the control system is high, and charge measurement may be affected by zero-point 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 guarantee system stability. The feedback tracking control strategy based on inverse compensation is currently the most widely used control strategy. This strategy first models the hysteresis phenomenon of the PEA, then constructs the inverse model of the model, and customizes a compensator through the inverse hysteresis model to eliminate hysteresis nonlinearity. Finally, a feedback control method is used to achieve closed-loop control of the system. Compared with the first two methods, this strategy has more advantages. However, due to the potential errors in the hysteresis inverse model modeling, coupled with the influence of external disturbances, advanced feedback control algorithms are needed to ensure the tracking accuracy and robustness of the system.

[0005] In summary, to address the error issues arising from modeling and approximate linearization, this invention employs robust control principles to design a high-precision controller. The loop-shaping design procedure, proposed by McFarlane and Glover, balances dynamic performance and robust stability by appropriately selecting loop compensators, and uses standard left coprime factorization to ensure the stability of the closed-loop system. However, this controller suffers from two drawbacks that limit its widespread application in industrial production. First, designers struggle to obtain suitable open-loop weight functions, requiring repeated experimentation to find a function that satisfies both the desired dynamic performance and maximizes the generalized robust stability margin. Second, the resulting optimal controller is an unstructured high-order controller, which is difficult to implement and maintain in industrial production. Therefore, finding a weight function that balances performance and robustness, along with a low-order robust controller that ensures closed-loop stability and exhibits good performance, is a worthy research challenge. Summary of the Invention

[0006] The purpose of this invention is to provide a hysteresis modeling and loop-forming structured control method for piezoelectric actuators in piezoelectric positioning systems with severe hysteresis nonlinear characteristics. This method yields a loop-forming fixed structure controller with good performance characteristics and robust stability, enabling high-precision positioning and tracking control of the PEA and effective suppression of modeling uncertainties and external disturbances.

[0007] To achieve the above objectives, the present invention provides the following solution:

[0008] A hysteresis modeling and loop shaping structured control method for piezoelectric actuators includes:

[0009] Step 1: Construct a piezoelectric actuator model using the Hammerstein structure;

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

[0011] Step 3: Based on the uncertainty description, set open-loop amplitude constraints, as well as the controller's structure and parameter vector search domain;

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

[0013] Step 5: Construct inequality constraints for the loop shaping system under the open-loop amplitude constraints;

[0014] Step 6: Under the inequality constraints, iteratively solve the first optimization problem and the second optimization problem until the preset iteration conditions are met, and obtain the optimal weight function and the desired controller. The first optimization problem is a generalized eigenvalue minimization problem, used to solve for the optimal weight function; the second optimization problem is a non-convex optimization problem, used to solve for the desired controller.

[0015] Step 7: If no solution to the first optimization problem and the second optimization problem is 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 repeat steps 4-6 until the optimal weight function and the desired controller are obtained.

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

[0017] The piezoelectric actuator is modeled using the Hammerstein structure, and the static hysteresis nonlinear module in the Hammerstein structure is described using the Bouc-Wen model. The dynamic linear module in the piezoelectric actuator is identified using the Hankel matrix method.

[0018] The hysteresis nonlinearity characteristics of the piezoelectric actuator are obtained based on the static hysteresis nonlinearity module and the dynamic linearity module, and the hysteresis nonlinearity characteristics are eliminated by using the Bouc-Wen inverse model, thus completing the construction of the piezoelectric actuator model.

[0019] Optionally, the static hysteresis nonlinear module in the Hammerstein structure can be described using the Bouc-Wen model 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, and k, d, n, ρ, σ are undetermined parameters controlling the shape of the hysteresis loop. The derivative of the hysteresis component, It is the derivative of the input voltage.

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

[0023]

[0024] Where P0 is the dynamic linear module transfer function and s is the complex domain operator.

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

[0026]

[0027] Where, δ ν P1 represents the piezoelectric actuator model after hysteresis inverse compensation, P2 represents 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 constraints for constructing the loop shaping system under the open-loop amplitude constraint are:

[0029] P0 * (jω i )P0(jω i )<|s U (jω i )| 2 Λ(ω i ).

[0030] |s L (jω i )| 2 Λ(ω i )>P0 * (jω i )P0(jω i )

[0031] Among them, P0 * (jω i P0(jω) is the complex conjugate transpose of the frequency response of a dynamic linear system. i ) represents the frequency response of a dynamic linear system, s U (jω i ) represents the frequency response of the open-loop amplitude constraint upper bound, sL (jω i ) represents the frequency response of the open-loop amplitude constraint lower bound, Λ(ω) i ) for W -* (jω i W -1 (jω i ), W -* (jω i W is the reciprocal of the complex conjugate transpose of the frequency response of the weight function. -1 (jω i ) is the reciprocal of the frequency response of the weight function.

[0032] Optionally, the first optimization problem is:

[0033]

[0034] Where P0(jω) is the frequency response of the dynamic linear system, and Λ(ω) is the frequency response of W. j -* (jω)W j -1 (jω), W j -* (jω) is the reciprocal of the complex conjugate transpose of the frequency response of the weight function obtained in the j-th iteration, W j -1 (jω) is the reciprocal of the frequency response of the weight function obtained in the j-th iteration, γ is the reciprocal of the robust stability margin, and K j-1 (jω) is the optimal structured controller obtained in the (j-1)th 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 Let T be a closed-loop system consisting of controllers obtained from the j-th iteration. zw(q) represents the closed-loop system composed of the controller parameters obtained from the j-th iteration. For the uncertainty described by v-gap, Re(λ) i (A cl (q))) represents the real part of the eigenvalues ​​in the closed-loop system state matrix, q is the parameter vector in the controller, and D is the search domain of the controller parameters.

[0042] The beneficial effects of this invention are as follows:

[0043] The method proposed in this 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 with other robust control methods.

[0044] This invention quantifies the PEA modeling error under the Hammerstein structure using the v-gap metric theory, effectively reflecting the uncertainties present in hysteretic nonlinear modules and dynamic modules. Compared to previous uncertainty descriptions, the v-gap metric is simpler to calculate, provides a more accurate description between systems, and has a clear frequency domain concept. The uncertainty described by the v-gap metric is equivalent to the coprime factor uncertainty description, and the stability conditions described by the small gain theorem still apply.

[0045] The structured controller (H) designed in this invention ∞ Loop Shaping Structured Control,H ∞ _LSSC) can effectively solve the standard H ∞ The shortcomings of the LS controller design were addressed, and a structured controller and weight function were finally obtained that can meet the predetermined control performance requirements while maximizing the generalized robust stability margin, thereby achieving high-precision positioning and tracking control of PEA and effective suppression of modeling uncertainties and external disturbances. Attached Figure Description

[0046] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

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

[0048] Figure 2 The figures show the hysteresis characteristic curves of the Bouc-Wen model and the piezoelectric actuator in this embodiment of the invention.

[0049] Figure 3 The Hankel singular value of the dynamic linear module in this embodiment of the invention;

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

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

[0052] Figure 6 The v-gap metric values ​​in the frequency domain of the nominal system and the identification system in this embodiment of the invention;

[0053] Figure 7 Standard H of the embodiments of the present invention ∞ Loop forming control block diagram;

[0054] Figure 8 These are the open-loop singular values ​​of the optimized forming system and the unformed system in this embodiment of the invention;

[0055] Figure 9 This refers to the v-gap metric and generalized robust stability margin in the frequency domain of this invention.

[0056] Figure 10 The following are single-frequency tracking control diagrams according to embodiments of the present invention: (a) is a comparison diagram of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 20Hz; (b) is a comparison diagram of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 40Hz; (c) is a comparison diagram of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 80Hz; and (d) is a comparison diagram of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 100Hz.

[0057] Figure 11 The above 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 1μm and a frequency of 20 / 40 / 80Hz; and (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 1μm and a frequency of 40 / 80 / 100Hz. Detailed Implementation

[0058] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0059] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be 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 piezoelectric actuators, including:

[0061] Step 1: Construct a piezoelectric actuator model using the Hammerstein structure;

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

[0063] Step 3: Based on the uncertainty description, set open-loop amplitude constraints, as well as the controller's structure and parameter vector search domain;

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

[0065] Step 5: Construct inequality constraints for the loop shaping system under open-loop amplitude constraints;

[0066] Step 6: Under inequality constraints, iteratively solve the first optimization problem and the second optimization problem until the preset iteration conditions are met, and obtain the optimal weight function and the desired controller. The first optimization problem is a generalized eigenvalue minimization problem, used to solve for the optimal weight function; the second optimization problem is a non-convex optimization problem, used to solve for the desired controller.

[0067] Step 7: If no solution to the first and second optimization problems is obtained until the preset iteration conditions are met, return to step 3 to reset the open-loop amplitude constraints and the parameter vector search domain of the controller, and repeat steps 4-6 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 with other robust control methods.

[0069] The following is combined with Figures 1-9 The method proposed in this embodiment will be described in detail, specifically including the following steps:

[0070] Step 1: Model the PEA using the Hammerstein structure, as shown in the block diagram below. Figure 1 As shown.

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

[0072] The piezoelectric actuator is modeled using the Hammerstein structure, and the static hysteresis nonlinear module in the Hammerstein structure is described using the Bouc-Wen model. The dynamic linear module in the piezoelectric actuator is identified using the Hankel matrix method.

[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 hysteresis nonlinear characteristics are eliminated by using the Bouc-Wen inverse model, thus completing 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, which is not limited to odd symmetry. Its expression is as follows:

[0075]

[0076] Where 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 is controlled by reasonably selecting the undetermined parameters k, d, n, ρ, and σ. The derivative of the hysteresis component, It is the derivative of the input voltage.

[0077] Since piezoelectric actuators can be approximated as static hysteresis systems in the low-frequency range, based on the input / output experimental data in the low-frequency range, the relative error (RE) between the actual output and the model output of the piezoelectric actuator is set as the performance index. The Bouc-Wen model parameters are identified by minimizing the RE using an optimization algorithm. Bouc-Wen modeling is performed using the measured input / output data of the actuator excited by a 1Hz signal. The Bouc-Wen model parameters obtained by GA optimization are shown below:

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

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

[0080] Specifically, the Hankel matrix identification method is used to identify the dynamic linear modules that have had their nonlinear modules compensated for.

[0081] A: Suppose the dynamic linear model for piezoelectric actuator identification is as follows:

[0082]

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

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

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

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

[0087] B: A pseudo-random signal is used as the input signal to the system to be identified. The input and output signals of the system are collected, and the autocorrelation and cross-correlation of the input and output are defined as follows:

[0088]

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

[0090] C: Based on the relationship between impulse response and correlation function:

[0091]

[0092] For a sufficiently large N >> n, we can assume g(NT + lT) ≈ 0, l ≥ 0. To determine the system parameters A, B, C, D, we construct the Hankel matrix H. Simultaneously, based on the relationship between the impulse response and the system matrix, we construct the following Hankel matrix:

[0093]

[0094] D: The order n of the system to be determined dThe order of the system is determined by the singular value decomposition of the Hankel matrix. The Hankel matrix constructed from experimental data is usually a non-singular matrix. The singular values ​​of the Hankel matrix are arranged in ascending order, and the order of the system is selected based on the abrupt changes in the singular values. Singular value decomposition is performed on the Hankel matrix H:

[0095]

[0096] Where U1, U2, V1, and V2 are orthogonal matrices and U1 = [u1…u2… ... r ],V1=[v1…v r ], Σ1 and Σ2 are diagonal matrices with decreasing diagonal elements, and here σ (Σ1), Let n represent the minimum singular value of Σ1 and the maximum singular value of Σ2, respectively. The dimension of Σ1 is equal to the dimension n of the state matrix A. d .

[0097] Construct a dimension of n d Hankel matrix H1:

[0098]

[0099] E: Obtain the parameter matrices A, B, and C in 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 appendix. Figure 3 As shown, the final transfer function P0 of order 2 is obtained:

[0103]

[0104] Where s is a complex field operator.

[0105] The obtained model frequency response comparison is attached. Figure 4 As shown.

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

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

[0108]

[0109] Where, δ ν Let P1 be the PEA system after hysteresis inverse compensation, P2 be the linear system identified by the Hankel matrix method, j represent the imaginary unit, and ω represent the angular frequency.

[0110] The v-gap values ​​obtained from multiple experiments are shown in the appendix. Figure 6 As shown.

[0111] Step 3: Set open-loop amplitude constraints based on the uncertainty description, as well as the search domain for the controller's structure and parameter vectors.

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

[0113]

[0114] Where s is a complex field operator.

[0115] The controller is configured as a standard PID controller, and its structure is given by the following formula:

[0116]

[0117] Among them, K P For proportional gain, K I For integral gain, K D Let A be the differential gain, τ be the time constant of the filter applied to the differential action, and A be the differential gain. K Let B be the state matrix. K Given the input matrix, C K For the output matrix, D K This is for direct matrix transmission.

[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 realm q = [-5 -100 -10 -5].

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

[0120] Set the initial weight function W0 (e.g., W0 = 1) and the precision coefficient E. r Let j = 1, where j represents the iteration number, and the optimal H will be designed according to the standard of the unformed system P0. ∞ The _LS controller is set as the initial controller K0, and the corresponding robust stability margin b0(P0,K0) is calculated.

[0121]

[0122] Where I is the identity matrix.

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

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

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

[0126] P0 * (jω i )P0(jω i )<|s U (jω i )| 2 Λ(ω i ).

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

[0128] |s L (jω i )| 2 Λ(ω i )>P0 * (jω i )P0(jω i )

[0129] Among them, P0 * (jω i P0(jω) is the complex conjugate transpose of the frequency response of a dynamic linear system. i ) represents the frequency response of a dynamic linear system, s U (jω i ) represents the frequency response of the open-loop amplitude constraint upper bound, s L (jω i ) represents the frequency response of the open-loop amplitude constraint lower bound, Λ(ω) i ) for W-* (jω i W -1 (jω i ), W -* (jω i W is the reciprocal of the complex conjugate transpose of the frequency response of the weight function. -1 (jω i ) is the reciprocal of the frequency response of the weight function.

[0130] The optimization problem of maximizing b(P,K) is transformed into minimizing γ in the following equation under the constraints of the two inequalities mentioned above. 2 The problem of minimizing generalized eigenvalues ​​(GEVP):

[0131]

[0132] Where P0(jω) is the frequency response of the dynamic linear system, and Λ(ω) is the frequency response of W. j -* (jω)W j -1 (jω), W j -* (jω) is the reciprocal of the complex conjugate transpose of the frequency response of the weight function obtained in the j-th iteration, W j -1 (jω) is the reciprocal of the frequency response of the weight function obtained in the j-th iteration, γ is the reciprocal of the robust stability margin, and K j-1 (jω) is the optimal structured controller obtained in the (j-1)th iteration.

[0133] By calculating the generalized eigenvalue minimization problem described by this set of LMIs at each frequency point, the shaped amplitude curve located at S can be obtained. U (s) and S L Within the range defined by (s), it can simultaneously maximize the robust stability margin b(P). j ,K j The matrix parameters Λ(ω) are then obtained by fitting a stable minimum phase transfer function to (Λ(ω)). 1 / 2 This allows us to obtain the desired weight function W in the design. j (s).

[0134] Set P j (s)=W j (s)P0(s), where P j Let be the shaped system obtained in the j-th iteration.

[0135] Step 6: Under the inequality constraints, iteratively solve the first and second optimization problems until the preset iteration conditions are met, and obtain the optimal weight function and the desired controller.

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

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

[0138]

[0139] The corresponding structured controller is the solution to the following non-convex, non-smooth 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 Let T be a closed-loop system consisting of controllers obtained from the j-th iteration. zw (q) represents the closed-loop system composed of the controller parameters obtained from the j-th iteration. To represent the uncertainty described by v-gap, this embodiment incorporates the δ calculated above. v (P1,P2), Re(λ) i (A)) represents the real part of the eigenvalues ​​of matrix A. In this embodiment, Re(λ) i (A cl (q))) represents the real part of the eigenvalues ​​in the closed-loop system state matrix, q is the parameter vector in the controller, and D is the search domain of the controller parameters.

[0146] The structured controller K, which minimizes γ, is solved using a genetic algorithm based on the obtained weight function. j We obtain the corresponding minimum value of γ, denoted as b. j (P j ,K j )=γ -1 .

[0147] Step 7: If no solution to the first and second optimization problems is obtained until the preset iteration conditions are met, return to step 3 to reset the open-loop amplitude constraints and the parameter vector search domain of the controller, and repeat steps 4-6 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 If the iteration ends, exit; otherwise, set j = j + 1 and return to step 5.

[0149] By iterating through the two optimization problems described above, we can eventually obtain a solution that makes b... j (P j ,K j Maximize the weight function W(s) and the controller K(s), and finally b j (P j ,K j It may not be optimal, but it can guarantee b. j (P j ,K j If the problem is not solved 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, using the aforementioned weight function and controller iterative optimization algorithm, the weight transfer function can be obtained as follows:

[0151]

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

[0153] 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 The value of ) = 0.663 satisfies the robust stabilization condition. Its generalized robust stability margin and the uncertainty described by v-gap in the frequency domain are described as follows: Figure 9 As shown.

[0156] In the experiment, the tracking trajectory r(t) was selected as a sinusoidal signal of different frequencies with an amplitude of 1 μm. Sinusoidal tracking signals of 20 Hz, 40 Hz, 80 Hz, and 100 Hz were input respectively, and the tracking effect was as follows: Figure 10 As shown, (a) is a comparison of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 20Hz; (b) is a comparison of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 40Hz; (c) is a comparison of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 80Hz; and (d) is a comparison of tracking control experimental curves of different control schemes under a single-frequency sinusoidal signal with a reference input amplitude of 1μm and a frequency of 100Hz.

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

[0158] Table 1 shows the loop shaping structured control method under open-loop amplitude constraints for different frequency inputs (H). ∞ _LSSC), H ∞ _LS control method and H ∞ The relative error and root mean square error corresponding to the control method.

[0159] Table 1

[0160]

[0161] Depend on Figure 10 and Figure 11 It can be seen that the loop forming structured control method (H) under open-loop amplitude constraints 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. As can be seen from the tracking error comparison in Table 1, H... ∞ _LSSC method, H ∞The control performance of the LS control method is better than that of the H method. ∞ Control. And H ∞ _LSSC and H ∞ The reason why _LS control effects are similar is because H ∞ The _LS controller uses H ∞ The same weight function after optimization of the LSSC control method, after iterative optimization, results in a weight function with a frequency range of 10... -2 ≤ω≤10 5 Within rad / s, it satisfies the preset upper and lower bounds of the open-loop amplitude. And H ∞ The final optimized γ value of the LSSC control method is 1.5099, H ∞ The optimal controller obtained by the _LS method has a γ value of 1.2147. Since a γ value less than 5 does not significantly alter its closed-loop shape, the performance characteristics mapped by the open-loop amplitudes of both methods are very similar. However, H... ∞ The structured controller of LSSC adopts a standard PID controller structure, and its order is less than H. ∞ The high-order controller obtained by the _LS method is easier to implement and maintain in engineering. Ultimately, it is determined by H. ∞ The generalized robust stability margin b(P,K) of the controller obtained by iterative optimization using the LSSC control method PID With a value of 0.663, it is generally considered that the controller has good robustness when b(P,K)>0.2~0.3. Therefore, the designed structured controller can effectively suppress errors introduced by modeling and external disturbances, inheriting the standard H... ∞ Robustness and performance characteristics under the LS framework. Based on the above experimental results, the loop-forming structured control method under open-loop amplitude constraints proposed in this invention can track multi-frequency sinusoidal signals in real time and efficiently. The resulting low-order controller is easy to implement and adjust in engineering applications, and its overall control quality is superior to the other two control methods used for comparison.

[0162] The method proposed in this embodiment quantifies the PEA modeling error under the Hammerstein structure using the v-gap metric theory, effectively reflecting the uncertainties present in hysteretic nonlinear modules and dynamic modules. Compared to previous uncertainty descriptions, the v-gap metric is simpler to calculate, provides a more accurate description between systems, and has a clear frequency domain concept. The uncertainty described by the v-gap metric is equivalent to the coprime factor uncertainty description, and the stability conditions described by the small gain theorem still apply.

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

[0164] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A hysteresis modeling and loop shaping structured control method for piezoelectric actuators, characterized in that, include: Step 1: Construct a piezoelectric actuator model using the Hammerstein structure, including: The piezoelectric actuator is modeled using the Hammerstein structure, and the static hysteresis nonlinear module in the Hammerstein structure is described using the Bouc-Wen model. The dynamic linear module in the piezoelectric actuator is identified using the Hankel matrix method. The hysteresis nonlinearity characteristics of the piezoelectric actuator are obtained based on the static hysteresis nonlinearity module and the dynamic linearity module, and the hysteresis nonlinearity characteristics are eliminated by using the Bouc-Wen inverse model, thus completing the construction of the piezoelectric actuator model. Step 2, based on The uncertainty of the piezoelectric actuator model is described by the following method: ; in, For the reason The uncertainty described This is a piezoelectric actuator model after hysteresis inverse compensation. Dynamic linear modules identified by the Hankel matrix method Represented as the imaginary unit, Expressed as angular frequency; Step 3: Based on the uncertainty description, set open-loop amplitude constraints, as well as the controller's structure and parameter vector search domain; Step 4: Set the initial weight function, accuracy coefficients, and initial controller; Step 5: Construct inequality constraints for the loop shaping system under the open-loop amplitude constraint; Step 6: Under the inequality constraints, iteratively solve the first optimization problem and the second optimization problem until the preset iteration conditions are met, and obtain the optimal weight function and the desired controller. The first optimization problem is a generalized eigenvalue minimization problem, used to solve for the optimal weight function; the second optimization problem is a non-convex optimization problem, used to solve for the desired controller. Step 7: If no solution to the first optimization problem and the second optimization problem is 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 repeat steps 4-6 until the optimal weight function and the desired controller are obtained.

2. The hysteresis modeling and loop shaping structured control method for piezoelectric actuators according to claim 1, characterized in that, The static hysteresis nonlinear module in the Hammerstein structure is described by the Bouc-Wen model as follows: ; in, This is the input voltage of the piezoelectric actuator. This refers to the output displacement of the piezoelectric actuator. For hysteresis components, To control the shape of the hysteresis loop, the undetermined parameters, The derivative of the hysteresis component, It is the derivative of the input voltage.

3. The hysteresis modeling and loop shaping structured control method for piezoelectric actuators according to claim 1, characterized in that, The dynamic linear module in the piezoelectric actuator is identified using the Hankel matrix method as follows: ; in, Transfer functions for dynamic linear modules For complex domain operators.

4. The hysteresis modeling and loop shaping structured control method for piezoelectric actuators according to claim 1, characterized in that, The inequality constraints for constructing the loop shaping system under the open-loop amplitude constraint are as follows: ; ; in, This is the complex conjugate transpose of the frequency response of a dynamic linear system. For the frequency response of a dynamic linear system, The frequency response is the upper bound of the open-loop amplitude constraint. The frequency response is the lower bound of the open-loop amplitude constraint. for , The reciprocal of the complex conjugate transpose of the frequency response of the weight function is given. It is the reciprocal of the frequency response of the weight function.

5. The hysteresis modeling and loop shaping structured control method for piezoelectric actuators according to claim 1, characterized in that, The first optimization problem is: ; in, For the frequency response of a dynamic linear system, for , Let be the reciprocal of the complex conjugate transpose of the frequency response of the weight function obtained in the j-th iteration. Let be the reciprocal of the frequency response of the weight function obtained in the j-th iteration. It is the reciprocal of the robust stability margin. This is the optimal structured controller obtained in the (j-1)th iteration.

6. The hysteresis modeling and loop shaping structured control method for piezoelectric actuators according to claim 5, characterized in that, The second optimization problem is: ; in, For a closed-loop system consisting of controllers obtained from the j-th iteration, For a closed-loop system consisting of controller parameters obtained from the j-th iteration, For the uncertainty described by v-gap, Let q be the real part of the eigenvalues ​​in the closed-loop system state matrix, q be the parameter vector in the controller, and D be the search domain of the controller parameters.

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