Precise compound control method of piezoelectric driving structure

Abstract

The application discloses a precision compound control method of a piezoelectric driving structure, characterized by the following steps: firstly, an experiment platform is built and a voltage-displacement curve of the piezoelectric driving structure is measured; secondly, a cascade model is established and an inverse model of the cascade model is solved; thirdly, a model predictive controller is established to compensate for dynamic disturbance and residual error of the piezoelectric driving structure in real time; then, an iterative learning controller based on the inverse model and the model predictive controller is established to inversely compensate for input voltage according to actual error; finally, convergence and robustness of the iterative learning controller are analyzed, and filter parameters of the iterative learning controller are determined according to the analysis result, so as to improve driving precision and trajectory tracking precision of the piezoelectric driving structure. The method has the advantages that compound control is adopted, problems such as insufficient tracking precision and phase lag caused by single feedforward control are avoided, stable and accurate control response is realized, and driving precision, trajectory tracking capability and trajectory tracking precision of the piezoelectric driving structure are improved.

Description

Technical Field

[0001] This invention relates to the field of control technology for piezoelectric drive structures, and more particularly to a precision composite control method for piezoelectric drive structures. Background Technology

[0002] Traditional rigid components are increasingly unable to meet practical application requirements. The demand for high-speed performance and low energy consumption in industrial applications is the primary purpose of using lightweight robots and manufacturing flexible robotic manipulators. Flexible robots can transmit force and energy through deformation and vibration. Due to their operational and environmental requirements, flexible robots often utilize smart materials. Among these smart materials, piezoelectric materials have become the preferred component for achieving precision actuation, deformation, and vibration control in conjunction with flexible structures due to their advantages such as rapid response, ease of integration, and high driving precision. Piezoelectric materials possess advantages such as high displacement resolution, fast response time, and high flexibility, making them invaluable in structural health monitoring, vibration control, and structural actuation. Piezoelectric actuation structures generate strain under an electric field to achieve high-resolution and high-precision positioning. However, due to the combined effects of ferroelectricity and electrostriction, piezoelectric actuation structures exhibit hysteresis, resulting in a biased nonlinear hysteresis characteristic in the displacement. This means that the output displacement has a certain phase delay relative to the input voltage and is biased relative to the zero starting line. This is an inherent characteristic of the material, which leads to a decrease in the trajectory tracking accuracy of the piezoelectric actuation structure, hindering its precision control.

[0003] To mitigate the hysteresis phenomenon in piezoelectric drive structures, some systems incorporate hysteresis models into their control systems. However, traditional hysteresis models (e.g., the Prandtl-Ishlinski model (PI) and the Bouc-Wen model) can only describe symmetrical hysteresis characteristics. Even with improvements to the hysteresis model, such as cascading dead-zone operators into the PI model, the problems of cumbersome modeling and computational complexity remain. Furthermore, commonly used feedback control methods, such as PID control, suffer from insufficient accuracy, phase lag, and system instability due to the inherent strong nonlinearity of the piezoelectric drive structure. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a precision composite control method for piezoelectric drive structures, which can overcome the inherent bias nonlinear hysteresis characteristics of piezoelectric drive structures and improve the driving accuracy and trajectory tracking accuracy of piezoelectric drive structures.

[0005] The technical solution adopted by this invention to solve the above-mentioned technical problems is as follows: a precision composite control method for a piezoelectric drive structure, comprising the following specific steps:

[0006] (1) Build an experimental platform and measure the voltage-displacement curve of the piezoelectric drive structure;

[0007] (2) Establish a cascade model using the measured voltage-displacement curve, and ensure the uniqueness of the inverse model by constraining the derivative value of the polynomial model, and solve for the inverse model of the cascade model.

[0008] (3) Establish a model predictive controller to compensate for the dynamic disturbances and residual errors of the piezoelectric drive structure in real time, so as to improve the displacement accuracy of the piezoelectric drive structure;

[0009] (4) Establish an iterative learning controller based on the inverse model and the model predictive controller, and compensate the input voltage inversely according to the actual error to further reduce the steady-state error of the output displacement of the piezoelectric drive structure;

[0010] (5) By performing convergence and robustness analysis on the iterative learning controller, the filter parameters of the iterative learning controller are determined based on the analysis results. A zero-phase filter is established, and the phase lag is eliminated and the timing relationship of the signal is accurately maintained through the zero-phase filter, so as to improve the driving accuracy and trajectory tracking accuracy of the piezoelectric drive structure.

[0011] Furthermore, the process of establishing the cascade model in step (2) is as follows:

[0012] The mathematical form of the Bouc-Wen model is:

[0013]

[0014] Where: k is the discrete time, and Y(k) is the total output displacement of the piezoelectric drive structure. The parameter z(k) is between 0 and 1, used to control the proportion of linear and nonlinear components. K is the stiffness coefficient of the piezoelectric drive structure, and θ, β, and γ are parameters controlling the hysteresis characteristics. These are the internal variables and their derivatives that describe the displacement hysteresis behavior of the piezoelectric driven structure. The input voltage of the Bouc-Wen model is... The derivatives of the input voltages for the Bouc-Wen model are all time-dependent variables;

[0015] The polynomial model uses polynomials as saturation operators to characterize bias hysteresis properties. The mathematical expression for the integer-order polynomial model is:

[0016]

[0017] Where: m represents the polynomial order, m is an integer, and S[u](k) is the output displacement of the polynomial model. This is the weight vector of the multinomial model. This represents the weight values ​​of the multinomial model. For the operator vectors of the polynomial model, For the operator of the polynomial model, that is, the input voltage of the polynomial model;

[0018] The model is cascaded with the polynomial model first and the Bouc-Wen model second. The cascaded model is represented as follows:

[0019]

[0020] in: The output displacement of the nonlinear hysteresis model is obtained by using the input voltage and output displacement from the measured voltage-displacement curves, and then obtaining the coefficients of the cascaded model, namely α, K, and z(k), through the least squares method. ,θ,β,γ and Thus obtain .

[0021] Furthermore, the method for solving the inverse model in step (2) is as follows:

[0022] The derivative solution of the polynomial model is always greater than or less than 0 as a solution constraint to ensure that the inverse solution of the polynomial model is unique. Using the direct inversion method, through a cascaded sequence of the Bouc-Wen inverse model first and the polynomial inverse model second, the output voltage of the inverse model is obtained as follows:

[0023]

[0024] in: This represents the inverse model of the cascaded model. The output voltage of the inverse model. The desired displacement.

[0025] Furthermore, in step (3), the process of establishing the model predictive controller is as follows:

[0026] (3-1) Establish the discretized state increment space model of the piezoelectric drive structure, which is in the following form:

[0027]

[0028] Where: k is the discrete time, y(k) is the output displacement of the piezoelectric drive structure, and Δu(k) is the change in the input voltage of the piezoelectric drive structure between the current time and the previous time. Represents the state matrix, Represents the input matrix, The output matrix is ​​represented by x(k), which is the state variable of the piezoelectric drive structure at the current moment, x(k+1) is the state variable of the piezoelectric drive structure at the next moment, and Δx(k) represents the change in the state variable between the current moment and the previous moment.

[0029] (3-2) Using the discretized state increment space model as the prediction model, the predicted value of the output displacement is predicted, and the predicted output displacement sequence is expressed as:

[0030]

[0031] in: For the predicted output displacement sequence, From the system state matrix Composed of power terms, its function is to map the current state variable x(k) of the piezoelectric drive structure to the predicted output of the next Np steps, D The lower triangular matrix is ​​formed by the dynamic response coefficients of the piezoelectric drive structure. To control the increment matrix, This means using the voltage change at time k to predict k+N. P Output displacement at time N p To predict the step size, N c To control the step size, and ;

[0032] (3-3) Based on the discretized state increment space model, the cost function is constructed as follows:

[0033]

[0034] Where: the superscript T denotes matrix transpose, and E=YY d , representing the predicted output displacement error sequence. , This means using the voltage change at time k to predict k+N. P The expected output displacement at time Y d This represents the desired output displacement sequence of the piezoelectric drive structure. and All are adjustable diagonal weight matrices, n r J is the cost function of the model predictive controller, used to select the number of modes of the state variables.

[0035] (3-4) Substituting the predicted output displacement sequence into the cost function, and considering the input constraints of the model predictive controller, the constraints are as follows:

[0036]

[0037] in: and These are the lower and upper limits of the input voltage. and These are the minimum and maximum values ​​of the input voltage change. and Here, k+n-1 represents the lower and upper limits of the output displacement; u(k+n-1) represents the voltage of the current calculation step; Δu(k+n-1) represents the voltage increment of the current calculation step; y(k+n-1) represents the output displacement of the current calculation step; and n represents an increasing natural number sequence.

[0038] (3-5) Model predictive control employs rolling optimization, using quadratic programming to obtain the control increment matrix ∆U, and taking the first value ∆u(k) as the control quantity at the next time step, and so on to achieve the rolling optimization process, denoted as:

[0039]

[0040] Where: K I The integral control gain is given by u(k), where u(k) represents the input voltage of the piezoelectric drive structure at the current moment, and u(k-1) represents the input voltage of the piezoelectric drive structure at the previous moment. This represents the expected output displacement at the current moment.

[0041] Furthermore, in step (4), the process of establishing the iterative learning controller is as follows:

[0042] (4-1) Sample the actual output displacement of the piezoelectric drive structure, and then use the desired output displacement. With actual output displacement The difference between them is used as the tracking error. ,Right now: ,

[0043] in: This represents the actual output displacement during the i-th iteration. This represents the tracking error at the i-th iteration, and the tracking error is stored in memory;

[0044] (4-2) Filter the tracking error, that is: ,

[0045] Where: L represents the learning filter L, ζ i (k) represents the tracking error through the learning filter L;

[0046] (4-3) The filtered tracking error is superimposed on the current input voltage signal to generate a new input voltage signal for the next control iteration, i.e.:

[0047] ,

[0048] Where: i is the number of iterations, and k is the discrete time. Let be the input voltage during the i-th iteration. λ represents the input voltage at the (i+1)th iteration, 0 < λ ≤ 1 represents the adjustable learning gain, and Q represents the Q-filter.

[0049] Furthermore, in step (5), the process of performing convergence and robustness analysis on the iterative learning controller is as follows:

[0050] Assuming the inverse model is fully compensated, the discrete Fourier domain representation of the dynamic model of the piezoelectric driven structure in the iterative learning controller is obtained as follows:

[0051]

[0052] in: It is the identity matrix. Let G represent the initial state of the piezoelectric drive structure, and let G, H, and C represent the state discrete matrix, control discrete matrix, and output discrete matrix of the discrete incremental space model of the piezoelectric drive structure, respectively. This represents the form of the output voltage of the static hysteresis nonlinear component in the z-domain during the i-th iteration. Let Q(z), L(z), and y represent the form of the output voltage of the static hysteresis nonlinearity in the z-domain at the (i+1)th iteration. d (z) represent the Q filter, the L learning filter, and y, respectively. d (k) in the z-domain; z denotes the z-domain; superscript Indicates the current discrete-time step number;

[0053] when When the value is less than 1 and the number of iterations is i→∞, the domain is transformed from the z-domain to the frequency domain, i.e., z=e jω The convergence condition is:

[0054] ,

[0055] Where: ω c The cutoff frequency chosen for the Q filter, ω is the digital angular frequency, j is the imaginary unit, and e is the natural constant. jω This indicates a conversion to the frequency domain; C(e jw IG) -1 H represents the dynamic model of the piezoelectric driven structure under zero initial conditions; and it can be seen from the equation that the inverse model of the piezoelectric driven structure is the optimal choice for the learning filter L(z) to achieve maximum bandwidth and minimum phase. Therefore, the obtained learning filter L(z) is:

[0056] ;

[0057] And obtain the final converged output voltage of the iterative learning controller. Represented as:

[0058] ,

[0059] Under zero initial conditions, the final convergent output voltage can be expressed as:

[0060] ,

[0061] The error term in the iterative learning controller for:

[0062]

[0063] As can be seen from the formula, Q(z) and L(z) determine the tracking error. When Q(z) is an ideal filter with Q(z)→1, the tracking error is... Approaching 0, a zero-phase digital filtering method is used for Q(z) to establish a zero-phase filter for the input voltage of the i-th iteration. With the filtered tracking error ζ i (k) Perform forward time-domain filtering and reverse time-domain filtering sequentially to eliminate the input voltage in the (i+1)th iteration. Phase lag and precise maintenance The timing relationship of the signal is considered, and a Butterworth filter is used. The Butterworth filter is expressed in pole-zero form as follows:

[0064]

[0065] Where: G(s) represents the filter transfer function, Let ρ be the order of the Butterworth filter, s be the complex frequency variable, and ρ be the pole index. Let ξ represent the position of the pole in the complex plane when the cutoff frequency is 1, and let ξ represent the pole's index, where ξ = 1, 2, 3, ... , Represents the complex coordinates of the corresponding pole.

[0066] Compared with the prior art, the advantages of the present invention are:

[0067] (1) This method adopts composite control, which avoids problems such as insufficient tracking accuracy and phase lag caused by single feedforward control, and achieves a smooth and accurate control response. It also improves the driving accuracy, trajectory tracking capability and trajectory tracking accuracy of the piezoelectric drive structure, making it suitable for complex control tasks and high-precision application scenarios.

[0068] (2) Based on the Bouc-Wen model, this method introduces a polynomial model to deal with the bias problem of piezoelectric structures. The Bouc-Wen model cascaded polynomial model adopts a pure analytical model. By constraining the polynomial derivative, it is guaranteed that it has a unique inverse solution, which has significant advantages in solving the inverse model and identifying parameters.

[0069] (3) This method overcomes the shortcomings of traditional feedforward inverse compensation in real time by establishing a model predictive controller and, under the premise of cascaded feedforward compensation, based on the real-time output displacement dynamic compensation voltage, thus significantly improving the control accuracy. At the same time, the model predictive controller has excellent disturbance suppression capability and can effectively cope with external interference.

[0070] (4) This method establishes an iterative learning controller, analyzes its convergence and robustness characteristics, and proposes a zero-phase filter based on the convergence condition of iterative learning control to ensure that the piezoelectric drive structure achieves asymptotic convergence and robust stability.

[0071] (5) This method comprehensively adopts model predictive control and iterative learning control. Iterative learning control learns historical periodic errors to eliminate repetitive static deviations. At the same time, it uses rolling optimization based on model predictive control to compensate for dynamic disturbances and residual errors in real time, further improving control accuracy. Attached Figure Description

[0072] Figure 1 The voltage-displacement curve of the piezoelectric drive structure obtained by this invention is shown below.

[0073] Figure 2 This is a block diagram of the model prediction-iterative learning composite control of the present invention;

[0074] Figure 3 This is a comparison of the experimental results of the piezoelectric drive structure of the present invention after 5 iterations under a quasi-static constant amplitude trajectory.

[0075] Figure 4 The diagram shows the composite control tracking results of the piezoelectric drive structure of the present invention at different frequencies.

[0076] Figure 5 The graph shows the multi-frequency tracking error curve of the piezoelectric drive structure of the present invention after five iterations of learning control. Detailed Implementation

[0077] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0078] like Figure 1-3 As shown, a precision composite control method for a piezoelectric drive structure includes the following specific steps:

[0079] (1) Build an experimental platform and measure the voltage-displacement curve of the piezoelectric drive structure;

[0080] The measurement of the voltage-displacement curve (i.e., the input voltage versus output displacement curve) of the piezoelectric drive structure is a mature technology in this field and will not be elaborated here.

[0081] (2) A cascade model is established using the measured voltage-displacement curves. The uniqueness of the inverse model is ensured by constraining the derivative values ​​of the polynomial model. The inverse model of the cascade model is then obtained by solving the problem. Specifically:

[0082] The mathematical form of the Bouc-Wen (BW) model is:

[0083] (1)

[0084] Where: k is the discrete time, and Y(k) is the total output displacement of the piezoelectric drive structure. The parameter z(k) is between 0 and 1, used to control the proportion of linear and nonlinear components. K is the stiffness coefficient of the piezoelectric drive structure, and θ, β, and γ are parameters controlling the hysteresis characteristics. These are the internal variables and their derivatives that describe the displacement hysteresis behavior of the piezoelectric driven structure. The input voltage of the Bouc-Wen model is... The derivatives of the input voltages for the Bouc-Wen model are all time-dependent variables;

[0085] The polynomial model uses polynomials as saturation operators to characterize bias hysteresis properties. The mathematical expression for the integer-order polynomial model is:

[0086] (2)

[0087] Where: m represents the polynomial order, m is an integer, and S[u](k) is the output displacement of the polynomial model. This is the weight vector of the multinomial model. This represents the weight values ​​of the multinomial model. For the operator vectors of the polynomial model, For the operator of the polynomial model, that is, the input voltage of the polynomial model;

[0088] The model is cascaded with the polynomial model first and the Bouc-Wen model second. The cascaded model is represented as follows:

[0089] (3)

[0090] in: The output displacement of the nonlinear hysteresis model is obtained by using the input voltage and output displacement from the measured voltage-displacement curves, and then obtaining the coefficients of the cascaded model, namely α, K, and z(k), through the least squares method. ,θ,β,γ and Thus obtain ;

[0091] To compensate for the bias hysteresis nonlinearity, the inverse model of the cascaded model needs to be obtained as the feedforward controller. The solution method for the inverse model is as follows:

[0092] The derivative solution of the polynomial model is always greater than or less than 0 as a solution constraint to ensure that the inverse solution of the polynomial model is unique. Using the direct inversion method, through a cascaded sequence of the Bouc-Wen inverse model first and the polynomial inverse model second, the output voltage of the inverse model is obtained as follows:

[0093] (4)

[0094] in: This represents the inverse model of the cascaded model. The output voltage of the inverse model. The desired displacement;

[0095] (3) Establish a model predictive controller to compensate for the dynamic disturbances and residual errors of the piezoelectric drive structure in real time, so as to improve the displacement accuracy of the piezoelectric drive structure; the process of establishing the model predictive controller is as follows:

[0096] (3-1) Establish the discretized state increment space model of the piezoelectric drive structure, which is in the following form:

[0097] (5)

[0098] Where: k is the discrete time, y(k) is the output displacement of the piezoelectric drive structure, and Δu(k) is the change in the input voltage of the piezoelectric drive structure between the current time and the previous time. Represents the state matrix, Represents the input matrix, The output matrix is ​​represented by x(k), which is the state variable of the piezoelectric drive structure at the current moment, x(k+1) is the state variable of the piezoelectric drive structure at the next moment, and Δx(k) represents the change in the state variable between the current moment and the previous moment.

[0099] (3-2) Using the discretized state increment space model as the prediction model, the predicted value of the output displacement is predicted, and the predicted output displacement sequence is expressed as:

[0100] (6)

[0101] in: For the predicted output displacement sequence, From the system state matrix Composed of power terms, its function is to map the current state variable x(k) of the piezoelectric drive structure to the predicted output of the next Np steps, D The lower triangular matrix is ​​formed by the dynamic response coefficients of the piezoelectric drive structure. To control the increment matrix, This means using the voltage change at time k to predict k+N. P Output displacement at time N p To predict the step size, N c To control the step size, and ;

[0102] (3-3) Based on the discretized state increment space model, the cost function is constructed as follows:

[0103] (7)

[0104] Where: the superscript T denotes matrix transpose, and E=YY d , representing the predicted output displacement error sequence. , This means using the voltage change at time k to predict k+N. P The expected output displacement at time Y d This represents the desired output displacement sequence of the piezoelectric drive structure. and All are adjustable diagonal weight matrices, n r J is the cost function of the model predictive controller, used to select the number of modes of the state variables.

[0105] (3-4) Substituting the predicted output displacement sequence into the cost function, and considering the input constraints of the model predictive controller, the constraints are as follows:

[0106] (8)

[0107] in: and These are the lower and upper limits of the input voltage. and These are the minimum and maximum values ​​of the input voltage change. and Here, k+n-1 represents the lower and upper limits of the output displacement; u(k+n-1) represents the voltage of the current calculation step; Δu(k+n-1) represents the voltage increment of the current calculation step; y(k+n-1) represents the output displacement of the current calculation step; and n represents an increasing natural number sequence.

[0108] (3-5) Model predictive control employs rolling optimization, using quadratic programming to obtain the control increment matrix ∆U, and taking the first value ∆u(k) as the control quantity at the next time step, and so on to achieve the rolling optimization process, denoted as:

[0109] (9)

[0110] Where: K I The integral control gain is given by u(k), where u(k) represents the input voltage of the piezoelectric drive structure at the current moment, and u(k-1) represents the input voltage of the piezoelectric drive structure at the previous moment. This represents the expected output displacement at the current moment;

[0111] (4) Establish an iterative learning controller based on the inverse model and the model predictive controller, and compensate the input voltage inversely according to the actual error to further reduce the steady-state error of the output displacement of the piezoelectric drive structure;

[0112] The process of establishing the iterative learning controller is as follows:

[0113] (4-1) Sample the actual output displacement of the piezoelectric drive structure, and then use the desired output displacement. With actual output displacement The difference between them is used as the tracking error. ,Right now:

[0114] (10)

[0115] in: This represents the actual output displacement during the i-th iteration. This represents the tracking error at the i-th iteration, and the tracking error is stored in memory;

[0116] (4-2) Filter the tracking error, that is:

[0117] (11)

[0118] Where: L represents the learning filter L, ζ i (k) represents the tracking error through the learning filter L;

[0119] (4-3) The filtered tracking error is superimposed on the current input voltage signal to generate a new input voltage signal for the next control iteration, i.e.:

[0120] (12)

[0121] Where: i is the number of iterations, and k is the discrete time. Let be the input voltage during the i-th iteration. λ represents the input voltage at the (i+1)th iteration, 0 < λ ≤ 1 represents the adjustable learning gain, and Q represents the Q-filter.

[0122] (5) By performing convergence and robustness analysis on the iterative learning controller, the filter parameters of the iterative learning controller are determined based on the analysis results, a zero-phase filter is established, and the phase lag is eliminated and the timing relationship of the signal is accurately maintained through the zero-phase filter, so as to improve the driving accuracy and trajectory tracking accuracy of the piezoelectric drive structure.

[0123] The process of performing convergence and robustness analysis on the iterative learning controller is as follows:

[0124] Assuming the inverse model is fully compensated, the discrete Fourier domain representation of the dynamic model of the piezoelectric driven structure in the iterative learning controller is obtained as follows:

[0125] (13)

[0126] in: It is the identity matrix. Let G represent the initial state of the piezoelectric drive structure, and let G, H, and C represent the state discrete matrix, control discrete matrix, and output discrete matrix of the discrete incremental space model of the piezoelectric drive structure, respectively. This represents the form of the output voltage of the static hysteresis nonlinear component in the z-domain during the i-th iteration. Let Q(z), L(z), and y represent the form of the output voltage of the static hysteresis nonlinearity in the z-domain at the (i+1)th iteration. d (z) represent the Q filter, the L learning filter, and y, respectively. d (k) in the z-domain; z denotes the z-domain; superscript Indicates the current discrete-time step number;

[0127] when When the value is less than 1 and the number of iterations is i→∞, the domain is transformed from the z-domain to the frequency domain, i.e., z=e jω The convergence condition is:

[0128] (14)

[0129] Where: ω c The cutoff frequency chosen for the Q filter, ω is the digital angular frequency, j is the imaginary unit, and e is the natural constant. jω This indicates a conversion to the frequency domain; C(e jw IG) -1H represents the dynamic model of the piezoelectric drive structure under zero initial conditions; and from equation (14), it can be seen that the inverse model of the piezoelectric drive structure is the best choice for the learning filter L(z) to achieve maximum bandwidth and minimum phase. Therefore, the obtained learning filter L(z) is:

[0130] (15)

[0131] And obtain the final converged output voltage of the iterative learning controller. Represented as:

[0132] (16)

[0133] Under zero initial conditions, the final convergent output voltage can be expressed as:

[0134] (17)

[0135] The error term in the iterative learning controller for:

[0136] (18)

[0137] As can be seen from equation (18), Q(z) and L(z) determine the tracking error. When Q(z) is an ideal filter with Q(z)→1, the tracking error is... Approaching 0, a zero-phase digital filtering method is used for Q(z) to establish a zero-phase filter for the input voltage of the i-th iteration. With the filtered tracking error ζ i (k) Perform forward time-domain filtering and reverse time-domain filtering sequentially to eliminate the input voltage in the (i+1)th iteration. Phase lag and precise maintenance The timing relationship of the signal is considered. This composite processing method achieves zero-phase response while preserving the amplitude-frequency characteristics, effectively suppressing the time-domain offset phenomenon caused by traditional single-pass filtering. The filter used is a Butterworth filter, which is expressed in pole-zero form as follows:

[0138] (19)

[0139] Where: G(s) represents the filter transfer function, Let ρ be the order of the Butterworth filter, s be the complex frequency variable, and ρ be the pole index. Let ξ represent the position of the pole in the complex plane when the cutoff frequency is 1, and let ξ represent the pole's index, where ξ = 1, 2, 3, ... , Represents the complex coordinates of the corresponding pole.

[0140] The following are the results verifying this method:

[0141] Trajectory tracking experiments were conducted on the piezoelectric driven structure. Before each iteration, residual stress in the piezoelectric driven structure had to be eliminated to ensure zero initial deformation, guaranteeing strict initial conditions for each iteration. The tracking performance of the composite controller (i.e., inverse model + model predictive controller + iterative learning controller) was first tested under constant-amplitude sinusoidal and constant-amplitude triangular trajectories. The excitation frequency of the two quasi-static signals was set to 0.1Hz. The tracking trajectory results for 5 iterations are shown below. Figure 3 As shown, compared with the feedforward compensator (i.e., the cascaded model), the composite control of this method further improves the trajectory tracking accuracy of the piezoelectric drive structure under quasi-static conditions.

[0142] To verify the tracking accuracy of the piezoelectric drive structure at higher frequencies, tracking experiments were conducted on variable-amplitude sinusoidal trajectories with frequencies of 1.5, 2.0, 3.0, and 4.0 Hz. The tracking results after the 5th iteration are shown below. Figure 4 As shown in the figure, the output displacement of the piezoelectric drive structure maintains a low tracking error at different frequencies, and achieves high-precision tracking of variable amplitude sinusoidal trajectories at different frequencies.

[0143] To verify the effectiveness of the iterative learning controller, amplitude-varying sine waves of other frequencies were added, and the performance of the hysteresis model was quantitatively evaluated using relative error (RE) and root mean square error (RMSE). Figure 5 As shown, the tracking error of the composite controller was reduced by a maximum of 47.60% and an average of 39.74% under different trajectories. The experimental results indicate that this method significantly reduces the tracking error of the piezoelectric drive structure across the entire exploration frequency range, thus improving the tracking performance of the piezoelectric drive structure.

[0144] The scope of protection of this invention includes, but is not limited to, the above embodiments. The scope of protection is defined by the claims. Any substitutions, modifications, or improvements to this technology that are easily conceived by those skilled in the art fall within the scope of protection of this invention.

Claims

1. A precision compound control method of a piezoelectric driving structure, characterized by The method comprises the following specific steps: (1) build an experimental platform and measure the voltage-displacement curve of the piezoelectric driving structure; (2) establish a cascade model by using the measured voltage-displacement curve, and ensure the uniqueness of the inverse model by constraining the derivative value of the polynomial model, and solve to obtain the inverse model of the cascade model; The establishment process of the cascade model is: The mathematical form of the Bouc-Wen model is: where: k is the discrete time, Y(k) is the total output displacement of the piezoelectric actuation structure, is a parameter between 0 and 1, used to control the proportion of linear and nonlinear components, K is the stiffness coefficient of the piezoelectric actuation structure, θ, β and γ are all parameters to control the hysteresis characteristics, z(k) and are the internal variables and their derivatives, respectively, describing the displacement hysteresis behavior of the piezoelectric actuation structure, is the input voltage of the Bouc-Wen model, is the derivative of the input voltage of the Bouc-Wen model, all are variables with respect to time; The polynomial model uses a polynomial as a saturation operator to represent the bias hysteresis characteristic, and the mathematical expression of the integer-order polynomial model is: wherein: m represents a polynomial order, m is an integer, S[u](k) is an output displacement of the polynomial model, is a weight vector of the polynomial model, represents a weight value of the polynomial model, is an operator vector of the polynomial model, is an operator of the polynomial model, i.e. an input voltage of the polynomial model; The polynomial model is cascaded in front of the Bouc-Wen model, and the cascade model is represented as: wherein: is the output displacement of the nonlinear hysteresis model, the input voltage and the output displacement in the measured voltage-displacement curve are used to obtain the coefficients of the cascade model, i.e. α, K, z(k), , θ, β, γ and , by the least square method, so as to obtain ; (3) establish a model predictive controller to compensate for the dynamic disturbance and residual error of the piezoelectric driving structure in real time, so as to improve the displacement accuracy of the piezoelectric driving structure; (4) establish an iterative learning controller based on the inverse model and the model predictive controller, and inversely compensate for the input voltage according to the actual error, so as to further reduce the steady-state error of the output displacement of the piezoelectric driving structure; (5) analyze the convergence and robustness of the iterative learning controller, determine the filter parameters of the iterative learning controller according to the analysis result, establish a zero-phase filter, eliminate the phase lag through the zero-phase filter, and accurately maintain the timing relationship of the signal, so as to improve the driving accuracy and trajectory tracking accuracy of the piezoelectric driving structure.

2. The method of claim 1, wherein the method comprises: The solving method of the inverse model in the step (2) is: ​ The derivative solution of the polynomial model is always greater than 0 or always less than 0, which is used as a solving constraint to ensure that the inverse solution of the polynomial model is unique, and the direct inverse method is adopted, the inverse model is cascaded in front of the Bouc-Wen inverse model and behind the polynomial inverse model, and the output voltage of the inverse model is: wherein: represents an inverse model of the cascade model, is an output voltage of the inverse model, is a desired displacement.

3. The method of claim 1, wherein the method further comprises: determining a first control signal for the first piezoelectric actuator; determining a second control signal for the second piezoelectric actuator; and determining a third control signal for the third piezoelectric actuator. In the step (3), the establishment process of the model predictive controller is: (3-1) establish a discretized state increment space model of the piezoelectric driving structure, which is in the form of: Wherein: k is discrete time, y(k) is the output displacement of the piezoelectric driving structure, and Δu(k) is the change of the input voltage of the piezoelectric driving structure at the current time and the previous time, denotes a state matrix, denotes an input matrix, denotes an output matrix, x(k) is the state variable of the piezoelectric driving structure at the current time, x(k+1) is the state variable of the piezoelectric driving structure at the next time, and Δx(k) represents the change of the state variable at the current time and the previous time; (3-2) take the discretized state increment space model as a prediction model to predict the predicted value of the output displacement, and the predicted output displacement sequence is represented as: in: For the predicted output displacement sequence, From the system state matrix Composed of power terms, its function is to map the current state variable x(k) of the piezoelectric drive structure to the predicted output of the next Np steps, D The lower triangular matrix is ​​formed by the dynamic response coefficients of the piezoelectric drive structure. To control the increment matrix, This means using the voltage change at time k to predict k+N. P Output displacement at time N p To predict the step size, N c To control the step size, and ; (3-3) based on the discretized state increment space model, construct a cost function as: wherein: superscript T denotes matrix transpose, E = Y - Y d represents the predicted output displacement error sequence, , represents the expected output displacement at k+N P time using the voltage variation at k time, Y d represents the expected output displacement sequence of the piezoelectric driving structure, and are both adjustable diagonal weight matrices, n r is the modal number of selected state variables, and J is the cost function of the model predictive controller. (3-4) substitute the predicted output displacement sequence into the cost function, and constrain the input constraint condition of the model predictive controller, the constraint condition is: wherein: and are lower and upper limits of the input voltage, and are minimum and maximum values of the input voltage variation, and are lower and upper limits of the output displacement; k+n-1 represents the current calculation step, u(k+n-1) represents the voltage at the current calculation step, Δu(k+n-1) represents the voltage increment at the current calculation step, y(k+n-1) represents the output displacement at the current calculation step, and n represents an increasing natural number series. (3-5) the model predictive control adopts rolling optimization, and the control increment matrix is obtained by using quadratic programming, and the first value is taken as the control quantity at the next moment, and the rolling optimization process is realized in turn, which is denoted as: wherein: K I is the integral control gain, u(k) represents the input voltage of the piezoelectric driving structure at the current time, u(k-1) represents the input voltage of the piezoelectric driving structure at the previous time, represents the desired output displacement at the current time.

4. The method of claim 1, wherein the piezoelectric driving structure is a piezoelectric actuator. In the step (4), the establishment process of the iterative learning controller is: (4-1), sampling the actual output displacement of the piezoelectric driving structure, and then taking the difference between the expected output displacement and the actual output displacement as the tracking error , that is: , wherein: represents the actual output displacement at the i-th iteration, represents the tracking error at the i-th iteration, and storing the tracking error in a memory; (4-2), filtering the tracking error, i.e.: , wherein: L denotes a learning filter L, ζ i (k) denotes a tracking error through the learning filter L; (4-3) superimpose the filtered tracking error and the current input voltage signal to generate a new input voltage signal for the next control iteration, that is: ， where: i is the iteration number, k is the discrete time, is the input voltage at the i-th iteration, is the input voltage at the i+1-th iteration, 0<λ≤1 is the adjustable learning gain, and Q represents a Q filter.

5. The method of claim 1, wherein the method further comprises: determining a plurality of control signals for the plurality of piezoelectric actuators based on the plurality of control parameters; and applying the plurality of control signals to the plurality of piezoelectric actuators. In the step (5), the process of analyzing the convergence and robustness of the iterative learning controller is: Assuming that the inverse model is completely compensated, the discrete Fourier domain representation of the dynamic model of the piezoelectric driving structure in the iterative learning controller is obtained: wherein: is the identity matrix, is the initial state of the piezoelectric actuation structure, G, H, C are the state discrete matrix, the control discrete matrix and the output discrete matrix of the piezoelectric actuation structure respectively after discretization, represents the form of the output voltage of the static hysteresis nonlinear part in the z domain at the i-th iteration, represents the form of the output voltage of the static hysteresis nonlinear part in the z domain at the i+1-th iteration, Q(z), L(z), and y d (z) are the Q filter, the L learning filter and the y d (k) in the z domain; z represents the z domain; the superscript represents the current discrete time step; When <1, and the iteration number is i→∞, the conversion from z domain to frequency domain is z=e jω The convergence condition is: ， where: ω c is the cutoff frequency chosen for the Q filter, ω is the digital angular frequency, j is the imaginary unit, e is the natural constant, e jω represents the conversion to the frequency domain; C(e jw I-G) -1 H represents the dynamic model of the piezoelectric driving structure under zero initial conditions; and from the equation, it can be known that the inverse model of the piezoelectric driving structure is the best choice for the learning filter L(z) to realize the maximum bandwidth and the minimum phase, and thus the learning filter L(z) obtained is: ； and the final convergent output voltage of the iterative learning controller is obtained is represented as: ， Under the condition of zero initial value, the finally converged output voltage is represented as: ， Error term in iterative learning controller is: From the formula, Q(z) and L(z) determine the tracking error, when Q(z) is an ideal filter of Q(z)→1, the tracking error tends to 0, the zero-phase digital filter method is adopted for Q(z), a zero-phase filter is established, and the input voltage of the i-th iteration is filtered with the filtered tracking error ζ i (k) the forward time domain filtering and the reverse time domain filtering are executed in sequence to eliminate the phase lag of the input voltage of the i+1-th iteration and accurately maintain the timing relationship of the signal The filter adopts a Butterworth filter, and the Butterworth filter is expressed in the form of zero-pole as follows: where: G(s) represents the filter transfer function, is the Butterworth filter order, s is the complex frequency variable, and p is the pole index, represents the location of the pole in the complex plane for a cutoff frequency of 1, and ξ represents the pole location index, with ξ = 1, 2, 3,..., , represents the complex coordinate of the corresponding pole.

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