Piezoelectric driving positioning table robust resonance control method based on generalized internal model architecture
By employing a robust resonant control method based on a generalized internal model architecture, and utilizing zero-pole cancellation and internal model control PID tuning algorithm, the contradiction between robustness and performance in the resonant control of a piezoelectric-driven positioning stage is resolved, achieving rapid suppression and high-performance control in the resonant frequency band.
Patent Information
- Application Number
- CN202511261373.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-09-05
- Publication Date
- 2025-12-05
AI Technical Summary
There is a trade-off between robustness and performance in piezoelectric-driven positioning stages in resonant control, and existing methods are insufficient in terms of resonance suppression and positioning performance.
A robust resonant control method based on a generalized internal model architecture is adopted. Through zero-pole cancellation direct resonance compensation, combined with internal model control PID tuning algorithm and robust controller design, the accurate suppression of resonance and performance guarantee are achieved.
It achieves rapid suppression in the resonant frequency band, avoids phase lag problems, ensures high-performance control of the positioning stage under nominal operating conditions, and improves robustness and positioning accuracy.
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Figure CN121069733A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of piezoelectric driving positioning table motion control, and particularly relates to a piezoelectric driving positioning table robust resonance control method based on a generalized internal model architecture. BACKGROUND
[0002] The piezoelectric driving positioning table is a precision motion mechanism directly driven by piezoelectric ceramics, which has high positioning accuracy of sub-micron or even nanometer level and fast response speed of microsecond level, and has very important significance for ultra-precision manufacturing fields such as semiconductor lithography. However, due to the resonance phenomenon caused by the inherent frequency of piezoelectric ceramics, the piezoelectric driving positioning table has a resonance characteristic of amplifying output near a specific frequency. The resonance characteristic will cause uncontrolled mechanical vibration in the positioning process, causing the positioning accuracy to decrease or even causing mechanical structure fatigue damage. In order to avoid the influence of the resonance characteristic, the actual safe working bandwidth needs to be limited to less than one-third of the inherent frequency, which restricts the performance of the piezoelectric ceramic positioning table in ultra-high speed positioning.
[0003] The current resonance control method is based on the accurate modeling of the resonance peak to achieve gain suppression at a specific frequency without affecting the gain at other frequencies. However, in actual applications, due to the change of working conditions, the position and amplitude of the resonance peak may shift to a certain extent, which reduces the resonance suppression effect of this method, i.e. the robustness is weak. Another method considers the possible shift of the resonance peak and directly suppresses the frequency band near the resonance peak to ensure the resonance suppression effect under different working conditions. However, due to the decrease of the gain in the full frequency band, it also reduces the positioning performance of the piezoelectric driving positioning table. Therefore, the current resonance control method has a contradiction between robustness and performance. SUMMARY
[0004] In order to solve the contradiction between robustness and performance of the piezoelectric driving positioning table resonance control method, the present application proposes a piezoelectric driving positioning table robust resonance control method based on a generalized internal model architecture. This method directly compensates the piezoelectric driving positioning table as a whole through zero-pole cancellation, designs a nominal controller in the outer loop of the control loop to ensure performance for the nominal model assuming complete compensation, and designs a robust controller in the inner loop of the control loop to ensure resonance suppression effect for the part not completely compensated as an uncertain disturbance, thereby solving the contradiction between robustness and performance of the piezoelectric driving positioning table resonance control method. The present application is mainly used for processing the high-performance control problem of the piezoelectric driving positioning table under the nonlinear resonance characteristic.
[0005] The present application is realized by the following technical scheme, and the present application proposes a piezoelectric driving positioning table robust resonance control method based on a generalized internal model architecture, and the method specifically comprises: S1, establishing a dynamic response model of the piezoelectric driving positioning table through a frequency response analysis identification method; S2, calculating a feedforward resonance suppression controller based on a zero-pole cancellation resonance suppression method; S3, taking a nominal model with complete resonance peak suppression as a controlled object, and calculating a nominal controller through an internal model control PID tuning algorithm; S4, fitting a controlled object multiplicative uncertainty based on difference model frequency response data of an actual object and a nominal model; S5, solving a standard robust suboptimal problem to obtain a robust controller for robust resonance control.
[0006] Further, the S1 comprises: Considering the delay generated by the internal calculation of the digital controller, the output interface of the frequency sweeper is connected to the No. 1 ADC interface of the digital controller through a cable, and the output from the No. 1 DAC interface of the digital controller is directly output without any processing in the digital controller; Considering the delay generated by the transmission of the control quantity calculated by the digital controller to the piezoelectric drive positioning table, the No. 1 DAC interface of the digital controller is connected to the control input interface of the driver of the piezoelectric drive positioning table through a cable; Considering the delay generated by the sensor feedback signal of the piezoelectric drive positioning table, the sensor of the piezoelectric drive positioning table is connected to the No. 2 ADC interface of the digital controller through a cable, and the output from the No. 2 DAC interface of the digital controller is directly output without any processing in the digital controller, and the No. 2 DAC interface of the digital controller is connected to the input interface of the frequency sweeper through a cable; According to the frequency response corresponding table of the piezoelectric drive positioning table at different frequency sampling points, linear identification is performed using the following model:
[0007] Wherein, represents the open-loop transfer function of the piezoelectric drive positioning table, represents a complex frequency variable, represents the order of the numerator of the transfer function, represents the order of the denominator of the transfer function, represents the coefficient of each term of the numerator of the transfer function, represents the coefficient of each term of the numerator of the transfer function, represents the delay time; In the system identification process, different model orders and delay times are selected, and the differences between the identification results and the actual data are compared to select a set of parameters with the best fitting effect to obtain the open-loop transfer function of the piezoelectric drive positioning table .
[0008] Further, the S2 comprises: Open-loop function of piezoelectric driving positioning stage according to linear identification The part describing the resonance characteristics of the piezoelectric driving positioning stage is analyzed as follows:
[0009] wherein, represents the sum of all transfer functions describing the resonance characteristics of the piezoelectric driving positioning stage, represents the total number of resonance peaks of the piezoelectric driving positioning stage, and the denominator part of the second-order transfer function describes the resonance peak, represents the damping corresponding to the resonance peak, represents the resonance frequency corresponding to the resonance peak, and the numerator part of the second-order transfer function describes the anti-resonance peak, represents the damping corresponding to the anti-resonance peak, represents the resonance frequency corresponding to the anti-resonance peak; Resonance characteristic transfer function of piezoelectric driving positioning stage The feedforward resonance controller is calculated by the direct zero-pole cancellation resonance suppression method as follows:
[0010] wherein, represents the transfer function of the feedforward resonance suppression controller, represents the inverse of the resonance characteristic transfer function of the piezoelectric driving positioning stage.
[0011] Further, the S3 comprises: In order to facilitate the calculation of the nominal controller by using the internal model control PID tuning algorithm, the nominal model is designed as follows:
[0012] wherein represents the transfer function of the nominal model of the piezoelectric driving positioning stage after resonance compensation, and compared with the open-loop transfer function of the piezoelectric driving positioning stage the resonance link and the pure time delay link are completely eliminated, and the minimum phase characteristic is obtained; Due to the minimum phase characteristic of the nominal model, the corresponding internal model controller is designed as follows:
[0013] wherein, represents the transfer function of the internal model controller, represents the time constant of the filter, represents the order of the filter, and the order thereof should not be less than the relative order of the nominal model to ensure that the internal model controller is physically realizable; According to the performance index, select the appropriate filter time constant And filter order Get the internal model controller .
[0014] According to the internal model control standard model, the equivalent forward controller As follows:
[0015] Further, the S3 also includes: The internal model controller Approximated as a PI controller using Taylor expansion method:
[0016] Where The equivalent forward controller After multiplying the factor The purpose is to avoid The function is defined when expanding at. PI controller transfer function The controller proportional term coefficient The controller integral term coefficient The controller overall gain, the coefficient is solved as follows:
[0017] Where The constant term of the denominator polynomial in the nominal model transfer function The first term of the denominator polynomial in the nominal model transfer function The constant term of the numerator polynomial in the nominal model transfer function The first term of the numerator polynomial in the nominal model transfer function The constant term of the denominator polynomial in the nominal model transfer function The first term of the denominator polynomial in the nominal model transfer function The constant term of the numerator polynomial in the nominal model transfer function The first term of the numerator polynomial in the nominal model transfer function Substitute the above formula, and adjust the controller overall gain To the appropriate size according to the performance index, get the nominal controller .
[0018] Further, the S4 includes: complete hardware configuration according to step S1; after completing the hardware configuration, deploy two computing channels in the digital controller respectively; channel 1 receives the data read by the digital controller No. 1 ADC interface, and the data is processed by the feedforward resonance suppression controller The output of channel 1 is obtained by reading data from the 2nd DAC interface through a sensor after the piezoelectric drive positioning table is driven by the driver input drive voltage output from the 1st DAC interface of the digital controller; channel 2 also receives data read from the 1st ADC interface of the digital controller, and the output of channel 2 is obtained through the nominal model The difference model between the actual object and the nominal model is deployed by subtracting the output of channel 2 from the output of channel 1 and outputting from the 2nd DAC interface of the digital controller; A sine sweep signal with sufficient frequency sampling points in a suitable frequency range is generated by a frequency sweeper, and the displacement output of the difference model between the actual object and the nominal model corresponding to the sine sweep signal is recorded by the frequency sweeper, the frequency response at the current frequency sampling point is calculated, and a corresponding table of the frequency response of the difference model between the actual object and the nominal model at different frequency sampling points is obtained.
[0019] Further, the S4 further comprises: According to the corresponding table of the frequency response of the difference model between the actual object and the nominal model at different frequency sampling points, the uncertainty is estimated according to the following model:
[0020] wherein represents an additive uncertainty weight function, represents an estimate of the additive uncertainty of the actual object after compensation by the feedforward resonance suppression controller , represents the order of the numerator of the additive uncertainty weight function, represents the order of the denominator of the additive uncertainty weight function, represents the coefficient of each order term of the numerator of the additive uncertainty weight function, represents the coefficient of each order term of the numerator of the additive uncertainty weight function; The model order and the transfer function coefficient are adjusted so that the additive uncertainty weight function satisfies the condition that the amplitude of the additive uncertainty weight function is not less than the gain of the difference model between the actual object and the nominal model; The additive uncertainty weight function is converted to a multiplicative uncertainty weight function in the following manner:
[0021] wherein represents the multiplicative uncertainty weight function.
[0022] Further, the S5 comprises: determining the output weight function and the amplitude limiting weight function ; the output weight function is the measure of the degree of disturbance rejection of the system, the clipping weight function is set for adjusting the output of the robust controller, the clipping weight function has a high-pass characteristic.
[0023] Further, the S5 further comprises: respectively calculating the nominal model , the multiplicative uncertainty weight function , the output weight function , the nominal controller , the clipping weight function The state space implementation of the clipping weight function is as follows:
[0024] wherein is the nominal model The state space implementation of the corresponding state variable, is its corresponding coefficient matrix with appropriate dimensions, is the multiplicative uncertainty weight function The state space implementation of the corresponding state variable, is its corresponding coefficient matrix with appropriate dimensions, is the output weight function The state space implementation of the corresponding state variable, is its corresponding coefficient matrix with appropriate dimensions, is the nominal controller The state space implementation of the corresponding state variable, is its corresponding coefficient matrix with appropriate dimensions, is the clipping weight function The state space implementation of the corresponding state variable, is its corresponding coefficient matrix with appropriate dimensions, is the control output of the augmented object, is the disturbance input of the augmented object, is the input of the robust controller, is the output of the robust controller, denotes the output of the nominal controller , denotes the total control amount applied to the actual object, denotes the output of the nominal model ; According to the above state space implementation, the state space implementation of the generalized object is as follows: .
[0025] Further, the S5 further comprises: According to the above state space implementation, by solving the following standard The robust suboptimal problem can obtain the robust controller:
[0026] Wherein represents a lower linear fractional transformation, represents a robust controller, represents a given infinite norm index; Verify whether the following condition is met, if not, adjust the multiplicative uncertainty weight function , output the weight function And the limiting amplitude weight function Until the requirements are met:
[0027] Wherein, represents the transfer function with disturbance input As output, the control output As output; Check whether the closed-loop transfer function Achieve the desired resonance suppression effect, if not, adjust the multiplicative uncertainty weight function , output the weight function And the limiting amplitude weight function Until the requirements are met; The robust controller Is reduced by using the reduced order algorithm, and the reduced order robust controller ; further controller deployment, finally complete the piezoelectric drive positioning table robust resonance control based on generalized internal model architecture.
[0028] The present application has the beneficial effects that: 1、The present application contains a direct zero-pole cancellation method for piezoelectric drive positioning table resonance suppression, which can directly construct reverse zero-pole based on the identification model to realize direct cancellation of resonance mode, has the advantage of realizing rapid suppression of high-frequency resonance without relying on feedback mechanism, and avoids the phase lag problem introduced by traditional notch filter.
[0029] 2、In order to ensure the theoretical performance of nominal performance, the present application embeds the nominal model into the control structure based on the internal model control method to form a feedforward-feedback composite control, combines the closed-loop bandwidth target to realize the systematic setting of classical PID parameters by using Taylor expansion method, so that the system has the characteristics of second-order optimal tracking without overshoot under nominal working condition, and overcomes the blindness of traditional trial and error method parameter setting.
[0030] 3. In view of the contradiction between performance and robustness in the resonance control method, the present application constructs the zero-pole cancellation residual as a multiplicative uncertainty model, adopts The robust control method designs the inner loop controller to compensate for unmodeled dynamics, while ensuring the nominal performance and realizing accurate suppression of resonance peaks. BRIEF DESCRIPTION OF DRAWINGS
[0031] Figure 1 A flowchart of the piezoelectric driving positioning table robust resonance control method based on the generalized internal model architecture of the present application; Figure 2 A robust augmented object configuration diagram based on the generalized internal model architecture; Figure 3 An actual deployment architecture diagram of the piezoelectric driving positioning table robust resonance control method based on the generalized internal model architecture; Figure 4 A comparison diagram of the piezoelectric driving positioning table identification model and the actual model in the embodiment; Figure 5 A comparison diagram of the actual model and the nominal model of the piezoelectric driving positioning table after resonance suppression in the embodiment; Figure 6 A comparison diagram of the closed loop Bode plot of the nominal controller and the actual model of the piezoelectric driving positioning table after resonance suppression, and the closed loop Bode plot of the nominal controller and the nominal model of the piezoelectric driving positioning table after resonance suppression in the embodiment; Figure 7 A comparison diagram of the piezoelectric driving positioning table multiplicative uncertainty and its fitted upper bound in the embodiment; Figure 8 A Bode plot of the output weight function set in the embodiment; Figure 9 A Bode plot of the amplitude limiting weight function set in the embodiment; Figure 10 A closed loop control loop Bode plot after deployment of the resonance control method based on the generalized internal model architecture in the embodiment; Figure 11 A piezoelectric driving positioning table tracking diagram after inputting a 10Hz reference trajectory in the embodiment; Figure 12 A piezoelectric driving positioning table tracking error diagram after inputting a 10Hz reference trajectory in the embodiment; Figure 13 A piezoelectric driving positioning table tracking diagram after inputting a 700Hz reference trajectory in the embodiment; Figure 14 A piezoelectric driving positioning table tracking error diagram after inputting a 700Hz reference trajectory in the embodiment. DETAILED DESCRIPTION
[0032] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative effort belong to the scope of protection of the present application.
[0033] As shown in Figure 1 The present application proposes a robust resonance control method for a piezoelectric driving positioning table based on a generalized internal model architecture, including the following steps: Step S1: establishing a dynamic response model of the piezoelectric driving positioning table by a frequency response analysis identification method, including: First, to ensure the accuracy of modeling of the piezoelectric driving positioning table, the piezoelectric driving positioning table should be identified according to the hardware configuration connection of the closed loop in its entirety, so as to completely evaluate the influence of the hardware configuration.
[0034] Considering the delay caused by internal calculation of the digital controller, the output interface of the frequency sweeper is connected to the No. 1 ADC interface of the digital controller through a cable, and the output is directly output from the No. 1 DAC interface of the digital controller without any processing in the digital controller.
[0035] Considering the delay caused by transmission of the control quantity calculated by the digital controller to the piezoelectric driving positioning table, the No. 1 DAC interface of the digital controller is connected to the control input interface of the driver of the piezoelectric driving positioning table through a cable.
[0036] Considering the delay caused by the sensor feedback signal of the piezoelectric driving positioning table, the sensor of the piezoelectric driving positioning table is connected to the No. 2 ADC interface of the digital controller through a cable, and the output is directly output from the No. 2 DAC interface of the digital controller without any processing in the digital controller, and the No. 2 DAC interface of the digital controller is connected to the input interface of the frequency sweeper through a cable.
[0037] In the above hardware configuration description, the numbers of the ADC / DAC interfaces of the digital controller are only used to distinguish the positions of the interfaces, and do not represent the actual interface numbers.
[0038] After the hardware configuration is completed, the frequency sweeper generates a sinusoidal sweep signal with sufficient frequency sampling points in a suitable frequency range to the piezoelectric driving positioning table, and records the displacement output of the piezoelectric driving positioning table under the sweep signal by the frequency sweeper, calculates the frequency response at the current frequency sampling point, and obtains a corresponding table of the frequency responses of the piezoelectric driving positioning table at different frequency sampling points.
[0039] According to the above corresponding table of the frequency responses of the piezoelectric driving positioning table at different frequency sampling points, linear identification is performed by using an identification tool according to the following model:
[0040] wherein, represents the open-loop transfer function of the piezoelectric driving positioning stage, represents a complex frequency variable, represents the order of the numerator of the transfer function, represents the order of the denominator of the transfer function, represents the coefficient of each term of the numerator of the transfer function, represents the coefficient of each term of the numerator of the transfer function, represents the time delay.
[0041] In the above linear system identification process, different model orders and time delays are selected, and the differences between the identification results and the actual data are compared, and the set of parameters with the best fitting effect is selected to obtain the open-loop transfer function of the piezoelectric driving positioning stage .
[0042] Step S2: based on the zero-pole cancellation resonance suppression method, a feedforward resonance suppression controller is calculated and obtained, comprising: First, according to the open-loop function of the piezoelectric driving positioning stage obtained by linear identification , the part describing the resonance characteristics of the piezoelectric driving positioning stage is analyzed as follows:
[0043] wherein, represents the sum of all transfer functions describing the resonance characteristics of the piezoelectric driving positioning stage, represents the total number of resonance peaks of the piezoelectric driving positioning stage, and the denominator part of the second-order transfer function describes the resonance peak, represents the damping corresponding to the resonance peak, represents the resonance frequency corresponding to the resonance peak, and the numerator part of the second-order transfer function describes the anti-resonance peak, represents the damping corresponding to the anti-resonance peak, represents the resonance frequency corresponding to the anti-resonance peak.
[0044] According to the resonance characteristic transfer function of the piezoelectric driving positioning stage , the feedforward resonance controller can be calculated by the direct zero-pole cancellation resonance suppression method as follows:
[0045] wherein, represents the transfer function of the feedforward resonance suppression controller, represents the inverse of the resonance characteristic transfer function of the piezoelectric driving positioning stage.
[0046] Step S3: take the nominal model with the assumption of complete suppression of the resonance peak as the controlled object, calculate the nominal controller through the internal model control PID tuning algorithm, including: To facilitate the calculation of the nominal controller by the internal model control PID tuning algorithm, the nominal model is designed as follows:
[0047] Among them represents the transfer function of the nominal model of the piezoelectric drive positioning table after resonance compensation, compared with the open-loop transfer function of the piezoelectric drive positioning table completely eliminates the resonance link and the pure delay link, and has the minimum phase characteristic.
[0048] Due to the minimum phase characteristic of the nominal model, the corresponding internal model controller can be designed as follows:
[0049] Among them, represents the transfer function of the internal model controller, represents the time constant of the filter, represents the order of the filter, which should not be less than the relative order of the nominal model to ensure that the internal model controller is physically realizable.
[0050] According to the performance index, select the appropriate filter time constant and the filter order , to get the internal model controller ; According to the internal model standard model, the equivalent forward controller is as follows: .
[0051] The internal model controller generally has the same order as the controlled object, which is difficult to deploy, so the Taylor expansion method is further used to approximate the internal model controller to a PI controller:
[0052] Among them represents the equivalent complex function obtained by multiplying the equivalent forward controller and the factor , which is to avoid the condition that the function is defined when expanding at ; represents the transfer function of the PI controller, represents the coefficient of the proportional term of the controller, represents the coefficient of the integral term of the controller, represents the overall gain of the controller, and the coefficient is solved as follows:
[0053] wherein represents the constant term of the denominator polynomial in the nominal model transfer function represents the linear term of the denominator polynomial in the nominal model transfer function represents the constant term of the numerator polynomial in the nominal model transfer function represents the linear term of the numerator polynomial in the nominal model transfer function represents the constant term of the denominator polynomial in the nominal model transfer function represents the linear term of the denominator polynomial in the nominal model transfer function represents the constant term of the numerator polynomial in the nominal model transfer function represents the linear term of the numerator polynomial in the nominal model transfer function
[0054] Substitute the above formula into the performance index to adjust the overall gain of the controller to a suitable size to obtain the nominal controller . .
[0055] Step S4: Based on the difference model frequency response data of the actual object and the nominal model, the controlled object multiplicative uncertainty is fitted, including: First, complete the hardware configuration according to the method in step S1.
[0056] After completing the hardware configuration, two calculation channels are deployed in the digital controller. Channel 1 receives data read by the digital controller 1st ADC interface, and after passing through the feedforward resonance suppression controller , the output is output from the digital controller 1st DAC interface, and after passing through the driver and inputting the piezoelectric drive positioning table, the data is read again from the 2nd DAC interface through the sensor to obtain the output of channel 1. Channel 2 also receives data read by the digital controller 1st ADC interface, and after passing through the nominal model , the output of channel 2 is obtained. The output of channel 1 is subtracted from the output of channel 2, and the output is output from the digital controller 2nd DAC interface to complete the deployment of the difference model between the actual object and the nominal model.
[0057] A sinusoidal sweep signal with sufficient frequency sampling points in a suitable frequency range is generated by a frequency sweeper, and the displacement output of the difference model between the actual object and the nominal model corresponding to the sweep signal is recorded by the frequency sweeper. The frequency response at the current frequency sampling point is calculated to obtain a corresponding table of the frequency response of the difference model between the actual object and the nominal model at different frequency sampling points.
[0058] According to the above corresponding table of the frequency response of the difference model between the actual object and the nominal model at different frequency sampling points, the uncertainty is estimated according to the following model:
[0059] wherein represents an additive uncertainty weight function, represents a nominal model compensated by a feedforward resonance suppression controller the compensated estimation of the additive uncertainty of the actual object, represents the order of the numerator of the additive uncertainty weight function, represents the order of the denominator of the additive uncertainty weight function, represents the coefficient of each order term of the numerator of the additive uncertainty weight function, represents the coefficient of each order term of the numerator of the additive uncertainty weight function.
[0060] To ensure the robust stability of the system, the model order and the transfer function coefficient are adjusted so that the additive uncertainty weight function satisfies the condition that the amplitude of the difference between the actual object and the nominal model is not less than the gain of the difference model.
[0061] To facilitate the design of the robust controller, the additive uncertainty weight function is converted into a multiplicative uncertainty weight function in the following manner:
[0062] wherein represents the multiplicative uncertainty weight function.
[0063] Step S5: solving the standard robust suboptimal problem to obtain a robust controller for robust resonance control, including: Figure 2 To configure a graph for the robust augmented object based on the generalized internal model architecture, it is necessary to first determine the output weight function and the amplitude limiting weight function .
[0064] The output weight function is a measure of the degree of interference suppression of the system, and in the present application, the residual resonance characteristics after compensation by the feedforward resonance suppression controller are taken as interference, thereby achieving the purpose of resonance suppression.
[0065] Due to the limitation of the waterbed effect of the linear system, the output weight function should have a low-pass characteristic, and the output weight function gain at a specific resonance peak should be increased to achieve accurate suppression of the resonance peak.
[0066] To prevent the control output of the robust controller from exceeding the allowable range of the actuator, the amplitude limiting weight function is set to adjust the output of the robust controller, and the amplitude limiting weight function should have a high-pass characteristic.
[0067] According toFigure 2 The configurations are used to calculate the nominal model respectively. Multiplicative uncertainty weight function Output weight function Nominal controller Amplitude limiting weight function The state space is implemented as follows:
[0068] in For nominal model The state space implements the corresponding state variables. This is its corresponding coefficient matrix with appropriate dimensions. For multiplicative uncertainty weighting functions The state space implements the corresponding state variables. This is its corresponding coefficient matrix with appropriate dimensions. For output weight function The state space implements the corresponding state variables. This is its corresponding coefficient matrix with appropriate dimensions. For nominal controller The state space implements the corresponding state variables. This is its corresponding coefficient matrix with appropriate dimensions. For the amplitude limiting weight function The state space implements the corresponding state variables. This is its corresponding coefficient matrix with appropriate dimensions. To augment the control output of the object, To augment the object's interference input, For the input of the robust controller, For the output of the robust controller, Indicates the nominal controller The output, This represents the total amount of control applied to the actual object. Representation of the nominal model The output.
[0069] Based on the above state space implementation, we can derive the generalized object. The state space is implemented as follows:
[0070] Based on the above state space implementation, the following criteria are solved. A robust controller can be derived from a robust suboptimal problem:
[0071] in represents a lower linear fractional transformation, represents a robust controller, represents a given infinity norm index.
[0072] To ensure the stability of the system, verify whether the following condition is met, if not, re-adjust the multiplicative uncertainty weight function , output the weight function and the clipping weight function until the requirements are met:
[0073] wherein, represents a transfer function with the disturbance input as the output and the control output as the output.
[0074] Check whether the closed-loop transfer function achieves a suitable resonance suppression effect, if not, re-adjust the multiplicative uncertainty weight function , output the weight function and the clipping weight function until the requirements are met.
[0075] The order of the robust controller obtained by this method is often greater than the order of the controlled object, and needs to be reduced in actual arrangement, and a suitable order reduction algorithm is used to reduce the robust controller to obtain the reduced robust controller .
[0076] Finally, according to the control architecture shown in Figure 3 , the controller is deployed, and the piezoelectric driving positioning table robust resonance control based on the generalized internal model architecture is completed.
[0077] Embodiment: First, an actual model of a piezoelectric ceramic positioning table is established, which is a three-order system composed of an inertia link and a resonance link, and the specific transfer function is as follows:
[0078] By adding disturbances to the resonance frequency and damping coefficient parameters of the resonance link, the situation that cannot be accurately identified in the real identification process is simulated, and the specific transfer function of the identification model of the piezoelectric ceramic positioning table is as follows:
[0079] The comparison chart of the identification model and the actual model is shown in Figure 4 .
[0080] Based on the identification model, the specific transfer function of the feedforward resonance suppressor is as follows:
[0081] The feedforward resonance suppressor is connected in series with the actual model of the piezoelectric ceramic positioning table as the actual model of the piezoelectric ceramic positioning table after resonance suppression .The feedforward resonance suppressor is connected in series with the identification model of the piezoelectric ceramic positioning table as the nominal model of the piezoelectric ceramic positioning table after resonance suppression .Obviously, the nominal model is the result of assuming that the feedforward resonance suppressor completely compensates for the resonance link, and the specific transfer function is as follows:
[0082] The comparison chart of the actual model and the nominal model of the piezoelectric drive positioning table after resonance suppression is shown in Figure 5 .
[0083] Based on the nominal model of the piezoelectric ceramic positioning table after resonance suppression described above, a nominal controller is further designed. The filter time constant and the relative order are selected, and the corresponding coefficients of the PID controller and are calculated. The gain of the entire PID controller is further adjusted to achieve the desired nominal performance, and the specific transfer function of the nominal controller is finally calculated as follows:
[0084] The comparison chart of the Bode diagram of the closed loop formed by the nominal controller and the actual model of the piezoelectric drive positioning table after resonance suppression and the Bode diagram of the closed loop formed by the nominal controller and the nominal model of the piezoelectric drive positioning table after resonance suppression is shown in Figure 6 .
[0085] According to the multiplicative uncertainty calculated from the actual model of the piezoelectric ceramic positioning table after resonance suppression and the nominal model of the piezoelectric ceramic positioning table after resonance suppression, the upper bound is fitted, and the calculation method is as follows:
[0086] The transfer function of the fitted upper bound is as follows:
[0087] A comparison of multiplicative uncertainty and its upper bound for fitting is shown in the figure below. Figure 7 As shown.
[0088] Based on the above, a robust controller will be further designed. .according to Figure 2 The configuration also requires determining the output weight function. and amplitude limiting weight function Only then can the robust controller be solved. The specific transfer function of the designed weight function is as follows:
[0089]
[0090] The Bode plot of the output weight function is as follows Figure 8 As shown, the Bode plot of the amplitude limiting weight function is as follows: Figure 9 As shown.
[0091] By solving the standard The robust suboptimal problem yields the following specific transfer function for the robust controller:
[0092] Verify robust stability metrics:
[0093] Verify the closed-loop transfer function Its Bode plot is as follows Figure 10 As shown, resonance suppression in a specific frequency band is achieved while maintaining the gain of other frequency bands, thus meeting the design requirements.
[0094] A robust controller with reduced order can be obtained through a suitable order reduction algorithm. The specific transfer function is as follows:
[0095] Based on the above solution results, a closed-loop trajectory tracking test was performed. The reference trajectory is as follows:
[0096] in The frequency of the reference trajectory signal, This is the simulation step size. In this simulation, we select... , 0.00004s, the simulation results are as follows Figure 11-14 As shown, where Figure 11 The piezoelectric-driven positioning stage tracks the image after inputting the above 10Hz reference trajectory. Figure 12 The piezoelectric-driven positioning stage tracking error diagram is generated after inputting the above 10Hz reference trajectory.Figure 13 Figure 7 is a tracking diagram of the piezoelectric ceramic positioning table after inputting the above-mentioned 700Hz reference trajectory, Figure 14 Figure 8 is a tracking error diagram of the piezoelectric ceramic positioning table after inputting the above-mentioned 700Hz reference trajectory. The effect of good tracking in other frequency bands while maintaining resonance suppression is achieved.
[0097] The piezoelectric driving positioning table robust resonance control method based on the generalized internal model architecture is described in detail above, and the principle and implementation mode of the present application are described by applying specific examples. The above example is only used to help understand the method and core idea of the present application. Meanwhile, for those skilled in the art, according to the idea of the present application, the specific implementation mode and application range will be changed, and the above description should not be understood as a limitation of the present application.
Claims
1. A robust resonance control method for a piezoelectric driving positioning stage based on a generalized internal model architecture, characterized in that, The method is specifically: S1, a dynamic response model of the piezoelectric drive positioning table is established by a frequency response analysis identification method; S2, a feedforward resonance suppression controller is calculated based on a zero-pole cancellation resonance suppression method; S3, a nominal controller is calculated by an internal model control PID tuning algorithm, with the assumption that the resonance peak is completely suppressed; S4, a controlled object multiplicative uncertainty is fitted based on the frequency response data of the difference model between the actual object and the nominal model; S5, solving criterion robust suboptimal problem, obtaining a robust controller for robust resonance control.
2. The method of claim 1, wherein, The S1 comprises: Considering the delay caused by the internal calculation of the digital controller, the output interface of the frequency sweeper is connected to the No. 1 ADC interface of the digital controller through a cable, and no processing is performed in the digital controller, and the output is directly output from the No. 1 DAC interface of the digital controller; Considering the delay caused by the transmission of the control quantity calculated by the digital controller to the piezoelectric drive positioning table, the No. 1 DAC interface of the digital controller is connected to the control input interface of the driver of the piezoelectric drive positioning table through a cable; Considering the delay caused by the sensor feedback signal of the piezoelectric drive positioning table, the sensor of the piezoelectric drive positioning table is connected to the No. 2 ADC interface of the digital controller through a cable, and no processing is performed in the digital controller, and the output is directly output from the No. 2 DAC interface of the digital controller, and the No. 2 DAC interface of the digital controller is connected to the input interface of the frequency sweeper through a cable; According to the frequency response corresponding table of the piezoelectric drive positioning table at different frequency sampling points, linear identification is performed by using an identification tool according to the following model: wherein, Gp(z) represents an open-loop transfer function of a piezoelectric driving positioning stage, Gp(z) represents a complex frequency variable, Gp(z) represents a numerator order of a transfer function, Gp(z) represents a denominator order of a transfer function, Gp(z) represents a coefficient of each term of a numerator of a transfer function, Gp(z) represents a coefficient of each term of a numerator of a transfer function, Gp(z) represents a time delay; Selecting different model order in system identification process and time delay , comparing the difference between identification result and actual data, selecting the best fitting parameter set to get the open-loop transfer function of piezoelectric driving positioning stage .
3. The method of claim 2, wherein, The S2 comprises: The open loop function of the piezoelectric driving positioning table obtained according to linear identification The part describing the resonance characteristics of the piezoelectric driving positioning table obtained by analysis is as follows: wherein represents the sum of all transfer functions describing the resonance behavior of the piezo-driven positioning stage, represents the total number of resonance peaks of the piezo-driven positioning stage, the denominator part of the second order transfer function describes a resonance peak, represents the damping corresponding to the resonance peak, represents the resonance frequency corresponding to the resonance peak, the numerator part of the second order transfer function describes an anti-resonance peak, represents the damping corresponding to the anti-resonance peak, represents the resonance frequency corresponding to the anti-resonance peak; According to the piezoelectric drive positioning table resonance characteristic transfer function The feedforward resonance controller is calculated by a direct zero-pole cancellation resonance suppression method as follows: wherein represents a transfer function of a feedforward resonance suppression controller, represents an inverse of a transfer function of a piezoelectric drive positioning stage resonance characteristic.
4. The method of claim 3, wherein, The S3 comprises: In order to calculate the nominal controller by using the internal model control PID tuning algorithm, the nominal model is designed as follows: wherein represents the transfer function of the nominal model of the piezoelectric driving positioning table after resonance compensation, compared with the open-loop transfer function of the piezoelectric driving positioning table completely eliminates the resonance link and the pure delay link, and has the minimum phase characteristic; Due to the minimum phase characteristic of the nominal model, the corresponding internal model controller is designed as follows: wherein, G(s) represents the transfer function of the internal model controller, T represents the time constant of the filter, N represents the filter order, which should not be less than the relative order of the nominal model to ensure that the internal model controller is physically realizable; Selecting appropriate filter time constant according to performance index and filter order to obtain internal model controller ; According to the internal model control standard model, an equivalent forward controller As shown below: .
5. The method of claim 4, wherein, The S3 further comprises: The inner model controller is approximated as a PI controller using Taylor expansion method: where represents the equivalent forward controller multiplied by the factor The equivalent complex function is obtained after multiplication, whose purpose is to circumvent the condition that the function is defined at the expansion point; represents the PI controller transfer function, represents the controller proportional term coefficient, represents the controller integral term coefficient, represents the controller overall gain, the coefficient is solved as follows: wherein represents the constant term of the denominator polynomial in the nominal model transfer function represents the linear term of the denominator polynomial in the nominal model transfer function represents the constant term of the numerator polynomial in the nominal model transfer function represents the linear term of the numerator polynomial in the nominal model transfer function Substitute the above equation into the performance index to adjust the overall gain of the controller to the appropriate size, resulting in a nominal controller .
6. The method of claim 5, wherein, The S4 includes: completing hardware configuration according to step S1; after completing hardware configuration, respectively deploying two computing channels in the digital controller; channel 1 receives data read by the digital controller 1st ADC interface, and the data is output from the digital controller 1st DAC interface after passing through a feedforward resonance suppression controller , input drive pressure and then input drive positioning platform through a sensor to obtain the output of channel 1 by re-reading data from the 2nd DAC interface; channel 2 also receives data read by the digital controller 1st ADC interface, and the data is output after passing through a nominal model to obtain the output of channel 2; after subtracting the output of channel 2 from the output of channel 1, the output is output from the digital controller 2nd DAC interface to complete the deployment of the difference model between the actual object and the nominal model; A sine sweep signal with sufficient frequency sampling points in a suitable frequency interval is generated by the frequency sweeper, and the displacement output of the difference model between the actual object and the nominal model corresponding to the sweep signal is recorded by the frequency sweeper, the frequency response at the current frequency sampling point is calculated, and the frequency response corresponding table of the difference model between the actual object and the nominal model at different frequency sampling points is obtained.
7. The method of claim 6, wherein, The S4 further comprises: According to the frequency response corresponding table of the difference model between the actual object and the nominal model at different frequency sampling points, the uncertainty estimation is performed according to the following model: wherein denotes an additive uncertainty weight function, denotes an estimate of the additive uncertainty of the actual plant after the feedforward resonance suppression controller denotes an estimate of the additive uncertainty of the actual plant after the compensation, denotes the order of the numerator of the additive uncertainty weight function, denotes the order of the denominator of the additive uncertainty weight function, denotes the coefficients of the terms of the numerator of the additive uncertainty weight function, denotes the coefficients of the terms of the numerator of the additive uncertainty weight function; by adjusting the model order and transfer function coefficients such that the additive uncertainty weight function satisfies the condition that its magnitude is not less than the gain of the difference model between the actual plant and the nominal model; The additive uncertainty weight function is converted into a multiplicative uncertainty weight function in the following manner: wherein denotes a multiplicative uncertainty weight function.
8. The method of claim 7, wherein, The S5 comprises: determining an output weight function and clipping the weight function ; the output weight function is a measure of the degree of disturbance rejection by the system, the clipping weight function is used to adjust the output of the robust controller, the clipping weight function has a high-pass characteristic.
9. The method of claim 8, wherein, The S5 further comprises: Respective nominal models , multiplicative uncertainty weight function , output weight function , nominal controller , clipping weight function The state-space realization of the clipping weight function is as follows: wherein is a nominal model State space realization corresponding to the state variable is its corresponding coefficient matrix of appropriate dimension, is a multiplicative uncertainty weight function State space realization corresponding to the state variable is its corresponding coefficient matrix of appropriate dimension, is an output weight function State space realization corresponding to the state variable is its corresponding coefficient matrix of appropriate dimension, is a nominal controller State space realization corresponding to the state variable is its corresponding coefficient matrix of appropriate dimension, is a clipping weight function State space realization corresponding to the state variable is its corresponding coefficient matrix of appropriate dimension, is the control output of the augmented plant, is the disturbance input of the augmented plant, is the input to the robust controller, is the output of the robust controller, denotes the output of the nominal controller , denotes the total control amount applied to the actual plant, denotes the output of the nominal model . From the above state-space realization, the generalized plant is obtained as follows: 。 10. The method of claim 9, wherein, The S5 further comprises: According to the above state space realization, the robust controller is obtained by solving the following standard The robust suboptimal problem yields the robust controller: wherein represents a lower linear fractional transformation, represents a robust controller, represents a given infinity norm index; Verify if the following conditions are met, if not, re-adjust the multiplicative uncertainty weight function , output weight function and clipping weight function until the requirements are met: wherein, represents a transfer function with the disturbance input as input and the control output as output; Test closed loop transfer function , whether a suitable resonance suppression effect is achieved, and if not, readjusting the multiplicative uncertainty weight function , output weight function and clipping weight function until requirements are met; The robust controller is reduced by a reduction algorithm to obtain a reduced robust controller ; and further controller deployment is performed to finally complete the robust resonance control of the piezoelectric driving positioning table based on the generalized internal model architecture.