Modal controller generation method, medium, and apparatus for a nanopositioning platform

Abstract

The present application relates to the technical field of ultra-precision machining control, and particularly relates to a modal controller generation method for a nano-positioning platform, a medium and equipment. The present application performs zero-pole decomposition on a transfer function of a controlled object, separates stable zero points, unstable zero points, non-resonant poles and resonant poles, constructs a desired inner loop closed-loop transfer function, retains the stable zero points, the unstable zero points and the non-resonant poles, and replaces the resonant poles as a whole with high-damping desired poles, inversely solves to obtain a modal controller structure, introduces zero-pole cancellation constraints to eliminate controller instability caused by unstable zero points, introduces order constraints to ensure causal realizability, applies constraints with a damping ratio greater than 0.1 to the desired poles and the introduced new zero points respectively, and outputs an inner loop modal controller. The present application realizes simultaneous suppression of multiple resonant peaks, gets rid of dependence on a low-order model, ensures stability, realizability and smoothness of a control signal of the controller, and significantly improves control bandwidth and dynamic performance.

Description

Technical Field

[0001] This invention relates to the field of ultra-precision machining control technology, and in particular to a method, medium, and device for generating modal controllers for nanometer positioning platforms. Background Technology

[0002] With the rapid development of ultra-precision manufacturing, scanning probe microscopy, and biological cell manipulation, the demand for high-speed and high-precision motion control of nanopositioning platforms is becoming increasingly urgent. Nanopositioning platforms (such as piezoelectric ceramic driven scanners and constant-stress electromagnetic actuator driven positioning stages) typically have complex high-order dynamic characteristics. Their mechanical structures contain multiple low-damped resonant modes, which are easily excited during high-speed scanning, leading to system oscillation and instability, and severely limiting control bandwidth and positioning accuracy.

[0003] To suppress resonance, existing technologies often employ inner-loop modal controllers (such as positive position feedback (PPF) and positive velocity-position feedback (PVPF). However, these traditional modal controllers have the following significant drawbacks:

[0004] Limited order, unable to handle multiple resonance peaks: Traditional modal controllers are usually second- or third-order structures, and their design relies on low-order approximation models of the controlled object. For high-order systems with multiple resonance peaks (corresponding to multiple low-damped poles), the low-order controllers lack sufficient degrees of freedom and cannot suppress all resonance peaks simultaneously, resulting in residual oscillations still existing.

[0005] Strong dependence on accurate low-order models: Existing methods require obtaining accurate low-order transfer function models of the controlled object, while the dynamic characteristics of actual nano-positioning platforms are extremely complex. Model downgrading inevitably introduces mismatch, resulting in poor damping control and poor system robustness.

[0006] Limited control bandwidth: Due to the inability to effectively suppress multiple resonance peaks, the closed-loop bandwidth of the system is usually much lower than the first-order resonant frequency, making it difficult to meet the requirements of high-speed and high-precision applications.

[0007] Therefore, there is an urgent need for a modal controller generation method that can handle high-order complex dynamic characteristics, does not depend on low-order models, and can simultaneously suppress multiple resonance peaks. Summary of the Invention

[0008] To address one of the aforementioned technical problems, the present invention adopts the following technical solution:

[0009] According to one aspect of the present invention, a method for generating a modal controller for a nanometer positioning platform is provided, comprising the following steps:

[0010] Transfer function of the controlled object Perform zero-pole decomposition to isolate stable zeros. Unstable zeros Non-resonant poles and resonant poles ;

[0011] Based on the decomposition results, construct the desired inner-loop closed-loop transfer function. And the modal controller is obtained by inverse solving. The structural form; reserve and and will Replace with a desired pole with high damping characteristics. At the same time, a new zero point is introduced. ; The following conditions must be met:

[0012] ;

[0013] The following conditions must be met: ;

[0014] Introducing zero-pole cancellation constraints and order constraints, for Perform stability and feasibility design to obtain an implementable form. The zero-pole cancellation constraint is used to eliminate This leads to controller instability; the order constraint is used to ensure... Causal realizability; The implementable form satisfies the following conditions:

[0015] ;

[0016] right and Constraints with damping ratios greater than 0.1 are applied to output the final inner-loop modal controller.

[0017] According to a second aspect of the present invention, a non-transitory computer-readable storage medium is provided, which stores a computer program that, when executed by a processor, implements the above-described method for generating a modal controller for a nanopositioning platform.

[0018] According to a third aspect of the present invention, an electronic device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described method for generating a modal controller for a nanopositioning platform.

[0019] This invention has at least one of the following beneficial effects:

[0020] First, this invention utilizes the transfer function of the controlled object. Perform zero-pole decomposition to isolate stable zeros. Unstable zeros Non-resonant poles and resonant poles Based on the decomposition results, the desired inner-loop closed-loop transfer function is constructed. ,Will Replace with a desired pole with high damping characteristics. .

[0021] "Low damping" refers to a pole damping ratio less than 0.1. Such poles cause spikes (i.e., resonance peaks) on the system's frequency response curve, which is a major factor limiting the control bandwidth. "High damping," on the other hand, refers to a pole damping ratio greater than 0.1. Traditional modal controllers are typically low-order controllers with second- or third-order structures. When the controlled object has a large number of resonant poles (e.g., four, six, or more), traditional methods cannot simultaneously suppress all resonance peaks due to insufficient degrees of freedom in the controller order. This invention accurately identifies all poles through zero-pole decomposition. (Regardless of their quantity), and During the construction process, Retain it, and keep all of it. Replace the entire unit with a high-damping one. Therefore, regardless of the number of resonance peaks in the controlled object or the high order of its dynamics, this method can suppress all resonance peaks simultaneously, thus breaking free from the dependence of traditional methods on low-order models and overcoming the technical bottleneck of being unable to handle multiple resonance peaks due to the limitation of controller order.

[0022] Second, this invention introduces zero-pole cancellation constraints and order constraints to... Perform stability and feasibility design to obtain an implementable form. .

[0023] Zero-pole cancellation constraint is used to eliminate The resulting controller instability ensures that the designed modal controller itself is stable; the order constraint is used to guarantee... The causal realizability ensures that the controller is physically implementable. These two constraints work together to enable the method to stably and reliably design practically applicable modal controllers without relying on accurate low-order models.

[0024] Third, this invention addresses and Constraints with a damping ratio greater than 0.1 are applied respectively.

[0025] right The constraints ensure that the replaced closed-loop poles have rapidly decaying dynamic characteristics, resulting in a smooth and oscillating system response; The constraints prevent the controller itself from generating high-frequency oscillating control signals, thus avoiding unsmooth control signals. The combination of these two factors ensures that the final output inner-loop modal controller effectively suppresses resonance in the controlled object without introducing new dynamic problems.

[0026] In summary, the steps of this invention work together in a progressive manner to simultaneously suppress multiple resonance peaks, achieve universal applicability to high-order systems, and ensure the stability and feasibility of the controller itself. This significantly improves the control bandwidth and dynamic performance of the nano-positioning platform and overcomes the technical defects of traditional modal controllers, such as limited order, reliance on low-order models, and inability to handle multiple resonance peaks. Attached Figure Description

[0027] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0028] Figure 1 This is a flowchart of a modal controller generation method for a nano-positioning platform provided in an embodiment of the present invention.

[0029] Figure 2 This is a structural block diagram of the high-bandwidth dual-loop closed-loop control system in an embodiment of the present invention.

[0030] Figure 3 This is a schematic diagram of the stability circle constraint based on the Nyquist plot in an embodiment of the present invention.

[0031] Figure 4 This is a convergence curve of the differential evolution algorithm in an embodiment of the present invention.

[0032] Figure 5 This is a comparison diagram of the closed-loop frequency response of the system before and after optimization in an embodiment of the present invention.

[0033] Figure 6 The figure shows the results of the sine wave tracking experiment of the optimized closed-loop system in this embodiment of the invention.

[0034] Figure 7 The figure shows the results of the triangular wave tracking experiment of the optimized closed-loop system in this embodiment of the invention.

[0035] Figure 8 A flowchart illustrating a method for optimizing control parameters for a nano-positioning platform, as provided in an embodiment of the present invention. Detailed Implementation

[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0037] As a first possible embodiment of the present invention, a method for generating a modal controller for a nano-positioning platform is provided. The technical problem addressed by this method is that traditional low-order modal controllers (such as positive position feedback (PPF) and positive velocity position feedback (PVPF)) are typically second- or third-order structures. When the controlled object (i.e., the nano-positioning platform) has multiple low-damped resonance peaks (i.e., multiple resonant poles with a damping ratio less than 0.1), the low-order controller, due to insufficient degrees of freedom, cannot simultaneously suppress all resonance peaks, leading to oscillations in the system during high-speed scanning and limiting the control bandwidth. This embodiment achieves simultaneous suppression of any number of resonance peaks by designing a modal controller structure with adaptive order.

[0038] This method acquires the open-loop frequency response data of the controlled object and identifies its transfer function, thereby designing an inner-loop modal controller capable of suppressing multiple resonant peaks. The core idea is to first identify all resonant poles of the original system, then replace them entirely with high-damping poles specified by the designer, ensuring the controller's stability and physical realizability through mathematical constraints. Figure 1 As shown, the method includes the following steps:

[0039] S1: Obtain the open-loop frequency response data of the controlled object, and identify the transfer function of the controlled object based on the frequency response. .

[0040] S1 specifically includes:

[0041] S1.1: By inputting band-limited white noise into the controlled object, the open-loop frequency response data of the controlled object within the Nyquist frequency range is obtained.

[0042] White noise is a random signal with a flat power spectral density in the frequency domain, which can be understood as an excitation with "equal energy at all frequencies". When such white noise is input to the controlled object, all the inherent modes of the system (including those high-order resonant modes that are easily ignored) will be uniformly excited, thus reflecting complete dynamic information in the output response. In practice, a control framework can be built using dSPACE's hardware ControlBox and MATLAB's Simulink, for example, by adjusting the sampling period of the control system. Set to 0.05 ms (corresponding to a sampling frequency of 20 kHz). Apply a white noise signal with a certain energy to the controlled object. This signal is approximately uniformly distributed within a limited bandwidth and is sufficient to fully excite the resonant modes of each order of the system. Synchronously collect the input voltage signal and the output displacement signal through the sensor supporting the controlled object, and save the data in real time through the ControlDesk software. The frequency response data obtained in this way can completely cover the resonant modes of each order of the system, providing a reliable basis for subsequent accurate modeling.

[0043] S1.2: Based on this open-loop frequency response data, use the transfer function identification function to fit the input and output data, and select the model order with the highest fitting degree as the discrete transfer function of the controlled object.

[0044] In the MATLAB environment, the frequency response characteristics of the system can be obtained first through spectral analysis (such as using the spafdr function), and then different combinations of the numerator and denominator orders (such as from the 2nd order to the 8th order, but not limited to this) can be tried for transfer function identification (using the tfest function), through the frequency response data, and record the identification results (such as data files, truncation time, optimal order, fitting degree) in a log file. The transfer function obtained in this way can accurately describe the resonant pole positions and damping ratios of the system, providing an accurate target for subsequent pole replacement.

[0045] S2: For the transfer function of the controlled object Perform zero-pole decomposition to separate the stable zeros , unstable zeros , non-resonant poles and resonant poles .

[0046] In the present invention, the symbol is used to represent the set or concept of zeros and poles. In specific formulas, in order to clarify that they are polynomial functions of the variable , they are usually written in a bracketed form, such as . The two essentially refer to the same object, but only have different expression forms: with representing the explicit function form (such as ), and without serving as a code name. In this specification, the form with is always used in formulas to ensure mathematical rigor, and can be omitted for brevity in text descriptions. For example, we can say "the number of stable zeros is 4", while in the formula it is written as .

[0047] S2 specifically includes:

[0048] S2.1: Through calculation The modulus of all zeros is used to classify zeros with a modulus less than 1 as... Zeros with a modulus greater than or equal to 1 are classified as .

[0049] In discrete control systems, the magnitude of a zero determines its impact on system stability. Zeros with a magnitude less than 1 are called "stable zeros," which do not cause system instability and can therefore be safely retained in controller design to preserve some of the original system's dynamic characteristics. Zeros with a magnitude greater than or equal to 1 are called "unstable zeros," which would cause instability in the controller if they appeared in it. Therefore, they cannot be directly retained and must be precisely canceled out by subsequent zero-pole cancellation equations. It is precisely because of this distinction that this invention can preserve the beneficial characteristics of the original system without sacrificing stability.

[0050] S2.2: Through calculation The damping ratio of all poles is used to classify complex poles with a damping ratio less than 0.1 as... Real poles and complex poles with a damping ratio greater than or equal to 0.1 are classified as... .

[0051] The damping ratio of a pole describes the system's response characteristics in the mode corresponding to that pole. "Resonant poles" (damping ratio < 0.1) cause sharp peaks (resonance peaks) in the system's frequency response curve. When the input signal contains components of that frequency, the output will be significantly amplified, producing violent oscillations. These types of poles are precisely the ones that need to be eliminated. "Non-resonant poles" (damping ratio < 0.1) The system already exhibits good damping characteristics and a smooth response, making it safe to retain. Through this classification, the present invention clarifies "what needs to be removed and what can be retained," providing direction for subsequent controller design.

[0052] After pole-zero decomposition, the transfer function It can be uniformly represented as:

[0053]

[0054] To facilitate understanding, a simplified example can be given: Suppose that the transfer function of a controlled object is obtained through identification, and after decomposition, it contains 2 unstable zeros, 4 stable zeros, 3 non-resonant poles, and 4 resonant poles (corresponding to 2 resonant peaks). The number of poles in the actual system may be different, but the decomposition method and subsequent design process are completely consistent.

[0055] Regarding the relationship between the number of poles and zeros and the order of a polynomial: the number of poles and zeros is equal to the number of zeros in the polynomial. or The order (highest power). For example, if there are 4 stable zeros, then It can be represented as That is, a 4th-order polynomial. Similarly, the number of poles determines... The order of the polynomial. Once the order is determined, the coefficients of the polynomial (such as...) These are the specific numerical values, obtained through the identification process. In the subsequent controller design, we also need to determine the desired pole polynomial. and the new zero-point polynomial The order of the controller, and the choice of these orders, will affect the complexity of the controller and the number of adjustable parameters.

[0056] S3: Based on the decomposition results, construct the desired inner-loop closed-loop transfer function. And the modal controller is obtained by inverse solving. The structural form. The desired inner-loop closed-loop transfer function. The definition of is:

[0057]

[0058] In order to retain the beneficial characteristics of the original system (stable zeros, unstable zeros, and non-resonant poles) while eliminating the harmful characteristics (resonant poles), this invention designs the desired closed-loop transfer function in the following form:

[0059]

[0060] in, It is a designer-defined desired pole polynomial, in which the damping ratio of all poles is greater than 0.1 (i.e., high damping characteristic). This is a new zero-point polynomial introduced by the modal controller, which will also be forced to be highly damped in subsequent designs. Note the original resonant poles. exist It completely disappears in the denominator and is... Replacement. This means that no matter how many resonant poles (i.e., how many resonant peaks) the original system has, as long as they are all placed into... Then use a high-damping one A complete replacement can eliminate all resonance peaks at once. This is the core reason why this method can handle any number of resonance peaks.

[0061] In order to achieve the desired Reverse-pull controller The following derivation can be made: From Taking the reciprocal gives

[0062]

[0063] Transpose to obtain

[0064]

[0065] Will and Substituting the expression into the equation, and after performing common denominator and simplification operations, we obtain:

[0066]

[0067] The molecule in this expression contains It is this difference term that causes the original resonant pole to be replaced by the desired pole. Intuitively, if Exactly equal to Then the controller It will be zero (no control effect), but in fact we are adjusting... and This difference is made non-zero and satisfies subsequent constraints, thereby achieving active damping control.

[0068] S4: Introduce zero-pole cancellation constraints and order constraints for... Perform stability and feasibility design to obtain an implementable form. .

[0069] Observe the above The expression contains in its denominator (Unstable zeros). If this controller is implemented directly, it will contain unstable poles, causing the controller output to diverge. Therefore, this term must be eliminated. The method to eliminate this term is to make the zeros in the molecule... Include As a factor, that is, there exists a polynomial. Make:

[0070]

[0071] This is the "zero-pole cancellation constraint". Through this equation, Precisely canceled, controller The denominator no longer contains unstable zeros, thus ensuring the stability of the controller itself.

[0072] In this equation, the known quantities are: and The unknown quantity is and In order for the equation to have a solution, The order must be at least equal to The order of (because enough degrees of freedom are needed to match the equation). For example, if If there are two zeros, then you can choose The order of the is 2, thus the equation has a solution. This is another manifestation of the universality of this method: no matter how many unstable zeros there are, it can always be solved by choosing a sufficiently high order . Make the equation solvable.

[0073] Furthermore, the controller must be physically realizable, meaning the denominator order must not be lower than the numerator order. This necessitates the introduction of an "order constraint":

[0074]

[0075] The first inequality guarantees the controller denominator order The order of the numerator (because the denominator is) molecule is The second inequality guarantees the final closed-loop system. denominator order The molecular order, i.e. the closed-loop transfer function, is also causal.

[0076] These constraints not only guarantee physical realizability but also determine the order of the controller polynomials. Once the order is determined, the number of coefficients in each polynomial is also determined (an nth-order polynomial has...). (e.g., coefficients). For example, if If the order is 2, then it can be written as ,in These are coefficients to be determined. Similarly, If its order is 6, then it contains 7 coefficients. These coefficients will be finalized during subsequent optimization.

[0077] To illustrate this further, let's continue with the previous example: assuming... , .Pick Then, according to the zero-pole cancellation equation, it can be deduced that A certain relationship must be satisfied. Furthermore, this can be solved using the order constraint inequality. .Pick Then we can get These order choices are not unique, but there exists a feasible region. Once the order is determined, the specific form of the controller polynomial can be written (e.g., The coefficients of the polynomial are then determined by subsequent optimization algorithms (see embodiments 2 and 3).

[0078] Finally, after zero-pole cancellation and order constraints, the modal controller... The implementable form can be simplified as follows:

[0079]

[0080] S5: Yes and Constraints with damping ratios greater than 0.1 are applied to output the final inner-loop modal controller.

[0081] Although step S4 has yielded a general form of the controller structure, it contains... (Expected extreme point) and The exact location of the poles (new zeros) has not yet been determined. To ensure the stability of the system response and the smoothness of the control signal, it is mandatory that the damping ratio of all roots (i.e., poles) of these two sets of polynomials be greater than 0.1. A damping ratio greater than 0.1 means that the transient response corresponding to these poles decays rapidly and will not produce sustained oscillations.

[0082] right Applying this constraint directly ensures that the main dynamic modes of the closed-loop system are underdamped but sufficiently stable, eliminating the amplification effect caused by the original resonance peak.

[0083] right Apply this constraint to prevent the controller itself from introducing new high-frequency oscillations; otherwise, the control voltage will fluctuate at high frequencies, which may damage the piezoelectric actuator or excite unmodeled high-frequency modes.

[0084] Through the above five steps, the final output is an inner-loop modal controller that can be practically applied in digital control systems. The controller has the following characteristics: (1) It can simultaneously suppress any number of resonance peaks (because all (1) All were replaced); (2) The controller itself is stable (through zero-pole cancellation); (3) It is physically realizable (through order constraints); (4) The control signal is smooth (through the adjustment of the control signal). (Damping constraints). This allows subsequent tracking control to be performed on a flat controlled object, thereby achieving higher bandwidth and accuracy.

[0085] As a second possible embodiment of the present invention, a joint optimization method for inner and outer loops for a nano-positioning platform is also provided. The technical problem addressed by this method is that, based on the modal controller designed in Embodiment 1, although the system has eliminated resonance peaks, if the parameters of the outer loop tracking controller (such as the proportional gain and integral gain of the P controller) are tuned separately, they will produce a coupling effect with the inner loop modal controller, resulting in the actual closed-loop bandwidth and tracking accuracy not reaching the theoretical optimum. Traditional sequential design (designing the inner loop first and then the outer loop) ignores this coupling and can only obtain suboptimal performance. This embodiment unifies the inner and outer loop parameters as optimization variables and uses a differential evolution algorithm to simultaneously seek optimization, obtaining the globally optimal combination of control parameters.

[0086] This method is applied to a dual-loop closed-loop control system consisting of an inner-loop modal controller and an outer-loop PI tracking controller. The structure of this dual-loop closed-loop control system is as follows: Figure 2 The method described above obtains globally optimal control parameters through joint parameter optimization to improve the system's tracking accuracy and dynamic performance. The method includes the following steps:

[0087] S6: Using the desired pole parameters of the inner-loop modal controller and the parameters of the outer-loop PI tracking controller as optimization variables, establish an optimization problem with the objective function of minimizing the maximum deviation between the amplitude-frequency response and the 0dB line of the dual-loop closed-loop control system in the target frequency band, and including multiple types of nonlinear constraints.

[0088] Parameter vector to be optimized It includes four key parameters: the proportional gain of the outer-loop PI tracking controller. and integral gain and the desired damping ratio of the inner loop modal controller. and expected natural frequency These parameters were chosen as optimization variables because they directly determine the dynamic behavior of the two-loop closed-loop control system: proportional gain. Integral gain affects response speed Eliminating steady-state error, desired damping ratio Controlling the degree of resonance suppression, the desired natural frequency Define the position of the dominant closed-loop mode. By jointly adjusting these four types of parameters, the system's bandwidth, stability, and tracking accuracy can be optimized simultaneously. (It should be noted that the logarithm of the desired poles equals the number of resonance peaks of the controlled object. For example, if two resonance peaks are identified, there are two pairs of desired poles (i.e., two...) and 2 (The specific quantity depends on the actual system; this is just an example and does not constitute a limitation.)

[0089] objective function The maximum deviation between the amplitude-frequency response of the closed-loop system and the 0dB line, i.e., within the frequency range. Within ], for the double-loop closed-loop transfer function The maximum absolute deviation between the amplitude-frequency response and the 0dB line is calculated. Its expression is:

[0090]

[0091] in The maximum natural frequency of the controlled object itself (the open-loop system without an inner-loop modal controller and an outer-loop PI tracking controller). The bandwidth factor is usually taken as... This indicates that the target bandwidth covers the main dynamic range of the system.

[0092] The specific definitions of the four types of nonlinear constraints are given in S8.1 of the third embodiment. These constraints are introduced to ensure that the optimization result is not only mathematically optimal but also meets the safety boundaries of practical engineering: stability constraints prevent system instability, damping constraints avoid severe oscillations, high-frequency amplitude constraints suppress noise amplification, and stability margin constraints ensure robustness to model errors. The absence of any one of these constraints may lead to problems with the controller during actual operation (e.g., high-frequency howling, severe overshoot in step response, etc.).

[0093] The search range for each parameter is set as follows based on its physical meaning and system characteristics:

[0094] ;

[0095] .

[0096] In this embodiment, the damp function is used to extract the passed function of the controlled object. The natural frequency is used to identify all poles of the system and calculate their corresponding natural frequencies, from which the maximum frequency is selected. This maximum natural frequency is primarily used to set the desired natural frequency of the closed-loop system. The search range is defined to ensure that the optimization process can focus on the vicinity of the system's main resonant modes.

[0097] S7: The differential evolution algorithm is used to synchronously iterate and optimize the variables to obtain the globally optimal parameter combination.

[0098] The parameters for the differential evolution algorithm can be set as follows: population size Set to 300, the mutation factor Set the crossover probability to 0.8. The value is set to 0.8, and the maximum number of iterations is set to 400. The differential evolution algorithm is adopted. Mutation strategy, for the first Generational mutation, the way mutated individuals are generated is as follows:

[0099]

[0100] The process involves randomly selecting three distinct individuals from the current population. The first individual is used as the basis vector, and the result is a mutation factor multiplied by the product of the differences between the other two individuals. After mutation, parameters exceeding the search range are truncated. The crossover operation employs a binomial crossover strategy. First, a random forced mutation bit is generated. For each optimized parameter, if the generated random number is less than the crossover probability or the current dimension equals the forced mutation bit, the experimental individual takes the parameter value of the mutated individual in that dimension; otherwise, it takes the parameter value of the original individual. Boundary processing is also performed on the experimental individuals after crossover.

[0101] Based on unconstrained objective function value (See S9 in the third embodiment) A greedy selection is performed. If the objective function value of the experimental individual is less than that of the original individual, the experimental individual replaces the original individual. Furthermore, if the objective function value of the current individual is less than the globally optimal objective function value, the globally optimal individual is updated. This iterative update continues until the termination condition is met, and the optimal parameter combination is output.

[0102] This embodiment overcomes the performance loss caused by neglecting coupling effects in traditional sequential design by placing the desired pole parameters of the inner-loop modal controller and the parameters of the outer-loop PI controller in the same optimization vector and using a differential evolution algorithm for synchronous iteration. Specifically, joint optimization allows the damping parameters of the inner loop and the gain parameters of the outer loop to work together: for example, the inner loop can appropriately relax the suppression of the resonance peak (while still satisfying the constraints) to obtain a higher bandwidth, while the outer loop uses high gain to further reduce residual resonance and improve tracking accuracy. Experimental data show (see the subsequent effect verification section for details) that the parameter combination obtained by joint optimization can make the closed-loop bandwidth exceed the first-order resonant frequency of the open loop, and the tracking error is reduced compared with the traditional method. That's all. Furthermore, the global search capability of the differential evolution algorithm avoids the blindness of manual trial and error, significantly improving design efficiency and parameter optimization.

[0103] As a third possible embodiment of the present invention, a method for optimizing control parameters for a nano-positioning platform is also provided. The technical problem addressed by this method is that, in the joint optimization process of Embodiment Two, constraints such as stability, damping, high frequency, and stability margin are typically considered together with the objective function. The traditional approach is to add constraints to the objective function in the form of a penalty function; however, when the optimization algorithm cannot find a feasible solution, the designer cannot determine which constraint is too stringent. This embodiment designs a penalty coefficient with a significant gradient difference, enabling the minimum objective function value recorded during the optimization process to indicate the type of constraint violated, thereby achieving automatic feedback diagnosis of constraint rationality.

[0104] This method transforms a constrained optimization problem by constructing a problem and introducing a penalty factor. It then uses the numerical range of the objective function value to determine the type of constraint violation, thereby achieving automatic diagnosis and correction of constraints and improving the efficiency and success rate of parameter tuning. Figure 8 As shown, the method includes the following steps:

[0105] S8: Construct a constrained optimization problem for a two-loop closed-loop control system. This optimization problem involves minimizing the maximum deviation of the amplitude-frequency response of the two-loop closed-loop control system from the 0 dB line within the target frequency band as the performance index. Four types of nonlinear constraints are used as the boundaries of the feasible region.

[0106] S8.1: The specific definitions of the four types of nonlinear constraints are as follows.

[0107] Stability constraints: Requirements for modal controllers, the inner-loop closed-loop system formed after adding modal controllers, and the addition of... In the dual-loop closed-loop control system following the controller, all pole magnitudes are less than 1. In discrete systems, a pole magnitude less than 1 is a necessary and sufficient condition for system stability; therefore, a triple stability check mechanism is required. (Penalty term) The definition of is:

[0108]

[0109] Damping characteristic constraint: The damping ratio of all conjugate poles of the two-loop closed-loop control system must be greater than or equal to 0.1. The damping ratio of the two-loop closed-loop control system is calculated using the damping function. The damping ratio of all poles must be no less than 0.1 to ensure good dynamic characteristics of the system response and avoid significant overshoot and oscillation. Penalty term. The definition of is:

[0110]

[0111] High-frequency amplitude constraint: The modal controller's amplitude response must be less than a preset threshold (60dB in this embodiment) at all frequency points within the Nyquist frequency range. A sufficient number of frequency points are selected within the Nyquist frequency range to calculate the inner-loop modal controller... The amplitude response at these frequency points must be less than 60 dB at all frequencies to prevent the modal controller from amplifying noise at high frequencies and to avoid exciting high-frequency resonant modes of the system. (Penalty term) The definition of is:

[0112]

[0113] Stability margin constraints: such as Figure 3 As shown, the stability circle criterion based on the Nyquist plot requires that the Nyquist curve of the system's open-loop transfer function lies completely outside the stability circle. A minimum phase margin is set. and minimum gain margin Based on this, the center parameters of the stability circle can be calculated. and radius parameters The center and radius are calculated using the following formula:

[0114]

[0115]

[0116] Calculate the open-loop frequency response for each frequency point of the open-loop transfer function. The distance to the center of the stability circle is always greater than or equal to the radius of the stability circle. Penalty items The definition of is:

[0117]

[0118] In this embodiment, the amplitude margin requirement is taken as follows: Phase margin is The parameters of the stability circle can be confirmed as follows: The necessary and sufficient condition for a system to meet the stability margin requirement is that the Nyquist curve lies entirely outside the stability circle for all frequency points, thus ensuring that the system has a sufficient robust stability margin.

[0119] S9: Introducing a penalty factor transforms the constrained optimization problem into an unconstrained optimization problem. Unconstrained objective function. The following conditions must be met:

[0120]

[0121] in These are penalty terms for stability, damping characteristics, high-frequency amplitude, and stability margin, respectively. This represents the corresponding penalty coefficient.

[0122] S9.1: Performance Indicators The upper bound of the range of values ​​of is denoted as Each penalty coefficient satisfies And the difference between adjacent penalty coefficients satisfies In this embodiment, take All penalty coefficients are greater than The upper bound of the range of values ​​for is defined, and the difference between any two penalty coefficients is greater than this upper bound, so that different constraint violations correspond to different magnitudes of . Value. Because the penalty coefficient is on a much larger order of magnitude than... The possible values ​​of , any individual that violates the constraint (its At least one more The feasible individual will be assigned an unconstrained objective function value much larger than that of the feasible individual, thus being naturally eliminated in the selection operation of the optimization algorithm. This transformation method allows complex constrained optimization problems to directly apply efficient heuristic optimization algorithms without the need to develop specialized constraint handling techniques.

[0123] S10: Use optimization algorithms to optimize control parameters Perform iterative optimization to find the solution that makes Minimize the optimal combination of parameters.

[0124] In this embodiment, the optimization algorithm employs differential evolution. Differential evolution uses a binomial crossover strategy to generate test individuals and is based on an unconstrained objective function value. A greedy selection process is employed to retain the better individuals. For specific operational details, please refer to S7 in the second embodiment.

[0125] S11: Record the minimum objective function value during the optimization process. ,according to The type of constraint that was violated is determined by the numerical range in which the range is located.

[0126] when At a certain penalty coefficient When the corresponding increment interval is reached, the first... Class constraints are bottleneck constraints that the current system cannot satisfy; they are corrected by modifying the constraints of the corresponding constraint terms. It should be noted that because there is a fixed difference between the penalty coefficients, if... Much greater than the maximum penalty coefficient (e.g.) If multiple constraints are violated simultaneously, then the satisfaction of each constraint should be checked one by one, rather than just diagnosing a single constraint.

[0127] To clearly demonstrate the judgment logic, refer to Table 1 for comparison and judgment (the actual values ​​can be adjusted according to the specific problem; this is only an example for this embodiment).

[0128] Table 1

[0129]

[0130] For example, if convergence Then it falls The interval specifies the damping characteristic constraints (requiring damping ratios at all poles). If the constraint threshold cannot be met under the current system's physical limits, the designer should lower the threshold to 0.08 or 0.09 and re-optimize. This may violate both stability and damping constraints simultaneously, requiring adjustment of stability-related parameters first. Under normal circumstances (where all constraints are satisfied),... It should be much less than 5000, for example, 1.826 in the subsequent effect verification example of this invention, in which case no constraint adjustment is required.

[0131] This embodiment designs a gradient that is much larger than... By using a penalty coefficient, this invention transforms a constrained optimization problem into an unconstrained optimization problem while preserving the diagnosability of constraint violations. Specific beneficial effects include: First, ensuring the feasibility of the solution: because the penalty coefficient is orders of magnitude larger than... Any individual violating the constraints will be naturally eliminated in differential evolutionary selection, and the final convergent solution must satisfy all constraints (if the feasible region is not empty). Second, it provides a basis for constraint diagnosis: using the fixed difference of the penalty coefficient ( This causes different constraints to be violated. If the values ​​fall within non-overlapping ranges, bottleneck constraints can be clearly identified by looking up a table, and reasonable inferences can be made based on the numerical ranges when multiple constraints are violated simultaneously. Third, it improves engineering debugging efficiency: designers can modify the thresholds of only one or a few constraints based on the diagnostic results, and then rerun the optimization, typically within [timeframe]. Feasible constraint boundaries can be found within a few iterations, significantly shortening the parameter tuning cycle. Furthermore, this penalty function framework does not depend on the specific details of differential evolution and can be ported to other global optimization methods such as genetic algorithms and particle swarm optimization, demonstrating good versatility and extensibility.

[0132] Example of effect verification

[0133] To verify the effectiveness and superiority of the technical solutions constituted by the above three embodiments, a complete experimental verification is conducted below using a specific nano-positioning platform as an example. The controlled object used in this verification example is a piezoelectric ceramic-driven nano-positioning platform, whose dynamic characteristics exhibit complex dual-resonance peak characteristics (the first-order resonance peak is approximately 1050Hz, and the second-order resonance peak is approximately 1400Hz), which are difficult to accurately describe using low-order models. By implementing the methods described in embodiments 1, 2, and 3, optimal controller parameters were obtained, and performance evaluations were performed in the frequency and time domains.

[0134] First, following step S1 of the first embodiment, band-limited white noise (sampling frequency 20kHz, duration 10 seconds) is input to the controlled object to obtain open-loop frequency response data, and the 8th order transfer function is identified using the tfest function. Through zero-pole decomposition (S2 of the first embodiment), four stable zeros, two unstable zeros, three non-resonant poles, and four resonant poles (corresponding to two resonant peaks) were obtained. Then, the inner-loop modal controller structure was designed according to S3-S5 of the first embodiment, and joint parameter optimization was performed based on the second and third embodiments. During the optimization process, the unconstrained objective function... The penalty coefficient in the middle is taken ,like Figure 4 As shown, the differential evolution algorithm converges after 400 iterations.

[0135] The optimal parameters obtained after optimization are: For the desired damping ratio and desired natural frequency, respectively Minimum objective function value The value is much less than 5000, indicating that all constraints are satisfied and there is no need to adjust the constraint threshold.

[0136] Substitute the optimized parameters into the system and draw the open-loop system. Inner loop closed-loop system and dual-loop closed-loop control system Bode plot. (Comparison) Figure 5 It can be seen that the original open-loop system has obvious resonance peaks at approximately 1050Hz and 1400Hz (amplitude amplification exceeding 10dB). However, after adding the designed modal controller, the resonance peaks of the inner-loop closed-loop system are completely suppressed, and the amplitude-frequency response is flat. Furthermore, after adding a PI tracking controller, the -3dB bandwidth of the dual-loop closed-loop control system reaches 932.6Hz, exceeding the first-order resonant frequency of the open-loop system (approximately 1050Hz), proving the effectiveness of this method in bandwidth enhancement.

[0137] In terms of time-domain performance, a step response test was conducted on the dual-loop closed-loop control system. The settling time was approximately 0.002 s with no overshoot, indicating that the system has a fast dynamic response and good damping characteristics.

[0138] To further quantify the tracking accuracy advantages of this invention, a comparative experiment was conducted between the method of this invention and a traditional two-degree-of-freedom control method. The input signals are respectively... Calculate the maximum tracking error percentage using triangular and sine waves. and root mean square tracking error percentage The experimental results are shown in Table 2 (triangular wave) and Table 3 (sine wave).

[0139] Table 2

[0140]

[0141] At the same time, such as Figure 7 As shown, the method of the present invention outperforms the comparative method in terms of tracking error of triangular wave signals at all frequencies, with a particularly significant improvement in the low-frequency band.

[0142] Table 3

[0143]

[0144] At the same time, such as Figure 6 As shown, the tracking error of the sine wave signal by the method of the present invention is significantly smaller than that of the two-degree-of-freedom control method, and the waveform fidelity is higher.

[0145] As can be seen from the data in Tables 2 and 3, the method of this invention outperforms the traditional two-degree-of-freedom control method under all test frequencies and waveform conditions. The reduction in tracking error is particularly significant in the low-frequency range (25Hz and 50Hz): the maximum reduction in tracking error reaches [value missing]. The root mean square tracking error was reduced by 100%. At high frequencies (100Hz), the method of this invention still maintains certain performance advantages, with a maximum reduction in tracking error of [missing value]. The root mean square tracking error was reduced by Experimental results show that the dual-loop control method proposed in this invention can effectively improve the tracking accuracy of the system, and is especially suitable for applications with high requirements for low-frequency signal tracking performance. The method of this invention exhibits smaller tracking errors and better waveform fidelity at all test frequencies.

[0146] This invention achieves joint optimization design of high-bandwidth controller parameters based on frequency domain response data, which can be effectively applied to nanoscale positioning platforms with complex high-order dynamic characteristics, achieving high closed-loop control bandwidth exceeding the system's first-order resonant frequency and excellent dynamic tracking performance. The results demonstrate that the method of this invention can efficiently generate high-precision lookup table files, significantly improving processing efficiency and system stability while ensuring nanoscale machining quality, exhibiting good engineering feasibility and promising prospects for industrial application.

[0147] Furthermore, although the steps of the method in this disclosure are described in a specific order in the accompanying drawings, this does not require or imply that the steps must be performed in that specific order, or that all the steps shown must be performed to achieve the desired result. Additional or alternative steps may be omitted, multiple steps may be combined into one step, and / or a step may be broken down into multiple steps.

[0148] From the above description of the embodiments, those skilled in the art will readily understand that the exemplary embodiments described herein can be implemented by software or by combining software with necessary hardware. Therefore, the technical solutions according to the embodiments of this disclosure can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (such as a CD-ROM, USB flash drive, external hard drive, etc.) or on a network, including several instructions to cause a computing device (such as a personal computer, server, mobile terminal, or network device, etc.) to execute the methods according to the embodiments of this disclosure.

[0149] In an exemplary embodiment of this disclosure, an electronic device capable of implementing the above-described method is also provided.

[0150] Those skilled in the art will understand that various aspects of the present invention can be implemented as systems, methods, or program products. Therefore, various aspects of the present invention can be specifically implemented in the following forms: entirely in hardware, entirely in software (including firmware, microcode, etc.), or in a combination of hardware and software, collectively referred to herein as “circuit,” “module,” or “system.”

[0151] An electronic device according to this embodiment of the invention. The electronic device is merely an example and should not be construed as limiting the functionality or scope of the embodiments of the invention.

[0152] Electronic devices are manifested in the form of general-purpose computing devices. Components of an electronic device may include, but are not limited to: at least one processor, at least one memory, and buses connecting different system components (including memory and processor).

[0153] The memory stores program code that can be executed by a processor, causing the processor to perform the steps described in the "Exemplary Methods" section above, according to various exemplary embodiments of the present invention.

[0154] The storage may include readable media in the form of volatile storage, such as random access memory (RAM) and / or cache memory, and may further include read-only memory (ROM).

[0155] The storage may also include programs / utilities having a set (at least one) of program modules, including but not limited to: an operating system, one or more applications, other program modules, and program data, each or some combination of these examples may include an implementation of a network environment.

[0156] A bus can represent one or more of several bus architectures, including a memory bus or memory controller, a peripheral bus, a graphics acceleration port, a processor, or a local bus that uses any of the various bus architectures.

[0157] The electronic device can also communicate with one or more external devices (e.g., keyboards, pointing devices, Bluetooth devices, etc.), one or more devices that enable a user to interact with the electronic device, and / or any device that enables the electronic device to communicate with one or more other computing devices (e.g., routers, modems, etc.). This communication can be performed via input / output (I / O) interfaces. Furthermore, the electronic device can communicate with one or more networks (e.g., local area networks (LANs), wide area networks (WANs), and / or public networks, such as the Internet) via a network adapter. The network adapter communicates with other modules of the electronic device via a bus. It should be understood that, although not shown in the figures, other hardware and / or software modules can be used in conjunction with the electronic device, including but not limited to: microcode, device drivers, redundant processors, external disk drive arrays, RAID systems, tape drives, and data backup storage systems.

[0158] In exemplary embodiments of this disclosure, a computer-readable storage medium is also provided, on which a program product capable of implementing the methods described above is stored. In some possible embodiments, various aspects of the present invention may also be implemented as a program product comprising program code that, when the program product is run on a terminal device, causes the terminal device to perform the steps of the various exemplary embodiments of the present invention described in the "Exemplary Methods" section above.

[0159] The program product may employ any combination of one or more readable media. A readable medium may be a readable signal medium or a readable storage medium. A readable storage medium may be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of readable storage media (a non-exhaustive list) include: electrical connections having one or more wires, portable disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof.

[0160] Computer-readable signal media may include data signals propagated in baseband or as part of a carrier wave, carrying readable program code. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. A readable signal medium may also be any readable medium other than a readable storage medium, capable of sending, propagating, or transmitting programs for use by or in conjunction with an instruction execution system, apparatus, or device.

[0161] The program code contained on the readable medium may be transmitted using any suitable medium, including but not limited to wireless, wired, optical fiber, RF, etc., or any suitable combination thereof.

[0162] Program code for performing the operations of this invention can be written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Java and C++, and conventional procedural programming languages ​​such as C or similar languages. The program code can execute entirely on the user's computing device, partially on the user's device, as a standalone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server. In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0163] Furthermore, the accompanying drawings are merely illustrative of the processes included in the method according to exemplary embodiments of the present invention and are not intended to be limiting. It is readily understood that the processes shown in the above drawings do not indicate or limit the temporal order of these processes. Additionally, it is readily understood that these processes may be executed synchronously or asynchronously, for example, in multiple modules.

[0164] It should be noted that although several modules or units for the device used to perform actions have been mentioned in the detailed description above, this division is not mandatory. In fact, according to embodiments of this disclosure, the features and functions of two or more modules or units described above can be embodied in one module or unit. Conversely, the features and functions of one module or unit described above can be further divided and embodied by multiple modules or units.

[0165] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for generating a modal controller for a nanometer positioning platform, characterized in that, Includes the following steps: Transfer function of the controlled object Perform zero-pole decomposition to isolate stable zeros. Unstable zeros Non-resonant poles and resonant poles ; Based on the decomposition results, construct the desired inner-loop closed-loop transfer function. And the modal controller is obtained by inverse solving. The structural form; reserve and and will Replace with a desired pole with high damping characteristics. At the same time, a new zero point is introduced. ; The following conditions must be met: ； The following conditions must be met: ; Introducing zero-pole cancellation constraints and order constraints, for Perform stability and feasibility design to obtain an implementable form. The zero-pole cancellation constraint is used to eliminate This leads to controller instability; the order constraint is used to ensure... Causal realizability; The implementable form satisfies the following conditions: ； right and Constraints with damping ratios greater than 0.1 are applied to output the final inner-loop modal controller.

2. The method according to claim 1, characterized in that, In the transfer function of the controlled object Before performing pole-zero decomposition, the following also includes: Obtain the open-loop frequency response data of the controlled object, and identify the transfer function of the controlled object based on the frequency response. .

3. The method according to claim 2, characterized in that, Obtain the open-loop frequency response data of the controlled object, and identify the transfer function of the controlled object based on the frequency response. ,include: By inputting band-limited white noise into the controlled object, the open-loop frequency response data of the controlled object within the Nyquist frequency range is obtained; Based on the loop frequency response data, the input and output data are fitted using a transfer function identification function, and the model order with the highest fit is selected as the discrete transfer function of the controlled object.

4. The method according to claim 1, characterized in that, The zero-pole decomposition includes: Through calculation The modulus of all zeros is used to classify zeros with a modulus less than 1 as... Zeros with a modulus greater than or equal to 1 are classified as ; Through calculation The damping ratio of all poles is used to classify complex poles with a damping ratio less than 0.1 as... Real poles and complex poles with a damping ratio greater than or equal to 0.1 are classified as... .

5. The method according to claim 1, characterized in that, The zero-pole cancellation constraint satisfies the following condition: ； The order constraint satisfies the following condition: 。 6. The method according to claim 1, characterized in that, The modal controller further includes: Based on the inner-loop modal controller, an outer-loop PI tracking controller is added, which together with the inner-loop modal controller constitutes a dual-loop closed-loop control system.

7. The method according to claim 6, characterized in that, The method further includes: The desired pole parameters of the inner loop modal controller and the parameters of the outer loop PI tracking controller are used together as optimization variables to establish an optimization problem with the objective function of minimizing the maximum deviation between the amplitude-frequency response and the 0dB line of the dual-loop closed-loop system in the target frequency band, and including multiple types of nonlinear constraints. The differential evolution algorithm is used to perform synchronous iterative optimization on the optimization variables to obtain the globally optimal parameter combination.

8. The method according to claim 7, characterized in that, The optimization variables include: The proportional gain of the outer loop PI tracking controller and integral gain And the damping ratio of the desired pole of the inner loop modal controller. and natural frequency ; The search range is set according to the maximum natural frequency of the open-loop system of the controlled object; The objective function The following conditions must be met: ； in, The frequency response of a dual-loop closed-loop system. The maximum natural frequency of the open-loop system. This is the bandwidth factor; The nonlinear constraints include: stability constraints, damping characteristic constraints, high-frequency amplitude constraints, and stability margin constraints based on the Nyquist stability circle criterion.

9. A non-transitory computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements a modal controller generation method for a nanopositioning platform as described in any one of claims 1 to 8.

10. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements a modal controller generation method for a nanopositioning platform as described in any one of claims 1 to 8.

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