PID parameter optimization method for reducing motion control position deviation uncertainty

Abstract

The application relates to a PID parameter optimization method for reducing motion control position deviation uncertainty, which comprises the following steps: firstly, determining the PID parameter optimization form and constraint conditions, and constructing an optimization objective function for the motion control position deviation uncertainty; secondly, establishing a Gaussian process regression proxy model between the PID parameters and the optimization objective function; thirdly, determining a next group of control parameters according to an expected enhancement acquisition function and measuring the corresponding optimization objective function value; finally, optimizing the PID parameters by using a double loop, wherein the inner loop uses randomly set initial control parameters to iteratively execute and obtain inner loop optimization control parameters, the outer loop is used for starting the inner loop optimization process multiple times, and finally, the final tuning parameter value is calculated by using non-convex scenario optimization; the selection of the optimization objective function of the application considers the average position deviation and the position deviation uncertainty, the optimization method adopts a double loop Bayesian optimization mode, and the method has the advantages of not being prone to local optimization and good consistency of optimization results under the action of uncertainty disturbance.

Description

Technical Field

[0001] This invention relates to the field of motion control technology for CNC machine tool feed axes, and in particular to a PID parameter optimization method for reducing the uncertainty of motion control position deviation. Background Technology

[0002] The feed system is a core component of CNC machine tools. During the machining process of parts, there are disturbances with random characteristics such as cutting force, motor harmonic force, and friction. These disturbances acting on the feed system will cause the position deviation of its motion control to also be random, thus affecting the machining accuracy and surface quality of the machined parts.

[0003] CNC machine tool feed systems often employ PID controllers, and the selection of their control parameters significantly impacts the motion control performance of the feed system, including its ability to suppress random disturbances. Currently, scholars both domestically and internationally have conducted extensive research on PID controller parameter tuning, resulting in various methods, including those based on predictive control, pole placement, and linear quadratic regulators. However, all these methods require an accurate mathematical model of the controlled object in the feed system. Model-based controller parameter tuning must consider the influence of uncertainties in the feed system's service state, thus exhibiting a degree of conservatism and limiting improvements in controller performance. Meanwhile, based on the frequency and time domain performance indicators during operation, the mode of adjusting or secondary optimization of controller parameters using data-driven methods has been widely applied and studied. However, such methods currently only focus on the field of deterministic motion accuracy optimization. At the same time, existing data-driven control parameter optimization methods, such as the cascade controller parameter tuning method disclosed in the paper "KHOSRAVI M, BEHRUNANI VN, MYSZKOROWSKI P, et al. Performance-Driven Cascade Controller Tuning With Bayesian Optimization[J / OL]. IEEE Transactions on Industrial Electronics,2022, 69(1): 1032-1042. DOI:10.1109 / TIE.2021.3050356.", adopt Bayesian optimization method, which is prone to getting trapped in local optima. Under the same set of initial control parameters, due to the many random components such as disturbance uncertainty and parameter uncertainty of the controlled object itself, the optimization results may be inconsistent. Summary of the Invention

[0004] To address the shortcomings of the existing technologies, this invention provides a PID parameter optimization method for reducing the uncertainty of motion control position deviation. The selection of the objective function takes into account both the average position deviation and the position deviation uncertainty. The optimization method adopts a dual-loop Bayesian optimization approach, which has the advantages of being less prone to getting trapped in local optima and having good consistency of optimization results under the influence of uncertainty disturbances.

[0005] To achieve the above objectives, the technical solution of the present invention is as follows:

[0006] A PID parameter optimization method for reducing the uncertainty of motion control position deviation includes the following steps:

[0007] S1. Based on the optimization form and constraints of the PID control parameters, construct an optimization objective function for the uncertainty of the motion control position deviation; whereby the PID control parameters include the proportional coefficient. Integral coefficient Differential coefficients The objective function is selected to optimize the control system when it runs according to the displacement command trajectory, and the weighted linear combination of the first norm of the motion control position deviation uncertainty and the first norm of the average position deviation is used.

[0008] S2. Based on the optimization objective function given in step S1, calculate the optimization objective function value corresponding to the initial PID parameter set from the measured position deviation data, and establish a Gaussian process regression surrogate model between the PID parameters and the optimization objective function.

[0009] S3. Based on the surrogate model established in step S2, determine the next set of control parameters by maximizing the acquisition function and experimentally measure the value of the optimization objective function under the control parameters. Select the desired enhancement function for the acquisition function.

[0010] S4. Optimize the PID parameters using a dual-loop system. The inner loop uses randomly set initial control parameters for Bayesian optimization. Steps S2 and S3 are executed iteratively to obtain the optimized control parameters for the inner loop. The outer loop involves repeatedly starting the inner loop optimization process. Finally, the final tuning parameter values ​​are obtained using non-convex scenario optimization calculation.

[0011] Step S1 specifically involves:

[0012] The optimized form of the PID control parameters is shown in formula (1):

[0013] (1)

[0014] in, To optimize the objective function, we express the mapping relationship between the uncertainty of the feed system position deviation, the average position deviation, and the combination of servo parameters. This is a vector of PID parameter combinations, i.e. , This represents the domain of all feasible PID parameter combinations constrained by system constraints. For PID parameter optimization results, Input constraint functions for the control system;

[0015] The objective function for optimization is constructed as shown in formula (2):

[0016] (2)

[0017] in, and These are the weighting coefficients; and These are the first-order norms of the average position deviation and the position deviation uncertainty, respectively, obtained by sampling the given PID parameters. The position deviation data of the lower control system when repeatedly running the displacement command trajectory is obtained and calculated. The calculation formula is shown in formula (3) and (4):

[0018] (3)

[0019] (4)

[0020] in, This represents the total number of sampling points for the position deviation during a single operation of the displacement command trajectory. , The first The average position deviation and position deviation uncertainty at each sampling time are calculated using formulas (5), (6), and (7):

[0021] (5)

[0022] (6)

[0023] (7)

[0024] in, , , The control system is the first The displacement command trajectory is repeated for the [number]th time. The actual position, commanded position, and position deviation at each sampling moment The number of times the displacement command trajectory is repeatedly run with the same control parameters.

[0025] Step S2 specifically involves: Recording the set of control parameters and corresponding objective function values ​​that have been experimentally measured as follows: ,in, The control parameters that have been experimentally measured are written in matrix form. , For control parameters The corresponding measured optimization objective function values ​​are written in matrix form. ;

[0026] Establish a Gaussian process regression surrogate model between the control parameters and the optimization objective function, denoted as [the control parameters to be predicted]. The corresponding predicted value of the optimization objective function is If the surrogate model predicts the objective function value, it follows a normal distribution, as shown in formula (8):

[0027] (8)

[0028] in, , Let be the mean and variance of the predicted distribution, respectively, and their calculation formulas are shown in formulas (9)(10)(11)(12)(13):

[0029] (9)

[0030] (10)

[0031] (11)

[0032] (12)

[0033] (13)

[0034] in, To optimize the variance of the measurement error of the objective function, the following approach is adopted. ; The covariance function of the Gaussian process is expressed as a squared exponential function. , For its parameters, use .

[0035] Step S3 specifically involves determining the next set of control parameters based on the agent model established in step S2 by maximizing the acquisition function. The specific process is expressed in the form of formula (14):

[0036] (14)

[0037] in, To obtain the function, we use the expected enhancement function and define its utility function. As shown in formula (15):

[0038] (15)

[0039] The formula for obtaining the function is shown in formula (16):

[0040]

[0041] (16)

[0042] in, , These are the probability distribution function and probability density function of the standard normal distribution, respectively.

[0043] The next set of control parameters is solved using a grid search algorithm. And the optimization objective function value corresponding to the next set of control parameters is measured experimentally. ,Will Add to collection middle.

[0044] Step S4 specifically involves optimizing the PID parameters using a dual-loop system. The inner loop uses randomly set initial control parameters for Bayesian optimization, iteratively executing steps S2 and S3 to obtain the optimized control parameters for the inner loop. The outer loop involves repeatedly initiating the inner loop optimization process, with the number of iterations for the traditional Bayesian optimization of the inner loop being [number missing]. The outer loop is randomly set with initial parameters, and the inner loop is started a certain number of times. The extended double-loop Bayesian optimization problem can be expressed in the form of formula (17):

[0045] (17)

[0046] in, This represents the scenario when the inner loop performs a traditional Bayesian optimization on the control system, i.e., the set of uncertainties in the control system; These are slack variables; For penalty weighting;

[0047] Solve the scenario optimization problem represented by formula (17) to obtain the final optimized controller parameters. The calculation formula is shown in formulas (18)(19)(20):

[0048] (18)

[0049] (19)

[0050] (20)

[0051] in, For the outer ring The first Bayesian optimization is initiated, and the inner-loop Bayesian optimization is performed. The next set of control parameters is determined by formula (14) in step S3 during the next iteration.

[0052] The present invention also includes:

[0053] A system includes a processor capable of running the PID parameter optimization method described above for reducing the uncertainty of motion control position deviation.

[0054] An apparatus comprising:

[0055] Memory: Used to store the computer program for the PID parameter optimization method for reducing the uncertainty of motion control position deviation;

[0056] Processor: Used to implement the PID parameter optimization method for reducing the uncertainty of motion control position deviation when executing the computer program.

[0057] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the PID parameter optimization method for reducing the uncertainty of motion control position deviation.

[0058] Compared with the prior art, the present invention has the following beneficial effects and innovative points:

[0059] (1) Using the weighted linear combination of motion control position deviation uncertainty and the first norm of average position deviation as the optimization objective function, the control parameters for position deviation uncertainty of motion control system can be optimized.

[0060] (2) Extending the traditional data-driven Bayesian optimization method into a dual-loop scenario optimization method can effectively improve the problem that the traditional Bayesian optimization method is prone to getting trapped in local optima, and that the optimization results may not be consistent under the same set of initial control parameters due to the random components such as disturbance uncertainty and parameter uncertainty of the controlled object itself.

[0061] In summary, this invention not only takes into account both the mean position deviation and uncertainty control of the motion control system in the design of the optimized objective function, but also introduces a dual-loop scenario optimization mechanism in the optimization method, which has higher robustness and practicality. Attached Figure Description

[0062] Figure 1 This is a flowchart of step four of the present invention.

[0063] Figure 2 This is a displacement command trajectory diagram of this embodiment.

[0064] Figure 3This is a comparison chart of the position deviation uncertainty and average position deviation when the PID parameters tuned according to the technical solution of this invention and the parameters tuned by a commercial CNC system are used to execute the displacement command of this embodiment. Figure 3 In the figure, (a) represents the average position deviation under different controller parameters; Figure 3 (b) represents the position deviation uncertainty under different controller parameters. Detailed Implementation

[0065] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0066] Taking a linear motor CNC machine tool feed system as an example, the motor back EMF constant is: The winding inductance is The resistance is The position feedback sensor has a resolution of The grating ruler.

[0067] The present invention provides a PID parameter optimization method for reducing the uncertainty of motion control position deviation, comprising the following steps:

[0068] S1. Based on the optimization form and constraints of the PID control parameters, construct an optimization objective function for the uncertainty of the motion control position deviation; whereby the PID control parameters include the proportional coefficient. Integral coefficient Differential coefficients The objective function is selected to optimize the control system when it runs according to the displacement command trajectory, and the weighted linear combination of the first norm of the motion control position deviation uncertainty and the first norm of the average position deviation is used.

[0069] Further, step S1 specifically involves establishing the PID parameter optimization form as shown in formula (1):

[0070] (1)

[0071] in, To optimize the objective function, we express the mapping relationship between the uncertainty of the feed system position deviation, the average position deviation, and the combination of servo parameters. This is a vector of PID parameter combinations, i.e. , Based on the system manual and actual field tests, the domain of all feasible PID parameter combinations constrained by system constraints is determined as follows: , For PID parameter optimization results, To provide the control system with input constraint functions, in this embodiment, the maximum thrust current of the linear motor is: The constraint function is ,in This is the maximum thrust current of the linear motor feed system when it runs along the target trajectory.

[0072] Furthermore, in step S1, the objective function is selected as a weighted linear combination of the positional deviation uncertainty and the first norm of the average positional deviation. Therefore, the objective function is constructed as shown in formula (2):

[0073] (2)

[0074] and These are weighting coefficients, which can be adjusted according to actual needs; here, we choose weights w1=1 and w2=100, therefore, it is expressed as:

[0075] (2)

[0076] in, and These are the first-order norms of the average position deviation and the position deviation uncertainty, respectively, obtained by sampling the given PID parameters. The position deviation data of the lower control system repeatedly running the displacement command trajectory is obtained and calculated. In this embodiment, the displacement command trajectory is an S-shaped acceleration / deceleration curve displacement trajectory, such as... Figure 2 As shown. and The calculation formulas are shown in formulas (3) and (4):

[0077] (3)

[0078] (4)

[0079] in, This represents the total number of sampling points for the positional deviation during a single run of the target trajectory. , The first The average position deviation and position deviation uncertainty at each sampling time are calculated using formulas (5), (6), and (7):

[0080] (5)

[0081] (6)

[0082] (7)

[0083] in, , , The control system is the first The displacement command trajectory is repeated for the [number]th time. The actual position, commanded position, and position deviation at each sampling moment The number of times the command trajectory is repeatedly run with the same control parameters is taken in this embodiment. .

[0084] S2. Based on the optimization objective function given in step S1, calculate the optimization objective function value corresponding to the initial PID parameter set from the measured position deviation data, establish a Gaussian process regression surrogate model between the PID parameters and the optimization objective function, and realize the prediction of the optimization objective function of the control system under unknown control parameters.

[0085] Further, step S2 specifically involves: denoting the set of control parameters and corresponding optimized objective function values ​​that have been experimentally measured as follows: ,in Write the control parameters that have been measured experimentally in matrix form. , For control parameters The corresponding measured optimization objective function values ​​are written in matrix form. ;

[0086] Furthermore, a Gaussian process regression surrogate model is established between the known control parameters and the optimization objective function, denoted as [the control parameters to be predicted]. The corresponding predicted value of the optimization objective function is If the surrogate model predicts the objective function value, it follows a normal distribution, as shown in formula (8):

[0087] (8)

[0088] in, , Let be the mean and variance of the predicted distribution, respectively, and their calculation formulas are shown in formulas (9), (10), (11), (12), and (13):

[0089] (9)

[0090] (10)

[0091] (11)

[0092] (12)

[0093] (13)

[0094] in, To optimize the variance of the measurement error of the objective function, the following approach is adopted. ; The covariance function of the Gaussian process is expressed as a squared exponential function. , For its parameters, use .

[0095] S3. Based on the surrogate model established in step S2, determine the next set of control parameters by maximizing the acquisition function and experimentally measure the value of the optimization objective function under the control parameters. Select the desired enhancement function for the acquisition function to achieve a balance between exploration and utilization within the feasible region of the parameters and improve optimization efficiency.

[0096] Furthermore, step S3 specifically involves determining the next set of control parameters based on the surrogate model established in step S2 by maximizing the acquisition function. The specific process is expressed in the form of formula (14):

[0097] (14)

[0098] in, To obtain the function, we use the expected enhancement function and define its utility function. As shown in formula (15):

[0099] (15)

[0100] The formula for obtaining the function is shown in formula (16):

[0101]

[0102] (16)

[0103] in, , These are the probability distribution function and probability density function of the standard normal distribution, respectively.

[0104] Furthermore, a grid search algorithm is used to solve for the next set of control parameters. And the target optimization function value corresponding to the next set of control parameters is measured experimentally. ,Will Add to collection middle.

[0105] S4. Considering that traditional Bayesian optimization methods are prone to getting trapped in local optima, and that the controlled object in the control system itself has many random components such as disturbance uncertainty and parameter uncertainty, the results of traditional Bayesian optimization under the same set of initial control parameters may also be inconsistent. Therefore, this invention uses a dual-loop to optimize PID parameters. The inner loop uses randomly set initial control parameters for Bayesian optimization and iteratively executes steps S2 and S3 to obtain the inner loop optimized control parameters. The outer loop involves repeatedly starting the inner loop optimization process. Finally, the final tuning parameter values ​​are obtained by using non-convex scenario optimization calculation.

[0106] Furthermore, the process is as follows: Figure 1 As shown, step S4 specifically involves extending the traditional Bayesian optimization method into a dual-loop optimization method. The inner loop uses randomly set initial control parameters for Bayesian optimization, iteratively executing steps S2 and S3 to obtain the inner loop optimization control parameters. The outer loop uses a multi-loop Bayesian optimization algorithm. Let the number of iterations for the traditional Bayesian optimization of the inner loop be... The outer loop is randomly set with initial parameters to initiate Bayesian optimization for a certain number of iterations. In this embodiment, the number of iterations for the inner-loop Bayesian optimization is selected as [number]. The number of Bayesian optimizations initiated on the outer loop is The extended double-loop Bayesian optimization problem can be expressed in the form of formula (17):

[0107] (17)

[0108] in, This represents the scenario when the inner loop performs a traditional Bayesian optimization on the control system, i.e., the set of uncertainties in the control system; These are slack variables; For penalty weights.

[0109] Furthermore, by solving the scenario optimization problem represented by formula (17), the final optimized controller parameters are obtained. The calculation formula is shown in formulas (18)(19)(20):

[0110] (18)

[0111] (19)

[0112] (20)

[0113] in, For the outer ring The first Bayesian optimization is initiated, and the inner-loop Bayesian optimization is performed. The next set of control parameters is determined by formula (14) in step S3 during the next iteration.

[0114] In this embodiment, the optimized PID control parameters obtained according to the technical solution described in this invention are: The results were compared with the PID parameter tuning function of a commercial CNC system. The tuning result of the commercial CNC system for the PID parameters of the linear motor feed system in this embodiment is as follows: Using this set of parameters as the baseline parameters, the average position deviation and position deviation uncertainty of the two sets of control parameters under the command displacement in this embodiment were experimentally measured. Figure 3 As shown, compared to the reference controller parameters, the parameters tuned according to the technical solution of this invention improve the average position deviation and position deviation uncertainty of the feed system. The peak value of the position deviation uncertainty of the feed system is reduced by... Reduce to During the acceleration and deceleration phases, the average position deviation from Reduce to The maximum value of the average position deviation can be improved by about 50%, indicating that the PID parameter optimization technology proposed in this invention can effectively improve the uncertainty of system position deviation and average position deviation.

[0115] The present invention also includes:

[0116] A system includes a processor capable of running the PID parameter optimization method described above for reducing the uncertainty of motion control position deviation.

[0117] An apparatus comprising:

[0118] Memory: Used to store the computer program for the PID parameter optimization method for reducing the uncertainty of motion control position deviation;

[0119] Processor: Used to implement the PID parameter optimization method for reducing the uncertainty of motion control position deviation when executing the computer program.

[0120] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the PID parameter optimization method for reducing the uncertainty of motion control position deviation.

Claims

1. A PID parameter optimization method for reducing the uncertainty of motion control position deviation, characterized in that, Includes the following steps: S1. Based on the optimization form and constraints of the PID control parameters, construct an optimization objective function for the uncertainty of the motion control position deviation; whereby the PID control parameters include the proportional coefficient. Integral coefficient Differential coefficients The objective function is selected to optimize the control system when it runs according to the displacement command trajectory, and the weighted linear combination of the first norm of the motion control position deviation uncertainty and the first norm of the average position deviation is used. S2. Based on the optimization objective function given in step S1, calculate the optimization objective function value corresponding to the initial PID parameter set from the measured position deviation data, and establish a Gaussian process regression surrogate model between the PID parameters and the optimization objective function. S3. Based on the surrogate model established in step S2, determine the next set of control parameters by maximizing the acquisition function and experimentally measure the value of the optimization objective function under the control parameters. Select the desired enhancement function for the acquisition function. S4. Optimize the PID parameters using a dual-loop system. The inner loop uses randomly set initial control parameters for Bayesian optimization. Steps S2 and S3 are executed iteratively to obtain the optimized control parameters for the inner loop. The outer loop involves repeatedly starting the inner loop optimization process. Finally, the final tuning parameter values ​​are obtained using non-convex scenario optimization calculation.

2. The PID parameter optimization method for reducing the uncertainty of motion control position deviation according to claim 1, characterized in that, Step S1 specifically involves: The optimized form of the PID control parameters is shown in formula (1): （1） in, To optimize the objective function, we express the mapping relationship between the uncertainty of the feed system position deviation, the average position deviation, and the combination of servo parameters. This is a vector of PID parameter combinations, i.e. , This represents the domain of all feasible PID parameter combinations constrained by system constraints. For PID parameter optimization results, Input constraint functions for the control system; The objective function for optimization is constructed as shown in formula (2): （2） in, and These are the weighting coefficients; and These are the first-order norms of the average position deviation and the position deviation uncertainty, respectively, obtained by sampling the given PID parameters. The position deviation data of the lower control system when repeatedly running the displacement command trajectory is obtained and calculated. The calculation formula is shown in formula (3) and (4): （3） （4） in, This represents the total number of sampling points for the position deviation during a single operation of the displacement command trajectory. , The first The average position deviation and position deviation uncertainty at each sampling time are calculated using formulas (5), (6), and (7): （5） （6） （7） in, , , The control system is the first The displacement command trajectory is repeated for the [number]th time. The actual position, commanded position, and position deviation at each sampling moment The number of times the displacement command trajectory is repeatedly run with the same control parameters.

3. The PID parameter optimization method for reducing the uncertainty of motion control position deviation according to claim 1, characterized in that, Step S2 specifically involves: Recording the set of control parameters and corresponding objective function values ​​that have been experimentally measured as follows: ,in The control parameters that have been experimentally measured are written in matrix form. , For control parameters The corresponding measured optimization objective function values ​​are written in matrix form. ; Establish a Gaussian process regression surrogate model between the control parameters and the optimization objective function, denoted as [the control parameters to be predicted]. The corresponding predicted value of the optimization objective function is If the surrogate model predicts the objective function value, it follows a normal distribution, as shown in formula (8): （8） in, , Let be the mean and variance of the predicted distribution, respectively, and their calculation formulas are shown in formulas (9)(10)(11)(12)(13): （9） （10） （11） （12） （13） in, To optimize the variance of the measurement error of the objective function, the following approach is adopted. ; The covariance function of the Gaussian process is expressed as a squared exponential function. , For its parameters, use .

4. The PID parameter optimization method for reducing the uncertainty of motion control position deviation according to claim 1, characterized in that, Step S3 specifically involves determining the next set of control parameters based on the agent model established in step S2 by maximizing the acquisition function. The specific process is expressed in the form of formula (14): （14） in, To obtain the function, we use the expected enhancement function and define its utility function. As shown in formula (15): （15） The formula for obtaining the function is shown in formula (16): （16） in, , These are the probability distribution function and probability density function of the standard normal distribution, respectively. The next set of control parameters is solved using a grid search algorithm. And the optimization objective function value corresponding to the next set of control parameters is measured experimentally. ,Will Add to collection middle.

5. The PID parameter optimization method for reducing the uncertainty of motion control position deviation according to claim 1, characterized in that, Step S4 specifically involves optimizing the PID parameters using a dual-loop system. The inner loop uses randomly set initial control parameters for Bayesian optimization, iteratively executing steps S2 and S3 to obtain the optimized control parameters for the inner loop. The outer loop involves repeatedly initiating the inner loop optimization process, with the number of iterations for the traditional Bayesian optimization of the inner loop being [number missing]. The outer loop is randomly set with initial parameters, and the inner loop is started a certain number of times. The extended double-loop Bayesian optimization problem can be expressed in the form of formula (17): （17） in, This represents the scenario when the inner loop performs a traditional Bayesian optimization on the control system, i.e., the set of uncertainties in the control system; These are slack variables; For penalty weighting; Solve the scenario optimization problem represented by formula (17) to obtain the final optimized controller parameters. The calculation formula is shown in formulas (18)(19)(20): （18） （19） （20） in, For the outer ring The first Bayesian optimization is initiated, and the inner-loop Bayesian optimization is performed. The next set of control parameters is determined by formula (14) in step S3 during the next iteration.

6. A system, characterized in that, Includes a processor capable of running the PID parameter optimization method for reducing motion control position deviation uncertainty as described in any one of claims 1-5.

7. A device, characterized in that, include: Memory: for storing the computer program of the PID parameter optimization method for reducing the uncertainty of motion control position deviation as described in any one of claims 1-5; Processor: Used to implement the PID parameter optimization method for reducing the uncertainty of motion control position deviation as described in any one of claims 1-5 when executing the computer program.

8. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the PID parameter optimization method for reducing the uncertainty of motion control position deviation as described in any one of claims 1-5.

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