Control parameter online calibration device and method of engine control system

Abstract

The invention discloses a control parameter online calibration device and method of an engine control system, and relates to the technical field of engine control. The device is characterized in that an upper computer is connected with a controller through a communication module; the controller is respectively connected with the direct current motor driving module and the electronic throttle body; the direct current motor driving module is connected with the electronic throttle body; the upper computer is used for monitoring the state of each input pin and each output pin of the controller, receiving calibration index measurement parameters, calculating calibration sampling point data by applying a control parameter online calibration method of an engine control system, and sending the calibration sampling point data to the controller; the controller is used for transmitting motor operation parameters to the direct current motor driving module by applying a PID algorithm according to the calibrated sampling point data, and receiving a throttle valve opening signal returned by the electronic throttle valve body; the direct-current motor driving module is used for controlling the electronic throttle body to work according to the motor operation parameters. The online calibration period of the control parameters of the engine control system is shortened.

Description

Technical Field

[0001] The present invention relates to the technical field of engine control, and in particular to an online calibration device and method for control parameters of an engine control system. Background Art

[0002] The engine control system has strong nonlinearity, strong coupling, strong dynamics and other characteristics, so it cannot be described by a simple mathematical expression, which makes it difficult to optimize the control parameters. The engine control parameter calibration problem mainly requires optimizing the control parameters of the controller of the subsystem based on the control requirements of the relevant engine control subsystem, combining experimental data with statistical optimization methods. In addition, according to statistical research on the application of controllers in industry, PID is used as a controller in about 90% of products. Therefore, the discussion of engine control parameter calibration in this invention mainly focuses on the situation when PID is used as the controller of the engine control system subsystem.

[0003] Traditional control parameter calibration methods rely on a comprehensive analysis of the interactions between control parameters under different working conditions, and repeated debugging tests under multiple working conditions to determine the optimal control parameter settings that can meet specific requirements. However, this method faces significant implementation difficulties. First, the constant changes in test conditions make it difficult to accurately calibrate the calibration process. Common test bench systems are difficult to simulate real dynamic working conditions, resulting in deviations between test results and actual application scenarios. Secondly, frequent dynamic tests not only affect the maintenance and life of the bench equipment itself, but also greatly reduce the repeatability of the calibration process due to the complexity of the operation. This control parameter tuning method is based on some assumptions about the control object model and the required output. The system characteristics are extracted by analysis or graphics to select the controller settings. Since it is a control parameter calibration based on assumptions, further adjustments are almost required in actual use, and the control effect of the tuned control parameters is poor when the system characteristics do not meet the relevant assumptions.

[0004] The online calibration methods of control parameters are divided into model-based online calibration methods and non-model-based online calibration methods. Non-model-based online calibration methods, that is, in the process of control parameter calibration, do not require real-time modeling of the control object. This type of online calibration method generally has the characteristics of long parameter calibration cycle and poor performance when the working conditions and environment change greatly.

[0005] At present, patent literature on online automatic calibration systems for engine control parameters generally focuses on the structure and function of the automatic calibration system, including key components such as the control unit, acquisition unit and optimization unit. The main function of the control unit is to output the control parameters of the actuator to adjust the operating state of the engine. The acquisition unit is responsible for monitoring and recording the performance response of the engine, such as key indicators such as torque, fuel consumption rate and emissions. In addition, the function of the optimization unit is to analyze and compare the performance of the engine under different control parameter configurations, so as to identify and determine the optimal control parameter configuration. The design concept of this automatic calibration system is based on the need to improve engine performance and efficiency. By integrating control, acquisition and optimization functions, the system can achieve instant adjustment and optimization of engine parameters, which is particularly important for meeting increasingly stringent fuel efficiency and emission standards. The application of automatic calibration technology not only improves the accuracy and efficiency of the engine adjustment process, but also paves the way for further intelligence and automation of engine management systems. These patents demonstrate the application prospects and potential of automatic calibration technology in modern engine technology, and emphasize the core role of comprehensive automation systems in improving engine performance. However, the calibration cycle required is long and there are many sampling points, so it is difficult to meet the rapidity requirements of online calibration.

[0006] In modern engineering applications, especially in the online calibration process of engine control systems, optimization algorithms play a core role. The two main commonly used optimization algorithms include meta-heuristic algorithms and extreme value search algorithms. These algorithms have their own advantages and disadvantages and are suitable for different application scenarios and requirements.

[0007] Metaheuristic algorithms, such as genetic algorithms, particle swarm optimization (PSO) and simulated annealing, are a class of algorithms that seek optimal solutions by imitating natural processes. The main advantage of these algorithms is that they have strong global search capabilities and can effectively jump out of local optimal solutions and find global optimal solutions. This makes metaheuristic algorithms particularly suitable for solving complex optimization problems with large problem spaces and many local minima. However, this type of algorithm also has some disadvantages, such as high computational cost, slow convergence speed, and complex algorithm parameter settings that require a lot of experiments to determine.

[0008] Extreme value search algorithms, including golden section search and binary search, are methods of locating the optimal solution by gradually narrowing the search range. The advantages of this type of algorithm are simple implementation and fast convergence speed, especially when the objective function is unimodal and continuous, it can quickly and effectively find the extreme point. However, the main disadvantage of extreme value search algorithms is that they rely on the unimodality of the problem, perform poorly for multi-peak problems, and are prone to fall into local optimal solutions.

[0009] In practical applications, when the above algorithm solves the non-convex optimization calibration problem of control parameters, there are disadvantages such as long solution time, many required sampling points, and inability to find the global optimal point. Summary of the Invention

[0010] The object of the present invention is to provide an on-line calibration device and method for control parameters of an engine control system, which can shorten the on-line calibration cycle of control parameters.

[0011] To achieve the above object, the present invention provides the following solutions:

[0012] An on-line calibration device for control parameters of an engine control system, the on-line calibration device includes: an electronic throttle body, a DC motor drive module, a controller, a communication module, and a host computer;

[0013] The host computer is connected to the controller through the communication module; the controller is respectively connected to the DC motor drive module and the electronic throttle body; the DC motor drive module is connected to the electronic throttle body;

[0014] The host computer is used to monitor the states of each input and output pin of the controller, and receive calibration index measurement parameters, and calculate calibration sampling point data according to the calibration index measurement parameters by using the on-line calibration method for control parameters of the engine control system, and send the calibration sampling point data to the controller; the calibration index measurement parameters are calculated by the controller according to the throttle opening signal of the electronic throttle body; the on-line calibration method for control parameters of the engine control system is a grid adaptive Bayesian on-line calibration algorithm;

[0015] The controller is used to send motor operation parameters to the DC motor drive module according to the calibration sampling point data by using the PID algorithm, and receive the throttle opening signal returned by the electronic throttle body; the motor operation parameters include motor speed and motor rotation direction;

[0016] The DC motor drive module is used to control the operation of the electronic throttle body according to the motor operation parameters.

[0017] Optionally, the host computer includes an edge computing device and a controller host computer;

[0018] Both the edge computing device and the controller host computer are connected to the controller;

[0019] The edge computing device is used to receive calibration index measurement parameters, and calculate calibration sampling point data according to the calibration index measurement parameters by using the grid adaptive Bayesian on-line calibration algorithm, and send the calibration sampling point data to the controller;

[0020] The controller host computer is used to monitor the status of each input and output pin of the controller.

[0021] An online calibration method for control parameters of an engine control system is applied to the above-mentioned online calibration device for control parameters of the engine control system. The online calibration method comprises:

[0022] Setting the feasible domain of the control parameters and the initial sampling point values ​​of the control parameters; the control parameters are the proportional parameters, differential parameters and integral parameters of the PID algorithm;

[0023] According to the mean function, the covariance function and the initial sampling point value of the control parameter, the prior Gaussian process model and the Bayesian sampling strategy are applied to determine the next sampling point value of the control parameter;

[0024] According to the value of the next sampling point of the control parameter, the posterior Gaussian process model is applied to determine the value of the objective function; the objective function is the absolute value integral of the difference between the throttle opening of the electronic throttle body under the action of the control parameter and the target opening;

[0025] Determining whether the value of the objective function satisfies a preset convergence condition;

[0026] When the value of the objective function does not meet the preset convergence condition, the posterior Gaussian process model is updated by applying a method based on maximizing the logarithmic marginal probability, and the next sampling point value of the control parameter is input into the updated posterior Gaussian process model to obtain the updated value of the objective function, and return to the step of "determining whether the value of the objective function meets the preset convergence condition";

[0027] When the value of the objective function meets the preset convergence condition, the value of the next sampling point of the control parameter is used as the online calibration result.

[0028] Optionally, the mean function is μ(x; c)≡c;

[0029] Among them, x is the calibration parameter, that is, the three parameters of PID; c is a constant.

[0030] Optionally, the covariance function is:

[0031]

[0032] Among them, cov f is the covariance of different calibration parameters, x i 、x j are the i-th and j-th components of the calibration parameters, l 1 , ..., l D is the kernel length along each dimension, α is the shape parameter; D represents the data dimension; is the variance of the calibration parameter.

[0033] Optionally, the objective function is:

[0034]

[0035] Among them, μ(x) is the mean function, κ(x, x′) is the covariance kernel function, f(x) represents the objective function value, x and x′ are two sets of calibration parameter data points, E represents the mean operation, R p Represents the p-dimensional real number space.

[0036] Optionally, the updated posterior Gaussian process model is:

[0037]

[0038] Among them, x * ,f(x * ) are the calibration parameter data value and the corresponding objective function value of the next sampling point, μ * and κ * are the updated mean function matrix and covariance kernel function matrix respectively, X, Y are the calibration parameters of the previous sampling point and the obtained objective function data, μ(x * ) is the mean value of the calibration parameters of the next sampling point.

[0039] Optionally, the Bayesian sampling strategy is obtained by optimizing the acquisition function using a grid adaptive search algorithm.

[0040] Optionally, the sampling function is an acquisition function of a lower confidence bound.

[0041] Optionally, the preset convergence condition is Any one of;

[0042] Among them, Δ mesh Represents the actual mesh size tolerance, M size is the mesh size tolerance value, which determines the accuracy of the calibration parameters; n and n+1 represents the objective function of the nth and n+1th evaluations, F c is the minimum improvement of the objective function, which depends on the accuracy requirement of the calibration index; T no is the actual number of iterations without significant changes in the objective function value, I s is the maximum number of iterations without significant changes in the objective function value; F eval and T are the actual number of objective function evaluations and the actual number of algorithm iterations, respectively. max with I max are the maximum number of evaluations of the objective function and the maximum number of iterations of the algorithm; Cv The threshold value is required for calibration.

[0043] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0044] The present invention provides an online calibration device and method for control parameters of an engine control system. The present invention is an automatic online calibration device for control parameters of an electronic throttle subsystem in an engine control system. The sampling process of online calibration of electronic throttle control parameters is guided by Bayesian inference and Bayesian decision theory. The control parameters are continuously updated by Bayesian optimization algorithm in a fixed throttle following condition, and the calibration objective function is solved to realize automatic online calibration of control parameters. The present invention uses Bayesian optimization algorithm in the online calibration process of control parameters, uses grid adaptive algorithm to optimize sampling function to solve non-convex optimization problems in the calibration process, and uses Gaussian process model as a proxy model, which greatly shortens the online calibration cycle of control parameters and reduces the sampling points required for calibration. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0046] Figure 1 A schematic diagram of the overall scheme for physical test verification provided in Example 1 of the present invention;

[0047] Figure 2 This is a flow chart of the grid adaptive optimization algorithm;

[0048] Figure 3 It is a schematic diagram of the online calibration algorithm simulation verification process;

[0049] Figure 4 This is a schematic diagram of the Bayesian optimization process;

[0050] Figure 5 This is a schematic diagram of the grid adaptation algorithm flow;

[0051] Figure 6 Schematic diagram of the main program flow of the electronic throttle test platform control;

[0052] Figure 7 A schematic diagram of the process of online calibration of control parameters using the online calibration method provided by the present invention. DETAILED DESCRIPTION

[0053] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0054] The object of the present invention is to provide an online calibration device and method for control parameters of an engine control system, aiming to shorten the online calibration period of the control parameters.

[0055] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0056] Example 1

[0057] like Figure 1 As shown, the control parameter online calibration device of the engine control system in this embodiment includes: an electronic throttle body, a DC motor drive module, a controller, a communication module and a host computer.

[0058] The host computer is connected to the controller via the communication module; the controller is connected to the DC motor drive module and the electronic throttle body respectively; the DC motor drive module is connected to the electronic throttle body.

[0059] The host computer is used to monitor the status of each input and output pin of the controller, and to receive calibration index measurement parameters, and to calculate calibration sampling point data based on the calibration index measurement parameters by applying the online calibration method of the control parameters of the engine control system, and to send the calibration sampling point data to the controller; the calibration index measurement parameters are calculated by the controller based on the throttle opening signal of the electronic throttle body; the online calibration method of the control parameters of the engine control system is a grid adaptive Bayesian online calibration algorithm.

[0060] The controller is used to apply the PID algorithm to send motor operating parameters to the DC motor drive module according to the calibration sampling point data, and receive the throttle opening signal returned by the electronic throttle body; the motor operating parameters include motor speed and motor rotation direction.

[0061] The DC motor driving module is used to control the operation of the electronic throttle body according to the motor operating parameters.

[0062] As a specific implementation, the host computer includes an edge computing device and a controller host computer.

[0063] The edge computing device and the controller host computer are both connected to the controller.

[0064] The edge computing device is used to receive calibration index measurement parameters, and calculate calibration sampling point data using a grid adaptive Bayesian online calibration algorithm according to the calibration index measurement parameters, and send the calibration sampling point data to the controller.

[0065] The controller host computer is used to monitor the status of each input and output pin of the controller.

[0066] In practical applications, the present invention relates to the technical field of online calibration of control parameters of an engine control system, and in particular to an online calibration device for control parameters based on grid adaptive Bayesian optimization. The present invention uses a grid adaptive Bayesian optimization algorithm that incorporates an extended Kalman filter as an online calibration algorithm for an online calibration device for control parameters of an engine control system, and the device includes an electronic throttle body, an L298N DC motor driver board, a MircoAutoBox II controller, an engine controller host computer, a VectorVN1630ACAN communication device, a PC or an edge computing device.

[0067] Among them, the computer algorithm software platform MATLAB / SIMULINK runs on a PC or edge computing device, and transmits the calibration index calculation related parameters with the MicroAutoBox II controller through CAN communication technology (the message IDs are set to 90 and 91 respectively). At the same time, the latest sampling points are sent through the grid adaptive Bayesian online calibration algorithm to control the electronic throttle system in the next cycle and collect its calibration indexes. The MicroAutoBox II platform outputs the control signal of the H-bridge drive circuit by running discrete PID control, and collects the throttle body sensor signal.

[0068] The electronic throttle body includes a throttle valve plate, a DC drive motor, a return spring and a Hall position sensor; the L298N driver board is composed of four metal oxide semiconductor field effect transistors (MOSFETs), which are divided into two groups to control a direction of the motor respectively, and the forward and reverse rotation of the motor is realized by controlling the conduction of different combinations of MOS tubes. At the same time, the H-bridge PWM drive circuit also allows the speed of the motor to be controlled by adjusting the duty cycle of the PWM (pulse width modulation) signal; the MircoAutoBox II controller, as the controller of the electronic throttle, collects the throttle opening and calculates the control signal, and communicates with the computer algorithm software platform (the PC or edge computing device communicates via the CAN bus); the engine controller host computer is used to monitor the status of each input and output pin of the MircoAutoBox II controller in real time and observe the CAN communication status and the control effect of the current iteration of the algorithm; the Vector VN1630A CAN communication device is used to transmit the sampling points of the grid adaptive Bayesian optimization algorithm on the edge computing device or PC to the MircoAutoBox II controller, and send the calibration index calculation related signals from the MircoAutoBox II controller to the PC. Technical solutions such as Figure 1 shown.

[0069] In order to make the purpose, technical solutions and advantages of the present invention clearer, the following will be combined with Figure 1 The embodiments of the present invention are described in further detail.

[0070] The main process of this example is to give the initial throttle controller control parameters in the MATLAB / SIMULINK software on the PC or edge computing device, and the host computer detects the control effect; at this time, the parameters related to the calibration index calculation are transmitted to the PC or edge computing device through VectorVN1630A, and the grid adaptive Bayesian optimization algorithm is run on the PC or edge computing device to select the next set of control parameter sampling points, and the control parameters are updated through CAN communication. At the same time, the corresponding control index calculation parameters required for the next iterative optimization are collected (the message IDs are set to 90 and 91 respectively). After several rounds of optimization, the best PID parameters suitable for throttle control can be found. The best PID parameters will be sent to the electronic throttle controller and saved in the controller for subsequent throttle control.

[0071] Example 2

[0072] The present invention further provides an online calibration method for control parameters of an engine control system, which is applied to the online calibration device for control parameters of an engine control system described in Embodiment 1. The online calibration method comprises:

[0073] Step S1: Setting the feasible domain of control parameters and the initial sampling point values ​​of the control parameters; the control parameters are the proportional parameters, differential parameters and integral parameters of the PID algorithm.

[0074] Step S2: According to the mean function, the covariance function and the initial sampling point value of the control parameter, the a priori Gaussian process model and the Bayesian sampling strategy are applied to determine the next sampling point value of the control parameter. The Bayesian sampling strategy is obtained by optimizing the acquisition function using a grid adaptive search algorithm.

[0075] Step S3: According to the next sampling point value of the control parameter, the posterior Gaussian process model is applied to determine the value of the objective function; the objective function is the absolute value integral of the difference between the throttle opening of the electronic throttle body under the control parameter and the target opening.

[0076] Step S4: Determine whether the value of the objective function satisfies a preset convergence condition.

[0077] Step S5: When the value of the objective function does not meet the preset convergence condition, the posterior Gaussian process model is updated by applying a method based on maximizing the logarithmic marginal probability, and the next sampling point value of the control parameter is input into the updated posterior Gaussian process model to obtain the updated value of the objective function, and return to step S4.

[0078] Step S6: When the value of the objective function satisfies a preset convergence condition, the value of the next sampling point of the control parameter is used as an online calibration result.

[0079] In the method of this embodiment, first, the corresponding parameters required for the calculation of the control index are collected, and the calibration index, that is, the size of the objective function, is solved. Then, the sampling function is designed based on the Bayesian sampling strategy, and the grid adaptive search algorithm is used to search the optimal value of the sampling function to determine the next set of optimal control parameter sampling points. The iteration is repeated until the improvement of the calibration index is not obviously reflected in the actual control effect or when other algorithm convergence conditions are reached.

[0080] The basic operation process of the grid adaptive Bayesian optimization algorithm is as follows: Figure 2 As shown in the figure, for the online calibration of electronic throttle control parameters, after designing the discrete PID controller, the optimal sampling point design is performed for the controller control parameters to be optimized (proportional gain, integral gain, differential gain), and iterates continuously until the optimal parameter combination is found. The specific implementation steps are as follows:

[0081] Step 1: Determine the feasible domain [LB, UB] of the parameters to be calibrated and give the initial calibration point x 0 And initialize the algorithm.

[0082] Step 2: Train the prior Gaussian process model based on the mean function μ(x) and the covariance function κ(x, x′) combined with the initial point and the random initial sampling point information generated by the random sequence.

[0083] Step 3: Based on the probability information of the Gaussian process model, optimize the sampling strategy to obtain the next suitable sampling point x n+1 .

[0084] Step 4: For the new sampling point x n+1 Sampling is performed to obtain the objective function value y n+1 .

[0085] Step 5: Determine whether the algorithm convergence conditions are met. If so, the parameter values ​​and objective function values ​​corresponding to the incumbent points are output as the online calibration results; if not, the posterior Gaussian process model is updated using the updated data set D based on the method of maximizing the logarithmic marginal probability and the third step is executed.

[0086] In practical applications, taking the PID parameters of throttle control as an example, Figure 3 and Figure 4 As shown, the Bayesian optimization process is as follows:

[0087] Step 1: Use the three initial parameters of PID, Kp, Ki and Kd, as the control parameter values ​​for throttle control. The tracking signal used for calibration test is a 4s sinusoidal signal in the form of 1.474sin(2πt)+0.096. The measurement parameter is the throttle position. Then, the objective function value is counted. The objective function is set as the absolute value integral of the error (IAE). The controller controls the throttle position to track the sinusoidal signal. After the operation is completed, the three PID parameters are used as input and IAE is used as output to train the Gaussian process model by the maximum likelihood method, that is, to find the parameter θ (the mean function and covariance function are parameterized by θ) so that the probability of the actual function value is maximized. In practice, the method of maximizing the log-likelihood function is usually used, and the maximum likelihood estimation can be achieved through gradient ascent. The trained Gaussian process model is used for subsequent IAE prediction.

[0088] Step 2: Using the Bayesian sampling strategy, select the next PID parameter group and use it for the experiment.

[0089] Step 3: Evaluate the actual absolute error integral (IAE) of the throttle position tracking effect of this set of parameters, add this set of test data to the training data set of the Gaussian process, and update the Gaussian process model.

[0090] The above three steps are repeated until the convergence condition is reached, and the optimization is considered to be completed. The best parameter group obtained by optimization is the final calibration result.

[0091] The implementation process of the present invention applies Bayesian optimization proxy model selection, Gaussian process parameter update, Gaussian process model prediction principle, acquisition function design and optimization, algorithm convergence condition design, controller and control strategy, and algorithm hyperparameter design for electronic throttle control parameter calibration problem.

[0092] 1. Surrogate model selection for Bayesian optimization.

[0093] The core of the Bayesian optimization algorithm is to use prior knowledge and Bayesian probability and decision theory to determine the next sampling action. Therefore, the form of the proxy model in the Bayesian optimization algorithm needs to be able to reflect certain probabilistic characteristics. The role of the proxy model is to establish a preliminary model of the objective function from known prior knowledge, and to continuously modify it through each iterative sampling to obtain a posterior model that better reflects the characteristics of the objective function.

[0094] The proxy models commonly used for Bayesian optimization are divided into parameter proxy models and non-parameter proxy models, where the parameter proxy model needs to have explicit parameters directly related to the objective function. This form of proxy model is called a parameter proxy model. Updating the posterior of the proxy model is actually equivalent to updating the posterior probability of its explicit parameters. It is usually used for problems with a small search space and a relatively simple calibration objective function. It is more appropriate to use a non-parameter model as a proxy model for the control parameter calibration problem of the engine control system studied in this application. This model has no explicit parameters or its parameters are not directly related to the objective function. Among the non-parameter proxy models, the more popular ones are the random forest model and the Gaussian process model. The random forest model is generally only applied to the discretized parameter space, requires a large amount of data for each iteration, and cannot optimize its parameters based on the gradient. Therefore, the prediction proxy model of this application adopts the Gaussian process (GP) model. GP has a unique and important feature: it does not generate point estimate predictions like other machine learning models, but uses probability distribution for prediction. Predictions in the form of probability distribution or probability prediction are the key to achieving Bayesian optimization, which can quantify the uncertainty in the prediction, thereby improving the risk-return trade-off in decision-making.

[0095] A Gaussian process model is a random process model that is defined if and only if for any finite number of points x drawn from the variable x 1 , ..., x n Its joint probability density function p(f(x 1 ), ..., f(x n)) all conform to the joint Gaussian distribution, this random process is called a Gaussian process as shown in the formula, and its characteristics are completely determined by its mean and covariance function.

[0096]

[0097] Among them, M is a multivariate Gaussian distribution, θ is a hyperparameter of the GP model structure, Θ represents the range of parameter variation, p(f|θ) is the predicted probability distribution, p(y|x, θ) is the prior distribution, and μ x is the mean function, cov f is the covariance function.

[0098]

[0099] There are three key points in the process of modeling using the Gaussian process model: using the mean function to control the expected behavior of the GP, using the covariance function to control the smoothness of the GP, and using gradient descent to learn the optimal hyperparameters of the GP. These contents will be explained below.

[0100] 1) Mean function and covariance function design.

[0101] The mean function and the covariance function are two core components of the Gaussian process (GP). By making a specific selection of the mean or covariance function, the prior knowledge for the GP can be effectively specified, so this application needs to select and design the mean function and the covariance function.

[0102] 2) Mean function design.

[0103] When the characteristics of the objective function to be optimized are clear and its variation with the variable to be optimized is simple, the form of the mean function can be directly specified. The main influence of the prior mean on predictions and the optimization strategies based on these predictions lies in the principle of Bayesian inference. However, making inferences without sufficiently clear prior knowledge can be a dangerous thing. Therefore, in Bayesian optimization, the prior mean function is usually regarded as a constant function, as shown in the following equation (3):

[0104] μ(x; c)≡c (3)

[0105] The value of the constant c is usually chosen to be 0 when the distribution of the objective function is unknown, and then it is updated by maximizing the log marginal likelihood using the characteristic descent algorithm through the initial point data and subsequent sampling data.

[0106] 3) Covariance function design.

[0107] Although the mean function defines the expectation of the overall behavior of the objective function, the covariance function or kernel function of the GP plays a more complex role. It expresses the relationship between data points in the definition domain and controls the structure and smoothness of the GP. When its length scale is set relatively large, the prediction of the model will become smoother and less uncertain. If it is set relatively small, it will cause the predicted value of the GP model to change faster and the uncertainty to increase. The size of the shape parameter will also affect the uncertainty of the predicted value.

[0108] The selection of its parameters is usually complicated, so it is necessary to use the gradient descent method to configure and update its parameters by maximizing the log marginal likelihood. In addition, different types of covariance functions also play a vital role in the flexibility and practicality of GP modeling. Among them, the RBF kernel is mainly used to model infinitely differentiable functions. However, since the objective function of the calibration problem usually performed cannot meet such strict requirements, it is more appropriate to use the Matérn kernel. The Matérn 5 / 2 kernel is usually the preferred kernel function for Bayesian optimization, because it can ensure that the GP has a certain smoothness, so that the Bayesian optimization algorithm has sufficient stability, and also make the GP have sufficient variability, making it more realistic when modeling the optimization calibration problem in reality. Its specific form is shown in formula (4).

[0109] For any set of n points [x 1 , ..., x n ], its covariance matrix Σ is defined as:

[0110]

[0111] In the above formula, cov f is the covariance of different calibration parameters, x i 、x j are the i-th and j-th components of the calibration parameters, l 1 , ..., l D is the kernel length along each dimension, α is the shape parameter; D represents the data dimension; is the variance of the calibration parameter.

[0112] 2. Gaussian process parameter update.

[0113] After determining the two main components of the Gaussian process, the mean function and the covariance function, it is necessary to continuously update the prior model of the Gaussian process after collecting the sampling points determined by the Bayesian decision. The hyperparameters of the mean function and the hyperparameters of the covariance function together constitute the hyperparameters θ of the Gaussian process model. In the process of updating the model, the gradient descent method is mainly used to maximize the log marginal likelihood, thereby ensuring the fit of the GP to the sampled data pairs, as shown in formula (5).

[0114]

[0115] Among them, μ and Σ are the mean vector and covariance matrix respectively, N is the covariance matrix of the observation noise, n is the amount of data, and x and y are the input and output vectors of the data respectively.

[0116] 3. Gaussian process model prediction principle.

[0117] In the Bayesian optimization process, after the Gaussian process proxy model is updated, model prediction is required to evaluate each candidate sampling point according to the acquisition function. When performing Gaussian process model prediction, since each set of calibration parameters corresponds to the normal distribution of one of its objective functions, when considering the entire function space, the objective function random variable {f(x)} conforms to the following formula.

[0118]

[0119] Among them, μ(x) is the mean function of the objective function, κ(x, x′) is the covariance kernel function of the objective function, f(x) represents the objective function value, x and x′ are two sets of data points for calibration parameters, E represents the mean operation, and Rp represents the p-dimensional real space.

[0120] Using the parameter set X=[x 1 , ..., x N ] T The set of measurement evaluation values ​​corresponding to X is Y = [y 1 , ..., y N ] T Carry out x * The Gaussian process prediction of the parameter group is obtained by solving the conditional probability through the joint high-dimensional Gaussian distribution of the training data and the prediction point, as shown in formula (9).

[0121]

[0122] Among them, x * ,f(x * ) are the calibration parameter data value and the corresponding objective function value of the next sampling point, μ * and κ *are the updated mean function matrix and covariance kernel function matrix respectively, X, Y are the calibration parameters of the previous sampling point and the obtained objective function data, μ(x * ) is the mean of the calibration parameters of the next sampling point; μ(X) is the mean of the calibration parameters of the previous sampling point; κ(X,X) is the variance of the historical sampling points; κ(x * ,X) is the covariance between the next sampling point and the historical sampling point; κ(X,x * ) is the covariance between the historical sampling point and the next sampling point; κ(x * ,x * ) is the variance of the next sampling point.

[0123] 4. Acquisition function design and optimization.

[0124] The proxy model and the acquisition function are the two main components of the Bayesian optimization process. This application designs the corresponding sampling function and selects the optimization algorithm for the calibration problem of the engine control system. Because it is impossible to know the results of the observations before sampling, the sampling optimization strategy must take this uncertainty into account. Since the results of each observation not only have a direct impact on the current situation, but also form the basis for all future decisions, the sequential nature of the optimization further exacerbates the importance of this uncertainty. When formulating an effective sampling optimization strategy, it is necessary to solve this uncertainty in some way.

[0125] The acquisition function quantifies the value of each sampling point given a predicted probability distribution, and determines the location of the next sampling point by finding the point that maximizes the sampling value. The Bayesian optimization principle shows that the selection of sampling points is an iterative process. In each iteration of the loop, the GP is trained based on the data observed from the target, and the sampling strategy is evaluated on the GP to obtain sampling recommendations. Observations are made at the recommended sampling points, new points are added to the training data, and the whole process is repeated until the algorithm cutoff condition is reached.

[0126] 1) Acquisition function design.

[0127] In the Bayesian optimization process, commonly used acquisition functions include the probability improvement based acquisition function (PI), the expected improvement based acquisition function (EI), the upper confidence bound based acquisition function (UCB) and the lower confidence bound based acquisition function (LCB).

[0128] The probability-based improved acquisition function (PI) takes the probability that the observed point may find a better objective function value than the incumbent point as the evaluation criterion, which can be expressed as:

[0129]

[0130] Among them, f *is the current optimal value corresponding to the incumbent point, μ(x) is the predicted mean corresponding to the current posterior Gaussian process, σ(x) is the predicted variance corresponding to the current posterior Gaussian process, Φ is the cumulative probability density of the standard normal distribution, D is the currently collected data set, x is the data point, α PI (x) is the acquisition function based on probability improvement.

[0131] When using the PI acquisition function, we can only evaluate the probability of improvement of the objective function without caring about the amount of improvement. This may lead to an overly conservative search strategy and ignore areas where the improvement may be large but the probability of improvement is low. Therefore, when using this acquisition function for Bayesian optimization, the optimization often falls into a local optimum.

[0132] The acquisition function based on expected improvement (EI) not only considers the possible improvement probability of the observation point, but also takes the degree of improvement into consideration. Its expression is:

[0133]

[0134] Among them, a EI (x; D) represents the acquisition function based on expectation improvement, μ(x) is the predicted mean corresponding to the current posterior Gaussian process, σ(x) is the predicted variance corresponding to the current posterior Gaussian process, Φ is the cumulative probability density of the standard normal distribution, D is the currently collected data set, and x is the data point.

[0135] This sampling strategy balances exploration and exploitation, but is not suitable for the presence of noise.

[0136] There is no essential difference between the acquisition function based on the upper confidence bound (UCB) and the acquisition function based on the lower confidence bound (LCB). The upper and lower confidence bounds of the posterior Gaussian process are used as evaluation criteria to determine the position of the next sampling point. This strategy can still make better sampling decisions in the case of large noise. For the control system calibration problem studied in this application, the objective function value is usually minimized, so it is more reasonable to choose LCB, which is expressed as follows:

[0137]

[0138] where β l =2ln(D t 2π 2 / (6δ)), D is the input data space, t is the time, v, δ are empirical parameters. According to the research of Srinivas et al., the parameters v and δ take the recommended values ​​of 0.2 and 0.1 to better balance exploration and utilization. If two data points have the same prediction mean but different prediction standard deviations, the data point with higher uncertainty will get a higher evaluation value; if two data points have the same prediction standard deviation but different prediction means, the data point with higher mean will get a higher evaluation value. Where vβ t The smaller vβ is, the more the LCB strategy is inclined to exploit; on the contrary, t The larger it is, the more the LCB strategy tends to explore.

[0139] 2) Acquisition function optimization.

[0140] In Bayesian optimization, the acquisition function is used to guide how to select the next sampling point, and the optimization method of the acquisition function is the direct link in determining the sampling point. The acquisition function found by the optimization method determines the next sampling point according to the maximum evaluation value calculated by the current posterior Gaussian process, and iterates and optimizes the solution after continuously updating the posterior Gaussian process. Finally, the sampling point corresponding to the evaluation value that cannot be improved sufficiently is the optimal solution of Bayesian optimization, and its value is the optimal value of Bayesian optimization. Therefore, the optimization method of the acquisition function is very important in Bayesian optimization, which directly determines whether the global optimal solution can be found.

[0141] However, the acquisition function is usually non-convex and multi-peaked, and sometimes even more difficult to optimize than the objective function itself. Fortunately, since the acquisition function is calculated based on the probability information predicted by the Gaussian process, its evaluation cost is much lower than the objective function. Common acquisition function optimization algorithms include the gradient-based quasi-Newton method (L-BFGS-B), pattern search algorithm, meta-heuristic algorithm, etc. For gradient-based optimization algorithms, it is often impossible to find the global optimal solution when dealing with acquisition function optimization problems with non-convex and multi-peak characteristics. The convergence speed of the meta-heuristic algorithm is slow and the algorithm can only design the convergence conditions based on the number of iterations and the objective function value. This method is very inconvenient to apply to the online calibration problem of the engine control system, because in the online calibration problem of the engine control system, not only the calibration cycle is required to be high, but also the calibration accuracy, parameter accuracy, sampling times, etc. need to be considered in the convergence condition design. In addition, in order to avoid a series of safety problems caused by the calibration parameters exceeding the feasible domain during the actual calibration process, it is necessary to constrain the parameter range while optimizing. Therefore, this application uses the grid adaptive search algorithm (MADS) in the pattern search algorithm. Compared with the direct search algorithm, it can perform intensive searches in multiple directions in the search space through vector sets, rather than only searching along the grid direction. This feature determines its fast optimization speed and strong ability to find the global optimal solution.

[0142] The MADS algorithm first initializes the initial point and its initial grid parameters, and then iterates to find the next sampling point that meets the improvement standard. The iterative search is divided into two parts, including the global search in the Search stage, and the local refinement search in the Poll stage after the optimal candidate point (i.e., sampling point) is not found in the Search stage. If the next best candidate point is found during the search process, the sampling point is updated and the grid parameters are increased to speed up the convergence speed. Otherwise, the grid parameters are reduced for refinement search. When the grid parameters are less than the set grid size tolerance, the algorithm terminates. The pseudo code of the optimization process is as follows: Figure 5 shown.

[0143] 5. Design of algorithm convergence conditions.

[0144] In view of the control parameters and control MAP calibration problems of the engine control system studied in this application, the following convergence criteria are designed in combination with the Bayesian optimization algorithm principle, in combination with the accuracy requirements of the calibration parameters, the accuracy requirements of the calibration indicators, the evaluation cost of the experiment, and other issues in the calibration process:

[0145]

[0146] Among them, Δ mesh Represents the actual mesh size tolerance, M sizeis the mesh size tolerance value, which determines the accuracy of the calibration parameters; n and n+1 represents the objective function of the nth and n+1th evaluations, F c is the minimum improvement of the objective function, which depends on the accuracy requirement of the calibration index; T no is the actual number of iterations without significant changes in the objective function value, I s F is the maximum number of iterations without significant changes in the objective function value, which can achieve rapid convergence of the algorithm; eval and T are the actual number of objective function evaluations and the actual number of algorithm iterations, respectively. max with I max They are the maximum number of evaluations of the objective function and the maximum number of iterations of the algorithm, which depends on the experimental cost that can be tolerated for the calibrated problem; C v To calibrate the required threshold, it is necessary to design it according to the specific calibration problem. When the algorithm meets one of the above convergence conditions, the algorithm ends. The convergence parameter design needs to be designed through the algorithm's hyperparameter influence experiment on the specific calibration problem.

[0147] 6. Controller and control strategy.

[0148] 1) Design of discrete PID controller for electronic throttle.

[0149] Since the control process of the actual controller belongs to sampling control, the deviation calculation at the sampling moment is required before the control quantity can be calculated. This section designs a discrete PID controller. However, since the position PID control algorithm needs to accumulate the error term for each control signal output calculation, when there is a problem with the measurement signal acquisition, the calculated control signal may change significantly, thereby damaging the driving DC motor of the electronic throttle system. Therefore, the following incremental PID controller is designed:

[0150]

[0151] Among them, k p , k i and k d are the three control parameters of PID, k i =k p / T I , k d =k p T D , T is the sampling period, k is the kth sampling point, e(k), e(k-1), e(k-2) are the error terms of the throttle position at the kth, k-1, k-2th moments respectively, Δu is the control voltage increment, and u is the control voltage value.

[0152] At the same time, the integrator clamping method is used to prevent integral saturation. When the control signal reaches the actuator saturation voltage limit and the error is in the same direction as the control signal, integral separation is performed. This algorithm can avoid the control quantity staying in the saturation zone for a long time to ensure the accuracy and speed of electronic throttle valve opening control.

[0153] 2) Control strategy.

[0154] Based on the selection of the drive circuit and the frequency design of the PWM speed regulation signal, the following is carried out according to the relevant requirements of the electronic throttle control system: Figure 6 The electronic throttle test platform shown controls the main program.

[0155] In the main control program of the electronic throttle test platform, the sensor voltage signal is measured through the ADC module and the actual angle value is calculated through the sensor characteristic curve. After the discrete PID control outputs the duty cycle of the PWM signal, it first passes through the threshold judgment part of the duty cycle and then finally determines the duty cycle of the control signal to be output. This is because the sensor and the ADC module will produce measurement errors during the measurement process and process errors during the actual operation. These errors will cause the DC drive motor to frequently switch forward and reverse if the threshold judgment module is not set, resulting in rapid vibration of the throttle body.

[0156] 7. Design algorithm hyperparameters for electronic throttle control parameter calibration problem.

[0157] For the convergence criteria designed in this application, most of the parameters need to be determined through specific analysis of the applied calibration problem. This application designs the convergence parameters as shown in Tables 1, 2 and 3 for the calibration accuracy, parameter accuracy, and evaluation cost of the control parameter calibration problem of the electronic throttle control system.

[0158] Table 1 Algorithm general hyperparameter statistics

[0159]

[0160]

[0161] Table 2 Algorithm convergence hyperparameter statistics

[0162]

[0163] Table 3 Convergence parameter statistics for the control parameter calibration problem of the electronic throttle control system

[0164] Converging Hyperparameters symbol Parameter Value Mesh size tolerance <![CDATA[M size ]]> 0.1 Minimum improvement of objective function <![CDATA[F c ]]> 0.00698 Maximum number of objective function evaluations <![CDATA[F max ]]> 200 Maximum number of iterations <![CDATA[I max ]]> 100 Calibration index requirements <![CDATA[C v ]]> 0.0698

[0165] Among them, since the influence of the calibrated control parameters on the control effect can be ignored when the variation range is less than 0.1, the grid size tolerance M size For example, 0.1 can be selected; and the calibration index requires C v The minimum improvement of the objective function F c The design of C can be based on the offline calibration convergence parameters of the control parameters in Chapter 2. The IAE calculated when the tracking error is less than 1 degree each time under the tracking condition within 4 seconds is selected as C v The value of F c Then take the IAE calculated when the tracking error is 0.1 degrees each time (IAE = ∫ 0 τ|e(t)|), sampling time T s =0.001s; the maximum number of evaluations and the maximum number of iterations are calculated according to the specific calibration cost, and F max =200,I max = 100. Finally, based on Figure 7 The online calibration process shown is used to calibrate the control parameters online.

[0166] In practical applications, the online calibration method of control parameters is continuously carried out during the operation of the system. It uses the information provided by real-time data to calibrate the control parameters. This method optimizes the calibration results by continuously monitoring and adapting to changes in the environment and the system, and can adapt to changes in the characteristics of the environment and the system. Among them, the non-model-based online calibration method of control parameters solves the problem that the calibration accuracy is overly dependent on the model accuracy, but it still requires a large amount of experimental sampling, and this method is less applicable when the sampling cost is high. Therefore, in order to reduce the number of sampling times, the present invention is based on the Bayesian strategy and the Gaussian process probability modeling principle, and guides the sampling process in the calibration process, so as to ensure the calibration requirements of the control parameters while sampling as little as possible. An online calibration algorithm based on grid adaptive Bayesian optimization is proposed and a corresponding online calibration device for control parameters is developed. In a fixed throttle following condition, the control parameters are continuously updated by the Bayesian optimization algorithm, and the calibration objective function is solved to realize the automatic online calibration of the control parameters. The present invention uses a Bayesian optimization algorithm in the process of online calibration of control parameters, utilizes a grid adaptive algorithm to optimize the sampling function to solve the non-convex optimization problem in the calibration process, and uses a Gaussian process model as a proxy model, which greatly shortens the online calibration period of the control parameters and reduces the sampling points required for calibration. In addition, this method also introduces an extended Kalman filter technology to further improve the accuracy of the proxy model and the anti-interference ability of the algorithm. Therefore, the present invention not only improves the automation level of online calibration of control parameters, but also significantly improves the anti-interference ability of online calibration, shortens the calibration period, and reduces unnecessary sampling points under the premise of meeting the calibration indicators by integrating Bayesian optimization and extended Kalman filter algorithms, providing a more efficient and reliable technical solution for the calibration of control parameters of the engine control system.

[0167] The present invention has the following advantages:

[0168] 1. Compared with traditional calibration methods, it can be achieved without human intervention, reducing the uncertainty caused by the calibration personnel's operation, improving the reliability of the calibration results, and enhancing the degree of automation of the calibration process.

[0169] 2. The designed grid-adaptive Bayesian optimization algorithm can quickly solve complex non-convex optimization problems and ensure global optimality. It performs well in terms of convergence speed and accuracy of search results, greatly improving the real-time performance of online calibration.

[0170] 3. Compared with non-model-based online calibration algorithms, it greatly reduces unnecessary sampling points and shortens the online calibration cycle.

[0171] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0172] The principles and implementation methods of the present invention are described in this article using specific examples. The description of the above embodiments is only used to help understand the method and core idea of ​​the present invention. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. An online calibration device for control parameters of an engine control system, characterized in that: The online calibration device comprises: an electronic throttle body, a DC motor drive module, a controller, a communication module and a host computer; The host computer is connected to the controller via the communication module; the controller is connected to the DC motor drive module and the electronic throttle body respectively; the DC motor drive module is connected to the electronic throttle body; The host computer is used to monitor the status of each input and output pin of the controller, and receive calibration index measurement parameters, and calculate calibration sampling point data according to the calibration index measurement parameters by applying the control parameter online calibration method of the engine control system, and send the calibration sampling point data to the controller; the calibration index measurement parameters are calculated by the controller according to the throttle opening signal of the electronic throttle body; the control parameter online calibration method of the engine control system is a grid adaptive Bayesian online calibration algorithm; The controller is used to apply the PID algorithm to send the motor operating parameters to the DC motor drive module according to the calibration sampling point data, and receive the throttle opening signal returned by the electronic throttle body; the motor operating parameters include the motor speed and the motor rotation direction; The DC motor driving module is used to control the operation of the electronic throttle body according to the motor operating parameters.

2. The control parameter online calibration device of the engine control system according to claim 1, characterized in that: The host computer includes an edge computing device and a controller host computer; The edge computing device and the controller host computer are both connected to the controller; The edge computing device is used to receive calibration index measurement parameters, and calculate calibration sampling point data by applying a grid adaptive Bayesian online calibration algorithm according to the calibration index measurement parameters, and send the calibration sampling point data to the controller; The controller host computer is used to monitor the status of each input and output pin of the controller.

3. An online calibration method for control parameters of an engine control system, characterized in that: The control parameter online calibration device for the engine control system according to any one of claims 1 to 2, wherein the online calibration method comprises: Setting the feasible domain of the control parameters and the initial sampling point values ​​of the control parameters; the control parameters are the proportional parameters, differential parameters and integral parameters of the PID algorithm; According to the mean function, the covariance function and the initial sampling point value of the control parameter, the prior Gaussian process model and the Bayesian sampling strategy are applied to determine the next sampling point value of the control parameter; According to the value of the next sampling point of the control parameter, the posterior Gaussian process model is applied to determine the value of the objective function; the objective function is the absolute value integral of the difference between the throttle opening of the electronic throttle body under the action of the control parameter and the target opening; Determining whether the value of the objective function satisfies a preset convergence condition; When the value of the objective function does not meet the preset convergence condition, the posterior Gaussian process model is updated by applying a method based on maximizing the logarithmic marginal probability, and the next sampling point value of the control parameter is input into the updated posterior Gaussian process model to obtain the updated value of the objective function, and return to the step of "determining whether the value of the objective function meets the preset convergence condition"; When the value of the objective function meets the preset convergence condition, the value of the next sampling point of the control parameter is used as the online calibration result.

4. The method for online calibration of control parameters of an engine control system according to claim 3, characterized in that: The mean function is μ(x; c)≡c; Among them, x is the calibration parameter, that is, the three parameters of PID; c is a constant.

5. The method for online calibration of control parameters of an engine control system according to claim 3, characterized in that: The covariance function is: Among them, cov f is the covariance of different calibration parameters, x i 、x j are the i-th and j-th components of the calibration parameters, l1,...,l D is the kernel length along each dimension, α is the shape parameter; D represents the data dimension; is the variance of the calibration parameter, r 2 (x i , x j ) represents x i and x j The sum of squared distances; x id is the data of the dth dimension of the ith component of the calibration parameter, x jd The data of the d-th dimension of the j-th component of the calibration parameter.

6. The method for online calibration of control parameters of an engine control system according to claim 3, characterized in that: The objective function is: Where μ(x) is the mean function of the objective function, κ(x, x′) is the covariance kernel function of the objective function, f(x) represents the objective function value, x and x′ are two sets of data points for calibration parameters, E represents the mean operation, and R p Represents the p-dimensional real number space.

7. The method for online calibration of control parameters of an engine control system according to claim 3, characterized in that: The updated posterior Gaussian process model is: m * =κ(x * ,X)(κ(X,X)) -1 (Y-μ(X))+μ(x * ); k * =κ(x * ,x * )-k(x * ,X)(κ(X,X)) -1 k(X,x * ); Among them, x * ,f(x * ) are the calibration parameter data value and the corresponding objective function value of the next sampling point, μ * and κ * are the updated mean function matrix and covariance kernel function matrix respectively, X, Y are the historical sampling point calibration parameters and the obtained objective function data, μ(x * ) is the mean of the calibration parameters of the next sampling point; μ(X) is the mean of the calibration parameters of the historical sampling points; κ(X,X) is the variance of the historical sampling points; κ(x * ,X) is the covariance between the next sampling point and the historical sampling point; κ(X,x * ) is the covariance between the historical sampling point and the next sampling point; κ(x * ,x * ) is the variance of the next sampling point.

8. The method for online calibration of control parameters of an engine control system according to claim 3, characterized in that: The Bayesian sampling strategy is obtained by optimizing the acquisition function using a grid adaptive search algorithm.

9. The method for online calibration of control parameters of an engine control system according to claim 8, characterized in that: The sampling function is the acquisition function of the lower confidence bound.

10. The method for online calibration of control parameters of an engine control system according to claim 3, characterized in that: The preset convergence condition is: Any one of; Among them, Δ mesh Represents the actual mesh size tolerance, M size is the mesh size tolerance value, which determines the accuracy of the calibration parameters; n and n+1 represents the objective function of the nth and n+1th evaluations, F c is the minimum improvement of the objective function, which depends on the accuracy requirement of the calibration index; T no is the actual number of iterations without significant changes in the objective function value, I s is the maximum number of iterations without significant changes in the objective function value; F eval and T are the actual number of objective function evaluations and the actual number of algorithm iterations, respectively. max with I max are the maximum number of evaluations of the objective function and the maximum number of iterations of the algorithm; C v The threshold value is required for calibration.

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