A time delay position tracking control method and system for a permanent magnet synchronous linear motor

Abstract

The application relates to the technical field of permanent magnet synchronous motors, in particular to a time delay position tracking control method and system for a permanent magnet synchronous linear motor. A PMSLM kinematic model considering parameter uncertainty, friction and external disturbance is established in a quadrature axis coordinate system, a position tracking error dynamic model is derived through high-order expansion and equivalent processing of the same order, a monotonic error transformation function is constructed, a nonlinear sliding mode error dynamics is introduced, and equivalent control law design of expected error convergence characteristics is completed. A system sampling period is used to collect historical delay input signals, and system lumped uncertainty is estimated online based on time delay assumption. A time delay estimation error is defined, a sliding mode variable is introduced, and an adaptive gain correction term is designed to suppress the influence of the time delay estimation error on the dynamic performance of the system. The application realizes high-precision and low-jitter position tracking control of the PMSLM servo system under the conditions of uncertainty disturbance and time delay, and optimizes the servo accuracy of the permanent magnet synchronous linear motor.

Description

Technical Field

[0001] This invention relates to the field of permanent magnet synchronous motor technology, and in particular to a time-delay position tracking control method and system for a permanent magnet synchronous linear motor. Background Technology

[0002] In recent years, permanent magnet synchronous linear motors (PMSLMs) have been widely used in high-speed, high-precision linear servo applications such as industrial robots, CNC machine tools, 3D printing, and semiconductor manufacturing due to their advantages of high precision, high dynamic response, and simple structure. However, a PMSLM is a highly nonlinear and strongly coupled complex system, and its model often contains uncertainties. Furthermore, under complex reciprocating motion conditions, parameter mismatch, nonlinear friction, thrust fluctuations, cable disturbances, and load fluctuations directly affect the PMSLM itself, leading to a decrease in position tracking accuracy.

[0003] To overcome these problems and achieve ideal servo control performance, many advanced robust control strategies have been proposed in recent years. Among them, sliding mode control, due to its strong robustness to internal parameter disturbances and external disturbances and its simple implementation, has been applied to some extent in PMSLM servo systems. However, the implementation of traditional sliding mode control depends on the dynamic model and parameters of the motor, and in order to overcome disturbances, a high switching gain is often required, which can cause control chattering and lead to large steady-state errors in system position tracking. Summary of the Invention

[0004] This invention overcomes the shortcomings of the prior art and provides a time-delay position tracking control method and system for permanent magnet synchronous linear motors.

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

[0006] The first aspect of this invention provides a time-delay position tracking control method for a permanent magnet synchronous linear motor, comprising the following steps:

[0007] S1: Construct the PMSLM kinematic equations under orthogonal axial coordinates considering different types of disturbances in the servo system. Perform equivalent processing of the same order based on the higher-order extension of the PMSLM kinematic equations to obtain the equivalent order model. Based on the equivalent order model, derive the position tracking error of the permanent magnet synchronous linear motor servo to obtain the position error dynamic model of PMSLM.

[0008] S2: Construct a monotonic error transformation function based on the desired control requirements of the motor. Substitute the servo dynamic data of the position tracking error generated by the PMSLM servo system into the derivative of the monotonic error transformation function to generate a nonlinear sliding mode dynamic model. Based on the position error dynamic model, use the nonlinear sliding mode dynamic model to perform convergence derivation and design to obtain the equivalent convergence control law.

[0009] S3: Collect historical servo delay input signals from the sampling time period of the servo system. Based on the time delay assumption, push forward the equivalent order model to time t0 and then replace the collected delay signal queue data into the system. Estimate the lumped uncertainty of the system time delay online and obtain the uncertainty estimate based on time delay estimation when the PMSLM servo system is disturbed.

[0010] S4: Define the time delay estimation error and introduce the sliding diaphragm variable. Suppress the time delay estimation error between the sliding diaphragm variable and the ideal response speed of the system by adjusting the correction gain of the adaptive gain law. Design an adaptive correction term for the delay estimation error.

[0011] S5: The equivalent convergent control law, the uncertainty estimate based on time delay estimation, and the adaptive correction term are summed to obtain the position tracking optimization control law of the PMSLM servo system.

[0012] Preferably, step S1 specifically includes the following steps:

[0013] The position and mass parameters of the mover of the permanent magnet synchronous linear motor, as well as the uncertainty parameters of different servo system disturbances, are obtained. Among them, the different servo system disturbances include load changes, nonlinear friction, and thrust fluctuations.

[0014] By dynamically coupling the mover position information, mover mass parameters and various uncertainty parameters with perturbation observation, a PMSLM kinematic equation considering different types of perturbations of the servo system in orthogonal axial coordinates is established.

[0015] The unknown state variables of the PMSLM servo system are injected into the explicit extended state into the PMSLM kinematic equations. An equivalent higher-order extension of the PMSLM servo system is performed by detecting and analyzing the time-varying rate trend of the explicit extended state. The higher-order extension observed in the final gain of the PMSLM kinematic equations is then compressed into a perturbation estimate of the same order, yielding the equivalent order model of the PMSLM servo system. The specific formula is as follows:

[0016] ;

[0017] In the formula: For the gain to be designed, This represents nonlinear lumped time-varying dynamics, which includes parameter uncertainty, thrust fluctuation, and nonlinear friction.

[0018] Define the position tracking error of the permanent magnet synchronous linear motor servo, and import the position tracking error into the equivalent order model for derivation to obtain the position error dynamic model of PMSLM.

[0019] Preferably, the unknown state variable terms of the PMSLM servo system are injected into the explicit extended state into the PMSLM kinematic equations. An equivalent higher-order extension of the PMSLM servo system is performed by detecting and analyzing the time-varying rate trend of the explicit extended state. The higher-order extension observed in the final gain of the PMSLM kinematic equations is then compressed into a perturbation estimate of the same order, resulting in an equivalent-order model of the PMSLM servo system. This process specifically includes the following steps:

[0020] Obtain the unknown state variable terms of the PMSLM servo system, construct and introduce an explicit extended state based on the unknown state variable terms in the PMSLM kinematic equations, detect the real-time uncertain changes of the servo system after the intervention of the unknown terms, and obtain the time-varying rate of the explicit extended state.

[0021] A high-frequency time-varying rate threshold is set for the system's rapid disturbance or nonlinear lumped time-varying uncertainty. If the time-varying rate exceeds the high-frequency time-varying rate threshold, then the explicit expansion state is subjected to higher-order expansion processing during the above-mentioned introduction process to generate a higher-order ESO gain observer for PMSLM servo kinematics.

[0022] Based on nonlinear lumped time-varying dynamics, the equivalent compression concept of bandwidth matching and gain reconstruction is introduced. By using the equivalent compression concept, the different high-order expansion states output by the high-order ESO gain observer are equivalently mapped and compressed into the second-order perturbation estimate of the PMSLM kinematic equation, thus obtaining the equivalent order model of the PMSLM servo system.

[0023] Preferably, step S2 specifically includes the following steps:

[0024] Obtain the desired control requirements for position tracking error of permanent magnet synchronous linear motors applied in different servo operating scenarios, and formulate the performance boundary function and asymmetric error constraint interval for guiding error directional convergence based on the desired control requirements;

[0025] By combining the performance boundary function and the asymmetric error constraint interval, the tracking error at each position is mapped to an unconstrained variable through error normalization, and the desired monotonic error transformation function is constructed. The specific formula is as follows:

[0026] ;

[0027] The equivalent control disturbance parameters and corresponding prior control gain when the PMSLM servo system generates position tracking error are obtained. The equivalent control disturbance parameters and prior control gain are bound as servo system dynamics terms, and the error dynamics space is constructed based on the first derivative of the position tracking error.

[0028] The desired monotonic error transformation function is differentiated in the error dynamics space. By differentiation, the second-order error system of the PMSLM kinematic equations is reduced to a first-order sliding mode variable. Substituting the servo system dynamics term into the first-order sliding mode variable, the nonlinear sliding mode dynamics model is obtained, with the specific formula as follows:

[0029] ;

[0030] In the formula: , , and The positive coefficients of the desired error dynamics to be designed are... ;

[0031] An error convergence control law is established, and the nonlinear sliding mode dynamics model is imported into the error convergence control law to derive the closed-loop convergence of the sliding mode error state. An equivalent control term for the desired nonlinear position tracking error dynamics is designed and generated.

[0032] Substituting the equivalent control term into the position error dynamics model of the PMSLM, the equivalent convergence control law for the servo position tracking error of the permanent magnet synchronous linear motor is obtained.

[0033] Preferably, step S3 specifically includes the following steps:

[0034] The PMSLM kinematic equations are algebraically rearranged to eliminate the explicit acceleration dependence of the system's lumped uncertainty perturbation, generating PMSLM kinematic equations containing lumped uncertainty.

[0035] The time delay assumption is introduced, which assumes that the lumped uncertainty of the PMSLM servo system satisfies the following within a short-term variable delay interval:

[0036] ;

[0037] Select the sampling time period of the servo system, synchronously build a delay buffer empty stack, collect the input signal of historical servo delay online according to the sampling time period and transfer and store it to the delay buffer empty stack, and generate a variable delay signal buffer queue.

[0038] The equivalent order model of the PMSLM servo system is pushed forward to time t0. The PMSLM kinematic equations containing lumped uncertainty are replaced by a variable delay signal buffer queue at time t-t0 to approximate the time delay information of the servo control input before the current uncertain disturbance of the system. A series of digital variable delay estimation terms of system sampling are obtained.

[0039] A series of digital variable delay estimation terms are substituted into the time delay assumption to calculate the delay estimation error, and the stable chaotic index of the online estimated position tracking error is output.

[0040] When the stable chaos index is less than 1, it is determined that the time delay estimation has reached the convergence state, the online time delay estimation operation is terminated, and the uncertainty estimate based on the time delay estimation is obtained when the PMSLM servo system is disturbed.

[0041] Preferably, step S4 specifically includes the following steps:

[0042] By combining the equivalent convergent control law with the control synthesis conversion of the permanent magnet synchronous linear motor servo position tracking error of the aforementioned uncertainty estimate, the time-delay position tracking control law is obtained, and the specific formula is as follows:

[0043] ;

[0044] Define a sub-term for time delay estimation error, perform system error analysis on the time delay position tracking control law based on the equivalent order model using equivalent disturbance input, and construct a dynamic model for the delay estimation error, the specific formula of which is:

[0045] ;

[0046] By substituting the time delay estimation error sub-term into the time delay estimation error dynamic model and solving it, the specific time delay estimation error value is obtained, and the specific formula is as follows:

[0047] ;

[0048] In the formula: This represents the time delay estimation error value;

[0049] Based on the desired control requirements of permanent magnet synchronous linear motors, gain shaping indices related to position tracking errors under different servo operating conditions are shaped, and sliding mode variables are determined based on the gain shaping indices.

[0050] By performing a variable structure integral with dynamic gain on both sides of the nonlinear sliding mode dynamics model, the ideal response speed of the PMSLM servo system is obtained.

[0051] An adaptive gain law is introduced to suppress the time delay estimation error between the sliding mode variable and the ideal response speed. The dynamic gain is smoothed in the adaptive gain law, and the correction gain is adjusted online to obtain an adaptive correction term for the delay estimation error.

[0052] A second aspect of the present invention provides a time-delay position tracking control system for a permanent magnet synchronous linear motor, applicable to any of the time-delay position tracking control methods for permanent magnet synchronous linear motors described in any one of the claims. The system specifically includes:

[0053] A time delay estimation module is used to estimate the uncertainty of the PMSLM servo system based on time delay estimation online, eliminating most of the continuous nonlinear errors;

[0054] The desired position tracking error dynamics calculation module is responsible for designing the desired nonlinear position tracking error dynamics and the equivalent control law.

[0055] An adaptive correction module for time delay estimation error is provided. This module is responsible for suppressing the time delay estimation error between the sliding variable and the ideal response speed of the system by adjusting the correction gain through an adaptive gain law, and for designing an adaptive correction term for the delay estimation error.

[0056] This invention addresses the technical deficiencies in the prior art, and its beneficial technical effects are as follows:

[0057] This invention proposes a time-delay position tracking control method for PMSLM. Compared with the traditional linear PMSLM robust position control method, this method ensures system robustness while avoiding dependence on the precise dynamic model and parameters of the motor. It also avoids control chattering and noise caused by high switching gain, thus greatly improving the position tracking accuracy of PMSLM servo. Attached Figure Description

[0058] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other embodiments can be obtained from these drawings without creative effort.

[0059] Figure 1 A flowchart of the first method of a time-delay position tracking control method for a permanent magnet synchronous linear motor is shown;

[0060] Figure 2 The position tracking error curve of a traditional linear sliding mode is shown.

[0061] Figure 3 The position tracking error curve of the nonlinear sliding mode of the present invention is shown;

[0062] Figure 4 The control input current diagram of a conventional linear sliding mode is shown.

[0063] Figure 5 The control input current diagram of the nonlinear sliding mode of the present invention is shown;

[0064] Figure 6 A system framework diagram of a time-delay position tracking control system for a permanent magnet synchronous linear motor is shown. Detailed Implementation

[0065] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other.

[0066] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.

[0067] The first aspect of this invention provides a time-delay position tracking control method for a permanent magnet synchronous linear motor, such as... Figure 1 As shown, it includes the following steps:

[0068] S1: Construct the PMSLM kinematic equations under orthogonal axial coordinates considering different types of disturbances in the servo system. Perform equivalent processing of the same order based on the higher-order extension of the PMSLM kinematic equations to obtain the equivalent order model. Based on the equivalent order model, derive the position tracking error of the permanent magnet synchronous linear motor servo to obtain the position error dynamic model of PMSLM.

[0069] S2: Construct a monotonic error transformation function based on the desired control requirements of the motor. Substitute the servo dynamic data of the position tracking error generated by the PMSLM servo system into the derivative of the monotonic error transformation function to generate a nonlinear sliding mode dynamic model. Based on the position error dynamic model, use the nonlinear sliding mode dynamic model to perform convergence derivation and design to obtain the equivalent convergence control law.

[0070] S3: Collect historical servo delay input signals from the sampling time period of the servo system. Based on the time delay assumption, push forward the equivalent order model to time t0 and then replace the collected delay signal queue data into the system. Estimate the lumped uncertainty of the system time delay online and obtain the uncertainty estimate based on time delay estimation when the PMSLM servo system is disturbed.

[0071] S4: Define the time delay estimation error and introduce the sliding diaphragm variable. Suppress the time delay estimation error between the sliding diaphragm variable and the ideal response speed of the system by adjusting the correction gain of the adaptive gain law. Design an adaptive correction term for the delay estimation error.

[0072] S5: The equivalent convergent control law, the uncertainty estimate based on time delay estimation, and the adaptive correction term are summed to obtain the position tracking optimization control law of the PMSLM servo system.

[0073] Preferably, step S1 specifically includes the following steps:

[0074] The position and mass parameters of the mover of the permanent magnet synchronous linear motor, as well as the uncertainty parameters of different servo system disturbances, are obtained. Among them, the different servo system disturbances include load changes, nonlinear friction, and thrust fluctuations.

[0075] By dynamically coupling the mover position information, mover mass parameters, and various uncertainty parameters with perturbation observations, PMSLM kinematic equations considering different types of perturbations in the servo system under orthogonal axial coordinates are established. The specific formulas are as follows:

[0076] ;

[0077] In the formula: This is the electromagnetic thrust coefficient. To control the input current, It is the mass of the mover. The coefficient of viscous friction, Let be the Coulomb friction coefficient. For the position of the mover, For thrust fluctuation, For load thrust, For unmodeled dynamics, These are standard symbolic functions; where the subscript "o" represents the nominal value. 、 、 and Represents the change value of motor parameters. =1.5π , The pole pitch of the motor. For permanent magnet flux linkage;

[0078] The unknown state variables of the PMSLM servo system are injected into the explicit extended state into the PMSLM kinematic equations. An equivalent higher-order extension of the PMSLM servo system is performed by detecting and analyzing the time-varying rate trend of the explicit extended state. The higher-order extension observed in the final gain of the PMSLM kinematic equations is then compressed into a perturbation estimate of the same order, yielding the equivalent order model of the PMSLM servo system. The specific formula is as follows:

[0079] ;

[0080] In the formula: For the gain to be designed, This represents nonlinear lumped time-varying dynamics, which includes parameter uncertainty, thrust fluctuation, and nonlinear friction.

[0081] Define the position tracking error of the permanent magnet synchronous linear motor servo, and import the position tracking error into the equivalent order model for derivation to obtain the position error dynamic model of PMSLM.

[0082] It should be noted that by using the PMSLM kinematic equations in orthogonal axial coordinates, which include servo system disturbances, the multi-source uncertainties in the servo system and the motor's own dynamics are explicitly enumerated and co-coupled for modeling. This identifies and integrates the fundamental state variables and major sources of uncertainty affecting PMSLM performance, enabling the system to simultaneously reflect electromagnetic drive characteristics and disturbance effects, achieving systematic coverage of key disturbance sources in the PMSLM servo system. By incorporating load changes, nonlinear friction, and thrust fluctuations into a unified control framework from the source, this eliminates the structural fragmentation problem caused by separate modeling of different disturbances in traditional dynamic methods. This provides a more realistic and consistent system foundation for subsequent robust control transitions and control design under actual operating conditions. From the specific formulas of the PMSLM kinematic equations, it is easy to observe that the PMSLM position servo system is a second-order system. Therefore, this method maps the high-order uncertainties that are difficult to accurately model in PMSLM kinematics to equivalent disturbance terms of the same order, achieving the observation and estimation of complex disturbances without increasing the system control order. The nonlinear lumped time-varying dynamics can be defined as:

[0083] ;

[0084] This avoids system order inflation and parameter complexity, effectively reducing reliance on precise motor parameters and prior knowledge of disturbances. It makes the servo coupling relationship between control input and disturbance clearer and more logical, resulting in high controllability. This provides a low-complexity, robust model foundation for subsequent time delay estimation, sliding mode, or adaptive control. Furthermore, the specific definition formula for the position tracking error of the permanent magnet synchronous linear motor servo is as follows:

[0085] ;

[0086] In the formula: The motor position is given a value. Through further derivation and calculation, the specific formula for the position error dynamic model of the PMSLM servo system is:

[0087] ;

[0088] As can be seen from the above equations of the position error dynamics model, the PMSLM position error dynamics based on the equivalent order model eliminates motor parameters and nonlinear terms. It is a model-free single-input single-output structure, which allows the time delay controller design to focus directly on the error convergence speed and steady-state accuracy, and significantly reduces the controller's sensitivity to changes in motor physical parameters.

[0089] Preferably, the unknown state variable terms of the PMSLM servo system are injected into the explicit extended state into the PMSLM kinematic equations. An equivalent higher-order extension of the PMSLM servo system is performed by detecting and analyzing the time-varying rate trend of the explicit extended state. The higher-order extension observed in the final gain of the PMSLM kinematic equations is then compressed into a perturbation estimate of the same order, resulting in an equivalent-order model of the PMSLM servo system. This process specifically includes the following steps:

[0090] Obtain the unknown state variable terms of the PMSLM servo system, construct and introduce an explicit extended state based on the unknown state variable terms in the PMSLM kinematic equations, detect the real-time uncertain changes of the servo system after the intervention of the unknown terms, and obtain the time-varying rate of the explicit extended state.

[0091] A high-frequency time-varying rate threshold is set for the system's rapid disturbance or nonlinear lumped time-varying uncertainty. If the time-varying rate exceeds the high-frequency time-varying rate threshold, then the explicit expansion state is subjected to higher-order expansion processing during the above-mentioned introduction process to generate a higher-order ESO gain observer for PMSLM servo kinematics.

[0092] Based on nonlinear lumped time-varying dynamics, the equivalent compression concept of bandwidth matching and gain reconstruction is introduced. By using the equivalent compression concept, the different high-order expansion states output by the high-order ESO gain observer are equivalently mapped and compressed into the second-order perturbation estimate of the PMSLM kinematic equation, thus obtaining the equivalent order model of the PMSLM servo system.

[0093] It should be noted that during the equivalent transformation of kinematic models of the same order, the unknown disturbances of the system considered by the PMSLM kinematic equations in orthogonal coordinates can only exist as implicit terms. That is, load changes, frictional abrupt changes, or thrust fluctuations can only be regarded as noise, and it is impossible to distinguish between slowly varying uncertainties and rapid disturbances. This makes it difficult for the system to estimate the "value" of the disturbance, thus making it difficult to observe the dynamic behavior of the uncertainty and to determine whether the disturbance is changing drastically. This leads to a lag in the response of the time delay controller to sudden operating conditions, which can easily cause equivalent drift with insufficient robustness or excessive conservatism. Therefore, this method introduces an explicit extended state into the PMSLM kinematic equations. This explicit extended state can explicitly model the global total disturbance of the PMSLM position servo system, further capture the unknown quantities of nonlinear terms and external disturbances, and transform them into observable states. This allows for real-time reflection of the evolution of system uncertainties, advancing the PMSLM time delay control from "static disturbance compensation" to the stage of "online perception and response to dynamic disturbance behavior". This breaks through the limitation of traditional observers that can only estimate physical states and endows them with the ability to actively sense the lumped uncertainty of the system. If the intrinsic change (time-varying rate) of the explicit extended state is too rapid or contains high-frequency components (exceeding the high-frequency time-varying rate threshold of rapid system disturbances or nonlinear lumped time-varying uncertainties), it indicates that the system disturbance dynamics are showing a rapid or drastic abrupt change trend. Since the PMSLM position servo system is a second-order system, and the order of the explicit extended state is relatively fixed in the equivalent process of the same order (second order), this can easily lead to a series of serious performance imbalances, such as aggressive observation (over-observation) under steady-state conditions and insufficient observation (lagging) under severe conditions. This results in the time delay controller design having to compromise between rapid disturbance suppression and steady-state accuracy. Therefore, this method further introduces a higher-order derivative of the extended state to perform the extended operation. Higher-order observations only intervene when the system disturbance is severe, avoiding the noise amplification problem caused by high-order ESOs throughout the entire time period. This significantly improves the error tracking accuracy for strong time-varying disturbances, time-varying uncertainties, rapid load changes, and unmodeled higher-order dynamics in rapid disturbance scenarios.

[0094] It should be noted that the designed high-order ESO gain observer is mainly used for synchronous online estimation of system state and disturbance state, establishing a closed-loop information channel of "PMSLM servo system - observer - controller" to achieve the condition-triggered adaptive extension effect of the observer structure in the PMSLM servo system. Furthermore, since the output of the high-order ESO gain observer is difficult to use directly for control, the system order passively increases, and the time-delay controller structure becomes bloated, making it difficult to accurately achieve equivalent control of the time-delay position tracking dynamics. Therefore, in the high-order ESO gain observation, this method extracts the equivalent disturbance component that plays a dominant role in the control input through an equivalent compression mechanism, equating the contribution of multiple extended states to the control law to a low-order disturbance estimate. This transforms the multi-state disturbance estimation into a single-state equivalent disturbance, compressing and reconstructing the high-order ESO into a low-order number. This reduces the servo system order to the same order as the second-order system while retaining the accuracy of rapid high-order disturbance capture and estimation, thus effectively maintaining the original second-order system structure of the PMSLM. This method introduces an unknown disturbance sensing mechanism with explicit extended state terms, and uses the concept of equivalent compression to map the high-order observation results to the same-order disturbance estimates. While ensuring strong observation capability for fast nonlinear disturbances, it avoids system order expansion and increased control complexity, thus achieving high robustness and feasibility of the PMSLM servo system.

[0095] Preferably, step S2 specifically includes the following steps:

[0096] Obtain the desired control requirements for position tracking error of permanent magnet synchronous linear motors applied in different servo operating scenarios, and formulate the performance boundary function and asymmetric error constraint interval for guiding error directional convergence based on the desired control requirements;

[0097] By combining the performance boundary function and the asymmetric error constraint interval, the tracking error at each position is mapped to an unconstrained variable through error normalization, and the desired monotonic error transformation function is constructed. The specific formula is as follows:

[0098] ;

[0099] The equivalent control disturbance parameters and corresponding prior control gain when the PMSLM servo system generates position tracking error are obtained. The equivalent control disturbance parameters and prior control gain are bound as servo system dynamics terms, and the error dynamics space is constructed based on the first derivative of the position tracking error.

[0100] The desired monotonic error transformation function is differentiated in the error dynamics space. By differentiation, the second-order error system of the PMSLM kinematic equations is reduced to a first-order sliding mode variable. Substituting the servo system dynamics term into the first-order sliding mode variable, the nonlinear sliding mode dynamics model is obtained, with the specific formula as follows:

[0101] ;

[0102] In the formula: , , and The positive coefficients of the desired error dynamics to be designed are... ;

[0103] An error convergence control law is established, and the nonlinear sliding mode dynamics model is imported into the error convergence control law to derive the closed-loop convergence of the sliding mode error state. An equivalent control term for the desired nonlinear position tracking error dynamics is designed and generated.

[0104] Substituting the equivalent control term into the position error dynamics model of the PMSLM, the equivalent convergence control law for the servo position tracking error of the permanent magnet synchronous linear motor is obtained.

[0105] It should be noted that, in order to achieve high accuracy and rapid convergence of the position tracking error in PMSLM, this method designs the desired nonlinear position tracking error dynamics, namely a nonlinear sliding mode dynamics model. However, during the dynamics design process, it was found that only requiring the error to approach zero, without limiting the convergence speed, overshoot, or transient performance, leads to several problems. First, the control effect varies significantly with the operating conditions, and the error convergence target is unclear. Second, the same error constraint is usually used for servo conditions such as start-stop, reciprocating, or heavy load of permanent magnet synchronous linear motors, which can easily lead to large overshoot or slow convergence in local operating conditions, making it difficult to reflect the differentiated requirements of different operating conditions. Moreover, in actual systems, the safety margin and performance requirements of the forward and reverse errors are often different, which is not friendly to asymmetrical loads or unidirectional constraint scenarios. To address this, this method proposes a performance boundary function for directional convergence of the guided error and an asymmetric error constraint interval based on the desired control requirements. This explicitly transforms the performance requirements under different servo operating conditions into performance constraints. The introduction of the asymmetric error constraint interval precisely describes the actual engineering requirements. Based on this, a performance boundary function is constructed to allow the guided error to converge in a specified direction and rate—the desired monotonic error transformation function. This explicitly defines the maximum allowable range of error variation over time. Error convergence is strictly limited within the predetermined contraction boundary throughout the entire servo control process, allowing independent adjustment of the convergence speed and steady-state accuracy. This effectively avoids undesirable servo transients such as slow convergence and lack of convergence accuracy, achieving a performance constraint-guided effect throughout the entire control process. The desired monotonic error transformation function is a monotonic and invertible constraint term, possessing the advantage of not introducing additional unstable factors and eliminating the inherent disconnect between performance constraints and the control law.

[0106] It should be noted that the first-order sliding mode variable is a single mapping of the position tracking error and its derivative through a linear combination, ensuring that the error converges synchronously when the sliding mode variable approaches zero. Injecting the equivalent control disturbance parameters and prior control gain as servo system dynamics into the first-order sliding mode variable creates a consistent influence channel of the control input on the sliding mode variable. The evolution of the position tracking error is directly adjusted by the servo system dynamics, making the control design closer to actual operating conditions and achieving structural isolation of system uncertainties, which is beneficial for suppressing lumped uncertainties in subsequent system sliding mode design. From the formula of the nonlinear sliding mode dynamics model, it can be observed that when... = When =1, the position tracking error of the permanent magnet synchronous linear motor servo is: This would then evolve into traditional linear error dynamics; otherwise, the proposed nonlinear error dynamics can be equivalent to a mass-nonlinear damping-nonlinear spring system. By directly embedding the nonlinear error dynamics into the error convergence control law, ensuring that the closed-loop system converges according to a preset trajectory, and then substituting the equivalent control term into the position error dynamics model of the PMSLM, the specific formula of the equivalent convergence control law is as follows:

[0107] ;

[0108] This enables high-precision position tracking of the PMSLM servo system under complex disturbances and multi-servo operating conditions. By introducing a nonlinear sliding mode error dynamics design based on a performance boundary error transformation mechanism, this method achieves fast, smooth, and robust position control of the PMSLM servo system under multiple operating conditions while strictly constraining the evolution of position tracking errors. This ensures high precision and rapid convergence of the PMSLM position tracking error, overcoming the technical gaps and defects of traditional linear position tracking error dynamics, which struggle to balance dynamic performance and steady-state accuracy.

[0109] Preferably, step S3 specifically includes the following steps:

[0110] The PMSLM kinematic equations are algebraically rearranged to eliminate the explicit acceleration dependence of the system's lumped uncertainty perturbation, generating PMSLM kinematic equations containing lumped uncertainty.

[0111] The time delay assumption is introduced, which assumes that the lumped uncertainty of the PMSLM servo system satisfies the following within a short-term variable delay interval:

[0112] ;

[0113] Select the sampling time period of the servo system, synchronously build a delay buffer empty stack, collect the input signal of historical servo delay online according to the sampling time period and transfer and store it to the delay buffer empty stack, and generate a variable delay signal buffer queue.

[0114] The equivalent order model of the PMSLM servo system is pushed forward to time t0. The PMSLM kinematic equations containing lumped uncertainty are replaced by a variable delay signal buffer queue at time t-t0 to approximate the time delay information of the servo control input before the current uncertain disturbance of the system. A series of digital variable delay estimation terms of system sampling are obtained.

[0115] A series of digital variable delay estimation terms are substituted into the time delay assumption to calculate the delay estimation error, and the stable chaotic index of the online estimated position tracking error is output.

[0116] When the stable chaos index is less than 1, it is determined that the time delay estimation has reached the convergence state, the online time delay estimation operation is terminated, and the uncertainty estimate based on the time delay estimation is obtained when the PMSLM servo system is disturbed.

[0117] It should be noted that, in order to accurately estimate the uncertainty of the PMSLM system, this method uses time-delay dynamics for approximation. However, since the lumped uncertainty term explicitly depends on acceleration, the acceleration signal usually needs to be obtained by second differentiation. Differentiation easily amplifies noise components, leading to severe noise phenomena, often requiring additional filtering. This undoubtedly significantly increases the burden and accumulated error of servo system delay estimation, making it difficult to directly and accurately estimate time delay online. To address this, this method algebraically rearranges the PMSLM kinematic equations, separating the lumped uncertainty from the explicit acceleration term without changing the physical meaning of the system. This constructs an equivalent dynamic expression that facilitates delay estimation, avoiding the direct use of acceleration signals containing high noise and improving the numerical stability of time delay estimation. The basic idea of ​​time-delay dynamics is to assume... It can be continuous or segmented continuous for a sufficiently small time interval t0. The value at time t will be infinitely close to the value at time t. The value at time t-t0 is used to introduce a time delay assumption, thus assuming that the lumped uncertainty term changes slowly in a short period of time. This establishes a continuous approximation relationship and feasibility estimation between the current unknown disturbance and its historical values, relaxing the high dependence on the fixed sampling period. In the actual digital implementation process, the delay time is generally selected as the system sampling time. By collecting the servo delay input signal at time t-t0 in the system time and performing stack interpolation buffering, the signal includes the historical servo state of the motor and the historical control information input, thus providing reliable and dynamically adjustable historical servo delay data (variable delay signal buffer queue) for flexible selection of different delay lengths and approximation of unknown disturbances, improving the adaptability of online uncertainty estimation to sampling jitter and variable delay. This is achieved by pushing the equivalent order model forward to time t0, specifically the forward formula:

[0118] ;

[0119] Traditional time-delay estimation dynamics cannot be mapped to the actual servo control input of a permanent magnet synchronous linear motor, leading to a severe disconnect between disturbance estimation and system input. To address this, this method uses time-delay data to replace the current uncertainty term, thereby approximating the unknown disturbance. This establishes a crucial link between historical control data and current disturbance estimation, eliminating acceleration measurement errors and noise amplification issues caused by numerical differentiation, and improving the smoothness and reliability of online disturbance estimation. The PMSLM uncertainty estimate is:

[0120] ;

[0121] From the above formula, it can be seen that to obtain The estimated value only requires knowing the current input value and linear acceleration at the previous time step of PMSLM, and the shorter the delay time t0, the higher the estimation accuracy. The stable chaotic index measures the stability and convergence of the online time delay estimation process. Through stability analysis, it is easy to see that the convergence condition for time delay estimation is: That is, when the stable chaos index is less than 1, blind continuous estimation is avoided, and the risk of continuous accumulation of noise, chattering and delay is reduced when estimating uncertainty.

[0122] Preferably, step S4 specifically includes the following steps:

[0123] By combining the equivalent convergent control law with the control synthesis conversion of the permanent magnet synchronous linear motor servo position tracking error of the aforementioned uncertainty estimate, the time-delay position tracking control law is obtained, and the specific formula is as follows:

[0124] ;

[0125] Define a sub-term for time delay estimation error, perform system error analysis on the time delay position tracking control law based on the equivalent order model using equivalent disturbance input, and construct a dynamic model for the delay estimation error, the specific formula of which is:

[0126] ;

[0127] By substituting the time delay estimation error sub-term into the time delay estimation error dynamic model and solving it, the specific time delay estimation error value is obtained, and the specific formula is as follows:

[0128] ;

[0129] In the formula: This represents the time delay estimation error value;

[0130] Based on the desired control requirements of the permanent magnet synchronous linear motor, a gain shaping index related to the position tracking error is shaped under different servo operating conditions. Based on this gain shaping index, a sliding mode variable is determined. The specific formula for the sliding mode variable is as follows:

[0131] ;

[0132] By performing a variable structure integral with dynamic gain on both sides of the nonlinear sliding mode dynamics model, the ideal response speed of the PMSLM servo system is obtained, as shown in the following formula:

[0133] ;

[0134] An adaptive gain law is introduced to suppress the time delay estimation error between the sliding mode variable and the ideal response speed. The dynamic gain is smoothed in the adaptive gain law, and the correction gain is adjusted online to obtain an adaptive correction term for the delay estimation error.

[0135] It should be noted that, because the servo system cannot accurately distinguish between modeling errors and sudden disturbances, the source of delay estimation error is unclear. The controller still relies on high-gain robust terms, and in this case, the time-delay controller can only passively amplify the gain, leading to deviations in position tracking error correction. Furthermore, the estimation error remains only qualitative, and the correction strategy is set based on experience, making error correction prone to extreme situations such as over-compensation or under-compensation. Therefore, this method constructs a master control law model describing how position tracking estimation error affects system convergence by combining the equivalent convergent control law and the uncertainty estimate, namely, a time-delay position tracking control law. The delay estimation error is no longer an implicit term; it is explicitly compensated for the impact of the previous moment's disturbance through the current control input, achieving simultaneous execution of error dynamics shaping and disturbance compensation, significantly reducing the long-term dependence on high switching gain or strong robust terms. From the formula of the delay estimation error dynamics model, it can be seen that if... If the position error can be accurately estimated using uncertainty estimates, then the time-delay estimation error dynamic model is equivalent to the nonlinear sliding mode dynamic model. This means that under the action of the time-delay position tracking control law, the PMSLM system can achieve the desired position error convergence trajectory. However, in reality, even for a small sampling period t0, it is not possible to accurately estimate the position error. Because inherent measurement noise and nonlinear friction inevitably lead to estimation errors, the definition of the time delay estimation error sub-term is as follows: The delay estimation error is explicitly solved from the abstract model to obtain the adaptive adjustment error. This enables online, computable, and feedback-enabled targeted quantification of delay estimation errors, providing precise input for the delay estimation correction strategy. Simultaneously, this method quantifies the current system operating state and the "demand intensity" of control for uncertainty compensation based on gain shaping indices for different servo operating scenarios, constructing a sliding mode variable. This sliding mode variable can perceive the magnitude and trend of system error, rather than blindly increasing or decreasing control gain with a fixed compensation intensity, thus improving the accuracy and reliability of delay estimation correction.

[0136] It should be noted that, in order to prevent excessive chattering and because traditional fixed gain cannot accommodate different disturbance intensities, this method introduces an adaptive gain law. Based on this adaptive gain law, the variable structure correction gain is automatically adjusted according to the error evolution to smoothly suppress the delay estimation error, achieving dynamic matching between the error magnitude and the control strength. Compensation is enhanced for large errors and automatically weakened for small errors. By combining the two formulas for sliding mode variables and the ideal response speed of the PMSLM servo system, the following can be obtained: From this equation, it can be observed that the sliding mode variable is the error between the ideal speed and the actual speed of the PMSLM. Therefore, in order to suppress the time delay estimation error, the design formula for the adaptive delay estimation correction term is as follows:

[0137] ;

[0138] In the formula: For adaptive correction coefficients, >0;

[0139] This effectively avoids overcompensation or undercompensation issues caused by fixed gain, reducing control chattering and enhancing stability. When the error in time delay estimation increases, it leads to an increase in the tracking accuracy and sliding mode variable of the PMSLM. As can be seen from the second term of the adaptive delay estimation correction term formula, the adaptive correction coefficient also increases synchronously, thus overcoming the influence of the estimation error. After introducing the correction term, the optimized time delay estimation position tracking controller is:

[0140] ;

[0141] This method optimizes the time delay estimation position tracking controller by designing an adaptive delay estimation correction term. While ensuring the fast and smooth convergence of the PMSLM servo system, it effectively suppresses the impact of time delay estimation error on control performance and significantly improves the robustness of the servo system.

[0142] Based on the above embodiments, performance data examples of the nonlinear time-delay position tracking control of the present invention are further provided:

[0143] On a PMSLM servo system platform, the method of this invention and the traditional linear sliding mode method were compared. The PMSLM motor parameters used are as follows: mover winding phase resistance of 3.62Ω, quadrature-axis and direct-axis inductance of 0.004H, permanent magnet flux linkage of 0.0924 Wb, viscous friction coefficient of 9.36 Ns / m, pole pitch of 0.0237m, and mover mass of 1.9kg. During the experiment, the motor operated in position loop mode, given a sinusoidal position trajectory with a period of 1s and an amplitude of 0.1m. The position tracking results are as follows. Figures 2-5 As shown.

[0144] from Figure 2 , 3 It can also be seen that the maximum steady-state error of the traditional linear sliding mode is 0.74 mm, while the maximum steady-state error of the method of this invention is 0.065 mm, clearly demonstrating that the method of this invention has better position tracking performance. This is because the time delay estimation term in the method of this paper effectively compensates for the lumped disturbances present in the system, and the integral term in the designed sliding mode variable can further reduce the steady-state error of the system. Furthermore, from... Figure 4 , 5 It can be seen that the control input current of traditional linear sliding mode Severe chattering and noise are observed in the SMC's control law, which includes discontinuous sign function terms and requires a sufficiently large switching gain to overcome system uncertainties. This not only hinders further improvement in tracking accuracy but also introduces significant motor noise. In contrast, the control input current of the method described in this invention... The fluttering phenomenon has been significantly reduced.

[0145] A second aspect of the present invention provides a time-delay position tracking control system for a permanent magnet synchronous linear motor, such as... Figure 6 As shown, the time-delay position tracking control method for a permanent magnet synchronous linear motor, applied to any one of the claims, specifically includes:

[0146] A time delay estimation module is used to estimate the uncertainty of the PMSLM servo system based on time delay estimation online, eliminating most of the continuous nonlinear errors;

[0147] The desired position tracking error dynamics calculation module is responsible for designing the desired nonlinear position tracking error dynamics and the equivalent control law.

[0148] An adaptive correction module for time delay estimation error is provided. This module is responsible for suppressing the time delay estimation error between the sliding variable and the ideal response speed of the system by adjusting the correction gain through an adaptive gain law, and for designing an adaptive correction term for the delay estimation error.

[0149] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A time-delay position tracking control method for a permanent magnet synchronous linear motor, characterized in that, Includes the following steps: S1: Construct the PMSLM kinematic equations under orthogonal axial coordinates considering different types of disturbances in the servo system. Perform equivalent processing of the same order based on the higher-order extension of the PMSLM kinematic equations to obtain the equivalent order model. Based on the equivalent order model, derive the position tracking error of the permanent magnet synchronous linear motor servo to obtain the position error dynamic model of PMSLM. S2: Construct a monotonic error transformation function based on the desired control requirements of the motor. Substitute the servo dynamic data of the position tracking error generated by the PMSLM servo system into the derivative of the monotonic error transformation function to generate a nonlinear sliding mode dynamic model. Based on the position error dynamic model, use the nonlinear sliding mode dynamic model to perform convergence derivation and design to obtain the equivalent convergence control law. S3: Collect historical servo delay input signals from the sampling time period of the servo system. Based on the time delay assumption, push forward the equivalent order model to time t0 and then replace the collected delay signal queue data into the system. Estimate the lumped uncertainty of the system time delay online and obtain the uncertainty estimate based on time delay estimation when the PMSLM servo system is disturbed. S4: Define the time delay estimation error and introduce a sliding mode variable. Suppress the time delay estimation error between the sliding mode variable and the ideal response speed of the system by adjusting the correction gain of the adaptive gain law. Design an adaptive correction term for the delay estimation error. S5: The equivalent convergent control law, the uncertainty estimate based on time delay estimation, and the adaptive correction term are summed together to obtain the position tracking optimization control law of the PMSLM servo system. Specifically, S2 includes the following steps: Obtain the desired control requirements for position tracking error of permanent magnet synchronous linear motors applied in different servo operating scenarios, and formulate the performance boundary function and asymmetric error constraint interval for guiding error directional convergence based on the desired control requirements; By combining the performance boundary function and the asymmetric error constraint interval, the tracking error at each position is mapped to an unconstrained variable through error normalization, and the desired monotonic error transformation function is constructed. The specific formula is as follows: ； The equivalent control disturbance parameters and corresponding prior control gain when the PMSLM servo system generates position tracking error are obtained. The equivalent control disturbance parameters and prior control gain are bound as servo system dynamics terms, and the error dynamics space is constructed based on the first derivative of the position tracking error. The desired monotonic error transformation function is differentiated in the error dynamics space. By differentiation, the second-order error system of the PMSLM kinematic equations is reduced to a first-order sliding mode variable. Substituting the servo system dynamics term into the first-order sliding mode variable, the nonlinear sliding mode dynamics model is obtained, with the specific formula as follows: ； In the formula: and The positive coefficients of the desired error dynamics to be designed are... ; An error convergence control law is established, and the nonlinear sliding mode dynamics model is imported into the error convergence control law to derive the closed-loop convergence of the sliding mode error state. An equivalent control term for the desired nonlinear position tracking error dynamics is designed and generated. Substituting the equivalent control term into the position error dynamics model of the PMSLM, the equivalent convergence control law for the servo position tracking error of the permanent magnet synchronous linear motor is obtained.

2. The time-delay position tracking control method for a permanent magnet synchronous linear motor according to claim 1, characterized in that, S1 specifically includes the following steps: The position and mass parameters of the mover of the permanent magnet synchronous linear motor, as well as the uncertainty parameters of different servo system disturbances, are obtained. Among them, the different servo system disturbances include load changes, nonlinear friction, and thrust fluctuations. By dynamically coupling the mover position information, mover mass parameters and various uncertainty parameters with perturbation observation, a PMSLM kinematic equation considering different types of perturbations of the servo system in orthogonal axial coordinates is established. The unknown state variables of the PMSLM servo system are injected into the explicit extended state into the PMSLM kinematic equations. An equivalent higher-order extension of the PMSLM servo system is performed by detecting and analyzing the time-varying rate trend of the explicit extended state. The higher-order extension observed in the final gain of the PMSLM kinematic equations is then compressed into a perturbation estimate of the same order, yielding the equivalent order model of the PMSLM servo system. The specific formula is as follows: ； In the formula: For the gain to be designed, This represents nonlinear lumped time-varying dynamics, which includes parameter uncertainty, thrust fluctuation, and nonlinear friction. Define the position tracking error of the permanent magnet synchronous linear motor servo, and import the position tracking error into the equivalent order model for derivation to obtain the position error dynamic model of PMSLM.

3. The time-delay position tracking control method for a permanent magnet synchronous linear motor according to claim 2, characterized in that, The unknown state variable terms of the PMSLM servo system are injected into the explicit extended state into the PMSLM kinematic equations. By detecting and analyzing the time-varying rate trend of the explicit extended state, an equivalent higher-order extension of the PMSLM servo system is performed. The higher-order extension observed in the final gain of the PMSLM kinematic equations is compressed into a perturbation estimate of the same order, thus obtaining the equivalent order model of the PMSLM servo system. The specific steps include: Obtain the unknown state variable terms of the PMSLM servo system, construct and introduce an explicit extended state based on the unknown state variable terms in the PMSLM kinematic equations, detect the real-time uncertain changes of the servo system after the intervention of the unknown terms, and obtain the time-varying rate of the explicit extended state. A high-frequency time-varying rate threshold is set for the system's rapid disturbance or nonlinear lumped time-varying uncertainty. If the time-varying rate exceeds the high-frequency time-varying rate threshold, then the explicit expansion state is subjected to higher-order expansion processing during the above-mentioned introduction process to generate a higher-order ESO gain observer for PMSLM servo kinematics. Based on nonlinear lumped time-varying dynamics, the equivalent compression concept of bandwidth matching and gain reconstruction is introduced. By using the equivalent compression concept, the different high-order expansion states output by the high-order ESO gain observer are equivalently mapped and compressed into the second-order perturbation estimate of the PMSLM kinematic equation, thus obtaining the equivalent order model of the PMSLM servo system.

4. The time-delay position tracking control method for a permanent magnet synchronous linear motor according to claim 1, characterized in that, S3 specifically includes the following steps: The PMSLM kinematic equations are algebraically rearranged to eliminate the explicit acceleration dependence of the system's lumped uncertainty perturbation, generating PMSLM kinematic equations containing lumped uncertainty. The time delay assumption is introduced, which assumes that the lumped uncertainty of the PMSLM servo system satisfies the following within a short-term variable delay interval: ； Select the sampling time period of the servo system, synchronously build a delay buffer empty stack, collect the input signal of historical servo delay online according to the sampling time period and transfer and store it to the delay buffer empty stack, and generate a variable delay signal buffer queue. The equivalent order model of the PMSLM servo system is pushed forward to time t0. The PMSLM kinematic equations containing lumped uncertainty are replaced by a variable delay signal buffer queue at time t-t0 to approximate the time delay information of the servo control input before the current uncertain disturbance of the system. A series of digital variable delay estimation terms of system sampling are obtained. A series of digital variable delay estimation terms are substituted into the time delay assumption to calculate the delay estimation error, and the stable chaotic index of the online estimated position tracking error is output. When the stable chaos index is less than 1, it is determined that the time delay estimation has reached the convergence state, the online time delay estimation operation is terminated, and the uncertainty estimate based on the time delay estimation is obtained when the PMSLM servo system is disturbed.

5. The time-delay position tracking control method for a permanent magnet synchronous linear motor according to claim 1, characterized in that, S4 specifically includes the following steps: By combining the equivalent convergent control law with the control synthesis conversion of the permanent magnet synchronous linear motor servo position tracking error of the aforementioned uncertainty estimate, the time-delay position tracking control law is obtained, and the specific formula is as follows: ； Define a sub-term for time delay estimation error, perform system error analysis on the time delay position tracking control law based on the equivalent order model using equivalent disturbance input, and construct a dynamic model for the delay estimation error, the specific formula of which is: ； By substituting the time delay estimation error sub-term into the time delay estimation error dynamic model and solving it, the specific time delay estimation error value is obtained, and the specific formula is as follows: ； In the formula: This represents the time delay estimation error value; Based on the desired control requirements of permanent magnet synchronous linear motors, gain shaping indices related to position tracking errors under different servo operating conditions are shaped, and sliding mode variables are determined based on the gain shaping indices. By performing a variable structure integral with dynamic gain on both sides of the nonlinear sliding mode dynamics model, the ideal response speed of the PMSLM servo system is obtained. An adaptive gain law is introduced to suppress the time delay estimation error between the sliding mode variable and the ideal response speed. The dynamic gain is smoothed in the adaptive gain law, and the correction gain is adjusted online to obtain an adaptive correction term for the delay estimation error.

6. A time-delay position tracking control system for a permanent magnet synchronous linear motor, applied to the time-delay position tracking control method for a permanent magnet synchronous linear motor as described in any one of claims 1-5, the system specifically comprising: A time delay estimation module is used to estimate the uncertainty of the PMSLM servo system based on time delay estimation online, eliminating most of the continuous nonlinear errors; The desired position tracking error dynamics calculation module is responsible for designing the desired nonlinear position tracking error dynamics and the equivalent control law. An adaptive correction module for time delay estimation error is provided. This module is responsible for suppressing the time delay estimation error between the sliding mode variable and the ideal response speed of the system by adjusting the correction gain through an adaptive gain law, and for designing an adaptive correction term for the delay estimation error.

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