A fixed-time adaptive tracking control method for stochastic high-order nonlinear multi-time-delay systems

Abstract

The present application relates to the technical field of electromechanical control, and especially relates to a fixed-time adaptive tracking control method for a random high-order nonlinear multi-time-delay system, which is applied to electromechanical equipment and comprises the following steps: a high-order dynamic model containing time-varying time delay, unmodeled dynamics and noise is established; coordinate transformation is implemented to calculate tracking errors at all levels, an error auxiliary signal is constructed to offset input time delay; a neural network is used to approximate nonlinear terms, and a hyperbolic tangent function is introduced to eliminate zero discontinuity and avoid algebraic loops; a Lyapunov-Krasovskii functional is constructed to compensate for state time delay, and a bounded decaying signal generated by a dynamic mechanism is used to suppress unmodeled dynamics; control laws and adaptive laws at all levels are recursively designed to generate control commands; parameters are constrained according to fixed-time convergence conditions, and are converted into driving signals. The present application can eliminate tracking errors at all levels within a fixed time, ensure that signals in a closed-loop system are bounded, and solve the problem that trajectory convergence time is uncontrollable under multiple disturbances.

Description

Technical Field

[0001] This invention relates to the field of electromechanical control technology, and in particular to a fixed-time adaptive tracking control method for stochastic high-order nonlinear multi-time-delay systems. Background Technology

[0002] With the development of industrial automation technology, the requirements for trajectory tracking accuracy and safety of electromechanical equipment in complex operating environments are constantly increasing. During physical operation, the acquisition of sensor signals and the issuance of commands by actuators inevitably generate time-varying state delays and input delays, and the wear of mechanical transmission components introduces unmodeled dynamics. Currently, most control schemes for electromechanical equipment with high-order nonlinearity and random noise interference adopt adaptive control strategies based on asymptotic stability theory.

[0003] However, traditional asymptotic stability control strategies have a fatal technical flaw: the convergence time of the control system is highly dependent on the initial physical values ​​of the electromechanical equipment system's state vector. When the electromechanical equipment faces extreme initial position deviations at startup, or encounters strong random noise and unknown dynamic disturbances during controlled operation, traditional asymptotic stability controllers cannot provide a definite upper bound on the error convergence time. This aforementioned flaw causes the mechanical body of the electromechanical equipment to be unable to complete trajectory tracking within a predictable, absolutely safe time threshold, easily leading to prolonged physical phase distortion, and even causing the drive system to diverge and become uncontrollable. Summary of the Invention

[0004] To overcome the above shortcomings, this invention provides a fixed-time adaptive tracking control method for stochastic high-order nonlinear multi-delay systems. It aims to improve the problem that traditional electromechanical trajectory control, which mostly adopts asymptotic stabilization strategies, suffers from errors that cannot be eliminated as expected and are prone to divergence under complex operating conditions due to the convergence time being highly dependent on the initial state of the system.

[0005] This invention provides the following technical solution: a fixed-time adaptive tracking control method for stochastic high-order nonlinear multi-time-delay systems, applied to trajectory tracking control of electromechanical equipment, comprising the following steps:

[0006] S1. Obtain the state parameters of the electromechanical equipment, establish a high-order dynamic model that includes time-varying time delay, unmodeled dynamics and random noise, extract the state vector of the high-order dynamic model, and set the constraints.

[0007] S2. Perform coordinate transformation on the state vector to calculate the tracking error at each level, and construct an error auxiliary signal to offset the input time delay. The error auxiliary signal is limited by the control command and the delay term of the control command.

[0008] S3. The nonlinear terms in the higher-order dynamic model are approximated by a neural network to obtain the network output. The hyperbolic tangent function is introduced to eliminate the zero discontinuity of the network output to avoid algebraic loops.

[0009] S4. Construct a Lyapunov-Krasovskii functional to compensate for state time delays, and use a bounded decay signal generated by a dynamic mechanism to suppress the unmodeled dynamics.

[0010] S5. Based on the tracking errors at each stage, the network output, the Lyapunov-Krasovskii functional, and the bounded decay signal, recursively design the virtual control law and the parameter adaptive law at each stage; combine the final stage tracking error, the final stage parameter adaptive law, and the error auxiliary signal to generate the control command.

[0011] S6. Based on the fixed-time convergence condition, update the network weights and constrain the controller parameters using the adaptive laws of the parameters at each level to ensure that the closed-loop system signal is bounded.

[0012] S7. The control command is converted into a drive signal and output to the actuator of the electromechanical equipment to drive the tracking reference signal and eliminate the tracking errors at each level within a fixed time.

[0013] By adopting the above technical solution, integrating multiple time delay compensation and fixed-time convergence conditions, and then outputting continuous and bounded control commands, the problem of traditional electromechanical trajectory control, which mostly adopts asymptotic stabilization strategies, being highly dependent on the initial state of the system for convergence time, is that errors cannot be eliminated as expected under complex working conditions and are prone to divergence.

[0014] The present invention has the following beneficial effects:

[0015] 1. In this invention, by integrating multiple time delay compensations and fixed-time convergence conditions, continuous and bounded control commands are output, thereby improving the problem that traditional electromechanical trajectory control mostly adopts asymptotic stabilization strategies. Since the convergence time is highly dependent on the initial state of the system, errors cannot be eliminated as expected under complex working conditions and are prone to divergence.

[0016] 2. In this invention, by constructing an error auxiliary signal to cancel the input time delay, the phase delay of command communication transmission is eliminated, thereby improving the conventional execution mechanism that is mostly used in electromechanical control without feedforward compensation. Since it is impossible to dynamically quantify and cancel the command lag deviation, the physical execution components are prone to phase distortion and high-frequency chattering.

[0017] 3. In this invention, by using bounded attenuation signals to suppress unmodeled dynamics, the energy expansion of unknown physical disturbances is offset, thereby improving the problem that traditional electromechanical control, which mostly relies on high-precision deterministic parameter models, is prone to trajectory divergence when facing unknown frictional resistance due to the difficulty in dynamically measuring sudden disturbances such as mechanical gear wear. Attached Figure Description

[0018] Figure 1 This is a flowchart of a fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system proposed in this invention;

[0019] Figure 2 This is a detailed flowchart of the dynamic model construction and error feedforward generation of a fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system proposed in this invention.

[0020] Figure 3 This is a flowchart illustrating the neural network approximation and functional disturbance rejection mechanism analysis of a fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system proposed in this invention.

[0021] Figure 4 The flowchart shows the adaptive law generation and fixed-time energy convergence definition of the fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-delay system proposed in this invention.

[0022] Figure 5 This is a flowchart of the hardware physical execution control closed-loop application logic of a fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-delay system proposed in this invention.

[0023] Figure 6 This is a physical model diagram of a two-degree-of-freedom mass-spring-damped system for a fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system proposed in this invention.

[0024] Operating Condition 1: Simulation Verification under the First Set of Initial Parameters

[0025] Figure 7 This is a comparison diagram of the continuous running trajectory of the system state variable tracking reference signal under the operating condition of the present invention;

[0026] Figure 8 This is a timing diagram of the control command input under operating condition one of the present invention;

[0027] Figure 9 This is a diagram illustrating the convergence process of neural network adaptive weight estimation under condition one of the present invention.

[0028] Figure 10 This is a bounded periodic evolution diagram of the system under the first operating condition of this invention without modeling the dynamics;

[0029] Working Condition 2: Simulation Verification under the Second Set of Initial Parameters

[0030] Figure 11 This is a comparison diagram of the continuous running trajectory of the system state variable tracking reference signal under the second working condition of the present invention;

[0031] Figure 12 This is a timing diagram of the control command input under operating condition two of the present invention;

[0032] Figure 13 This is a diagram illustrating the convergence process of neural network adaptive weight estimation under condition two of the present invention.

[0033] Figure 14 This is a bounded periodic evolution diagram of the system under condition two of the present invention, without modeling the dynamics. Detailed Implementation

[0034] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0035] Example 1:

[0036] In a first embodiment of the present invention, a fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system is provided, which is applied to the trajectory tracking control of electromechanical equipment, such as... Figures 1-2 As shown, it includes the following steps:

[0037] S1. Obtain the state parameters of the electromechanical equipment, establish a high-order dynamic model that includes time-varying time delay, unmodeled dynamics and random noise, extract the state vector of the high-order dynamic model, and set the constraints.

[0038] Furthermore, in S1, the constraints are set as follows:

[0039] Obtain the unknown dynamic disturbance of the high-order dynamic model, and set the unknown dynamic disturbance to satisfy the upper bound constraint bound composed of the first uncertain non-negative smooth function and the second unknown monotonically increasing non-negative smooth function.

[0040] Acquire the reference signal of the electromechanical equipment, and set the reference signal and its derivatives to be smooth and bounded functions.

[0041] Obtain the state delay and input delay of the high-order dynamic model, and set the absolute value of the state delay, the derivative of the state delay, the absolute value of the input delay, and the derivative of the input delay to have limited upper bounds.

[0042] Specifically, at the data input / output level, physical sensors collect the displacement and velocity physical quantities of the electromechanical equipment in real time, converting them into system state vectors that are input to the controller's processing unit. Based on this, the system establishes a dynamic model and sets constraint boundaries, outputting the controlled system state vector and system output vector to the downstream coordinate transformation and functional compensation modules as reference data for calculating tracking errors at each stage.

[0043] The specific expression for the established dynamic model of the non-strict feedback stochastic high-order nonlinear multi-delay system is as follows:

[0044] ;

[0045] ;

[0046] ;

[0047] ;

[0048] in Let be the system state vector of the electromechanical equipment, and its expansion form is: . This is the system output vector. This represents the unmodeled dynamics of electromechanical equipment caused by physical wear or sudden changes in external load. This refers to an unknown continuous local Lipschitz nonlinear function that describes the unmodeled dynamic evolution. For higher-order parameters, and set To characterize the higher-order properties of the system. and It is an unknown continuous nonlinear function. It is an unknown continuous nonlinear function vector. The lag vector represents the system state. This refers to unknown dynamic disturbances existing within the system. Represents independent standard Brownian motion, used to characterize random noise in the physical environment at the algebraic level. This is the final control instruction issued to the executing agency. This refers to the time-varying delay during the transmission of the input signal. (Subscript) For system dimension variables, satisfying .

[0049] To ensure that the adaptive back-reasoning algorithm is theoretically solvable and physically safe under harsh conditions, S1 sets three core constraints on the model.

[0050] The first constraint establishes an analytical boundary for unknown dynamic disturbances, and its expression is:

[0051] ;

[0052] in For the first uncertain nonnegative smooth function, Let it be the second unknown monotonically increasing non-negative smooth function. This represents the Euclidean norm of the variable. The technical purpose of this setting is to cut off the unrestricted energy exchange path between the unmodeled dynamics and the system's main state. The sudden changes in frictional force in electromechanical systems during actual operation are unpredictable. Locking the disturbance within the upper bound of the function composed of the state vector and the unmodeled dynamics prevents the controller from outputting infinite gain when compensating for extreme disturbances, providing a low-level mechanism to prevent divergence in the subsequent derivation of the global Lyapunov energy function.

[0053] The second constraint requires that the reference signal of the electromechanical equipment and its derivatives be smooth and bounded functions. This signal is denoted as... The derivation of the reverse control algorithm essentially requires repeated differentiation calculations of the virtual control laws at each level. If the desired reference signal has a step jump or a non-differentiable inflection point, the differentiation calculation will produce a value that tends towards infinity, directly leading to overload and burnout of the physical actuator motor. Constraining the boundedness of its derivatives eliminates the computational singularity risk caused by sudden trajectory jumps.

[0054] The third constraint imposes an extreme value limit on the rate of change of time delay in signal transmission, and the constraint relationship is as follows:

[0055] ;

[0056] ;

[0057] ;

[0058] ;

[0059] in The state is time-varying and time-delayed. This is the preset maximum allowable time delay boundary, and its value is a constant and strictly greater than 0. The constraint limit representing the rate of change of state time delay. These represent the constraint limits for the rate of change of the input time delay. Both of these rate of change limits are strictly limited to a range of values ​​less than 1. This constraint eliminates the phenomenon of out-of-order overlapping of data packets caused by extreme communication congestion. Locking the rate of change within a range of less than 1 not only ensures that physical communication data packets arrive at the actuator in chronological order, but more importantly, it guarantees that the time derivative term of the improved Lyapunov-Krasovskii functional constructed in step S4 can strictly maintain its negative definite property when performing differentiation and algebraic scaling, thus upholding the mathematical foundation for verifying the stability of the closed-loop system.

[0060] S2. Perform coordinate transformation on the state vector to calculate the tracking error at each stage, and construct an error auxiliary signal to cancel the input time delay. The error auxiliary signal is limited by the control command and the delay term of the control command.

[0061] Furthermore, in S2, coordinate transformation is performed on the state vector to calculate the tracking error at each stage, including:

[0062] Extract the first-level state variables from the state vector, calculate the difference between the first-level state variables and the reference signal, and generate the first-level tracking error.

[0063] Extract intermediate-level state variables from the state vector, calculate the difference between the intermediate-level state variables and the adjacent previous-level virtual control law, and generate intermediate-level tracking error;

[0064] Extract the final-level state variables from the state vector, and then combine and perform a joint operation on the final-level state variables, the penultimate-level virtual control law, and the error auxiliary signal to generate the final-level tracking error.

[0065] In S2, constructing the error auxiliary signal to cancel the input time delay includes:

[0066] The control command is extracted based on the higher-order exponential mapping term formed by the highest order of the system as the control input term at the current moment;

[0067] The delay term of the extracted control command is based on the higher-order exponential mapping term formed by the highest order of the system as the delay control input term;

[0068] Extract the linear function term and the odd-order higher-order function term of the error auxiliary signal;

[0069] By performing a joint difference operation on the current control input, the delayed control input, the linear function, and the odd-order higher-order function, a differential evolution equation for the dynamic evolution of the constraint error auxiliary signal is generated.

[0070] Specifically, the core of step S2 is to transform the original physical state tracking problem into an error convergence space and construct a feedforward compensation channel to address the communication delay at the input end. The data input / output flow of this step is as follows: the computation module receives the system state vector and desired reference signal established in the preceding steps as input benchmarks. After internal algebraic transformations and differential equation operations, it outputs the dimensionality-reduced tracking error sequences at each level, along with dynamically updated error auxiliary signals. These generated error data directly serve as the computational foundation for the subsequent neural network approximation module and the inverse control solution module.

[0071] The error coordinate transformation implemented on the state vector is specifically composed of the following set of algebraic equations:

[0072] ;

[0073] ;

[0074] ;

[0075] in This represents the first level of tracking error. This represents the tracking error of each intermediate stage in the series connection. This represents the final stage tracking error of the system. These are the extracted first-level state variables. This represents the expected reference signal pre-set in the preceding sequence. For each intermediate level of state variable, the subscript independent variable takes the value of a positive integer sequence from level two to the non-final level. Defined as the intermediate virtual control law of the previous level. These are the extracted final-level state variables. This is the virtual control law calculated from the penultimate level. The error auxiliary signal constructed for this step.

[0076] The aforementioned coordinate transformation strips away the physical absolute coordinate system of the electromechanical equipment's mechanical body. The first-level transformation directly compares the deviation between the initial state of the system and the reference signal. The intermediate-level transformation separates the deviation between the current state and the theoretical control target, thereby maintaining the progressively decreasing order recursive calculation of the adaptive back-calculation algorithm. The final-level coordinate transformation forcibly incorporates an error auxiliary signal, injecting an offset in advance into the highest-order operational dimension of the system, reserving an algebraic cancellation interface in advance for subsequent calculations of the final-level control law, which is unaffected by time delays.

[0077] To compensate for the transmission delay during the control command issuance process, step S2 constructs the differential evolution equation of the error auxiliary signal based on the error accumulation effect of the command deviation:

[0078] ;

[0079] in The time derivative of the error auxiliary signal. Defined as the control command derived at the current moment. This represents a delay control command that is hindered by unknown time-varying delays in the transmission link. For specific time-varying and time-delay variables. The highest-order parameter of the system, which characterizes the higher-order properties of the electromechanical model, is used as an exponential mapping term in the equation to amplify higher-order errors. This is a linear damping adjustment parameter, and its specific numerical condition is set to a positive real number that is strictly greater than 1. This is a higher-order damping adjustment parameter, and its numerical condition is defined as a positive real number that is strictly greater than 0.

[0080] This differential equation establishes a venting channel for time delay deviation quantification. Conventional back-calculation solution systems are prone to recursion interruption or numerical explosion when encountering control-end lag. This step directly extracts the current instruction's higher-order mapping and the delayed instruction's higher-order mapping for difference comparison, using the difference as the driving source term of the differential equation. A negative definite damped loop is constructed by relying on linear function terms in conjunction with odd-order higher-order function terms. This algebraic structure dynamically collects the phase distortion accumulated in the closed-loop link due to the input time delay. After solving this differential evolution equation and injecting its calculation result into the aforementioned final-stage tracking error, the system can directly cancel the positive interference energy caused by lag using algebraic rules when subsequently verifying energy attenuation conditions, completely eliminating the potential operating conditions of electromechanical actuator chattering or even divergence induced by control lag.

[0081] like Figure 3 As shown in Figure S3, the nonlinear terms in the higher-order dynamic model are approximated by a neural network to obtain the network output, and the hyperbolic tangent function is introduced to eliminate the zero discontinuity of the network output to avoid algebraic loops.

[0082] Furthermore, in S3, a neural network is used to approximate the nonlinear terms in the higher-order dynamic model to obtain the network output. A hyperbolic tangent function is introduced to eliminate the zero-point discontinuities in the network output and avoid algebraic loops, including:

[0083] By performing an inner product operation between the network weights and basis function vectors of a neural network, a basic approximation signal for the nonlinear term is constructed.

[0084] The denominator of the positioning control derivation includes singular correlation terms for tracking errors at each stage;

[0085] The algebraic ratio of each level of tracking error to a preset minimum constant is used as the independent variable and input into the hyperbolic tangent function to generate an even-power smoothing factor.

[0086] By using an even-power smoothing factor to multiply the singular correlation terms, the multiplied singular correlation terms are integrated with the basic approximation signal to generate a network output without zero singularity.

[0087] Specifically, the core technical function of step S3 is to handle the nonlinear unknown disturbances in the high-order nonlinear dynamic model and completely eliminate the inherent zero-point singularity trap in the derivation of the back-reasoning control law from the underlying algebraic logic, so as to ensure the safe issuance and continuous execution of control commands for electromechanical equipment.

[0088] Specifically, the data input / output process in this step involves the algorithm unit receiving the system state variables and tracking errors at various levels from the front-end module as raw input data. This input data enters the radial basis function neural network solving engine to perform forward mapping and nonlinear approximation operations, outputting a continuous network signal that has undergone hyperbolic tangent smoothing. This output signal is directly transmitted to the virtual control law recursion module as an algebraic compensation benchmark for eliminating nonlinear friction and unknown damping terms in the physical model.

[0089] For the unknown nonlinear terms contained within the higher-order dynamic model, the module extracts the dynamically updated network weights of the neural network and performs a vector dot product operation with the pre-calibrated basis function vectors to construct the basic approximation signal for the corresponding nonlinear term. Its basic algebraic structure is expressed as follows:

[0090] ;

[0091] in Defined as the system's first The nonlinear fundamental approximation signal of the order. This represents the dynamic weight vector that the neural network adaptively adjusts. This represents the radial basis function vector. This refers to the network input variable matrix assembled from the state variables of the electromechanical equipment system and the desired reference trajectory. (Superscript) This represents performing a matrix transpose operation.

[0092] Electromechanical equipment systems inherently possess high-order coupling characteristics with non-strict feedback. When reversing the control commands using traditional inverse calculation methods, singular correlation terms easily arise in the denominators of each level of the recursive equations, with the tracking errors at each level being present. Once the motion trajectory of the physical actuator closely matches the theoretical target, causing the tracking error to approach the zero coordinate point, the algebraic denominator of the singular correlation term immediately becomes zero. This algebraic phenomenon will trigger an algebraic loop deadlock where the control gain diverges towards infinity, directly leading to overload and damage to the physical actuator.

[0093] To overcome this algebraic discontinuity technical barrier, the algorithm extracts the tracking error values ​​at each level and performs algebraic division with a preset minimal constant. The calculated real ratio is then input as the independent variable into the hyperbolic tangent function module to generate a specific even-power smoothing factor. The corresponding mathematical generative relation is:

[0094] ;

[0095] in This represents the even-power smoothing factor of the solution output. As a hyperbolic tangent operator, it is forcibly assigned an even-degree mathematical property of the fourth power to ensure the absolute smoothness and differentiability of the function curve when it passes through the zero of the two-dimensional coordinate system. This refers to the corresponding level of tracking error. This is a preset minimum constant parameter, which specifically defines the narrowness of the effective range for smooth transition near the zero point.

[0096] After obtaining the aforementioned smoothing factor, the module uses this even-power smoothing factor to perform multiplicative modulation on the previously located singular correlation terms. Multiplicative modulation utilizes the absolute convergence property of the smoothing factor at zeros to forcibly cancel the divergence trend of the singular correlation terms. Subsequently, the multiplicative-modulated singular correlation terms are algebraically multiplied and integrated with the fundamental approximation signal to output a network output without zero-point singularities throughout the entire time domain. The final integration operation formula is expressed as:

[0097] ;

[0098] in This represents the network output that has been processed and possesses global continuity. This represents a specific singular correlation term in the denominator of the control derivation above, which includes tracking errors at various levels. The integrated solution of the multiplicative modulation ensures that the output signal can be safely embedded into the iterative closed loop of the subsequent fixed-time control law.

[0099] like Figure 3 As shown in Figure S4, construct a Lyapunov-Krasovskii functional to compensate for state delays, and use the bounded decay signal generated by the dynamic mechanism to suppress unmodeled dynamics.

[0100] Furthermore, in S4, a Lyapunov-Krasovskii functional is constructed to compensate for state delays, and a bounded decaying signal generated by a dynamic mechanism is used to suppress unmodeled dynamics, including:

[0101] An exponential decay coefficient is set to generate a time-domain integral kernel. The square term of the nonlinear state coupling function corresponding to the state delay in the higher-order dynamic model is multiplied by the time-domain integral kernel. Time-domain integration is performed in the dynamic integration interval to generate a Lyapunov-Krasovskii functional to cancel the state delay energy.

[0102] The non-negative state envelope constraint term is constructed based on the absolute value of the higher-order polynomial of the state vector. The linear attenuation coefficient is linearly superimposed with the negative product term of the attenuation signal variable, the non-negative state envelope constraint term, and the preset adjustment factor to establish a dynamic compensation differential equation.

[0103] Solve the dynamic compensation differential equation to generate a bounded attenuation signal.

[0104] Specifically, in the implementation of step S4, the data input / output flow of the control unit is as follows: the algorithm calculation module receives the electromechanical equipment state vector and time-varying state delay parameters from upstream as the calculation reference input, and performs integral superposition and ordinary differential equation solving within the module. After the solution is completed,

[0105] For time-varying and time-delay states, the improved Lyapunov-Krasovskii functional integral expression is constructed as follows:

[0106] ;

[0107] in Representative regarding the first The Lyapunov-Krasovskii functional component constructed for the stage subsystem to compensate for state lag energy. The set exponential decay coefficient is defined as a positive real number that is strictly greater than 0. This is the current running time. This represents the time-domain integral variable. The supremum of the state-delay derivative is defined, and its range is forcibly locked within the range of real numbers greater than 0 and strictly less than 1. This represents the time-varying time delay quantity corresponding to the system state. This represents the nonlinear state coupling function corresponding to the state delay.

[0108] The aforementioned integral structure plays a fundamental control role in preventing the divergence of hysteresis energy. In high-order nonlinear electromechanical systems containing multiple time-varying delays, historical state data can easily induce phase lag and oscillation surges in the closed-loop feedback channel. The functional construction process uses the squared term of the nonlinear state coupling function as the main body of the integral kernel, forcing the unknown nonlinear hysteresis terms that may cause divergence to be transformed into a non-negative energy storage form. The exponential term containing a set decay rate forces the accumulated energy to release exponentially over time. In the subsequent derivation of the control law and the solution of the system energy derivative, this functional can directly generate specific negative definite algebraic terms, using the sign cancellation rule to cancel out the redundant nonlinear disturbances caused by the hysteresis of state variables.

[0109] For unmodeled dynamics, the system constructs a dynamic mechanism to generate a bounded attenuation signal, and the established dynamic compensation differential equation takes the following form:

[0110] ;

[0111] in This represents the first derivative of a bounded decaying signal with respect to time. That is, the bounded decaying signal variable entity generated by the operation. The linear attenuation coefficient is a predefined linear attenuation coefficient, and its value is a positive real number greater than zero. Defined as a non-negative state envelope constraint term, its algebraic reconstruction logic is to extract the absolute values ​​of higher-order polynomials of the electromechanical equipment state vector and perform positive superposition. This is a preset positive real number adjustment factor.

[0112] Solving this differential equation cuts off the path to actuator failure caused by unmodeled dynamics. Since unmodeled dynamics derived from mechanical transmission wear often lack precise analytical expressions, this step abandons the conventional approach of directly estimating disturbance values ​​and instead establishes a dynamically changing energy envelope boundary. The non-negative state envelope constraint term tracks the surge trend of the current system state vector, while the negative product of the linear decay coefficient and the decay signal variable provides forced damping that compels the signal to converge towards the origin. The calculated bounded decay signal is directly injected into the negative definite energy boundary structure of the global closed-loop system, serving as a time-varying safety threshold to suppress the sudden disturbance expansion caused by various unmodeled dynamics.

[0113] like Figure 4 As shown in Figure S5, based on the tracking errors at each stage, network output, Lyapunov-Krasovskii functional, and bounded decay signal, virtual control laws and adaptive laws for each stage of parameters are recursively designed; control commands are generated by combining the final stage tracking error, the final stage adaptive law for parameters, and the error auxiliary signal.

[0114] Furthermore, in S5, the recursive design of virtual control laws and adaptive laws for parameters at each level includes:

[0115] Obtain the feedback gain coefficients at each level, the first exponential polynomial and the second exponential polynomial of the current level tracking error, and construct the feedback stabilization term;

[0116] Extract the squared terms of the basis function vectors of the neural network, the product of the third exponential polynomial of the current stage tracking error and the network weights, and construct the neural network compensation term;

[0117] The feedback stabilizing term and the neural network compensation term are superimposed and mapped using fractional radicals to generate the virtual control laws of each level in the current stage.

[0118] By combining adaptive gain adjustment, the even-order exponential polynomial of the current stage tracking error, and the squared terms of the basis function vector, a network driving term is constructed. Then, by combining an anti-drift suppression term containing first-order and third-order parameter correction components, the adaptive laws for each level of the current stage are generated.

[0119] In S5, the control commands generated by combining the final stage tracking error, the final stage parameter adaptive law, and the error auxiliary signal include:

[0120] Obtain the final stage feedback gain term and the exponential polynomial of the final stage tracking error, and construct the final stage reference stability term;

[0121] The final-level network compensation term is constructed by combining the squared terms of the basis function vectors corresponding to the adaptive law of the final-level parameters.

[0122] The linear amplification term and the nonlinear odd-power amplification term of the error auxiliary signal are extracted to generate the time-delay feedforward compensation signal;

[0123] The final-stage reference stability term, the final-stage network compensation term, and the time-delay feedforward compensation signal are added together, and the result of the addition is subjected to fractional-order radical mapping dimensionality reduction processing to output control commands.

[0124] Specifically, step S5 is the core convergence computation center of the control algorithm. The algorithm computation unit receives all series tracking errors generated in the preceding sequence, the approximation signal extracted by the neural network, the functional integral term used to compensate for state lag, and the bounded decay signal to suppress sudden disturbances as basic data inputs. After algebraic fusion and differentiation dimensionality reduction by the back-reasoning module, the determined control command entity is output to the physical interface at the end of the system, and the dynamically updated weights of each hidden layer are synchronously output to the neural network.

[0125] For the non-final-stage subsystems of electromechanical equipment, the system recursively solves for the virtual control laws and parameter adaptive laws at each stage. The first and second exponential polynomials for the current-stage tracking error configuration are extracted, and combined with the set feedback gain coefficients at each stage, the baseline feedback boundary for preventing system divergence is established. The squared terms of the basis function vectors of the neural network are extracted and multiplied with the current network weight estimates to construct neural network compensation terms to cancel uncertainties such as electromechanical friction. After the above terms are superimposed and integrated, fractional radical mapping is performed, and the mathematical equations for the intermediate-stage virtual control laws are derived as follows:

[0126] ;

[0127] in For the first Virtual control laws at the level of [level]. and These represent the first feedback gain coefficient and the second feedback gain coefficient, which are pre-calibrated and are greater than 0, respectively. The tracking error corresponding to the current calculation level. This represents the maximum constant value among all levels and orders of the electromechanical system. This corresponds to a specific level of system hierarchy. This is a set smoothing constant greater than 0. This represents the weight estimate of the current network layer. The corresponding extracted neural network basis function vectors, superscript This is the matrix transpose symbol. The outer wrapper... This is the fractional radical mapping term.

[0128] To prevent the weights of the neural network from drifting excessively and without restraint under severe random noise interference, an anti-drift suppression mechanism is introduced into the calculation process to generate adaptive laws for the parameters of each layer:

[0129] ;

[0130] in This is the time derivative of the weight estimator. This is an adaptively adjusted gain whose value is limited to the range of positive real numbers. The two algebraic terms at the end of the right side of the equal sign together constitute the anti-drift suppression term, where... For the set linear term parameter correction component, These are the parameter correction components for the cubic term, both of which are fixed constants greater than 0.

[0131] When the control logic is recursively passed to the final output of the system, the final physical control command is derived by integrating the final stage state characteristics and the time-delay feedforward mechanism. The final stage reference stability term is constructed using the final stage feedback gain combined with the final stage tracking error exponential polynomial. The final stage network compensation term is reconstructed based on the basis function vector pointed to by the final stage parameter adaptive law. The error auxiliary signal obtained in the preceding stage is linearly amplified and then amplified by odd-order higher powers, and the two are combined to generate a time-delay feedforward compensation signal specifically for crossing the input delay time barrier. These computational components are integrated and the final control command is output through dimensionality reduction mapping.

[0132] ;

[0133] in This is to issue control commands directly to the actuator driver hardware. and This is the positive definite feedback gain term of the final stage, which is greater than 0. This represents the final stage tracking error. Represents the entity of the error auxiliary signal. and These are the linear amplification gain and the nonlinear odd-power amplification gain set for the signal, respectively. It represents the highest order of the electromechanical system. and Each corresponds to a weight variable and a basis function vector in the final stage of the neural network. This is the final stage smoothing adjustment parameter.

[0134] The aforementioned control law solution architecture establishes a self-healing multi-source disturbance rejection calculation rule at the bottom layer of the electromechanical system. Addressing the inherent state dimension misalignment dilemma in high-order systems, each level of virtual control law utilizes a fractional-order radical mapping mechanism to perform nonlinear dimensionality reduction, forcibly compressing the complex high-order error manifold into a unified-dimensional energy judgment space. The cubic weight correction component introduced in the anti-drift suppression term provides strong nonlinear centripetal pull-back damping when the network evaluation value approaches the explosion threshold. Direct algebraic stitching of the time-delay feedforward compensation signal during the final-stage instruction generation essentially injects an equal amount of bias instruction energy in the opposite direction before uncontrollable chattering occurs in the physical execution components, severing the chain of physical lag and system collapse that is easily triggered by time-varying delays in the electromechanical equipment.

[0135] like Figure 4 As shown, S6, based on the fixed-time convergence condition, uses the adaptive law of each level of parameters to update the network weights and constrain the controller parameters to ensure that the closed-loop system signal is bounded;

[0136] Furthermore, in S6, based on the fixed-time convergence condition, the network weights are updated using adaptive laws for each level of parameters, and the controller parameters are constrained to ensure that the closed-loop system signal is bounded, including:

[0137] The global system energy function is formed by aggregating the exponential polynomial energy terms corresponding to the tracking errors at each level, the squared energy terms corresponding to the network weight estimation errors, the Lyapunov-Krasovskii functional, and the algebraic energy terms associated with the bounded decaying signal.

[0138] Calculate the stochastic differential operator of the global system energy function and perform algebraic scaling to generate a negative definite convergence inequality containing the fractional-order nonlinear components, integer-order nonlinear components, and bounded constant components of the global system energy function.

[0139] Based on the mapping relationship between the coefficients of each term in the negative definite convergence inequality and the control gain and adaptive adjustment parameters, the values ​​of the controller parameters are constrained in reverse, and the closed-loop convergence time threshold is limited to be independent of the initial value of the state vector.

[0140] Specifically, the steps of updating parameters and constraining the system based on fixed-time convergence conditions are the central criteria for determining the global stability and convergence of the algorithm. This technical module receives tracking errors at each level, network output weights, functional integral compensation terms, and bounded decay signals as input streams for the underlying computation. The arithmetic unit internally performs parametric inequality scaling and algebraic mapping solutions based on the set energy decay criteria. The control gain range and adaptive parameter boundary values ​​obtained from the solution are output inversely to the downstream system controller, directly solidified, and used to guide the physical drive level of the actuator.

[0141] To quantify the energy dissipation trend of the entire electromechanical equipment during controlled operation, the algorithm aggregates various error data to construct a global system energy function:

[0142] ;

[0143] in This represents the global system energy function. This represents the total dimensional order of the dynamic model of electromechanical equipment. Corresponding to the Level of system state tracking error. This is the even-order parameter for the energy term of the exponential polynomial, and its value must be defined as an even number greater than 0. These are preset adaptive adjustment parameters and are strictly limited to the range of positive real numbers. The variable representing the network weight estimation error is actually the algebraic difference between the ideal neural network weights and the current adaptive iterative output weights. The Lyapunov-Krasovskii functional component corresponding to the time-delay energy accumulation in the compensated state. This represents the bounded decaying signal variable representing the unmodeled dynamics of the suppression system. This aggregation operation brings together all the dispersed mechanical kinematic deviations, approximator estimation deviations, and residual energy from unknown frictional disturbances within the system into a single scalar computational space.

[0144] For controlled electromechanical objects containing complex time-varying delays and random noise interference, the computing unit uses Itō's lemma to perform stochastic differential operator differentiation on the above global system energy function, and derives negative definite convergent inequalities using Young's inequality principle:

[0145] ;

[0146] in This represents the stochastic differential operator corresponding to the global system energy function. This represents the coefficient of the first convergence constant. It is a fractional-order nonlinear component exponent, whose value is forcibly locked within an open interval between 0 and positive 1. This represents the set second convergence constant coefficient. The exponent of the nonlinear component is of integer order and is defined as a constant that is strictly greater than 1. This represents the bounded constant component that integrates the extreme values ​​of approximation errors from various networks with the upper bound of the physical limits of external bounded perturbations. Here, the scaling solution skips the extremely complex high-order nonlinear dynamic evolution process, reducing it to a strictly controlled algebraically decaying geometric boundary condition.

[0147] Based on the algebraic structure of the constructed negative definite convergence inequality, the computation module derives the parameter mapping rules between the constant coefficients and the actual control gain. The value space of the internal gain of the system controller is forcibly limited to the solution set where both the first and second convergence constant coefficients are positive. The convergence time threshold of the closed-loop system thus exhibits a clear algebraic boundary constraint requirement:

[0148] ;

[0149] in This represents the upper limit of the absolute time threshold for the closed-loop system to complete trajectory convergence. This step completely severs the dependence of the final convergence time on the initial physical state values ​​of the system from the underlying algebraic laws. Regardless of how extreme the geometric displacement deviation or velocity vector jump of the electromechanical equipment is at the moment the control command is initiated, the controller continuously applies a forced constraint mechanism to force various multi-level tracking errors to converge within the absolute time threshold, through online iterative updates of the weights using adaptive laws. Before reaching its destination, the signal completely dissipates, ensuring that the signal amplitude of all closed-loop feedback nodes is confined within a physically safe and bounded operating area.

[0150] like Figure 5 As shown, S7 converts the control command into a drive signal and outputs it to the actuator of the electromechanical equipment to drive the tracking reference signal and eliminate tracking errors at all levels within a fixed time.

[0151] Furthermore, in S7, the control commands are converted into drive signals and output to the actuators of the electromechanical equipment to drive the tracking reference signal and eliminate tracking errors at all levels within a fixed time, including:

[0152] Extract real-time logical operation values ​​of control commands;

[0153] The signal mapping mechanism is used to convert real-time logic operation values ​​into drive signals that match the physical interface of the actuator hardware of electromechanical equipment.

[0154] The drive signal is continuously sent to the actuator of the electromechanical equipment, which drives the mechanical body of the electromechanical equipment to generate a corresponding dynamic response trajectory.

[0155] The actual output state variables corresponding to the dynamic response trajectory are collected in real time by sensors. The actual output state variables are continuously compared with the reference signal through closed-loop feedback. Within a fixed convergence time threshold independent of the initial value of the state vector, the actual output state variables are forced to approach the reference signal until the tracking errors at all levels are eliminated.

[0156] Specifically, at the data input / output level, the underlying hardware microprocessor receives the control command logic values ​​derived from the upstream algorithm as digital input. After processing by the internal digital-to-analog converter interface, the controller outputs matching analog voltage or current drive signals to the external electromechanical equipment actuator. Under the excitation of the drive signal, the electromechanical equipment generates a real mechanical response trajectory. External physical sensors capture the physical quantities of this motion trajectory, convert them into feedback state variables, and re-input them to the controller's error calculation unit, thus completing the information loop.

[0157] By extracting the real-time logical operation values ​​of control commands and performing hardware conversion using a signal mapping mechanism, a transmission channel between the virtual algebraic space and the real energy space is established. The control commands calculated internally by the microprocessor are merely discrete data variables in registers, lacking the actual physical energy to drive mechanical components. By configuring the signal amplification factor to match the hardware physical interface through the mapping mechanism, the abstract control logic is converted into the physical quantity of the driving torque of a servo motor or the control pressure of a hydraulic component, ensuring a seamless transfer of algorithmic computing power to mechanical power.

[0158] The drive signal is continuously sent to drive the electromechanical equipment to generate a dynamic response. Combined with real-time sensor data acquisition and closed-loop comparison, a physical defense against external uncertain environmental disturbances is established. During actual operation, the electromechanical equipment inevitably encounters unknown disturbances such as sudden load changes or increased transmission friction resistance. By utilizing external high-precision sensors to transmit actual output state variables at high frequency, and comparing these with a predetermined reference signal using continuous difference analysis, the system actuator is forced to dynamically utilize previously derived and accumulated anti-disturbance compensation energy for physical displacement correction based on the measured end-to-end deviation.

[0159] By forcing the actual output state variable to approximate the reference signal within a fixed convergence time threshold until all kinds of errors are eliminated, this control scheme demonstrates extremely strong time deterministic boundary protection. The convergence constraint of this physical tracking error satisfies the following mathematical boundary equation:

[0160] ;

[0161] in The actual output state variables of electromechanical equipment represented by real-time sampling from physical sensors. This represents the desired reference signal preset by the system. The absolute value of the difference between the two represents the magnitude of the physical tracking deviation of the electromechanical system at the current moment. This represents the minimum steady-state error limit allowed by objective hardware factors such as sensor sampling resolution and mechanical gear dead zone, and its actual value is defined as a very small positive real number. It is a continuous time variable in the controlled operation process of the electromechanical system. This is a fixed convergence time threshold derived based on the convergence inequality of the global energy function.

[0162] Based on this convergence limit, regardless of how far the actual physical position of the electromechanical equipment deviates from the desired trajectory at startup, i.e., how extreme the initial state vector value is, the system controller has the execution constraint to forcibly pull it back to the controlled trajectory. The actuator will, before the system reaches its absolute time node, exert its limited physical work to forcibly compress the actual motion deviation to within the minimum steady-state error limit. Even in harsh operating environments with multiple superpositions of high-order nonlinear oscillations, data transmission delays, and spatial random noise, the electromechanical body can still complete the precise alignment and shaping of the preset trajectory on time, completely eliminating the engineering application shortcomings of conventional asymptotic stabilization schemes, such as long and unpredictable convergence times.

[0163] Example 2:

[0164] This invention focuses on the trajectory tracking and control of complex electromechanical equipment in harsh environments, such as multi-joint robots in marine operations or heavy industrial robotic arms. In these scenarios, the equipment control system faces extremely demanding physical obstacles, resulting in significant technical defects and operational risks. Limited by physical distance and communication bandwidth, the process of acquiring state parameters and issuing drive commands inevitably generates time-varying delays in both state and input, easily accumulating phase distortion in the feedback loop and triggering mechanical oscillations. The electromechanical body, during dynamic operation, is accompanied by unpredictable gear friction, sudden load changes, and random external environmental disturbances, constituting severe unmodeled dynamics and random noise. Existing control strategies, when dealing with such non-strict feedback high-order nonlinear dynamic models, cannot effectively decouple multiple time delays and nonlinear coupling disturbances. The control law derivation process is prone to falling into algebraic loop traps with singular denominators, directly leading to numerical explosions in drive commands and hardware damage. Furthermore, traditional control schemes are mostly based on asymptotic stability mechanisms, and their actual convergence time is highly dependent on the initial physical spatial position of the equipment. This makes it impossible to guarantee that the electromechanical equipment eliminates tracking errors within an absolutely fixed time independent of the initial state, thus failing to meet the practical control requirements of modern industry for actuators with high precision, high safety, and time determinism. To solve these problems, this invention provides a fixed-time adaptive tracking control method for stochastic high-order nonlinear multi-time-delay systems, the structure of which is as follows: Figure 1 As shown. The specific implementation process of this method is as follows:

[0165] A high-order dynamic model incorporating multiple time delays and random noise is established by extracting state parameters to define the boundary of the system control solution. Coordinate transformation is implemented to extract tracking errors, and an error auxiliary signal is introduced via feedforward to directly offset the phase distortion caused by input lag in the underlying communication link. For unknown nonlinear characteristics within the model, a neural network is used for online approximation, and a hyperbolic tangent function is incorporated into the computational derivation to smooth out the signal discontinuity at the origin, fundamentally eliminating the algebraic loop deadlock that is easily encountered in the back-reasoning recursive solution. To address mechanical oscillations caused by the superposition of state lag and unmodeled dynamics, a functional is constructed to absorb historical lag energy, and a bounded decay signal generated by a dynamic mechanism is used to suppress sudden physical disturbances. The above compensation components are fused step-by-step through adaptive back-reasoning logic to generate the final control command. The controller dynamically constrains the underlying parameters based on fixed-time convergence conditions, converts the command into a drive signal, and sends it to the actuator, forcing the electromechanical body's trajectory to precisely conform to the reference target within an absolute time threshold completely independent of the initial spatial position, eliminating the risk of divergence and loss of control of the electromechanical system under strong interference environments.

[0166] Example 3:

[0167] To verify the engineering effectiveness and control robustness of the fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-delay system provided by this invention, this embodiment provides a cross-comparison of multiple sets of operating data under operating condition one containing a first set of initial physical parameters and operating condition two containing a second set of extreme initial physical parameters.

[0168] like Figure 6 As shown, this embodiment selects a two-degree-of-freedom mass-spring-damping system as the mechanical body of the electromechanical equipment for verifying the control method. Figure 6 The first mass block is involved. Second mass block Spring stiffness coefficient And damping elements that generate unmodeled frictional disturbances The constructed electromechanical equipment body strictly corresponds to the high-order dynamic model established in step S1, thus establishing the physical work carrier of the control algorithm in real industrial applications.

[0169] like Figures 7 to 10 As shown, Figures 7 to 10 This is a physical simulation waveform of the electromechanical equipment under operating condition 1. Figure 7 This is a comparison chart of the continuous running trajectories of the system state variables tracking the reference signal under operating condition one. Under the operating environment with a first set of initial spatial deviation coordinates, it represents the running trajectory of the actual output state variables. It is able to approach and completely match the trajectory representing the desired reference signal before a preset absolute time point, even with a large initial deviation. . Figure 8 This is a timing diagram for the control command input under operating condition one. It represents the numerical trajectory of the control command. No divergent jumps towards infinity occurred, and the overall amplitude was forcibly constrained within the power range allowed by the physical servo mechanism. This verifies from the underlying algebraic architecture that the introduction of the hyperbolic tangent function to perform smoothing operations eliminates algebraic loop deadlocks. Figure 9 This diagram illustrates the convergence process of adaptive weight estimation for a neural network under condition one. The trajectory represents the weight estimator. and It exhibits a smooth and definite convergence trend, without any unlimited parameter calculation drift. Figure 10 This is a bounded periodic evolution diagram of the unmodeled dynamics of the system under condition one. The trajectory represents the unmodeled dynamics. The results show strictly limited bounded fluctuations, verifying that functional integration and dynamic disturbance rejection mechanisms establish an absolute safety boundary for the signal energy of the closed-loop feedback node.

[0170] like Figures 11 to 14 As shown, Figures 11 to 14 The waveforms represent the physical simulation of the electromechanical equipment under condition two. To verify the algorithm's independence from the initial state, condition two forces a change in the electromechanical equipment model to enter the second set of extreme initial spatial poses. Figure 11 This is a comparison chart of the continuous running trajectory of the system state variables tracking the reference signal under operating condition two. The running trajectory is shown under completely new extreme initial parameters. It can still achieve trajectory tracking before the preset absolute time node that is completely consistent with working condition one. A precise fit. Figure 12 The timing diagram and numerical trajectory of the control command input under operating condition two. Continue to maintain physical continuity and signal boundedness. Figure 13 The diagram shows the convergence process and trajectory of the neural network adaptive weight estimation under condition two. and It still converges to the expected algebraic boundary even when encountering different sudden disturbances. Figure 14 The trajectory is the unmodeled, dynamically bounded periodic evolution diagram of the system under condition two. It remains within the preset energy envelope limit.

[0171] By cross-comparing the two sets of objective operational physical data from operating conditions one and two, it was found that the actual output state variables of the system, regardless of the initial deviation, could complete trajectory correction before reaching the upper bound of the time limit defined by the system's global energy scalar function. Multi-condition cross-validation directly confirmed the technical judgment in step S7 regarding eliminating errors within a fixed convergence time threshold independent of the initial value of the state vector. Facing the multiple superpositions of random noise in physical space and high-frequency communication time delays, the control method provided by this invention possesses a fundamental execution constraint that forces electromechanical equipment to complete high-precision trajectory shaping within a defined absolute time boundary.

[0172] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A fixed-time adaptive tracking control method for a random high-order nonlinear multi-time-delay system, applied to trajectory tracking control of electromechanical equipment, characterized in that, Includes the following steps: S1. Obtain the state parameters of the electromechanical equipment, establish a high-order dynamic model that includes time-varying time delay, unmodeled dynamics and random noise, extract the state vector of the high-order dynamic model, and set the constraints. S2. Perform coordinate transformation on the state vector to calculate the tracking error at each level, and construct an error auxiliary signal to offset the input time delay; wherein, the error auxiliary signal is obtained by solving the differential evolution equation that limits its dynamic evolution, and the differential evolution equation is constructed according to the following terms: the extracted control command based on the higher-order exponential mapping term formed by the highest order of the system as the control input term at the current time, the extracted delay term of the control command based on the higher-order exponential mapping term formed by the highest order of the system as the delay control input term, the linear function term of the extracted error auxiliary signal, and the odd-order higher-order function term of the extracted error auxiliary signal; S3. The nonlinear terms in the higher-order dynamic model are approximated by a neural network to obtain the network output. The hyperbolic tangent function is introduced to eliminate the zero discontinuity of the network output to avoid algebraic loops. S4. Construct a Lyapunov-Krasovskii functional to compensate for state time delays, and use a bounded decay signal generated by a dynamic mechanism to suppress the unmodeled dynamics. S5. Based on the tracking errors at each stage, the network output, the Lyapunov-Krasovskii functional, and the bounded decay signal, recursively design the virtual control law and the parameter adaptive law at each stage; combine the final stage tracking error, the final stage parameter adaptive law, and the error auxiliary signal to generate the control command. S6. Based on the fixed-time convergence condition, update the network weights and constrain the controller parameters using the adaptive laws of the parameters at each level to ensure that the closed-loop system signal is bounded. S7. The control command is converted into a drive signal and output to the actuator of the electromechanical equipment to drive the tracking reference signal and eliminate the tracking errors at each level within a fixed time. 2.The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S1, the set constraint conditions include: Obtain the unknown dynamic disturbance of the higher-order dynamic model, and set the unknown dynamic disturbance to satisfy the upper bound constraint limit composed of the first uncertain non-negative smooth function and the second unknown monotonically increasing non-negative smooth function. Acquire the reference signal of the electromechanical equipment, and set the reference signal and its derivatives to be smooth and bounded functions. Obtain the state delay and input delay of the higher-order dynamic model, and set the absolute value of the state delay, the derivative of the state delay, the absolute value of the input delay, and the derivative of the input delay to have limited upper bounds.

3. The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S2, the calculation of tracking errors at each stage by performing coordinate transformation on the state vector includes: Extract the first-level state variables from the state vector, calculate the difference between the first-level state variables and the reference signal, and generate the first-level tracking error. Extract the intermediate-level state variables from the state vector, calculate the difference between the intermediate-level state variables and the adjacent previous-level virtual control law, and generate the intermediate-level tracking error. Extract the final-level state variables from the state vector, and then combine and perform a joint operation on the final-level state variables, the penultimate-level virtual control law, and the error auxiliary signal to generate the final-level tracking error.

4. The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S2, the differential evolution equation that defines its dynamic evolution is specifically as follows: In the formula The time derivative of the error auxiliary signal; This is an error auxiliary signal; The control command at the current moment; For delayed control commands that are hindered by time-varying delays; For time-varying, time-delay variables; This is the highest-order parameter of the system; The linear damping adjustment parameter must be strictly greater than 1; It is a higher-order damping adjustment parameter that is strictly greater than 0.

5. The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S3, the method of using a neural network to approximate the nonlinear terms in the higher-order dynamic model to obtain the network output, and introducing a hyperbolic tangent function to eliminate the zero-point discontinuities of the network output to avoid algebraic loops, includes: By performing an inner product operation between the network weights and basis function vectors of the neural network, a basic approximation signal for the nonlinear term is constructed. The denominator of the positioning control derivation includes singular correlation terms for the tracking errors at each level; The algebraic ratio of the tracking errors at each level to the preset minimum constant is used as the independent variable and input into the hyperbolic tangent function to generate an even-power smoothing factor. The singular correlation term is multiplied and modulated using the even-power smoothing factor. The multiplied and modulated singular correlation term is then integrated with the basic approximation signal to generate the network output without zero singularity.

6. The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S4, the construction of the Lyapunov-Krasovskii functional to compensate for state delays, and the suppression of the unmodeled dynamics using a bounded decaying signal generated by a dynamic mechanism, includes: An exponential decay coefficient is set to generate a time-domain integral kernel. The square term of the nonlinear state coupling function corresponding to the state time delay in the higher-order dynamic model is multiplied by the time-domain integral kernel. Time-domain integration is performed in the dynamic integration interval to generate the Lyapunov-Krasovskii functional used to offset the state lag energy. A non-negative state envelope constraint term is constructed based on the absolute value of the higher-order polynomial of the state vector. The linear attenuation coefficient is linearly superimposed with the negative product term of the attenuation signal variable, the non-negative state envelope constraint term, and the preset adjustment factor to establish a dynamic compensation differential equation. The bounded attenuation signal is generated by solving the dynamic compensation differential equation.

7. The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S5, the recursive design of virtual control laws and adaptive laws for parameters at each stage includes: Obtain the feedback gain coefficients at each level, the first exponential polynomial and the second exponential polynomial of the current level tracking error, and construct the feedback stabilization term; Extract the squared terms of the basis function vector of the neural network, the product of the third exponential polynomial of the current level tracking error and the network weights, and construct the neural network compensation term; The feedback stabilizing term and the neural network compensation term are superimposed and mapped using fractional radicals to generate the virtual control laws of each level at the current level. By combining the adaptive adjustment gain, the even-order exponential polynomial of the current stage tracking error, and the squared term of the basis function vector, a network driving term is constructed. Then, by combining the anti-drift suppression term, which includes the first-order parameter correction component and the cubic parameter correction component, the adaptive law of each level of the current stage is generated.

8. The fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S5, the step of generating the control command by combining the final-stage tracking error, the final-stage parameter adaptive law, and the error auxiliary signal includes: Obtain the final stage feedback gain term and the exponential polynomial of the final stage tracking error, and construct the final stage reference stability term; The final-stage network compensation term is constructed by multiplying the squared terms of the basis function vectors corresponding to the adaptive law of the final-stage parameters. The linear amplification term and the nonlinear odd-power amplification term of the error auxiliary signal are extracted to generate a time-delay feedforward compensation signal; The final-stage reference stabilization term, the final-stage network compensation term, and the time-delay feedforward compensation signal are added together, and the result of the addition is subjected to fractional-order radical mapping dimensionality reduction processing to output the control command.

9. A fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S6, the step of updating network weights and constraining controller parameters using the adaptive laws of each level of parameters based on the fixed-time convergence condition to ensure that the closed-loop system signal is bounded includes: The global system energy function is formed by aggregating the exponential polynomial energy terms corresponding to the tracking errors at each level, the squared energy terms corresponding to the network weight estimation errors, the Lyapunov-Krasovskii functional, and the algebraic energy terms associated with the bounded decay signal. Calculate the stochastic differential operator of the global system energy function and perform algebraic scaling to generate a negative definite convergence inequality containing the fractional-order nonlinear components, integer-order nonlinear components and bounded constant components of the global system energy function. Based on the mapping relationship between the coefficients of each term in the negative definite convergence inequality and the control gain and adaptive adjustment parameters, the values ​​of the controller parameters are constrained in reverse, and the closed-loop convergence time threshold is limited to be independent of the initial value of the state vector.

10. A fixed-time adaptive tracking control method for a stochastic high-order nonlinear multi-time-delay system according to claim 1, characterized in that, In S7, the step of converting the control command into a drive signal and outputting it to the actuator of the electromechanical equipment, driving the tracking reference signal, and eliminating the tracking errors at each level within a fixed time includes: Extract the real-time logical operation values ​​of the control commands; The real-time logic operation values ​​are converted into drive signals that match the physical interface of the actuator hardware of the electromechanical equipment using a signal mapping mechanism. The drive signal is continuously sent to the actuator of the electromechanical equipment to drive the mechanical body of the electromechanical equipment to generate a corresponding dynamic response trajectory; The actual output state variables corresponding to the dynamic response trajectory are collected in real time by sensors. The actual output state variables are continuously compared with the reference signal through closed-loop feedback. Within a fixed convergence time threshold independent of the initial value of the state vector, the actual output state variables are forced to approach the reference signal until the tracking errors at each level are eliminated.

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