Non-singular preset time sliding mode tracking control method for uncertain nonlinear system

Abstract

The invention belongs to the technical field of sliding mode tracking control, and provides an uncertain nonlinear system nonsingular preset time sliding mode tracking control method, which comprises the following steps: establishing an uncertain nonlinear system model, defining a tracking error variable, and establishing an error dynamic model according to the uncertain nonlinear system model and the tracking error variable; based on the error dynamic model, according to a comparison result of an absolute value of the tracking error variable and a preset error threshold value, a piecewise function is adopted to establish a sliding mode surface; setting a preset convergence time upper limit value and a self-adaptive gain value, and determining a reaching law of the self-adaptive preset time according to the preset convergence time upper limit value, the self-adaptive gain value and the sliding mode surface; and establishing a sliding mode controller according to the sliding mode surface and the reaching law, and executing a sliding mode tracking control task by using the sliding mode controller. According to the scheme, parameter design is simplified, setting difficulty is reduced, meanwhile, stable convergence of all variables of a closed-loop system within preset time is ensured, and high-precision tracking of an expected trajectory is achieved.

Description

Technical Field

[0001] This invention relates to the field of sliding mode tracking control technology, and in particular to a non-singular preset time sliding mode tracking control method for uncertain nonlinear systems. Background Technology

[0002] In fields such as industrial control and robotics control, the dynamic characteristics of many controlled objects exhibit uncertain nonlinear features. These systems commonly suffer from parameter perturbations, external disturbances, and unmodeled dynamics, making it difficult for traditional linear control methods to meet high-precision tracking requirements. Sliding mode control, due to its strong robustness, has become one of the commonly used solutions for controlling uncertain nonlinear systems.

[0003] However, traditional sliding mode control has several drawbacks. Specifically, on the one hand, when the tracking error approaches zero, traditional sliding surfaces are prone to the peculiar phenomenon of infinite derivatives, leading to abnormal controller output and making it impossible to achieve stable control under small errors. On the other hand, most sliding mode controls can only guarantee convergence in finite time, but the convergence time depends on the initial state of the system and cannot pre-specify the upper limit of the convergence time, making it difficult to meet the strict time constraints required in industrial scenarios. Furthermore, the convergence rate of traditional reaching laws such as constant velocity and exponential reaching laws is fixed, resulting in slow convergence under large errors and easy chattering under small errors, making it difficult to balance speed and stability. In addition, some sliding mode controls do not fully consider compensation for the total system disturbance, and the tracking accuracy decreases significantly when facing strong uncertainties.

[0004] It is evident that traditional sliding mode control schemes suffer from technical problems of insufficient reliability and accuracy due to shortcomings such as uncontrollable convergence time, poor adaptability of approach rate, insufficient robustness to uncertain disturbances, and singularity. Summary of the Invention

[0005] This invention provides a non-singular preset time sliding mode tracking control method for uncertain nonlinear systems, which solves the shortcomings of traditional sliding mode control schemes in terms of both reliability and accuracy.

[0006] This invention provides a nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems, comprising: An uncertain nonlinear system model is established, and a tracking error variable is defined. Based on the uncertain nonlinear system model and the tracking error variable, an error dynamic model is established. Based on the aforementioned error dynamic model, and according to the comparison result between the absolute value of the tracking error variable and the preset error threshold, a piecewise function is used to establish the sliding surface; Set a preset upper limit value for convergence time and an adaptive gain value, and determine the approach law of the adaptive preset time based on the preset upper limit value for convergence time, the adaptive gain value, and the sliding surface; Based on the sliding surface and the reaching law, a sliding mode controller is established, and the sliding mode controller is used to perform the sliding mode tracking control task.

[0007] According to the nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, an uncertain nonlinear system model is established, including: Define the state variables and input / output data of an uncertain nonlinear system, wherein the state variables are used to describe the dynamic characteristics of the uncertain nonlinear system; Based on the dynamic relationship corresponding to the known nonlinear dynamic characteristics of the uncertain nonlinear system obtained in advance, a known nonlinear smooth function is established; The uncertainties of the pre-obtained system parameters and external environmental disturbances are integrated into a total disturbance term; Based on the state variables and input / output data, the known nonlinear smooth function, and the total disturbance term, an uncertain nonlinear system model is constructed.

[0008] According to the nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, an error dynamic model is established based on the uncertain nonlinear system model and the tracking error variable, including: Substituting the tracking error variable into the uncertain nonlinear system model yields an error dynamic model that includes only the tracking error variable, the total disturbance term, and the related term of the known nonlinear smooth function.

[0009] According to the non-singular preset-time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, based on the error dynamic model, and according to the comparison result of the absolute value of the tracking error variable and the preset error threshold, a sliding surface is established using a piecewise function, including: Based on the aforementioned error dynamic model, determine the absolute value of the tracking error variable; When the absolute value of the tracking error variable is greater than the preset error threshold, an exponential function term is used to establish the sliding surface; When the absolute value of the tracking error variable is less than or equal to the preset error threshold, a quadratic function term is used to establish the sliding surface.

[0010] According to the nonsingular preset-time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, based on the preset upper limit of convergence time, the adaptive gain value, and the sliding surface, a convergence law for an adaptive preset time is determined, including: Based on the preset upper limit of convergence time and the sliding surface, the adjustment rate of the adaptive gain value is calculated; The adaptive gain value is adjusted according to the adjustment rate to obtain the adjusted adaptive gain value; Input the sliding surface into a sign function with an exponent to obtain the first function value; Multiplying the first function value by the negative of the adjusted adaptive gain value yields the approach law for the adaptive preset time.

[0011] According to the non-singular preset time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, the adjustment rate of the adaptive gain value is: ; The approach law of the adaptive preset time is: ; in, Indicates the adaptive gain value Adjustment rate, This indicates the preset exponent parameter. T di This indicates the preset upper limit of convergence time. s i Indicates the sliding surface. Indicates the synovial surface s i The derivative of is the law of convergence.

[0012] According to the nonsingular preset-time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, a sliding mode controller is established based on the sliding surface and the reaching law, including: Based on the sliding surface and the approach rate, the error dynamic driving term, the system uncertainty compensation term, and the sliding mode approach constraint term are determined respectively. The core summation term is obtained by adding the error dynamic driving term, the system uncertainty compensation term, and the sliding mode approach constraint term. The inverse function of the known nonlinear smooth function of the highest-order term of the system is determined, and the core summation term is multiplied by the inverse function to obtain the sliding mode controller.

[0013] According to the non-singular preset time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, the error dynamic driving term is: ; The system uncertainty compensation term is: ; The sliding mode convergence constraint term is: ; in, This represents the dynamic driver of error. Represents tracking error variable The derivative, This represents the derivative of the first function determined based on the piecewise function in the sliding surface. This represents the derivative of the second function, determined based on the piecewise function, in the sliding surface. Indicates the preset convergence time. , , , All of these represent preset parameters. This represents the system uncertainty compensation term. Represents a known nonlinear term representing the highest-order term of the system. This represents the upper bound estimate of the total disturbance term. Indicates the desired reference signal y d of i First derivative, This represents the sliding mode convergence constraint term.

[0014] According to the nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, the method further includes: The stability of the sliding mode controller was verified during the approach phase and the sliding phase, and the stability verification results were obtained.

[0015] According to the nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems provided by the present invention, the method further includes: In the van der Bohr system and the two-bar linkage robotic arm system, different initialization parameters were set, and the sliding mode controller was simulated and verified in multiple scenarios based on the initialization parameters.

[0016] The non-singular preset-time sliding mode tracking control method for uncertain nonlinear systems provided by this invention, through a systematic design logic, combines multiple advantages such as preset-time convergence, non-singularity, strong robustness, and low design complexity. Specifically, the error dynamic model derived from the uncertain nonlinear system model provides a precise target for the control strategy; the sliding surface constructed by piecewise functions fundamentally avoids the controller singularity problem under small errors, while adapting to the requirements of rapid convergence under large errors and smooth transition under small errors; at the same time, the incorporation of a preset convergence time upper limit and an adaptive gain reaching law not only achieves flexible specification of the convergence time, but also dynamically adjusts the convergence rate according to the system error, effectively suppressing chattering; finally, the sliding mode controller constructed by combining the sliding surface and the reaching law simplifies parameter design and reduces tuning difficulty, while ensuring stable convergence of all variables in the closed-loop system within a preset time, achieving high-precision tracking of the desired trajectory, and strongly resisting parameter perturbations and external disturbances. It is applicable to various engineering scenarios such as robotic arms and drones, and has significant practical and engineering application value. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in this 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 some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0018] Figure 1 This is a flowchart illustrating the non-singular preset time sliding mode tracking control method for uncertain nonlinear systems provided in this embodiment of the invention. Figure 2 This is a schematic diagram of trajectory tracking under different initial states; Figure 3 This is a schematic diagram of tracking errors under different initial states; Figure 4 This is a control input diagram; Figure 5 This is a schematic diagram of trajectory tracking and error under different convergence times; Figure 6 It is a schematic diagram comparing the actual trajectory of the joint with the target trajectory under different initial conditions; Figure 7 This is a schematic diagram of the tracking error of the robotic arm under different initial states; Figure 8 These are schematic diagrams of robotic arm control inputs under different initial states; Figure 9 This is a schematic diagram of robotic arm trajectory tracking with different convergence times; Figure 10 This is a schematic diagram of the tracking error of a robotic arm with different convergence times; Figure 11 These are schematic diagrams of the sliding surface response at different convergence times; Figure 12 This is a schematic diagram of the control inputs for the robotic arm with different convergence times. Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0020] The following is combined with Figures 1 to 12 This invention describes the detailed scheme of the non-singular preset time sliding mode tracking control method for uncertain nonlinear systems provided by embodiments of the present invention.

[0021] like Figure 1As shown in the embodiments of the present invention, the non-singular preset time sliding mode tracking control method for uncertain nonlinear systems mainly includes the following steps: Step 110: Establish an uncertain nonlinear system model and define the tracking error variable. Based on the uncertain nonlinear system model and the tracking error variable, establish an error dynamic model.

[0022] In this embodiment, an uncertain nonlinear system model is first established, clarifying the system's state variables, inputs and outputs, and the total disturbance term including external disturbances and parameter uncertainties. Then, a tracking error variable is defined to derive the error dynamic model. Simultaneously, the core assumptions of bounded expected signal and its derivative, and bounded uncertain disturbances are clarified, providing a theoretical foundation for subsequent control design.

[0023] Step 120: Based on the error dynamic model, and according to the comparison results of the absolute value of the tracking error variable and the preset error threshold, a piecewise function is used to establish the sliding surface.

[0024] This embodiment overcomes the limitation of traditional sliding surfaces, which are prone to controller singularities due to errors approaching zero. It constructs the sliding surface using piecewise functions, using exponential functions to achieve fast convergence when there are large errors, and switching to quadratic functions when there are small errors. This ensures that the sliding surface is continuously differentiable at the switching boundary, fundamentally avoiding singularities, while retaining the convergence characteristics of the preset time.

[0025] Step 130: Set the preset upper limit of convergence time and the adaptive gain value. Based on the preset upper limit of convergence time, the adaptive gain value and the sliding surface, determine the approach law of the adaptive preset time.

[0026] Understandably, in response to the shortcomings of the fixed convergence rate of traditional convergence laws, this embodiment introduces a preset upper limit value for convergence time and designs a convergence law with an adaptive gain value, which enables it to dynamically and autonomously adjust the convergence rate according to system errors. This ensures that the system can quickly reach the sliding surface within a preset time and enhances its adaptability to changes in system state.

[0027] Step 140: Based on the sliding surface and the reaching law, establish a sliding controller and use the sliding controller to perform the sliding tracking control task.

[0028] This embodiment combines the sliding surface and the reaching law obtained above to construct a sliding mode controller with a simplified number of parameters. By controlling the input first to act on the highest order error, and then driving the errors of each order to converge in sequence, the lowest order error is finally made to approach zero within a preset time. While ensuring control performance, this greatly reduces the difficulty of parameter tuning and the complexity of controller design.

[0029] In one embodiment, establishing an uncertain nonlinear system model specifically includes: On the one hand, the state variables and input / output data of the uncertain nonlinear system are defined, where the state variables are used to describe the dynamic characteristics of the uncertain nonlinear system.

[0030] On the other hand, based on the dynamic relationship corresponding to the known nonlinear dynamic characteristics of the uncertain nonlinear system obtained in advance, a known nonlinear smooth function is established.

[0031] On the other hand, the uncertainties of the pre-obtained system parameters and external environmental disturbances are integrated into a total disturbance term.

[0032] Finally, based on the state variables and input / output data, the known nonlinear smooth function, and the total disturbance term, an uncertain nonlinear system model is constructed.

[0033] In this embodiment, the nth-order uncertain nonlinear system model can be represented as follows: (1); in, For system state variables, For system input, For system output, Given a nonlinear smooth function, This is the total disturbance term, which includes external disturbances and the effects of uncertainty.

[0034] In one embodiment, an error dynamic model is established based on an uncertain nonlinear system model and tracking error variables, specifically including: Substituting the tracking error variable into the uncertain nonlinear system model yields an error dynamic model that includes only the tracking error variable, the total disturbance term, and the related term of the known nonlinear smooth function.

[0035] In this embodiment, a tracking error variable is defined. in, This is a reference signal.

[0036] Furthermore, by substituting the aforementioned tracking error variables into the uncertain nonlinear system model, we can obtain the error dynamic model, as follows: (2); in, Indicates the first i Tracking error variables e i ( t The time derivative of ) Indicates the system's first i State variables x i ( t The time derivative of ) Indicates the desired reference signaly d ( t )of i First derivative, Represents a known nonlinear smooth function of the highest-order term of the system. This represents the highest-order known nonlinear term of the system.

[0037] The control objective of this embodiment is to ensure that the system output can track the desired reference signal, while the tracking error variable needs to converge to zero within a preset time, and all other system states are bounded, thereby achieving high-precision trajectory tracking of uncertain nonlinear systems.

[0038] Consider uncertain nonlinear systems If there exists a Lyapunov function satisfy: (3); in, , , , , If the nonlinear system is pre-time stable, then the upper bound of the convergence time is... .

[0039] For uncertain nonlinear systems If there exists a Lyapunov function satisfy: (4); in, , , Then, the uncertain nonlinear system is assumed to be time-stable, and the upper bound of the convergence time is... .

[0040] Therefore, the following two assumptions exist: Assumption 1: The desired reference signal and its derivative are known and bounded.

[0041] Assumption 2: The total interference term is bounded and satisfies ,in It is a known positive constant.

[0042] For ease of explanation, all variables in the subsequent formulas of this embodiment are... t This will be omitted.

[0043] In one embodiment, based on the error dynamic model, and according to the comparison result of the absolute value of the tracking error variable and the preset error threshold, a piecewise function is used to establish the sliding surface, specifically including: First, based on the error dynamic model, determine the absolute value of the tracking error variable.

[0044] In one scenario, when the absolute value of the tracking error variable is greater than a preset error threshold, an exponential function term is used to establish the sliding surface.

[0045] In another scenario, when the absolute value of the tracking error variable is less than or equal to a preset error threshold, a quadratic function term is used to establish the sliding surface.

[0046] In practical applications, the general form of a sliding surface with a non-singular preset time is as follows: (5); in, , , Represents a symbolic function.

[0047] In the general form of the above sliding surface expression, the first function Typically designed as To ensure convergence. However, when the tracking error parameter approaches zero, the equivalent control requires differentiating the sliding surface, and The derivative is .because This will lead to The value tends towards infinity, thus triggering a controller singularity. This singularity can lead to system instability and increase control chattering.

[0048] To eliminate this singularity and maintain convergence properties, this embodiment will The piecewise function can be designed as follows: (6); (7); in, , , , , , Let be an arbitrarily small constant. It is an adjustable parameter. sgn .

[0049] It is not difficult to see that when , At that time, it was still adopted ;when Then switch to Analysis of the piecewise function shows that Switching boundaries It remains continuous and differentiable.

[0050] Second function Essentially, it's an acceleration function designed for sliding surfaces, based on... The trend, its functions are as follows: when , It is an exponential function, grows rapidly, and is suitable for fast convergence under large errors; when hour, It exhibits a quadratic function, with a more gradual growth rate.

[0051] To accelerate the convergence speed of the sliding mode and ensure that the system reaches the designed non-singular preset time sliding surface within a preset time, this embodiment proposes an adaptive preset time approach law.

[0052] In one embodiment, the adaptive convergence time convergence law is determined based on a preset upper limit value, an adaptive gain value, and a sliding mode surface, specifically including: First, the adjustment rate of the adaptive gain value is calculated based on the preset upper limit of convergence time and the sliding surface. Then, the adaptive gain value is adjusted according to the adjustment rate to obtain the adjusted adaptive gain value.

[0053] Simultaneously, the sliding surface is input into a sign function with an exponent to obtain the first function value.

[0054] Finally, the first function value is multiplied by the negative of the adjusted adaptive gain value to obtain the approach law of the adaptive preset time.

[0055] In this embodiment, the adjustment rate of the adaptive gain value is: (8); The approach law for adaptive preset time is: (9); in, Indicates the adaptive gain value The adjustment rate, and satisfy , This indicates the preset exponent parameter. , T di This indicates the preset upper limit of convergence time. , s i Indicates the sliding surface. Indicates the synovial surface s i The derivative of is the law of convergence.

[0056] In one embodiment, a sliding mode controller is established based on the sliding surface and the reaching law, specifically including: First, based on the sliding surface and the approach rate, the error dynamic driving term, the system uncertainty compensation term, and the sliding mode approach constraint term are determined respectively.

[0057] Then, the error dynamic driving term, the system uncertainty compensation term, and the sliding mode convergence constraint term are added together to obtain the core summation term.

[0058] Finally, the inverse function of the known nonlinear smooth function of the highest-order term of the system is determined, and the core summation term is multiplied by the inverse function to obtain the sliding mode controller.

[0059] In this embodiment, the sliding mode controller is designed by combining the sliding surface with the reaching law as follows: (10); The error dynamic driving term is: (11); The system uncertainty compensation term is: (12); The sliding mode convergence constraint term is: (13); in, This represents the dynamic driver of error. Represents tracking error variable The derivative, This represents the derivative of the first function determined based on the piecewise function in the sliding surface. This represents the derivative of the second function, determined based on the piecewise function, in the sliding surface. Indicates the preset convergence time. , , , All of these represent preset parameters. This represents the system uncertainty compensation term. Represents a known nonlinear term representing the highest-order term of the system. This represents the upper bound estimate of the total disturbance term. Indicates the desired reference signal y d of i First derivative, This represents the sliding mode convergence constraint term. It represents the inverse function of a known nonlinear smooth function of the highest-order term of the system.

[0060] For uncertain nonlinear systems, control input u First, through the sliding surface Acting on the highest order error This ensures that it converges to zero within a preset time. Simultaneously, the sliding surface... sequential drive Convergence. This process proceeds gradually until the lowest-order error is reached. It converges to zero within a preset time. In this way, the system output can achieve high-precision tracking of the desired trajectory.

[0061] In one embodiment, the above-described nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems may further include: The stability of the sliding mode controller was verified during the approach and sliding phases, and the stability verification results were obtained.

[0062] Since traditional sliding mode control typically involves two phases—the approach phase and the sliding phase—the stability analysis of the entire control method in this embodiment is as follows: Under the action of the adaptive preset time convergence law, the tracking error variable Able to be in the preset time The inner reaches of the non-singular preset time sliding surface can be proven using the following Lyapunov function: (14); right Taking the derivative, we get: (15); Therefore, Able to be in the preset time Once the surface reaches the sliding surface, the proof of the approach phase is complete.

[0063] During the sliding phase, for the uncertain nonlinear system, under assumptions 1 and 2, considering the characteristics of the sliding mode controller, the closed-loop system is pre-time stable, and the following conclusions hold: First, system output Able to track the desired reference signal And track error variables It can converge to zero within a preset time.

[0064] Second, all signals within the closed-loop system can converge to zero within a preset time.

[0065] In the proof, the following Lyapunov function can be designed: (16); Further differentiation yields: (17); when , Then the following conditions are met: (18); The relevant parameters satisfy: (19); (20); (twenty one); Furthermore, it exists that: (twenty two); Among them, parameters b 1 and b 2 respectively satisfy: (twenty three); (twenty four); (25); (26); when , Then the following conditions are met: (27); According to Lyapunov stability theory, the tracking error variable will converge exponentially to zero.

[0066] Combined with The analysis shows that the tracking error variable will be within a preset time. The conclusion is that it converges to zero. Because... Since its derivatives are continuous and bounded, the above results indicate that the tracking error variable... Its derivative is also bounded, by the relation It can be seen that, It can converge to a bounded region within a preset time. Therefore, the expression for the sliding mode controller is bounded within the preset time. In summary, all signals within the closed-loop system can converge to zero within the preset time.

[0067] In one embodiment, the above-described nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems may further include: Different initialization parameters were set in the van der Bohr system and the two-bar linkage robotic arm system, and the sliding mode controller was simulated and verified in multiple scenarios based on the initialization parameters.

[0068] To verify the effectiveness of the non-singular preset time sliding mode control strategy proposed in this embodiment, a numerical system and a robotic arm system simulation experiment were conducted on a computer. MATLAB was used as the simulation platform, and the differential equations were solved using the ODE45 algorithm. The simulation time step was set to 0.05 seconds, and the controller parameters were set to... , , , .

[0069] First, we will demonstrate the effectiveness of the method provided in this embodiment using the van der Bohr system, as follows: (28); Then, the dynamic model of the system tracking error is given as follows: (29); To analyze trajectory tracking performance under different initial values, this embodiment defines three sets of initial states, namely: , , The control parameters are set to s and s, s, the expected trajectory is defined as: (30); At the same time, external interference is introduced. To evaluate the system's anti-interference capability, simulation results based on MATLAB are as follows: Figure 2 , 3 As shown in Figure 4, where, Figure 2 The trajectory tracking scenarios with different initial states are shown. Figure 3 The tracking error under different initial conditions is shown. Figure 4 The fluctuations in the control input are shown.

[0070] like Figure 2 and Figure 3 As shown, the three initial states represent the tracking tasks starting from different initial states and their corresponding system tracking errors. Simulation results show that, under different initial conditions, the non-singular preset time sliding mode control strategy proposed in this embodiment can effectively overcome the initial deviation, and the system state converges rapidly within the preset 2-second timeframe, maintaining high-precision tracking performance throughout. Furthermore, from... Figure 4 It can be seen that although a certain control torque needs to be applied in the initial stage of control, the control input tends to be stable after the system enters steady state, and no obvious oscillations occur.

[0071] To analyze the impact of different preset times on the trajectory tracking performance of the control strategy proposed in this embodiment, the initial system conditions are set as follows: The expected trajectory is defined as: (31); Considering four different preset times, namely 0.8s, 1.5s, 2s, and 3s, the parameters... and The settings are as follows: ; ; ; ; With all other simulation conditions and controller parameters remaining unchanged, the numerical simulation results are as follows: Figure 5 As shown, Figure 5 The trajectory tracking and error under different convergence times are shown. The analysis results of the overall system state trajectory tracking and tracking error show that the control strategy proposed in this embodiment can stably and accurately achieve trajectory tracking within different preset time periods.

[0072] Consider the following robotic arm system model: (32); According to the system dynamics model, the tracking error is defined as: (33); Therefore, its error dynamic model is obtained as follows: (34); in, This is the joint angle position vector. and These are the angular velocity and angular acceleration vectors, respectively. For an uncertain perturbation vector, It is a positive definite inertial matrix. The inertia matrix, The gravity vector To control the torque.

[0073] To further verify the applicability of the control strategy proposed in this embodiment, four sets of initial joint angles were generated, and these angles are distributed in... Within the rad interval. Specifically, the four initial conditions are: Case 1: [1.3; 0.5] rad; Case 2: [2;2.3] rad; Case 3: [-0.8; -0.6] rad; Case 4: [-1.8; -1.6] rad; The desired trajectory during simulation is set as follows: (35); The physical parameters of the robotic arm model are shown in Table 1, and the convergence time parameter is set to... , The preset total duration is .

[0074] Table 1 Physical parameters of the robotic arm

[0075] Figure 6 The comparison between the actual joint trajectory and the target trajectory under different initial conditions is shown, such as... Figure 6 As shown, although the initial angle deviations of each case differed significantly, all trajectories converged rapidly in the initial stage and closely tracked the target trajectory within about 2 seconds. Subsequently, throughout the simulation process, the actual trajectory always closely followed the target periodic waveform without any obvious overshoot or oscillation.

[0076] Figure 7 The convergence process of the tracking error was further quantified. In all cases, the error curve showed a convergence trend from the initial deviation and converged to zero within a predetermined time of 2 seconds. Figure 6 and Figure 7 The simulation results show that, under different initial joint angles, the non-singular preset time sliding mode control strategy proposed in this embodiment can effectively drive the actual joint angle of the robotic arm to converge to the desired trajectory within a preset time of 2 seconds, and ensure continuous high-precision tracking.

[0077] from Figure 8 It can be seen that a large torque is required in the initial stage of control under various conditions to suppress the initial error, which is the primary requirement for achieving rapid convergence. Once the system enters steady state, the control torque exhibits smooth periodic fluctuations without violent oscillations. This proves that the control strategy provided in this embodiment successfully suppresses chattering, which is difficult to avoid in traditional sliding mode control, while ensuring rapid dynamic response.

[0078] To analyze the impact of different predetermined times on the trajectory tracking performance of the control strategy proposed in this embodiment, the initial state of the robotic arm was set. The desired trajectory remains unchanged, while the convergence time is rad. Four preset convergence times were considered: 1s, 1.5s, 2s, and 3s, corresponding to the parameters of Cases 1-4 and Cases 1-4, respectively. and Set them to: ; ; ; ; The desired trajectory is set as follows: (36); With all other simulation conditions and controller parameters remaining unchanged, the numerical simulation results show... Figure 9 , Figure 10 , Figure 11 as well as Figure 12 In, among them, Figure 9 The robot arm trajectory tracking results with different convergence times are shown. Figure 10 The tracking error of the robotic arm with different convergence times is shown. Figure 11 The sliding surface response at different convergence times is shown. Figure 12 The robot arm control input data with different convergence times are shown.

[0079] Figure 9 In the simulation, the desired trajectory exhibits a periodic sinusoidal waveform. The responses of both joints of the robotic arm start from their initial values ​​and rapidly approach the desired trajectory within their respective preset time intervals. Although the preset time intervals differ across the four cases, both joints of the robotic arm achieve high-precision tracking within their respective preset time intervals.

[0080] like Figure 10 As shown, in all four cases, the tracking error rapidly decayed from its initial value to zero and converged within their respective preset timeframes; however, the convergence rates differed significantly. In Case 1, the error curve decayed the fastest, approaching zero within approximately 1 second, demonstrating excellent transient performance. Cases 2 and 3 converged slightly slower, stabilizing at around 1.5 and 2 seconds respectively, while Case 4 converged the slowest, reaching a steady state at 3 seconds. All four cases exhibited a consistent error convergence trend and showed no significant chattering.

[0081] Figure 11 The convergence of the sliding surface under different convergence times is shown. All four curves converge within their respective preset times without severe chattering, proving the effectiveness of the adaptive preset time approach law provided in this embodiment, which suppresses chattering while ensuring fast convergence.

[0082] It is worth noting that a shorter preset convergence time significantly increases the initial control torque. This is because the system needs to overcome inertia and tracking errors in a shorter time. Figure 12 This has been clearly verified. Therefore, in practical engineering applications, we should not blindly pursue fast theoretical convergence performance, but rather, based on the specific operational task and combined with comprehensive analysis, finely adjust and determine a convergence time that both meets the performance indicators and is practically feasible.

[0083] In summary, this embodiment proposes a novel nonsingular preset-time sliding mode control strategy to solve the problem of accurate trajectory tracking in uncertain nonlinear systems. The core innovation of this scheme lies in the design of a nonsingular preset-time sliding surface, fundamentally avoiding singular phenomena in the control process; the proposal of an adaptive preset-time approaching law, which dynamically adjusts the convergence rate based on the tracking error within a preset time frame; and the achievement of high-precision tracking performance and strong robustness through a simplified controller structure. Simulation results show that the proposed strategy can achieve rapid convergence, asymptotically approach zero tracking error, and eliminate chattering. This achievement enriches the development of preset-time control theory and provides new research ideas for related fields.

[0084] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A nonsingular preset time sliding mode tracking control method for an uncertain nonlinear system, characterized in that, include: An uncertain nonlinear system model is established, and a tracking error variable is defined. Based on the uncertain nonlinear system model and the tracking error variable, an error dynamic model is established. Based on the aforementioned error dynamic model, and according to the comparison result between the absolute value of the tracking error variable and the preset error threshold, a piecewise function is used to establish the sliding surface; Set a preset upper limit value for convergence time and an adaptive gain value, and determine the approach law of the adaptive preset time based on the preset upper limit value for convergence time, the adaptive gain value, and the sliding surface; Based on the sliding surface and the reaching law, a sliding mode controller is established, and the sliding mode controller is used to perform the sliding mode tracking control task.

2. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 1, characterized in that, Establishing a model for an uncertain nonlinear system includes: Define the state variables and input / output data of an uncertain nonlinear system, wherein the state variables are used to describe the dynamic characteristics of the uncertain nonlinear system; Based on the dynamic relationship corresponding to the known nonlinear dynamic characteristics of the uncertain nonlinear system obtained in advance, a known nonlinear smooth function is established; The uncertainties of the pre-obtained system parameters and external environmental disturbances are integrated into a total disturbance term; Based on the state variables and input / output data, the known nonlinear smooth function, and the total disturbance term, an uncertain nonlinear system model is constructed.

3. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 2, characterized in that, Based on the uncertain nonlinear system model and the tracking error variable, an error dynamic model is established, including: Substituting the tracking error variable into the uncertain nonlinear system model yields an error dynamic model that includes only the tracking error variable, the total disturbance term, and the related term of the known nonlinear smooth function.

4. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 1, characterized in that, Based on the aforementioned error dynamic model, and according to the comparison result between the absolute value of the tracking error variable and the preset error threshold, a piecewise function is used to establish the sliding surface, including: Based on the aforementioned error dynamic model, determine the absolute value of the tracking error variable; When the absolute value of the tracking error variable is greater than the preset error threshold, an exponential function term is used to establish the sliding surface; When the absolute value of the tracking error variable is less than or equal to the preset error threshold, a quadratic function term is used to establish the sliding surface.

5. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 1, characterized in that, Based on the preset upper limit of convergence time, the adaptive gain value, and the sliding surface, the adaptive preset time convergence law is determined, including: Based on the preset upper limit of convergence time and the sliding surface, the adjustment rate of the adaptive gain value is calculated; The adaptive gain value is adjusted according to the adjustment rate to obtain the adjusted adaptive gain value; Input the sliding surface into a sign function with an exponent to obtain the first function value; Multiplying the first function value by the negative of the adjusted adaptive gain value yields the approach law for the adaptive preset time.

6. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 5, characterized in that, The adjustment rate of the adaptive gain value is: ； The approach law of the adaptive preset time is: ； in, Indicates the adaptive gain value Adjustment rate, This indicates the preset exponent parameter. T di This indicates the preset upper limit of convergence time. s i Indicates the sliding surface. Indicates the synovial surface s i The derivative of is the law of convergence.

7. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 1, characterized in that, Based on the sliding surface and the reaching law, a sliding mode controller is established, including: Based on the sliding surface and the approach rate, the error dynamic driving term, the system uncertainty compensation term, and the sliding mode approach constraint term are determined respectively. The core summation term is obtained by adding the error dynamic driving term, the system uncertainty compensation term, and the sliding mode approach constraint term. The inverse function of the known nonlinear smooth function of the highest-order term of the system is determined, and the core summation term is multiplied by the inverse function to obtain the sliding mode controller.

8. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 7, characterized in that, The error dynamic driving term is: ； The system uncertainty compensation term is: ； The sliding mode convergence constraint term is: ； in, This represents the dynamic driver of error. Represents tracking error variable The derivative, This represents the derivative of the first function determined based on the piecewise function in the sliding surface. This represents the derivative of the second function, determined based on the piecewise function, in the sliding surface. Indicates the preset convergence time. , , , All of these represent preset parameters. This represents the system uncertainty compensation term. Represents a known nonlinear term representing the highest-order term of the system. This represents the upper bound estimate of the total disturbance term. Indicates the desired reference signal y d of i First derivative, This represents the sliding mode convergence constraint term.

9. The non-singular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 1, characterized in that, The method further includes: The stability of the sliding mode controller was verified during the approach phase and the sliding phase, and the stability verification results were obtained.

10. The nonsingular preset time sliding mode tracking control method for uncertain nonlinear systems according to claim 1, characterized in that, The method further includes: In the van der Bohr system and the two-bar linkage robotic arm system, different initialization parameters were set, and the sliding mode controller was simulated and verified in multiple scenarios based on the initialization parameters.

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