Dynamic event triggering preset time fault-tolerant control method considering non-affine fault

Abstract

The invention discloses a dynamic event triggering preset time fault-tolerant control method considering a non-affine fault. The method comprises the following steps: carrying out online approximation on an unknown nonlinear term in a dynamic decoupling model by adopting a fuzzy logic system to obtain an optimized system model; constructing a segmented preset time performance function, and introducing an instruction filter based on a dynamic surface control technology to process a virtual control signal of the constructed adaptive virtual control law so as to obtain a filtering error; constructing a dynamic signal compensation mechanism based on the filtering error in combination with a sectional preset time performance function, and obtaining a compensated final tracking error; and constructing a dynamic event triggering mechanism to obtain a final fault-tolerant controller, and realizing dynamic event triggering preset time fault-tolerant control considering the non-affine fault according to the final fault-tolerant controller. According to the method, the technical problems of uncontrollability of convergence time, influence of actuator faults and nonlinearity on control performance and limitation of communication and computing resources in the existing method are solved.

Description

A dynamic event-triggered preset-time fault-tolerant control method considering non-affine faults Technical Field

[0001] This invention relates to the field of automatic control technology, and in particular to a dynamic event-triggered preset time fault-tolerant control method that considers non-affine faults. Background Technology

[0002] With the development of industrial intelligence, nonlinear systems such as robotic arms are widely used in precision assembly, medical surgery, and other scenarios with extremely high requirements for timeliness and precision. In practical applications, three major challenges are faced: 1. Uncontrollability of convergence time: Traditional asymptotic stability control (such as PID and backstepping) theoretically requires infinite time to converge, while the convergence time of finite-time control often depends on the initial state of the system and is difficult to specify in advance by the user.

[0003] 2. Actuator failure and nonlinearity: Actuators may develop non-affine faults such as dead zone, saturation or gain loss during long-term operation, and the system itself has unknown nonlinear dynamics such as friction and load changes, which seriously affect control performance.

[0004] 3. Communication and computing resource limitations: Traditional digital control systems employ time-triggered mechanisms, resulting in the transmission and computation of a large amount of redundant data. While existing static event-triggered mechanisms can reduce communication volume, the fixed thresholds make it difficult to balance the large error tolerance during the dynamic response phase with the high precision requirements during the steady-state phase.

[0005] Therefore, there is an urgent need for an advanced control method that can simultaneously solve the problems of preset time convergence, non-affine fault tolerance, and dynamic resource scheduling. Summary of the Invention

[0006] This invention provides a dynamic event triggering preset time fault-tolerant control method that considers non-affine faults, in order to overcome the above-mentioned technical problems.

[0007] To achieve the above objectives, the technical solution of this invention is: a dynamic event-triggered preset-time fault-tolerant control method considering non-affine faults, comprising the following steps: S1: establishing a second-order dynamic model considering unknown nonlinear terms and actuator non-affine faults; and using the mean value theorem to decouple the actuator non-affine faults in the second-order dynamic model into affine form, obtaining a dynamic decoupling model; S2: using a fuzzy logic system to approximate the unknown nonlinear terms in the dynamic decoupling model online, obtaining an optimized system model; S3: constructing a piecewise preset-time performance function, based on dynamic surface control technology, processing the virtual control signal of the constructed adaptive virtual control law by introducing an instruction filter to obtain the filtering error; based on the filtering error combined with the piecewise preset-time performance function, constructing a dynamic signal compensation mechanism to obtain the compensated final tracking error; S4: constructing a dynamic event triggering mechanism, based on the dynamic event triggering mechanism and according to the optimized system model and the final tracking error, designing an intermediate control law by introducing a robust gain term to compensate for the actuator's non-affine fault tolerance; and based on the intermediate control law combined with... The gain function yields the final fault-tolerant controller, and based on the final fault-tolerant controller, dynamic event-triggered preset-time fault-tolerant control considering non-affine faults is implemented.

[0008] Furthermore, S1 specifically includes the following steps: S11: Establishing a second-order dynamic model considering unknown nonlinear terms and non-affine faults of the actuator as follows:

[0009] In the formula: Represents the system states of a second-order dynamic model of a cascaded structure, each state... Indicates the previous level state Virtual control input; Represents the system's total state vector ; Represents the full state subvectors and ; Represents the nth state variable The derivative; Indicates system control input; Indicates system output; Represents an unknown, smooth, nonlinear dynamic term; Indicates unknown external disturbances and , Indicates the upper bound of external disturbances; Represented by the time distribution function Activated burst non-affine fault item; S12: The sudden non-affine fault term in the second-order dynamic model is decomposed using the mean value theorem as follows:

[0010]

[0011] In the formula: The affine part representing the fault dynamics; Indicates time-varying gain; Indicates a state between 0 and A point value between; S13: Based on steps S11 to S12, obtain the nth subsystem in the second-order dynamic model as:

[0012]

[0013] ,

[0014] S14: Based on S13, the second-order dynamic model is rewritten to obtain the dynamic decoupling model as follows: .

[0015] Furthermore, S2 specifically includes the following steps: S21: Define the fuzzy basis function vector of the fuzzy logic system as... and , Indicates the first There are fuzzy basis functions, and the weight vector is... and The fuzzy basis function The expression is:

[0016] In the formula: Indicates the number of fuzzy rules; Indicates input variables For fuzzy sets The membership function; S22: Based on the fuzzy basis function vector and weight vector, the unknown nonlinear term in the dynamic decoupling model is approximated online as follows:

[0017] In the formula: Let represent the fuzzy approximation error of the i-th subsystem, and , express The upper bound; Indicates transpose; S23: Based on S22, the dynamic decoupling model is rewritten as an optimized system model as follows: .

[0018] Furthermore, S3 specifically includes the following steps: S31: Constructing a piecewise preset time performance function as follows:

[0019] In the formula: This indicates the user-defined preset convergence time. S32: Based on dynamic surface control technology, an instruction filter is introduced to obtain the virtual control signal of the adaptive virtual control law to be designed. A tracking error compensation mechanism is constructed based on the virtual control signal to obtain the filtering error.

[0020]

[0021]

[0022]

[0023] In the formula: Indicates tracking error; This represents the desired reference trajectory or the desired output signal; Indicates the filtering error; Represents dynamic surface signals; This represents a time constant that is greater than zero. This represents the virtual control signal output by the adaptive virtual control law designed in step i-1. , express , The initial value; S33: Based on the piecewise preset time performance function, the final tracking error after compensation is obtained according to the filtering error, which includes: S331: For the first subsystem in the second-order dynamic model: Define the initial compensation tracking error. for:

[0024]

[0025] In the formula: Indicates a compensation signal; express The first derivative; Indicate design parameters; This indicates the next compensation signal; based on the tracking error after initial compensation. Combining tracking error Construct the adaptive virtual control law corresponding to step 1. for:

[0026] In the formula: This represents the adaptive law; Indicate design parameters; express The first derivative; S332: for the second-order dynamic model, the first derivative; Subsystems Define the tracking error after intermediate compensation. for:

[0027]

[0028] In the formula: express The first derivative; Indicates the first The compensation signal for the step; based on the tracking error after intermediate compensation. Combining tracking error Construct the adaptive virtual control law corresponding to step i for:

[0029] In the formula: This represents the adaptive law; Indicate design parameters; express The first derivative; S333: for the second-order dynamic model, the first derivative; Subsystem: Based on the tracking error compensation mechanism of S32, the final tracking error after compensation is obtained by combining steps S331 to S332. for:

[0030] .

[0031] Furthermore, S4 specifically includes the following steps: S41: Define the control signals of the system controller. With control input The measurement error between them is:

[0032] In the formula: S42: Based on the measurement error, a dynamic event triggering mechanism is constructed as follows:

[0033]

[0034]

[0035] In the formula: Indicates the time of triggering The system controller's control signal sample value; Indicates the trigger time; Indicates a dynamic threshold variable; express The first derivative, i.e., the attenuation rate; , Indicates the design parameters and 0 < θ < 1. ; This represents the design parameters that determine the attenuation rate; S43: Based on the dynamic event triggering mechanism, by introducing time-varying parameters, the initial value of the dynamic threshold variable is represented; , and and The measurement error The description is as follows:

[0036] S44: Combining the formulas in S41 and S43, we obtain the following derivation:

[0037]

[0038] In the formula: This indicates a disturbance caused by a dynamic event triggering mechanism; , Indicate design parameters and , S44: Combine the optimized system model obtained in S23 with the derivation relationship obtained in S44, and then combine them with the final tracking error after compensation in S32. get:

[0039]

[0040] In the formula: express The first derivative; Indicates intermediate variables; Represents the total equivalent perturbation of the dynamic event triggering mechanism and ; S45: Indicates design constants; introduces a robust gain term to compensate for non-affine actuator faults and design intermediate control laws.

[0041] In the formula: This represents the robust gain term used to compensate for non-affine faults in the actuator, and ; Represents a small positive constant; This represents the design parameters; and the adaptive law of the intermediate control law. Designed as follows:

[0042] S46: Based on the intermediate control law, combined with The gain function yields the final fault-tolerant controller as follows:

[0043] In the formula: express Gain function; This represents the intermediate control law; and the adaptive law of the final fault-tolerant controller. Designed as follows:

[0044] In the formula: Indicates the design parameters; based on the final fault-tolerant controller, dynamic event-triggered preset-time fault-tolerant control considering non-affine faults is implemented.

[0045] This invention discloses a dynamic event-triggered preset-time fault-tolerant control method considering non-affine faults, with the following advantages: 1. Convergence time is preset and stable across the entire time domain: By introducing a piecewise preset-time performance function, this invention not only ensures that the system error is within the user-specified finite time... The convergence also cleverly solves the singularity problem of traditional preset time control after the terminal moment, achieving stable control in the infinite time domain.

[0046] 2. Strong fault tolerance: To address potential non-affine faults (such as dead zone and saturation) in the actuator, this invention utilizes the median theorem for decoupling and combination. Gain technology effectively addresses the unknown control direction and gain loss caused by faults, enhancing the robustness of the system.

[0047] 3. Low computational complexity: By introducing a command filter using dynamic surface control (DSC) technology, the "differential explosion" problem caused by repeatedly differentiating the virtual control law in the traditional backstepping method is avoided, and the influence of filtering error on control accuracy is eliminated through a dynamic error compensation mechanism.

[0048] 4. Communication resource saving: A dynamic event triggering mechanism with dynamic attenuation threshold is designed. Compared with a fixed threshold, this mechanism can dynamically adjust the triggering frequency according to the system status, which significantly reduces the number of control signal updates while ensuring control performance and saving communication resources. Attached Figure Description

[0049] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0050] Figure 1 is a flowchart of the dynamic event triggering preset time fault-tolerant control method of the present invention considering non-affine faults; Figure 2 is a comparison diagram of the trajectory tracking performance of the robotic arm system in this embodiment; Figure 3 is a comparison diagram of the tracking error response of the robotic arm system in this embodiment; Figure 4 is a curve diagram of the actual control input in this embodiment; Figure 5 is a curve diagram of the actual control input of the prior art (Solution [1]) in this embodiment; Figure 6 is a distribution diagram of the event triggering interval of the dynamic event triggering mechanism in this embodiment; Figure 7 is a distribution diagram of the event triggering interval of the static event triggering mechanism in this embodiment. Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of 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, 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.

[0052] This embodiment provides a dynamic event-triggered preset time fault-tolerant control method considering non-affine faults, as shown in Figure 1. Specifically, it includes the following steps: S1: Establishing a second-order dynamic model considering unknown nonlinear terms and actuator non-affine faults; and using the mean value theorem to decouple the actuator non-affine faults in the second-order dynamic model into an affine form to obtain a decoupled dynamic model. Specifically, this includes the following steps: S11: Establishing a second-order dynamic model considering unknown nonlinear terms and actuator non-affine faults as follows: (1) In the formula: This represents the system state of a second-order dynamical model corresponding to a cascaded structure with strict feedback, where each state... Indicates the previous level state Virtual control input; Represents the system's total state vector ; Represents the full state subvectors and ; Represents the nth state variable The derivative; Indicates system control input; Indicates system output; Represents an unknown, smooth, nonlinear dynamic term; Indicates unknown external disturbances and , Indicates the upper bound of external disturbances; Represented by the time distribution function Activated burst non-affine fault item; This indicates the time of occurrence of the non-affine fault; the model shows that the control input during the fault... Its non-affine form renders standard affine control design methods inapplicable; S12: To handle fault items The non-affine nature of the fault is addressed by using the Mean Value Theorem (MVT) to decompose the sudden non-affine fault term in the second-order dynamic model, i.e., for any... The nonlinear fault function can be decomposed into: (2) (3) Where: The affine part representing the fault dynamics; Indicates time-varying gain; Indicates a state between 0 and A point value between; Assumption 1: It is bounded, and there exists a known constant. Make and S13: Substituting equation (2) into equation (1) yields the nth subsystem in the second-order dynamic model as: (4)

[0053] , (5) Total control gain in this embodiment Used to handle the transition between normal operation and fault operation; S14: This embodiment transforms the non-affine fault system into a structure by restating the description, wherein the control input It is affine in form and has a time-varying, strictly non-zero gain. This facilitates controller design, namely, rewriting the second-order dynamic model based on S13 to obtain the dynamic decoupling model as follows: (6).

[0054] S2: A fuzzy logic system is used to approximate the unknown nonlinear terms in the dynamic decoupling model online to obtain an optimized system model. This includes the following steps: S21: A fuzzy logic system (FLS) is used to approximate the unknown function. To address system uncertainty, among which The representation is defined in compact sets A continuous function on, in this example, is... Define the fuzzy basis function vector of the fuzzy logic system as follows: and , Indicates the first There are fuzzy basis functions, and the weight vector is... and By employing a single-point fuzzifier, a product inference engine, and a center-average defuzzifier, the fuzzy basis function is... It can be represented as: (7) Where: Indicates the number of fuzzy rules; Indicates input variables For fuzzy sets Membership function; Lemma 1: Let It is defined on compact sets A continuous function on ; then, for any constant There exists an FLS approximator such that: (8) Where: Represents a fuzzy basis function vector. S22: Using Lemma 1, the unknown nonlinear terms in the dynamic decoupling model can be approximated online based on the fuzzy basis function vector and the weight vector as follows: (9) Where: Let represent the fuzzy approximation error of the i-th subsystem, and , express The upper bound; Indicates transpose; S23: Using a fuzzy logic system, i.e., based on S22, the dynamic decoupling model is rewritten into an optimized system model as follows: (10).

[0055] S3: Construct a piecewise preset time performance function. Based on dynamic surface control (DSC) technology, the virtual control signal of the constructed adaptive virtual control law is processed by introducing an instruction filter to obtain the filtering error. Based on the filtering error and the piecewise preset time performance function, a dynamic signal compensation mechanism is constructed to obtain the final tracking error after compensation. In this embodiment, a novel control scheme is proposed by combining dynamic surface control (DSC), PTC, and adaptive backstepping mechanism. To solve the "computational complexity explosion" and boundary problems of filter error in PTC, the specific steps include: S31: Construct the piecewise preset time performance function as follows: (11) Where: This indicates the user-defined preset convergence time. This represents a positive design parameter; in this embodiment, a piecewise preset time performance function is introduced. Furthermore, based on Dynamic Surface Control (DSC) technology, a command filter is introduced to process the virtual control signal, mapping the system tracking error to a preset time domain. This is done to mitigate the impact of time scale functions. To mitigate the computational complexity arising from differentiation, a piecewise preset time performance function was designed; since most existing PT control methods only... It is valid within the scope of the application and cannot be extended to other applications. This embodiment uses time Then cleverly used Replaced positive design parameters This extends the time domain to infinity ([0, +∞); S32: Based on dynamic surface control technology, an instruction filter is introduced to obtain the virtual control signal of the adaptive virtual control law to be designed. A tracking error compensation mechanism is constructed based on the virtual control signal to obtain the filtering error. (12) (13) (14) (15) Where: Indicates tracking error; This represents the desired reference trajectory or the desired output signal; Indicates the filtering error; Represents dynamic surface signals; This represents a time constant that is greater than zero. This represents the virtual control signal output by the adaptive virtual control law designed in step i-1. , express , The initial value; S33: Based on the piecewise preset time performance function, the final tracking error after compensation is obtained according to the filtering error, which includes: S331: For the first subsystem in the second-order dynamic model: Define the initial compensation tracking error. for: (16) (17) Where: Indicates a compensation signal. Dynamic design for handling filter errors ; express The first derivative; Indicate design parameters; Indicates the next compensation signal ( In this embodiment, the following are considered: Candidate function : (18) The time derivative is: (19) will Substituting the compensation dynamics (17) into (19): (20) Young's inequality is used to handle disturbances and approximation errors: (21) In order to achieve the preset time Internally stable system, based on the tracking error after initial compensation Combining tracking error Construct the adaptive virtual control law corresponding to step 1. for:

[0056] In the formula: This represents the adaptive law; Indicate design parameters; express The first derivative; Substitute equations (21), (22), and (23) into... In the expression: (24) S332: For the second-order dynamic model, the first Subsystems Define the tracking error after intermediate compensation. for: (25) (26) Where: express The first derivative; Indicates the first The compensation signal for the step; consider the following Candidate function : (27) The time derivative is: (28) will Substituting the compensation dynamics (26) into (28): (29) Next, similar to step 1, Young's inequality is used to handle the disturbance and approximation error: (30) For the i-th intermediate subsystem in the system (where i=2,...,n-1), in order to achieve recursive stability of the state and handle the dynamic coupling effects passed down from the previous subsystem, this embodiment is based on the compensated tracking error. Combining tracking error Construct the general adaptive virtual control law corresponding to step i for: (31) (32) Where: This represents the adaptive law; Indicate design parameters; express The first derivative; substituting equations (30), (31), and (32) into... In the expression: (33)S333: For the second-order dynamic model, the first Subsystem: Based on the tracking error compensation mechanism of S32, the final tracking error after compensation is obtained by combining steps S331 to S332. for: (34) (35)

[0057] In this implementation, since this is the final step, there are no subsequent filter errors or compensation signals. The dynamic design is used to counteract the coupling from the previous step.

[0058] S4: Construct a dynamic event triggering mechanism. Based on this mechanism, and according to the optimized system model and the final tracking error, design an intermediate control law by introducing a robust gain term to compensate for non-affine fault tolerance of the actuator; based on the intermediate control law combined with... The gain function yields the final fault-tolerant controller. Based on the final fault-tolerant controller, dynamic event-triggered preset-time fault-tolerant control considering non-affine faults is implemented. Specifically, the following steps are included: S41: Define the control signals of the system controller. With control input The measurement error between them is: (36) Where: Indicates measurement error; S42: In this embodiment, based on the amplitude of the control signal, the relative threshold strategy allows for different update times and control accuracies. The event-triggered control strategy, i.e., constructing a dynamic event-triggered mechanism based on the measurement error, is as follows: (37) (38) (39) Where: Indicates the time of triggering The system controller's control signal sample value; Indicates the trigger time; unlike static thresholds, Indicates a dynamic threshold variable; express The first derivative, i.e., the attenuation rate; , Indicates the design parameters and 0 < θ < 1. ; This represents the design parameters that determine the attenuation rate; This represents the initial value of the dynamic threshold variable; S43: Based on the dynamic event triggering mechanism, i.e. (38), by introducing time-varying parameters. , and and The measurement error The description is as follows: (40)S44: Combining the formulas of S41 and S43, we obtain the following derivation: (41) (42) Where: This represents a disturbance caused by a dynamic event triggering mechanism; the disturbance is bounded and satisfies... , Indicate design parameters and , S44: Combine the optimized system model obtained in S23 with the derivation relationship obtained in S44, and then combine them with the final tracking error after compensation in S32. get: (43) (44) Where: express The first derivative; Indicates intermediate variables; Represents the total equivalent perturbation of the dynamic event triggering mechanism and ; Indicates design constants; selection Candidate function : (45) The time derivative is: (46)S45: Introduce a robust gain term for compensating for non-affine faults in the actuator to design the intermediate control law: (47) Where: This represents the robust gain term used to compensate for non-affine faults in the actuator, and This is used to cover the trigger error boundary; Represents a small positive constant; This represents the design parameters; and the adaptive law of the intermediate control law. Designed as follows: (48)S46: Based on the intermediate control law, combined with The gain function yields the final fault-tolerant controller as follows: (49) Where: express Gain function; This represents the intermediate control law; and the adaptive law of the final fault-tolerant controller. Designed as follows: (50) Where: Indicates the design parameters; based on the final fault-tolerant controller, dynamic event-triggered preset-time fault-tolerant control considering non-affine faults is implemented.

[0059] In this embodiment Type function It has the following properties: (51) (52) Where: Indicates the upper limit of the integration interval; The gain and switching frequency of the type function are both infinite; in the embodiment, it is selected as... As Functions of type ; Lemma 2: Consider functions defined on the interval Smooth functions on and , making For all It holds true if the following inequalities are satisfied: (53) of which Represents positive constants, α represents positive design parameters, and time-varying parameters. Located in a closed interval ,but and exist The boundedness of the above is guaranteed.

[0060] In this embodiment, equations (47), (48), and (50) are substituted into Time derivative: (51) To deal with the hyperbolic tangent term, we introduce Lemma 3: for any variable and normal numbers The following inequalities hold: (52) where κ≈ 0.2785, applied The robustness property (52) is obtained as follows: (53) of which and took advantage of the conditions Therefore, the final inequality becomes: (54) This form is consistent with the results of steps 1 to n-1, ensuring a unified stability analysis framework.

[0061] This embodiment is based on the backstepping control method framework, combined with... The gain function is used to design virtual control laws and intermediate control laws, and a robust term is introduced to compensate for non-affine actuator faults and errors caused by event triggering. A dynamic event-triggered control strategy is designed, defining triggering conditions that include dynamic attenuation parameters. The actual control input acting on the actuator is only updated when the change in the control signal calculated by the controller exceeds the dynamic threshold; otherwise, the control value of the previous moment is maintained.

[0062] This embodiment also includes stability analysis: Theorem 1: The closed-loop system consists of the controlled object (6) and the reference trajectory It consists of a DSC filter (15) and incorporates adaptive laws (23), (32) and (48) as well as control laws (22), (31) and (47) constrained by a dynamic event triggering mechanism. Under the premise of satisfying assumption 1, the following characteristics are ensured: 1. Boundedness and convergence: It ensures that all closed-loop signals remain semi-globally consistent and eventually bounded (SGUUB). In addition, the compensated tracking error ; at the preset time Previously driven to a small set of residuals around the origin.

[0063] 2. No Zeno Behavior: The proposed dynamic event triggering mechanism successfully avoids Zeno behavior.

[0064] The proof process in this embodiment is as follows: The proof consists of three parts: through... The analysis included comprehensive stability analysis, boundedness analysis of tracking errors, and elimination of Zeno behavior.

[0065] (1) Stability analysis of tracking error: considering the overall The candidate function V is defined as follows: Sum of functions: Its time derivative is the sum of the derivatives of each step:

[0066] Substituting the derivative expressions obtained in steps 1 to n, specifically equations (20), (29), and (54), into (55), it can be observed that the virtual control terms designed in adjacent steps cancel out the cross-coupling terms. Furthermore, applying the property of the hyperbolic tangent function (Lemma 3), i.e. The total time derivative of V satisfies:

[0067] make To minimize the control gain, and define the lumped disturbance term. ,in It is the upper bound of the neural network approximation error. It is the upper bound of external disturbances. Is related to trigger parameters The relevant constants. Inequality (56) can be simplified to: (57) Among them, The energy function representing the tracking error.

[0068] (2) Boundedness and convergence analysis: In order to prove boundedness, within the time interval Integrating (57) above, and using Lemma 2 combined with the decoupled backstepping mechanism, confirms that the involved The boundedness of the integral term of a function, specifically:

[0069] This indicates , And the adaptive parameters remain consistent and eventually bounded. Therefore, it can be inferred that the signal... , , Both are bounded.

[0070] Next, we prove the convergence of the tracking error. By rewriting (57), we obtain: (59) Integrating both sides of (59), and noting that all closed-loop signals are bounded, we get:

[0071] This embodiment takes into account a preset time scaling function. The definition, obviously when Therefore, to ensure the boundedness of the integral, the tracking error... It must converge to a compact small set near the origin. Note that... and It is bounded, therefore the original tracking error can be known. It meets the preset performance requirements.

[0072] (3) Elimination of Zeno behavior: Finally, it is proved that the developed controller has semi-global event separation characteristics, that is, there exists a normal number. So that adjacent trigger times satisfy Based on the triggering conditions, for The measurement error is Its time derivative is:

[0073] Based on the structure of the control law, Depends on the system state, the adaptive law, and its derivatives (e.g.) and Given that the preceding analysis has verified the boundedness of all closed-loop signals, a positive constant can be determined. , making the inequality Established.

[0074] Note , and when The event is triggered at a certain time, from which we can conclude: (62) Therefore, the lower bound of the execution interval Must meet This indicates that the mechanism can effectively avoid Zeno behavior, even when the control signal approaches zero, by introducing a positive parameter. It also guarantees a non-zero minimum trigger interval. This characteristic ensures the feasibility of physical implementation.

[0075] To verify the effectiveness of the method described in this embodiment, a simulation verification is performed using a single-link robotic arm system as an example. The dynamic model of the single-link robotic arm system is described as follows: (63) Where q is the joint angle, w is the angular velocity, and u(t) is the control input torque. This represents the fault and disturbance terms. m is the mass of the connecting rod. Let g be the length of the connecting rod, g be the acceleration due to gravity, and B be the coefficient of viscous friction. Let be the moment of inertia.

[0076] The system physical parameters are set as follows: connecting rod mass m = 1.0 kg, connecting rod length... = 1.0m, gravitational acceleration q = 9.8m / The coefficient of viscous friction B = 1.0 N·m·s / rad, and the moment of inertia J = 1.0 kg·m·s / rad. .

[0077] Given a reference trajectory signal: .

[0078] Initial system state: .

[0079] The controller simulation parameters are set as follows: preset convergence time = 5.0s; Controller gain = 5, = 5; Filter time constant = 0.02s; Adaptive gain = 2, = 0.017; Leakage parameter = 1, fuzzy system width = 0.8; Robustness term parameter = 5.0, = 0.5; Dynamic event triggering parameter =0.15, = 0.25, =1.0, =5.0; Control input limit = 80 N·m.

[0080] Fault setting: When the system runs for t = 10s, a non-affine abrupt fault occurs, and the actuator efficiency drops to 60% accompanied by deviation disturbance. Computer simulation was conducted using MATLAB software, and the simulation results are shown in Figures 2 to 5.

[0081] In addition, to demonstrate the superiority of the method proposed in this embodiment in this work, simulations were performed and compared with the event-triggered preset time control scheme proposed in the literature "Event-triggered control for nonlinear uncertain systems via a prescribed-time approach" (hereinafter referred to as the scheme [1]); Figures 2 to 3 show the system trajectory tracking and error curves. It can be seen from the figures that the controller designed in this invention enables the system output to be within the preset time. The internal tracking of the reference signal is accurate, and the convergence speed is significantly faster than that of the scheme [1]. The tracking error converges to near zero. After the fault occurs at t=10s, the error fluctuation is minimal and it quickly recovers to stability, while the image of the scheme [1] is unstable after experiencing a non-affine fault.

[0082] Figures 4 and 5 show the controller input signal curves. It can be seen from the figures that the control torque in this example always remains within the physical limit range without any spikes. The control torque of scheme [1] reaches the limit after encountering a non-affine fault. Figures 6 and 7 show the event trigger interval curves. It can be seen from the figures that the dynamic event triggering mechanism adjusts the communication frequency according to the system state. The number of triggers is small in steady state, which effectively saves communication resources and avoids Zeno behavior.

[0083] 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 or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A dynamic event-triggered preset-time fault-tolerant control method considering non-affine faults, characterized in that, Specifically, the following steps are included: S1: Establish a second-order dynamic model considering unknown nonlinear terms and actuator non-affine faults; and use the mean value theorem to decouple the actuator non-affine faults in the second-order dynamic model into affine form to obtain a dynamic decoupling model; S2: Use a fuzzy logic system to approximate the unknown nonlinear terms in the dynamic decoupling model online to obtain an optimized system model; S3: Construct a piecewise preset time performance function, and based on dynamic surface control technology, introduce an instruction filter to process the virtual control signal of the constructed adaptive virtual control law to obtain the filtering error; A dynamic signal compensation mechanism is constructed based on filtering error and a piecewise preset time performance function to obtain the final tracking error after compensation. S4: Construct a dynamic event triggering mechanism. Based on this mechanism, and according to the optimized system model and the final tracking error, design an intermediate control law by introducing a robust gain term to compensate for non-affine fault tolerance of the actuator; based on the intermediate control law combined with... The gain function yields the final fault-tolerant controller, and based on the final fault-tolerant controller, dynamic event-triggered preset-time fault-tolerant control considering non-affine faults is implemented.

2. The dynamic event-triggered preset time fault-tolerant control method considering non-affine faults according to claim 1, characterized in that, The S1 specifically The following steps are included: S11: Establish a second-order dynamic model considering unknown nonlinear terms and non-affine faults of the actuator as follows: In the formula: Represents the system states of a second-order dynamic model of a cascaded structure, each state... Indicates the previous level state Virtual control input; Represents the system's total state vector ； Represents the full state subvectors and ； Represents the nth state variable The derivative; Indicates system control input; Indicates system output; Represents an unknown, smooth, nonlinear dynamic term; Indicates unknown external disturbances and ， Indicates the upper bound of external disturbances; Represented by the time distribution function Activated burst non-affine fault item; S12: The sudden non-affine fault term in the second-order dynamic model is decomposed using the mean value theorem as follows: In the formula: The affine part representing the fault dynamics; Indicates time-varying gain; Indicates a state between 0 and A point value between; S13: Based on steps S11 to S12, obtain the nth subsystem in the second-order dynamic model as: ， S14: Based on S13, the second-order dynamic model is rewritten to obtain the dynamic decoupling model as follows: 。 3. The dynamic event-triggered preset time fault-tolerant control method considering non-affine faults according to claim 2, characterized in that, The S2 specifically The steps include: S21: Define the fuzzy basis function vector of the fuzzy logic system as... and ， Indicates the first There are fuzzy basis functions, and the weight vector is... and The fuzzy basis function The expression is: In the formula: Indicates the number of fuzzy rules; Indicates input variables For fuzzy sets The membership function; S22: Based on the fuzzy basis function vector and weight vector, the unknown nonlinear term in the dynamic decoupling model is approximated online as follows: In the formula: Let represent the fuzzy approximation error of the i-th subsystem, and ， express The upper bound; Indicates transpose; S23: Based on S22, the dynamic decoupling model is rewritten as an optimized system model as follows: 。 4. The dynamic event-triggered preset time fault-tolerant control method considering non-affine faults according to claim 3, characterized in that, The S3 specifically The following steps are included: S31: Construct a piecewise preset time performance function as follows: In the formula: This indicates the user-defined preset convergence time. S32: Based on dynamic surface control technology, an instruction filter is introduced to obtain the virtual control signal of the adaptive virtual control law to be designed. A tracking error compensation mechanism is constructed based on the virtual control signal to obtain the filtering error. In the formula: Indicates tracking error; This represents the desired reference trajectory or the desired output signal; Indicates the filtering error; Represents dynamic surface signals; This represents a time constant that is greater than zero. This represents the virtual control signal output by the adaptive virtual control law designed in step i-1. , express , The initial value; S33: Based on the piecewise preset time performance function, the final tracking error after compensation is obtained according to the filtering error, which includes: S331: For the first subsystem in the second-order dynamic model: Define the initial compensated tracking error. for: In the formula: Indicates a compensation signal; express The first derivative; Indicate design parameters; This indicates the next compensation signal; based on the tracking error after initial compensation. Combining tracking error Construct the adaptive virtual control law corresponding to step 1. for: In the formula: This represents the adaptive law; Indicate design parameters; express The first derivative; S332: for the second-order dynamic model, the first derivative; Subsystems Define the tracking error after intermediate compensation. for: In the formula: express The first derivative; Indicates the first The compensation signal for the step; based on the tracking error after intermediate compensation. Combining tracking error Construct the adaptive virtual control law corresponding to step i for: In the formula: This represents the adaptive law; Indicate design parameters; express The first derivative; S333: for the second-order dynamic model, the first derivative; Subsystem: Based on the tracking error compensation mechanism of S32, the final tracking error after compensation is obtained by combining steps S331 to S332. for: 。 5. The dynamic event-triggered preset time fault-tolerant control method considering non-affine faults according to claim 4, characterized in that, S4 specifically includes the following steps: S41: Define the control signals of the system controller. With control input The measurement error between them is: In the formula: S42: Based on the measurement error, a dynamic event triggering mechanism is constructed as follows: In the formula: Indicates the time of triggering The system controller's control signal sample value; Indicates the trigger time; Indicates a dynamic threshold variable; express The first derivative, i.e., the attenuation rate; , Indicates the design parameters and 0 < θ < 1. ； This represents the design parameters that determine the attenuation rate; S43: Based on the dynamic event triggering mechanism, by introducing time-varying parameters, the initial value of the dynamic threshold variable is represented; , and and The measurement error The description is as follows: S44: Combining the formulas in S41 and S43, we obtain the following derivation: In the formula: This indicates a disturbance caused by a dynamic event triggering mechanism; ， Indicate design parameters and ， S44: Combine the optimized system model obtained in S23 with the derivation relationship obtained in S44, and then combine them with the final tracking error after compensation in S32. get: In the formula: express The first derivative; Indicates intermediate variables; Represents the total equivalent perturbation of the dynamic event triggering mechanism and ； S45: Indicates design constants; introduces a robust gain term to compensate for non-affine actuator faults and design intermediate control laws. In the formula: This represents the robust gain term used to compensate for non-affine faults in the actuator, and ； Represents a small positive constant; This represents the design parameters; and the adaptive law of the intermediate control law. Designed as follows: S46: Based on the intermediate control law, combined with The gain function yields the final fault-tolerant controller as follows: In the formula: express Gain function; This represents the intermediate control law; and the adaptive law of the final fault-tolerant controller. Designed as follows: In the formula: Indicates the design parameters; based on the final fault-tolerant controller, dynamic event-triggered preset-time fault-tolerant control considering non-affine faults is implemented.

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