Fixed-time preset-performance ship rolling chaos suppression fault-tolerant control method
By designing the final fault-tolerant control law using a fuzzy logic system and an inverse stepping framework, the problems of rapid convergence and dynamic performance constraints under actuator failure in ship roll control are solved. This achieves high-precision, fast-convergence ship roll suppression, improving the stability and safety of ships in complex sea conditions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- DALIAN MARITIME UNIVERSITY
- Filing Date
- 2026-01-23
- Publication Date
- 2026-04-21
AI Technical Summary
Traditional ship roll control methods struggle to simultaneously guarantee transient performance and steady-state accuracy when dealing with model uncertainties, external disturbances, and actuator failures. In particular, they cannot effectively address the issues of rapid convergence and dynamic performance constraints under partial actuator failure conditions.
A fuzzy logic system with fixed-time preset performance is adopted. By constructing a fuzzy state observer to approximate the unknown nonlinear function, a controlled system model containing a health state matrix is designed. Finally, the fault-tolerant control law is designed using the backstepping framework to achieve high-precision and fast-convergence control of the ship's roll motion.
Even under various abnormal operating conditions such as actuator failure, the system can still remain stable, and the tracking error is constrained within the preset performance range, which significantly improves the safety and reliability of ship navigation and enhances the system's convergence speed and robustness.
Smart Images
Figure CN121900469A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of ship motion control and stability assurance, and in particular to a fixed-time preset performance ship roll chaos suppression and fault-tolerant control method. Background Technology
[0002] When ships navigate in complex sea states, their roll motion exhibits significant nonlinear characteristics, easily inducing chaotic motion under specific parameter conditions. In severe cases, this can lead to large rolls or even capsizing risks. Traditional roll control methods (such as classical feedback or gain shaping control) have limited robustness and convergence speed when dealing with model uncertainties, external disturbances, and actuator failures, making it difficult to simultaneously guarantee transient performance and steady-state accuracy. In recent years, pre-defined performance control and fuzzy backstepping methods have gained attention in nonlinear system control, but they still cannot effectively solve the problems of rapid convergence and dynamic performance constraints for chaotic roll motion of ships, especially under conditions of partial actuator failure. Summary of the Invention
[0003] This invention provides a fixed-time preset performance ship roll chaos suppression and fault-tolerant control method to overcome the problems of rapid convergence and dynamic performance constraints under actuator partial failure conditions.
[0004] To achieve the above objectives, the technical solution of the present invention is as follows: A fixed-time preset performance ship roll chaos suppression and fault-tolerant control method, characterized by comprising: S1. Establish a mathematical model of the ship's rolling motion; S2. Based on the mathematical model of ship rolling motion, construct a controlled system model that includes unknown nonlinear functions and a health state matrix; S3. Based on the controlled system model, and by introducing a fuzzy logic system as an approximator for the unknown nonlinear function, a fuzzy state observer is constructed to output the state estimate of the mathematical model of the ship's rolling motion. S4. Define the tracking error based on the desired roll angle and the state estimate; design a fixed-time preset performance function to generate the time-varying constraint boundary of the tracking error; S5. Based on the tracking error and time-varying constraint boundary, design the final fault-tolerant control law through the backstepping framework to provide stable ship roll suppression capability.
[0005] Furthermore, the mathematical model for the ship's rolling motion is as follows: (1) In the formula, This refers to the roll angle; The excitation amplitude is a dimensionless parameter. The excitation frequency is a parameter. and This is the roll attenuation coefficient; and The coefficient of dimensionless restoring torque; The natural frequency of ship rolling; The current moment; This refers to the roll speed; This is the roll acceleration.
[0006] Furthermore, the steps for constructing a controlled system model that includes unknown nonlinear functions and a health state matrix include: S21. Introduce control inputs and state variables, and set the state variables... , Based on the mathematical model of ship rolling motion, the following expression is obtained: (2) In the formula, for The first derivative with respect to time, i.e., the roll angular velocity; for The first derivative with respect to time, i.e., the roll acceleration; It is an unknown nonlinear function; Let be the health state matrix of the actuator, and ,in, The health status of the i-th executor is given by i, where i is the index of the executor and n is the total number of executors. For the controller input; S22. Introducing the observer gain, equation (2) is rewritten as a controlled system model containing an unknown nonlinear function and a health state matrix, expressed as: (3) In the formula, Let be the state vector, and ; The derivative of the state vector; It is a measurable variable; and All are observer gain matrices; The distribution matrix of nonlinear terms; and This is the observer gain.
[0007] Furthermore, the fuzzy state observer is constructed as follows: (4) In the formula, To approximate a fuzzy logic system with an unknown nonlinear function, where, For parameter estimation vectors; Here, the state estimate is given by the mathematical model of the ship's rolling motion, where... This is an estimated value for the roll angle. This is an estimated value for the roll rate; The derivative of the state estimate of the mathematical model of the ship's rolling motion; This is the observation matrix.
[0008] Furthermore, based on the desired roll angle and the state estimate, the expression for the tracking error is defined as follows: (5) In the formula, and For error variables; The desired roll angle; The desired roll rate; Design a fixed-time preset performance function to generate time-varying constraint boundaries for tracking errors. The fixed-time preset performance function is as follows: (6) In the formula, The time-varying constraint boundary for tracking error; , , , All of these are design parameters, among which, To stabilize the boundary, As the initial boundary, For shape parameters, Let be the convergence rate parameter, and , , , ; This is the preset convergence time.
[0009] Furthermore, the specific process of designing the final fault-tolerant control law includes: S501. Based on the tracking error and time-varying constraint boundary, construct the transformation function, the expression of which is: (7) In the formula, These are design parameters used to adjust the scaling ratio of the transformation function; For tracking error The time-varying constraint boundary, i.e. ; For tracking error The transformation variable; S502. Taking the derivative of the transformation function, we obtain the following expression: (8) In the formula, and The parameter is time-varying, and ; for The derivative; for The derivative; for The derivative; S503. Using the backstepping framework, select the following Lyapunov function: (9) In the formula, Lyapunov functions of observation error and tracking error; Lyapunov function of observation error; S504. Differentiating equation (9) yields the following expression: (10) In the formula, for The derivative; for The derivative; S505. Design the following virtual control law: (11) In the formula, For the design of positive constant parameters, and ; For virtual control laws; S506. Substituting equation (11) into equation (10), we obtain the following expression: (12) S507. Design a first-order low-pass filter with a virtual control law to output a substitution signal, expressed as follows: (13) In the formula, These are the design parameters for a first-order low-pass filter, and ; As a substitute signal; S508, according to equation (13), and defining the tracking error of the first-order low-pass filter. And rewrite equation (12) as follows: (14) In the formula, For tracking error The transformation variable; S509, Based on the backstepping framework Taking the derivative, we get the following expression: (15) S510, Set For an unknown nonlinear function, i.e. Approximation through fuzzy logic systems This yields the following expression: (16) In the formula, The optimal parameters are unknown. To minimize the fuzzy approximation error; S511. Substituting equation (16) into equation (15), we obtain the following expression: (17) S512. Using the backstepping framework, select the following Lyapunov function: (18) In the formula, To synthesize Lyapunov functions; S513. Differentiating equation (18), we obtain the following expression: (19) S514. Based on equations (17) and (19), the final fault-tolerant control law is obtained. ,in, , and It is expressed as follows: (20) (twenty one) (twenty two) In the formula, For design parameters, and In order to Provides negative feedback damping; This represents the minimum performance lower bound for the health state of the actuator.
[0010] Beneficial effects: This invention provides a fixed-time preset performance ship roll chaos suppression and fault-tolerant control method. By constructing a fuzzy state observer to approximate the unknown nonlinear function in the system and designing a controlled model including a health state matrix, it can effectively compensate for the effects of model uncertainty, external disturbances, and partial failure or malfunction of actuators. The designed final fault-tolerant control law ensures that the system can still maintain stability and preset control performance under various abnormal operating conditions such as actuator failure, which greatly improves the safety and reliability of ship navigation. By modeling the nonlinear dynamics of ship roll and combining it with a pre-set performance function at a specified time and a fuzzy fault-tolerant strategy, high-precision and fast-convergence control of ship roll motion is achieved. Compared with traditional methods, this control strategy can effectively suppress chaotic oscillations even under actuator failure conditions, and the tracking error is always constrained within the preset performance range. The system's convergence speed, robustness, and fault tolerance are significantly improved, which is of great value for enhancing ship roll stability and navigation safety under complex sea conditions. Attached Figure Description
[0011] 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.
[0012] Figure 1 This is a flowchart of the fault-tolerant control method of the present invention; Figure 2 The Lyapunov exponent is used in the mathematical model of ship roll motion in this embodiment of the invention. Figure 3 This is a comparison diagram of roll angles in an embodiment of the present invention; Figure 4 This is a comparison diagram of roll angular velocities in embodiments of the present invention; Figure 5 System state in an embodiment of the present invention With observation status curve; Figure 6 System state in an embodiment of the present invention With observation status curve; Figure 7 This refers to the control input curve in an embodiment of the present invention; Figure 8 Tracking error in embodiments of the present invention curve; Figure 9 Tracking error in embodiments of the present invention curve. Detailed Implementation
[0013] 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.
[0014] This embodiment provides a fixed-time preset performance ship roll chaos suppression and fault-tolerant control method, such as... Figure 1 As shown, it includes: S1. Establish a mathematical model of the ship's rolling motion; S2. Based on the mathematical model of ship rolling motion, construct a controlled system model that includes unknown nonlinear functions and a health state matrix; S3. Based on the controlled system model, and by introducing a fuzzy logic system as an approximator for the unknown nonlinear function, a fuzzy state observer is constructed to output the state estimate of the mathematical model of the ship's rolling motion. S4. Define the tracking error based on the desired roll angle and the state estimate; design a fixed-time preset performance function to generate the time-varying constraint boundary of the tracking error; S5. Based on the tracking error and time-varying constraint boundary, design the final fault-tolerant control law through the backstepping framework to provide stable ship roll suppression capability.
[0015] Preferably, the mathematical model for the ship's rolling motion is as follows: (1) In the formula, This refers to the roll angle; The excitation amplitude is a dimensionless parameter. The excitation frequency is a parameter. and This is the roll attenuation coefficient; and The coefficient of dimensionless restoring torque; The natural frequency of ship rolling; The current moment; This refers to the roll speed; This is the roll acceleration.
[0016] In this embodiment, the parameters of the ship's roll motion mathematical model are directly calculated from the ship's main dimensions, mass distribution, and stability data, or derived through empirical formulas. Taking a patrol boat as an example, the boat is 49 meters long, 7.4 meters wide, has a full-load displacement of 341.14 tons, an initial metacentric height of 0.987 meters, a full-load draft of 2.317 meters, and a total mass moment of inertia of 269.42 tons / meter. The relevant parameters obtained through calculation are shown in Table 1. Table 1 Parameters of the Chaotic Mathematical Model for Ship Roll
[0017] Lyapunov exponents for solving a chaotic mathematical model of ship roll using the Jacobi method, such as Figure 2 As shown, the value of the Lyapunov exponent changes over time, and the final Lyapunov exponent of the iteration result is... , , The largest Lyapunov exponent is positive, indicating that the ship rolling chaos mathematical model exhibits chaotic phenomena.
[0018] Preferably, the steps for constructing a controlled system model that includes an unknown nonlinear function and a health state matrix include: S21. Introduce control inputs and state variables, and set the state variables... , Based on the mathematical model of ship rolling motion, the following expression is obtained: (2) In the formula, for The first derivative with respect to time, i.e., the roll angular velocity; for The first derivative with respect to time, i.e., the roll acceleration; It is an unknown nonlinear function; Let be the health state matrix of the actuator, and ,in, The health status of the i-th executor is given by i, where i is the index of the executor and n is the total number of executors. For the controller input; S22. Introducing the observer gain, equation (2) is rewritten as a controlled system model containing an unknown nonlinear function and a health state matrix, expressed as: (3) In the formula, Let be the state vector, and ; The derivative of the state vector; It is a measurable variable; and All are observer gain matrices; The distribution matrix of nonlinear terms; and This is the observer gain.
[0019] Preferably, a fuzzy logic system is used. Approximating an unknown nonlinear function And assume ,in, For positive constants; using the controlled system model, the fuzzy state observer is constructed as follows: (4) In the formula, To approximate a fuzzy logic system with an unknown nonlinear function, where, For parameter estimation vectors; Here, the state estimate is given by the mathematical model of the ship's rolling motion, where... This is an estimated value for the roll angle. This is an estimated value for the roll rate; The derivative of the state estimate of the mathematical model of the ship's rolling motion; This is the observation matrix.
[0020] In a specific embodiment, the stability of the fuzzy state observer is verified, and the specific steps are as follows: S301. Define the observation error as follows: (5) In the formula, This is the observation error; S302. Based on equations (4) and (5), the following expression is obtained: (6) In the formula, For parameter error, and ; Let be the fuzzy approximation error vector, and ,in, This is the first fuzzy approximation error. This is the second fuzzy approximation error; S303. Define the following Lyapunov functions: (7) In the formula, Lyapunov function of observation error; It is a symmetric positive definite matrix; Substituting equation (6) into equation (7), we get: (8) In the formula, For fuzzy basis function vectors in The value of ; S304. Define the following expression using Young's inequality: (9) (10) S305. Substituting equations (9) and (10) into equation (8), we obtain the following expression: (11) In the formula, For the final convergence performance, and ; This is the error boundary, and ; By selecting a suitable fuzzy logic system for approximation using equation (11), it can be guaranteed that... Smaller, while selecting an appropriate matrix Make If it is large enough, the observation error can be concluded to converge to a small neighborhood containing the origin.
[0021] Preferably, based on the desired roll angle and the state estimate, the expression for the tracking error is defined as follows: (12) In the formula, and For error variables; The desired roll angle; The desired roll rate; Design a fixed-time preset performance function to generate time-varying constraint boundaries for tracking errors. The fixed-time preset performance function is as follows: (13) In the formula, The time-varying constraint boundary for tracking error; , , , All of these are design parameters, among which, To stabilize the boundary, As the initial boundary, For shape parameters, Let be the convergence rate parameter, and , , , ; This is the preset convergence time.
[0022] Specifically, If the tracking error satisfies the following property: Always in Within the range, then in hour, The transient performance can be guaranteed; in hour, Guaranteed to be bound by Within the specified interval, the system's steady-state performance is satisfied; The physical meaning and mechanism of action of each design parameter are as follows: Parameter Decision made exist The constant value of the parameter indirectly determines the final convergence region of the preset performance function, thereby determining the steady-state convergence accuracy of the system tracking error; The initial value determines the initial envelope range of the preset performance function; therefore, the proposed method requires the initial tracking error of the system to satisfy... Convergence time Depend on , , Together they determine the system's convergence speed, that is ; also Decide The slope of the envelope's contraction indirectly affects the system's overshoot characteristics.
[0023] Preferably, the specific process of designing the final fault-tolerant control law includes: S501. Based on the tracking error and time-varying constraint boundary, construct the transformation function, the expression of which is: (14) In the formula, These are design parameters used to adjust the scaling ratio of the transformation function; For tracking error The time-varying constraint boundary, i.e. ; For tracking error The transformation variable; S502. Taking the derivative of the transformation function, we obtain the following expression: (15) In the formula, and The parameter is time-varying, and ; for The derivative; for The derivative; for The derivative; S503. Using the backstepping framework, select the following Lyapunov function: (16) In the formula, Lyapunov functions of observation error and tracking error; Lyapunov function of observation error; S504. Differentiating equation (16) yields the following expression: (17) In the formula, for The derivative; for The derivative; S505. Design the following virtual control law: (18) In the formula, For the design of positive constant parameters, and ; For virtual control laws; S506. Substituting equation (18) into equation (17), we obtain the following expression: (19) S507. Design a first-order low-pass filter with a virtual control law to output a substitution signal, expressed as follows: (20) In the formula, These are the design parameters for a first-order low-pass filter, and ; As a substitute signal; S508, according to equation (20), and defining the tracking error of the first-order low-pass filter. And rewrite equation (19) as follows: (twenty one) In the formula, For tracking error The transformation variable; S509. Using Young's inequality, equation (22) is scaled and the expression is: ;(twenty three) S510. Substituting equation (23) into equation (22), we obtain the following expression: ;(twenty four) S511, To calm down Based on the backstepping framework Taking the derivative, we get the following expression: (25) S512, Let For an unknown nonlinear function, i.e. Approximation through fuzzy logic systems This yields the following expression: (26) In the formula, The optimal parameters are unknown. To minimize the fuzzy approximation error; S513. Substituting equation (26) into equation (25), we obtain the following expression: (27) S514. Using the backstepping framework, select the following Lyapunov function: (28) In the formula, To synthesize Lyapunov functions; S515. Differentiating equation (27), we obtain the following expression: (29) S516. Based on equations (27) and (29), the final fault-tolerant control law is obtained. ,in, , and It is expressed as follows: (30) (31) (32) In the formula, For design parameters, and In order to Provides negative feedback damping; This represents the minimum performance lower bound for the health state of the actuator.
[0024] In a specific embodiment, the stability of the final fault-tolerant control law is verified, and the specific steps are as follows: S601. Substituting equations (30), (31), and (32) into equation (29), we obtain the following expression: (33) S602. Using Young's inequality, we obtain the following expression: (34) S603. Substituting equation (34) into equation (33), we obtain the following expression: (35) In the formula, for Upper boundary; Design parameters, to represent The attenuation coefficient in the derivative of a Lyapunov function; in, The adaptive law is defined as: (36) S604. Substituting equation (36) into equation (35), we obtain the following expression: (37) S605. Using Young's inequality, we obtain the following expression: (38) (39) (40) S606. Substituting equations (38), (39), and (40) into equation (37), we obtain the following expression: (41) S607, Set According to the final steps of the Lyapunov stability analysis, equation (41) is transformed into: (42) In the formula, Let be the total energy function of all errors; The upper bound of the total perturbation, and ; According to equation (42), it can be proved that all signals in the control of the ship's rolling motion are consistent and eventually bounded, and have In the design of virtual controllers and actual controllers, selecting appropriate design parameters can make the target parameter values smaller, thereby enabling the tracking error to converge to a smaller neighborhood containing the origin.
[0025] In this embodiment, to verify the effectiveness of the final fault-tolerant control law and evaluate its performance, a comparative simulation experiment was conducted with the fuzzy adaptive state feedback algorithm. The ship in Table 1 was used as the simulation object for the effectiveness verification. To ensure a fair comparison, each control parameter was kept consistent in the simulation experiment. In the simulation experiment, the design parameters were set as follows: Expected value of roll angle The initial value of the roll angle Initial value of roll angular velocity , , , , , , , , , , , , , , , ; Simulation results are as follows Figures 3 to 9 As shown; where, Figure 3 and Figure 4 This indicates that, compared with the fuzzy adaptive state feedback algorithm, the final fault-tolerant control law of this invention has significant advantages in suppressing ship roll motion: it not only effectively reduces the oscillation amplitude of roll angle and roll velocity, but also significantly shortens the convergence time, and the system state converges to the vicinity of the equilibrium point faster and more smoothly. Figure 5 The tracking performance of the fuzzy state observer proposed in this invention for the roll angle was verified. The observed state highly coincides with the actual roll angle within a short time, and the error is strictly limited to within a certain range. Within the error band, it is proven that its estimation accuracy for directly measurable state quantities meets the control requirements; Figure 6 The estimation performance of the fuzzy state observer for stateless state variables was further verified. The observed values in the figure can converge quickly and closely track the real dynamic change trajectory. Figure 7 To control the input curve, namely the roll angle change curve of the patrol boat, the initial stage has a large amplitude. This is to quickly respond to the initial chaotic deviation of the ship's roll and efficiently suppress chaotic motion. As the system roll angle and roll angular velocity gradually converge to the desired state, the control input quickly transitions to a stable state, which not only avoids excessive consumption of control capability, but also reflects the algorithm's ability to smoothly adjust under actuator failure conditions. from Figure 8 and Figure 9 As can be seen, the tracking error is always limited to within the two envelopes, satisfying the performance constraints. .exist hour, The transient performance is guaranteed; in hour, The steady-state performance is guaranteed, thus enabling preset performance control of the roll motion at a specified time. For performance evaluation, in order to conduct quantitative analysis and comparison, Mean Absolute Error (MAE), Mean Integrated Absolute (MIA), Mean Total Variation (MTV), and performance metrics were selected. As performance indicators, the specific numerical results are shown in Table 2: Table 2 Comparison of Simulation Results Data
[0026] As shown in Table 2, the simulation results demonstrate that, compared to the fuzzy adaptive state feedback algorithm, the fault-tolerant control method of this invention achieves significant improvements in all performance indicators. Decreased by 0.68%, and These figures decreased by 79.54% and 76.25% respectively. The efficiency was reduced by 34.2%. As can be seen from the above results, the fault-tolerant control method of the present invention is superior to the fuzzy adaptive state feedback algorithm in terms of output response time, energy consumption related to rudder angle, smoothness, and system convergence speed.
[0027] The present invention has the following beneficial effects: This invention discloses a fixed-time preset performance ship roll chaos suppression fault-tolerant control method. By constructing a fuzzy state observer to approximate the unknown nonlinear function in the system and designing a controlled model including a health state matrix, it can effectively compensate for the effects of model uncertainty, external disturbances, and partial failure or malfunction of actuators. The designed final fault-tolerant control law ensures that the system can still maintain stability and the preset control performance under various abnormal conditions such as actuator failure, which greatly improves the safety and reliability of ship navigation. By modeling the nonlinear dynamics of ship roll and combining it with a pre-set performance function at a specified time and a fuzzy fault-tolerant strategy, high-precision and fast-convergence control of ship roll motion is achieved. Compared with traditional methods, this control strategy can effectively suppress chaotic oscillations even under actuator failure conditions, and the tracking error is always constrained within the preset performance range. The system's convergence speed, robustness, and fault tolerance are significantly improved, which is of great value for enhancing ship roll stability and navigation safety under complex sea conditions.
[0028] 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 fixed-time preset performance ship roll chaos suppression and fault-tolerant control method, characterized in that, include: S1. Establish a mathematical model of the ship's rolling motion; S2. Based on the mathematical model of ship rolling motion, construct a controlled system model that includes unknown nonlinear functions and a health state matrix; S3. Based on the controlled system model, and by introducing a fuzzy logic system as an approximator for the unknown nonlinear function, a fuzzy state observer is constructed to output the state estimate of the mathematical model of the ship's rolling motion. S4. Define the tracking error based on the desired roll angle and the state estimate; design a fixed-time preset performance function to generate the time-varying constraint boundary of the tracking error; S5. Based on the tracking error and time-varying constraint boundary, design the final fault-tolerant control law through the backstepping framework to provide stable ship roll suppression capability.
2. The fixed-time preset performance ship roll chaos suppression and fault-tolerant control method according to claim 1, characterized in that, The mathematical model for the ship's rolling motion is as follows: （1） In the formula, This refers to the roll angle; The excitation amplitude is a dimensionless parameter. The excitation frequency is a parameter. and This is the roll attenuation coefficient; and The coefficient of dimensionless restoring torque; The natural frequency of ship rolling; The current moment; This refers to the roll speed; This is the roll acceleration.
3. The fixed-time preset performance ship roll chaos suppression and fault-tolerant control method according to claim 2, characterized in that, The steps for constructing a controlled system model that includes an unknown nonlinear function and a health state matrix include: S21. Introduce control inputs and state variables, and set the state variables... , Based on the mathematical model of ship rolling motion, the following expression is obtained: （2） In the formula, for The first derivative with respect to time, i.e., the roll angular velocity; for The first derivative with respect to time, i.e., the roll acceleration; It is an unknown nonlinear function; Let be the health state matrix of the actuator, and ,in, The health status of the i-th executor is given by i, where i is the index of the executor and n is the total number of executors. For the controller input; S22. Introducing the observer gain, equation (2) is rewritten as a controlled system model containing an unknown nonlinear function and a health state matrix, expressed as: （3） In the formula, Let be the state vector, and ; The derivative of the state vector; It is a measurable variable; and All are observer gain matrices; The distribution matrix of nonlinear terms; and This is the observer gain.
4. A fixed-time preset performance ship roll chaos suppression and fault-tolerant control method according to claim 3, characterized in that, The fuzzy state observer is constructed as follows: （4） In the formula, To approximate a fuzzy logic system with an unknown nonlinear function, where, For parameter estimation vectors; Here, the state estimate is given by the mathematical model of the ship's rolling motion, where... This is an estimated value for the roll angle. This is an estimated value for the roll angular velocity; The derivative of the state estimate of the mathematical model of the ship's rolling motion; This is the observation matrix.
5. A fixed-time preset performance ship roll chaos suppression and fault-tolerant control method according to claim 4, characterized in that, Based on the desired roll angle and the state estimate, the expression for the tracking error is defined as follows: （5） In the formula, and For error variables; The desired roll angle; The desired roll rate; Design a fixed-time preset performance function to generate time-varying constraint boundaries for tracking errors. The fixed-time preset performance function is as follows: （6） In the formula, The time-varying constraint boundary for tracking error; , , , All of these are design parameters, among which, To stabilize the boundary, As the initial boundary, For shape parameters, Let be the convergence rate parameter, and , , , ; This is the preset convergence time.
6. A fixed-time preset performance ship roll chaos suppression and fault-tolerant control method according to claim 5, characterized in that, The specific process of designing the final fault-tolerant control law includes: S501. Based on the tracking error and time-varying constraint boundary, construct the transformation function, the expression of which is: （7） In the formula, These are design parameters used to adjust the scaling ratio of the transformation function; For tracking error The time-varying constraint boundary, i.e. ; For tracking error The transformation variable; S502. Taking the derivative of the transformation function, we obtain the following expression: （8） In the formula, and The parameter is time-varying, and ; for The derivative; for The derivative; for The derivative; S503. Using the backstepping framework, select the following Lyapunov function: （9） In the formula, Lyapunov functions of observation error and tracking error; Lyapunov function of observation error; S504. Differentiating equation (9) yields the following expression: （10） In the formula, for The derivative; for The derivative; S505. Design the following virtual control law: （11） In the formula, For the design of positive constant parameters, and ; For virtual control laws; S506. Substituting equation (11) into equation (10), we obtain the following expression: ；（12） S507. Design a first-order low-pass filter with a virtual control law to output a substitution signal, expressed as follows: （13） In the formula, These are the design parameters for a first-order low-pass filter, and ; As a substitute signal; S508, according to equation (13), and defining the tracking error of the first-order low-pass filter. And rewrite equation (12) as follows: （14） In the formula, For tracking error The transformation variable; S509, Based on the backstepping framework Taking the derivative, we get the following expression: ；（15） S510, Set For an unknown nonlinear function, i.e. Approximation through fuzzy logic systems This yields the following expression: （16） In the formula, The optimal parameters are unknown. To minimize the fuzzy approximation error; S511. Substituting equation (16) into equation (15), we obtain the following expression: ；（17） S512. Using the backstepping framework, select the following Lyapunov function: （18） In the formula, To synthesize Lyapunov functions; S513. Differentiating equation (18), we obtain the following expression: ；（19） S514. Based on equations (17) and (19), the final fault-tolerant control law is obtained. ,in, , and It is expressed as follows: （20） （21） （22） In the formula, For design parameters, and In order to Provides negative feedback damping; This represents the minimum performance lower bound for the health state of the actuator.