A ship bow shaking chaos phenomenon nested iterative sliding mode control method

Abstract

The application discloses a ship bow shaking chaos phenomenon nested iteration sliding mode control method, through the established ship straight sailing chaos model; the nested iteration sliding mode controller is constructed, the dynamic surface technology is combined to simplify the design process, the heading stability and the rudder angle response optimization are realized; the model perturbation and the sea wave disturbance are effectively coped with, and the bow shaking chaos control precision and the robustness are obviously improved. Compared with the traditional method, the application is better in the heading convergence speed, the steady-state error suppression and the rudder angle regulation sensitivity, effectively solves the problems that the traditional sliding mode control has large chattering, slow response and poor adaptability under complex disturbance, and has important engineering value for guaranteeing the ship heading stability and the steering safety under complex sea conditions.

Description

Technical Field

[0001] This invention relates to the field of ship motion control technology, and in particular to a nested iterative sliding mode control method for ship bow rolling chaos. Background Technology

[0002] Large ships are prone to chaotic bow roll under the combined effects of wind and wave disturbances, nonlinear hydrodynamics, and parameter perturbations, leading to a surge in energy consumption, course deviation, and even loss of control. Traditional sliding mode control, PID control, or backstepping methods often suffer from problems such as large chattering, slow response, and poor adaptability to unknown disturbances when dealing with such high-order nonlinear and strongly disturbed systems. In recent years, iterative sliding mode control has shown good potential in stabilizing complex systems due to its successive approximation characteristics. However, its application to suppressing bow roll chaos in ships still requires solving engineering challenges such as high-order differential explosion, estimation of unknown disturbances, and rudder angle saturation. Therefore, an advanced control strategy is needed that can simultaneously achieve chaos suppression, rapid heading convergence, and smooth rudder angle output under model uncertainty and complex sea conditions. Summary of the Invention

[0003] This invention discloses a nested iterative sliding mode control method for chaotic phenomena of ship bow rolling, in order to overcome the above-mentioned technical problems.

[0004] To achieve the above objectives, the technical solution of the present invention is as follows: A nested iterative sliding mode control method for chaotic phenomena in ship bow rolling includes the following steps: S1: Establish a chaotic dynamic model of ship bow roll; S2: Based on the ship's bow roll chaotic dynamics model, obtain the current bow angle, bow angular velocity, and bow angular acceleration state variables to construct the first sliding surface; S3: Based on the first sliding surface, construct a second sliding surface to establish a virtual control law and obtain the desired angular acceleration; S4: Based on the desired angular acceleration, obtain the nested iterative sliding mode control law, and then obtain the final rudder angle command based on dynamic surface technology to achieve control of the ship.

[0005] Furthermore, the method for establishing the chaotic dynamics model of the ship's bow roll is as follows: First, a second-order nonlinear Nomoto model of ship motion is established: (1)

[0006] In the formula: The angular velocity of the bow; , , These are all intermediate calculation parameters; The second derivative of the turning angular velocity; The first derivative of the turning angular velocity is the state variable of the turning angular acceleration; For rudder angle; All are parameters of the second-order Nomoto model; All are nonlinear Nomoto parameters; for The first derivative; The second-order nonlinear Nomoto model of ship motion is rewritten to obtain the following chaotic dynamic model of ship bow roll: (2) In the formula: The disturbance caused by sea waves to the ship; The frequency of ocean waves; Indicates time.

[0007] Furthermore, S2 includes: make (3) The chaotic dynamics model of ship bow rolling is then transformed into differential equation form: (4) In the formula: These are all intermediate calculation parameters; express The first derivative; express The first derivative; express The first derivative; Indicates the ship's heading angle; make , ,have to: (5) in,

[0008]

[0009]

[0010] In the formula: and These are all intermediate calculation parameters; Define the first error variable: (6) In the formula: Indicates the first error variable; Indicates the desired course; but, (7) In the formula: Indicates the first sliding surface; , All represent the parameters of the first sliding surface; express The first derivative; This represents the first derivative of the desired heading.

[0011] Furthermore, the method used to obtain the desired angular acceleration is as follows: First, construct the second sliding surface as follows: (8) In the formula: All are parameters of the second sliding surface; Indicates the second sliding surface; express The first derivative; = (9) In the formula: The second derivative represents the desired heading; Based on the second sliding surface, the first Lyapunov function is obtained as follows: (10) In the formula: This represents the first Lyapunov function; right Differentiate, and we get (11) (12) In the formula: express The first derivative; for The first derivative; For stabilization parameters; Equation (11) is expressed as: (13) Therefore, the virtual control law is established as follows: (14) In the formula: Indicates the desired angular acceleration; This represents the hyperbolic secant function.

[0012] Furthermore, the method used to obtain the final rudder angle command is as follows: Based on the desired angular acceleration, obtain the second error variable: (15) In the formula: It is the second error variable; Based on the second error variable, the third sliding surface is obtained as follows: (16) In the formula: Indicates the third sliding surface; , All represent the parameters of the third sliding surface; express The first derivative; express The first derivative; Based on the third sliding surface, obtain the second Lyapunov function: (17) (18) In the formula: This represents the second Lyapunov function; Set the fixing parameters for the third sliding surface; Substituting equation (16) into equation (18), we obtain the preliminary nested iterative sliding mode control law as follows: (19) In the formula: Indicates a preliminary controller; Indicates unknown items in the system; Transforming the initial nested iterative sliding mode control law (19) yields: (20) In the formula: Indicates the equivalent control input; (twenty one) In the formula: This indicates the control output at the previous minimum time step; Indicates a very small time interval; Substituting (18) into equation (21), we get: (twenty two) Substituting equation (22) into equation (20), we get: (twenty three) Introducing dynamic surface technology for processing The term, expressed as: (twenty four) In the formula: These are filter parameters; This represents an estimate of the third sliding surface; After transforming equation (24), we get: (25) In the formula: express The first derivative; Substituting equation (25) into equation (23), we get: (26) make = Equal, that is (27) The rudder angle input is obtained by combining equation (4), that is: (28) make The final rudder angle command is obtained: (29) In the formula: R represents the intermediate variable of the controller.

[0013] Beneficial Effects: This invention provides a nested iterative sliding mode control method for ship bow roll chaos. It utilizes an established ship straight-line chaotic model; constructs a nested iterative sliding mode controller; and simplifies the design process by combining dynamic surface technology, achieving heading stability and rudder angle response optimization. It effectively addresses model perturbations and wave disturbances, significantly improving the accuracy and robustness of bow roll chaos control. Compared to traditional methods, this invention demonstrates superior performance in heading convergence speed, steady-state error suppression, and rudder angle adjustment sensitivity. It effectively solves the problems of large chattering, slow response, and poor adaptability inherent in traditional sliding mode control under complex disturbances, and has significant engineering value for ensuring ship heading stability and maneuvering safety in complex sea conditions. Attached Figure Description

[0014] 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.

[0015] Figure 1 This is a flowchart of the ship bow roll chaotic nested iterative sliding mode control method of the present invention; Figure 2 This is the state change curve of the ship under yaw chaos control in an embodiment of the present invention; Figure 3 This is the control input variation curve of the ship under yaw chaos control in an embodiment of the present invention. Detailed Implementation

[0016] 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.

[0017] This embodiment introduces a chaotic nested iterative sliding mode control method for ship bow rolling, including the following steps: Figure 1 As shown: S1: Establish a chaotic dynamic model of ship bow roll; Preferably, the chaotic dynamics model of the ship's bow roll is established as follows: First, a second-order nonlinear Nomoto model of ship motion is established: (1)

[0018] In the formula: The angular velocity of the bow; , , These are all intermediate calculation parameters; The second derivative of the turning angular velocity; The first derivative of the turning angular velocity is the state variable of the turning angular acceleration; For rudder angle; All are parameters of the second-order Nomoto model; All of these are nonlinear Nomoto parameters, which can be obtained by least squares identification algorithm through ship turning tests; for The first derivative; Specifically, when the ship is sailing in a straight line and environmental disturbances are weak, the steering gear does not need to intervene frequently to maintain the course. In this case, the second-order nonlinear Nomoto model of ship motion is rewritten to obtain the following chaotic dynamic model of ship bow roll: (2) In the formula: The disturbance caused by sea waves to the ship; The frequency of the waves is denoted as . Under normal circumstances, the disturbance torque caused by waves experienced by a ship during navigation can be approximated by a combination of sine and cosine curves. Indicates time; S2: Construct the first sliding surface based on the chaotic dynamics model of the ship's bow roll; Preferably, S2 includes: make (3) The chaotic dynamics model of ship bow rolling is then transformed into differential equations, i.e., by applying rudder angles. To overcome the effects of chaos: (4) In the formula: These are all intermediate calculation parameters; express The first derivative; express The first derivative; express The first derivative; Indicates the ship's heading angle; Define the heading error and the rate of change of the error, and then construct the first sliding surface as follows: Specifically, to facilitate controller design, let , Substituting into equation (4), we get: (5) in,

[0019]

[0020]

[0021] In the formula: and These are all intermediate calculation parameters; Define the first error variable: (6) In the formula: Indicates the first error variable; Indicates the desired course; Then, the nonlinear first sliding surface of the design error variable is: (7) In the formula: Indicates the first sliding surface; , All represent the parameters of the first sliding surface; express The first derivative; This represents the first derivative of the desired heading.

[0022] S3: Based on the first sliding surface, construct a second sliding surface to establish a virtual control law and obtain the desired angular acceleration. ; Preferably, the method used to obtain the desired angular acceleration is as follows: First, construct the second sliding surface as follows: (8) In the formula: All are parameters of the second sliding surface; Indicates the second sliding surface; express The first derivative; = (9) In the formula: The second derivative represents the desired heading; Based on the second sliding surface, the first Lyapunov function is obtained as follows: (10) In the formula: This represents the first Lyapunov function; right Differentiate, and we get (11) In the formula: express The first derivative; for The first derivative; To make equation (11) satisfy the first Lyapunov stability condition, let... (12) In the formula: For stabilization parameters; Equation (11) is expressed as: (13) Substituting equation (9) into equation (12), a virtual control law is established, as follows: (14) In the formula: Indicates the desired angular acceleration; This represents the hyperbolic secant function.

[0023] S4: Based on the desired angular acceleration The nested iterative sliding mode control law is obtained, and combined with dynamic surface technology, the final rudder angle command is obtained. Preferably, S4 includes: Based on the desired angular acceleration, obtain the second error variable: (15) In the formula: It is the second error variable; Specifically, to construct the functional relationship between the sliding surface and the control input, the third sliding surface is further obtained based on the second error variable as follows: (16) In the formula: Indicates the third sliding surface; , All represent the parameters of the third sliding surface; express The first derivative; express The first derivative; Based on the third sliding surface, obtain the second Lyapunov function: (17) In the formula: This represents the second Lyapunov function; Similarly, to make equation (17) stable, then: (18) In the formula: Set the fixing parameters for the third sliding surface; Substituting equation (16) into equation (18), we obtain the preliminary nested iterative sliding mode control law as follows: (19) In the formula: Indicates a preliminary controller; Indicates unknown items in the system; Specifically, although the first error variable (6) asymptotically converges and eventually approaches zero under the influence of equation (19), and the system is stable, equation (19) clearly contains unknown terms. and The controller cannot be directly derived, therefore further design of the control law is still required. The preliminary nested iterative sliding mode control law (19) is transformed to obtain... (20) In the formula: Indicates the equivalent control input, and ; This includes system perturbation terms and unknown disturbance terms. Consider the system input. It should be a smooth, bounded function that changes little in a short time, therefore it adopts... right To make an estimate, among which The equivalent controller is then represented as: (twenty one) In the formula: This indicates the control output at the previous minimum time step; Indicates a very small time interval; Substituting the system stability condition (18) into equation (21), we get: (twenty two) Substituting equation (22) into equation (20), we get: (twenty three) To simplify the controller design steps below and considering its complexity, dynamic surface techniques are introduced. Item, introducing new variables And as The output of the first-order low-pass filter is expressed as: (twenty four) In the formula: These are filter parameters; This represents an estimate of the third sliding surface; After transforming equation (24), we get: (25) In the formula: express The first derivative; Substituting equation (25) into equation (23), we get: (26) make = Equal, that is (27) Solve for the final rudder angle transfer function and update the state variables in S1 in real time to achieve closed-loop chaos suppression and heading maintenance.

[0024] The rudder angle input is obtained by combining equation (4), that is (28) make , about The transfer function is used to obtain the final rudder angle command: (29) In the formula: R represents the intermediate variable of the controller.

[0025] This embodiment uses the 180,000-ton large bulk carrier "FRONTIER BONANZA" as the research object for simulation verification. The ship's parameters are shown in Table 1.

[0026] Table 1. Parameters of the large bulk carrier "FRONTIER BONANZA"

[0027] The guidance and controller parameters for this embodiment are shown in Table 2: Table 2 Guidance and controller parameters in this embodiment

[0028] First, the control effect of the nested iterative sliding mode algorithm under the influence of ship yaw chaos is simulated. Then, in equation (4)... Let the initial values ​​of the ship's heading, turning angular velocity, and turning angular acceleration be... Initial value of variable The value is 0. To simulate the ship's straight-ahead state, the desired heading value is set to 0. The simulation lasted 150 seconds. No control was applied for the first 70 seconds, after which steering was initiated to maintain course stability. The simulation results are as follows: Figure 2 As shown in the figure, the red dashed line represents the comparison control algorithm, and the blue solid line represents the control algorithm of this embodiment.

[0029] Depend on Figure 3 It is known that the algorithm in this embodiment can effectively control the chaotic phenomenon of ship bow roll. Regarding the control of the ship's heading angle, both algorithms exhibit fluctuations in the early stages, but the fluctuation amplitude of the algorithm in this embodiment is relatively small, and it converges more quickly within 50-60 seconds. After convergence and stabilization, there is a small static error but no chattering, while the comparative algorithm exhibits fluctuations. In terms of the turning angular velocity, the algorithm in this embodiment converges effectively after intervention, with a convergence speed comparable to the comparative algorithm. After stabilization, there is no static error or chattering, but the initial stage of intervention shows a larger fluctuation amplitude. Regarding the rate of change of turning angular velocity, the algorithm in this embodiment experiences large fluctuations at the moment of intervention, followed by rapid stabilization, while the comparative algorithm exhibits milder fluctuations. After stabilization, the algorithm in this embodiment maintains a low level without significant chattering. In summary, the algorithm in this embodiment has advantages in controlling the chaotic phenomenon of ship bow roll, particularly in terms of heading convergence speed and stabilization state. Although the fluctuation amplitude is large in some stages, the overall control effect is superior.

[0030] Regarding equivalent control input, the comparative algorithm exhibits minimal fluctuations upon system intervention, maintaining a near-zero level with a smooth initial state. In contrast, the algorithm in this embodiment experiences large pulse-like fluctuations upon intervention followed by a rapid decline, demonstrating a strong initial impact. After rudder angle conversion, the comparative algorithm's corresponding rudder angle exhibits a small-range steady-state oscillation around 2°, demonstrating its strong stability and directional tendency in rudder angle control. Conversely, the algorithm in this embodiment exhibits dynamic periodic fluctuations around 0°, indicating that its rudder angle control strategy possesses higher response sensitivity and adjustment freedom, enabling real-time and precise adjustments near zero based on system status to achieve more accurate control objectives.

[0031] By inputting the rudder angle into the ship's bow roll chaotic dynamics model, the state variables in the ship's bow roll chaotic dynamics model are updated in real time, thereby achieving closed-loop chaos suppression and course maintenance.

[0032] This embodiment presents a nested iterative sliding mode control method for ship bow roll chaos. It utilizes an established ship straight-course chaotic model; constructs a nested iterative sliding mode controller; and simplifies the design process by combining dynamic surface technology to achieve heading stability and rudder angle response optimization. Simulation verification is performed on the 180,000-ton large bulk carrier "FRONTIER BONANZA," comparing the proposed control strategy with existing sliding mode control algorithms. This embodiment effectively addresses model perturbations and wave disturbances by introducing a multi-order sliding surface iterative mechanism and an adaptive estimation strategy, significantly improving the accuracy and robustness of bow roll chaos control. Compared to traditional methods, the control strategy of this invention outperforms traditional methods in terms of heading convergence speed, steady-state error suppression, and rudder angle adjustment sensitivity. It effectively solves the problems of large chattering, slow response, and poor adaptability in traditional sliding mode control under complex disturbances, and has significant engineering value for ensuring ship heading stability and maneuvering safety in complex sea conditions.

[0033] 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 ship bow shake chaos phenomenon nested iterative sliding mode control method, characterized in that, Includes the following steps: S1: Establish a chaotic dynamic model of ship bow roll; S2: Based on the ship's bow rolling chaotic dynamics model, obtain the bow turning angular velocity and bow turning angular acceleration state variables to construct the first sliding surface; S3: Based on the first sliding surface, construct a second sliding surface to establish a virtual control law and obtain the desired angular acceleration; S4: Based on the desired angular acceleration, obtain the nested iterative sliding mode control law, and then obtain the final rudder angle command based on dynamic surface technology to achieve control of the ship.

2. The chaotic phenomenon nested iterative sliding mode control method for bow shake of a ship according to claim 1, characterized in that, The method for establishing the chaotic dynamic model of the ship's bow roll is as follows: First, a second-order nonlinear Nomoto model of ship motion is established: （1） In the formula: The angular velocity of the bow; , , These are all intermediate calculation parameters; The second derivative of the turning angular velocity; The first derivative of the turning angular velocity is the state variable of the turning angular acceleration; For rudder angle; All are parameters of the second-order Nomoto model; All are nonlinear Nomoto parameters; for The first derivative; The second-order nonlinear Nomoto model of ship motion is rewritten to obtain the following chaotic dynamic model of ship bow roll: （2） In the formula: The disturbance caused by sea waves to the ship; The frequency of ocean waves; Indicates time.

3. The nested iterative sliding mode control method for chaotic ship bow rolling phenomena according to claim 2, characterized in that, S2 includes: make (3) The chaotic dynamics model of ship bow rolling is then transformed into differential equation form: （4） In the formula: These are all intermediate calculation parameters; express The first derivative; express The first derivative; express The first derivative; Indicates the ship's heading angle; make , ,have to: （5） in, In the formula: and These are all intermediate calculation parameters; Define the first error variable: （6） In the formula: Indicates the first error variable; Indicates the desired course; but, （7） In the formula: Indicates the first sliding surface; , All represent the parameters of the first sliding surface; express The first derivative; This represents the first derivative of the desired heading.

4. The nested iterative sliding mode control method for chaotic ship bow rolling phenomena according to claim 3, characterized in that, The method used to obtain the desired angular acceleration is as follows: First, construct the second sliding surface as follows: （8） In the formula: All are parameters of the second sliding surface; Indicates the second sliding surface; express The first derivative; = （9） In the formula: The second derivative represents the desired heading; Based on the second sliding surface, the first Lyapunov function is obtained as follows: （10） In the formula: This represents the first Lyapunov function; right Differentiate, and we get （11） （12） In the formula: express The first derivative; for The first derivative; For stabilization parameters; Equation (11) is expressed as: （13） Therefore, the virtual control law is established as follows: （14） In the formula: Indicates the desired angular acceleration; This represents the hyperbolic secant function.

5. The nested iterative sliding mode control method for chaotic ship bow rolling phenomena according to claim 3, characterized in that, The method used to obtain the final rudder angle command is as follows: Based on the desired angular acceleration, obtain the second error variable: （15） In the formula: It is the second error variable; Based on the second error variable, the third sliding surface is obtained as follows: （16） In the formula: Indicates the third sliding surface; , All represent the parameters of the third sliding surface; express The first derivative; express The first derivative; Based on the third sliding surface, obtain the second Lyapunov function: （17） （18） In the formula: This represents the second Lyapunov function; Set the fixing parameters for the third sliding surface; Substituting equation (16) into equation (18), we obtain the preliminary nested iterative sliding mode control law as follows: （19） In the formula: Indicates a preliminary controller; Indicates unknown items in the system; Transforming the initial nested iterative sliding mode control law (19) yields: （20） In the formula: Indicates the equivalent control input; （21） In the formula: This indicates the control output at the previous minimum time step; Indicates a very small time interval; Substituting (18) into equation (21), we get: （22） Substituting equation (22) into equation (20), we get: （23） Introducing dynamic surface technology for processing The term, expressed as: （24） In the formula: These are filter parameters; This represents an estimate of the third sliding surface; After transforming equation (24), we get: （25） In the formula: express The first derivative; Substituting equation (25) into equation (23), we get: （26） make = Equal, that is （27） The rudder angle input is obtained by combining equation (4), that is: （28） make The final rudder angle command is obtained: （29） In the formula: R represents the intermediate variable of the controller.

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