A model-free finite-time saturation control method for wave compensation platform
By employing a model-free finite-time saturation control method and designing a controller using a hyperbolic tangent function and adaptive parameter terms, the problem of actuator saturation in a three-degree-of-freedom wave compensation platform was solved, achieving system stability and high-precision displacement compensation, which is superior to traditional PID control.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- OCEAN UNIV OF CHINA
- Filing Date
- 2026-05-27
- Publication Date
- 2026-06-30
Smart Images
Figure CN122308420A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of control technology for three-degree-of-freedom wave compensation platforms at sea, and for example to a model-free finite-time saturation control method for wave compensation platforms. Background Technology
[0002] Due to the disturbances caused by wind and waves in the marine environment, ships are subject to six degrees of freedom: roll, pitch, yaw, swell, roll, and heave. Employing a wave compensation platform can effectively mitigate the impact of ship motion and establish a relatively stable operating environment. With advancements in ship mooring systems and dynamic positioning technology, the impact of roll, pitch, and heave degrees of freedom is increasingly prioritized in offshore construction scenarios. Furthermore, due to the need for expensive sensors to measure roll, pitch, and yaw, as well as a relatively large number of actuators, the six-degree-of-freedom wave compensation platform (SWCP) is gradually being replaced by the more cost-effective three-degree-of-freedom compensation platform due to its high cost.
[0003] Existing technologies include: acquiring deck motion information via IMU, deriving reference trajectories based on inverse kinematics, and employing PD control; a dual-loop control strategy based on velocity feedforward, which designs a multi-degree-of-freedom velocity feedforward compensator to decouple motion disturbances from the base platform, while transforming the original dynamic model into a linear parameterized form and using adaptive laws to estimate key parameters; a controller based on the Beetle Antennae Search (BAS) algorithm to enhance trajectory tracking capabilities, and introducing a Radial Basis Function Neural Network (RBFNN) to improve anti-interference capabilities; and the establishment of a complete multi-degree-of-freedom compensated dynamic model, which overcomes the uncertainty of model parameters by adding robust terms to the normal inverse dynamic model, enabling the system to meet the desired performance under parameter uncertainty, while combining a Kalman filter to construct the optimal equivalent input disturbance, suppressing the destructive effect of sensor noise on disturbance estimation, and improving estimation accuracy.
[0004] These existing technologies are all based on the accurate dynamics of the system. Although adaptive control can be used to correct disturbances caused by inaccurate modeling, as the application scenarios of the compensation platform gradually expand, even the parameters such as the mass and moment of inertia of the upper platform and actuator of the compensation platform cannot be obtained with a relatively accurate initial value for dynamic modeling.
[0005] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this application, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention
[0006] To provide a basic understanding of some aspects of the disclosed embodiments, a brief summary is given below. This summary is not intended as a general commentary, nor is it intended to identify key / important components or describe the scope of protection of these embodiments, but rather as a prelude to the detailed description that follows.
[0007] This disclosure provides a model-free finite-time saturation control method for a wave compensation platform to achieve precise motion control of the wave compensation platform.
[0008] In some embodiments, the model-free finite-time saturation control method for the three-degree-of-freedom wave compensation platform includes: S10, analyzing the inverse kinematics of the compensation platform to construct a dynamic model of the compensation platform; S20, introducing a control gain matrix based on the hyperbolic tangent function. Core state vector and adaptive parameter terms S30, design the expression for the model-free controller of the compensation platform; S30, represent the positive definite Lyapunov function using the dynamic model; based on the expression of the positive definite Lyapunov function and the controller, and combined with the first Lyapunov function... The stability of the closed-loop system of the controller is analyzed using the first lemma, the first hypothesis, and the second hypothesis. The first lemma is the hyperbolic tangent smooth sign function error bound lemma. The first hypothesis is that the disturbance caused by the waves to the ship is bounded, the unknown disturbance inside the compensation platform is bounded, and the load mass of the compensation platform is finite. The second hypothesis is that the expected trajectory of the compensation platform and its derivatives are continuous and bounded. S40 represents the auxiliary variable; S40 sets the second Lyapunov function. Differentiate the second Lyapunov function and introduce the second lemma to analyze the convergence of the tracking error of the compensation platform; wherein, the second lemma is the lemma of the positive term sum and power sum inequality; S50 represents the position error vector; S50 sets the third Lyapunov function. Combined with the terminal sliding surface Auxiliary variables The third lemma analyzes the finite-time convergence of the terminal sliding surface; wherein, the third lemma is the finite-time stability criterion lemma of the Lyapunov inequality for nonlinear systems; S60, the control parameters of the controller are adjusted according to preset rules.
[0009] The wave compensation platform model-free finite-time saturation control method provided in this disclosure can achieve the following technical effects: 1. By utilizing the bounded smoothness property of the hyperbolic tangent function, the expression for the compensation platform is designed by introducing the control gain matrix, core state vector, and adaptive parameter terms. This effectively avoids actuator saturation, ensures hardware safety and the stability and robustness of the closed-loop system, and solves the dilemma when the compensation platform cannot obtain relatively accurate initial values for dynamic modeling. 2. Adjust the controller parameters using preset rules to ensure that the actuator does not saturate and affect the stability of the system; 3. The stability of the controller within the closed-loop system, the convergence of the tracking error of the compensation platform, and the finite-time convergence of the terminal sliding surface were verified. This ensures the global stability of the model-free controller within the closed-loop system and the high accuracy and reliability of the displacement compensation platform in actual operation, guaranteeing that the tracking performance is superior to traditional model-free control methods such as PID. By applying this method to a three-degree-of-freedom wave compensation platform, the ship motion caused by ocean waves can be effectively offset, providing a stable operating platform for offshore construction and other scenarios.
[0010] The above general description and the description below are exemplary and illustrative only and are not intended to limit this application. Attached Figure Description
[0011] One or more embodiments are illustrated by way of example with reference to the accompanying drawings. These illustrations and drawings do not constitute a limitation on the embodiments. Elements having the same reference numerals in the drawings are shown as similar elements. The drawings are not to be scaled. And wherein: Figure 1 This is a schematic diagram of a model-free finite-time saturation control method for a wave compensation platform provided in an embodiment of this disclosure; Figure 2 This is a schematic diagram illustrating the motion description of a three-degree-of-freedom wave compensation platform provided in an embodiment of this disclosure; Figure 3 This is a schematic diagram comparing the compensation effects of a model-free finite-time saturation control method and a PID control method for a wave compensation platform under sinusoidal disturbances, according to an embodiment of this disclosure. Figure 4 This is a schematic diagram of the tracking error results of a model-free finite-time saturation control method and a PID control method for a wave compensation platform under sinusoidal disturbances, provided in an embodiment of this disclosure. Figure 5 This is a schematic diagram comparing the stochastic wave compensation effects of a model-free finite-time saturation control method and a PID control method for a wave compensation platform, as provided in this embodiment of the disclosure. Figure 6 This is a schematic diagram of the tracking error results under random ocean waves using a model-free finite-time saturation control method and a PID control method for a wave compensation platform provided in this embodiment. Detailed Implementation
[0012] To provide a more detailed understanding of the features and technical content of the embodiments of this disclosure, the implementation of the embodiments of this disclosure will be described in detail below with reference to the accompanying drawings. The accompanying drawings are for illustrative purposes only and are not intended to limit the embodiments of this disclosure. In the following technical description, for ease of explanation, several details are used to provide a full understanding of the disclosed embodiments. However, one or more embodiments may still be implemented without these details. In other cases, well-known structures and devices may be simplified in their depiction to simplify the drawings.
[0013] The terms "first," "second," etc., used in the specification and accompanying drawings of this disclosure are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate for the embodiments of this disclosure described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion.
[0014] Unless otherwise stated, the term "multiple" means two or more.
[0015] In this embodiment of the disclosure, the character " / " indicates that the objects before and after it are in an "or" relationship. For example, A / B means: A or B.
[0016] The term "and / or" describes an association between objects, indicating that three relationships can exist. For example, A and / or B means: A or B, or A and B.
[0017] The term "correspondence" can refer to an association or binding relationship. The correspondence between A and B means that there is an association or binding relationship between A and B.
[0018] Combination Figure 1 As shown, this disclosure provides a model-free finite-time saturation control method for a wave compensation platform, including: S10, the inverse kinematics of the compensation platform is analyzed, and a dynamic model of the compensation platform is constructed.
[0019] S20, based on the hyperbolic tangent function, introduces the control gain matrix, core state vector and adaptive parameter terms, and designs the expression of the model-free controller of the compensation platform.
[0020] S30, using a dynamic model to represent the positive definite Lyapunov function; based on the expressions of the positive definite Lyapunov function and the controller, and combined with the first Lyapunov function... The stability of the closed-loop system of the controller is analyzed using the first lemma, the first assumption, and the second assumption. The first lemma is the hyperbolic tangent smooth sign function error bound lemma. The first assumption is that the disturbance caused by the waves to the ship is bounded, the unknown disturbance inside the compensation platform is bounded, and the load mass of the compensation platform is finite. The second assumption is that the expected trajectory of the compensation platform and its derivatives are continuous and bounded. This represents an auxiliary variable.
[0021] S40, Set the second Lyapunov function Differentiate the second Lyapunov function and introduce the second lemma to analyze the convergence of the tracking error of the compensation platform; where the second lemma is the lemma of the positive term sum and power sum inequality. This represents the position error vector.
[0022] S50, set the third Lyapunov function Combined with the terminal sliding surface Auxiliary variables The third lemma analyzes the finite-time convergence of the terminal sliding surface; the third lemma is the finite-time stability criterion lemma of the Lyapunov inequality for nonlinear systems.
[0023] S60 adjusts the controller parameters according to preset rules.
[0024] First, execute S10 to analyze the inverse kinematics of the compensation platform and construct its dynamic model. Specifically, this includes: S11, firstly, the inverse kinematics of a typical electrically driven three-degree-of-freedom wave compensation platform is analyzed to obtain the ideal actuator length trajectory. To perform mathematical analysis of the physical model of the compensation platform, a coordinate system is first established, such as... Figure 2 As shown, and It is the position vector pointing from the origin of the local coordinate system of the upper and lower platforms to the corresponding connecting joint. The coordinate systems of the lower platform, upper platform, and actuator are respectively represented as: , ,and ,in, . α , β , γ These represent the Euler angles about the X, Y, and Z axes of a fixed coordinate system, respectively, with reference to the fixed axis. XYX The Euler angle representation, and the corresponding rotation matrix can be expressed as: (1) Due to the physical characteristics of the rotating joint, there exists a constraint force, the direction of which is determined by a vector. By definition, the three-degree-of-freedom wave compensation platform has the following constraints: (2) in, Defined as: (3) Based on the coordinate system transformation relationship, the upper platform rotary joint is in the fixed coordinate system. The middle can be represented as: (4) in, This represents the line vector pointing from the origin of the fixed coordinate system to the origin of the moving coordinate system. The solution to equation (2) yields... Substituting it into equation (1), the transformation matrix It can be represented as: (5) in, Represents the sine function , Represents the cosine function .
[0025] When the control objective is to maintain stability on the upper platform, the actuator's linear vector... It can be obtained from the following formula: (6) in, and These are the position vectors pointing from the origin of the local coordinate system of the upper and lower platforms to the corresponding connecting joints.
[0026] Furthermore, the velocity vector of the actuator can be expressed as: (7) in, Jacobian matrix representing the mapping between actuator velocity and generalized velocity.
[0027] For parallel robots like the three-degree-of-freedom motion compensation platform, the dynamic model of S12 conforms to the following form: (8) in This represents the position vector of the compensation platform. , , These respectively indicate the heave, roll, and pitch positions of the compensation platform; , , These represent the mass matrix, Coriolis force matrix, and gravity vector, respectively. This indicates the force exerted by the ship's motion on the compensation platform. It is the output force of the actuator of the compensation platform. This indicates time-varying disturbances caused by system friction, etc. Let represent the Jacobian matrix that maps the motion of the upper platform relative to the deck to the motion of the actuator. Meanwhile, model (8) possesses the following properties: 1) Mass matrix It is positive definite; 2) Matrix It is an antisymmetric matrix, and it exists for any n-order square matrix. Make .
[0028] S13, Lemma and Assumption.
[0029] The first lemma, the error bound lemma for the hyperbolic tangent smoothing sign function, states that for any positive real smoothing parameter and any real variable, the difference between the absolute value of the real variable and the product of the real variable and the hyperbolic tangent function whose independent variable is the arbitrary real variable divided by the arbitrary positive real smoothing parameter, is non-negative and has an upper bound. This upper bound is the product of a fixed proportionality constant and the arbitrary positive real smoothing parameter. The fixed proportionality constant is approximately 0.2785. Its formula is expressed as follows: For any real number, satisfy and Then the following inequalities exist: (9) in, , It is a natural constant.
[0030] The second lemma, the inequality lemma between positive sums and power sums, states that for any finite number of positive definite functions, their arithmetic sum is not less than the square root of their sum of squares; and for any exponent c between 0 and 1, the sum of the c powers of this set of positive definite functions is not less than the c power of their arithmetic sum. Its formula is as follows: Assumption If the function is positive definite, then the following inequality is satisfied: (10) in, .
[0031] The third lemma, the finite-time stability criterion lemma of the Lyapunov inequality for nonlinear systems, states that if a nonlinear system with initial conditions has a positive definite Lyapunov function satisfying specific derivative constraints, then the system is a semi-global finite-time stable system, and both the positive definite Lyapunov function with specific derivative constraints and its convergence time have upper bounds determined by the parameters and the initial conditions. Its formula is expressed as follows: For a nonlinear system And it has a positive constant. , , If there exists a positive definite Lyapunov function It must satisfy the following form: (11) Therefore, the entire system is actually a semi-global finite-time stable system, and satisfy: (12) Here Meanwhile, the convergence time satisfies: (13) First assumption: The disturbance caused by ocean waves to the ship is bounded, and the unknown disturbances inside the three-degree-of-freedom wave compensation platform are also bounded, and the load mass of the wave compensation platform is finite.
[0032] Second assumption: The expected trajectory of the compensation platform and its first and second derivatives are both continuous and bounded.
[0033] Then, S20 is executed, based on the hyperbolic tangent function, introducing the control gain matrix, core state vector, and adaptive parameter terms to design the expression for the model-free controller of the compensation platform. Specifically, this includes: S21, the final control rate output to the compensation platform actuator is obtained by subtracting two parts: taking the ratio of the core state vector to the preset first state gain constant as the independent variable, the hyperbolic tangent function value is obtained; the hyperbolic tangent function value is multiplied by the preset first positive definite diagonal parameter matrix and the negative value is taken; finally, the adaptive parameter term is subtracted from this negative value to obtain the final control output that can limit actuator saturation. Its formula is expressed as equation (14): (14) in, To compensate for the actuator output force of the platform; This is the first positive definite diagonal parameter matrix; This represents the hyperbolic tangent function, which, when applied to a vector, takes the hyperbolic tangent for each component of the vector. The core state vector; These are adaptive parameter terms used to improve system robustness; The first state gain constant, .
[0034] S22 introduces an adaptive parameter term to improve system robustness. Its rate of change over time consists of two parts: the hyperbolic tangent function value is calculated using the ratio of the core state vector to a preset second state gain constant as the independent variable. The negative value of multiplying the hyperbolic tangent function value by a preset second positive definite diagonal parameter matrix is taken as the first part; the negative value of multiplying the preset third positive definite diagonal parameter matrix by the adaptive parameter term itself is taken as the second part. Its formula is expressed as equation (15): (15) in, The time rate of change of the adaptive parameter; This is the second positive definite diagonal parameter matrix; The gain constant for the second state. ; It is the third positive definite diagonal parameter matrix.
[0035] S23, the position error vector is equal to the difference between the actual position vector of the compensation platform and the desired trajectory. The velocity signal error vector is equal to the difference between the actual velocity signal and the desired velocity signal. The core state vector is equal to the difference between the actual velocity signal of the compensation platform and the reference velocity signal. Its formula is expressed as equation (16): (16) in, This is the system position error vector; This represents the actual position vector of the compensation platform; To compensate for the platform's expected trajectory; This is the system velocity signal error vector; This represents the actual velocity vector of the compensation platform; To compensate for the platform's desired velocity vector.
[0036] The system reference velocity signal is calculated as follows: it equals the desired velocity signal minus the product of the fourth positive definite diagonal parameter matrix and the absolute value of the position error vector raised to a preset power, and then minus the product of the preset fifth positive definite diagonal parameter matrix and the auxiliary variable. The formula is expressed as equation (17): (17) in, This is the system reference speed; This is the fourth positive definite diagonal parameter matrix; Let be the system position error vector. Represents the position error vector Take the absolute value of each component and calculate The new vector formed by exponentiation; The fractional power parameter that determines the convergence rate of the position signal error; It is the fifth positive definite diagonal parameter matrix.
[0037] S24, As an auxiliary variable in the design, the time derivative of this auxiliary variable is calculated as follows: the product of the preset sixth positive definite diagonal parameter matrix and the terminal sliding surface, plus the sign function value of the terminal sliding surface. Its formula is expressed as equation (18): (18) in, For the terminal sliding surface; The sixth positive definite diagonal parameter matrix, For the terminal sliding surface vector Find the vectorization operation of the sign function for each component.
[0038] S25, after performing a preset power operation on the absolute value of the position error vector, compare it with the preset fourth positive definite diagonal parameter matrix. Multiply the results and add the product to the velocity signal error vector to construct the terminal sliding surface of the system. Its formula is expressed as equation (19): (19) Thus, through the design of S20, on the one hand, the bounded smoothness characteristic of the hyperbolic tangent function is utilized to strictly limit the control force output to the actuator within the physically permissible threshold range, effectively avoiding actuator saturation and ensuring hardware safety and closed-loop system stability. On the other hand, through the dynamic adjustment of adaptive parameter terms, it can compensate for lumped disturbances caused by waves and unmodeled dynamics within the platform in real time, enabling the control system to maintain strong robustness even without obtaining precise dynamic model parameters of the platform (i.e., without a model). In addition, the terminal sliding surface constructed in conjunction with auxiliary variables provides a structural basis for the subsequent rapid convergence of position tracking errors within a finite time, significantly improving the tracking accuracy and response speed of wave compensation.
[0039] Then, S30 is executed, using the dynamic model to represent the positive definite Lyapunov function; based on the expressions of the positive definite Lyapunov function and the controller, and combined with the first Lyapunov function... Based on the first lemma, the first hypothesis, and the second hypothesis, the stability of the controller within the closed-loop system is analyzed. Specifically, this includes: S31, For the dynamic model shown in equation (8), construct the state vector. ,in Then the dynamic model of the compensation platform is rewritten as: (20) in, This refers to the lumped disturbance of the system.
[0040] S32, first consider the positive definite Lyapunov function: (twenty one) Represents state variables The final boundedness threshold, obtained by taking its derivative, is: (twenty two) S33, by utilizing the relationship between the mass matrix and the Coriolis force matrix, and substituting equation (14) into equation (22), a differential equation can be obtained. This differential equation reflects the operational logic of the differential equation of the time derivative of the positive definite Lyapunov function: the time derivative is equal to the sum of the first part and the second part; where the first part is the negative of the product of the transpose of the core state vector and the first positive definite diagonal parameter matrix and the hyperbolic tangent function, and the independent variable of the hyperbolic tangent function is the ratio of the core state vector to the first state gain constant; the second part is the negative of the product of the transpose of the core state vector and the system equivalent deviation term, and the system equivalent deviation term is obtained by subtracting the adaptive parameter term and the system lumped disturbance from the nominal dynamics term. Specifically, the differential equation is shown in equation (23): (twenty three) Among them, the nominal dynamics term Regarding adaptive parameter terms The boundedness of exists under the following conditions: (twenty four) S34, its upper bound can be obtained by solving this differential equation. satisfy: .
[0041] S35. Based on the first and second assumptions, since both system parameters and disturbances are bounded, the following inequality operation logic exists: the difference between the negative of the adaptive parameter term and the system lumped disturbance is strictly less than the preset upper bound of the nominal dynamics term. Upper bound of adaptive parameter terms With system lumped disturbances The difference between the three in the upper bound. That is, it clearly exists. .
[0042] Furthermore, according to the first lemma, by scaling the differential equation for the time derivative of the positive definite Lyapunov function, the following inequality operation logic is obtained: the time derivative is less than or equal to the sum of three terms; the first term is the negative of the product of the first positive definite diagonal parameter matrix and the norm of the core state vector; the second term is the product of the fixed proportionality constant (approximately equal to 0.2785) and the first state gain constant; the third term is the product of the norm of the core state vector and the algebraic difference between the upper bound of the nominal dynamics term, the upper bound of the adaptive parameter term, and the upper bound of the system lumped disturbance. The specific inequality operation logic is shown in equation (25): (25) S36, to ensure For asymptotic convergence, a sufficient parameter needs to be set. Make In addition, let a new first Lyapunov function be defined as follows: Differentiate it, and substitute the result into equations (18) and (19) and scale them to obtain the inequality operation logic reflecting the boundary of the time derivative of the function: under the premise that the norm of the auxiliary variable is greater than or equal to the position error convergence threshold, the time derivative is less than or equal to: the negative of half of the product of the smallest eigenvalue of the sixth positive definite diagonal parameter matrix and the fifth positive definite diagonal parameter matrix and the square of the norm of the auxiliary variable. Specifically, as shown in equation (26): (26) In this process, an intermediate parameter is introduced. It is equal to the product of the eigenvalues of the sixth positive definite diagonal parameter matrix, the norm of the auxiliary variables, and the maximum norm of the core state vector; the position error convergence threshold is equal to the ratio of twice the intermediate parameter to the minimum eigenvalues of the sixth and fifth positive definite diagonal parameter matrices, i.e. , ,therefore It is asymptotically convergent, and its upper bound can be expressed as This provides a prerequisite for subsequently determining the global stability of the closed-loop system.
[0043] Thus, through the analysis and derivation of S30, based on rigorous Lyapunov stability theory and the smooth sign function lemma, the global stability of the designed model-free controller within the closed-loop system was rigorously proven mathematically. The core technical benefits of this step are twofold: firstly, it successfully defines the convergence boundaries of the core state vector and auxiliary variables, providing solid theoretical support for the system to effectively resist external wave interference and internal lumped disturbances; secondly, it fundamentally ensures that the wave compensation platform can maintain absolute safety and robust operation of the physical system even under complex conditions of severe sea conditions and highly uncertain model parameters, completely eliminating the engineering risks of system divergence or even runaway damage.
[0044] Then execute S40 to set the second Lyapunov function. Differentiating the second Lyapunov function and introducing the second lemma to analyze the convergence of the tracking error of the compensation plateau. Specifically, this includes: S41, with Let the second Lyapunov function be defined as follows: The derivative of this function is calculated as follows: the time derivative is equal to the product of the norm of the position error vector and the norm of its time derivative. Its algebraic expansion is equivalent to the negative sum of the products of the specific powers of the absolute values of each component of the position error vector and the corresponding elements of the fourth positive definite diagonal parameter matrix, plus the product of the transpose of the position error vector and the terminal sliding surface. Further scaling using the matrix eigenvalue and norm inequality, the time derivative is less than or equal to a definite polynomial upper bound, which consists of the sum of two terms: the first term is the negative of the product of the largest eigenvalue of the fourth positive definite diagonal parameter matrix and the sum of the specific fractional powers of the squares of each component of the position error vector; the second term is the upper bound of the norm of the terminal sliding surface. The product of the norm of the position error vector. The derivative is shown in equation (27): (27) Applying the second lemma to the first term of the upper bound of the polynomial obtained from the derivative, we obtain the final inequality operation logic reflecting the convergence boundary of the tracking error: provided that the norm of the position error vector is greater than or equal to the preset position error convergence threshold, the time derivative of the second Lyapunov function is less than or equal to: the negative of half of the product of the largest eigenvalue of the fourth positive definite diagonal parameter matrix and a specific power of the norm of the position error vector. This final inequality operation logic is shown in equation (28): (28) S42, based on the time derivative of the second Lyapunov function Determine the convergence of the tracking error of the compensation platform. Specifically, this includes: From equation (28), it can be seen that the position error vector is asymptotically convergent and bounded, and its boundedness threshold is... According to the terminal sliding surface Position error vector The boundedness of the equation and equation (19) can be obtained. The conclusion of boundedness can also be obtained from equations (17) and (18), which also yield the derivatives of the auxiliary variables. System reference speed and its derivative The conclusion is that it is asymptotically convergent and bounded.
[0045] Thus, through the analysis and derivation in S40, and utilizing the second Lyapunov function combined with specific inequality lemmas, the asymptotic convergence and global boundedness of the system's position tracking error were rigorously proven mathematically. The core technical effect of this step is twofold: firstly, it provides a solid theoretical foundation for the wave compensation platform's accurate trajectory following capability under complex disturbances, ensuring high precision and reliability of displacement compensation in actual operations; secondly, it theoretically guarantees that all core state variables in the closed-loop system (such as auxiliary variables and their derivatives, system reference velocity, etc.) remain bounded and controlled, thereby clearing the way for proving the finite-time rapid convergence of the terminal sliding surface in subsequent steps (S50) and laying the essential prerequisite for system stability.
[0046] Then execute S50 to set the third Lyapunov function. Combined with the terminal sliding surface Auxiliary variables The third lemma analyzes the finite-time convergence of the terminal sliding surface. Specifically, it includes: S51, with terminal sliding surface Set a third Lyapunov function for the independent variable. . Indicates the terminal sliding surface The final boundedness threshold. Combined with the terminal sliding surface. The auxiliary variable shown in equation (18) Taking the derivative of the third Lyapunov function, we obtain the operational logic reflecting the time derivative of the function: the time derivative is expanded into a sum of three algebraic terms; the first term is the negative of the product of the transpose of the terminal sliding surface, the fifth positive definite diagonal parameter matrix, the sixth positive definite diagonal parameter matrix, and the terminal sliding surface; the second term is the negative sum of the products of the absolute values of the diagonal elements of the fifth positive definite diagonal parameter matrix and the corresponding components of the terminal sliding surface; the third term is the product of the transpose of the terminal sliding surface and the time derivative of the core state vector; where the terminal sliding surface is equal to the core state vector minus the product of the fifth positive definite diagonal parameter matrix and the auxiliary variables. The specific operational logic is shown in equation (29): (29) in, .
[0047] S52, based on the derivative result and the pre-selected fifth positive definite diagonal parameter matrix. The convergence time of the terminal sliding surface is obtained. Specifically, this includes: S521, based on the proven boundedness of all state variables within the closed-loop system, guarantees... The existence of , and also the existence of according to the inequality properties of matrix norms. ,in, , and yes The elements within this matrix lead to the upper bound of the time derivative of the third Lyapunov function: this time derivative is less than or equal to a negative product, which is determined by the smallest eigenvalue of the fifth positive definite diagonal parameter matrix. Norm of the terminal sliding surface and the upper bound of the time derivative of the core state vector This is jointly determined. Specifically, the upper bound of the time derivative of the third Lyapunov function is shown in equation (30): (30) S522, the pre-selected fifth positive definite diagonal parameter matrix satisfies the assumption of specific constraints. Furthermore, based on the derivative of equation (30), the upper bound of the time derivative of the terminal sliding surface norm is further derived: the time derivative of this norm is less than or equal to the negative of the difference between the product of the smallest eigenvalue of the fifth positive definite diagonal parameter matrix and the terminal sliding surface norm, minus the upper bound of the time derivative of the core state vector. Specifically, the upper bound of the time derivative of the terminal sliding surface norm is shown in equation (31): (31) S523, by integrating both sides of the inequality on the upper bound of the time derivative of the terminal sliding surface norm in the time domain, the convergence time upper bound of the terminal sliding surface is finally calculated. This convergence time upper bound is equivalent to the ratio of the product of the initial value of the terminal sliding surface and the minimum eigenvalue of the fifth positive definite diagonal parameter matrix with the norm of the terminal sliding surface, minus the upper bound of the time derivative of the core state vector. Specifically, it is expressed as shown in equation (32): (32) S53, under the premise that all states in the closed-loop system are bounded and the terminal sliding surface converges in finite time, when the system stabilizes to the terminal sliding surface (i.e., the terminal sliding surface value is always zero), the dynamic evolution equation of the position error vector satisfies: for the third Lyapunov function, its time derivative is less than or equal to a specific algebraic polynomial bound, which is equivalent to the product of a negative proportionality constant and a specific fractional power of the third Lyapunov function itself, plus a positive constant term bound. Among them, the dynamic evolution equation of the position error vector is expressed as equation (33): (33) S54, using the dynamic evolution equation of the position error vector and the third lemma, determines the finite-time convergence of the terminal sliding surface. Specifically, this includes: S541, based on the dynamic evolution equation of the position error vector, constructs a fourth Lyapunov function with half of the square of the position error vector as the variable.
[0048] S542, the time derivative of the fourth Lyapunov function is obtained, and the result of the derivative is substituted into the dynamic evolution equation (33) of the position error vector to obtain the substitution result.
[0049] S543, based on the substitution result of S542, can be derived as follows: If the substitution result is less than or equal to the product of the specific fractional power of the fourth Lyapunov function (which is determined by a preset parameter and lies between 0 and 1) and a negative proportionality coefficient, then this differential inequality result perfectly matches the criterion for semi-global finite-time stability of nonlinear systems in the third lemma (i.e., equivalent to the constant in the third lemma). (The specific case where it equals zero). Therefore, based on the criterion of the third lemma, it can be concluded that the position error vector will converge to a steady state in a finite time, and its theoretical upper limit of convergence time can be directly calculated by mapping from the general formula for convergence time in the third lemma. Position error vector It converges in a finite time. Convergence to .in, and This is a proportional constant corresponding to the dynamic characteristics of the system. It is an adjustment constant between 0 and 1.
[0050] In summary, it can be determined that the position error vector of the terminal sliding surface can converge to a steady state within a finite time. The convergence time is set to... .
[0051] Thus, through in-depth analysis and derivation of S50, combined with the terminal sliding surface, auxiliary variables, and finite-time stability criteria for nonlinear systems, it was rigorously proven mathematically that the system's terminal sliding surface and position tracking error will inevitably converge to a steady state within a definite and finite time. The core technical effect of this step is twofold: firstly, it breaks the limitation of traditional control algorithms that can only achieve theoretically infinite asymptotic convergence, endowing the wave compensation platform with extremely fast dynamic tracking response speed and transient error recovery capability; secondly, it ensures that when facing harsh and variable actual sea conditions, the system can rapidly eliminate displacement deviations within a very short and theoretically calculable time window, accurately lock onto and follow the desired displacement compensation trajectory, thereby providing strong real-time and high-precision theoretical guarantees for high-requirement and high-standard offshore construction operations, constituting the core technical barrier of this control method.
[0052] Finally, S60 is executed to adjust the controller's control parameters according to preset rules. These control parameters include: the first positive definite diagonal parameter matrix. Second positive definite diagonal parameter matrix Third positive definite diagonal parameter matrix Fourth positive definite diagonal parameter matrix Fifth positive definite diagonal parameter matrix The sixth positive definite diagonal parameter matrix Specifically, this includes: (1) The design of control parameters should first ensure the saturation constraint of the actuator, then , , satisfy: ,in, and These are the minimum and maximum values output by the actuator, respectively. (2) This pertains to the main control gain. This is an adaptive compensation gain. Set it to a sufficiently large value. To compensate for system disturbances and adaptive term fluctuations. Priority adjustment. This leads to error convergence and chatter less than the chatter threshold (no obvious chatter). If errors due to uncertainty still exist after adjustment, then further adjustment is needed. Increase and supplement robustness; (3) and Both factors jointly determine the error convergence rate; increasing either can accelerate convergence. Adjustments should be made based on the dynamic characteristics of the desired trajectory to avoid overshoot due to excessively rapid convergence. (4) This is an integral gain, initially set to a small preset value, and then slowly increased based on the convergence of the tracking error. (5) The purpose of this is to improve the transient response of the system; it is set to the minimum value. It can be appropriately increased. However, no major adjustments are needed to avoid affecting system stability due to excessively large values.
[0053] In this way, the S60's preset control parameter adjustment rules enable systematic and decoupled tuning of complex nonlinear controller parameters. On one hand, these rules pre-set the actuator's physical output limits as the constraint boundary for the core gain parameters, fundamentally eliminating the risk of system instability caused by control saturation. On the other hand, through step-by-step coordinated tuning of the main control gain and adaptive compensation gain, high-frequency flutter is effectively suppressed while maximizing robustness against strong wave disturbances. Furthermore, the refined configuration of error convergence rate and transient characteristic parameters allows the wave compensation platform to achieve the optimal balance between rapidly tracking the target trajectory and maintaining stable system operation. Overall, these preset rules significantly improve the operability and comprehensive compensation efficiency of this model-free control method in actual offshore operating environments.
[0054] In summary, the control method provided by the embodiments of this disclosure can achieve the following beneficial effects: 1. By utilizing the bounded smoothness property of the hyperbolic tangent function, and introducing the control gain matrix, core state vector, and adaptive parameter terms, the expression for the compensation platform is designed. This effectively avoids actuator saturation, ensures hardware safety and the stability and robustness of the closed-loop system, and solves the dilemma when the compensation platform cannot obtain relatively accurate initial values for dynamic modeling. 2. Adjust the controller parameters using preset rules to ensure that the actuator does not saturate and affect the stability of the system; 3. The stability of the controller within the closed-loop system, the convergence of the tracking error of the compensation platform, and the finite-time convergence of the terminal sliding surface were verified. This ensures the global stability of the model-free controller within the closed-loop system and the high accuracy and reliability of the displacement compensation platform in actual operation, guaranteeing that the tracking performance is superior to traditional model-free control methods such as PID. By applying this method to a three-degree-of-freedom wave compensation platform, the ship motion caused by ocean waves can be effectively offset, providing a stable operating platform for offshore construction and other scenarios.
[0055] To verify the effectiveness of this control method, a system was built as follows: Figure 2 The experimental platform shown is mainly composed of two parts: a lower six-degree-of-freedom motion platform used to simulate ship motion caused by ocean waves, and an upper inverted three-degree-of-freedom motion platform used to simulate the TWCP, with an unknown payload of non-uniform arrangement placed on top of the TWCP.
[0056] The experimental setup employed a motion measurement unit (FDISYSTMES-EPSILON2) to measure ship motion; this sensor can simultaneously acquire three degrees of freedom: roll, pitch, and heave. Due to experimental limitations, a low-cost IMU (FDISYSTMES-DETA30) and a wire displacement sensor were used to test the compensation effect, respectively, to measure the compensated roll, pitch, and heave data. The control core of the experimental setup was an NI myrio-1900. The actuator position and velocity information were directly read from the encoder integrated into the TWCP servo motor. After calculation, the control input was sent to the servo motor to compensate for the ship's motion.
[0057] The effectiveness of the proposed method was verified by comparing it with the classic model-free control method using PID control. Two sets of experiments were conducted to test the results under sinusoidal disturbance and random ocean wave disturbance conditions, respectively.
[0058] Figure 3 and Figure 5The experimental results under two different perturbations are presented, with the RMS values of the compensated effects shown in the upper left corner. Under sinusoidal perturbations, compared with traditional methods, the compensation effects of this method on the three degrees of freedom of heave, roll, and pitch are improved by 54.43%, 58.82%, and 58.82%, respectively. Under random wave perturbations, the heave effect remains basically the same, while the compensation effects on roll and pitch are improved by 60.43% and 57.96%, respectively.
[0059] Figure 4 and Figure 6 The upper figure shows the tracking error of this method, and the lower figure shows the tracking error of the PID method. Only one actuator is selected to represent the tracking performance of the controller. The figure shows that the tracking accuracy is improved by 85.22% and 84.78% under sinusoidal disturbances and random disturbances, respectively.
[0060] In conclusion, the experimental results above demonstrate the superiority of this method.
[0061] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.
[0062] While the specific embodiments of the present invention have been described above, they are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.
Claims
1. A model-free finite-time saturated control method for a wave-compensated platform, characterized in that, include: S10, Analyze the inverse kinematics of the compensation platform and construct a dynamic model of the compensation platform; S20, based on hyperbolic tangent function, introduce control gain matrix, core state vector and adaptive parameter term , design the expression of the model-free controller of the compensation platform; S30, using the dynamic model to represent the positive definite Lyapunov function; based on the expression of the positive definite Lyapunov function and the controller, and combined with the first Lyapunov function... The stability of the closed-loop system of the controller is analyzed using the first lemma, the first hypothesis, and the second hypothesis. The first lemma is the hyperbolic tangent smooth sign function error bound lemma. The first hypothesis is that the disturbance caused by the waves to the ship is bounded, the unknown disturbance inside the compensation platform is bounded, and the load mass of the compensation platform is finite. The second hypothesis is that the expected trajectory of the compensation platform and its derivatives are continuous and bounded. Indicates auxiliary variables; S40, Set the second Lyapunov function Differentiate the second Lyapunov function and introduce the second lemma to analyze the convergence of the tracking error of the compensation platform; wherein, the second lemma is the lemma of the positive term sum and power sum inequality; Represents the position error vector; S50, set the third Lyapunov function Combined with the terminal sliding surface The auxiliary variables The third lemma analyzes the finite-time convergence of the terminal sliding surface; wherein, the third lemma is the finite-time stability criterion lemma of the Lyapunov inequality for nonlinear systems; S60, adjust the control parameters of the controller according to preset rules.
2. The model-free finite-time saturation control method for a wave compensation platform according to claim 1, characterized in that, S20 includes: The final control rate of the actuator output to the compensation platform is obtained by subtracting two parts: the core state vector and the first state gain constant. Using the ratio as the independent variable, find the value of the hyperbolic tangent function; then compare the hyperbolic tangent function value with the first positive definite diagonal parameter matrix. Multiply and take the negative value, and finally subtract the adaptive parameter from the negative value to obtain the final control output that can limit actuator saturation; The time rate of change of the adaptive parameter consists of two parts: the core state vector and the second state gain constant. Using the ratio as the independent variable, calculate the value of the hyperbolic tangent function; then compare the hyperbolic tangent function value with the second positive definite diagonal parameter matrix. The negative value of the multiplication is taken as the first part; the third positive definite diagonal parameter matrix is used. The negative value multiplied by the adaptive parameter term itself is used as the second part; System reference speed signal Equal to the desired velocity vector Subtract the fourth positive definite diagonal parameter matrix The product of the absolute value of the position error vector to a predetermined power, minus the fifth positive definite diagonal parameter matrix. The product with the auxiliary variable; The logic for calculating the time derivative of the auxiliary variable is as follows: the sixth positive definite diagonal parameter matrix. The product of the terminal sliding surface and the sign function value of the terminal sliding surface; The absolute value of the position error vector is raised to a predetermined power, multiplied by the fourth positive definite diagonal parameter matrix, and the product is then multiplied by the velocity signal error vector. They are added together to construct the terminal sliding surface of the system.
3. The model-free finite-time saturation control method for a wave compensation platform according to claim 1, characterized in that, In step S30, representing the positive definite Lyapunov function using the dynamic model includes: The dynamic model is as follows: ， in, This represents the position vector of the compensation platform. , , These respectively represent the heave, roll, and pitch positions of the compensation platform; , , These represent the mass matrix, Coriolis force matrix, and gravity vector, respectively. This indicates the force exerted by the ship's motion on the compensation platform. It is the actuator output force of the compensation platform. This indicates time-varying disturbances caused by system friction. The Jacobian matrix represents the mapping from the motion of the upper platform relative to the deck to the motion of the actuator; For the aforementioned dynamic model, construct the state vector. ,in The dynamic model is then rewritten as: ， in, For system lumped disturbances; The positive definite Lyapunov function is: ， in, Represents the core state vector The final boundedness threshold.
4. The model-free finite-time saturation control method for a wave compensation platform according to claim 3, characterized in that, In step S30, the expression based on the positive definite Lyapunov function and the controller, combined with the first Lyapunov function... Based on the first lemma, the first hypothesis, and the second hypothesis, the stability of the controller within the closed-loop system is analyzed, including: Differentiating the positive definite Lyapunov function and substituting the actuator output force of the compensation platform into the derivative using the relationship between the mass matrix and the Coriolis force matrix, yields the differential equation operation logic reflecting the time derivative of the positive definite Lyapunov function: this time derivative is equal to the sum of the first part and the second part; wherein, the first part is the negative of the product of the transpose of the core state vector with the first positive definite diagonal parameter matrix and the hyperbolic tangent function, the independent variable of the hyperbolic tangent function being the ratio of the core state vector to the first state gain constant; the second part is the negative of the product of the transpose of the core state vector with the system equivalent deviation term, the system equivalent deviation term being obtained by subtracting the adaptive parameter term and the system lumped disturbance from the nominal dynamics term, wherein the nominal dynamics term... ,in, Let be the desired velocity vector of the compensation platform; Let be the desired acceleration vector of the compensation platform; Solving the differential equation yields the upper bound of the adaptive parameter term. ; Based on the first and second assumptions, since both system parameters and disturbances are bounded, the following inequality operation logic exists: the difference between the negative of the adaptive parameter term and the lumped disturbance of the system is strictly less than the preset upper bound of the nominal dynamics term. Upper bound of adaptive parameter terms and upper bound of system lumped disturbance The differences between the three; By scaling the differential equation for the time derivative of the positive definite Lyapunov function according to the first lemma, the following inequality operation logic is obtained: the time derivative is less than or equal to the sum of three terms; the first term is the negative of the product of the first positive definite diagonal parameter matrix and the norm of the core state vector; the second term is the product of the fixed proportionality constant and the first state gain constant; the third term is the product of the norm of the core state vector and the algebraic difference between the upper bound of the nominal dynamics term, the upper bound of the adaptive parameter term, and the upper bound of the system lumped disturbance. The derivative of the first Lyapunov function is taken, and the result is substituted into the expression of the auxiliary variable and the expression of the terminal sliding surface. After scaling, the inequality operation logic reflecting the boundary of the time derivative of the function is obtained: provided that the norm of the auxiliary variable is greater than or equal to the position error convergence threshold, the time derivative is less than or equal to: the negative of half of the product of the smallest eigenvalue of the sixth positive definite diagonal parameter matrix and the fifth positive definite diagonal parameter matrix and the square of the norm of the auxiliary variable. In this process, an intermediate parameter is introduced. It is equal to the product of the eigenvalues of the sixth positive definite diagonal parameter matrix, the norm of the auxiliary variable, and the maximum norm of the core state vector; the position error convergence threshold Equal to: twice the intermediate parameter, and the ratio of the minimum eigenvalues of the sixth positive definite diagonal parameter matrix and the fifth positive definite diagonal parameter matrix; thus proving that the auxiliary variable and its related dynamics are asymptotically convergent and bounded.
5. The model-free finite-time saturation control method for a wave compensation platform according to claim 1, characterized in that, In step S40, the introduction of the second lemma to analyze the convergence of the tracking error of the compensation platform includes: The operational logic for the derivative of the second Lyapunov function is as follows: the time derivative of the function is equal to the product of the norm of the position error vector and the norm of its time derivative; its algebraic expansion is equivalent to: the negative sum of the products of the specific powers of the absolute values of each component of the position error vector and the corresponding elements of the fourth positive definite diagonal parameter matrix, plus the product of the transpose of the position error vector and the terminal sliding surface; further, by scaling through the matrix eigenvalue and norm inequality, the time derivative of the second Lyapunov function is less than or equal to a definite polynomial upper bound, which is composed of the sum of two terms: the first term is the negative of the product of the largest eigenvalue of the fourth positive definite diagonal parameter matrix and the sum of the specific fractional powers of the squares of each component of the position error vector, and the second term is the norm upper bound of the terminal sliding surface. The product of the norm of the position error vector; Applying the second lemma to the first term of the polynomial upper bound of the derivative of the second Lyapunov function, we obtain the final inequality operation logic that reflects the convergence boundary of the tracking error: provided that the norm of the position error vector is greater than or equal to the position error convergence threshold, the time derivative of the second Lyapunov function is less than or equal to: the negative of half of the product of the largest eigenvalue of the fourth positive definite diagonal parameter matrix and a specific power of the norm of the position error vector. Based on the characteristic that the time derivative of the second Lyapunov function is strictly bounded by a negative upper bound, it is determined that the tracking error of the compensation platform has asymptotic convergence.
6. The model-free finite-time saturation control method for a wave compensation platform according to claim 5, characterized in that, In S50, the combined terminal sliding surface The auxiliary variables and the third lemma are used to analyze the finite-time convergence of the terminal sliding surface, including: Combining the terminal sliding surface and the auxiliary variables, the time derivative of the third Lyapunov function is obtained. The algebraic expansion of this time derivative is equivalent to the sum of three terms: the first term is the negative of the product of the transpose of the terminal sliding surface, the fifth positive definite diagonal parameter matrix, the sixth positive definite diagonal parameter matrix, and the terminal sliding surface; the second term is the negative summation of the products of the absolute values of the diagonal elements of the fifth positive definite diagonal parameter matrix and the corresponding components of the terminal sliding surface; the third term is the product of the transpose of the terminal sliding surface and the time derivative of the core state vector. Wherein, the terminal sliding surface is equal to the core state vector minus the product of the fifth positive definite diagonal parameter matrix and the auxiliary variable; The convergence time of the terminal sliding surface is obtained based on the derivative of the third Lyapunov function and the fifth positive definite diagonal parameter matrix. Under the premise that all states in the closed-loop system are bounded and the terminal sliding surface converges in finite time, when the system stabilizes at the terminal sliding surface, the dynamic evolution equation of the position error satisfies the following inequality operation logic: the time derivative of the third Lyapunov function is strictly limited by an equivalent algebraic polynomial upper bound, which is composed of the product of a negative proportionality coefficient and a specific fractional power of the third Lyapunov function itself, and the addition of a positive constant term; The finite-time convergence of the terminal sliding surface is determined by the dynamic evolution equation of the position error vector and the third lemma.
7. The model-free finite-time saturation control method for a wave compensation platform according to claim 6, characterized in that, The step of obtaining the convergence time of the terminal sliding surface based on the derivative of the third Lyapunov function and the fifth positive definite diagonal parameter matrix includes: Based on the boundedness of all state variables in the closed-loop system, the upper bound of the time derivative of the core state vector is determined. Combining the inequality property of matrix norms, the upper bound of the time derivative of the third Lyapunov function is derived: the time derivative of the third Lyapunov function is less than or equal to the negative of a specific algebraic expression, which is: the product of the smallest eigenvalue of the fifth positive definite diagonal parameter matrix and the norm of the terminal sliding surface minus the difference between the upper bound of the time derivative of the core state vector and the product of the norm of the terminal sliding surface. Based on the assumption that the fifth positive definite diagonal parameter matrix satisfies specific constraints, namely that the product of the minimum eigenvalue of the fifth positive definite diagonal parameter matrix and the norm of the terminal sliding surface is greater than the upper bound of the time derivative of the core state vector, and combined with the derivative result of the third Lyapunov function, the upper bound of the time derivative of the norm of the terminal sliding surface is further derived: the time derivative of the norm of the terminal sliding surface is less than or equal to the negative of the difference between the product of the minimum eigenvalue of the fifth positive definite diagonal parameter matrix and the norm of the terminal sliding surface and the upper bound of the time derivative of the core state vector. Integrating both sides of the inequality for the upper bound of the time derivative of the norm of the terminal sliding surface in the time domain, the upper bound of the convergence time of the terminal sliding surface is finally calculated. The upper bound of the convergence time is equivalent to the ratio of the initial value of the terminal sliding surface to the product of the minimum eigenvalue of the fifth positive definite diagonal parameter matrix and the norm of the terminal sliding surface, minus the upper bound of the time derivative of the core state vector.
8. The model-free finite-time saturation control method for a wave compensation platform according to claim 7, characterized in that, The determination of the finite-time convergence of the terminal sliding surface using the dynamic evolution equation of the position error vector and the third lemma includes: Based on the dynamic evolution equation of the position error vector, a fourth Lyapunov function is constructed with half of the square of the position error vector as the variable. The time derivative of the fourth Lyapunov function is obtained, and the derivative result is substituted into the dynamic evolution equation of the position error vector to obtain the substitution result. If the substitution result is less than or equal to the product of a specific fractional power of the fourth Lyapunov function and a negative proportionality coefficient, it matches the criterion for semi-global finite-time stability of the nonlinear system in the third lemma, thus determining that the position error vector of the terminal sliding surface can converge to a steady state in a finite time.
9. A model-free finite-time saturation control method for a wave compensation platform according to any one of claims 1 to 8, characterized in that, The first lemma includes: for any positive real number smoothing parameter and any real number variable, the difference between the absolute value of the arbitrary real number variable and the product of the arbitrary real number variable and the hyperbolic tangent function with the arbitrary real number variable divided by the arbitrary positive real number smoothing parameter as the independent variable is non-negative and has an upper bound, and the upper bound is the product of a fixed proportionality constant and the arbitrary positive real number smoothing parameter; The second lemma includes: for any finite number of positive definite functions, their arithmetic sum is not less than the square root of their sum of squares; and for any exponent c between 0 and 1, the sum of the c powers of the set of positive definite functions is not less than the c power of their arithmetic sum. The third lemma states that if a nonlinear system with initial values has a positive definite Lyapunov function that satisfies a specific derivative constraint, then the system is a semi-global finite-time stable system, and both the positive definite Lyapunov function with the specific derivative constraint and the convergence time have upper bounds determined by the parameters and the initial values.
10. A model-free finite-time saturation control method for a wave compensation platform according to any one of claims 1 to 8, characterized in that, The control parameters include: a first positive definite diagonal parameter matrix. Second positive definite diagonal parameter matrix Third positive definite diagonal parameter matrix Fourth positive definite diagonal parameter matrix Fifth positive definite diagonal parameter matrix The sixth positive definite diagonal parameter matrix S60 includes: , , satisfy: ,in, and These are the minimum and maximum values of the actuator output, respectively; adjust first. This ensures that the error converges and the chatter is less than the chatter threshold. If errors due to uncertainty still exist after adjustment, then further adjustment is needed. Increase; Adjustments are made based on the dynamic characteristics of the desired trajectory; The initial value is set to a preset value, and then increased according to the convergence of the tracking error. Set to the minimum value.