Adaptive neural time-specified trajectory tracking control method for underactuated ships with input saturation

Abstract

The present invention discloses an adaptive neural specified time trajectory tracking control method for an under-actuated ship with input saturation. By combining a preset performance control method with finite time theory and introducing an additional control method and a saturation compensation mechanism, the offline customization of tracking convergence time, position error and speed error is guaranteed, and the under-actuated vector design is realized; the virtual control law is designed by designing an auxiliary dynamic filter to avoid the differential inflation phenomenon; by considering internal and external disturbances and actuator saturation conditions, the ship longitudinal control law and the ship steering control law are designed for the under-actuated ship based on an adaptive neural network and a single parameter learning algorithm, and the defects in design and control performance between the adaptive neural network, actuator saturation compensation technology and specified performance control are coordinated. In addition, the present invention can ensure that the ship tracking error is limited to a predefined range within the specified time, overcome the influence of input saturation on the tracking error convergence performance, and make the position error and speed error in the tracking control reach the specified range within the predefined convergence time.

Description

Technical Field

[0001] The present invention relates to the technical field of ship tracking control based on new generation information technology, and in particular to an adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation. Background Art

[0002] As an important carrier for the development of today's marine economy, the research and development of intelligent / unmanned ships has once again been pushed to a new height. The motion control problems of underactuated ships can be mainly divided into two categories: stabilization control and track tracking control. The essence of stabilization control is to find a suitable control law to stabilize the position and heading angle of the ship to near the equilibrium point. Depending on whether the tracking state of the ship is independent of time, track tracking control can be divided into trajectory tracking and path tracking control problems. Trajectory tracking requires that the ship can reach the specified reference position at a certain moment, and the change in the ship's state is strictly related to time. Path tracking is the tracking of geometric positions without considering time. In engineering practice, the implementation of ship trajectory tracking tasks is more difficult than path tracking control. This problem is one of the mainstream research problems in the field of ship motion control and is also the theory focused on in this research.

[0003] With the continuous development of modern ship technology and new generation information technology, the requirements for ship maneuverability, stability and energy efficiency are becoming increasingly higher, and the optimization and improvement of these performances are closely related to ship motion control. Considering that in practice, most ships are underactuated ships, only two input variables, rudder and propeller, are needed to achieve the three degrees of freedom of the ship's horizontal forward movement, drift and bow pitch. Therefore, based on the new generation of information industry technology, an effective control scheme for underactuated ship motion is proposed to improve its steady-state and transient performance, which has broad application prospects. However, from the perspective of underactuated ship control design, the tracking control of underactuated ships faces the following major challenges:

[0004] 1) From the perspective of system characteristics, the underactuated ship is a typical nonholonomic constraint system with typical nonlinear, strong coupling, nonholonomic constraints, and complex uncertainties, which brings huge challenges to the control design.

[0005] 2) Ship actuators always have some constraints, such as input saturation constraints. Once these physical constraints are ignored in system design, they may lead to poor closed-loop performance or even system instability.

[0006] 3) During navigation, all ships are inevitably subject to internal and external uncertainties. Due to the complexity of the marine environment, external environmental disturbances such as wind, waves, and currents are known as external uncertainties. Ship characteristics such as mass and moment of inertia may change, which inevitably leads to perturbations in ship model parameters. Furthermore, due to modeling techniques and the complex structure of the ship itself, ship mathematical models have unmodeled dynamics. These model parameter perturbations and unmodeled dynamics are known as internal uncertainties. Consequently, accurate ship model knowledge cannot be obtained during design.

[0007] 4) To ensure safe navigation, tracking error must be kept within a pre-defined bounded range. Within the vessel's maneuverability, the user can define the vessel's tracking stabilization time and tracking control accuracy based on the operating environment and practical requirements, thereby improving the motion control capabilities of surface vessels. Ultimately, this ensures safe navigation and enables efficient and high-precision tracking of the vessel's trajectory. Therefore, the ability to specify tracking error performance within a limited timeframe is a pressing challenge for intelligent ship safety. Summary of the Invention

[0008] The present invention provides an adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation to overcome the above technical problems.

[0009] In order to achieve the above object, the technical solution of the present invention is:

[0010] An adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation comprises the following steps:

[0011] S1: Establish a mathematical model of a three-degree-of-freedom underactuated MSV with a control input saturation function;

[0012] S2: Based on the mathematical model of the three-degree-of-freedom underactuated MSV, the position error variable and velocity error variable are defined according to the preset desired trajectory;

[0013] S3: constructing a predefined / specified time performance function, obtaining preset performance boundary conditions on the position error variable and the speed error variable, and based on the preset performance function boundary conditions, performing nonlinear transformation on the position error variable and the speed error variable according to the predefined / specified time performance function to obtain a position error conversion variable and a speed error conversion variable;

[0014] S4: Construct a virtual control law for trajectory tracking based on the position error conversion variable, and design an auxiliary dynamic filter based on the virtual control law to avoid the differential explosion caused by its derivative;

[0015] S5: Based on the mathematical model of the three-degree-of-freedom underactuated MSV and the speed error conversion variable, the additional control law and adaptive parameter control law of the ship trajectory tracking controller are constructed;

[0016] S6: constructing a ship trajectory tracking controller according to the auxiliary dynamic filter, the additional control law and the adaptive parameter control law; and the ship trajectory tracking controller includes a ship longitudinal control law and a ship steering control law;

[0017] S7: Implementing adaptive neural specified time trajectory tracking control for underactuated ships with input saturation based on the ship trajectory tracking controller.

[0018] Furthermore, the mathematical model of the three-degree-of-freedom underactuated MSV with the control input saturation function established in S1 includes the kinematic equations of the underactuated ship, the dynamic equations of the underactuated ship, and the control input saturation function;

[0019] The expression of the kinematic equation of the underactuated ship is:

[0020]

[0021] Where: x, y, ψ represent the longitudinal and transverse positions and yaw angle of the under-actuated ship in the geodetic / inertial coordinate system respectively; u, v, r represent the longitudinal, transverse speed and turning speed of the under-actuated ship in the appendage / hull coordinate system respectively;

[0022] The expression of the underactuated ship dynamic equation is:

[0023]

[0024] f u (υ)=m 22 vr-d 11 u

[0025] f v (υ)=(-m 11 ud 23 )rd 22 v

[0026] f v (υ)=(m 11 -m 22 )uv-d 32 vd 33 r

[0027] Where: υ=[uvr] T They represent the longitudinal, transverse and turning speeds of the underactuated ship in the appendage / hull coordinate system respectively; f = [f u f v f r] T Represents the nonlinear dynamic terms of the ship model in the longitudinal, transverse speed and steering direction; τ=[τ u 0 τ r ] T And τ u With τ r denote the longitudinal and steering control inputs of the underactuated ship respectively; τ w =[τ wu τ wv τ wr ] T And τ wu , τ wv , τ wr Respectively represent the external unknown time-varying environmental disturbances suffered in the longitudinal, lateral and steering directions; m 11 , m 22 , m 33 Represents the parameters in the inertia matrix; d 11 , d 22 , d 23 , d 32 , d 33 Represents the parameters in the nonlinear hydrodynamic damping matrix of the undership system;

[0028] The expression of the control input saturation function is:

[0029]

[0030] Where: τ Mj represents the maximum control force or torque provided by the underactuated ship propulsion system, and τ Mj >0, j = u, r, τ cj Represents the control input instruction, and the intermediate variable Δτ u =τ u -τ cu , Δτ r =τ r -τ cr .

[0031] Furthermore, the expression of the tracking error variable defined in S2 is

[0032]

[0033] Where: x e ,y e , ψ e Respectively represent the lateral position error, longitudinal position error and heading angle error; x d ,y d , ψ d Respectively represent the lateral position, longitudinal position and yaw angle of the preset desired trajectory;

[0034] The expression of the speed error variable is:

[0035]

[0036] Where: a1, a2, a3 are constants; γ=[γ u , γ v , γ r ] T The virtual control law α=[α u , α v , α r ] T The filtering form; tanh(·) represents the function used to constrain the additional control signals β1, β2, β3; u e , v e , r e They represent the longitudinal velocity error, lateral velocity error and turning velocity error of the underactuated ship respectively.

[0037] Furthermore, the S3 specifically includes the following steps:

[0038] S31: Construct a predefined / specified time performance function, whose expression is

[0039]

[0040] Where:

[0041] l* represents the design parameters; T represents the custom predefined convergence time; f

[0042] Indicates the custom convergence control accuracy, that is, time t is greater than or equal to T f When the system error converges to d range; and ρ(t) satisfies ρ(0) represents the initial value of the predefined / specified time performance function;

[0043] S32: When When , the position error is defined as e η =η-η d ;

[0044] where e η =[e η，1 , e η，2 , e η，3 ] T =[x e ,y e , ψ e ] T , η=[x,y,ψ] T , ηd =[x d ,y d , ψ d ] T ;

[0045] Define the speed error as e υ , where e υ =[e υ，1 , e υ，2 , e υ，3 ] T =[u e , v e , r e ] T ;

[0046] Then, the preset performance boundary conditions for the position error variable and the speed error variable are obtained according to the predefined / specified time performance function;

[0047] The preset performance boundary condition of the position error variable is

[0048] -ρ η，1 <x e <ρ η，1

[0049] -ρ η，2 <y e <ρ η，2

[0050] -ρ η，3 <ψ e <ρ η，3

[0051] Where: ρ η，i =ρ(t) and i=1, 2, 3;

[0052] The preset performance boundary condition of the speed error variable is

[0053] -ρ υ，1 <u e <ρ υ，1

[0054] -ρ υ，2 <v e <ρ υ，2

[0055] -ρ υ，3 <r e <ρ υ，3

[0056] Where: ρ v，i =ρ(t) and i=1, 2, 3;

[0057] S33: performing nonlinear transformation on the position error variable and the speed error variable according to a predefined / specified time performance function to obtain a tracking error conversion variable and a speed error conversion variable;

[0058] The expression of the position error conversion variable is:

[0059]

[0060] Where: e η，i (i=1, 2, 3) represents the position error e η The i-th element of η，i (i=1, 2, 3) represents the position error conversion variable and satisfies

[0061] The expression of the speed error conversion variable is:

[0062]

[0063] Where: e υ，i (i=1, 2, 3) represents the speed error e υ The i-th element of υ，i (i=1, 2, 3) represents the speed error conversion variable and satisfies

[0064] Furthermore, the S4 specifically includes the following steps:

[0065] S41: Construct a virtual control law for trajectory tracking based on the position error conversion variable, which is expressed as

[0066]

[0067] And the vector expression of the trajectory tracking virtual control law is

[0068]

[0069] In the formula: A=diag(a1, a2, a3), β=[tanhβ1, tanhβ2, tanhβ3] T ; k = diag[k1, k1, k2], k1 and k2 are constants; Ψ η =diag[Ψ η，1 ,Ψ η，2 ,Ψ η3 ],φ η =diag[φ η，1 ,φ η，2 ,φ η，3 ],ρ η =diag[ρ η，1, ρ η，2 , ρ η，3 ];Ψ η，i ,φ η，i represents the intermediate parameter variable, i = 1, 2, 3 and α t , t = u, v, r represent the longitudinal speed virtual control law, the lateral speed virtual control law and the steering speed virtual control law respectively; represents η d The first derivative of ; J(ψ) represents the rotation matrix that transforms the information between the hull coordinate system and the earth coordinate system;

[0070] S42: According to the virtual control law, an auxiliary dynamic filter is designed to avoid differential explosion caused by derivation of the virtual control law, and the expression of the auxiliary dynamic filter is:

[0071]

[0072] Where: γ i represents the first-order auxiliary dynamic filter; μ ft represents the time constant; γ ι (0) represents the first-order auxiliary dynamic filter γ ι The initial value of α t (0) represents the virtual control law α t The initial value of

[0073] And define the filtering error as e ft =γ i -a t , according to the filtering error, the first-order auxiliary dynamic filter γ l The derivative of

[0074] Furthermore, the additional control law and the adaptive parameter control law of the ship trajectory tracking controller are constructed based on the mathematical model of the three-degree-of-freedom underactuated MSV and the speed error conversion variable as described in S5;

[0075] The expression of the additional control law of the ship trajectory tracking controller is:

[0076]

[0077]

[0078] Where: μ u , μ r , k4 are all constants; β1, β3 represent virtual control signals that cannot be realized due to the actuator saturation constraint effect; β2 represents an additional control signal used to provide lateral stability to solve the under-actuation problem under saturation conditions; Represent the first-order derivatives of β1, β2, and β3 respectively; Ψ v，i ,φ v，i represents the intermediate parameter variable, i = 1, 2, 3 and Θ v represents the adaptive parameters of the ship trajectory tracking controller, and Θ v =max{||W v *T ||,|ε2+τ wv / m 22 |}; Represents Θ v The estimated value of Φ v (s) is Φ v (Z v ) represents the intermediate parameter variable, and Φ v (Z v )=||s v (Z v )||+|φ υ，2 ζ υ，2 z υ，2 |+1;Z v =[u,V,r,ρ υ，2 , v e ] T Represents the input vector of the neural network NN model; s v (Z v )=[s v1 (Z v ),…,s vn (Z v )] T Represents the radial basis function vector of the neural network NN model; Represents the weight vector of the neural network NN model; ζ υ，2 represents the design parameter; ε2 represents the approximation error;

[0079] The expression of the adaptive parameter control law of the ship trajectory tracking controller is:

[0080]

[0081] Where: Θ u ,Θ v ,Θ r represents the adaptive parameters of the ship trajectory tracking controller; Represent Θ respectively u ,Θ v ,Θ r estimated value of; Respectively The first derivative of b u , b v , b r , lu , l v , l r Both represent design constants; Φ u (S), Φ v (S), Φ r (S) is Φ u (Z u ), Φ v (Z v ), Φ r (Z r ) represent intermediate parameter variables, Θ u =max{||W u *T ||,|ε1+τ wu / m 11 |,1},Φ u (Z u )=||s u (Z u )||+|φ υ，1 ζ υ，1 z υ，1 |+1, Z u =[u,v,r,ρ υ，1 ,u e ] T Represents the input vector of the neural network NN model, s u (Z u )=[s u1 (Z u ),…,s ub (Z u )] T Represents the radial basis function vector of the neural network NN model, Represents the weight vector of the neural network NN model; ε1 represents the approximation error; ζ υ，1 represents the design parameter; Θ r =max{||W r *T ||,|ε3+τ wr / m 33 |};Z r =[u,V,r,ρ υ，3 , r e ] T Represents the input vector of the neural network NN model; Φ r (Z r )=||s r (Z r )||+|φ υ，3 ζ υ，3 z υ，3 |+1,s r (Z r )=[s r1(Z r ),…,s rn (Z r )] T is the radial basis function vector of the neural network NN, The weight vector of the neural network NN model; ε3 represents the approximation error; ζ υ，3 Represents design parameters.

[0082] Furthermore, the ship trajectory tracking controller constructed in S6 includes the ship longitudinal control law and the ship steering control law;

[0083] The expression of the ship longitudinal control law is:

[0084]

[0085] The expression of the ship steering control law is:

[0086]

[0087] Beneficial effects: The present invention provides an adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation. By combining a preset performance control method with finite time theory and introducing an additional control method and a saturation compensation mechanism, the tracking convergence time, position error, and velocity error can be customized offline, thus realizing the vectored design of the underactuated ship. The "hard" nonlinear problem caused by actuator deficiency and saturation is solved by adopting additional control and an auxiliary dynamic filter. Compared with the dynamic auxiliary system method, the virtual control law designed by the auxiliary dynamic filter avoids the differential inflation phenomenon. By considering internal and external disturbances and actuator saturation conditions, the longitudinal control law and the steering control law of the underactuated ship are designed based on an adaptive neural network and a single parameter learning algorithm. The design and control performance defects of the adaptive neural network, the actuator saturation compensation technology, and the specified performance control are coordinated. In addition, the present invention can ensure that the ship tracking error is limited to a predefined range within the specified time, overcome the influence of input saturation on the tracking error convergence performance, and make the position error and velocity error in the tracking control reach the specified range within the predefined convergence time, greatly improving the advantages of fast finite time convergence speed and strong robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0089] Figure 1 This is a flow chart of the adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation according to the present invention;

[0090] Figure 2 This is a core flow chart of the adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation in this embodiment;

[0091] Figure 3 This is a simulation diagram of the tracking performance in this embodiment;

[0092] Figure 4 : is a simulation diagram of the reference position and the actual position in this embodiment;

[0093] Figure 5 This is a control input simulation diagram of the underactuated ship in this embodiment;

[0094] Figure 6 : This is a ship speed simulation diagram of the under-actuated ship in this embodiment;

[0095] Figure 7 This is a simulation diagram of the ship position error in this embodiment;

[0096] Figure 8 This is a simulation diagram of the ship speed error in this embodiment;

[0097] Figure 9 This is a simulation diagram of the ship adaptive law in this embodiment. DETAILED DESCRIPTION

[0098] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0099] This embodiment provides an adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation, such as Figures 1 to 2 As shown, the following steps are included:

[0100] S1: Establish a mathematical model of a three-degree-of-freedom underactuated MSV with a control input saturation function;

[0101] Specifically, a mathematical model of a three-degree-of-freedom underactuated MSV with a control input saturation function is established, including the kinematic equations of the underactuated ship, the dynamic equations of the underactuated ship, and the control input saturation function;

[0102] The expression of the kinematic equation of the underactuated ship is:

[0103]

[0104] Where: x, y, ψ represent the longitudinal and transverse positions and yaw angle of the under-actuated ship in the geodetic / inertial coordinate system respectively; u, v, r represent the longitudinal, transverse speed and turning speed of the under-actuated ship in the appendage / hull coordinate system respectively;

[0105] The expression of the underactuated ship dynamic equation is:

[0106]

[0107]

[0108] Where: υ=[uvr] T They represent the longitudinal, transverse and turning speeds of the underactuated ship in the appendage / hull coordinate system respectively; f = [f u f v f r ] T Represents the nonlinear dynamic terms of the ship model in the longitudinal, transverse speed and steering direction; τ=[τ u 0 τ r ] T And τ u With τ r denote the longitudinal and steering control inputs of the underactuated ship respectively; τ w =[τ wu τ wv τ wr ] T And τ wu , τ wv , τ wr Respectively represent the external unknown time-varying environmental disturbances suffered in the longitudinal, lateral and steering directions; m 11 , m 22 , m 33 Represents the parameters in the inertia matrix; d 11 , d 22 , d 23 , d 32 , d 33 Represents the parameters in the nonlinear hydrodynamic damping matrix of the undership system;

[0109] Due to the structure and thruster configuration of the underactuated ship itself, there are natural limitations in its maneuverability. Additionally, due to physical constraints, the actuators of the underactuated ship inevitably have saturation constraints. Moreover, the saturation constraints of the actuators of the underactuated ship will inevitably cause the actuators to be unable to execute the expected command signal, making it difficult to ensure that the underactuated ship completes trajectory tracking within a specified time, that is, the control signal obeys the control input saturation function with the following saturation nonlinear constraints;

[0110] The expression of the control input saturation function is

[0111]

[0112] In the formula: τ Mj represents the maximum control force or torque provided by the propulsion system of the underactuated ship, and τ Mj > 0, j = u, r, τ cj represents the control input command, and the intermediate variable Δτ is denoted u = τ u - τ cu , Δτ r = τ r - τ cr ;

[0113] This embodiment also includes the following definitions and lemmas:

[0114] Definition 1. Practical finite-time stability: Consider the nonlinear system described as:

[0115]

[0116] In the formula, ξ ∈ R n represents the state variable of the underactuated ship control system, Ω0 represents a spherical domain containing the origin, and f(ξ) represents a continuous function; for any initial condition ξ0, if there exist constants a > 0 and an adjustment time function 0 < T(ξ0) < ∞ such that t ≥ T(ξ0, a), then the system (5) can be said to be practically finite-time stable.

[0117] Definition 2. Practical fixed-time stability: A practically finite-time stable system (5) can be said to be practically fixed-time stable if for ξ ∈ R n , there exist constants a > 0 and an adjustment time T(a) such that For holds.

[0118] Definition 3: Predefined practical finite-time stability: A practically finite-time stable system (5) can be said to be predefined-time stable if for ξ ∈ R n, there is a constant And the stabilization time is 0 <T u <∞, so that for Established, where T u It is predefined by the user offline.

[0119] Lemma 1. For the definition on a compact set Any nonlinear function β(Z) on R n →R, which can be obtained by the radial basis function vector W of the neural network *T s(Z) approximation, then

[0120] β(Z)=W *T s(Z)+ε (7)

[0121] Where: ε represents the approximation error of the neural network NN model, and satisfies and represents a constant; s(Z)=[s1(Z)…s l (Z)] T is the radial basis function vector of the neural network NN model; W * =[w1 * ,…,w1 * ] T is the weight vector of the neural network NN model; l>1 is the node of the neural network NN model; s h (Z) is the radial basis function vector of the neural network NN model, and h=1,…l,s h (Z) mathematical expression is

[0122]

[0123] Where: The center point vector value μ of the Gaussian function h =[μ h，1 ,…,μ h，l ] T ,ω h is the width of the Gaussian function; h is the node of the neural network NN model.

[0124] Lemma 2. Given any real variables M, N and a positive constant The following inequality holds

[0125]

[0126] Definition 4. If the smooth function ρ(t): R + →R + is a performance function, then:

[0127] 1)ρ(t) is a positive function and decreases.

[0128] 2)limt→∞ ρ(t)=ρ ∞ >0, where ρ ∞ It is a range.

[0129] According to Definition 2, this embodiment introduces the following definition of a predefined time performance function (PTPF):

[0130] Definition 5. If the smooth function ρ(t): R + →R + is a predefined time performance function, then it has the following properties:

[0131] The range and time constant T f Can be customized;

[0132] Considering Definitions 1-3, the predefined practical finite-time stability can be described as follows: The convergence time T u It is independent of the controller design parameters and initial conditions and is determined offline by the user.

[0133] In addition, to facilitate the design and analysis of the control scheme, this embodiment also provides the following assumptions:

[0134] Assumption 1: The external environmental disturbance τ suffered by the underactuated ship wu , τ wv , τ wr is an unknown time-varying disturbance and its first-order derivative is bounded, satisfying

[0135]

[0136] Where, the unknown limit of the external environmental disturbance is The unknown limit d of the derivative of the external environment disturbance wu >0,d wv >0,d wr >0;

[0137] Assumption 2: The expected trajectory of the ship η d =[x d ,y d , ψ d ] T and its first-order derivative Second-order derivative are all bounded;

[0138] Assumption 3 Nonlinear dynamics is unknown;

[0139] S2: Based on the mathematical model of the three-degree-of-freedom underactuated MSV, the tracking error variable and the velocity error variable are defined according to the preset desired trajectory;

[0140] Specifically, the expression of the tracking error variable is defined as

[0141]

[0142] Where: x e ,y e , ψ e Respectively represent the lateral position error, longitudinal position error and heading angle error; x d ,y d , ψ d Respectively represent the lateral position, longitudinal position and yaw angle of the preset desired trajectory;

[0143] The expression of the speed error variable is:

[0144]

[0145] Where: a1, a2, a3 are constants; γ=[γ u , γ v , γ r ] T The virtual control law α=[α u , α v , α r ] T The filtering form; tanh(·) represents the function used to constrain the additional control signals β1, β2, β3; u e , v e , r e They represent the longitudinal velocity error, lateral velocity error, and steering velocity error of the underactuated ship respectively;

[0146] S3: Constructing a predefined / specified time performance function, obtaining preset performance boundary conditions for the tracking error variable and the speed error variable, and performing nonlinear transformation on the tracking error variable and the speed error variable according to the predefined / specified time performance function based on the preset performance function boundary conditions to obtain a tracking error conversion variable and a speed error conversion variable, specifically comprising the following steps:

[0147] S31: Construct a predefined / specified time performance function according to Definition 3, whose expression is

[0148]

[0149] Where: l * represents the design parameters; T f Indicates a custom predefined convergence time; Indicates the custom convergence control accuracy, that is, time t is greater than or equal to T f When the system error converges to f range; and ρ(t) satisfies ρ(0) represents the initial value of the predefined / specified time performance function;

[0150] S32: According to the predetermined / specified performance control method, when When the position error is defined as e η =η-η d ;

[0151] where e η =[e η，1 , e η，2 , e η，3 ] T =[x e ,y e , ψ e ] T , η=[x,y,ψ] T , η d =[x d ,y d , ψ d ] T ;

[0152] According to the predetermined / specified performance control method, when The speed error is defined as e υ , where e υ =[e υ，1 , e υ，2 , e υ，3 ] T =[u e , v e , r e ] T ;

[0153] Wherein, Equation (14) has the limiting tracking error e η Here, four design parameters are used to describe the tracking control performance of the ship. and l * Represents the predefined convergence rate. Need to meet |e η，i (0)|<ρ η，i (0),|e υ，i (0)|<ρ υ，i (0). T fi It is the user-defined convergence setting time, which is used to input the convergence time to keep the input within the range. It should be pointed out that It should be set according to the actual requirements and performance of the ship. In practice, the performance of the ship is determined by performance indicators such as forward movement, turning, etc.

[0154] Then, the preset performance boundary conditions for the position error variable and the speed error variable are obtained according to the predefined / specified time performance function;

[0155] The preset performance boundary condition of the position error variable is

[0156]

[0157] Where: ρ η，i =ρ(t) and i=1, 2, 3; According to formula (14), when l * Very small, and e η (0) When bounded, the inequality |e η，i (0)|<ρ η，i (0) Established;

[0158] The preset performance boundary condition of the speed error variable is

[0159]

[0160] Where: ρ υ，i =ρ(t) and i=1, 2, 3; According to formula (17), when l * Very small, and e υ (0) When bounded, the inequality |e υ，i (0)|<ρ υ，i (0) Established;

[0161] S33: performing nonlinear transformation on the position error variable and the speed error variable according to a predefined / specified time performance function to obtain a position error conversion variable and a speed error conversion variable;

[0162] To ensure When the tracking error e η The following nonlinear transformation is introduced to satisfy equation (16), and the expression of the position error conversion variable is:

[0163]

[0164] Where: e η，i (i=1, 2, 3) represents the position error e η The i-th element of η，i (i=1, 2, 3) represents the position error conversion variable and satisfies

[0165] Therefore, when z η，i ∈L ∞ and|e η，i |<ρ η，i (0), Equation (16) holds; taking the derivative of the position error conversion variable, we can get

[0166]

[0167] Where: According to the above ρ η，i From the properties of , we can see that as long as (16) holds, there must be Ψ η，i >0;

[0168] Furthermore, Equation (19) can be written in the following vector form:

[0169]

[0170] In the formula Ψ η =diag[Ψ η，1 ,Ψ η，2 ,Ψ η，3 ],φ η =diag[φ η，1 ,φ η，2 ,φ η，3 ],

[0171] ρ η =diag[ρ η，1 , ρ η，2 , ρ η，3 ].

[0172] Furthermore, according to formula (13)

[0173]

[0174] In the formula, A = diag (a1, a2, a3), β = [tanhβ1, tanhβ2, tanhβ3] T ;

[0175] To ensure Speed ​​error e υ The following nonlinear transformation is introduced to satisfy equation (17), and the expression of the speed error conversion variable is:

[0176]

[0177] Where: e υ，i (i=1, 2, 3) represents the speed error e υ The i-th element of υ，i (i=1, 2, 3) represents the speed error conversion variable and satisfies

[0178] Therefore, when z υ，i ∈L ∞ and|e υ，i |<ρ υ，i (0), formula (17) holds.

[0179]

[0180] Where, According to the above ρ υ，i The properties of can be known. As long as (17) holds, there must be Ψ υ，i >0.

[0181] Furthermore, Equation (23) can be written in the following vector form:

[0182]

[0183] Where: Ψ υ =diag[Ψ υ，1 ,Ψ υ，2 ,Ψ υ，3 ],φ υ =diag[φ υ，1 ,φ υ，2 ,φ υ，3 ],ρ υ =diag[ρ υ，1 , ρ υ，2 , ρ υ，3 ],γ=[γ u , γ v , γ r ] T ;

[0184]

[0185] S4: Construct a virtual control law for trajectory tracking based on the position error conversion variable, and design an auxiliary dynamic filter based on the virtual control law to avoid the differential explosion caused by its derivative;

[0186] The specific steps include:

[0187] S41: Construct a virtual control law for trajectory tracking based on the position error conversion variable, which is expressed as

[0188]

[0189]

[0190] Where: η，i， φ η，i represents the intermediate parameter variable, i = 1, 2, 3 and k1, k2 represent design constants; α u , α u , α u They represent the longitudinal speed virtual control law, the lateral speed virtual control law, and the steering speed virtual control law respectively; Represents x d ,y d , ψ d The first derivative of

[0191] And the vector expression of the virtual control law of trajectory tracking is

[0192]

[0193] Where k = diag[k1, k1, k2], k1 and k2 are constants;

[0194] Substituting equation (29) into equation (21), we can obtain

[0195]

[0196] For the entire underactuated ship closed-loop tracking control system, consider the following control Lyapunov function

[0197]

[0198] Taking the derivative of V1, substituting Equation (30) into Equation (31), and using Lemma 2 and Young's inequality, we can obtain:

[0199]

[0200] According to Young's inequality, we can get

[0201]

[0202] According to Lemma 2, using Young's inequality, we can get:

[0203]

[0204] Where,

[0205] S42: According to the virtual control law, an auxiliary dynamic filter is designed to avoid differential explosion caused by derivation of the virtual control law, and the expression of the auxiliary dynamic filter is:

[0206]

[0207] Where: γ t represents the first-order auxiliary dynamic filter; μ ft represents the time constant; γ t (0) represents the first-order auxiliary dynamic filter γ i The initial value of α t (0) represents the virtual control law α t The initial value of

[0208] And define the filtering error as e ft =γ t -a t , according to the filtering error, the derivative of the first-order auxiliary dynamic filter γ,

[0209] In a specific embodiment, in order to avoid u Directly derive and introduce the following filter;

[0210]

[0211] Where, γ u is a first-order filter, α u With time constant μ fu Through the filter. Define a filtering error e fu =γ u -α u ,remember

[0212] According to formula (26), we can get

[0213]

[0214] Where A u (·) is a continuous function with a maximum value of M u .

[0215] Similarly, in order to avoid v Directly derive and introduce the following filter:

[0216]

[0217] Where, γ v is a first-order filter, α v With time constant μ fv Through the filter. Define a filtering error e fv =γ v -α v ,remember

[0218] According to formula (27), we can get

[0219]

[0220] Where A v (·) is a continuous function with a maximum value of M v .

[0221] In order to avoid the r Directly derive and introduce the following filter:

[0222]

[0223] Where, γ r is a first-order filter, α r With time constant μ fr Through the filter. Define a filtering error e fr =γ r -α r ,remember

[0224] Furthermore, according to formula (28), we can get

[0225]

[0226] Where A r (·) is a continuous function with a maximum value of M r ;

[0227] S5: Based on the mathematical model of the three-degree-of-freedom underactuated MSV and the speed error conversion variable, the additional control law and adaptive parameter control law of the ship trajectory tracking controller are constructed;

[0228] Specifically, the expression of the additional control law of the ship trajectory tracking controller is:

[0229]

[0230] Where: μ u , μ r , k4 are all constants; β1, β3 represent virtual control signals that cannot be realized due to the actuator saturation constraint effect; β2 represents an additional control signal used to provide lateral stability to solve the under-actuation problem under saturation conditions; Represent the first-order derivatives of β1, β2, and β3 respectively; Ψ υ，i ,φ υ，i represents the intermediate parameter variable, i = 1, 2, 3 and Θ v represents the adaptive parameters of the ship trajectory tracking controller, and Θ v =max{||W v *T ||,|ε2+τ wv / m 22 |}; Represents Θ v The estimated value of Φ v (s) is Φ v (Z v ) represents the intermediate parameter variable, and Φ v (Z v )=||s v (Z v )||+|φ υ，2 ζυ，2 z υ，2 |+1;Z v =[u,v,r,ρ υ，2 , v e ] T Represents the input vector of the neural network NN model; s v (Z v )=[s v1 (Z v ),…,s vn (Zv)] T Represents the radial basis function vector of the neural network NN model; Represents the weight vector of the neural network NN model; ζ υ，2 represents the design parameter; ε2 represents the approximation error;

[0231] In this embodiment, the following Lyapunov function is considered

[0232]

[0233] Taking the derivative of V2 and substituting equation (25) into equation (47), we can get

[0234]

[0235] According to formula (22), we can get

[0236]

[0237] Furthermore, formula (48) can be written as

[0238]

[0239] remember:

[0240]

[0241] According to Lemma 1, Equation (51) can be written as

[0242]

[0243] Where, Θ u =max{||W u *T ||,|ε1+τ wu / m 11 |,1},Φ u (Z u )=||s u (Z u )+|φ υ，1 ζ υ，1 z υ，1 |+1, Z u=[u,v,r,ρ υ，1 ,u e ] T is the input vector of the neural network NN model, s u (Z u )=[s u1 (Z u ),…,s un (Z u )] T is the radial basis function vector of the neural network NN model, ε1 is the approximation error.

[0244] Note: According to expressions (51) to (53) and assumption 3, this embodiment can obtain H u (Z u ), H v (Z v ) and H r (Z r ) is a composite function containing uncertain terms, disturbance terms, and known terms; Equations (54), (57), and (60) transform the composite function into a linear parameterized form, which only requires updating the unknown parameter Θ, reducing the computational burden of the system;

[0245] further, It can be written as:

[0246]

[0247] According to Lemma 1, Equation (52) can be written as

[0248]

[0249] Where Z v =[u,v,r,ρ υ，2 , v e ] T is the input vector of the neural network NN model, s v (Z v )=[s v1 (Z v ),…,s vn (Z v )] T is the radial basis function vector of the neural network NN model, ε2 is the approximation error. Where, Θ v =max{||W v *T ||,|ε2+τ wv / m 22 |},Φ v (Z v )=||s v(Z v )||+|φ υ，2 ζ υ，2 z υ，2 |+1.

[0250] further, It can be written as:

[0251]

[0252] According to Lemma 1, Equation (53) can be written as

[0253]

[0254] Where: Z r =[u,v,r,ρ υ，3 , r e ] T is the input vector of the neural network NN model, s r (Z r )=[s r1 (Z r ),…,s rn (Zr)] T is the radial basis function vector of the neural network NN model, ε3 is the approximation error. Where, Θ r =max{||W r *T ||,|ε3+τ wr / m 33 |},Φ r (v r )=||s r (Zr)||+|φ υ，3 ζ υ，3 z υ，3 |+1.

[0255] further, It can be written as:

[0256]

[0257] Substituting equations (55) to (62) into equation (50) yields:

[0258]

[0259] The expression of the adaptive parameter control law of the ship trajectory tracking controller is:

[0260]

[0261] Where: Θ u ,Θ v ,Θr represents the adaptive parameters of the ship trajectory tracking controller; Represent Θ respectively u ,Θ v ,Θ r estimated value of; Respectively The first derivative of b u , b v , b r , l u , l v , l r Both represent design constants; Φ u (s), Φ v (s), Φ r (s) is Φ u (Z u ), Φ v (Z v ), Φ r (Z r ) represent intermediate parameter variables, Θ u =max{||W u *T ||,|ε1+τ wu / m 11 |,1},Φ u (Z u )=||s u (Z u )||+|φ υ，1 ζ υ，1 z υ，1 |+1, Z u =[u,v,r,ρ υ，1 ,u e ] T Represents the input vector of the neural network NN model, s u (Z u )=[s u 1(Z u ),…,s un (Z u )] T Represents the radial basis function vector of the neural network NN model, Represents the weight vector of the neural network NN model; ε1 represents the approximation error; ζ υ，1 represents the design parameter; Θ r =max{||W r *T ||,|ε3+τ wr / m 33 |};Z r =[u,v,r,ρ υ，3 , r e ]T Represents the input vector of the neural network NN model; Φ r (Z r )=||s r (Z r )||+|φ υ，3 ζ υ，3 z υ，3 |+1,s r (Z r )=[s r1 (Z r ),…,s rn (Zr)] T is the radial basis function vector of the neural network NN, The weight vector of the neural network NN model; ε3 represents the approximation error; ζ υ，3 represents the design parameters;

[0262] S6: constructing a ship trajectory tracking controller according to the auxiliary dynamic filter, the additional control law and the adaptive parameter control law; and the ship trajectory tracking controller includes a ship longitudinal control law and a ship steering control law;

[0263] Specifically, the expression of the ship longitudinal control law is:

[0264]

[0265] The expression of the ship steering control law is:

[0266]

[0267] Substituting equations (67) and (68) into (63), we can obtain

[0268]

[0269] Consider the following Lyapunov function

[0270]

[0271] Where: and is the weight estimation error;

[0272] Taking the derivative of V3 and substituting equations (64) to (66) into equation (70), we can obtain

[0273]

[0274] According to Young's inequality, we can get:

[0275]

[0276]

[0277] Substituting equations (72) to (74) into (71), we can obtain

[0278]

[0279] According to Young's inequality, we can get:

[0280]

[0281] Substituting equations (76) to (78) into (75), we obtain

[0282]

[0283] In addition, the design parameters need to meet 3k3>k4, 3k5>k4;

[0284] S7: Implementing adaptive neural finite-time trajectory tracking control for underactuated ships with input saturation based on the ship trajectory tracking controller.

[0285] The stability analysis of this embodiment is as follows:

[0286] The specified performance trajectory tracking control scheme for underactuated ships can be summarized as the following theorem.

[0287] Theorem 1 Under Assumptions 1-3 and |e η，k (0)|<ρ η，i (0),|e v，i (0)|<ρ υ，i Under the initial condition of (0), the tracking control problem of underactuated ship is solved with unknown ship dynamics, unknown external disturbances and actuator input saturation. Aiming at the mathematical model (1)-(3) of nonlinear motion of underactuated ship, the predefined time performance function (14), virtual / intermediate control function (29), nonlinear transformation (18) and (22), adaptive law (64)-(66), auxiliary dynamic filter (37) and control law (44)-(46) of additional control signal are designed; at the same time, by selecting appropriate design parameters k1, k2, k3, k4, k5, a1, a2, a3, b u 、b v 、b r 、μ u 、μ r 、μ fu 、μ fv 、μ fr 、l u 、l v 、l r, forcing the ship to track along a given reference trajectory, and the trajectory tracking error can converge to a predetermined range within a predetermined time. At the same time, it ensures that all signals of the ship's closed-loop trajectory tracking control system are bounded.

[0288] Proof: Boundedness of all signals in underactuated ship tracking control system.

[0289] According to formula (79), we can get

[0290] 0≤V(t)≤d / c+[V(0)-d / c]e -ct (82)

[0291] Where V(0) is the initial value of V(t).

[0292] From equation (79), we can get that V(t) is uniformly ultimately bounded. Further, according to equations (31), (47), and (70), we can get that z η，1 、z η，2 、z η，3 、z v，1 、z v，2 、z v，3 、 e fu 、e fv and e fr is also bounded; in addition, according to and The boundedness and and It can be concluded that and is bounded; according to z η，1 、z η，2 、z η，3 Bounded, and Equation (18) shows that the position error x e 、y e , ψ e Bounded; according to z v，1 、z υ，2 、z υ，3 Bounded, and formula (22) can be obtained, the speed error u e 、v e 、r e Bounded; further, according to Assumption 2 and Equation (29), the virtual / intermediate control function α u , α v With α r is also bounded; then, according to the virtual / intermediate control function α u , α v With α r Bounded, e fu 、e fv and e frThe definition and boundedness of , the filtering form of the virtual control law γ u , γ v and γ r Bounded. According to u e 、v e 、r e , γ u , γ v and γ r The function tanh(·) is bounded, and Equation (13) shows that the ship's forward speed u and yaw angular velocity r are also bounded; due to the saturation constraint of the actuator and Δτ u =τ u -τ cu and Δτ r =τ r -τ cr , we can get Δτ u , Δτ r Bounded. According to Δτ u , Δτ r As the function cosh(·) is bounded, the additional control laws β1 and β3 are bounded. Furthermore, the control law τ in equations (67) and (68) is cu With τ cr is also bounded. Therefore, all signals in the closed-loop system are bounded. Furthermore, since z η，1 、z η，2 、z η，3 、z υ，1 、z υ，2 、z υ，3 Converges to a compact set According to formula (18), for |e η，i |<ρ η，i ,|e v，i |<ρ υ，i (i=1,2,3), inequality (16) holds. Therefore, due to the predefined function ρ η，i , ρ υ，i Characteristics, error e η，i 、e υ，i It can converge to a predefined residual set within a predefined time. The above analysis proves Theorem 1. The simulation analysis of this embodiment is as follows:

[0293] In order to verify the effectiveness of the ship trajectory tracking control law designed in this embodiment, CyberShip II is used as a test object for policy verification. The length of the ship is 1.255m, the width is 0.29m, the mass is 23.8kg, and the other parameters are m 11 =25.8kg, m 22 =33.8kg, m 33=2.760kg. d 11 =2.04995kg / s,d 22 =38.16752kg / s,d 23 =11.545kg / s,d 32 =-4.11775kg / s,d 33 =2.57kg / s. In order to show that the proposed control system can track both straight lines and curves, the reference speed is selected as follows: (1) 0≤t<40: u d =0.1, v d =0, r d =0; (2) 40≤t<80: u d =0.2, v d =0, r d =0; (3) 80≤t<120: u d =0.2, v d =0, r d =-0.1sin(πt / 20); (4)t≥120:u d =0.2, v d =0, r d = 0. The initial condition is η d =[0, 0, π / 4] T and η = [-1, -1, 0] T The actuator saturation limit is set to τ Mu =5N, τ Mr =3.5N. The number of nodes of the neural network basis function is l=20, and its node center μ i (i=1,…,20) are uniformly distributed on [-2,2]×[-2,2]×[-2,2]×[-2,2]×[-2,2], and all basis functions have width ω i (i=1,…,20) are all set to 1. The external environment disturbance is set to τ wu =0.75+sin(0.02t)+0.75sin(0.1t), τ wr =-0.5sin(0.09t+π / 3)-2sin(0.01t)τ wv =-0.45+sin(0.02t-π / 6)+0.75sin(0.03t). Other system parameters are set as follows: k1=0.1, k2=2.5, μ fu =μ fv =μ fr =0.5, a1=a2=a3=0.9, μ u =μ r =1.8, k3=k4=k5=0.001, b u =b v =br =0.006, l u =l r =0.0001, l v =0.0018, T fη，1 =T fη，2 =T fη，3 =13, T fυ，1 =T fυ，3 =12, T fυ，2 =13,

[0294] The simulation results under the control strategy proposed in this embodiment are as follows Figures 3 to 9 shown. Figure 3 Displays the tracking control performance in the xy plane, Figure 4 Displays the tracking performance of the position (x, y, ψ). Figure 3-4 It shows that the pre-specified time scheme proposed in this embodiment can force the underactuated ship to follow the reference trajectory x with satisfactory control performance. d and y d . Figure 5 The actual control input τ is plotted u and τ r The time curve of the control law τ u and τ r The calculated control instructions are bounded and reasonable. Figure 6 The ship tracking speed duration curve under the proposed control strategy is given. It can be seen from the figure that the ship speed is bounded. Figure 7-8 The tracking error curve is plotted. It can be seen from the figure that at the specified time T fη and T fυ Position error of the inner ship x e 、y e , ψ e and the speed error u e 、v e 、r e All of them can converge to the specified tracking accuracy range, and the results show that the control strategy designed in this embodiment can achieve offline presetting of tracking stability adjustment time and control accuracy by the user. Figure 9 It can be seen that the adaptive law The above simulation results show that under the proposed control strategy with preset performance, all signals in the entire closed-loop trajectory tracking control system of the ship are uniformly and ultimately bounded, and the offline predefined convergence speed and control accuracy of the underactuated ship tracking error can be achieved.

[0295] Compared with the prior art, the overall beneficial effects of the present invention are:

[0296] The control scheme proposed in this embodiment ensures offline custom design of tracking convergence time, position error, and velocity error, truly achieving finite-time convergence of the closed-loop control system. This scheme not only highlights the advantages of fast finite-time convergence and strong robustness, but also achieves pre-specified convergence time for under-actuated vessels, compared to schemes applicable only to fully-driven vessels.

[0297] This embodiment uses neural network technology and a single-parameter learning algorithm to overcome the compound uncertainty (both external and internal) caused by external environmental disturbances, perturbations of ship model parameters, and unmodeled dynamics. Compared to adaptive neural network methods, this solution only requires updating a single unknown parameter, reducing the computational burden on the system.

[0298] This embodiment uses additional control and an auxiliary dynamic filter to address the "hard" nonlinearity caused by actuator under-saturation and saturation. Compared to the dynamic auxiliary system approach, this solution avoids differential inflation by designing a virtual control law through the auxiliary dynamic filter. Furthermore, this solution does not require precise knowledge of the velocity.

[0299] The control objective of this embodiment is to design an adaptive neural network trajectory tracking control law τ for underactuated ships, taking into account internal and external disturbances and actuator saturation conditions. u and τ r , so that the ship follows the reference trajectory and all signals in the closed-loop tracking control system are consistent and ultimately bounded. The position error and velocity error in the tracking control reach the specified range within the user-defined convergence time.

[0300] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements 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. An adaptive neural specified time trajectory tracking control method for an underactuated ship with input saturation, characterized in that: The following steps are involved: S1: Establish a mathematical model of a three-degree-of-freedom underactuated MSV with a control input saturation function; The established mathematical model of the three-degree-of-freedom underactuated MSV with a control input saturation function includes an underactuated ship kinematic equation, an underactuated ship dynamic equation, and a control input saturation function; The expression of the kinematic equation of the underactuated ship is: Where: x, y, ψ represent the longitudinal and transverse positions and yaw angle of the under-actuated ship in the geodetic / inertial coordinate system respectively; u, v, r represent the longitudinal, transverse speed and turning speed of the under-actuated ship in the appendage / hull coordinate system respectively; The expression of the underactuated ship dynamic equation is: f u (υ)=m 22 vr-d 11 you f v (υ)=(-m 11 u-d 23 )r-d 22 v f r (υ)=(m 11 -m 22 )uv-d 32 v-d 33 r Where: υ=[uvr] T They represent the longitudinal, transverse and turning speeds of the underactuated ship in the appendage / hull coordinate system respectively; f = [f u f v f r ] T Represents the nonlinear dynamic terms of the ship model in the longitudinal, transverse speed and steering direction; τ=[τ u 0 τ r ] T And τ u With τ r denote the longitudinal and steering control inputs of the underactuated ship respectively; τ w =[τ wu τ wv τ wr ] T And τ wu ,τ wv ,τ wr Respectively represent the external unknown time-varying environmental disturbances suffered in the longitudinal, lateral and steering directions; m 11 ,m 22 ,m 33 Represents the parameters in the inertia matrix; d 11 ,d 22 ,d 23 ,d 32 ,d 33 Represents the parameters in the nonlinear hydrodynamic damping matrix of the underactuated ship system; The expression of the control input saturation function is: Where: τ Mj represents the maximum control force or torque provided by the underactuated ship propulsion system, and τ Mj >0, j=u,r,τ cj Represents the control input instruction, and the intermediate variable Δτ u =τ u -τ cu ,Δτ r =τ r -τ cr ; S2: Based on the mathematical model of the three-degree-of-freedom underactuated MSV, the tracking error variable and the velocity error variable are defined according to the preset desired trajectory; The expression of the tracking error variable defined in S2 is Where: x e ,y e ,ψ e Respectively represent the lateral position error, longitudinal position error and heading angle error; x d ,y d ,ψ d Respectively represent the lateral position, longitudinal position and yaw angle of the preset desired trajectory; The expression of the speed error variable is: Where: a1, a2, a3 are constants; γ=[γ u ,γ v ,γ r ] T The virtual control law α=[α u ,α v ,α r ] T The filtering form of tanh(·) represents the function used to constrain the additional control signals β1, β2, β3; u e ,v e ,r e They represent the longitudinal velocity error, lateral velocity error, and steering velocity error of the underactuated ship respectively; S3: constructing a predefined / specified time performance function, obtaining preset performance boundary conditions on the tracking error variable and the speed error variable, and based on the preset performance function boundary conditions, performing nonlinear transformation on the tracking error variable and the speed error variable according to the predefined / specified time performance function to obtain a position error conversion variable and a speed error conversion variable; The specific steps include: S31: Construct a predefined / specified time performance function, whose expression is Where: represents the design parameters; T f Indicates a custom predefined convergence time; Indicates the custom convergence control accuracy, that is, time t is greater than or equal to T f When the system error converges to range; and ρ(t) satisfies ρ(0) represents the initial value of the predefined / specified time performance function; S32: When When , the tracking error is defined as e η =η-η d ; among them η =[e η,1 ,e η,2 ,e η,3 ] T =[x e ,y e ,ψ e ] T ,η=[x,y,ψ] T ,or d =[x d ,y d ,ψ d ] T ; Define the speed error as e υ , where e υ =[e υ,1 ,e υ,2 ,e υ,3 ] T =[u e ,v e ,r e ] T ; Then, a preset performance boundary condition on the tracking error variable and the speed error variable is obtained according to a predefined / specified time performance function; The preset performance boundary condition of the tracking error variable is -r η,1 <x e <r η,1 -r η,2 <y e <r η,2 -r η,3 <ψ e <r η,3 Where: ρ η,i =ρ(t) and i=1,2,3; The preset performance boundary condition of the speed error variable is -r υ,1 <u e <r υ,1 -r υ,2 <v e <r υ,2 -r υ,3 <r e <r υ,3 Where: ρ υ,i =ρ(t) and i=1,2,3; S33: performing nonlinear transformation on the tracking error variable and the speed error variable according to a predefined / specified time performance function to obtain a position error conversion variable and a speed error conversion variable; The expression of the position error conversion variable is: Where: e η,i (i=1,2,3) represents the position error e η The i-th element of η,i (i=1,2,3) represents the position error conversion variable and satisfies The expression of the speed error conversion variable is: Where: e υ,i (i=1,2,3) represents the speed error e υ The i-th element of υ,i (i=1,2,3) represents the speed error conversion variable and satisfies S4: Construct a virtual control law for trajectory tracking based on the tracking error conversion variable, and design an auxiliary dynamic filter based on the virtual control law to avoid differential explosion caused by its derivative; The specific steps include: S41: Construct a virtual control law for trajectory tracking based on the position error conversion variable, which is expressed as And the vector expression of the trajectory tracking virtual control law is In the formula: A=diag(a1,a2,a3),β=[tanhβ1,tanhβ2,tanhβ3] T ; k = diag[k1,k1,k2], k1 and k2 are constants; Ψ η =diag[Ψ η,1 ,Ψ η,2 ,Ψ η,3 ],φ η =diag[φ η,1 ,φ η,2 ,φ η,3 ],ρ η =diag[ρ η,1 ,ρ η,2 ,ρ η,3 ];Ψ η,i ,φ η,i Represents the intermediate parameter variable, i=1,2,3 and α ι ,ι=u,v,r represent the longitudinal speed virtual control law, lateral speed virtual control law and steering speed virtual control law respectively; represents η d The first derivative of ; J(ψ) represents the rotation matrix that transforms the information between the hull coordinate system and the earth coordinate system; S42: According to the virtual control law, an auxiliary dynamic filter is designed to avoid differential explosion caused by derivation of the virtual control law, and the expression of the auxiliary dynamic filter is: Where: γ represents the first-order auxiliary dynamic filter; represents the time constant; γ ι (0) represents the first-order auxiliary dynamic filter γ ι The initial value of α ι (0) represents the virtual control law α ι The initial value of And define the filtering error as The derivative of the first-order auxiliary dynamic filter γ is recorded according to the filtering error S5: Based on the mathematical model of the three-degree-of-freedom underactuated MSV and the speed error conversion variable, the additional control law and adaptive parameter control law of the ship trajectory tracking controller are constructed; The expression of the additional control law of the ship trajectory tracking controller is: Where: μ u ,μ r , k4 are all constants; β1, β3 represent virtual control signals that cannot be realized due to the actuator saturation constraint effect; β2 represents the additional control signal used to provide lateral stability to solve the under-actuation problem under saturation conditions; Represent the first-order derivatives of β1, β2, and β3 respectively; Ψ υ,i ,φ υ,i Represents the intermediate parameter variable, i=1,2,3 and Θ v represents the adaptive parameters of the ship trajectory tracking controller, and Represents Θ v The estimated value of Φ v (s) is Φ v (Z v ) represents the intermediate parameter variable, and Φ v (Z v )=||s v (Z v )||+|φ υ,2 ζ υ,2 z υ,2 |+1;Z v =[u,v,r,ρ υ,2 ,v e ] T Represents the input vector of the neural network NN model; s v (Z v )=[s v1 (Z v ),…,s vn (Z v )] T Represents the radial basis function vector of the neural network NN model; Represents the weight vector of the neural network NN model; ζ υ,2 represents the design parameter; ε2 represents the approximation error; The expression of the adaptive parameter control law of the ship trajectory tracking controller is: Where: Θ u ,Θ v ,Θ r represents the adaptive parameters of the ship trajectory tracking controller; Represent Θ respectively u ,Θ v ,Θ r estimated value of; Respectively The first derivative of b u ,b v ,b r ,l u ,l v ,l r Both represent design constants; Φ v (s),Φ u (s),Φ r (s) is Φ v (Z v ),Φ u (Z u ),Φ r (Z r ) represent intermediate parameter variables, Φ u (Z u )=||s u (Z u )||+|φ υ,1 ζ υ,1 z υ,1 |+1, Z u =[u,v,r,ρ υ,1 ,u e ] T Represents the input vector of the neural network NN model, s u (Z u )=[s u1 (Z u ),…,s un (Z u )] T Represents the radial basis function vector of the neural network NN model, Represents the weight vector of the neural network NN model; ε1 represents the approximation error; ζ υ,1 represents the design parameters; Z r =[u,v,r,ρ υ,3 ,r e ] T Represents the input vector of the neural network NN model; Φ r (Z r )=||s r (Z r )||+|φ υ,3 ζ υ,3 z υ,3 |+1,s r (Z r )=[s r1 (Z r ),…,s rn (Z r )] T is the radial basis function vector of the neural network NN, The weight vector of the neural network NN model; ε3 represents the approximation error; ζ υ,3 represents the design parameters; S6: constructing a ship trajectory tracking controller according to the auxiliary dynamic filter, the additional control law, and the adaptive parameter control law; wherein the ship trajectory tracking controller includes a ship longitudinal control law and a ship steering control law; The expression of the ship longitudinal control law is: The expression of the ship steering control law is: Where: k3, k5 represent design parameters; S7: Implementing adaptive neural specified time trajectory tracking control for underactuated ships with input saturation based on the ship trajectory tracking controller.

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