Fixed-time ship trajectory tracking control method considering state limitation

Abstract

The invention discloses a fixed time ship trajectory tracking control method considering state limitation. The method comprises the following steps: S1, establishing an under-actuated ship mathematical model considering input saturation; s2, limiting the actual position of the ship based on the set first obstacle Lyapunov function; limiting the expected position of the ship based on a set second obstacle Lyapunov function; s3, defining a ship position tracking error, and designing a longitudinal expected speed and a transverse expected speed which can enable the ship position tracking error to converge; s4, defining a ship speed tracking error, and designing a first fixed time sliding mode surface and a second fixed time sliding mode surface according to the ship speed tracking error; and S5, designing a longitudinal control force and a yawing control moment which can enable the first fixed time sliding mode surface and the second fixed time sliding mode surface to converge so as to enable the ship speed tracking error to realize fixed time convergence and solve the problem that the convergence rate of the speed tracking error is low.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of ship trajectory control, and particularly relates to a fixed-time ship trajectory tracking control method considering state constraints. BACKGROUND

[0002] In recent years, with the continuous development of technology, the traffic density on the sea is increasing, which brings great challenges to navigation safety. The existing ship trajectory tracking control method has the following problems: (1) The preset performance control method is adopted, the tracking error is limited, the actual trajectory converges near the expected trajectory, but since the expected trajectory cannot be obtained in advance, the use of the preset performance control method is limited, and the safety during navigation cannot be ensured; (2) The increase of error convergence rate will inevitably be accompanied by the increase of control input, but since the controller has a saturation problem, the control input will have a certain limit, so how to achieve faster convergence speed under the condition that the control input has a limit is a technical problem. SUMMARY

[0003] The present application provides a fixed-time ship trajectory tracking control method considering state constraints to overcome the above technical problems.

[0004] In order to achieve the above purpose, the technical scheme of the present application is: A fixed-time ship trajectory tracking control method considering state constraints, the specific steps comprising: S1. Establishing an under-actuated ship mathematical model considering input saturation; S2. Setting a first barrier Lyapunov function and a second barrier Lyapunov function for limiting the state of the ship, and limiting the actual position of the ship in the under-actuated ship mathematical model based on the set first barrier Lyapunov function to obtain the limited actual position of the ship; limiting the set expected position of the ship based on the set second barrier Lyapunov function to obtain the limited expected position of the ship; S3. Defining a ship position tracking error based on the limited actual position of the ship and the limited expected position of the ship, and designing a longitudinal expected speed and a lateral expected speed that can make the ship position tracking error converge; S4. Defining a ship speed tracking error based on the actual speed of the ship in the under-actuated ship mathematical model, the longitudinal expected speed, the lateral expected speed, and the defined first and second finite-time auxiliary system variables, and designing a first fixed-time sliding mode surface and a second fixed-time sliding mode surface according to the ship speed tracking error; S5. Design the longitudinal control force and yaw control moment that can make the first fixed time sliding mode surface and the second fixed time sliding mode surface converge to realize the ship trajectory tracking.

[0005] Further, the established underactuated ship mathematical model considering input saturation is as follows: (1) wherein (2) wherein, and are the forward displacement, lateral displacement and heading angle of the ship, respectively; are the actual longitudinal velocity, actual lateral velocity and yaw angular velocity of the ship in the body coordinate system, respectively; and are the saturation-limited longitudinal control force and yaw control moment, respectively; is the inertia parameter; is the hydrodynamic parameter; , , is the unknown term of the ship; is the external unknown disturbance; is the upper bound of the control input; is the lower bound of the control input; is the actual control input.

[0006] Further, in S2, based on the set first barrier Lyapunov function, the actual longitudinal position and the lateral position of the underactuated ship mathematical model are limited, respectively, to obtain the limited actual longitudinal position and the lateral position , and the specific expression is: (3) wherein, and are the minimum and maximum longitudinal positions allowed by the ship, respectively, and are the minimum and maximum lateral positions allowed by the ship, respectively; based on the set second barrier Lyapunov function, the set expected longitudinal position and the lateral position are limited, respectively, to obtain the limited expected longitudinal position and the lateral position , and the specific expression is: (4).

[0007] Further, in S3, the ship position tracking error defined based on the limited ship actual position and the limited ship desired position includes a ship tracking longitudinal error and a lateral error , respectively, expressed as: (5) The longitudinal desired velocity and the lateral desired velocity are designed to achieve convergence, including: Assuming that and can converge, then and must be bounded, so the ship actual position , satisfies the following formula: (6) Therefore, the derivative of formula (5) is obtained as: (7) Assuming that the desired trajectory of the ship is smooth and has first and second derivatives, then the expressions of and are as follows: (8) According to formula (7), in order to make and converge, the first derivative of the ship actual position and is designed as: (9) In the formula, and are positive parameters to be designed; Combining formula (1) and formula (9), we get: (10) Define and as shown in formula (11): (11) According to formula (10) and (11), the desired longitudinal velocity and the desired lateral velocity are designed as: (12).

[0008] ​​Further, in S4, the ship speed tracking error is defined based on the actual ship speed, the longitudinal desired speed, the lateral desired speed, and the defined first and second finite-time auxiliary system variables in the underactuated ship mathematical model. The design of the first fixed-time sliding surface and the second fixed-time sliding surface based on the ship speed tracking error includes: For the desired longitudinal velocity and expected lateral velocity Taking the derivative and simplifying, we get: (13) in, and They are respectively and The first derivative; Define the longitudinal speed tracking error in ship speed tracking error. The expression is: (14) In the formula, As the first finite-time auxiliary variable; This refers to the ship's actual longitudinal speed; Based on ship longitudinal velocity tracking error The first fixed-time sliding surface of the design for: (15) In the formula, , and The positive parameters to be designed, and ; Define the ship's lateral speed tracking error in the ship speed tracking error. The expression is: (16) In the formula, It is the second finite-time auxiliary variable; This refers to the ship's actual lateral speed; Based on ship lateral velocity tracking error The second fixed-time sliding surface of the design for: (17) In the formula, , and These are the positive parameters that need to be designed.

[0009] Furthermore, in S5, the longitudinal control force and yaw control torque designed to enable the convergence of the first fixed-time sliding surface and the second fixed-time sliding surface include: S51. Design the longitudinal control force that can make the first fixed time sliding mode surface converge, including: Differentiate equation (15) and combine equation (1) and (14) to get: (18) Let (19) Use the first finite time auxiliary variable Convert equation (18) to get: (20) Use RBF neural network to estimate the unknown term The neural network output expression of the unknown term is: (21) In the formula, is the weight matrix of , is the Gaussian function used to estimate , is the error of estimating ; Assume that the external disturbance on the ship is a time-varying disturbance, bounded and unknown limit, and there is a first-order derivative, and combine equation (20) and (21) to design the longitudinal control force as follows: (22) In the formula, , and are positive design parameters; , is the upper bound of the composite disturbance , is the estimated value of ; At the same time, the adaptive law of design parameters and is shown in equation (23): (23) In the formula, , , , , and are positive design parameters; S52. Design the yaw control moment that can make the second fixed time sliding mode surface converge, including: Let (24) Then, combining with equation (13), we have whose derivative is given by equation (25): (25) Taking the derivative of equation (17) and combining with equations (1), (16) and (25), we have (26) where (27) Let (28) Using the second finite-time auxiliary variable Transforming equation (26), we have (29) Using the RBF neural network to estimate the unknown term , whose specific expression is (30) where is the weight matrix of , is the estimation of , is the Gaussian function used, is the error of the estimation of ; Assuming that the external disturbance experienced by the ship is time-varying, bounded and unknown, and has a first-order derivative, and combining equations (29) and (30), the yaw control moment is designed as follows: (31) where , and are positive design parameters, , is the upper bound of the compound disturbance , and is the estimated value of ; At the same time, the adaptive laws of design parameters and are given by equation (32): (32) where , , , ,​ and design parameters are positive, and respectively and estimated values.

[0010] Beneficial effects: Firstly, the first barrier Lyapunov function is used to limit the actual position of the ship in the mathematical model of the underactuated ship, the second barrier Lyapunov function is used to limit the set expected position of the ship, and the longitudinal expected speed and the lateral expected speed capable of making the ship position tracking error converge are designed to ensure the safety of sailing when the trajectory is unknown; secondly, the ship speed tracking error is defined, the first and second fixed time sliding mode surfaces are used, and the longitudinal control force and the bow shaking control moment are designed to make the ship speed tracking error achieve global fixed time convergence, and the problem of slow speed tracking error convergence rate is solved. BRIEF DESCRIPTION OF DRAWINGS

[0011] In order to more clearly illustrate the technical solutions of the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiment or prior art description. Obviously, the drawings in the following description are some embodiments of the present application, and other drawings can also be obtained by those skilled in the art without creative labor.

[0012] Figure 1 is the flow chart of the fixed time ship trajectory tracking control method considering state constraints in the present application; Figure 2 is the basic framework diagram of the fixed time ship trajectory tracking control method considering state constraints in the present application; Figure 3 is the expected trajectory and actual trajectory comparison diagram of the ship in the plane in the embodiment of the present application; Figure 4 is the actual longitudinal position curve diagram of the ship in the embodiment of the present application; Figure 5 is the actual lateral position curve diagram of the ship in the embodiment of the present application; Figure 6 is the limited ship longitudinal position tracking error curve diagram in the embodiment of the present application; Figure 7 is the limited ship lateral position tracking error curve diagram in the embodiment of the present application; Figure 8 is the ship longitudinal speed tracking error curve diagram in the embodiment of the present application; Figure 9 is the ship lateral speed tracking error curve diagram in the embodiment of the present application; Figure 10A ship longitudinal control force input curve diagram in the embodiment of the present application; Figure 11 A ship yaw control torque input curve diagram in the embodiment of the present application; Figure 12 A neural network estimation curve diagram in the embodiment of the present application; Figure 13 A composite disturbance upper bound estimation curve diagram in the embodiment of the present application; Figure 14 A finite time auxiliary system convergence curve diagram in the embodiment of the present application. DETAILED DESCRIPTION

[0013] To make the objects, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0014] The embodiment provides a fixed-time ship trajectory tracking control method considering state limitation, and specific steps include: Figure 1 and Figure 2 as shown, and the specific steps include: S1. establishing an under-actuated ship mathematical model considering input saturation; In specific embodiments, the established under-actuated ship mathematical model considering input saturation is in the following form: (1) wherein (2) In the formula, and are a forward displacement, a transverse drift displacement and a bow angle of the ship respectively; are an actual longitudinal velocity, an actual transverse velocity and a yaw angle velocity of the ship in an appendage coordinate system respectively; and are a saturated longitudinal control force and a saturated yaw control torque respectively; is an inertia parameter; is a hydrodynamic parameter; are unknown terms of the ship; is an unknown disturbance from outside; is an upper bound of a control input; is a lower bound of the control input; is an actual control input. ​​

[0015] This embodiment refers to the following lemma: Lemma 1: For any > 0, there exists the following inequality: (3) In the formula, is any variable, is any number.

[0016] Lemma 2: For any continuous smooth function defined on a compact set , there exists: (4) In the formula, is the approximation error, for all , there exists a vector > 0, and satisfies ; is the weight vector under ideal conditions, which is usually not easy to obtain, is an unknown variable, so it needs to be estimated; is the transpose of ;is a Gaussian function, the expression is as shown in formula (5): (5) In the formula: is the number of network nodes, ; is the input center vector; is the standard deviation of the Gaussian function. Lemma 3: Consider a dynamic system ,

[0017] where is the state of the dynamic system, and , is the equilibrium point of the dynamic system, if there exists a Lyapunov function satisfying formula (6), the dynamic system is practically fixed-time stable, and the stable time satisfies ; Where the constant , , , , , ; ; (6) In the formula: is the power of the Lyapunov function j ; is the Lyapunov function of q ; Lemma 4: If variables and , and constant , , , and , then the following inequality exists: (7).

[0018] S2. Set a first barrier Lyapunov function and a second barrier Lyapunov function for limiting the state of the ship, and limit the actual position of the ship in the underactuated ship mathematical model based on the set first barrier Lyapunov function to obtain a limited actual position of the ship; limit the set expected position of the ship based on the set second barrier Lyapunov function to obtain a limited expected position of the ship; In specific embodiments, in S2, the actual longitudinal position and the actual lateral position of the ship in the underactuated ship mathematical model are respectively limited based on the set first barrier Lyapunov function to obtain a limited actual longitudinal position and a limited actual lateral position , and the specific expressions are as follows: (8) In the formula, and are the minimum and maximum longitudinal positions allowed by the ship, and are the minimum and maximum lateral positions allowed by the ship; The set expected longitudinal position and the set expected lateral position of the ship are respectively limited based on the set second barrier Lyapunov function to obtain a limited expected longitudinal position and a limited expected lateral position , and the specific expressions are as follows: (9).

[0019] S3. Define a ship position tracking error based on the limited actual position of the ship and the limited expected position of the ship, and design a longitudinal expected speed and a lateral expected speed that can make the ship position tracking error converge; In specific embodiments, in S3, the ship position tracking error defined based on the limited actual position of the ship and the limited expected position of the ship includes a ship tracking longitudinal error and a lateral error , and is respectively expressed as: (10) Designing the ship position tracking error and Implementing convergent longitudinal and lateral desired velocities and , comprising: Assuming and can converge, then and must be bounded, so the ship actual position , satisfies the following equation: (11) Therefore, the derivative of equation (10) is obtained as: (12) Assuming the desired trajectory of the ship is smooth and has first and second derivatives, then the expressions of and are shown in equation (13): (13) According to equation (12), in order to make and converge, the first derivative of the ship actual position and is designed as: (14) wherein and are positive parameters to be designed; Combining equation (1) and equation (14), we have: (15) For convenience of description, define and as shown in equation (16): (16) According to equation (15) and (16), the desired longitudinal velocity and the desired lateral velocity are designed as: (17).

[0020] Specifically, when tends to , tends to , equation (12) can be simplified as: (18) Define Lyapunov function As shown in equation (19): (19) Derive equation (19) and substitute equation (18) to obtain: (20) Wherein, (21) According to equation (20), if converges to and converges to , then and can achieve convergence, which shows that the method proposed in the embodiment can achieve the limitation of the actual position and of the ship.

[0021] S4. Define the ship speed tracking error based on the actual speed of the underactuated ship in the mathematical model of the ship, the longitudinal desired speed, the lateral desired speed, and the defined first and second finite-time auxiliary system variables, and design the first fixed-time sliding mode surface and the second fixed-time sliding mode surface according to the ship speed tracking error; In specific embodiments, in S4, defining the ship speed tracking error based on the actual speed of the underactuated ship in the mathematical model of the ship, the longitudinal desired speed, the lateral desired speed, and the defined first and second finite-time auxiliary system variables, and designing the first fixed-time sliding mode surface and the second fixed-time sliding mode surface according to the ship speed tracking error includes: Derive the desired longitudinal speed and the desired lateral speed and simplify to obtain: (22) Wherein, and are the first derivatives of and respectively; In order to make the input in the design process of the ship controller not be limited, the first finite-time auxiliary variable is introduced in the embodiment, and the ship longitudinal speed tracking error in the ship speed tracking error is defined based on , and the expression is: (23) Based on the ship longitudinal speed tracking error a first fixed-time sliding mode surface designed is: (24) wherein, , and is a positive parameter to be designed, and ; define a ship lateral velocity tracking error in the ship velocity tracking error , the expression is: (25) based on the ship lateral velocity tracking error a second fixed-time sliding mode surface designed is: (26) wherein, , and is a positive parameter to be designed.

[0022] S5. Design the longitudinal control force and the yaw control moment capable of making the first fixed-time sliding mode surface and the second fixed-time sliding mode surface converge to realize the ship trajectory tracking.

[0023] In this embodiment, the longitudinal control force and the yaw control moment are designed in consideration of input saturation to make the actual speed of the ship tend to the desired speed, realizing the ship trajectory tracking. Specifically, in S5, the longitudinal control force and the yaw control moment capable of making the first fixed-time sliding mode surface and the second fixed-time sliding mode surface converge include: S51. Design the longitudinal control force capable of making the first fixed-time sliding mode surface converge, including: Differentiate formula (24) and combine formula (1) and (23) to obtain: (27) Let (28) use the first finite-time auxiliary variable to transform formula (27) to obtain: (29) Since the RBF neural network has a universal approximation function for any smooth function, the RBF neural network is used to estimate the unknown term in this embodiment. According to Lemma 2, the neural network output expression of the unknown term is as follows: (30) wherein, is a weight matrix, is an estimation of the Gaussian function used, is an error in estimating ; Assuming that the external disturbance experienced by the ship is time-varying, bounded and unknown, and has a first-order derivative, the longitudinal control force is designed as follows: (31) wherein, , and are positive design parameters; , is an upper bound of the compound disturbance , is an estimation value of ; At the same time, the adaptive laws of the design parameters and are shown as follows: (32) wherein, , , , , and are positive design parameters; S52. Design a bow yaw control moment capable of converging the second fixed-time sliding mode surface, comprising: For the convenience of subsequent writing, let (33) Then, combined with equation (22), we have , and its derivative is shown as follows: (34) Differentiate equation (26) and combine equations (1), (25) and (34) to obtain: (35) wherein, (36) In order to avoid the problem of "derivative explosion" caused by directly differentiating , a dynamic surface is designed to replace the differentiation process of , i.e., using ​replace The expression is as follows: (37) In the formula, For design parameters, and >0; make (38) In the formula, and These are the first derivatives of the first and second finite-time auxiliary variables, respectively; Using a second finite-time auxiliary variable Equation (35) is transformed to produce an unrestricted bow roll control torque. ,get: (39) Using RBF neural networks to analyze unknown terms The estimation is performed, and its specific expression is: (40) In the formula, for The weight matrix, For estimation The Gaussian function used To The error in the estimation; Assuming the external disturbances experienced by the ship The disturbance is time-varying, bounded, and its limit is unknown. It also has a first derivative. Therefore, according to equations (39) and (40), the yaw control torque is... The design is as follows: (41) In the formula, , and All are positive design parameters. , For composite disturbance The upper realm, for The estimated value; Meanwhile, design parameters and The adaptive law is shown in equation (42): (42) In the formula, , , , , and design parameters are positive, and are the estimated values of and respectively.

[0024] Specifically, the state-constrained fixed-time ship trajectory tracking control method proposed in the present application can effectively solve the problem of safe navigation with unknown desired trajectory and accelerate the convergence rate of error. Through Lyapunov theory, it is proved that all signals of the ship system are bounded, and the actual position of the ship is always within the limited range. The designed finite-time auxiliary system can effectively handle the nonlinear terms caused by input saturation, and by combining the fixed-time convergence theory to design the longitudinal control force and the yaw control moment, the speed tracking error can achieve fixed-time convergence in the approaching stage and the sliding stage.

[0025] In specific embodiments, in order to verify the stability of the ship controller designed in the present application, i.e. the longitudinal control force and the yaw control moment, the Lyapunov function shown in the following formula is defined: (43) In the formula, , , , ; Taking the derivative of both sides of formula (43) can obtain: (44) For the first term in formula (44), combined with formula (29) and (31), it can be obtained that: (45) For the second term in formula (44), combined with formula (39) and (41), it can be converted into the following form: (46) Combined with formula (45) and (46), formula (44) can be converted into the following formula: (47) According to Lemma 4, the in formula (47) can be converted into the following form: (48) Considering that the following inequality exists: (49) Combined with Young's inequality and formula (48), the in formula (47) can be converted into: (50) Combining formula (48), (50), Lemma 1 and Lemma 4, formula (47) is changed into: (51) wherein, (52) (53) (54) According to formula (51) and Lemma 3, the variable in can achieve fixed-time convergence in the approaching stage; when After achieving convergence in the approaching stage, since the designed first fixed-time sliding mode surface and the second fixed-time sliding mode surface have the characteristics of fixed-time convergence, the ship speed tracking error and can also achieve convergence in fixed time, so the ship speed tracking error and can achieve fixed-time convergence in the approaching stage and the sliding stage.

[0026] Specifically, in order to verify the effectiveness of the control method proposed in the present application, a computer simulation experiment is carried out on a patrol ship named BAY CLASS of Singapore Customs as a simulation object, the ship is 38 m long, the mass is kg, and other parameters are kg, kg, kg, kg / s, kg / s, kg / s.

[0027] In order to simulate the sea state, the external time-varying disturbance adopts formula (55): (55) The parameters are set as , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ; unknown term is set as , , ; set upper limit of input , ; neural network center vector selection and , 41 nodes are set; standard deviation of Gaussian function is set as and ; set sinusoidal desired trajectory as , ; select initial position and velocity of ship as .

[0028] Specifically, simulation results are shown in Figures 3 to 14 . It can be seen from Figure 3 that the control algorithm proposed in the application can make the ship complete trajectory tracking in the presence of disturbance, unknown term of model and input saturation, thereby verifying the effectiveness of the control algorithm. It can be seen from Figure 4 and Figure 5 that the actual longitudinal position and lateral position of the ship can always be within the range of the pre-limited range, ensuring the safety during navigation. Figure 6 and Figure 7 show the difference between the limited actual position and the limited desired position, from which it can be seen that although there is fluctuation, the ship tracking error can still converge at a relatively fast rate. Figure 8 and Figure 9 show the speed tracking error curve, from which it can be seen that the speed tracking error can quickly converge to the vicinity of 0 after fluctuation, showing the advantage of fixed time convergence. It can be seen from Figure 10 and Figure 11 that the control input is always within the pre-set limit, showing the limiting effect of the control algorithm proposed in the application on the control input. It can be found from Figure 12 and Figure 13 that the RBF neural network designed in the application can better approximate the unknown term of the model, and the adaptive law used can better estimate the upper bound of the compound disturbance. From Figure 14It can be seen that the finite time auxiliary system variable can realize convergence in finite time. In summary, the simulation experiment can verify the effectiveness and effect of the control algorithm.

[0029] Through the above comparative simulation experiment, the beneficial effects brought by the present application are summarized as follows: (1) For the problem of ship navigation safety and unknown desired trajectory, the first and second barrier Lyapunov functions are used to limit the actual position of the ship and the desired position of the ship respectively, the longitudinal desired speed and the lateral desired speed are designed to ensure that the barrier Lyapunov function is bounded, and the purpose of limiting the actual position is achieved; (2) For the problem of slow convergence rate, the first and second fixed time sliding mode surfaces are used, and the longitudinal control force and the yaw control moment are designed by combining the fixed time convergence theory, so that the ship speed tracking error realizes global fixed time convergence; at the same time, in the face of input saturation problem, the first and second finite time auxiliary system variables are designed to solve the nonlinear term caused by input saturation, and the finite time convergence is realized. (3) For the problem of unknown external disturbance and model unknown term, the neural network is used to estimate the model unknown term, the adaptive technology is used to estimate the upper bound of the composite disturbance composed of external disturbance and neural network estimation error, and the estimated value is fed back to the design process of longitudinal control force and yaw control moment to compensate the influence of disturbance.

[0030] Finally, it should be pointed out that: the above embodiments are only used to illustrate the technical solutions of the present application, and not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement to part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A fixed-time ship trajectory tracking control method considering state constraints, characterized in that, The specific steps include: S1. establishing an underactuated ship mathematical model considering input saturation; S2. setting a first barrier Lyapunov function and a second barrier Lyapunov function for limiting the state of the ship, and limiting the actual position of the ship in the underactuated ship mathematical model based on the set first barrier Lyapunov function to obtain a limited actual position of the ship; limiting the set expected position of the ship based on the set second barrier Lyapunov function to obtain a limited expected position of the ship; S3. defining a ship position tracking error based on the limited actual position of the ship and the limited expected position of the ship, and designing a longitudinal expected speed and a lateral expected speed that can make the ship position tracking error converge; S4. defining a ship speed tracking error based on the actual speed of the ship in the underactuated ship mathematical model, the longitudinal expected speed, the lateral expected speed, and the defined first and second finite-time auxiliary system variables, and designing a first fixed-time sliding mode surface and a second fixed-time sliding mode surface according to the ship speed tracking error; S5. designing a longitudinal control force and a yaw control moment that can make the first fixed-time sliding mode surface and the second fixed-time sliding mode surface converge to achieve ship trajectory tracking.

2. The fixed-time ship trajectory tracking control method with state constraints according to claim 1, wherein, The established underactuated ship mathematical model considering input saturation is as follows: (1) wherein (2) wherein, and are the forward displacement, the lateral drift displacement and the heading angle of the ship, respectively; are the actual longitudinal velocity, the actual lateral velocity and the yaw angular velocity of the ship in the body-fixed coordinate system, respectively; and are the saturation-limited longitudinal control force and the yaw control moment, respectively; is the inertia parameter; is the hydrodynamic parameter; , , are the unknown terms of the ship; is the external unknown disturbance; is the upper bound of the control input; is the lower bound of the control input; is the actual control input.

3. The fixed-time ship trajectory tracking control method with consideration of state constraints according to claim 2, characterized in that, In S2, the actual longitudinal position of the ship in the mathematical model of the underactuated ship is determined based on the Lyapunov function of the first obstacle. and horizontal position Each restriction is applied separately to obtain the actual longitudinal position of the ship after the restriction. and horizontal position The specific expression is: (3) wherein and is the minimum and maximum longitudinal position allowed for the vessel, and is the minimum and maximum lateral position allowed for the vessel; Based on the established second obstacle, the Lyapunov function is used to determine the desired longitudinal position of the ship. and horizontal position By applying restrictions separately, the desired longitudinal position of the ship after the restrictions are obtained. and horizontal position The specific expression is: (4)。 4. The fixed-time ship trajectory tracking control method with consideration of state constraints according to claim 3, characterized in that, In S3, a ship position tracking error defined based on the limited ship actual position and the limited ship desired position includes a ship tracking longitudinal error and a lateral error respectively represented as: (5) Designing to enable ship position tracking error and Implementing converging longitudinal desired speed and lateral desired speed comprising: Assume and converge, then and bounded, then the actual position of the ship , satisfies the following equation: (6) Therefore, the derivative of formula (5) is obtained as follows: (7) Assuming the desired trajectory of the ship is smooth, and there are first and second derivatives, then and and the expressions are as follows: (8) According to equation (7), to make and the first derivative of the design ship's actual position and is: (9) wherein and are positive parameters to be designed; Combining formula (1) and formula (9) can obtain: (10) Definitions and As shown in formula (11): (11) According to equations (10) and (11), the desired longitudinal velocity and the desired lateral velocity are designed to be: (12)。 5. The fixed-time ship trajectory tracking control method with consideration of state constraints according to claim 4, characterized in that, In S4, defining a ship speed tracking error based on the actual speed of the ship in the underactuated ship mathematical model, the longitudinal expected speed, the lateral expected speed, and the defined first and second finite-time auxiliary system variables, and designing a first fixed-time sliding mode surface and a second fixed-time sliding mode surface according to the ship speed tracking error include: For the desired longitudinal velocity and expected lateral velocity Taking the derivative and simplifying, we get: (13) wherein and respectively and the first derivative; Defining a ship longitudinal speed tracking error in a ship speed tracking error , the expression is: (14) wherein is a first finite time auxiliary variable; is the actual longitudinal speed of the ship; Based on ship longitudinal speed tracking error Designed first fixed time sliding mode surface For: (15) wherein , and are positive parameters to be designed, and ; Defining a ship lateral velocity tracking error in a ship velocity tracking error , the expression is: (16) wherein is a second finite-time auxiliary variable; is the actual lateral speed of the ship; Ship lateral velocity tracking error Second fixed-time sliding surface design For: (17) wherein , and are positive parameters to be designed.

6. The fixed-time ship trajectory tracking control method with consideration of state constraints according to claim 5, characterized in that, In S5, designing a longitudinal control force and a yaw control moment that can make the first fixed-time sliding mode surface and the second fixed-time sliding mode surface converge include: S51. designing a longitudinal control force that can make the first fixed-time sliding mode surface converge, including: Derivating formula (15) and combining formula (1) and (14) can obtain: (18) Let (19) Using a first finite-time auxiliary variable Transforming equation (18) gives: (20) The unknown term is estimated using an RBF neural network The neural network output expression for the unknown term is (21) wherein is a weight matrix, is an estimate of the Gaussian function used, is an error in estimating the estimate.​ Assuming external disturbances to the ship For time-varying disturbances, bounded and unknown bounds, and the existence of the first derivative, the longitudinal control force is designed as follows: (22) wherein , and are positive design parameters; , is an upper bound of the composite disturbance , is an estimate of . At the same time, the design parameters and of the adaptive law are given by equation (23) (23) wherein , , , , and are positive design parameters; S52. designing a yaw control moment that can make the second fixed-time sliding mode surface converge, including: Let (24) Then, combining with equation (13), we have whose derivative is given by equation (25): (25) Derivating formula (17) and combining formula (1), (16) and (25) can obtain: (26) wherein, (27) Let (28) Using a second finite-time auxiliary variable Transforming equation (26) gives: (29) Estimating unknown terms using RBF neural networks is given by (30) wherein is a weight matrix, is a weight matrix, is an estimate of the Gaussian function used, is an error in estimating the error; Assuming external disturbances acting on the ship For time-varying disturbances, bounded and unknown bounds, and the existence of the first derivative, the yaw control moment is designed as follows: (31) wherein , and are positive design parameters, , is an upper bound of the composite disturbance is an estimate of is an estimate of is an estimate of At the same time, the design parameters and of the adaptive law are given by equation (32): (32) wherein , , , , and are positive design parameters, and are the estimated values of and respectively.

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