Fixed-time preset performance trajectory tracking control method considering actuator characteristics

Abstract

The invention discloses a fixed time preset performance trajectory tracking control method considering actuator characteristics. The method comprises the following steps: constructing an under-actuated ship mathematical model considering input saturation and a dead zone; setting an expected trajectory of the ship, obtaining a trajectory tracking error based on the actual trajectory and the expected trajectory of the ship, defining a barrier Lyapunov function limiting error, and further designing an expected longitudinal velocity and an expected transverse velocity; calculating a speed tracking error, and designing a fixed time sliding mode surface according to the speed tracking error; estimating a nonlinear term by using an RBF neural network; and combining a fixed time convergence theory, designing a ship longitudinal control force and a yawing control moment based on the estimated nonlinear term and the fixed time sliding mode surface, and controlling the ship by using a ship longitudinal and ship yawing fixed time controller to realize trajectory tracking control of the ship. The method can improve the control precision, solves the problems of low convergence speed and instability, improves the control efficiency, and guarantees the tracking precision.

Description

Technical Field

[0001] The present invention relates to the technical field of ship trajectory control, and particularly relates to a fixed-time preset performance trajectory tracking control method considering actuator characteristics. Background Art

[0002] In recent years, with the continuous development of science and technology, more and more intelligent devices have been gradually used in military and civilian fields. As one of the intelligent devices, intelligent ships are widely used in multiple fields such as resource exploration, maritime patrol, and military activities, playing a crucial role. With the wide use of ships, the traffic flow has gradually increased, posing a huge challenge to navigation safety. By analyzing the recent research results, it is found that in the presence of external disturbances, enabling ships to achieve fast and safe trajectory tracking has always been a hot issue in the research field of control. Further considering actuator characteristics and model uncertainties on this basis is of great significance.

[0003] Ship actuators have a series of non-linear characteristics such as dead zones, saturation, and time delays. Ignoring these characteristics in the actual process will undoubtedly reduce the performance of the controller. However, considering these characteristics simultaneously will inevitably increase the design difficulty of the controller and make the controller structure more complex. Secondly, to achieve fast and safe trajectory tracking, it is necessary to accelerate the convergence rate of the error. However, accelerating the error convergence rate by adjusting the control gain will make the control input larger. Due to the saturation characteristics of ship actuators, there is a certain limit to the control input. Therefore, it is not very realistic to only accelerate the error convergence rate by adjusting the control gain. At the same time, the increase in the convergence rate will be accompanied by an increase in the overshoot of the system, affecting navigation safety. Therefore, how to enable ships to achieve fast and safe tracking under the condition that the control input has a limit is also a technical problem. In addition to the above problems that need to be considered, ships sailing at sea will also face the interference of external wind and waves, and there are also problems of model uncertainties in the ship mathematical model. Summary of the Invention

[0004] The present invention provides a fixed-time preset performance trajectory tracking control method considering actuator characteristics to overcome the technical problems that existing ship actuators have external disturbances, dead zones, and model uncertainties, and the convergence speed is slow when designing a controller for trajectory tracking, and it is difficult to enable ships to achieve fast and safe tracking under the condition that the control input has a limit.

[0005] To achieve the above object, the technical solution of the present invention is:

[0006] A fixed-time preset performance trajectory tracking control method considering actuator characteristics, comprising:

[0007] S1: Construct an underactuated ship mathematical model considering input saturation and dead zones;

[0008] S2: Set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, define a barrier Lyapunov function to limit the trajectory tracking error, obtain the limited trajectory tracking error, and design the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error;

[0009] S3: Introduce auxiliary variables, calculate the speed tracking error according to the actual speed and the desired speed of the ship, and design the first fixed-time sliding surface and the second fixed-time sliding surface according to the speed tracking error and the auxiliary variables;

[0010] S4: Use an RBF neural network to estimate the nonlinear terms in the underactuated ship mathematical model to obtain the estimated nonlinear terms;

[0011] S5: Combine the fixed-time convergence theory, and design the longitudinal control force and the yaw control moment of the ship based on the estimated nonlinear terms, the first fixed-time sliding surface and the second fixed-time sliding surface to represent the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship;

[0012] S6: Use the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship to control the ship, make the actual speed of the ship tend to the desired speed, and the speed tracking error converges, so as to realize the trajectory tracking control of the ship.

[0013] Construct an underactuated ship mathematical model considering input saturation and dead zone, as shown in formulas (1)-(3),

[0014]

[0015] In formula (1), x, y, ψ are the forward displacement, lateral drift displacement and heading angle of the ship respectively; u, v, r are the longitudinal speed, lateral speed and yaw angular velocity of the ship in the body-fixed coordinate system respectively; N(τ u ) is the saturated longitudinal control force, N(τ r ) is the saturated yaw control moment, as shown in formula (2) specifically; n(τ u ) is the input dead zone function of the saturated longitudinal control force, n(τ r ) is the input dead zone function of the saturated yaw control moment, as shown in formula (3) specifically; m 11 , m 22 , m 33 are inertial parameters; d 11 , d 22 , d 33 are hydrodynamic parameters; Δf u , Δf v , Δf rare the non - linear terms in the longitudinal, lateral, and yaw directions of the under - actuated ship mathematical model; τ wu , τ wv , τ wr is the unknown external disturbance; in Equation (2), κ is the control input gain, τ imax is the upper bound of the control input, τ imin is the lower bound of the control input, τ i is the actual control input; in Equation (3), br i > 0, bl i <0 are pre - set constant parameters.

[0016] Furthermore, set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the under - actuated ship mathematical model, define a barrier Lyapunov function to limit the trajectory tracking error, and obtain the limited trajectory tracking error. Design the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error, including:

[0017] S21. Set the desired trajectory of the ship, and obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the under - actuated ship mathematical model, as shown in Equation (4),

[0018]

[0019] where x d and y d are the longitudinal and transverse coordinates of the desired trajectory of the ship, x e and y e are the trajectory tracking errors of the longitudinal and lateral positions of the ship; x and y are the forward displacement and the cross - drift displacement of the ship, that is, the actual trajectory of the ship;

[0020] S22. Define a log - type barrier Lyapunov function to limit the trajectory tracking error, as shown in Equation (5),

[0021]

[0022] where ρ1, ρ2, ρ3, and ρ4 are positive design parameters, η zx and η zy represent the log - type barrier Lyapunov functions in the longitudinal and lateral directions of the ship; k z is the preset performance function, as shown in Equation (6),

[0023]

[0024] where α1, p and are positive design parameters, z(t) is an intermediate variable, as shown in Equation (7),

[0025]

[0026] where α2,T is a positive parameter to be designed, and t is time;

[0027] Derive the log-type barrier Lyapunov function as shown in formula (8),

[0028]

[0029] Design the derivative of the trajectory tracking error based on the derived log-type barrier Lyapunov function and to make the log-type barrier Lyapunov function converge as shown in formula (9),

[0030]

[0031] where c1 and c2 are positive design parameters;

[0032] S23. Obtain the derivative of the trajectory tracking error and the mathematical relationship with the underactuated ship mathematical model as shown in formula (10),

[0033]

[0034] Design the desired longitudinal speed and desired lateral speed based on formula (10) as shown in formula (11),

[0035]

[0036] where u d and v d represent the desired longitudinal speed and desired lateral speed of the ship, respectively.

[0037] Furthermore, introduce auxiliary variables, calculate the speed tracking error according to the actual speed and desired speed of the ship, and design the first fixed-time sliding surface and the second fixed-time sliding surface based on the speed tracking error and the auxiliary variables, including:

[0038] S31. Design the longitudinal speed tracking error according to the longitudinal speed and desired longitudinal speed of the ship, and introduce a first auxiliary variable into the longitudinal speed tracking error as shown in formula (12),

[0039] u e = u - u d - γ1 (12)

[0040] where u e represents the longitudinal speed tracking error, u represents the longitudinal speed, and u d$\upsilon_{d}$ represents the desired longitudinal speed, and $\gamma_{1}$ represents the first auxiliary variable;

[0041] Define the first fixed-time sliding surface according to the longitudinal speed tracking error, as shown in Equation (13):

[0042]

[0043] where $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, and $\theta$ are positive design parameters;

[0044] S32. Design the lateral speed tracking error based on the lateral speed and the desired lateral speed of the ship, and introduce a second auxiliary variable into the lateral speed tracking error, as shown in Equation (14):

[0045]

[0046] where e $\upsilon$ represents the lateral speed tracking error, $\upsilon$ represents the lateral speed of the ship, and $\upsilon_{d}$ d represents the desired lateral speed, and $\gamma_{2}$ represents the second auxiliary variable;

[0047] Define the second fixed-time sliding surface according to the lateral speed tracking error, as shown in Equation (15):

[0048]

[0049] where $s_{2}$ represents the second fixed-time sliding surface, and $\lambda_{4}$, $\lambda_{5}$, and $\lambda_{6}$ are positive design parameters.

[0050] Furthermore, use the RBF neural network to estimate the non-linear terms in the under-actuated ship mathematical model, including:

[0051] S41. Combine the under-actuated ship mathematical model and the longitudinal speed tracking error to take the derivative of the first fixed-time sliding surface, and obtain Equation (16):

[0052]

[0053] where $\dot{\gamma}_{1}$ represents the derivative of the first auxiliary variable, and the expression is as shown in Equation (17):

[0054]

[0055] Use Equation (17) to transform Equation (16), as shown in Equation (18):

[0056]

[0057] S42. Use the RBF neural network to estimate $\Delta f$ u as shown in Equation (19):

[0058] Δf u = W u T h(x)+o1(19)

[0059] In the formula, Δf u represents the longitudinal nonlinear term in the underactuated ship mathematical model, and W u represents the ideal weight matrix for estimating Δf by the RBF neural network, h(x) is a Gaussian function, and o1 is the approximation error of the RBF neural network for approximating Δf u ; u

[0060] S43. Differentiate the desired lateral velocity, as shown in formula (20),

[0061]

[0062] In the formula, f is an intermediate variable, and its expression is as shown in formula (21),

[0063]

[0064] In the formula, z xe is an intermediate variable, and its expression is as follows,

[0065]

[0066] Differentiate the derivative of the desired lateral velocity again, as shown in formula (22),

[0067]

[0068] Combine the underactuated ship mathematical model, the lateral velocity tracking error, and the second derivative of the desired lateral velocity to differentiate the second fixed-time sliding surface, and obtain formula (23),

[0069]

[0070] In the formula, △ is an intermediate variable, as shown in formula (24),

[0071]

[0072] Differentiate the second auxiliary variable, as shown in formula (25),

[0073]

[0074] S44. Use the RBF neural network to estimate Δf r as shown in formula (26),

[0075] Δf r ​= W r T h r (x) + o2 (26)

[0076] In the formula, Δf r represents the nonlinear term in the yaw direction of the underactuated ship mathematical model, and W r represents the ideal weight matrix for estimating Δf r using the RBF neural network, o2 is the approximation error of the RBF neural network for approximating Δf r and h r (x) is the Gaussian function for estimating Δf r .

[0077] Furthermore, combining the fixed-time convergence theory, based on the estimated nonlinear term, the first fixed-time sliding surface and the second fixed-time sliding surface, design the longitudinal control force and yaw control moment of the ship to represent the longitudinal fixed-time controller and the yaw fixed-time controller of the ship, including:

[0078] S51. Combining the fixed-time convergence theory, formula (18) and formula (19), design the longitudinal control force of the ship, as shown in formula (27),

[0079]

[0080] In the formula, c3, c4 and ε1 are all positive design parameters. Define the first composite disturbance φ1, φ1 = -o1 - n(τ u ) + τ wu ; μ1 is the upper bound of φ1, is the estimated value of μ1;

[0081] Design and adaptive laws, that is and Derive to obtain and values, as shown in formula (28),

[0082]

[0083] In the formula, η1, η2, σ1, σ2, σ3 and σ4 are positive design parameters;

[0084] S52. According to the fixed-time convergence theory, formula (23) and formula (26), design the yaw control moment of the ship, as shown in formula (29),

[0085]

[0086] where \(c_5\), \(c_6\) and \(\varepsilon_2\) are all positive design parameters, defining the second composite disturbance \(\varphi_2\), \(\varphi_2=-o_2 - n(\tau r )+\tau wr ; \(\mu_2\) is the upper bound of \(\varphi_2\), is the estimated value of \(\mu_2\);

[0087] Design and adaptive laws, that is and derivatives of to obtain and values, as shown in formula (30),

[0088]

[0089] where \(\eta_3\), \(\eta_4\), \(\sigma_5\), \(\sigma_6\), \(\sigma_7\) and \(\sigma_8\) are positive design parameters.

[0090] Beneficial effects: The present invention provides a fixed-time preset performance trajectory tracking control method considering actuator characteristics, having the following advantages:

[0091] (1) For unknown external disturbances, model uncertainties and dead zone problems, a neural network is used to estimate the model uncertainties, and an adaptive technique is used to estimate the upper bound of the composite disturbance composed of external disturbances, neural network estimation errors and dead zones, enhancing the robustness of the system and improving the control accuracy;

[0092] (2) For the problem of convergence speed, first, a controller is designed by combining the fixed-time convergence theory to make the speed tracking error achieve fixed-time convergence in the approaching stage; secondly, a fixed-time sliding surface is designed. By using the fixed-time sliding surface, when the error reaches or approaches the sliding surface, the speed error on the sliding surface will make the speed tracking error achieve fixed-time convergence in the sliding stage due to the action of the sliding surface; by designing a fixed-time sliding surface and a fixed-time controller through the fixed-time convergence theory, the problems of slow convergence speed and lack of stability can be solved, and the control efficiency can be improved; at the same time, facing the input saturation problem, an auxiliary variable is introduced to design a fixed-time sliding surface to solve the non-linear term caused by input saturation;

[0093] (3) For the problem of ship navigation safety, a barrier Lyapunov function is designed to limit the tracking error, and the preset performance function is set as the limit boundary, so that the system has good transient and steady-state performances and ensures the tracking accuracy. Description of the Drawings

[0094] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0095] Figure 1 Flowchart of the fixed-time preset performance trajectory tracking control method considering actuator characteristics provided by the present invention;

[0096] Figure 2 Basic framework diagram of the fixed-time preset performance trajectory tracking control considering actuator characteristics provided by the present invention;

[0097] Figure 3 Comparison diagram of the desired trajectory and the actual trajectory of a ship in a plane in an embodiment of the present invention;

[0098] Figure 4 Comparison diagram of the longitudinal position error effect of the simulation experiment in an embodiment of the present invention;

[0099] Figure 5 Comparison diagram of the lateral position error effect of the simulation experiment in an embodiment of the present invention;

[0100] Figure 6 Comparison diagram of the lateral velocity error of the simulation experiment in an embodiment of the present invention;

[0101] Figure 7 Comparison diagram of the longitudinal velocity error of the simulation experiment in an embodiment of the present invention;

[0102] Figure 8 Longitudinal control force input curve diagram of the simulation experiment in an embodiment of the present invention;

[0103] Figure 9 Yaw control torque input curve diagram of the simulation experiment in an embodiment of the present invention;

[0104] Figure 10 Neural network estimation curve diagram in an embodiment of the present invention;

[0105] Figure 11 Composite disturbance upper bound estimation curve diagram in an embodiment of the present invention. Detailed implementation manners

[0106] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0107] This embodiment provides a fixed-time preset performance trajectory tracking control method considering actuator characteristics, as Figure 1 shown, including:

[0108] S1: Construct an underactuated ship mathematical model considering input saturation and dead zone;

[0109] S2: Set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, define a barrier Lyapunov function to limit the trajectory tracking error, obtain the limited trajectory tracking error, and design the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error;

[0110] S3: Introduce auxiliary variables, calculate the speed tracking error according to the actual speed and the desired speed of the ship, and design the first fixed-time sliding surface and the second fixed-time sliding surface according to the speed tracking error and the auxiliary variables;

[0111] S4: Use an RBF neural network to estimate the non-linear terms in the underactuated ship mathematical model to obtain the estimated non-linear terms;

[0112] S5: Combine the fixed-time convergence theory, and design the longitudinal control force and the yaw control moment of the ship based on the estimated non-linear terms, the first fixed-time sliding surface, and the second fixed-time sliding surface to represent the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship;

[0113] S6: Use the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship to control the ship, make the actual speed of the ship tend to the desired speed, and the speed tracking error converges, so as to realize the trajectory tracking control of the ship.

[0114] Specifically, first construct an underactuated ship mathematical model considering input saturation and dead zone. The ship actuator has a series of non-linear characteristics such as dead zone, saturation, and time delay. Considering these characteristics, construct an underactuated ship mathematical model as the control object of the subsequent controller to achieve fast and safe trajectory tracking;

[0115] Secondly, set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, define a barrier Lyapunov function to limit the trajectory tracking error, and obtain the limited trajectory tracking error. Design the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error. Due to the saturation characteristic of the ship actuator, there will be a certain limit for the control input. An increase in the convergence rate will be accompanied by an increase in the overshoot of the system, which affects the navigation safety. The barrier Lyapunov function is used to limit the tracking error, and the preset performance function is set as the limiting boundary, so that the system has good transient and steady-state performance, ensures the tracking accuracy, and ensures the navigation safety.

[0116] Thirdly, introduce auxiliary variables, calculate the speed tracking error according to the actual speed and the desired speed of the ship, and design the first fixed-time sliding surface and the second fixed-time sliding surface according to the speed tracking error and the auxiliary variables. Combine the fixed-time convergence theory to design a controller and use the fixed-time sliding surface to achieve global fixed-time convergence. By using the fixed-time sliding surface, when the error reaches or approaches the sliding surface, the speed error on the sliding surface will make the speed tracking error also achieve fixed-time convergence in the sliding stage due to the action of the sliding surface. At the same time, facing the input saturation problem, auxiliary variables will be used to solve the nonlinear terms caused by input saturation. The fixed-time sliding surface eliminates the dependence on the initial conditions through mathematical design, provides predictable convergence performance, and is suitable for control systems with high dynamics and strong robustness requirements. Use an RBF neural network to estimate the nonlinear terms in the underactuated ship mathematical model to obtain the estimated nonlinear terms. Use a neural network to estimate the model uncertainties, and use an adaptive technique to estimate the upper bound of the composite disturbance composed of external disturbances, neural network estimation errors, and dead zones, which can significantly improve the control accuracy, robustness, and dynamic adaptability.

[0117] Combined with the fixed-time convergence theory, design the longitudinal control force and the yaw control moment of the ship based on the estimated nonlinear terms, the first fixed-time sliding surface, and the second fixed-time sliding surface to represent the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship. Design the fixed-time controller through the fixed-time convergence theory, which can solve the problems of slow convergence speed and instability, and improve the control efficiency.

[0118] Finally, use the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship to control the ship, make the actual speed of the ship tend to the desired speed, and the speed tracking error converges, so as to realize the trajectory tracking control of the ship.

[0119] In a specific embodiment, construct an underactuated ship mathematical model considering input saturation and dead zones, as shown in formulas (31)-(33).

[0120]

[0121]

[0122] In Equation (31), x, y, and ψ are the forward displacement, lateral drift displacement, and heading angle of the ship, respectively; u, v, and r are the longitudinal velocity, lateral velocity, and yaw angular velocity of the ship in the body-fixed coordinate system; N(τ i ) is the saturated longitudinal control force and yaw control moment, as specifically shown in Equation (32); n(τ i ) is the input dead zone function, as specifically shown in Equation (33); m 11 , m 22 , m 33 are the inertia parameters; d 11 , d 22 , d 33 are the hydrodynamic parameters; Δf u , Δf v , Δf r are the nonlinear terms in the longitudinal, lateral, and yaw directions of the underactuated ship mathematical model, respectively; τ wu , τ wv , τ wr are the external unknown disturbances;

[0123] In Equation (32), k is the control input gain, τ imax is the upper bound of the control input, τ imin is the lower bound of the control input, τ i is the actual control input;

[0124] In Equation (33), br i > 0, bl i <0 are the pre-set parameters.

[0125] This embodiment further includes the following preliminary knowledge:

[0126] 1) Lemma

[0127] Lemma 1: For any ε > 0, there exists the following inequality:

[0128] 0 ≤ |s| - stanh(s / ε) ≤ 0.2785ε

[0129] where s is any variable and ε is a constant;

[0130] Lemma 2: For any continuously smooth function f(x) defined on the compact set , there is

[0131]

[0132] where \(o\) is the approximation error. For all \(x\in\Omega\), there exists a vector \(o\) * \(>0\), and it satisfies \(|o|\leq o\); \(Wx\) is the weight vector under ideal conditions, which is usually not easy to obtain and is an unknown variable, so it needs to be estimated. \(h(x)\) is a Gaussian function, and its expression is as follows:

[0133]

[0134] where: \(j = 1,2,3,\cdots,n\), which is the number of network nodes; \(m\) j is the input center vector; \(b\) j is the standard deviation of the Gaussian function;

[0135] Lemma 3: Consider a dynamic system where \(x\in R\) n is the state of the system, and \(x_0\) is the equilibrium point of the system. If there exists a Lyapunov function \(V(x)\) that satisfies formula (34), then the system is actually fixed-time stable; and the stable time \(T\) r satisfies

[0136] where \(\chi_1>0\), \(\chi_2>0\), \(j>1\), \(0<q<1\), \(0<\chi<1\);

[0137]

[0138] Lemma 4: If \(\xi,\varPhi\in R\), \(\delta>0\), \(m>1\), \(n>1\), and \((m - 1)(n - 1)=1\), then there exists the following inequality:

[0139]

[0140] where \(\xi\), \(m\), \(n\), \(\varPhi\), and \(\delta\) are all constants;

[0141] 2) Assume

[0142] Assumption 1: The external disturbance received by the ship is a time-varying perturbation, which is bounded and has an unknown bound, and there is also a first-order derivative.

[0143] Assumption 2: It is assumed that the desired trajectory \((x\) d , \(y\) d ) of the ship is smooth and has first-order and second-order derivatives.

[0144] In this scheme, considering the problems of input saturation and dead zone, a mathematical model of an underactuated ship considering these two problems is constructed as the control object of the subsequent controller, which ensures that the ship can achieve path tracking control considering input saturation and dead zone and improves the tracking accuracy.

[0145] In a specific embodiment, a desired trajectory of a ship is set, a trajectory tracking error is obtained based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, and a barrier Lyapunov function is defined to limit the trajectory tracking error, and the limited trajectory tracking error is obtained. The scheme for designing the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error is as follows:

[0146] S21. Set the desired trajectory of the ship, and obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, as shown in formula (35):

[0147]

[0148] where x d and y d are the longitudinal and transverse coordinates of the desired trajectory of the ship, x e and y e are the trajectory tracking errors of the longitudinal and lateral positions of the ship; x and y are the forward displacement and the lateral drift displacement of the ship, that is, the actual trajectory of the ship;

[0149] S22. Define a log-type barrier Lyapunov function to limit the trajectory tracking error, as shown in formula (36):

[0150]

[0151] where ρ1, ρ2, ρ3, and ρ4 are positive design parameters, η zx and η zy represent the log-type barrier Lyapunov functions in the longitudinal and lateral directions of the ship; k z is a preset performance function, as shown in formula (37):

[0152]

[0153] where α1,p and are positive design parameters, z(t) is an intermediate variable, as shown in formula (38):

[0154]

[0155] where α2 and T are positive parameters to be designed, and t is time;

[0156] Derive the log-type barrier Lyapunov function, as shown in formula (39):

[0157]

[0158] Based on the derived log-type barrier Lyapunov function, design the derivative of the trajectory tracking error and Converge the log-type barrier Lyapunov function as shown in formula (40).

[0159]

[0160] where c1 and c2 are positive design parameters;

[0161] S23. Obtain the derivative of the trajectory tracking error and the mathematical relationship with the underactuated ship mathematical model as shown in formula (41).

[0162]

[0163] Design the desired longitudinal speed and the desired lateral speed based on formula (41) as shown in formula (42).

[0164]

[0165] where u d and v d represent the desired longitudinal speed and the desired lateral speed of the ship respectively;

[0166] Under the designed desired longitudinal speed and the desired lateral speed, η zx and η zy can achieve convergence, then the trajectory tracking error can satisfy formula (43).

[0167]

[0168] If formula (43) holds, the output limit can be satisfied;

[0169] When u approaches u d , and v approaches v d , the differentiated log-type barrier Lyapunov function can be simplified to formula (44).

[0170]

[0171] Define the Lyapunov function as shown in formula (45).

[0172]

[0173] Differentiate the Lyapunov function and substitute it into formula (44) to obtain formula (46).

[0174]

[0175] where is the defined variable as shown in formula (47).

[0176]

[0177] As can be seen from Equation (46), if u can approach u d and v can approach v d then η zx and η zy can achieve asymptotic convergence, and then the tracking errors x e and y e can be restricted.

[0178] Due to the saturation characteristics of the ship actuator, there are certain limits to the control input. An increase in the convergence rate will be accompanied by an increase in the overshoot of the system, which affects navigation safety. The barrier Lyapunov function is used to limit the tracking error, and the preset performance function is set as the limit boundary, so that the system has good transient and steady-state performance, ensuring tracking accuracy and navigation safety.

[0179] In a specific embodiment, an auxiliary variable is introduced. The scheme for calculating the speed tracking error according to the actual speed and the desired speed of the ship and designing the first fixed-time sliding surface and the second fixed-time sliding surface based on the speed tracking error and the auxiliary variable is as follows:

[0180] S31. Differentiate the desired longitudinal speed and the desired lateral speed, as shown in Equation (48),

[0181]

[0182] where z xe and z ye are intermediate variables, as shown in Equation (49),

[0183]

[0184] Design the longitudinal speed tracking error according to the longitudinal speed of the ship and the desired longitudinal speed, and introduce a first auxiliary variable into the longitudinal speed tracking error, as shown in Equation (50),

[0185] u e = u - u d - γ1 (50)

[0186] where u e represents the longitudinal speed tracking error, u represents the longitudinal speed, u d represents the desired longitudinal speed, and γ1 represents the first auxiliary variable;

[0187] Define the first fixed-time sliding surface according to the longitudinal speed tracking error, as shown in Equation (51),

[0188]

[0189] where λ1, λ2, λ3, and θ are positive design parameters, and θ > 1;

[0190] S32. Design a lateral velocity tracking error based on the lateral velocity and the desired lateral velocity of the ship, and introduce a second auxiliary variable into the lateral velocity tracking error, as shown in formula (52).

[0191]

[0192] where v e represents the lateral velocity tracking error, v represents the lateral velocity of the ship, and v d represents the desired lateral velocity, and γ2 represents the second auxiliary variable;

[0193] Define a second fixed-time sliding mode surface based on the lateral velocity tracking error, as shown in formula (53).

[0194]

[0195] where s2 represents the second fixed-time sliding mode surface, and λ4, λ5, and λ6 are positive design parameters.

[0196] Design a controller by combining the fixed-time convergence theory and using the fixed-time sliding mode surface to achieve global fixed-time convergence. By using the fixed-time sliding mode surface, when the error reaches the sliding mode surface or near it, the velocity error on the sliding mode surface will cause the velocity tracking error to achieve fixed-time convergence during the sliding phase due to the action of the sliding mode surface. At the same time, facing the input saturation problem, an auxiliary variable is used to solve the nonlinear term caused by input saturation. The fixed-time sliding mode surface eliminates the dependence on the initial conditions through mathematical design, provides predictable convergence performance, and is suitable for control systems with high dynamics and strong robustness requirements.

[0197] In a specific embodiment, the scheme for using an RBF neural network to estimate the nonlinear term in the underactuated ship mathematical model and obtaining the estimated nonlinear term is as follows:

[0198] S41. Differentiate the first fixed-time sliding mode surface by combining the underactuated ship mathematical model and the longitudinal velocity tracking error to obtain formula (54).

[0199]

[0200] where represents the derivative of the first auxiliary variable, and the expression is as shown in formula (55).

[0201]

[0202] Use formula (55) to transform formula (54), as shown in formula (56).

[0203]

[0204] S42. Since the RBF neural network has the universal approximation function for any smooth function, the RBF neural network is used to estimate Δf u According to Lemma 2, the output expression of the RBF neural network of Δf u is shown in Equation (57) as follows,

[0205] Δf u = W u T h(x)+o1 (57)

[0206] where Δf u represents the longitudinal nonlinear term in the underactuated ship mathematical model, W u represents the ideal weight matrix for the RBF neural network to estimate Δf u , h(x) is the Gaussian function, and o1 is the approximation error of the RBF neural network to approximate Δf u ;

[0207] S43. For the convenience of calculation, an intermediate variable f is defined. Let

[0208] Then the derivative of the desired lateral velocity can be expressed as Taking the derivative of the derivative of the desired lateral velocity again, as shown in Equation (58)

[0209]

[0210] To avoid the "differential explosion" problem caused by directly taking the derivative of f, a dynamic surface is designed to replace the f derivative process, and its expression is as follows,

[0211]

[0212] where T r > 0 is the design parameter;

[0213] Combining the underactuated ship mathematical model, the lateral velocity tracking error, and the second derivative of the desired lateral velocity, the second fixed-time sliding surface is differentiated to obtain Equation (60),

[0214]

[0216] where △ is the intermediate variable, as shown in Equation (61),

[0217]

[0218] Derive the second auxiliary variable as shown in formula (62).

[0219]

[0220] S44. Use the RBF neural network to estimate Δf r as shown in formula (63).

[0221] Δf r = W r T h r (x)+o2 (63)

[0222] In the formula, Δf r represents the nonlinear term in the yaw direction of the underactuated ship mathematical model, W r represents the ideal weight matrix for estimating Δf r using the RBF neural network, o2 is the approximation error of the RBF neural network for approximating Δf r , and h r (x) estimates Δf r which is a Gaussian function.

[0223] In this solution, since the ship is underactuated and there are only control inputs in the longitudinal and yaw directions, when using the RBF neural network to estimate the nonlinear terms in the underactuated ship mathematical model, only the nonlinear terms in the longitudinal and yaw directions need to be estimated, that is, Δf u and Δf r , and there is no need to estimate the lateral nonlinear term Δf v ; Using the RBF neural network to estimate the nonlinear terms of the model and using the adaptive technique to estimate the upper bound of the composite disturbance composed of external disturbances, neural network estimation errors, and dead zones can significantly improve the control accuracy, robustness, and dynamic adaptability.

[0224] In a specific embodiment, combining the fixed-time convergence theory, based on the estimated nonlinear terms, the first fixed-time sliding surface, and the second fixed-time sliding surface, the design of the longitudinal control force and yaw control moment of the ship is as follows:

[0225] S51. Combining the fixed-time convergence theory, formula (56), and formula (58), design the longitudinal control force of the ship as shown in formula (64).

[0226]

[0227] where \(c_3\), \(c_4\) and \(\varepsilon_1\) are all positive design parameters, defining the first composite disturbance \(\varphi_1\), \(\varphi_1=-o_1 - n(\tau u )+\tau wu ; \(\mu_1\) is the upper bound of \(\varphi_1\), is the estimated value of \(\mu_1\);

[0228] Design and adaptive laws, that is and derivatives to obtain and values, as shown in Equation (65)

[0229]

[0230] where \(\eta_1\), \(\eta_2\), \(\sigma_1\), \(\sigma_2\), \(\sigma_3\) and \(\sigma_4\) are positive design parameters;

[0231] S52. Design the yaw control moment according to the fixed-time convergence theory, Equation (60) and Equation (63), as shown in Equation (66),

[0232]

[0233] where \(c_5\), \(c_6\) and \(\varepsilon_2\) are all positive design parameters, defining the second composite disturbance \(\varphi_2\), \(\varphi_2=-o_2 - n(\tau r )+\tau wr ; \(\mu_2\) is the upper bound of \(\varphi_2\), is the estimated value of \(\mu_2\);

[0234] Design and adaptive laws, that is and derivatives of to obtain and values, as shown in Equation (67),

[0235]

[0236] where \(\eta_3\), \(\eta_4\), \(\sigma_5\), \(\sigma_6\), \(\sigma_7\) and \(\sigma_8\) are positive design parameters.

[0237] In this solution, a fixed-time controller is designed through the fixed-time convergence theory, which can solve the problems of slow convergence speed and insufficient stability, and improve the control efficiency.

[0238] In a specific embodiment, the scheme of using the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship to control the ship, making the actual speed of the ship tend to the desired speed and the speed tracking error converge, and then realizing the trajectory tracking control of the ship is:

[0239] As Figure 2 shown Figure 2 is the basic framework for controlling a ship using a longitudinal fixed-time controller and a yaw fixed-time controller for the ship. In the figure, x d , y d represent the desired positions of the ship longitudinally and laterally, i.e., the longitudinal and abscissa coordinates of the desired ship trajectory; x, y are the forward displacement and lateral drift displacement of the ship, i.e., the actual positions of the ship longitudinally and laterally; x e and y e represent the trajectory tracking errors longitudinally and laterally; η zx and η zy represent the log-type barrier Lyapunov function; u d , v d represent the desired longitudinal speed and desired lateral speed of the ship; u, v represent the longitudinal speed and lateral speed of the ship in the body-fixed coordinate system; γ1 and γ2 are the introduced first and second auxiliary variables; s1 and s2 represent the first and second fixed-time sliding surfaces respectively; τ u is the ideal longitudinal control force of the ship; τ r is the ideal yaw control moment; μ1 is the upper bound of the composite disturbance composed of τ wu , o1 and the dead zone n(N(τ u )); is the estimated value of μ1; μ2 is the upper bound of the composite disturbance composed of the external unknown disturbance τ wr , the neural network estimation error o2 and the dead zone n(N(τ r )); is the estimated value of μ2, Δf u and Δf r are the non-linear terms of the model;

[0240] First, the trajectory tracking errors x d , y d are obtained from the actual trajectory (x, y) of the ship and the desired trajectory (x e and y e ), and then the barrier Lyapunov function is used to limit the errors of η zx and η zy . The desired longitudinal speed u zx and the desired lateral speed v zy are designed through η d and η d ; then the first fixed-time sliding surface s1 is designed through the difference between the actual longitudinal speed u and the desired longitudinal speed u d and the first auxiliary variable γ1, and the lateral speed v and the desired lateral speed v dDesign the second fixed-time sliding surface \(s_2\) based on the difference between the first auxiliary variable \(\gamma_1\) and the second auxiliary variable \(\gamma_2\); finally, combine the adaptive law And use the backstepping method to design the longitudinal control force \(\tau\) u And the yaw control moment \(\tau_r\) r Make the sliding surfaces \(s_1\) and \(s_2\) converge to the vicinity of the stable region. After convergence, the velocity tracking error will slide on the sliding surfaces \(s_1\) and \(s_2\) to achieve its convergence, and then achieve trajectory tracking.

[0241] Verification:

[0242] To verify the stability of the controller designed in the present invention, define the Lyapunov function shown in Equation (68),

[0243]

[0244] In the formula,

[0245] Taking the derivative of both sides of Equation (68) gives

[0246]

[0247] For the first term in Equation (69) Combining Equations (56) and (64) gives

[0248]

[0249] For the second term in Equation (69) Combining Equations (60)-(63) and Equation (66) can be transformed into the following form

[0250]

[0251] Combining Equations (70) and (71), Equation (69) can be transformed into the form shown in the following Equation (72)

[0252]

[0253] According to Lemma 4, the in Equation (72) can be transformed into the following form

[0254]

[0255] Considering the following inequality

[0256]

[0257] Combining Young's inequality and Equation (74), then the in Equation (72) can be transformed into

[0258]

[0259] Combining Equation (73), (75) and Lemma 1, Equation (72) can be transformed into

[0260]

[0261] wherein,

[0262]

[0263] From Equation (76) and Lemma 3, it can be obtained that the variables in system V2 can achieve fixed-time convergence during the approaching stage;

[0264] When the system achieves convergence during the approaching stage, there will be certain bounds on the size and rate of change of the sliding surface. Therefore, the following equations exist

[0265]

[0266] wherein, and are bounded constants.

[0267] Construct the Lyapunov function as follows

[0268]

[0269] Taking the derivative of Equation (81) and combining with Lemma 1, we can obtain

[0270]

[0271] When the sliding surface parameter Equation (82) can be transformed into

[0272]

[0273] wherein,

[0274]

[0275] According to Equation (83) and Lemma 3, it can be obtained that the velocity tracking error can achieve fixed-time convergence on the sliding surface.

[0276] In summary, the velocity tracking error can achieve fixed-time convergence during both the approaching stage and the sliding stage, and thus achieve global fixed-time convergence.

[0277] To verify the effectiveness of the control scheme designed in the present invention, a computer simulation experiment was carried out with a patrol boat named BAY CLASS of the Singapore Customs as the simulation object. The length of the boat is 38 m, and the mass is m = 1.18×10 5 kg, and other parameters are m 11 = 1.2×105 kg, m 22 = 1.779×10 5 kg, m 33 = 6.36×10 7 kg, d 11 = 2.15×10 4 kg / s, d 22 = 1.47×10 5 kg / s, d 33 = 8.02×10 6 kg / s.

[0278] To verify the effectiveness of the controller of the present invention and its effect, next, simulation experiments will be carried out for comparison with other control algorithms. This simulation experiment is divided into 4 groups. The first group is the effect of using a linear sliding surface without considering input and output limitations. The specific controllers are shown in Eqs. (87), (88), (89), and (90). The second group is the effect of using a finite-time sliding surface without considering output limitations. The specific expressions are shown in Eqs. (90), (91), (92), (93), and (94). The third group is the effect of the controller of the present invention. The fourth group is the effect of using a saturation function to limit the control input. The specific expression of the saturation function is shown in Eq. (94), and the rest is the same as that of the present invention. To ensure the comparability of the control effects, the controller parameters used for the same controller are the same during the simulation experiment process.

[0279]

[0280]

[0281] To simulate the sea conditions, the external time-varying disturbance adopts Eq. (95),

[0282]

[0283] Set the parameters κ = 0.95, c1 = 9, c2 = 1, c3 = 1.2×10 3 , c4 = 7×10 2 , c5 = 2×10 4 , c6 = 7×10 3 , λ1 = 10, λ2 = 4, λ3 = 2, λ4 = 8, λ5 = 3, λ6 = 0.5, ε1 = 0.1, ε2 = 0.05, α1 = 2, α2 = 0.5, θ = 3, T = 20, T r = 0.1, p = 5, ρ1 = ρ2 = 15, ρ3 = ρ4 = 15, η1 = 7×10 5 , η2 = 7×10 3 , η3 = 1.3×10 3 , η4 = 3.3×102 , σ1 = 2×10 -21 , σ2 = 2×10 -8 , σ3 = 2×10 -8 , σ4 = 2×10 -6 , σ5 = 1×10 -20 , σ6 = 1×10 -2 , σ7 = 3×10 -26 , σ8 = 4×10 -7 , γ = 8, ζ = 1, k1 = 2, k2 = 1, k3 = 1, k4 = 1.

[0284] The uncertainty term of the model is set as Δf u = 0.2d 11 u 2 + 0.1d 11 u 3 , Δf v = 0.2d 22 v 2 + 0.1d 22 v 3 , Δf r = 0.2d 33 r 2 + 0.2d 33 r 3 . Set the input upper limit τ um = 6.2×10 6 N, τ rm = 4×10 8 (N·m). Set the dead zone as br u = -bl u = 2×10 4 , br r = -bl r = 5×10 5 .

[0285] Set the sine desired trajectory as x d = 300sin(0.03t), y d = 10t, select the initial position and velocity of the ship as (x0; y0; ψ0; u0; v0; r0) = (30; 0; 0; 0.1; 0; 0).

[0286] The simulation results are as Figures 3 to 11 shown:

[0287] It can be seen from Figure 3 that the control method of the present invention can enable the ship to complete trajectory tracking in the case of actuator saturation and dead zone, which proves the effectiveness of the control method. It can be seen from Figure 4 and Figure 5It can be seen that the longitudinal and lateral tracking errors of the ship under the action of the first group and the second group exceed the preset performance boundary in the transient stage, seriously affecting the ship safety. While the controllers designed in the third group and the fourth group can keep the ship tracking error within the preset performance boundary all the time, showing satisfactory transient and steady-state tracking performance. From Figures 6 to 7 it can be found that the speed tracking error under the action of the second group of controllers will oscillate around 30 s, while the method of the present invention can keep stable all the time after achieving tracking, having satisfactory robustness. The speed tracking errors of the third group and the fourth group are large and oscillate in the initial stage because the tracking error is large in the initial stage and the dead zone and saturation characteristics of the controller are considered. From Figure 8 and Figure 9 it can be seen that the control inputs of the first group all exceed the preset limits. Although the second group and the fourth group are saturated so that the inputs do not exceed the saturation value, there are respectively chattering and non-smooth input problems, and only the inputs of the third group are more in line with the actual requirements. From Figure 10 and Figure 11 it can be found that the neural network designed by the present invention can approximate the uncertain terms of the model well, and the adaptive law used can estimate the upper bound of the composite disturbance well. To sum up, this simulation experiment can verify the effectiveness of the control method of the present invention and the superiority of the saturation processing method adopted.

[0288] Through the comparative simulation experiment of the above invention and the existing research, the beneficial effects brought by the present invention are summarized as the following 3 points:

[0289] (1) Aiming at the problems of unknown external disturbances, model uncertain terms and dead zone, a neural network is used to estimate the model uncertain terms, and an adaptive technique is used to estimate the upper bound of the composite disturbance composed of external disturbances, neural network estimation errors and dead zone.

[0290] (2) Aiming at the problem of convergence speed, a controller is designed by combining the fixed-time convergence theory and a fixed-time sliding surface is used to achieve global fixed-time convergence. First, a controller is designed by combining the fixed-time convergence theory to make the speed tracking error achieve fixed-time convergence in the approaching stage; second, by using the fixed-time sliding surface, when the error reaches the sliding surface or near it, the speed error on the sliding surface will make the speed tracking error achieve fixed-time convergence in the sliding stage due to the action of the sliding surface. At the same time, facing the problem of input saturation, an auxiliary variable will be adopted to solve the non-linear term caused by input saturation.

[0291] (3) Aiming at the problem of ship navigation safety, a barrier Lyapunov function is designed to limit the tracking error, and the preset performance function is set as the limit boundary to make the system have better transient and steady-state performance and ensure the tracking accuracy.

[0292] The ship fixed-time preset performance trajectory tracking control method considering input saturation and dead zone proposed by the present invention can effectively solve the problems of the ship being under external disturbances, the actuator having dead zone and saturation, and the tracking error having limitations, and accelerate the convergence rate of the error. Through the Lyapunov theory, it is proved that all signals of the ship system are bounded, and the tracking error can converge within a fixed time. By comparing with various control strategies, it is verified that the control strategy of the present invention can handle the problems of input saturation and dead zone, keep the navigation trajectory within the pre-set limits all the time, and can achieve fixed-time convergence in both the approaching stage and the sliding stage, reaching global fixed-time convergence. The theoretical analysis and simulation experiment results show that the feasibility of the ship fixed-time preset performance trajectory tracking control method considering input saturation and dead zone proposed by the present invention is verified.

[0293] 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 them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some 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 various embodiments of the present invention.

Claims

1. A fixed-time preset performance trajectory tracking control method considering actuator characteristics, characterized in that Including: S1: Construct an underactuated ship mathematical model considering input saturation and dead zone; S2: Set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, define a barrier Lyapunov function to limit the trajectory tracking error, obtain the limited trajectory tracking error, and design the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error; S3: Introduce auxiliary variables, calculate the speed tracking error according to the actual speed and the desired speed of the ship, and design the first fixed-time sliding surface and the second fixed-time sliding surface according to the speed tracking error and the auxiliary variables; S4: Use an RBF neural network to estimate the nonlinear terms in the underactuated ship mathematical model to obtain the estimated nonlinear terms; S5: Combine the fixed-time convergence theory, and design the longitudinal control force and the yaw control moment of the ship based on the estimated nonlinear terms, the first fixed-time sliding surface and the second fixed-time sliding surface to represent the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship; S6: Use the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship to control the ship, make the actual speed of the ship tend to the desired speed, and the speed tracking error converges, so as to realize the trajectory tracking control of the ship.

2. The fixed-time preset performance trajectory tracking control method considering actuator characteristics according to claim 1, characterized in that Construct an underactuated ship mathematical model considering input saturation and dead zone, as shown in formulas (1)-(3); In Equation (1), x, y, and ψ are the forward displacement, transverse drift displacement, and heading angle of the ship, respectively; u, v, and r are the longitudinal velocity, lateral velocity, and yaw angular velocity of the ship in the appended body coordinate system; N(τ u ) is the saturated longitudinal control force, and N(τ r ) is the saturated yaw control moment, as specifically shown in Equation (2); n(τ u ) is the input dead zone function of the saturated longitudinal control force, and n(τ r ) is the input dead zone function of the saturated yaw control moment, as specifically shown in Equation (3); m 11 , m 22 , m 33 are the inertia parameters; d 11 , d 22 , d 33 are the hydrodynamic parameters; Δf u , Δf v , Δf r are the nonlinear terms in the longitudinal, lateral, and yaw directions of the underactuated ship mathematical model, respectively; τ wu , τ wv , τ wr are the external unknown disturbances; in Equation (2), κ is the control input gain, τ imax is the upper bound of the control input, τ imin is the lower bound of the control input, and τ i is the actual control input; in Equation (3), br i > 0, and bl i < 0 are the preset constant parameters.

3. The fixed-time preset performance trajectory tracking control method considering actuator characteristics according to claim 2, characterized in that Set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, define a barrier Lyapunov function to limit the trajectory tracking error, obtain the limited trajectory tracking error, and design the desired longitudinal speed and the desired lateral speed based on the limited trajectory tracking error, including: S21: Set the desired trajectory of the ship, obtain the trajectory tracking error based on the actual trajectory and the desired trajectory of the ship in the underactuated ship mathematical model, as shown in formula (4); where x d and y d are the longitudinal and transverse coordinates of the desired ship trajectory, and x e and y e are the trajectory tracking errors of the longitudinal and transverse positions of the ship; x and y are the forward displacement and lateral drift displacement of the ship, i.e., the actual trajectory of the ship; S22: Define a log-type barrier Lyapunov function to limit the trajectory tracking error, as shown in formula (5); where ρ1, ρ2, ρ3, and ρ4 are positive design parameters, η zx and η zy represent the log-type obstacle Lyapunov functions in the longitudinal and transverse directions of the ship; k z is a preset performance function, as shown in Equation (6). where α1,p and are positive design parameters, z(t) is an intermediate variable as shown in Equation (7). where α2,T is a positive design parameter, and t is time; Take the derivative of the log-type barrier Lyapunov function, as shown in formula (8); Design the derivative of the trajectory tracking error according to the log-type barrier Lyapunov function after derivation and Make the log-type barrier Lyapunov function converge, as shown in formula (9). where c1 and c2 are positive design parameters; S23. Obtain the derivative of the trajectory tracking error and the mathematical relationship with the underactuated ship mathematical model, as shown in formula (10). Design the desired longitudinal speed and the desired lateral speed based on formula (10), as shown in formula (11); where u d and v d represent the desired longitudinal speed and the desired lateral speed of the ship, respectively.

4. The fixed-time preset performance trajectory tracking control method considering actuator characteristics according to claim 3, characterized in that, Introduce auxiliary variables, calculate the speed tracking error according to the actual speed and the desired speed of the ship, and design the first fixed-time sliding surface and the second fixed-time sliding surface according to the speed tracking error and the auxiliary variables, including: S31: Design the longitudinal speed tracking error according to the longitudinal speed and the desired longitudinal speed of the ship, and introduce a first auxiliary variable into the longitudinal speed tracking error, as shown in formula (12); u e = u - u d -γ1(12) where, u e represents the longitudinal speed tracking error, u represents the longitudinal speed, and u d represents the desired longitudinal speed, and γ1 represents the first auxiliary variable; Define the first fixed-time sliding surface according to the longitudinal speed tracking error, as shown in formula (13); where λ1, λ2, λ3 and θ are positive design parameters; S32. Design the lateral velocity tracking error based on the lateral velocity and the desired lateral velocity of the ship, and introduce a second auxiliary variable into the lateral velocity tracking error, as shown in formula (14). where, v e represents the lateral velocity tracking error, v represents the lateral velocity of the ship, v d represents the desired lateral velocity, and γ2 represents the second auxiliary variable; Define a second fixed-time sliding surface according to the lateral velocity tracking error, as shown in formula (15). In the formula, s2 represents the second fixed-time sliding surface, and λ4, λ5, and λ6 are positive design parameters.

5. The fixed-time preset performance trajectory tracking control method considering actuator characteristics according to claim 4, characterized in that Use an RBF neural network to estimate the non-linear terms in the mathematical model of the under-actuated ship, including:[[]] S41. Combine the mathematical model of the under-actuated ship and the longitudinal velocity tracking error to take the derivative of the first fixed-time sliding surface, and obtain formula (16). Among them, represents the derivative of the first auxiliary variable, and the expression is as shown in formula (17). Use formula (17) to transform formula (16), as shown in formula (18). S42. Estimate Δf using an RBF neural network, as shown in Equation (19). u ​ Δf u = W u T h(x) + o1(19) where, Δf u represents the longitudinal nonlinear term in the underactuated ship mathematical model, W u represents the ideal weight matrix for the RBF neural network to estimate Δf u , h(x) is a Gaussian function, and o1 is the approximation error of the RBF neural network for approximating Δf u ; S43. Take the derivative of the desired lateral velocity, as shown in formula (20). In the formula, f is an intermediate variable, and the expression is as shown in formula (21). where z xe is an intermediate variable, and the expression is as follows: Take the derivative of the derivative of the desired lateral velocity again, as shown in formula (22). Combine the mathematical model of the under-actuated ship, the lateral velocity tracking error, and the second derivative of the desired lateral velocity to take the derivative of the second fixed-time sliding surface, and obtain formula (23). In the formula, △ is an intermediate variable, as shown in formula (24). Take the derivative of the second auxiliary variable, as shown in formula (25). S44. Estimate Δf using an RBF neural network, as shown in Equation (26). r ​ Δf r = W r T h r (x)+o2 (26) where, Δf r represents the nonlinear term in the yaw direction of the underactuated ship mathematical model, W r represents the ideal weight matrix for estimating Δf r using the RBF neural network, and o2 is the approximation error of the RBF neural network for approximating Δf r , and h r (x) is the Gaussian function for estimating Δf r .

6. The fixed-time preset performance trajectory tracking control method considering actuator characteristics according to claim 5, characterized in that Combine the fixed-time convergence theory, and design the longitudinal control force and yaw control moment of the ship based on the estimated non-linear terms, the first fixed-time sliding surface, and the second fixed-time sliding surface, so as to represent the longitudinal fixed-time controller of the ship and the yaw fixed-time controller of the ship, including:[[]] S51. Combine the fixed-time convergence theory, formula (18), and formula (19) to design the longitudinal control force of the ship, as shown in formula (27). wherein, c3, c4, and ε1 are all positive design parameters, defining a first composite disturbance φ1, φ1 = -o1 - n(τ u ) + τ wu ; μ1 is the upper bound of φ1, is the estimated value of μ1; Design and adaptive laws, i.e., and derivatives to obtain and values, as shown in Eq. (28), In the formula, η1, η2, σ1, σ2, σ3, and σ4 are positive design parameters. S52. Design the yaw control moment according to the fixed-time convergence theory, formula (23), and formula (26), as shown in formula (29). where \(c_5\), \(c_6\) and \(\varepsilon_2\) are all positive design parameters, and the second composite disturbance \(\varphi_2\) is defined as \(\varphi_2 = -o_2 - n(\tau r )+\tau wr ;\(\mu_2\) is the upper bound of \(\varphi_2\), is the estimated value of \(\mu_2\); Design and adaptive laws, i.e., and derivatives of and to obtain the values of In the formula, η3, η4, σ5, σ6, σ7, and σ8 are positive design parameters.

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