Unmanned surface vehicle recursive terminal sliding mode control method based on new disturbance observer

Abstract

This invention provides a recursive terminal sliding mode control method for unmanned surface vessels (USVs) based on a novel perturbation observer, relating to the field of USV technology. First, a fixed-time perturbation observer (FTDO) is designed to suppress model uncertainties and the adverse effects of external perturbations. Then, an adaptive fixed-time perturbation observer (AFTDO) is developed, whose adaptive law does not require prior knowledge of lumped perturbations. Unlike existing fast non-singular terminal sliding mode surfaces (FNTSMs) that use linear sliding mode (LSM) to avoid singularities, this invention innovatively constructs a recursive terminal sliding mode surface (RTSM), completely avoiding singularities through a recursive structure, while significantly improving convergence accuracy and speed near the equilibrium point. Based on the lumped perturbation estimate from AFTDO, a fixed-time recursive terminal sliding mode controller (FTRTSMC) is designed, strictly ensuring that the trajectory tracking error converges to zero within a fixed time (convergence time is independent of the initial state).

Description

Technical Field

[0001] This invention relates to the field of unmanned surface vessel technology, and in particular to a recursive terminal sliding mode control method for unmanned surface vessels based on a novel disturbance observer. Background Technology

[0002] In recent years, maritime safety and resource exploration have received increasing attention. With their high degree of autonomy, unmanned surface vessels (USVs) have shown great potential in missions such as maritime rescue, marine mapping, environmental monitoring, and geophysical exploration, and have been widely used in military and commercial fields.

[0003] However, various unknown disturbances exist in the actual marine environment, and the parameters of USV models are difficult to obtain accurately due to the limitations of sea trial conditions. Therefore, before designing a trajectory tracking controller, it is necessary to effectively identify internal and external disturbances of the system. Traditional disturbance identification methods, such as active disturbance rejection control, neural network approximation, and Kalman filtering, have been widely used, but they generally suffer from insufficient accuracy and susceptibility to getting trapped in local optima. To address this, disturbance observer technology has been proposed to improve the robustness and accuracy of the system by estimating unknown disturbances. Existing research has proposed schemes such as finite-time convergent observers and extended state observers (ESOs) for underactuated USVs, achieving navigation control and fault compensation, but the estimation accuracy and convergence performance are still limited. While fixed-time disturbance observers (FTDOs) can enhance the system's ability to suppress uncertainties, their estimation accuracy and convergence speed still have room for improvement.

[0004] Sliding mode control (SMC), proposed by Utkin in 1977, is widely used in USV control, motor drives, and power systems due to its strong robustness against model uncertainties and external disturbances. In recent years, with the development of fast finite / fixed-time convergence theory, researchers have proposed various fast nonsingular terminal sliding surfaces (FNTSMs) with finite or fixed-time stability, providing new approaches for high-precision trajectory tracking in USVs. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a recursive terminal sliding mode control method for unmanned surface vessels based on a novel disturbance observer;

[0006] A recursive terminal sliding mode control method for unmanned surface vessels based on a novel disturbance observer includes the following steps:

[0007] Step 1: Construct the kinematic and dynamic models of three-degree-of-freedom unmanned surface vessels (USVs);

[0008] Specifically, consider the following nonlinear system:

[0009] (1);

[0010] Where x(t) is the state variable at time t, f(x(t)) is a continuous nonlinear function, and x0 is the initial state of the nonlinear system;

[0011] Assume that the nonlinear system (1) has a unique solution for any initial state:

[0012] Assumption 1: Lumped perturbation D and its first derivative Second derivative Unknown and bounded, meaning there exist unknown positive constants δ1, δ2, δ3 satisfying ||D||≤δ1. ≤δ2, ≤δ3;

[0013] Assumption 2: The reference trajectory ηd and its first and second derivatives are available and bounded;

[0014] Lemma 1: For a nonlinear system, if there exists a positive definite Lyapunov candidate function V(x(t)) satisfying:

[0015] (2);

[0016] Where α, β, p, k, and γ are positive numbers, if pγ < 1 and kγ > 1, then the nonlinear system is stable in a fixed time, and the upper bound of the stable time is T. max satisfy:

[0017] (3);

[0018] Lemma 2: For a nonlinear system, if there exists a positive definite Lyapunov candidate function V(x(t)) and positive scalars pγ<1, kγ>1, , so that:

[0019] (4);

[0020] in Represents disturbances or uncertainties, when Then the nonlinear system is practically fixed-time stable, and the residual set of the solution is expressed as:

[0021] (5);

[0022] Where t represents time, and T is the actual convergence time; when the constant μ has 0 < μ < 1, the upper bound of the convergence time is T. max satisfy:

[0023] (6);

[0024] Lemma 3: For any real number z iLet i = 1, 2, ..., n, where n is the vector dimension and 0 < ζ1 < 1, ζ2 > 1. Then:

[0025] (7);

[0026] and:

[0027] (8);

[0028] The kinematic and dynamic model of the three-degree-of-freedom unmanned surface vessel (USV) is then modeled as follows:

[0029] (9);

[0030] Where η = [x, y, ψ] T This represents the position and heading angle of unmanned surface vessels (USVs) on a fixed inertial horizontal and vertical coordinate system on Earth. yes The time derivative, i.e., the velocity in the inertial coordinate system, is v'=[u,v,r]. T Let τ represent the sway velocity, yaw velocity, and angular rate in the Earth's fixed inertial coordinate system, respectively, and τ=[τ1,τ2,τ3]. T Let and d be the control inputs and satisfy d = [d1, d2, d3], respectively. T The bounded external disturbances are d1, d2, and d3, representing the sway disturbance, roll disturbance, and pitch moment disturbance, respectively; R(ψ) is a rotation matrix dependent on the heading angle ψ, used to transform the velocity between the inertial coordinate system and the body coordinate system; M is the inertial matrix, C(v) is the Coriolis and centripetal matrix, and D(v) is the damping matrix, expressed as follows:

[0031] , ;

[0032] , ;

[0033] in:

[0034] (10);

[0035] Here, m is the mass of USVs, and I z The yaw moment of inertia, symbol middle, = Let represent the corresponding hydrodynamic derivatives caused by the sway acceleration, the roll acceleration, and the heave angular acceleration, respectively. = These represent the corresponding hydrodynamic derivatives caused by the surge velocity, sway velocity, and pitch angular velocity, respectively. representing the square of the oscillation velocity, u 2 The resulting sway drag and sway velocity are multiplied by the absolute value of the velocity |v|v, and the sway drag and angular velocity are multiplied by the absolute value of the angular velocity |r|r, resulting in rotational damping torque. These represent the sway force caused by the |v|r term and the yaw moment caused by the |r|v term, respectively; x g This represents the longitudinal coordinate of the center of gravity in the ship's coordinate system; for ease of subsequent calculations, it is obtained through equivalent table transformation:

[0036] (11);

[0037] Among them, N0 -1 =R(ψ)M -1 B0 = C(v) + D(v), D = =[D1,D2,D3] T D represents the lumped disturbance, and D1, D2, and D3 represent the disturbance force in the sway direction, the disturbance force in the roll direction, and the disturbance torque in the head-sway direction, respectively.

[0038] Step 2: Assumption: The lumped disturbance is estimated in real time by adapting the fixed-time disturbance observer AFTDO. ;

[0039] Let x∈R n And a∈R, where R is the set of all real numbers, R n Let Sig be an n-dimensional vector space. a (x)=[∣x1∣ a sgn(x1),…,∣x n | a sgn(x n )] T , where sgn(∙) is the sign function, and diag(x) = diag(x1,…,x) is defined. n ) is the diagonal element composed of x i Construct a diagonal matrix; design the function sig a ( )=[sig( 1), sig( 2), sig( 3)] T Let sig be a sign function. Assume the lumped perturbation D and its first and second derivatives are unknown and bounded, i.e., they exist. satisfy And reference trajectory Its first and second derivatives are available and bounded;

[0040] The fixed-time perturbation observer (AFTDO) is specifically described below:

[0041] (12);

[0042] Among them, intermediate variables are defined. T , These represent the disturbance estimation errors in the sway direction, the lateral sway direction, and the head roll direction, respectively, and a new auxiliary dynamic equation is defined. And satisfy Z= ; This is an estimate of D. yes The second derivative with respect to time is acceleration. Represents the expected acceleration, represent First derivative, calculate The results are as follows:

[0043] (13);

[0044] in, For the disturbance estimation error, It is the first derivative of the intermediate construction variable Z with respect to time. It is the internal estimate of the intermediate variable Z by the observer. The first derivative with respect to time, k1=diag(k 11 ,k 12 ,k 13 ), k2=diag(k 21 ,k 22 ,k 23 ), k 11 k 12 k 13 k 21 k 22 k 23 k3, λ1, and λ2 are positive constants and satisfy 0 < λ2 < 1 and λ1 > 0.

[0045] For the USVs system in equation (11), when the assumptions are satisfied, It is observed through (12) that the perturbation estimation error will converge to zero within a fixed time T1, and the settling time satisfies:

[0046] (14);

[0047] in, , satisfy .

[0048] The sliding component k3sig(σ) in equation (12) is used as the differential term to compensate for lumped disturbances. Based on the inspiration from Hypothesis 1, k3 is chosen as the sliding gain; equivalent control is adopted. The derivative, after eliminating disturbances, satisfies |u| during the sliding mode phase. eq1 |=| |,∣u eq2 |=| |,∣u eq3 |=| |, in which yes The derivative, =[ ] T , It is equivalent control. If the equivalent control represents the sway direction, the equivalent control represents the roll direction, and the equivalent control represents the pitch direction, then the fixed-time disturbance observer AFTDO is adaptively implemented through a nonlinear low-pass filter, as follows:

[0049] (18);

[0050] in, For equivalent control A close approximation of, and , Let r1 and r2 represent the equivalent control tight approximation values ​​in the pitch, sway, and yaw directions, respectively. r1 and r2 are positive constants satisfying r1 > 0, r2 < 1, and r α It is a minimal time constant;

[0051] Design an adaptive law as follows:

[0052] (19);

[0053] Where k4, k5, r0 and β α It is a positive constant, satisfying 0 < β α <1 and r0>0; k 3i and r ai It is a time-varying positive scalar function, and ζ i It is a new error variable. It is a very small positive scalar that can be adjusted according to the estimation accuracy of the interference observer;

[0054] Combining equations (12), (17), and (18), we obtain the fixed-time perturbation observer AFTDO;

[0055] Step 3: Construct a recursive terminal sliding mode surface (RTSM) to solve the singularity problem of terminal sliding mode control;

[0056] Specifically, based on a disturbance observer, a recursive terminal sliding surface control scheme is designed, defining the tracking error of unmanned surface vessels (USVs) as e = η - η d =[e1,e2,e3] T e1, e2, and e3 represent the tracking errors in the sway direction, lateral sway direction, and pitch direction, respectively, where η d It is the reference trajectory vector; to ensure that the tracking error converges to zero quickly and to overcome the singularity problem, a new recursive terminal sliding surface (RTSM) is designed as follows:

[0057] (30);

[0058] Where γ and ρ are positive constants satisfying γ>1, 0<ρ<0.5, and α1=diag(α 11 ,α 12 ,α 13 ), α2=diag(α 21 ,α 22, α 23 ), β1=diag(β 11 ,β 12 ,β 13 ), β2=diag(β 21 ,β 22 ,β 23 ), where α 11 α 21 α 12 α 22 α 13 α 23 ,β 11 ,β 12 ,β 13 ,β 21 ,β 22 ,β 23 These are positive constants; S, s1, and s2 are the sliding surface variables, intermediate sliding surface variables, and bottom sliding surface variables, respectively, and S = [S1, S2, S3]. T , s1=[s 11 ,s 12 ,s 13 ] T , s2=[s 21 ,s 22 ,s 23 ] T Where S1, S2, and S3 represent the sliding surface variables in the longitudinal sway direction, the transverse sway direction, and the yaw direction, respectively; s 11,s 12 ,s 13 These represent the intermediate sliding surface variables in the longitudinal, transverse, and heave directions, respectively; s 21 ,s 22 ,s 23 These represent the bottom sliding surface variables in the longitudinal sway direction, the lateral sway direction, and the initial sway direction, respectively.

[0059] When the recursive terminal sliding mode (RTSM) slides on the sliding manifold, we have S=0, and thus:

[0060] (31);

[0061] Based on the derivation, we get:

[0062] (32) ;

[0063] The recursive terminal sliding surface (RTSM) has a two-layer recursive architecture:

[0064] First layer of sliding surface: This ensures that the principal error converges.

[0065] Second layer sliding surface: Eliminate singularities;

[0066] Lumped disturbances based on the fixed-time disturbance observer AFTDO A fixed-time recursive terminal sliding mode control (FTRTSMC) is designed to make the tracking error converge to zero within a fixed time, as detailed below:

[0067] Define the Lyapunov function:

[0068] (33);

[0069] Its time derivative is defined as:

[0070] (34);

[0071] According to Lemma 3:

[0072] (35);

[0073] in ξ1 and ξ2 are positive numbers that satisfy the following condition: ;

[0074] According to Lemma 1, s2 occurs at a fixed time T. aIt converges to zero and satisfies:

[0075] (36);

[0076] We derive that:

[0077] (37);

[0078] Choose the Lyapunov function V b :

[0079] (38);

[0080] V b The time derivative is:

[0081] (39);

[0082] Combining Lemma 3, we get:

[0083] (40);

[0084] According to Lemma 1, e has a fixed time T b Converging to zero:

[0085] (41);

[0086] in ξ3 and ξ4 are positive constants that satisfy the following condition:

[0087] ;

[0088] The final fixed-time recursive terminal sliding mode control (FTRTSMC) is as follows: (42);

[0089] in For fixed-time convergent terms, It is a positive constant and satisfies k1>k2>1;

[0090] Step 4: Implement recursive terminal sliding mode control of the unmanned surface vessel by outputting the control input vector τ.

[0091] On the other hand, the unmanned surface vessel recursive terminal sliding mode control method is implemented based on the unmanned surface vessel recursive terminal sliding mode control system, specifically including: a disturbance observation module, a recursive sliding mode generation module, a fixed time control module, and a disturbance observation module.

[0092] The disturbance observation module executes AFTDO based on a fixed-time disturbance observer and outputs data in real time. ;

[0093] The recursive sliding mode generation module: constructs a recursive terminal sliding mode surface (RTSM) calculation S;

[0094] The fixed-time control module: based on The control quantity τ is generated by S.

[0095] The disturbance observation module includes a signal processor and an adaptive gain regulator; the signal processor is used to calculate auxiliary variables. The adaptive gain regulator is used to dynamically adjust k. 3i To suppress Differential perturbation;

[0096] The beneficial effects of adopting the above technical solution are as follows:

[0097] This invention proposes a recursive terminal sliding mode control method for unmanned surface vessels based on a novel disturbance observer. The designed disturbance observer can handle external uncertain disturbances without prior knowledge and achieves fixed-time convergence of errors. To avoid singularities, a recursive terminal sliding surface (RTSM) is constructed, which improves convergence accuracy near the equilibrium point by eliminating singularities through a recursive structure. Subsequently, a sliding mode controller is designed to stabilize the closed-loop system, and the effectiveness of the method is verified through numerical simulation.

[0098] The technical effects of this method are as follows:

[0099] (1) The balance point tracking error was reduced to 10 -4 (1) The magnitude of the wave height is increased by about two levels; (2) The maximum deviation of the circular trajectory is ≤0.15m; (3) The anti-disturbance performance is enhanced, and the disturbance suppression rate is increased by 62% under the condition of wave height of 1.8 m, and the control of flutter is ≤0.8 N·m;

[0100] (4) The fixed convergence time is ≤8.7 s, which is about 50% shorter than the traditional method. Attached Figure Description

[0101] Figure 1 Overall flowchart of the recursive terminal sliding mode control method for unmanned surface vessels according to an embodiment of the present invention;

[0102] Figure 2 The coordinate system of unmanned surface vessels (USVs) in this embodiment of the invention;

[0103] Figure 3 Comparison diagram of xy-plane reference trajectory and actual trajectory in an embodiment of the present invention;

[0104] Figure 4 A schematic diagram of the reference and actual positions and headings of the xy plane in an embodiment of the present invention;

[0105] Where (a) is position x d The graph shows the change over time, with (b) representing the position y. d The graph shows the change over time, with (c) representing the heading ψ. d Graph showing changes over time;

[0106] Figure 5 A schematic diagram of position and heading tracking errors in an embodiment of the present invention;

[0107] Where (a) is the position error x e The graph shows the change over time, with (b) representing the position error y. e The graph shows the change over time, where (c) represents the heading error ψ. e Graph showing changes over time;

[0108] Figure 6 Speed ​​tracking error [u] in Embodiment 1 of the present invention e ,v e ,r e Curve graph;

[0109] Where (a) is the longitudinal velocity error u e The curve (b) shows the lateral velocity error v. e The graph (c) shows the angular velocity error r of the initial roll (rotation). e Line graph;

[0110] Figure 7 A comparison diagram of the reference trajectory and the actual trajectory on the xy plane in Embodiment 1 of the present invention;

[0111] Figure 8 Reference position, actual position, and heading diagram on the xy plane of Embodiment 1 of the present invention;

[0112] Where (a) is the desired x position x d Schematic diagram, (b) shows the desired y-position. d Schematic diagram, (c) represents the desired heading angle ψ d Schematic diagram.

[0113] Figure 9 Schematic diagram of position and heading tracking error in Embodiment 1 of the present invention;

[0114] Where (a) is the position error in the x-direction (xx) d (a) is a schematic diagram, and (b) shows the position error in the y-direction (yy). d ) Schematic diagram, (c) is the heading angle error (ψ-ψ d (Diagram)

[0115] Figure 10 Schematic diagram of speed tracking error in Embodiment 2 of the present invention;

[0116] Where (a) is the longitudinal velocity error u e Schematic diagram, (b) shows the lateral velocity error v e Schematic diagram, (c) represents the angular velocity error r of the initial rocking (rotation). e Schematic diagram.

[0117] Figure 11 A comparison diagram of the reference trajectory and the actual trajectory on the xy plane in Embodiment 2 of the present invention;

[0118] Figure 12 A schematic diagram of the reference position, actual position, and heading on the xy plane in Embodiment 2 of the present invention;

[0119] Where (a) is the desired x position x d Schematic diagram, (b) shows the desired y-position. d Schematic diagram, (c) represents the desired heading angle ψ d Schematic diagram;

[0120] Figure 13 Schematic diagram of position and heading tracking error in Embodiment 3 of the present invention;

[0121] Where (a) is the position error in the x-direction (xx) d (a) is a schematic diagram, and (b) shows the position error in the y-direction (yy). d ) Schematic diagram, (c) is the heading angle error (ψ-ψ d ) Schematic diagram;

[0122] Figure 14 Speed ​​tracking error [u] in Embodiment 3 of the present invention e ,v e ,r e ] Schematic diagram;

[0123] Where (a) is the longitudinal velocity error u e Schematic diagram, (b) shows the lateral velocity error v e Schematic diagram, (c) represents the angular velocity error r of the initial rocking (rotation). e Schematic diagram;

[0124] Figure 15 A comparison diagram of the reference trajectory and the actual trajectory on the xy plane in Embodiment 3 of the present invention;

[0125] Figure 16 A schematic diagram of the reference position, actual position, and heading on the xy plane in Embodiment 3 of the present invention;

[0126] Where (a) is the position x. d Schematic diagram, (b) shows the y-position. d Schematic diagram, (c) is the heading angle ψd Schematic diagram;

[0127] Figure 17 Schematic diagram of position and heading tracking error in Embodiment 3 of the present invention;

[0128] Where (a) is the position error in the x-direction (xx) d (a) is a schematic diagram, and (b) shows the position error in the y-direction (yy). d ) Schematic diagram, (c) is the heading angle error (ψ-ψ d ) Schematic diagram;

[0129] Figure 18 A schematic diagram of position and heading tracking errors in Embodiment 3 of the present invention.

[0130] Where (a) is the position error x in the x-direction. e The schematic diagram is shown in (b), where y is the position error in the y-direction. e (c) is a schematic diagram of the heading angle error ψ. Detailed Implementation

[0131] The specific implementation methods of this application will be further described in detail below with reference to the accompanying drawings and embodiments.

[0132] A recursive terminal sliding mode control method for unmanned surface vessels based on a novel disturbance observer, such as... Figure 1 As shown, it includes the following steps:

[0133] Step 1: Construct the kinematic and dynamic models of three-degree-of-freedom unmanned surface vessels (USVs);

[0134] Specifically, consider the following nonlinear system:

[0135] (1);

[0136] Where x(t) is the state variable at time t, f(x(t)) is a continuous nonlinear function, and x0 is the initial state of the nonlinear system;

[0137] Assume that the nonlinear system (1) has a unique solution for any initial state:

[0138] Assumption 1: Lumped perturbation D and its first order Second derivative Unknown and bounded, meaning there exist unknown positive constants δ1, δ2, δ3 ||D||≤δ1. ≤δ2, ≤δ3;

[0139] Assumption 2: The reference trajectory ηd and its first and second derivatives are available and bounded;

[0140] Lemma 1: For a nonlinear system (1), if there exists a positive definite Lyapunov candidate function V(x(t)) satisfying:

[0141] (2);

[0142] Where α, β, p, k, and γ are positive numbers, if pγ < 1 and kγ > 1, then the nonlinear system is stable in a fixed time, and the upper bound of the stable time is T. max satisfy:

[0143] (3);

[0144] Lemma 2: For a nonlinear system (1), if there exists a positive definite Lyapunov candidate function V(x(t)) and positive scalars pγ<1, kγ>1, Make:

[0145] (4);

[0146] in Represents disturbances or uncertainties, when The nonlinear system exhibits practical fixed-time stability, and the residual set of the solution is expressed as:

[0147] (5);

[0148] Where t represents time, and T is the actual convergence time; when the constant μ has 0 < μ < 1, the upper bound T of the convergence time satisfies:

[0149] (6);

[0150] Lemma 3: For any real number z i Let i = 1, 2, ..., n, where n is the vector dimension and 0 < ζ1 < 1, ζ2 > 1. Then:

[0151] (7);

[0152] (8);

[0153] The kinematic and dynamic model of the three-degree-of-freedom unmanned surface vessel (USV) is then modeled as follows:

[0154] (9);

[0155] Where η = [x, y, ψ] T This represents the position and heading angle of unmanned surface vessels (USVs) on a fixed inertial horizontal and vertical coordinate system on Earth. yes The time derivative, i.e., the velocity in the inertial coordinate system, is v'=[u,v,r].T Represent the sway velocity, roll velocity, and angular rate in the Earth's fixed inertial coordinate system, respectively (see...). Figure 2 ), τ=[τ1,τ2,τ3] T Let and d be the control inputs (forces and torques calculated and issued by the controller, acting directly on the hull of the USVs to make them move in a desired state (such as a specific trajectory, speed, or heading)) and satisfy d = [d1, d2, d3], respectively. T The bounded external disturbances are d1, d2, and d3, representing the sway disturbance, roll disturbance, and pitch moment disturbance, respectively; R(ψ) is a rotation matrix dependent on the heading angle ψ, used to transform the velocity between the inertial coordinate system and the body coordinate system; M is the inertial matrix, C(v) is the Coriolis and centripetal matrix, and D(v) is the damping matrix, expressed as follows:

[0156] , ;

[0157] , ;

[0158] in:

[0159] (10);

[0160] Here, m is the mass of USVs, and I z The yaw moment of inertia, symbol middle, = Let represent the corresponding hydrodynamic derivatives caused by the sway acceleration, the roll acceleration, and the heave angular acceleration, respectively. = These represent the corresponding hydrodynamic derivatives caused by the surge velocity, sway velocity, and pitch angular velocity, respectively. These represent the sway drag caused by the square of the sway velocity u², the sway drag caused by the absolute value of the sway velocity multiplied by the velocity |v|v, and the rotational damping torque caused by the absolute value of the angular velocity multiplied by the angular velocity |r|r, respectively. These represent the sway force caused by the |v|r term and the heave moment caused by the |r|v term, respectively. g This represents the longitudinal coordinate of the center of gravity in the ship's coordinate system; for ease of subsequent calculations, it is obtained through equivalent table transformation:

[0161] (11);

[0162] Among them, N0 -1 =R(ψ)M -1 B0 = C(v) + D(v), D = =[D1,D2,D3]T D represents the lumped disturbance, and D1, D2, and D3 represent the disturbance force in the sway direction, the disturbance force in the roll direction, and the disturbance torque in the head-sway direction, respectively.

[0163] Step 2: Assumption: The lumped disturbance is estimated in real time by adapting the fixed-time disturbance observer AFTDO. ;

[0164] x∈R n And a∈R, where R is the set of all real numbers, R n Let Sig be an n-dimensional vector space. a (x)=[∣x1∣ a sgn(x1),…,∣x n | a sgn(x n )] T , where sgn(∙) is the sign function, and diag(x) = diag(x1,…,x) is defined. n ) is the diagonal element composed of x i Construct a diagonal matrix; design the function sig a ( )=[sig( 1), sig( 2), sig( 3)] T Let sig be a sign function. Assume the lumped perturbation D and its first and second derivatives are unknown and bounded, i.e., they exist. satisfy And reference trajectory Its first and second derivatives are available and bounded. It is an estimate of D.

[0165] The fixed-time perturbation observer (AFTDO) is specifically described below:

[0166] (12);

[0167] Intermediate variables are defined to facilitate subsequent calculations. T , These represent the disturbance estimation errors in the sway direction, the lateral sway direction, and the head roll direction, respectively, and a new auxiliary dynamic equation is defined. And satisfy Z= ; yes The second derivative with respect to time is acceleration. Represents the expected acceleration, represent First derivative, calculate The results are as follows:

[0168] (13);

[0169] in, For the disturbance estimation error, It is the first derivative of the intermediate construction variable Z with respect to time. It is the internal estimate of the intermediate variable Z by the observer. The first derivative with respect to time, k1=diag(k 11 ,k 12 ,k 13 ), k2=diag(k 21 ,k 22 ,k 23 ), k 11 k 12 k 13 k 21 k 22 k 23 k3, λ1, and λ2 are positive constants and satisfy 0 < λ2 < 1 and λ1 > 0.

[0170] For the USVs system in equation (11), when the assumptions are satisfied, It is observed through (12) that the perturbation estimation error will converge to zero within a fixed time T1, and the settling time satisfies:

[0171] (14);

[0172] in, , satisfy .

[0173] By Lemma 2, we obtain: It can converge to zero within a fixed time, that is, it achieves fixed convergence of the disturbance observation error. Therefore, AFTDO in equation (12) is stable within a fixed time and the stable time satisfies equation (14).

[0174] The sliding mode component k3sig(σ) in (12) is used as the differential term to compensate for lumped disturbances. Inspired by Assumption 1, k3 is chosen as the sliding gain. However, the disturbance boundary δ2 is often unknown. To overcome this limitation, equivalent control is employed. Eliminating the derivative of the disturbance means that the switching signal in equation (12) must be averaged to maintain the sliding mode, which means that |u| must be satisfied during the sliding phase. eq1 |=| |,∣u eq2 |=| |,∣u eq3 |=| |, in which yes The derivative, =[ ] T , It is equivalent control. If the equivalent control represents the sway direction, the equivalent control represents the roll direction, and the equivalent control represents the pitch direction, then the fixed-time disturbance observer AFTDO is adaptively implemented through a nonlinear low-pass filter, as follows:

[0175] (18);

[0176] in, For equivalent control A close approximation of, and , Let r1 and r2 represent the equivalent control tight approximation values ​​in the pitch, sway, and yaw directions, respectively. r1 and r2 are positive constants satisfying r1 > 0, r2 < 1, and r α It is a very small time constant.

[0177] Subsequently, an adaptive law is designed as follows:

[0178] (19);

[0179] Where k4, k5, r0 and β α It is a positive constant, satisfying 0 < β α <1 and r0>0. k 3i and r ai It is a time-varying positive scalar function, and ζ i It is a new error variable. It is a tiny positive scalar that can be adjusted based on the accuracy of the interference observer estimation.

[0180] Combining equations (12), (17), and (18), we obtain the fixed-time perturbation observer AFTDO;

[0181] Step 3: Construct a recursive terminal sliding mode surface (RTSM) to solve the singularity problem of terminal sliding mode control;

[0182] Specifically, based on a disturbance observer, a recursive terminal sliding surface control scheme is designed, defining the tracking error of unmanned surface vessels (USVs) as e = η - η d =[e1,e2,e3] T e1, e2, and e3 represent the tracking errors in the sway direction, lateral sway direction, and pitch direction, respectively, where ηd It is the reference trajectory vector; to ensure that the tracking error converges to zero quickly and to overcome the singularity problem, a new recursive terminal sliding surface (RTSM) is designed as follows:

[0183] (30);

[0184] Where γ and ρ are positive constants satisfying γ>1, 0<ρ<0.5, and α1=diag(α 11 ,α 12 ,α 13 ), α2=diag(α 21 ,α 22, α 23 ), β1=diag(β 11 ,β 12 ,β 13 ), β2=diag(β 21 ,β 22 ,β 23 ), where α 11 α 21 α 12 α 22 α 13 α 23 ,β 11 ,β 12 ,β 13 ,β 21 ,β 22 ,β 23 These are positive constants. S, s1, and s2 are the sliding surface variables, intermediate sliding surface variables, and bottom sliding surface variables, respectively, and S = [S1, S2, S3]. T , s1=[s 11 ,s 12 ,s 13 ] T , s2=[s 21 ,s 22 ,s 23 ] T Where S1, S2, and S3 represent the sliding surface variables in the longitudinal sway direction, the transverse sway direction, and the yaw direction, respectively; s 11 ,s 12 ,s 13 These represent the intermediate sliding surface variables in the longitudinal, transverse, and heave directions, respectively; s 21 ,s 22 ,s 23 These represent the bottom sliding surface variables in the longitudinal sway direction, the lateral sway direction, and the initial sway direction, respectively.

[0185] Next, the stability of the recursive terminal sliding mode (RTSM) is proved using the Lyapunov method;

[0186] When the recursive terminal sliding mode (RTSM) slides on the sliding manifold, we have S=0, and thus:

[0187] (31);

[0188] Substituting (31) into (30), we get:

[0189] (32) ;

[0190] The recursive terminal sliding surface (RTSM) has a two-layer recursive architecture:

[0191] First layer of sliding surface: This ensures that the principal error converges.

[0192] Second layer sliding surface: Eliminate singularities;

[0193] Lumped disturbances based on the fixed-time disturbance observer AFTDO A fixed-time recursive terminal sliding mode control (FTRTSMC) is designed to make the tracking error converge to zero within a fixed time, as detailed below:

[0194] Define the Lyapunov function:

[0195] (33);

[0196] Its time derivative is defined as:

[0197] (34);

[0198] According to Lemma 3:

[0199] (35);

[0200] in ξ1 and ξ2 are positive numbers that satisfy the following condition: ;

[0201] According to Lemma 1, s2 occurs at a fixed time T. a It converges to zero and satisfies:

[0202] (36);

[0203] We derive that:

[0204] (37);

[0205] Choose the Lyapunov function V b :

[0206] (38);

[0207] V b The time derivative is:

[0208] (39);

[0209] Combining Lemma 3, we can obtain:

[0210] (40);

[0211] According to Lemma 1, e has a fixed time T b Converging to zero:

[0212] (41);

[0213] in ξ3 and ξ4 are positive constants that satisfy the following condition:

[0214] ;

[0215] Sliding surface variables s1, s2, and S are recursively associated with two layers of sliding surfaces. Once the second sliding surface is reached, the fixed-time convergence condition of s2 is satisfied. After a fixed time T... a Afterwards, s2 converges to zero, and the first sliding surface has been reached. Similarly, the fixed-time convergence condition for the error e is subsequently satisfied.

[0216] By designing sliding surfaces layer by layer, the system state gradually approximates each sliding surface, ultimately achieving the control objective. Due to the inherent characteristics of sliding mode control, RTSM (Recursive Terminal Sliding Mode) maintains strong robustness in the face of changes in system parameters or external disturbances.

[0217] Fixed-time nonsingular sliding mode (FTNSM) possesses superior characteristics such as fast convergence speed and high steady-state accuracy. However, most current FTNSM methods employ piecewise functions to avoid singularity issues, which may lead to unsmooth transitions between segments, affecting overall system performance and degrading response quality. This paper proposes a novel RTSM with a recursive structure that avoids singularity while achieving fixed-time convergence to zero.

[0218] The final fixed-time recursive terminal sliding mode control (FTRTSMC) is as follows: (42);

[0219] in For fixed-time convergent terms, It is a positive constant and satisfies k1>k2>1;

[0220] The stability analysis of the nonlinear system under the fixed-time recursive terminal sliding mode control (FTRTSMC) is as follows:

[0221] Through Lyapunov functions:

[0222] (43);

[0223] Its time derivative is calculated as follows:

[0224]

[0225] Substituting the control law of the recursive terminal sliding mode controller (RTSMC), we get:

[0226] (45);

[0227] According to the fixed-time perturbation observer AFTDO, the perturbation estimation error converges to zero after time T1, therefore:

[0228]

[0229] because and Then S can reach the RTSM surface within a fixed time. The steady-state time is bounded and satisfies:

[0230]

[0231] in It is a positive number that satisfies the following conditions:

[0232]

[0233] Based on the foregoing proof, once S converges to zero, e will also converge to zero. Therefore, this control law can stabilize the trajectory error over a fixed time; the complete convergence time is bounded by:

[0234] T≤T1+T2+T a +T b +T s (49);

[0235] Step 4: Implement recursive terminal sliding mode control of the unmanned surface vessel by outputting the control input vector τ.

[0236] The control input vector τ (i.e., thrust or torque) directly drives the dynamic system of the USVs.

[0237] Importance: τ is the execution variable that directly acts on USVs, ensuring that the system state (position, velocity) quickly tracks the reference trajectory. Simulation verification shows that the calculation of τ can effectively suppress disturbances and achieve high-precision control (see...). Figure 10 ).

[0238] Estimated disturbance It is the output of AFTDO, representing an estimate of the lumped disturbance (including external disturbances and system uncertainties).

[0239] Importance: As an intermediate output, it is used to compensate for disturbances in the control law (the term in formula (42)). This method emphasizes (Assumption 1) that the perturbation is unknown and bounded, and that FTDO converges within a fixed time, ensuring... Accurately estimating the true disturbance D improves control robustness and prevents trajectory deviation.

[0240] Sliding surface variables S, s1, s2: These are the output variables of the recursive terminal sliding surface RTSMS;

[0241] Importance: The sliding surface variable serves as the input to the control law (in formula (42)). The driving error converges rapidly. The RTSMS design ensures fixed-time convergence and has no singularity issues, which is superior to traditional sliding mode control.

[0242] Tracking error e and its derivative: This represents the tracking error in position and heading (η is the actual state, η...). d (This is the reference trajectory). Its derivative Used for calculating sliding surfaces and control inputs.

[0243] Importance: Error is the core input of the controller, which is determined by the dynamics of formula (37). Driven convergence. Simulations show that the error approaches zero within a fixed time interval (e.g., Figure 9 (This demonstrates the convergence of position and heading errors).

[0244] In addition, the control process involves other auxiliary outputs for state estimation and error correction. The mechanism is a closed-loop process, consisting of three stages: disturbance estimation, sliding mode control calculation, and execution.

[0245] Step S1: Perturbation Estimation Stage (AFTDO):

[0246] Objective: To estimate external disturbances (such as wind, waves, and water flow) and system uncertainties in real time to ensure control robustness.

[0247] process:

[0248] Input: USVs states η, ν and reference trajectory η d .

[0249] Output: Estimated disturbance .

[0250] Mechanism: AFTDO uses an intermediate variable σ and an adaptive gain k3. AFTDO converges in a fixed time (Lemma 2) and satisfies || -D||≤δ;

[0251] Importance: Disturbance compensation is crucial for control, preventing trajectory deviation. Simulation comparisons show that AFTDO is more effective than traditional observers. Figure 15-18 ).

[0252] Step S2: Sliding Mode Control Calculation Stage (RTSMC):

[0253] Objective: To calculate the control input τ so that the tracking error converges to zero within a fixed time.

[0254] process:

[0255] (1) Calculate the tracking error. Input the reference trajectory η d Given the actual state η, the output e = η - η d .

[0256] (2) Design the sliding surfaces. Calculate the recursive sliding surfaces S, s1, s2 based on the error e. This avoids singularities and ensures smooth convergence.

[0257] (3) Calculate the control input τ. Use the control law (Formula 42), combined with the error e and its derivative. Sliding surface variable s2 and estimated disturbance The core principle is to ensure stability through the Lyapunov function (Equation 43): Less than or equal to the convergent term. Equations 45-48 prove that the system at a fixed time T... s Convergence.

[0258] Output: Final control input τ.

[0259] Importance: RTSMC offers fast convergence and high accuracy. Compared to traditional sliding mode control (TISMC), RTSMC exhibits smaller overshoot and lower steady-state error. Figure 11-14 ).

[0260] Step S3: Execution and Closed-Loop Control Phase

[0261] Objective: To achieve trajectory tracking by applying τ-driven USVs dynamics.

[0262] process:

[0263] Input: The control input τ acts on the USVs dynamic equation (Equation 11): .

[0264] Output: Update the states η and ν of the USVs.

[0265] Closed-loop feedback: Measure the new state in real time and repeat the above steps.

[0266] Overall performance: Simulation results show that the control mechanism can achieve convergence within a fixed time (approximately 5-10 seconds). Figure 4 ,7), the upper bound of the total time is T≤T1+T2+T a +T b +T s (Formula 49).

[0267] Advantages:

[0268] Fixed-time convergence: The error converges within a predefined time, regardless of the initial state (Lemma 1-2).

[0269] Robustness: FTDO handles disturbances, and RTSMC suppresses chattering.

[0270] No singularity: Recursive sliding surface design avoids control failure.

[0271] On the other hand, the unmanned surface vessel recursive terminal sliding mode control method is implemented based on the unmanned surface vessel recursive terminal sliding mode control system, which includes a disturbance observation module, a recursive sliding mode generation module, and a fixed time control module.

[0272] The disturbance observation module employs an Adaptive Fixed-Time Disturbance Observer (AFTDO) to estimate external disturbances in real time. It includes a signal processor and an adaptive gain adjuster; the former calculates auxiliary variables, and the latter dynamically adjusts the gain k. 3i To suppress the perturbation differential term The module code execution frequency is ≥1 kHz.

[0273] The recursive sliding mode generation module calculates the sliding mode variable S based on the recursive terminal sliding surface (RTSM), with a recursive calculation depth of 3 layers.

[0274] The fixed-time control module generates the control quantity τ based on the disturbance estimate and the sliding mode variable S, thereby achieving fixed-time convergent control of the system.

[0275] The program execution flow is as follows: initialize the dynamic parameters M, C, and D of the USVs, read the sensor data η and v in real time, and output the control quantity τ to the thruster actuator after calculation by the above modules.

[0276] Example 1:

[0277] Reference trajectory Generated by the following systems:

[0278] Track 1:

[0279]

[0280] Track 2:

[0281]

[0282] The initial state (i.e., position and velocity) of the USVs is set as follows: External interference It is given by the following formula:

[0283]

[0284] For trajectory 2, the initial state of the USVs is set as follows: The external interference configuration is the same as that of trajectory 1.

[0285] Example 2:

[0286] Comparative experiments were conducted to more intuitively demonstrate the superior tracking performance of the proposed control strategy. Two sets of comparisons were set up: the first set compared the traditional sliding mode control (SMC) with the control scheme proposed in this paper; the second set compared the traditional disturbance observer with the disturbance observer proposed in this paper.

[0287] A comparison is made between the traditional integral sliding mode control (TISMC) method and the RTSMC scheme proposed in this paper.

[0288] To verify the advantages of the proposed approximation method, a comparative experiment was conducted. The traditional integral sliding mode control (TISMC) was designed as follows:

[0289]

[0290] in The parameters are selected as k1=k2=2, t1=1 / 3;

[0291] Example 3:

[0292] To verify the advantages of the proposed Adaptive Fixed-Time Disturbance Observer (AFTDO), a comparative experiment was conducted with the traditional disturbance observer.

[0293] Figure 15-18 Simulation results are presented. Through comparison... Figure 15-18 and Figure 7-9 As can be seen, when the prior information of the disturbance is unknown, the AFTDO proposed in this paper can effectively eliminate the influence of the disturbance and significantly improve the tracking accuracy of unmanned surface vessels (USVs), thereby achieving better control performance. In addition, this scheme can also guarantee error convergence within a fixed time.

[0294] Simulation study:

[0295] To verify the effectiveness of the developed scheme, numerical simulations were conducted using MATLAB / Simulink software. This embodiment uses a scaled-down CyberShipII model ship as the simulation case; its model parameters are shown in Table I, and the controller parameters shown in Table II were used for simulation analysis.

[0296] Table 1: Main parameters of the cybership II:

[0297] Table 2: CyberShip II:

[0298] The main parameters of the model:

[0299] Controller parameters: Para: parameter name, Val: parameter value;

[0300] Results of different trajectories:

[0301] Trajectory 1: Sine reference trajectory. Initial state and external disturbance are given.

[0302] Trajectory 2: Circular reference trajectory. Initial state is zero, perturbation is the same as Trajectory 1.

[0303] Figure 2-4 This demonstrates the tracking performance of trajectory 1.

[0304] like Figure 3 As shown, the solid line represents the actual trajectory of the unmanned surface vessel (USV) under the control strategy proposed in this paper, and the dashed line represents the expected trajectory of the USV, defined by a periodic function generated by formula (34). The two curves basically overlap, indicating that the controller can efficiently track complex trajectories, verifying the effectiveness of the control scheme in trajectory tracking.

[0305] like Figure 4 As shown in (a), the dashed line represents the desired position x. d In (b), the dashed line represents the desired position y. d In (c), the dashed line represents the desired heading angle ψ. d The solid line represents the actual output position and heading of the unmanned surface vessel. The actual curve closely matches the reference curve, indicating that the controller can accurately control both position and heading simultaneously, and exhibits small overshoot and fast convergence during dynamic response.

[0306] Figure 5-9The tracking performance of trajectory 2 demonstrates that USVs can track the desired trajectory with high precision and within a fixed time. Even with system uncertainties and external disturbances, RTSMC still exhibits superior tracking performance.

[0307] like Figure 5 As shown, (a), (b), and (c) represent the position error x in the x-direction, respectively. e =xx d y-direction position error e =yy d , heading angle error ψ e =ψ-ψ d The error rapidly approaches zero over time, proving that the system converges within a fixed time and has strong anti-interference ability, which is consistent with the fixed-time stability theory.

[0308] like Figure 6 As shown, the three red curves (a), (b), and (c) represent the velocity tracking error of the unmanned surface vessel in three degrees of freedom, u. e Longitudinal (forward / backward) speed error, v e Lateral (left / right) velocity error, r e Initial roll (rotation) angular velocity error. All error curves converge very quickly from the initial value and stabilize near zero, indicating that the designed controller can quickly and accurately command USVs to reach and maintain the desired speed, with excellent dynamic response performance.

[0309] like Figure 7 As shown, the dashed line represents the desired circular path (radius 1 meter) defined by formula (35), while the solid line represents the actual trajectory of the unmanned surface vessel under the proposed Adaptive Fixed-Time Disturbance Observer (AFTDO) Fast Terminal Sliding Mode Control (RTSMC) scheme. The two trajectories almost completely overlap. This is the most intuitive and powerful evidence that the overall control system (controller + observer) has extremely high trajectory tracking accuracy, and can strictly follow the desired path even with disturbances and initial deviations.

[0310] like Figure 8 As shown, the dashed lines in (a), (b), and (c) correspond to the expected x, respectively. d Position, y d Position and heading angle ψ d The solid lines correspond to the actual x-position, y-position, and heading angle ψ of the USVs output, respectively. This further confirms the excellent tracking performance in position and heading, with the actual signal closely matching the reference signal.

[0311] like Figure 9 As shown, curves (a), (b), and (c) represent the position error x in the x-direction, respectively. e y-direction position error eand heading angle error The curve converges quickly and smoothly to zero, and then stabilizes within a very small range of fluctuations, which demonstrates the stability, robustness and anti-interference capability of the control system.

[0312] like Figure 10 As shown, curves (a), (b), and (c) represent the longitudinal velocity error u of the unmanned surface vessel. e Lateral velocity error v e and the error of the initial angular velocity r e The comparison shows that the proposed RTSMC scheme has smaller velocity tracking error, faster convergence speed, and less steady-state ripple (chatter). This indicates that RTSMC can control the velocity of USVs more accurately and smoothly.

[0313] like Figure 11 As shown, the dashed line represents the target path that the unmanned surface vessel (USV) needs to track, which is preset by the program; the blue solid line represents the actual trajectory of the USV when using the traditional sliding mode control scheme (TISMC); the red solid line represents the actual trajectory of the USV when using the fast terminal sliding mode control (RTSMC) scheme proposed in this paper. This curve demonstrates the superior performance of the new scheme. It can be seen that it almost completely overlaps with the dashed line, which proves that the RTSMC scheme has extremely high tracking accuracy.

[0314] like Figure 12 As shown, the black dashed lines in (a), (b), and (c) represent the desired position x, respectively. d y d and heading signal ψ d The dashed line represents the x-coordinate of the desired trajectory. d y coordinates d and heading angle ψ d The blue solid line represents the actual output position (x, y) and heading (ψ) signals of the traditional control scheme (TISMC), while the red solid line represents the actual output position (x, y) and heading (ψ) signals of the proposed RTSMC scheme. The red solid line closely follows the red dashed line, indicating that both position and heading can achieve precise tracking with no delay and no steady-state error. This demonstrates a smooth control process, successful suppression of chattering, and a fast dynamic response with small overshoot. This is the most intuitive performance demonstration. The actual trajectory of RTSMC almost completely overlaps with the reference trajectory, while the actual trajectory of TISMC may have significant tracking lag or deviation. This directly proves that the proposed scheme has superior trajectory tracking accuracy.

[0315] like Figure 13 As shown, curves (a), (b), and (c) represent the position error x in the x-direction, respectively. e y-direction position errore and heading angle error ψ e The red curve represents the error curve of the proposed RTSMC scheme, which converges extremely quickly and stabilizes within a very small range near zero, indicating excellent convergence and stability. The blue curve represents the error curve of the traditional TISMC scheme, which converges more slowly and may not even converge completely to zero, instead fluctuating around a larger value, indicating poor tracking accuracy and steady-state error. This demonstrates that the control method proposed in this paper has smaller overshoot and superior tracking accuracy.

[0316] like Figure 14 As shown, curves (a), (b), and (c) represent the longitudinal velocity error u of the unmanned surface vessel. e Lateral velocity error v e and the error of the initial angular velocity r e The black dashed line represents the ideal zero-error line; the blue solid line represents the velocity error using the traditional observer scheme, which is larger and fluctuates more violently, indicating its limited anti-interference capability; the red solid line represents the velocity error using the AFTDO scheme presented in this paper. This error is smaller, converges faster, and stabilizes within a narrower range near zero. The overall performance of the red curve is far superior to that of the blue curve, proving that AFTDO can more accurately estimate and compensate for disturbances, thereby enabling the controller to output more precise control commands, allowing the USVs to keep up with the expected speed faster and more stably.

[0317] like Figure 15 As shown, the black dashed line represents the desired circular reference trajectory; the blue solid line represents the actual trajectory of the USVs when using the traditional observer scheme; and the red solid line represents the actual trajectory of the USVs when using the AFTDO scheme proposed in this paper. The red trajectory almost completely overlaps with the black trajectory, while the blue trajectory shows significant deviation and tracking lag, failing to perfectly reproduce the circular path. This demonstrates that the introduction of FTDO significantly improves the trajectory tracking accuracy of the system.

[0318] like Figure 16 As shown, the black dashed lines in (a), (b), and (c) represent the x-positions respectively. d , y position y d and heading angle (ψ) d The signal changes over time; the black dashed lines represent the actual output signal (x) of the traditional scheme. r y r , ψ r The red solid lines represent the actual output signals (x) of the AFTDO scheme presented in this paper. r y r , ψ rThe red solid line closely follows the black dashed line across the entire time axis, while the blue solid line exhibits significant phase lag and tracking error, particularly in heading angle tracking, where the difference may be even greater. This quantitatively confirms the advantages of the AFTDO scheme from the perspective of "signal tracking".

[0319] like Figure 17 As shown, the blue solid lines in (a), (b), and (c) represent the positional error x in the x-direction of the traditional scheme, respectively. e y-direction position error e and heading angle error ψ e The error is relatively large, convergence is slow, and it may continue to fluctuate. The red solid lines represent the x-direction position error x of the AFTDO scheme in this paper. e y-direction position error e and heading angle error ψ e The error converges rapidly and remains within a very small range. The amplitude of the red curve is much smaller than that of the blue curve and approaches zero much faster. This figure directly proves that the application of AFTDO significantly reduces all tracking errors, thereby achieving higher tracking accuracy.

[0320] like Figure 18 As shown, the blue curves in (a), (b), and (c) represent the x-direction position error of the control scheme of the traditional disturbance observer, respectively. e y-direction position error e and heading angle error ψ e The red curve represents the tracking error of the control scheme with an integrated adaptive fixed-time disturbance observer (AFTDO). Compared to the blue curve, the red curve is extremely smooth and closely follows the zero error line, which means that AFTDO has extremely high accuracy, excellent anti-interference ability, and effective suppression of chattering.

[0321] (1) Comparison with traditional integral sliding mode control (TISMC):

[0322] Figure 10-13 The TISMC results show that the proposed method has smaller RTSMC overshoot, faster convergence, and lower steady-state error, effectively suppressing chattering.

[0323] (2) Comparison with the literature perturbation observer:

[0324] Figure 14-17 The results of the proposed method show that the proposed AFTDO method outperforms the methods in the literature in terms of perturbation suppression and tracking accuracy, and also guarantees convergence within a fixed time.

[0325] Overall conclusion: The RTSMC algorithm is feasible, possessing fast tracking speed, high accuracy, and chatter-free control input.

Claims

1. A recursive terminal sliding mode control method for unmanned surface vessels based on an improved disturbance observer, characterized in that, Includes the following steps: Step 1: Construct the kinematic and dynamic models of three-degree-of-freedom unmanned surface vessels (USVs); Step 1 specifically involves considering the following nonlinear system: （1）； Where x(t) is the state variable at time t, f(x(t)) is a continuous nonlinear function, and x0 is the initial state of the nonlinear system; Assume that the nonlinear system (1) has a unique solution for any initial state: Assumption 1: Lumped perturbation D and its first derivative Second derivative Unknown and bounded, meaning there exist unknown positive constants δ1, δ2, and δ3 satisfying ||D||≤δ1. ≤δ2, ≤δ3; Assumption 2: Reference trajectory η d Its first and second derivatives are available and bounded; Lemma 1: For a nonlinear system, if there exists a positive definite Lyapunov candidate function V(x(t)) satisfying: （2）； Where α, β, p, k, and γ are positive numbers, if pγ < 1 and kγ > 1, then the nonlinear system is stable in a fixed time, and the upper bound of the stable time is T. max satisfy: （3）； Lemma 2: For a nonlinear system, if there exists a positive definite Lyapunov candidate function V(x(t)) and positive scalars pγ<1, kγ>1, , so that: （4）； in Represents disturbances or uncertainties, when Then the nonlinear system is practically fixed-time stable, and the residual set of the solution is expressed as: （5）； Where t represents time, and T is the actual convergence time; When the constant μ has 0 < μ < 1, the upper bound T of the convergence time satisfies: （6）； Lemma 3: For any real number z i Let i = 1, 2, ..., n, where n is the vector dimension and 0 < ζ1 < 1, ζ2 > 1. Then: （7）； and: （8）； The kinematic and dynamic model of the three-degree-of-freedom unmanned surface vessel (USV) is then modeled as follows: （9）； Where η = [x, y, ψ] T This represents the position and heading angle of unmanned surface vessels (USVs) on a fixed inertial horizontal and vertical coordinate system on Earth. yes The time derivative, i.e., the velocity in the inertial coordinate system, is v'=[u,v,r]. T Let τ represent the sway velocity, yaw velocity, and angular rate in the Earth's fixed inertial coordinate system, respectively, and τ=[τ1,τ2,τ3]. T Let and d be the control inputs and satisfy d = [d1, d2, d3], respectively. T The bounded external disturbances are d1, d2, and d3, representing the sway disturbance, roll disturbance, and pitch moment disturbance, respectively; R(ψ) is a rotation matrix dependent on the heading angle ψ, used to transform the velocity between the inertial coordinate system and the body coordinate system; M is the inertial matrix, C(v) is the Coriolis and centripetal matrix, and D(v) is the damping matrix, expressed as follows: ， ； ， ； in: (10)； Here, m is the mass of USVs, and I z The yaw moment of inertia, symbol middle, = Let these represent the corresponding hydrodynamic derivatives caused by the sway acceleration, the roll acceleration, and the heave acceleration, respectively. = These represent the corresponding hydrodynamic derivatives caused by the surge velocity, sway velocity, and pitch angular velocity, respectively. Let u² represent the sway drag caused by the square of the sway velocity, u², the sway drag caused by the absolute value of the sway velocity multiplied by the velocity |v|v, and the rotational damping torque caused by the absolute value of the angular velocity multiplied by the angular velocity |r|r, respectively. These represent the sway force caused by the |v|r term and the yaw moment caused by the |r|v term, respectively; x g This represents the longitudinal coordinate of the center of gravity in the ship's coordinate system; for ease of subsequent calculations, it is obtained through an equivalent table transformation: （11）； Among them, N0 -1 =R(ψ)M -1 B0 = C(v) + D(v), D = =[D1,D2,D3] T D represents the lumped disturbance, and D1, D2, and D3 represent the disturbance force in the sway direction, the disturbance force in the roll direction, and the disturbance torque in the head-sway direction, respectively. Step 2: Assume that the lumped disturbance is estimated in real time by adapting the fixed-time disturbance observer AFTDO. ; Step 3: Construct a recursive terminal sliding mode surface (RTSM) to solve the singularity problem of terminal sliding mode control; Step 4: Implement recursive terminal sliding mode control of the unmanned surface vessel by outputting the control input vector τ.

2. The recursive terminal sliding mode control method for unmanned surface vessels based on an improved disturbance observer according to claim 1, characterized in that, Step 2 specifically involves: Let x∈R n And a∈R, where R is the set of all real numbers, R n Let Sig be an n-dimensional vector space. a (x)=[∣x1∣ a sgn(x1),…,∣x n | a sgn(x n )] T , where sgn(∙) is the sign function, and diag(x) = diag(x1,…,x) is defined. n ) is the diagonal element composed of x i Construct a diagonal matrix; design the function sig a ( )=[sig( 1), sig( 2), sig( 3)] T Define sig as a sign function; assume that the lumped perturbation D and its first and second derivatives are unknown and bounded, i.e., there exists satisfy And reference trajectory Its first and second derivatives are available and bounded. It is an estimate of D; The fixed-time perturbation observer (AFTDO) is specifically described below: （12）； Intermediate variables are defined to facilitate subsequent calculations. T , These represent the disturbance estimation errors in the sway direction, the lateral sway direction, and the head roll direction, respectively, and a new auxiliary dynamic equation is defined. And satisfy Z= ; yes The second derivative with respect to time is acceleration. Represents the expected acceleration, represent First derivative, calculate The results are as follows: （13）； in, For the disturbance estimation error, It is the first derivative of the intermediate construction variable Z with respect to time. It is the first derivative of the observer's internal estimate of the intermediate variable Z with respect to time, k1=diag(k 11 ,k 12 ,k 13 ), k2=diag(k 21 ,k 22 ,k 23 ), k 11 k 12 k 13 k 21 k 22 k 23 k3, λ1, and λ2 are positive constants and satisfy 0 < λ2 < 1 and λ1 > 0. For the USVs system in equation (11), when the assumptions are satisfied, It is observed through (12) that the perturbation estimation error will converge to zero within a fixed time T1, and the settling time satisfies: （14）； in, , satisfy ; The sliding component k3sig(σ) in equation (12) is used as the differential term to compensate for lumped disturbances. Based on the inspiration from Hypothesis 1, k3 is chosen as the sliding gain; equivalent control is adopted. The derivative, after eliminating disturbances, satisfies |u| during the sliding mode phase. eq1 ∣=| |,∣u eq2 ∣=| |,∣u eq3 ∣=| |, in which yes The derivative of =[ ] T , It is equivalent control. The equivalent control represents the sway direction, the equivalent control represents the roll direction, and the equivalent control represents the pitch direction. The adaptive control of the fixed-time disturbance observer (AFTDO) is achieved through a nonlinear low-pass filter, as follows: （18）； in, For equivalent control A close approximation of, and , Let r1 and r2 represent the equivalent control tight approximation values ​​in the pitch, sway, and yaw directions, respectively. r1 and r2 are positive constants satisfying r1 > 0, r2 < 1, and r α It is a minimal time constant; Design an adaptive law as follows: （19）； Where k4, k5, r0 and β α It is a positive constant, satisfying 0 < β α <1 and r0>0; k 3i and r ai It is a time-varying positive scalar function, and ζ i It is a new error variable. It is a very small positive scalar that can be adjusted according to the estimation accuracy of the interference observer; Combining equations (12), (17) and (18), we obtain the fixed-time perturbation observer AFTDO.

3. The recursive terminal sliding mode control method for unmanned surface vessels based on an improved disturbance observer according to claim 2, characterized in that, Step 3 specifically involves: based on the disturbance observer, a recursive terminal sliding surface control scheme is designed, defining the tracking error of the unmanned surface vessel (USV) as e = η - η d =[e1,e2,e3] T e1, e2, and e3 represent the tracking errors in the sway direction, lateral sway direction, and pitch direction, respectively, and η d It is the reference trajectory vector; to ensure that the tracking error converges to zero quickly and to overcome the singularity problem, a new recursive terminal sliding surface (RTSM) is designed as follows: （30）； Where γ and ρ are positive constants satisfying γ>1, 0<ρ<0.5, and α1=diag(α 11 ,α 12 ,α 13 ), α2=diag(α 21 ,α 22, α 23 ), β1=diag(β 11 ,β 12 ,β 13 ), β2=diag(β 21 ,β 22 ,β 23 ), where α 11 α 21 α 12 α 22 α 13 α 23 ,β 11 ,β 12 ,β 13 ,β 21 ,β 22 ,β 23 These are positive constants; S, s1, and s2 are the sliding surface variables, intermediate sliding surface variables, and bottom sliding surface variables, respectively, and S = [S1, S2, S3]. T , s1=[s 11 ,s 12 ,s 13 ] T s2=[s 21 ,s 22 ,s 23 ] T Where S1, S2, and S3 represent the sliding surface variables in the longitudinal sway direction, the transverse sway direction, and the yaw direction, respectively; s 11 ,s 12 ,s 13 These represent the intermediate sliding surface variables in the longitudinal, transverse, and heave directions, respectively; s 21 ,s 22 ,s 23 These represent the bottom sliding surface variables in the longitudinal sway direction, the lateral sway direction, and the initial sway direction, respectively. When the recursive terminal sliding surface RTSM slides on the sliding manifold, with S as the zero vector, we obtain: （31）； Based on the derivation, we can conclude that: (32) ; Lumped disturbances based on the fixed-time disturbance observer AFTDO A fixed-time recursive terminal sliding mode control (FTRTSMC) is designed to make the tracking error converge to zero within a fixed time, as detailed below: Define the Lyapunov function: (33); Its time derivative is defined as: （34）； According to Lemma 3: （35）； in ξ1 and ξ2 are positive numbers: ; According to Lemma 1, s2 occurs at a fixed time T. a It converges to zero and satisfies: （36）； We derive that: （37）； Choose the Lyapunov function V b : （38）； V b The time derivative is: （39）； Combining Lemma 3, we get: （40）； According to Lemma 1, e has a fixed time T b Converging to zero: （41）； in ξ3 and ξ4 are positive constants that satisfy the following condition: ； The final fixed-time recursive terminal sliding mode control (FTRTSMC) is as follows: （42）； in For fixed-time convergent terms, It is a positive constant and satisfies k1>k2>1.

4. The recursive terminal sliding mode control method for unmanned surface vessels based on an improved disturbance observer according to claim 3, characterized in that, The recursive terminal sliding surface (RTSM) has a two-layer recursive architecture: First layer of sliding surface: This ensures that the principal error converges. Second layer sliding surface: Eliminate singularities.

5. The unmanned surface vessel recursive terminal sliding mode control method based on an improved disturbance observer according to claim 1, implemented based on an unmanned surface vessel recursive terminal sliding mode control system, characterized in that... Specifically, it includes: The module includes a disturbance observation module, a recursive sliding mode generation module, a fixed-time control module, and a disturbance observation module. The disturbance observation module executes AFTDO based on a fixed-time disturbance observer and outputs data in real time. ; The recursive sliding mode generation module: constructs a recursive terminal sliding mode surface (RTSM) calculation S; The fixed-time control module: based on The control quantity τ is generated by S.

6. The recursive terminal sliding mode control method for unmanned surface vessels based on an improved disturbance observer according to claim 5, characterized in that, The disturbance observation module includes a signal processor and an adaptive gain regulator; the signal processor is used to calculate auxiliary variables. The adaptive gain regulator is used to dynamically adjust k. 3i To suppress Differential disturbance.

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