All-drive surface ship tracking control method with time-varying control gain

Abstract

The invention provides an all-drive water surface ship tracking control method with time-varying control gain. The method comprises the following steps: establishing an all-drive water surface ship system model considering external interference; designing a time-varying control gain and non-time-varying control gain combined tracking controller by using a vector backstepping method; analyzing the asymptotic tracking performance and the practical tracking performance of the tracking controller by adopting a Lyapunov stability theory; and analyzing the boundaries of a control item containing a time-varying control gain and a control item containing a non-time-varying control gain in the tracking controller to realize the tracking control of the all-drive water surface ship. On one hand, a tracking control method comprising time-varying control gain is designed based on unknown bounded external interference, and is used for realizing the asymptotic tracking performance of the system and the stability of a closed-loop system; and on the other hand, the closed-loop performance of asymptotic tracking or practical tracking of the system can be realized only by adjusting the value of the time-varying control gain.

Description

Technical Field

[0001] The present invention relates to the field of intelligent tracking control of fully actuated surface ships, and particularly to a tracking control method for fully actuated surface ships with time-varying control gains. Background Art

[0002] Surface ships, as important carriers for humans to explore, develop, and utilize marine resources, have attracted great attention. Surface ships can generally be divided into two types: fully actuated surface ships and underactuated surface ships. Compared with underactuated surface ships, fully actuated surface ships have the characteristics of simpler structure, more flexible and precise maneuverability, and can effectively reduce accident risks. Therefore, the tracking control method for fully actuated surface ships has always been a research hotspot in the industry.

[0003] During the navigation or operation of surface ships at sea, they will inevitably be affected by external environmental disturbances such as sea winds, waves, and ocean currents. Since it is difficult to accurately evaluate and quantify the influence of external disturbances on surface ships, it is also difficult to achieve precise control of surface ships affected by external disturbances. Therefore, the control problem of surface ships affected by external environmental disturbances has always been an important research direction. Most existing methods mainly study the control problem of surface ships on the premise that external disturbances are known bounded disturbances, but in fact, it is very difficult to accurately obtain the bounds of external disturbances. On the other hand, in the current research on surface ship tracking control, different tracking controllers are designed based on different tracking control objectives (such as asymptotic tracking, practical tracking) to achieve the established tracking control performance, lacking a unified control scheme that integrates asymptotic tracking and practical tracking performance. Therefore, it is necessary to provide a tracking control method for fully actuated surface ships that combines time-varying control gains and non-time-varying control gains. Summary of the Invention

[0004] In view of the above-mentioned technical problems, a tracking control method for fully actuated surface ships with time-varying control gains is provided. The present invention proposes a tracking control method with time-varying control gains for a fully actuated surface ship system affected by unknown external disturbances. The control method in the present invention provides a unified tracking controller design method, and different tracking performances can be achieved only by adjusting the time-varying control gain parameters.

[0005] The technical means adopted by the present invention are as follows:

[0006] A tracking control method for fully actuated surface ships with time-varying control gains, comprising:

[0007] Establish a fully actuated surface ship system model considering external disturbances;

[0008] Design a tracking controller that combines time-varying control gains and non-time-varying control gains by using the vector backstepping method;

[0009] The asymptotic tracking performance and practical tracking performance of the tracking controller are analyzed using L'Hopital's rule;

[0010] The boundedness of the tracking controller is analyzed to achieve the tracking control of the fully actuated surface vessel.

[0011] Furthermore, the fully actuated surface vessel system model considering external disturbances is expressed as:

[0012]

[0013] where η = (x, y, ψ) T represents the position and attitude of the surface vessel in the earth coordinate system, (x, y) T represents the position of the surface vessel, ψ represents the yaw angle of the surface vessel, v = (v1, v2, v3) T represents the velocity of the surface vessel, (v1, v2) T represents the linear velocity of the surface vessel, v3 represents the yaw angular velocity, u = (u1, u2, u3) T represents the driving force and moment provided by the surface vessel power system, u1, u2, u3 represent the surge force, sway force and yaw moment respectively, the inertia matrix M is a positive definite symmetric matrix, M = diag{m 11 , m 22 , m 33},m ii > 0 (i = 1, 2, 3), R is the rotation matrix, satisfying R T R = RR T = I3, D(v) is the damping matrix, D(v) = diag{d 11 (v), d 22 (v), d 33 (v)}, where, d 11 (v) = -l 11 -l 12 |v1| - l 13 |v1| 2 , d ii (v) = -l i1 -l i2 |v2| - l i3 |v3| 2 (i = 2, 3); ξ represents the disturbance acceleration brought by the unknown bounded disturbance from the external environment to the surface vessel in the earth coordinate system, ξ = ξ(t) = (ξ1(t), ξ2(t), ξ3(t)) T , satisfying |ξ(t)| ≤ K0, K0 is a constant.

[0014] Furthermore, when designing the tracking controller that combines time-varying control gain and time-invariant control gain using the vector backstepping method, it specifically includes:

[0015] The given tracking signal η r =(x r , y r , ψ r ) T , introduce the following coordinate transformation:

[0016]

[0017] where α represents the virtual control;

[0018] Select Then:

[0019]

[0020] where V1 represents the first candidate Lyapunov function, is the derivative of V1;

[0021] According to the Young inequality and combined with R T R = I3, we get:

[0022]

[0023] Design the virtual control α = -R T C0z1, where the controller parameter matrix C0 = C1 + d1I3 / 2λ min (M), I3 = diag{1, 1, 1}, λ min (M) represents the minimum eigenvalue of the inertia matrix M, the controller parameter matrix C1 to be designed = diag{c 11 , c 12 , c 13}], c 1j > 0 (j = 1, 2, 3), and the controller design parameter d1 > 0;

[0024] Rewrite as:

[0025]

[0026] Let V = V1 + V2, Then:

[0027]

[0028] where V represents the Lyapunov function, V2 represents the second candidate Lyapunov function, S is a third-order matrix, S = (s ij ) 3×3and s 12 =-s 21 =1, s ii =0 (i = 1, 2, 3) and s 13 =s 31 =s 23 =s 32 =0;

[0029] It is deduced from Young's inequality that:

[0030]

[0031] where λ max (M) represents the maximum eigenvalue of the inertia matrix M, the time-varying control gain parameter γ ≥ 0, the adjustable parameter d0 > 0, and t represents time;

[0032] V is arranged into the following formula:

[0033]

[0034] The tracking controller u is expressed as:

[0035]

[0036] where u0 represents the control term containing the time-varying control gain, u1 represents the control term containing the non-time-varying control gain, the controller parameter matrix C2 = diag{c 21 , c 22 , c 23}, and c 2j > 0 (j = 1, 2, 3).

[0037] Furthermore, the asymptotic tracking performance and practical tracking performance of the tracking controller are analyzed by using Lyapunov stability theory, specifically including:

[0038] V is transformed by the tracking controller u into:

[0039]

[0040] where c = min{2c1, 2c2}, c i = min{c i1 , c i2 , c i3} (i = 1, 2);

[0041] Since m1 = min{1 / 2, λ min (M) / 2}, m2 = max{1 / 2, λ max (M) / 2}, and λ max (M) ≥ λmin (M) > 0, it follows that

[0042] A closed-loop system is obtained:

[0043]

[0044] According to It can be deduced that:

[0045]

[0046] Using L'Hopital's rule, we have:

[0047]

[0048] When γ > 0, Combined with the non-negativity of |z(t)| 2 we obtain The tracking controller has the asymptotic tracking performance for the tracking signal;

[0049] When γ = 0, That is d0 and c are independent adjustable parameters, and the tracking controller has the practical tracking performance for the tracking signal.

[0050] Furthermore, the boundedness analysis of the tracking controller specifically includes:

[0051] Analyze the boundedness of the control term u0 containing the time-varying control gain for the subsystem Introduce the Lyapunov function

[0052]

[0053] It is obtained that:

[0054]

[0055] Combined with L'Hopital's rule, it is further deduced that:

[0056]

[0057] Therefore, the control term u0 containing the time-varying control gain has boundedness;

[0058] The boundedness of the control term u1 containing the non-time-varying control gain is deduced from It is deduced that:

[0059]

[0060] In the closed-loop state, both z1 and z2 are bounded. Combined with η = z1 + ηr , and η r and boundedness, it is concluded that both η and v are bounded. Further, the control term u1 containing the time-invariant control gain is bounded.

[0061] Compared with the prior art, the present invention has the following advantages:

[0062] A full-driven surface ship tracking control method with time-varying control gain provided by the present invention includes: establishing a full-driven surface ship system model considering external disturbances; designing a tracking controller combining time-varying control gain and time-invariant control gain by using the vector backstepping method; analyzing the asymptotic tracking performance and practical tracking performance of the tracking controller by using L'Hopital's rule; analyzing the boundedness of the control term containing the time-varying control gain and the control term containing the time-invariant control gain in the tracking controller to achieve the tracking control of the full-driven surface ship. On the one hand, based on the unknown bounded external disturbances, the present invention designs a tracking control method containing time-varying control gain to achieve the asymptotic tracking performance of the system and the stability of the closed-loop system; on the other hand, by only adjusting the value of the time-varying control gain, the closed-loop performance of the system's asymptotic tracking or practical tracking can be achieved. For the above reasons, the present invention can be widely promoted in the fields of intelligent tracking control of full-driven surface ships, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order 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 following drawings are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0064] Figure 1 It is a flow chart of the full-driven surface ship tracking control method with time-varying control gain in the present invention.

[0065] Figure 2 It is a schematic diagram of the earth coordinate system and the body coordinate system of the surface ship in the present invention.

[0066] Figure 3 It is a schematic diagram of the decomposition of the external disturbance acceleration of the surface ship in the earth coordinate system in the body coordinate system in the present invention.

[0067] Figure 4 It is a trajectory diagram of the control input signal of the surface ship in the present invention when the time-varying control gain γ = 0.5.

[0068] Figure 5 It is a trajectory diagram of the tracking error of the surface ship in the present invention when the time-varying control gain γ = 0.5.

[0069] Figure 6 This is the tracking performance graph of the surface ship of the present invention under the condition that the time-varying control gain γ = 0.5.

[0070] Figure 7 This is the control input signal trajectory graph of the surface ship of the present invention under the condition that the time-varying control gain γ = 0.

[0071] Figure 8 This is the tracking error trajectory graph of the surface ship of the present invention under the condition that the time-varying control gain γ = 0.

[0072] Figure 9 This is the tracking performance graph of the surface ship of the present invention under the condition that the time-varying control gain γ = 0. Detailed implementation manners

[0073] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the drawings and in combination with the embodiments.

[0074] 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 only a part rather than all of the embodiments of the present invention. The description of at least one exemplary embodiment below is actually only illustrative and in no way restricts the present invention and its application or use. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present invention without creative efforts shall fall within the protection scope of the present invention.

[0075] It should be noted that the terms used here are only for describing the specific implementation manners and are not intended to limit the exemplary embodiments according to the present invention. As used here, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or their combinations.

[0076] Unless otherwise specifically stated, the relative arrangements of components and steps, numerical expressions, and numerical values set forth in these embodiments do not limit the scope of the present invention. At the same time, it should be clear that, for the sake of convenience in description, the dimensions of the various parts shown in the drawings are not drawn in actual proportional relationships. Technologies, methods, and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but where appropriate, such technologies, methods, and devices should be regarded as part of the authorization specification. In all the examples shown and discussed here, any specific values should be construed as merely exemplary and not as a limitation. Therefore, other examples of the exemplary embodiments may have different values. It should be noted that like reference numerals and letters denote like items in the following drawings, and thus, once an item is defined in one drawing, it does not need to be further discussed in subsequent drawings.

[0077] In the description of the present invention, it should be understood that the orientation or positional relationships indicated by orientation terms such as "front, rear, upper, lower, left, right", "lateral, vertical, perpendicular, horizontal", and "top, bottom" are generally based on the orientation or positional relationships shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description. Without contrary description, these orientation terms do not indicate and imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and thus should not be construed as limiting the protection scope of the present invention: The orientation terms "inside, outside" refer to the inside and outside relative to the contour of each component itself.

[0078] For the convenience of description, spatial relative terms such as "above...", "over...", "on the upper surface of...", "above" can be used here to describe the spatial positional relationship between a device or feature shown in the figure and other devices or features. It should be understood that the spatial relative terms are intended to include different orientations in use or operation in addition to the orientation described in the figure of the device. For example, if the device in the drawing is inverted, the device described as "above other devices or structures" or "over other devices or structures" will then be positioned as "below other devices or structures" or "under other devices or structures". Thus, the exemplary term "above..." can include both the orientations of "above..." and "below...". The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and corresponding interpretations should be made for the spatial relative descriptions used here.

[0079] In addition, it should be noted that the use of words such as "first", "second" to limit components is only for the convenience of distinguishing the corresponding components. Without otherwise stating, the above words have no special meanings, and thus should not be construed as limiting the protection scope of the present invention.

[0080] As Figure 1As shown in the figure, the present invention provides a tracking control method for a fully actuated surface ship with time-varying control gain, which is characterized by including:

[0081] Establish a fully actuated surface ship system model considering external disturbances;

[0082] Specifically, as a preferred implementation manner of the present invention, the fully actuated surface ship system model considering external disturbances is expressed as:

[0083]

[0084] where η = (x, y, ψ) T represents the position and attitude of the surface ship in the earth coordinate system, (x, y) T represents the position of the surface ship, ψ represents the yaw angle of the surface ship, v = (v1, v2, v3) T represents the speed of the surface ship, (v1, v2) T represents the linear speed of the surface ship's operation, v3 represents the yaw angular speed, u = (u1, u2, u3) T represents the driving force and torque provided by the surface ship's power system, u1, u2, u3 respectively represent the surge force, sway force and yaw torque, the inertia matrix M is a positive definite symmetric matrix, M = diag{m 11 , m 22 , m 33}, m ii > 0 (i = 1, 2, 3), R is a rotation matrix, satisfying R T R = RR T = I3, D(v) is a damping matrix, D(v) = diag{d 11 (v), d 22 (v), d 33 (v)}, where, d 11 (v) = -l 11 -l 12 |v1| - l 13 |v1| 2 , d ii (v) = -l i1 -l i2 |v2| - l i3 |v3| 2 (i = 2, 3); ξ represents the disturbance acceleration brought by the unknown bounded disturbance from the external environment to the surface ship in the earth coordinate system, as Figure 3 shown, ξ = ξ(t) = (ξ1(t), ξ2(t), ξ3(t)) T , satisfying |ξ(t)| ≤ K0, K0 is a constant.

[0085] Design a tracking controller that combines time-varying control gain and time-invariant control gain using the vector backstepping method;

[0086] Specifically, as a preferred embodiment of the present invention, the tracking controller designed by using the vector backstepping method to combine time-varying control gain and time-invariant control gain specifically includes:

[0087] The given tracking signal η r =(x r ,y r ,ψ r ) T , introduce the following coordinate transformation, as shown in Figure 2 :

[0088]

[0089] where α represents the virtual control;

[0090] Select Then:

[0091]

[0092] where V1 represents the first candidate Lyapunov function, is the derivative of V1; according to the Young inequality, and combined with R T R = I3, we get:

[0093]

[0094] Design the virtual control α = -R T C0z1, where the controller parameter matrix C0 = C1 + d1I3 / 2λ min (M), I3 = diag{1,1,1}, λ min (M) represents the minimum eigenvalue of the inertia matrix M, the controller parameter matrix C1 to be designed = diag{c 11 ,c 12 ,c 13}], c 1j > 0 (j = 1,2,3), the controller design parameter d1 > 0;

[0095] Rewrite as:

[0096]

[0097] Let V = V1 + V2, Then:

[0098]

[0099] where \(V\) represents the Lyapunov function, \(V_2\) represents the second candidate Lyapunov function, \(S\) is a third-order matrix, \(S=(s ij ) 3×3 and \(s 12 =-s 21 = 1\), \(s ii = 0 (i = 1, 2, 3)\) and \(s 13 =s 31 =s 23 =s 32 = 0;

[0100] It is deduced from the Young inequality that:

[0101]

[0102] where \(\lambda max (M)\) represents the maximum eigenvalue of the inertia matrix \(M\), the time-varying control gain parameter \(\gamma\geq0\), the adjustable parameter \(d_0>0\), and \(t\) represents time;

[0103] \(V\) is arranged into the following formula:

[0104]

[0105] The tracking controller \(u\) is expressed as:

[0106]

[0107] where \(u_0\) represents the control term containing the time-varying control gain, \(u_1\) represents the control term containing the non-time-varying control gain, the controller parameter matrix \(C_2 = diag\{c 21 ,c 22 ,c 23 \}\), \(c 2j >0 (j = 1, 2, 3).

[0108] The asymptotic tracking performance and practical tracking performance of the tracking controller are analyzed by using the Lyapunov stability theory;

[0109] Specifically, as a preferred embodiment of the present invention, the analysis of the asymptotic tracking performance and practical tracking performance of the tracking controller by using the Lyapunov stability theory specifically includes:

[0110] \(V\) is transformed by the tracking controller \(u\) into:

[0111]

[0112] where \(c = min\{2c_1, 2c_2\}\), \(c i =min\{c i1 ,c i2 ,c i3}(i = 1, 2);

[0113] Since m1 = min{1 / 2, λ min (M) / 2}, m2 = max{1 / 2, λ max (M) / 2}, and λ max (M) ≥ λ min (M) > 0, it follows that

[0114] The closed-loop system is obtained:

[0115]

[0116] The closed-loop system is practically stable.

[0117] According to it can be deduced that:

[0118]

[0119] Using L'Hopital's rule, we have:

[0120]

[0121] When γ > 0, and combined with the non-negativity of |z(t)| 2 we obtain The tracking controller has the asymptotic tracking performance for the tracking signal;

[0122] When γ = 0, that is d0 and c are independent adjustable parameters, and the tracking controller has the practical tracking performance for the tracking signal.

[0123] Analyze the boundedness of the control term containing the time-varying control gain and the control term containing the non-time-varying control gain in the tracking controller, and then prove the boundedness of the tracking controller to achieve the tracking control of the fully actuated surface ship.

[0124] Analyze the boundedness of the control term u0 containing the time-varying control gain for the subsystem Introduce the Lyapunov function

[0125]

[0126] It is obtained that:

[0127]

[0128] Combined with L'Hopital's rule, it is further deduced that:

[0129]

[0130] Therefore, the time-varying control term u0 is bounded;

[0131] The boundedness of the control term u1 containing the non-time-varying control gain is derived from as follows:

[0132]

[0133] In the closed-loop state, both z1 and z2 are bounded. Combining η = z1 + η r , and the boundedness of η r and , it is concluded that both η and v are bounded. Further, the non-time-varying control term u1 is bounded.

[0134] It should be noted that the designed controller u can be divided into a control term u0 containing a time-varying control gain and a control term u1 containing a non-time-varying control gain. In fact, both of them are bounded, that is, the designed controller u is bounded.

[0135] Embodiment

[0136] As Figure 1 shown, the present invention provides a full-driven surface ship tracking control method with a time-varying control gain. In this embodiment, the hull parameter data of the full-driven surface ship system, the unknown external disturbance acceleration, the tracking signal, and the control parameters are provided. Experiments are carried out under different values of the time-varying control gain, which proves that the full-driven surface ship tracking control method in the present invention has asymptotic tracking performance and practical tracking performance.

[0137] The hull parameter data of the full-driven surface ship system are as follows:

[0138] m 11 = 26, m 22 = 34, m 33 = 2.8

[0139] l 11 = -0.7, l 12 = -1.3, l 13 = -5.9, l 21 = 0.9, l 22 = -36, l 23 = l 31 = l 32 = 0, l 31 = 0.1,

[0140] The unknown external disturbance acceleration is set as:

[0141] ξ1(t) = 0.1sin(0.4t) + 0.25cos(0.8t)

[0142] ξ2(t) = 0.15sin(0.3t) + 0.25cos(0.6t)

[0143] ξ3(t) = 0.015cos(0.5t) + 0.01sint

[0144] Initial value:

[0145] η(0) = (0.3, -0.8, 1) T , v(0) = (0.3, 1.5, 0.4) T

[0146] Tracking signal:

[0147] η r1 = 0.2sin(0.1t) + 0.1cos(0.1t)

[0148] η r2 = 0.25sin(0.1t) - 0.15cos(0.1t)

[0149] η r3 = 0.3sin(0.1t) + 0.2cos(0.1t)

[0150] Control parameters:

[0151] c 11 = 4, c 12 = 4, c 13 = 0.3, c 21 = 4, c 22 = 4, c 23 = 0.3, d1 = 1, d0 = 1, γ = 0.5 or γ = 0

[0152] In the case of time-varying control gain γ = 0.5, Figure 4 denotes the system control input signal. Under the input signal shown in Figure 4 , Figure 5 and Figure 6 respectively show the tracking error and tracking performance of the system. From the tracking error curve, it is clearly shown that the designed control signal achieves the asymptotic tracking performance of the system.

[0153] In the case of time-varying control gain γ = 0, Figure 7 denotes the system control input signal. Under the input signal shown in Figure 7 , Figure 8 and Figure 9The tracking error and tracking performance of the system are respectively shown. From the tracking error curve diagram, it is clearly shown that the designed control signal realizes the practical tracking performance of the system.

[0154] 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 embodiments of the present invention.

Claims

1. A full-driven surface ship tracking control method with time-varying control gain, characterized in that, Including: Establish a fully actuated surface ship system model considering external disturbances; Design a tracking controller combining time-varying control gain and non-time-varying control gain using the vector backstepping method; Adopt Lyapunov stability theory to analyze the asymptotic tracking performance and practical tracking performance of the tracking controller; Analyze the boundedness of the tracking controller to achieve the tracking control of the fully actuated surface ship.

2. The full-actuated surface vessel tracking control method with time-varying control gain according to claim 1, characterized in that Express the fully actuated surface ship system model considering external disturbances as: where η = (x, y, ψ) T represents the position and attitude of the surface ship in the earth coordinate system, (x, y) T represents the position of the surface ship, ψ represents the yaw angle of the surface ship, v = (v1, v2, v3) T represents the speed of the surface ship, (v1, v2) T represents the linear speed of the surface ship's operation, v3 represents the yaw angular speed, u = (u1, u2, u3) T represents the driving force and moment provided by the surface ship's power system, u1, u2, u3 respectively represent the surge force, sway force and yaw moment, the inertia matrix M is a positive definite symmetric matrix, M = diag{m 11 , m 22 , m 33},m ii > 0 (i = 1, 2, 3), R is a rotation matrix, satisfying R T R = RR T = I3, D(v) is a damping matrix, D(v) = diag{d 11 (v), d 22 (v), d 33 (v)}, where, d 11 (v) = -l 11 -l 12 |v1| - l 13 |v1| 2 , d ii (v) = -l i1 -l i2 |v2| - l i3 |v3| 2 (i = 2, 3); ξ represents the disturbance acceleration brought by the unknown bounded disturbance from the external environment to the surface ship in the earth coordinate system, ξ = ξ(t) = (ξ1(t), ξ2(t), ξ3(t)) T , satisfying |ξ(t)| ≤ K0, K0 is a constant.

3. The full-actuated surface ship tracking control method with time-varying control gain according to claim 1, characterized in that The tracking controller designed by using the vector backstepping method to combine time-varying control gain and non-time-varying control gain specifically includes: The given tracking signal η r =(x r , y r , ψ r ) T , and introduce the following coordinate transformation: where α represents the virtual control; Select Then: wherein, V1 represents the first candidate Lyapunov function, is the derivative of V1; according to the Young inequality, and combined with R T R = I3, we get: Design the virtual control α = -R T C0z1, where the controller parameter matrix C0 = C1 + d1I3 / 2λ min (M), I3 = diag{1, 1, 1}, λ min (M) represents the minimum eigenvalue of the inertia matrix M, and the controller parameter matrix C1 to be designed = diag{c 11 , c 12 , c 13},c 1j > 0 (j = 1, 2, 3), and the controller design parameter d1 > 0; Convert to: Let V = V1 + V2, Then: where, V represents the Lyapunov function, V2 represents the second candidate Lyapunov function, S is a third-order matrix, S=(s ij ) 3×3 and s 12 =-s 21 =1, s ii =0 (i = 1, 2, 3) and s 13 =s 31 =s 23 =s 32 =0; It can be inferred from Young's inequality that: where, λ max (M) represents the maximum eigenvalue of the inertia matrix M, the time-varying control gain parameter γ ≥ 0, the adjustable parameter d0 > 0, and t represents time; Rearrange into the following formula: Express the tracking controller u as: Among them, u0 represents the control term containing the time-varying control gain, u1 represents the control term containing the time-invariant control gain, and the controller parameter matrix C2 = diag{c 21 , c 22 , c 23}, where c 2j > 0 (j = 1, 2, 3).

4. The full-actuated surface ship tracking control method with time-varying control gain according to claim 1, wherein The adoption of Lyapunov stability theory to analyze the asymptotic tracking performance and practical tracking performance of the tracking controller specifically includes: Converted by the tracking controller u into: where c = min{2c1, 2c2}, c i = min{c i1 , c i2 , c i3}(i = 1, 2); Since m1 = min{1 / 2, λ min (M) / 2}, m2 = max{1 / 2, λ max (M) / 2}, and λ max (M) ≥ λ min (M) > 0, it follows that Obtain the closed-loop system: According to it can be deduced that: Using L'Hopital's rule, we have: When γ > 0, and combined with the non-negativity of |z(t)| 2 we obtain the tracking controller has the asymptotic tracking performance for the tracking signal; When γ = 0, that is d0 and c are independent adjustable parameters, and the tracking controller has the practical tracking performance of tracking signals.

5. The full-drive surface ship tracking control method with time-varying control gain according to claim 1, characterized in that The analysis of the boundedness of the tracking controller specifically includes: Analyze the boundedness of the control term u0 containing the time-varying control gain for the subsystem Introduce a Lyapunov function It is concluded that: Combined with L'Hopital's rule, it is further deduced that: Therefore, the control term u0 containing time-varying control gain has boundedness; The boundedness of the control term u1 containing the time-invariant control gain is derived from as follows: In the closed-loop state, both z1 and z2 are bounded. Combining η = z1 + η r , and r and the boundedness of η, it is concluded that both η and v are bounded. Further, the control term u1 containing non-time-varying control gains is bounded.

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