An underactuated unmanned ship's sub-fixed time sliding mode control method

Abstract

The application discloses a kind of underactuated unmanned ship's sub fixed time sliding mode control method, comprising the following operational steps: S1: establish the mathematical model of underactuated unmanned ship under plane coordinate system;S2: design new sliding mode surface in combination with sub fixed time stability concept;S3: design the propelling force under sub fixed time stability concept;S4: design the yawing moment under sub fixed time stability concept.The application, the sub fixed time sliding mode control method proposed for underactuated unmanned ship, not only can improve the global navigation speed of unmanned ship, the key is to ensure that unmanned ship is navigated to specified position within fixed time, and the control scheme designed using sliding mode method can further improve the convergence speed of system.

Description

Technical Field

[0001] This invention relates to the field of control technology for underactuated vessels, and in particular to a sub-fixed-time sliding mode control method for an underactuated unmanned vessel. Background Technology

[0002] Unmanned operation is a key characteristic of intelligent ship development. Unmanned surface vessels (USVs) with autonomous driving capabilities can be used in numerous surface operations, such as maritime patrol, underwater surveying, surface cleanup, hydrological and water resource surveying, and maritime operations. Therefore, they have received increasing attention from research institutions, shipping companies, and governments in recent years. Currently, most ships navigating the surface are not equipped with side thrusters, meaning they cannot achieve lateral movement. When lateral movement is required, USVs generally rely on the thrust difference generated by their two main thrusters to first achieve yaw, and then navigate a curved path back to the predetermined position. This demonstrates that controlling an underactuated USV is far more complex than controlling a fully actuated system.

[0003] Sub-fixed-time stability is a stability concept that exhibits rapid convergence at both near and far equilibrium points without singular problems, and is currently in the preliminary research stage. Regarding sub-fixed-time control methods for underactuated unmanned surface vessels (USVs), only one patent has been published so far: Chinese Patent CN115047881A, entitled "A Sub-fixed-Time Backstepping Control Method for Underactuated Unmanned Surface Vessels," which includes the following steps: 1. Establishing a mathematical model of the underactuated USV in a planar coordinate system; 2. Designing a virtual control law under the sub-fixed-time stability concept; 3. Designing a USV controller under the sub-fixed-time stability concept.

[0004] The control method used in the aforementioned patent is a backstepping control method, which utilizes a designed virtual control law to achieve sub-fixed-time convergence of the unmanned vessel's position and state. However, the greatest advantage of the backstepping control method is that when dealing with control problems of high-order systems, it simplifies the system order, transforming high-order problems into low-order problems and reducing design difficulty. Since the unmanned vessel control system falls under the category of second-order system control, it is not a high-order system. Therefore, while the backstepping method can solve sub-fixed-time control problems of second-order systems, using a control method more suitable for second-order systems can further improve the unmanned vessel's control performance. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a sub-fixed-time sliding mode control method for underactuated unmanned surface vessels (USVs) that can improve the convergence speed of the USV control system and utilizes sliding mode control to design thrusters and yaw moments to enable the vessel to have sub-fixed-time stability characteristics.

[0006] The objective of this invention is achieved through the following technical solutions.

[0007] A sub-fixed-time sliding mode control method for an underactuated unmanned surface vessel includes the following steps:

[0008] S1: Establish a mathematical model of the underactuated unmanned surface vessel in a planar coordinate system;

[0009] S2: A novel sliding surface is designed by combining the concept of sub-fixed time stabilization;

[0010] S3: Propulsion under the concept of sub-fixed-time stability;

[0011] S4: Design yaw moment under the concept of sub-fixed time stability.

[0012] The mathematical model of the underactuated unmanned surface vessel in step S1 in the planar coordinate system is as follows:

[0013]

[0014]

[0015]

[0016]

[0017] Where, [x,y] T ∈R 2 The vector represents the ship's position in the geocentric coordinate system, u represents the unmanned vessel's speed in the sway direction, and v represents the unmanned vessel's speed in the yaw direction. Let represent the bow angle, i.e., the angle between the unmanned vessel's pitch velocity and the positive x-axis; r represent the bow angle angular velocity; m1, m2, and m3 represent the unmanned vessel's inertia including mass effects; d1, d2, and d3 represent the river hydrodynamic damping related to pitch, sway, and yaw angles, respectively; and τ represent the bow angle angular velocity. u and τ r This represents the propulsion and yaw moment of the unmanned vessel.

[0018] The specific mathematical form of the novel sliding surface designed in step S2, combining the concept of sub-fixed-time stability, is as follows:

[0019]

[0020]

[0021] Among them, the gain parameters k1 and k2 must satisfy k1 > 0 and k2 > 0, and the exponent parameter g must satisfy g > 2. When the unmanned vessel's position and velocity variables x, y, u, y are all on the sliding mode surface, i.e. s x =0 and s y=0, then these two sliding surfaces can guarantee that the unmanned vessel's position variables x and y converge in a second fixed time, that is, x and y converge asymptotically to 0, and that x and y converge to the vicinity of 0 within a time upper bound independent of the initial state value. The selection of a Lyapunov candidate function V1 has the following mathematical form:

[0022] V1(t)=x 2 +y 2

[0023] If we define an arbitrarily small positive constant ε to represent the desired precision, then regardless of how far the initial position of the state variable is from the equilibrium point, the state variable will always be within a fixed time T. max It converges internally to the range V1(t)≤ε, and defines V1(T) ε T in ) = ε ε Let T represent the convergence time. ε With T max Relationship and T max The expression is as follows:

[0024]

[0025] in,

[0026] The thrust controller and the concept of sub-fixed-time stabilization in step S3 are as follows:

[0027]

[0028] Wherein, the gain parameters k3 and l1 must satisfy k3 > 0 and l1 > 0, and the exponent parameters g2 and p must satisfy g2 > 1 and p ∈ (0, 1), then the controller can realize the sliding mode variable s x For fixed-time convergence, i.e., sliding mode variable s x It converges asymptotically to 0, and s x Within a time bound independent of the initial state, a Lyapunov candidate function V is selected. sx (t) has the following mathematical form:

[0029]

[0030] Then, choose an arbitrarily small constant ε to represent the desired precision, then regardless of s x How far is the initial position from the equilibrium point, s x All can be completed at a fixed time T xmax Converging to V sx Within the range (t)≤ε, and V is defined sx (T ε T in ) = ε εxLet T represent the convergence time. εx With T xmax Relationship and T xmax The expression is as follows:

[0031]

[0032] The bias torque controller in step S4 takes the following form:

[0033]

[0034] Wherein, the gain parameters k4 and k5 must satisfy k4 > 0 and k5 > 0, and the exponent parameter g3 must satisfy g3 > 1, then the controller can realize the sliding mode variable s y It converges in a fixed time.

[0035] Functions in the controller and The specific mathematical expression is as follows:

[0036]

[0037]

[0038]

[0039] Compared with the prior art, the advantages of the present invention are as follows: The present invention addresses the control problem of underactuated unmanned vessels by combining the sliding mode control method with the concept of sub-fixed time stability. It proposes a novel sub-fixed time sliding surface and a matching sub-fixed time thruster and yaw moment structure. This control scheme, based on the characteristic of sub-fixed time stability that can improve the convergence speed of near and far equilibrium point positions, also has the characteristics of sliding mode control that can further improve the system convergence speed.

[0040] This patent utilizes sliding mode control, a control method more suitable for second-order systems, to design a sub-fixed-time controller for an underactuated unmanned surface vessel. Compared to the backstepping method, the sliding mode method has several advantages. First, it divides the convergence region into two stages: the arrival stage and the coasting stage. The reaching law in the controller accelerates the arrival stage, while the sliding surface accelerates the coasting stage. By meticulously classifying the convergence region, parameters can be adjusted and stability analysis performed, thereby further improving the system's convergence speed. Second, the sliding mode control method exhibits better robustness and anti-interference capabilities. Depending on the stability requirements in actual situations, a variable structure term k·sign(s) can be added to the controller to further enhance system performance.

[0041] This invention presents a novel sliding surface and its controller designed by combining the concept of sub-fixed-time stability. It has the advantage of strong scalability and can incorporate approach terms such as k·sig(s) into the designed controller according to actual conditions. g 、k·s、k·sig(s) p The convergence properties are further improved by using functions such as [equation name missing]. Attached Figure Description

[0042] Figure 1 This is a schematic diagram of the ship in the geocentric inertial coordinate system in a fixed-time control method for underactuated unmanned vessels based on a novel sliding surface proposed in this invention.

[0043] Figure 2 The time curve of the ship position coordinates is provided for a fixed-time control method for underactuated unmanned vessels based on a novel sliding surface proposed in this invention.

[0044] Figure 3 The time curve of the ship position coordinates is provided for a fixed-time control method for underactuated unmanned vessels based on a novel sliding surface proposed in this invention.

[0045] Figure 4 The time curve of the sway velocity of a ship is provided by the underactuated unmanned vessel fixed-time control method based on a novel sliding surface proposed in this invention.

[0046] Figure 5 The time curve of the sway velocity of a vessel is provided by the underactuated unmanned vessel fixed-time control method based on a novel sliding surface proposed in this invention.

[0047] Figure 6 The propulsion time curve of a vessel based on a novel sliding surface-based underactuated unmanned vessel fixed-time control method is shown in this invention.

[0048] Figure 7 The time curve of the yaw moment of a vessel is presented in the present invention, which is based on a novel sliding surface-based underactuated unmanned vessel fixed-time control method. Detailed Implementation

[0049] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0050] Example

[0051] like Figure 1-7 As shown, S1: Establishing a mathematical model for underactuated ships.

[0052] Refer to the attached diagram in the instruction manual. Figure 1 As shown, the kinematic model of the unmanned surface vessel in the geocentric inertial coordinate system xOy is as follows:

[0053]

[0054]

[0055]

[0056]

[0057] Where, [x,y] T ∈R 2 The vector represents the ship's position in the geocentric coordinate system, u represents the unmanned vessel's speed in the sway direction, and v represents the unmanned vessel's speed in the yaw direction. Let represent the bow angle, i.e., the angle between the unmanned vessel's pitch velocity and the positive x-axis; r represent the bow angle angular velocity; m1, m2, and m3 represent the unmanned vessel's inertia including mass effects; d1, d2, and d3 represent the river hydrodynamic damping related to pitch, sway, and yaw angles, respectively; and τ represent the bow angle angular velocity. u and τ r This represents the propulsion and yaw moment of the unmanned vessel.

[0058] S2: Design a new sliding surface by combining the concept of sub-fixed time stabilization.

[0059] Considering the underactuated unmanned vessel described by mathematical models (1)-(4), a novel sliding surface under the concept of sub-fixed-time stability is designed as follows:

[0060]

[0061]

[0062] Among them, the gain parameters k1 and k2 must satisfy k1 > 0 and k2 > 0, and the exponent parameter g must satisfy g > 2. If the unmanned vessel's position and velocity variables x, y, u, y are all on the sliding mode surface, i.e. s x =0 and s y =0, then these two sliding surfaces can guarantee that the unmanned vessel's position variables x and y converge in a second fixed time. This concept of convergence has two meanings. First, it indicates that x and y converge asymptotically to 0. Second, it indicates that x and y converge to a neighborhood near 0 within a time upper bound independent of the initial state value. The specific mathematical description of this second meaning is as follows: If we choose a Lyapunov candidate function V1 with the following form:

[0063] V1(t)=x 2 +y 2 (7)

[0064] If we define an arbitrarily small positive constant ε to represent the desired precision, then regardless of how far the initial position of the state variable is from the equilibrium point, the state variable will always be within a fixed time T. maxIt converges internally to the range V1(t)≤ε, and defines V1(T) ε T in ) = ε ε Let T represent the convergence time. ε With T max Relationship and T max The expression is as follows:

[0065]

[0066] in,

[0067] As can be seen from formula (8), the upper bound of the convergence time T max It is independent of the initial state of the system. Although the distance between the initial state and the equilibrium point affects the convergence time T. ε The size, but T max The value of ε is only related to the accuracy constant ε and the control parameters, which is the main characteristic of fixed-time stability. In engineering practice, the magnitude of ε can be freely set according to the accuracy required by the system. If the value of ε is set smaller, it can be seen from formula (8) that the convergence time and its upper bound will also be smaller, which shows that formula (8) is consistent with the actual situation.

[0068] The following explains from a theoretical perspective when s x =0 and s y When = 0, the underactuated unmanned surface vessel system (1)-(4) is reasonably stable for a sub-fixed time. Let s x =0 and s y Substituting 0 into formulas (5) and (6) yields

[0069]

[0070]

[0071] Differentiating the candidate Lyapunov function in formula (7) yields:

[0072]

[0073] According to the inequality theorem, the above equation can be simplified to:

[0074] To illustrate the conclusion that the sliding surface system is sub-fixed-time stable from the above equation, we need to introduce the following criterion for sub-fixed-time stability:

[0075] Lemma 1: Consider the system: f(0) = 0, x ∈ R n , where x∈R n ,

[0076] f:U0→Rn Let the domain U containing the origin be an n-dimensional space R. n A function, 0∈R n surface

[0077] Let x0 be the zero vector and x0 be the initial state. If there exists a continuous positive definite function for this system...

[0078] V(x):R n →R satisfies the inequality:

[0079]

[0080] Here, α and g are positive constants and g > 1, an arbitrarily small positive real number ε is defined to satisfy ε ≤ V(x0), and the convergence time T is defined. ε V(x(T) represents ε If V(x) = ε, then the system can be called sub-fixed-time stable. This stability concept means that for any initial state V(x0), the system state x can converge to the region D = {x:V(x)≤ε} within a time upper bound T independent of x0, and the convergence time T is less than or equal to ε. ε The relationship between the time upper bound T and the specific expression for T are as follows:

[0081] As can be seen from the above lemma, inequality (11) has the same form as the decision condition (12) in the lemma. Therefore, according to this lemma, when s x =0 and s y When =0 always holds (i.e., the system state is stable on the sliding surface), system (1)-(4) is sub-fixed-time stable, that is, the unmanned vessel position variables x and y converge in sub-fixed-time, so x and y can converge in a fixed time T. max It converges inward to the region V1(t)≤ε, and the convergence time T ε With T max Relationship and T max The expression is

[0082]

[0083] S3: Design of a propulsion controller based on a sub-fixed-time stabilization concept

[0084] Considering the underactuated unmanned vessel described by mathematical models (1)-(4), design a propulsion controller τ. u as follows:

[0085]

[0086] Wherein, the gain parameters k3 and l1 must satisfy k3 > 0 and l1 > 0, and the exponent parameters g2 and p must satisfy g2 > 1 and p ∈ (0, 1), then the controller can realize the sliding mode variable s x For fixed-time convergence, the concept of convergence has two meanings. First, it indicates that the sliding mode variable s x It converges asymptotically to 0, which further indicates that s x The specific mathematical description of this second layer of meaning is: if a Lyapunov candidate function V is chosen, it can converge to a neighborhood near 0 within a time bound independent of the initial state. sx (t) has the following form:

[0087]

[0088] Then, choose an arbitrarily small constant ε to represent the desired precision, then regardless of s x How far is the initial position from the equilibrium point, s x All can be completed at a fixed time T xmax Converging to V sx Within the range (t)≤ε, and V is defined sx (T ε T in ) = ε εx Let T represent the convergence time. εx With T xmax Relationship and T xmax The expression is as follows:

[0089]

[0090] The following explains from a theoretical perspective how the thrust controller (13) can realize the sliding mode variable s x To justify the convergence at a fixed time. Regarding s in equation (5) x Taking the derivative, we get:

[0091]

[0092] in

[0093]

[0094] Substituting formula (16) into the derivative of the Lyapunov function (14), we get:

[0095]

[0096] Substituting the thrust controller (13) into formula (17) yields:

[0097]

[0098] As can be seen from Lemma 1, inequality (18) has the same form as the decision condition (12) in Lemma 1. Therefore, according to this lemma, the controller (13) can realize the sliding mode variable s. x For convergence in a fixed time, i.e., s x Able to complete in a fixed time T max Converging inward to region V sx (t)≤ε x Within, and the convergence time T xε With T xmax Relationship and T xmax The expression is:

[0099]

[0100] S4: Design of a yaw moment controller based on a sub-fixed-time stability concept

[0101] Considering the underactuated unmanned vessel described by mathematical models (1)-(4), design a yaw moment controller τ. r as follows:

[0102]

[0103] Among them, the gain parameters k4 and k5 must satisfy k4 > 0 and k5 > 0, and the exponent parameter g3 must satisfy g3 > 1. and The expression is:

[0104]

[0105]

[0106] Then the controller can realize the sliding mode variable s y It converges in a fixed time.

[0107] The following explains from a theoretical perspective how the thrust controller (19) can realize the sliding mode variable s y To justify the convergence at a fixed time. Regarding s in equation (6) y By taking the derivative and substituting it into formulas (1) and (3), we can obtain:

[0108]

[0109] Further differentiation based on formula (20) yields:

[0110]

[0111] Substituting formula (4) into (21) yields:

[0112]

[0113] Substituting the yaw moment controller (19) into formula (22) yields the following result:

[0114]

[0115] To prove that the system according to formula (23) is sub-fixed-time stable, the following lemma needs to be introduced.

[0116] Lemma 2: Consider the following general second-order system

[0117]

[0118] If α1>0, α2>0, and g>1 are constants, then system (24) is sub-fixed-time stable. This means that the variable x can asymptotically converge to 0. Define the variable. Let ε be an arbitrarily small constant, and then let T ε Represents the variable ||ξ(x(T) ε If the time corresponding to ||=ε is such that state x can converge to the region ||ξ(x)||≤ε within a fixed time T, and the convergence time T is such that... ε The following constraints must be satisfied:

[0119]

[0120] Where P is the correlation matrix, and a is a function of α1, α2, and g.

[0121] It is easy to see that system (23) has the same mathematical expression as formula (24) in Lemma 2. Therefore, according to Lemma 2, system (23) is sub-fixed-time stable and the sliding mode variable s y It can converge to the region near the sliding surface within a fixed time.

[0122] The effectiveness of the proposed algorithm is verified using simulation software. The proposed control schemes (13) and (19) are designed using the sliding mode method, hereinafter referred to as SSFTC (sliding-mode sub-fixed-time controllers). This section compares the SSFTC method with the backstepping sub-fixed-time control scheme (hereinafter referred to as BSFTC, backstepping sub-fixed-time controllers) based on the backstepping method, as published in CN115047881A. The thrust and yaw moment of the BSFTC design are as follows:

[0123]

[0124]

[0125] The definitions and values ​​of the functions and parameters are as described in the patent. The parameters of the unmanned vessel, initial state, and SSFTC control parameters are shown in Table 1 below:

[0126] Table 1 Model parameters, initial state, and control parameters

[0127]

[0128] The control performance of SSFTC and BSFTC is compared and analyzed below. The simulation results are shown in the attached figures in the manual. Figure 2-7 As shown. From Figure 2-5 It is evident that the SSFTC controller has significantly higher accuracy than the BSFTC controller, for example... Figure 2 The precision of the state variable x driven by SSFTC is 1×10⁻⁶. -3 The precision of the state variable x driven by BSFTC is 0.1m, indicating that SSFTC's precision is significantly better than BSFTC's. From... Figure 4 It can be seen that the accuracy of the longitudinal velocity variable u driven by SSFTC is 1×10⁻⁶. -5 m / s, while the accuracy of the longitudinal velocity variable u driven by BSFTC is 3×10 m / s. -3 Even at m / s, it is still clear that SSFTC's accuracy is significantly better than BSFTC's. Figure 3 and Figure 5 Similar results were found in [the text], so they will not be repeated here. Furthermore, from [the text]... Figure 6-7 It can be seen that the control output amplitude τ of SSFTC u and τ r It is also significantly smaller than BSFTC. In summary, the results show that SSFTC has superior control accuracy and speed compared to BSFTC, and requires less control input than BSFTC; therefore, SSFTC exhibits better control performance.

Claims

1. A sub-fixed-time sliding mode control method for an underactuated unmanned surface vessel, characterized in that, The following steps are included: S1: Establish a mathematical model of the underactuated unmanned surface vessel in a planar coordinate system; S2: Design the sliding surface by combining the concept of sub-fixed time stabilization; S3: Propulsion under the concept of sub-fixed-time stability; S4: Design yaw moment under the concept of sub-fixed-time stability; The specific mathematical form of the sliding surface designed in step S2, incorporating the concept of sub-fixed-time stability, is as follows: Where, [x,y] T ∈R 2 The vector represents the ship's position in the geocentric coordinate system, u represents the unmanned vessel's speed in the sway direction, and v represents the unmanned vessel's speed in the yaw direction. The bow angle represents the angle between the unmanned surface vessel's (USV) pitch velocity and the positive x-axis. Gain parameters k1 and k2 must satisfy k1>0 and k2>0, respectively, and the exponent parameter g1 must satisfy g1>2. This occurs when the USV's positions x and y, and velocity variables u and v, are all on the sliding mode surface, i.e., s... x =0 and s y =0, then these two sliding surfaces can guarantee that the unmanned vessel's position variables x and y converge in a second fixed time, that is, x and y converge asymptotically to 0, and that x and y converge to the vicinity of 0 within a time upper bound independent of the initial state value. The selection of a Lyapunov candidate function V1 has the following mathematical form: V1(t)=x 2 +y 2 If we define an arbitrarily small positive constant ε to represent the desired precision, then regardless of how far the initial position of the state variable is from the equilibrium point, the state variable will always be within a fixed time T. max It converges internally to the range V1(t)≤ε, and defines V1(T) ε T in ) = ε ε Let T represent the convergence time. ε With T max Relationship and T max The expression is as follows: in, 2. The sub-fixed-time sliding mode control method for an underactuated unmanned surface vessel according to claim 1, characterized in that, The mathematical model of the underactuated unmanned surface vessel in step S1 in the planar coordinate system is as follows: Where r represents the bow roll angular velocity, m1, m2, and m3 represent the inertia of the unmanned vessel including mass effects, d1, d2, and d3 represent the river hydrodynamic damping related to pitch, sway, and yaw angles, respectively, and τ u and τ r This represents the propulsion and yaw moment of the unmanned vessel.

3. The sub-fixed-time sliding mode control method for an underactuated unmanned surface vessel according to claim 2, characterized in that, The thrust controller and the concept of sub-fixed-time stabilization in step S3 are as follows: Wherein, the gain parameters k3 and l1 must satisfy k3>0 and l1>0, and the exponent parameters g2 and p must satisfy g2>1 and p∈(0,1), then the controller can realize the sliding mode variable s x For fixed-time convergence, i.e., sliding mode variable s x It converges asymptotically to 0, and s x Within a time bound independent of the initial state, a Lyapunov candidate function V is selected. sx (t) has the following mathematical form: Then, choose an arbitrarily small constant ε to represent the desired precision, then regardless of s x How far is the initial position from the equilibrium point, s x All can be completed at a fixed time T xmax Converging to V sx (t)≤ε x Within the range, and define V sx (T εx )=ε x T in εx Let T represent the convergence time. εx With T xmax Relationship and T xmax The expression is as follows:

4. The sub-fixed-time sliding mode control method for an underactuated unmanned surface vessel according to claim 3, characterized in that, The bias torque controller in step S4 takes the following form: Wherein, the gain parameters k4 and k5 must satisfy k4>0 and k5>0, and the exponent parameter g3 must satisfy g3>1, then the controller can realize the sliding mode variable s y It converges in a fixed time.

5. The sub-fixed-time sliding mode control method for an underactuated unmanned surface vessel according to claim 4, characterized in that, Functions in the controller and The specific mathematical expression is as follows:

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