Method for dynamic observation of heterogeneous usv-uav system bipartite sliding mode formation control based on preset time

Abstract

This invention provides a binary sliding mode formation control method for heterogeneous USV-UAV systems based on preset-time dynamic observation, comprising the following steps: S1, introducing auxiliary variables to convert the underactuated USV and UAV systems into fully driven systems, and constructing a heterogeneous USV-UAV system model based on a virtual leader-follower; S2, designing a preset-time uncertain dynamic observer for the followers of the heterogeneous USV-UAV system; S3, constructing a binary formation tracking error system for the heterogeneous USV-UAV system based on the virtual leader-follower model; S4, combining the uncertain dynamic observer and the binary formation tracking error system to design a preset-time distributed binary formation controller for the heterogeneous USV-UAV system based on sliding mode control. The beneficial effects of this invention are: the control method of this invention has high reliability and good robustness.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of unmanned system formation control, and particularly relates to a preset time dynamic observation heterogeneous USV-UAV system two-part sliding mode formation control method. BACKGROUND

[0002] Compared with a single ground, underwater, air and other multi-agent system, the sea-air heterogeneous USV-UAV system is extremely challenging in the coordination of formation in the sea. At the same time, due to the relative under-actuated characteristics of the USV and the UAV, the difficulty of the sea-air heterogeneous USV-UAV system coordination formation control is increased.

[0003] At present, the formation control of the heterogeneous USV-UAV system mainly adopts a virtual leader-follower method to design a robust adaptive and model predictive control algorithm. However, the existing USV-UAV system formation control literature is based on the cooperation between the nodes in the network, and in fact, in the complex network system in real life, there are not only cooperation between individuals but also competition. At present, there is no research on the two-part formation control of the heterogeneous USV-UAV system with cooperation and competition. For formation control, convergence speed is an important factor. The convergence time required by the finite time and fixed time control is affected by the initial state of the system and even the parameters. Considering some practical application scenarios that require the system to reach stability within a specified time, the preset time control strategy begins to enter people's sight. This time is independent of any other parameters and is completely set in advance according to the task requirements. Although the preset time control method has been widely used in USV or UAV multi-agent systems, it is relatively less used in heterogeneous USV-UAV systems. Therefore, in order to ensure the timeliness of the formation operation task, the application of the preset time control algorithm has very important significance.

[0004] Usually, the heterogeneous USV-UAV system is easily affected by sea surface wind, wave and other disturbances when performing formation tasks, thus affecting the dynamic characteristics of the USV-UAV system. In order to overcome the influence of uncertain dynamics or wind wave disturbance characteristics, the corresponding uncertain dynamics must be estimated and compensated. At present, the estimation of uncertain dynamics or disturbance mostly adopts disturbance observer or intelligent approximation algorithm. Similarly, the robustness and rapidity of uncertain dynamic compensation are the key indicators for measuring the superiority of the control algorithm, so it is necessary to adopt a preset time observation estimation method for uncertain dynamics. At present, the preset time disturbance observer is relatively less used in heterogeneous USV-UAV systems.

[0005] In summary, how to ensure that the heterogeneous USV-UAV system realizes two-part formation in a preset time is still a technical problem to be solved. SUMMARY

[0006] The application aims to provide a preset time dynamic observation-based heterogeneous USV-UAV system two-part sliding mode formation control method, consider the cooperative and competitive relationship of the heterogeneous USV-UAV system formation, and enhance the anti-disturbance ability when performing the formation task, and accelerate the uncertainty dynamic estimation characteristics of the controlled system.

[0007] To achieve the above application purposes, the application adopts the technical scheme, specifically a preset time dynamic observation-based heterogeneous USV-UAV system two-part sliding mode formation control method, including the following steps:

[0008] S1, auxiliary variables are introduced to convert the underactuated USV and UAV system into a full-drive system, and a virtual leader-follower-based heterogeneous USV-UAV system model is constructed;

[0009] S2, a preset time uncertainty dynamic observer is designed for the follower of the heterogeneous USV-UAV system, and the compound uncertainty dynamics composed of the uncertainty internal dynamics and external disturbance of the system are estimated and compensated in the preset time;

[0010] S3, a two-part formation tracking error system of the heterogeneous USV-UAV system is constructed according to the virtual leader-follower model;

[0011] S4, a sliding mode control-based heterogeneous USV-UAV system preset time distributed two-part formation controller is designed in combination with the uncertainty dynamic observer designed in S2 and the two-part formation tracking error system constructed in S3.

[0012] Further, step S1 is specifically:

[0013] S11: for the underactuated USV system, define (x si ,y si ) as the reference point of the i-th USV, then define a new output state variable p i (t)=[x si (t),y si (t)] T as

[0014]

[0015] In the formula, (x i (t),y i (t)) and θ i (t) are the position and heading angle of the i-th USV in the ground coordinate system; (u i (t),v i (t)) and r i (t) are the linear velocity and angular velocity, and γ i is a very small normal number representing the distance from the reference point to the center of mass.

[0016] The USV equivalent dynamics equation is

[0017]

[0018] where,

[0019] f xi (t) = (f iu (t) - v i (t) r i (t) - γ i r i 2 (t)) cos θ i (t) - (f iv (t) + u i (t) r i (t) + γ i f ir (t)) sin θ i (t)

[0020] f yi (t) = (f iu (t) - v i (t) r i (t) - γ i r i 2 (t)) sin θ i (t) + (f iv (t) + u i (t) r i (t) + γ i f ir (t)) cos θ i (t)

[0021]

[0022] where, τ ui (t), τ ri (t) are the lateral thrust and steering moment of the ith USV; m ui , m vi , m ri are determined by the moment of inertia and added mass; d ui , d vi , d ri are determined by the water flow damping effect; d eui (t), d evi (t), d eri (t) are the external disturbances received.

[0023] S12: for the underactuated UAV system, a new control variable τ xi , τyi τ zi Let be the virtual control variables for the i-th UAV's longitudinal, lateral, and height channels, respectively, and expressed as .

[0024]

[0025] The UAV displacement subsystem is:

[0026]

[0027] In the formula, J i (t)=diag{1 / m ai ,1 / m ai ,1 / m ai}, p i (t)=[x i (t),y i (t),z i (t)] T , Let u be the position and attitude variables of the i-th UAV; i1 (t) represents the total thrust; m ai d is mass; g is gravitational acceleration; d eai (t), a = x, y, z represents the external disturbances.

[0028] S13: Considering the different maneuverability of each USV and UAV in the heterogeneous USV-UAV system, a heterogeneous USV-UAV system model based on virtual leader-follower was constructed.

[0029] Consider a heterogeneous USV-UAV system consisting of N different USVs and UAVs. In the single USV / UAV system described in S11 and S12, graph theory is introduced... The represented communication topology constitutes the corresponding heterogeneous USV-UAV system. Wherein: Represents a set of nodes; Represents the set of directed edges; Representation diagram The adjacency matrix, a ij >0 indicates that node i and node j have a cooperative relationship, a ij <0 indicates that node i and node j are in competition; Graph Laplace matrix notation Defined as: if If j≠i, l ij =-a ij Let the leader node be v0. If the i-th follower can receive information from the leader, then b i >0, otherwise b i=0. Define a diagonal matrix. And define the matrix

[0030] In this system, all nodes are divided into two subsets. and Within each subset, adjacent nodes have a cooperative relationship, while nodes between two subsets have a competitive relationship. Define matrix D = diag{d1, d2, ..., d...} N}, if node d i =1; if node d i = -1, and all elements in matrix DAD = |A| are non-negative.

[0031] Considering only the formation in the XY plane, for the i-th USV and the i-th UAV, the new system state equation is defined as follows:

[0032]

[0033] In the formula, p i (t), For position and velocity variables; For external interference, satisfying ||D i (t)‖≤σ Di ,σ Di >0.

[0034] The virtual leader model is

[0035]

[0036] In the formula, p d (t), v d (t), For the virtual leader's position, speed, and control input. Acceleration It is a vector-valued function that relates to the actual internal physical state, and is therefore bounded, satisfying the following conditions:

[0037] Introducing continuously differentiable queuing vectors h(t) = [h1(t), h2(t), ..., h N (t)] T Let represent the desired formation of the followers in the heterogeneous USV-UAV system on the XY plane. For any follower with a bounded initial state and initial time t0, if ...

[0038]

[0039] The heterogeneous USV-UAV system (1) can achieve pre-set time binary formation, t0+Tp The time required for the system to implement binary formation can be designed according to the formation requirements.

[0040] Furthermore, step S2 specifically includes:

[0041] S21: Introduce the following time-varying function

[0042]

[0043] In the formula, the constant κ0>1 and t0>0 represent the initial time, and t p >0 represents the convergence time, T p =t0+t p To pre-set the convergence time;

[0044] S22: Design the following pre-defined time-uncertain dynamic observer.

[0045]

[0046] In the formula, E i (t)=v i (t)-χ i (t); Λ i (t)=F i (t)+D i (t) represents the uncertain internal dynamics F of the system. i (t) and external disturbance D i The composite uncertain dynamics of (t) are continuous and bounded, satisfying ||F i (t)||≤μ Fi μ Fi >0; Represents Λ i Estimate (t); It is a positive gain, and tanh(·) is the hyperbolic tangent function.

[0047] Furthermore, step S3 specifically includes:

[0048] Define the leader-following error in a heterogeneous USV-UAV system.

[0049]

[0050] make e1(t)=[e 11 (t) T e 12 (t) T … e 1N (t) T ] T e2(t)=[e 21(t) T e 22 (t) T … e 2N (t) T ] T . Let Kronecke product be represented. Then equation (3) can be written in vector form.

[0051]

[0052] Combining equations (1), (2), and (4), the bi-grouping tracking error system is as follows:

[0053]

[0054] Furthermore, step S4 specifically includes:

[0055] S41: Introduce the following time-varying function

[0056]

[0057] In the formula, constants κ1,κ2>1, T1 is the sliding time, and t0+T1 is the preset time for the system to reach the sliding surface.

[0058] S42: Design a preset time sliding surface based on the function Ψ(t).

[0059]

[0060] in,

[0061] S43: Based on the function Ψ(t) and The preset time-division sliding mode formation control rate is designed as follows:

[0062]

[0063] in,

[0064] u i (t)=u ai (t)+u bi (t)

[0065]

[0066] In the formula, the control gain k > 0.

[0067] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0068] (1) This invention introduces auxiliary variables to convert underactuated USV and UAV systems into fully driven systems, and designs a pre-set time uncertainty dynamic observer for heterogeneous USV-UAV systems for uncertain internal dynamics and external disturbances, which enhances the anti-disturbance capability during formation and accelerates the estimation characteristics of uncertain dynamics of the controlled system.

[0069] (2) Unlike existing USV-UAV system formations that only consider cooperative relationships between agents, this invention considers both cooperative and competitive relationships between USVs and UAVs. Under the structural equilibrium diagram, followers of heterogeneous USV-UAV systems can track leaders at precisely preset times, achieving binary formation, with the final positions being the same size but in opposite directions.

[0070] (3) Unlike the finite and fixed time control methods that rely on system parameters or even initial values, this invention designs a preset time binary sliding mode formation control rate to ensure that the heterogeneous USV-UAV system achieves binary formation at a preset time, which can be preset according to requirements. Attached Figure Description

[0071] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0072] Figure 1 The overall flowchart of the method provided by this invention;

[0073] Figure 2 This is a structural diagram of the heterogeneous USV-UAV system in this invention.

[0074] Figure 3 This is a diagram of the communication topology of the heterogeneous USV-UAV system in this invention;

[0075] Figure 4 This is a schematic diagram of the longitudinal, spatial, and lateral bipartite formation trajectories of followers under the steady-expected formation in this invention; wherein, (a) is a longitudinal top view of the steady-expected bipartite formation; (b) is a three-dimensional spatial diagram of the steady-expected bipartite formation; and (c) is a lateral trajectory diagram of the steady-expected bipartite formation.

[0076] Figure 5 This is a schematic diagram of the bi-grouping tracking error of the follower under the steady-expected formation in this invention; wherein, (a) is a schematic diagram of the bi-grouping tracking error in the steady-expected X direction; and (b) is a schematic diagram of the bi-grouping tracking error in the steady-expected Y direction.

[0077] Figure 6 This is a schematic diagram illustrating the estimated and actual dynamics of the follower under the steady-state expected formation in this invention.

[0078] Figure 7 This is a schematic diagram of the longitudinal, spatial, and lateral bipartite formation trajectory of the followers under the time-varying expected formation in this invention; wherein, (a) is a longitudinal top view of the time-varying expected bipartite formation; (b) is a three-dimensional spatial diagram of the time-varying expected bipartite formation; and (c) is a lateral trajectory diagram of the time-varying expected bipartite formation.

[0079] Figure 8 This is a schematic diagram of the bipartite formation tracking error of the follower under the time-varying expected formation in this invention; wherein, (a) is a schematic diagram of the bipartite formation tracking error in the time-varying expected X direction; and (b) is a schematic diagram of the bipartite formation tracking error in the time-varying expected Y direction.

[0080] Figure 9 This is a schematic diagram illustrating the estimated and actual dynamics of the follower under the time-varying expected formation in this invention. Figure 1 .

[0081] Figure 10 This is a schematic diagram illustrating the estimated and actual dynamics of the follower under the time-varying expected formation in this invention. Figure 2 . Detailed Implementation

[0082] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0083] Example 1

[0084] Reference Figure 1 The technical solution provided in this embodiment is: a binary sliding mode formation control method for heterogeneous USV-UAV systems based on preset time dynamic observation, including the following steps:

[0085] S1. Introduce auxiliary variables to convert underactuated USV and UAV systems into fully driven systems, and construct a heterogeneous USV-UAV system model based on virtual leader-follower;

[0086] S2. Design a time-determined dynamic observer for the heterogeneous USV-UAV system follower to estimate and compensate for the composite uncertainty dynamics of the system's uncertain internal dynamics and external disturbances at a predetermined time.

[0087] S3. Construct a binary formation tracking error system for heterogeneous USV-UAV systems based on the virtual leader-follower model;

[0088] S4. Combining the uncertain dynamic observer designed in S2 and the binary formation tracking error system constructed in S3, design a time-distributed binary formation controller for a heterogeneous USV-UAV system based on sliding mode control.

[0089] Specifically, step S1 is as follows:

[0090] S11: For underactuated USV systems, define (x si ,y si If p is used as the reference point for the i-th USV, then a new output state variable p is defined. i (t)=[x si (t),y si (t)] T for

[0091]

[0092] In the formula, (x i (t),y i (t)), θ i (t) represents the position and heading angle of the i-th USV in the ground coordinate system; (u) i (t),v i (t)), r i (t) represents the linear velocity and angular velocity, γ i It is a very small positive number, representing the distance from the reference point to the centroid.

[0093] The equivalent dynamic equation of USV is:

[0094]

[0095] in,

[0096] f xi (t)=(f iu (t)-v i (t)r i (t)-γ i r i 2 (t))cosθ i (t)-(f iv (t)+u i (t)r i (t)+γ i f ir (t))sinθ i (t),

[0097] f yi (t)=(f iu (t)-v i (t)r i (t)-γ i r i 2 (t))sinθ i (t)+(f iv (t)+ui (t)r i (t)+γ i f ir (t))cosθ i (t),

[0098]

[0099] In the formula, τ ui (t), τ ri (t) represents the lateral thrust and steering torque of the i-th USV; m ui m vi m ri Determined by the moment of inertia and added mass; d ui d vi d ri Determined by the water flow damping effect; d eui (t), d evi (t), d eri (t) represents the external disturbance received.

[0100] S12: For underactuated UAV systems, a new control variable τ is introduced. xi τ yi τ zi Let be the virtual control variables for the i-th UAV's longitudinal, lateral, and height channels, respectively, and expressed as .

[0101]

[0102] The UAV displacement subsystem is:

[0103]

[0104] In the formula, J i (t)=diag{1 / m ai ,1 / m ai ,1 / m ai}, p i (t)=[x i (t),y i (t),z i (t)] T , Let u be the position and attitude variables of the i-th UAV; i1 (t) represents the total thrust; m ai d is mass; g is gravitational acceleration; d eai (t), a = x, y, z represents the external disturbances.

[0105] S13: Considering the different maneuverability of each USV and UAV in the heterogeneous USV-UAV system, a heterogeneous USV-UAV system model based on virtual leader-follower was constructed.

[0106] Consider a heterogeneous USV-UAV system consisting of N different USVs and UAVs. In the single USV / UAV system described in S11 and S12, graph theory is introduced... The represented communication topology constitutes the corresponding heterogeneous USV-UAV system. Wherein: Represents a set of nodes; Represents the set of directed edges; Representation diagram The adjacency matrix, a ij >0 indicates that node i and node j have a cooperative relationship, a ij <0 indicates that node i and node j are in competition; Graph Laplace matrix notation Defined as: if j = i, If j≠i, l ij =-a ij Let the leader node be v0. If the i-th follower can receive information from the leader, then b i >0, otherwise b i =0. Define a diagonal matrix. And define the matrix

[0107] In this system, all nodes are divided into two subsets. and Within each subset, adjacent nodes have a cooperative relationship, while nodes between two subsets have a competitive relationship. Define matrix D = diag{d1, d2, ..., d...} N}, if node d i =1; if node d i = -1, and all elements in matrix DAD = |A| are non-negative.

[0108] Considering only the formation in the XY plane, for the i-th USV and the i-th UAV, the new system state equation is defined as follows:

[0109]

[0110] In the formula, p i (t), For position and velocity variables; For external interference, satisfying ||D i (t)‖≤σ Di ,σ Di >0.

[0111] The virtual leader model is

[0112]

[0113] In the formula, p d (t), v d (t), For the virtual leader's position, speed, and control input. Acceleration It is a vector-valued function that relates to the actual internal physical state, and is therefore bounded, satisfying the following conditions:

[0114] Introducing continuously differentiable queuing vectors h(t) = [h1(t), h2(t), ..., h N (t)] T Let represent the desired formation of the followers in the heterogeneous USV-UAV system on the XY plane. For any follower with a bounded initial state and initial time t0, if ... and The heterogeneous USV-UAV system (1) can achieve pre-set time binary formation, t0+T p The time required for the system to implement binary formation can be designed according to the formation requirements.

[0115] Specifically, step S2 is as follows:

[0116] S21: Introduce the following time-varying function

[0117]

[0118] In the formula, the constant κ0>1 and t0>0 represent the initial time, and t p >0 represents the convergence time, T p =t0+t p To pre-set the convergence time;

[0119] S22: Design the following pre-defined time-uncertain dynamic observer.

[0120]

[0121] In the formula, E i (t)=v i (t)-χ i (t); Λ i (t)=F i (t)+D i (t) represents the uncertain internal dynamics F of the system. i (t) and external disturbance D i The composite uncertain dynamics of (t) are continuous and bounded, satisfying ||Fi (t)||≤μ Fi μ Fi >0; Represents Λ i Estimate (t); It is a positive gain, and tanh(·) is the hyperbolic tangent function.

[0122] Theorem 1: Design a preset time dynamic observer (3) that can be used at a preset time t0+t p1 Accurate estimation of composite uncertain dynamics Λ i (t), and the observation error tends to zero.

[0123] The proof of Theorem 1 is as follows:

[0124] According to E i (t) Define and use equation (3), E i The derivative of (t) satisfies

[0125]

[0126] Define dynamic observation error as

[0127]

[0128] From equations (3) and (4), we can obtain

[0129]

[0130] Constructing Lyapunov functions And by taking the derivative

[0131]

[0132] When t∈[t0,t0+t] p1 When ψ(t) is in the range of ψ(t), according to the definition of ψ(t), equation (6) can be written as

[0133]

[0134] Multiplying both sides of inequality (7) achievable

[0135]

[0136] right Taking the derivative and combining it with equation (8), we can obtain

[0137]

[0138] Therefore, the function In [t0, t0+t p1 The expression is monotonically decreasing, which leads to the conclusion that... According to the definition of Ψ(t), we can obtain

[0139] When t∈[t0+t p1 When ,∞), since V(t) at t=t0+t p1 Continuous, we can obtain From equation (6), we can deduce In t∈[t0+t p1 This holds true for all x, ∞. Therefore, In t∈[t0+t p1 The condition holds true for all ,∞. According to the principle of function monotonicity, we can conclude that 0≤V(t)≤V(t0+t). p1 ) = 0. That is, when t∈[t0+t p1 When ,∞), V(t)≡0, Further observational errors exist.

[0140] Specifically, step S3 is as follows:

[0141] Define the leader-following error in a heterogeneous USV-UAV system.

[0142]

[0143] make e1(t)=[e 11 (t) T e 12 (t) T … e 1N (t) T ] T e2(t)=[e 21 (t) T e 22 (t) T … e 2N (t) T ] T . Let Kronecke product be represented. Then equation (9) can be written in vector form.

[0144]

[0145] Combining equations (1), (2), and (10), the bi-grouping tracking error system is as follows:

[0146]

[0147] Specifically, step S4 is as follows:

[0148] S41: Introduce the following time-varying function

[0149]

[0150] In the formula, constants κ1,κ2>1, T1 is the sliding time, and t0+T1 is the preset time for the system to reach the sliding surface.

[0151] S42: Design a preset time sliding surface based on the function Ψ(t).

[0152]

[0153] in,

[0154] S43: Based on the function Ψ(t) and The preset time-division sliding mode formation control rate is designed as follows:

[0155]

[0156] in,

[0157] u i (t)=u ai (t)+u bi (t)

[0158]

[0159] In the formula, the control gain k > 0.

[0160] Theorem 2: Under the preset time-division sliding mode formation control law (13), if the control gain satisfies Then the heterogeneous USV-UAV system at t=T p The preset time binary search grouping is implemented, and u i (t) remains bounded on [t0,∞).

[0161] The proof of the above theorem consists of three steps. A: Prove that the heterogeneous USV-UAV system can reach the sliding surface s(t) = 0 at t = t0 + T1; B: Prove that the system can reach the sliding surface s(t) = 0 at t = T p Implement binary sorting; C: Prove u i (t) remains bounded over the entire time interval [t0,∞).

[0162] A: Prove that under the control protocol (13), the heterogeneous USV-UAV system reaches the sliding surface s(t) = 0 at t = t0 + T1.

[0163] Differentiating with respect to the sliding mode variable yields Choose Lyapunov functions s T (t)s(t), differentiating, we get

[0164]

[0165] Based on the conclusion of Theorem 1, we can further derive...

[0166]

[0167] When t∈[t0,t0+T1), according to By definition, equation (14) can be written as

[0168]

[0169] Multiplying both sides of inequality (15) achievable

[0170]

[0171] right Taking the derivative and combining it with equation (16), we can obtain

[0172]

[0173] Therefore, the function It is monotonically decreasing on t∈[t0,t0+T1), therefore... according to From the definition, we can obtain

[0174] When t∈[t0+T1,t0+t p At time t = t0 + t, since V1(t) is at time t = t0 + t p Continuous, we can obtain From equation (14), we can deduce It holds true for all t∈[t0,∞), therefore it holds true for all t∈[t0+T1,t0+t). p )hour, This holds true consistently. According to the principle of function monotonicity, we can conclude that 0 ≤ V1(t) ≤ V(t0 + T1) = 0, that is, when t ∈ [t0 + T1, t0 + t...]. p When ), s(t)≡0.

[0175] Based on the above analysis, it can be seen that the heterogeneous USV-UAV system reaches the sliding surface s(t) = 0 at t = t0 + T1, and remains at [t0 + T1, t0 + T1]. p It slides along the sliding surface inside.

[0176] B: Prove that the errors e1(t) and e2(t) in the binary formation following the leader occur at t=T p =t0+t p The heterogeneous USV-UAV system converges to zero along the sliding surface at t=T p The two-part formation is achieved in time.

[0177] When t∈[t0+T1,t0+t p When ), since the sliding surface has been reached, then let Can make Constructing Lyapunov functions Taking the derivative of V2(t) gives

[0178]

[0179] Similar to the proof of A, we can further obtain

[0180] because It can be obtained

[0181] When t∈[t0+t p When ,∞), since e1(t) at t=t0+t p Time is continuous, and we can obtain... e1(t) = 0. Similarly, we can obtain Based on the definition of s(t), we can obtain Combining this with the conclusion in A, we can conclude that s(t) at t = t0 + t p The time intervals are continuous. Similar to the proof in A, according to equation (14), we know that 0 ≤ V1(t) ≤ V1(T) p ) = 0, that is, when t > T p When s(t) ≡ 0.

[0182] To prove that when t > T p When e1(t)≡0, e2(t)≡0. Construct the Lyapunov function.

[0183] When t∈[T] p When ,∞), Ψ(t)=1. From s(t)=e2(t)+e1(t)=0, we can get e2(t)=-e1(t). Differentiating V3(t) with respect to , we get Since V3(t) is in [T p Since V3(t) is monotonically decreasing on (∞), we know that 0 ≤ V3(t) ≤ V3(T). p ) = 0, that is, when t ≥ T p When t ≥ T, e1(t) = 0 always holds true. Also, since e2(t) = -e1(t), when t ≥ T... p When e2(t) = 0, the condition holds true.

[0184] From the binary formation tracking error system (11), we know that e1(t) = 0 is equivalent to And because It is neither singular nor strange, therefore Equivalent to p(t)-h(t)-p d (t)=0. Similarly, we can obtain that e2(t)=0 is equivalent to

[0185] Based on the above analysis, it can be seen that when t∈[T] p When p(t)-h(t)-p d (t)≡0, Therefore, the heterogeneous USV-UAV system at the preset time T p The two-part formation was achieved.

[0186] C: Prove that when t∈[t0,∞), u i (t) is always bounded.

[0187] Based on the sliding surface (12) and the preset time bisection sliding formation control law (13), it can be known that

[0188]

[0189] When t∈[t0,t0+T1),

[0190]

[0191] Given that κ1>1, it is easy to see that ‖u(t)‖ is bounded.

[0192] When t∈[t0+T1,t0+t p When ), according to the function The definitions of Ψ(t) are as follows: Equation (14) can be further written as

[0193] Similar to proof A, we have

[0194]

[0195] Then, substituting equation (20) into equation (18) yields...

[0196]

[0197] in, Therefore, when κ2>1, ||u(t)|| is bounded.

[0198] When t∈[t0+t P When t ∈ [t0+t), s(t) ≡ 0. From equation (18), we know that ||u(t)|| is true when t ∈ [t0+t]. P ,∞) is always bounded.

[0199] Based on the above analysis, it can be seen that u(t) is always bounded over the entire time interval [t0,∞).

[0200] The proof is complete.

[0201] Example 2

[0202] To verify the effectiveness of the time-division sliding mode formation control law designed in Example 1, a heterogeneous USV-UAV system consisting of a virtual leader (i=0), four follower USVs (i=1,2,5,6), and four follower UAVs (i=3,4,7,8) is considered in the Matlab / Simulink simulation environment. The corresponding communication topology is shown below. Figure 3 As shown, the set It can be divided into two subsets and D = diag{1,1,1,1,-1,-1,-1,-1,-1}. The interaction matrix between the leader and followers is B = diag{1,0,0,0,0,0,0,0}.

[0203] The parameters of the four follower USV and UAV systems are as follows:

[0204]

[0205] Let the external disturbance D(t) = 0.2sint. The parameters of the preset time-uncertain dynamic observer and the bipartite sliding mode formation controller are selected as follows: k=20, κ0=1.51, κ1=1.52, κ2=1.53.

[0206] Let the initial time of the heterogeneous USV-UAV system be t0 = 0s, the time to reach the sliding surface be t0 + T1 = 2s, and the time to achieve binary formation be T. p =4s.

[0207] Under a steady-state expected formation, the position and velocity variables of the virtual leader are:

[0208] p d (t) = [-1.5cos(t), 1.5sin(t)] T ,v d (t) = [1.5sin(t), 1.5cos(t)] T

[0209] Where t = [0, 12].

[0210] The initial position state of the follower in the heterogeneous USV-UAV system is randomly selected from [-1,1]×[-1,1].

[0211] The constant expected formation is

[0212] h1(t) = [0,0] T h2(t) = [0,1] T h3(t) = [1,1] T h4(t) = [1, 0] T ,

[0213] h5(t) = [0,0] T h6(t) = [0, =1] T h7(t) = [=1,=1] T h8(t) = [1, 0] T

[0214] Figure 4 The bipartite formation trajectories in the longitudinal, spatial, and lateral directions under the steady desired formation of a heterogeneous USV-UAV system. Figure 5 The variation process of the tracking error in a binary formation with a preset time is described. A comparison is made between the estimated dynamics of the composite uncertainty of the follower and the actual dynamics of the composite uncertainty. Figure 6 As shown.

[0215] Example 3

[0216] In a time-varying expected formation, the position and velocity variables of the virtual leader are:

[0217] p d (t) = [0.2t, 0.1t] T ,v d (t) = [0.2, 0.1] T

[0218] Where t = [0, 12].

[0219] Time-varying expected formation

[0220] h1(t) = [-0.1t, -0.1t] T h2(t) = [-0.1t, 0.1t] T h3(t) = [0.1t, 0.1t] T h4(t) = [0.1t, -0.1t] T ,

[0221] h5(t) = [0.1t, 0.1t] T h6(t) = [0.1t, -0.1t] T h7(t) = [-0.1t, -0.1t] T h8(t) = [-0.1t, 0.1t] T

[0222] All others are consistent with the steady-state expected formation. The corresponding bipartite formation trajectories of the heterogeneous USV-UAV system followers in the longitudinal, spatial, and lateral directions are as follows: Figure 7 As shown. The estimated dynamics and actual dynamics are as follows. Figure 8 As shown. Figure 9 The variation process of the tracking error in a binary formation with a preset time is described. A comparison is made between the estimated dynamics of the composite uncertainty of the follower and the actual dynamics of the composite uncertainty. Figure 10 As shown.

[0223] from Figure 4 and Figure 7 It can be seen that the heterogeneous USV-UAV system achieves binary formation at t=4s. (Subset) The positions of the four nodes (USV1, USV2, UAV3, UAV4) at t=4 are the same as the leader's position, while the subsets... The positions of the four nodes (USV5, USV6, UAV7, UAV8) at t=4 are the same size as the leader's position but opposite in direction, further confirming the conclusion of Theorem 2. Figure 5 and Figure 8 It can be seen that at t=4s, the binary formation tracking error of the eight followers in the heterogeneous USV-UAV system converges to zero, which is consistent with the theoretically analyzed time for achieving binary formation. From the two simulation examples above, it can be seen that the designed preset-time binary sliding mode formation control method can track the leader's trajectory well and can be applied to complex formation tasks requiring precise dwell time.

[0224] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A binary sliding mode formation control method for heterogeneous USV-UAV systems based on preset time dynamic observation, characterized in that, Includes the following steps: S1. Introduce auxiliary variables to convert underactuated USV and UAV systems into fully driven systems, and construct a virtual leader-based system. -A heterogeneous USV-UAV system model for followers; S2. Design a dynamic observer with preset time uncertainty for the follower of the heterogeneous USV-UAV system, and observe the system at a preset time. The complex uncertain dynamics, consisting of uncertain internal dynamics and external disturbances, are estimated and compensated. Step S2 includes the following steps: S21: Introduce the following time-varying function ； In the formula, constant , The initial time, To shorten the convergence time, To pre-set the convergence time; S22: Design the following pre-defined time-uncertain dynamic observer. ； In the formula, ; For including the uncertain internal dynamics of the system and external disturbances The composite uncertain dynamics, where the uncertain dynamics are continuous and bounded, satisfy... ; express The estimate; It is a positive gain, and ; It is the hyperbolic tangent function; S3. Construct a binary formation tracking error system for heterogeneous USV-UAV systems based on the virtual leader-follower model; S4. Combining the uncertain dynamic observer designed in step S2 and the binary formation tracking error system constructed in step S3, design a time-distributed binary formation controller for a heterogeneous USV-UAV system based on sliding mode control. Step S4 includes the following steps: S41: Introduce the following time-varying function ； In the formula, constant , For sliding time, The pre-set time for the system to reach the sliding surface; S42: Function-based The preset time sliding surface is designed as follows: ； in, ; S43: Function-based and The preset time-division sliding mode formation control rate is designed as follows: ； in, ； ； In the formula, control gain .

2. The binary sliding mode formation control method for heterogeneous USV-UAV systems based on preset time dynamic observation as described in claim 1, characterized in that, Step S1 includes the following steps: S11: For underactuated USV systems, define As the first For each reference point of a USV, a new output state variable is defined. for: ； In the formula, , For the first The position and heading angle of each USV in the ground coordinate system; , For linear velocity and angular velocity, This is a very small positive constant, representing the distance from the reference point to the centroid; The equivalent dynamic equation of the USV is: ； in, ; ； In the formula, , For the first The lateral thrust and steering torque of each USV; , , It is determined by the moment of inertia and the added mass; , , Determined by the water flow damping effect; , , External interference received; S12: Introducing new control variables for underactuated UAV systems. , , , respectively The virtual control variables for the longitudinal, lateral, and height channels of a UAV are represented as follows: ； The UAV displacement subsystem is: ； In the formula, , , , ; , For the first The position and attitude variables of each UAV; Total thrust; For quality; It is the acceleration due to gravity; External interference received; S13: Constructing a heterogeneous USV-UAV system model based on virtual leader-follower Heterogeneous USV-UAV system is composed of Composed of several different USVs and UAVs, the graph theory is introduced into the individual USV and UAV systems in steps S11 and S12. The represented communication topology constitutes the corresponding heterogeneous USV-UAV system, wherein: Represents a set of nodes; Represents the set of directed edges; Representation diagram The adjacency matrix, Represents a node and nodes The two parties have a cooperative relationship. Represents a node and nodes There is a competitive relationship; diagram Laplace matrix notation , defined as: if , ;like , The leader node is recorded as If the first If a follower receives information from a leader, then ,otherwise Define a diagonal matrix and define the matrix ; In this system, all nodes are divided into two subsets. and Within each subset, adjacent nodes have a cooperative relationship, while nodes between two subsets have a competitive relationship. A matrix is ​​defined as follows. If node , If node , And matrix All elements in the expression are non-negative; Formation on the XY plane, then for the first The USV and the first Given a UAV, define a new system state equation as follows: ； In the formula, For position and velocity variables; For external interference, satisfy , The virtual leader model is ； In the formula, , , Acceleration for the virtual leader's position, speed, and control input. It is a vector-valued function that relates to the actual internal physical state, and it is bounded, satisfying... ; Introducing continuously differentiable queuing vectors , Describe the desired formation of followers in a heterogeneous USV-UAV system in the XY plane, with bounded initial states and initial times for any follower. If satisfied and ; The heterogeneous USV-UAV system (1) then implements a pre-set time binary sorting formation. The time required for the system to implement binary formation.

3. The binary sliding mode formation control method for heterogeneous USV-UAV systems based on preset time dynamic observation as described in claim 1, characterized in that, Step S3 is as follows: Define the leader-following error in a heterogeneous USV-UAV system. ； make , , ; ； If we express the Kronecke product, then equation (3) can be written in vector form. ； Combining equations (1), (2), and (4), the bi-grouping tracking error system is as follows: 。

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