Multi-underwater vehicle virtual network elastic offset inclusion control method considering composite attack

Abstract

The invention discloses a multi-underwater vehicle virtual network elastic offset inclusion control method considering composite attacks, and belongs to the technical field of underwater vehicle control. The invention aims to solve the problem that the existing multi-underwater vehicle containing control method cannot cope with DoS attacks. Aiming at a multi-AUV system based on a following-pilot mode, the condition that the multi-AUV system suffers from DoS attack and unbounded actuator attack is considered, the offset is considered in the inclusion error, and the inclusion control consistency of the system is realized by designing a distributed elastic offset observer and a distributed elastic offset inclusion control algorithm. The method can effectively cope with composite attacks of DoS attacks and unbounded actuator attacks, and particularly, the elastic offset controller designed by the method not only can successfully resist external attacks, but also is wider in application range and can perform effective control.

Description

Technical Field

[0001] The present invention belongs to the technical field of underwater vehicle control, and particularly designs a multi-underwater vehicle virtual network elastic offset inclusion control method Background Art

[0002] With the continuous deepening of ocean development and scientific research, autonomous underwater vehicles (AUVs) have been widely used in the fields of ocean resource exploration, seabed terrain mapping, etc. The multi-AUV system, due to its collaborative ability, can complete complex tasks, such as large-area ocean monitoring, multi-target tracking, and task execution in complex situations, showing great application potential. However, the multi-AUV system faces many challenges in practical applications, such as difficult underwater communication, limited communication resources, and various communication interference attacks in military operations

[0003] The underwater communication environment is complex. The underwater acoustic channel has the characteristics of narrow bandwidth and low carrier frequency, and can only support a relatively low information transmission rate. This poses a great challenge to the cooperative control of AUV clusters. Compared with the centralized control system, the distributed control system has stronger robustness, flexibility, and scalability. For the leader-follower mode of the distributed multi-AUV system, the follower only needs to communicate with neighboring followers and some leaders, and the leader communicates with the surface mother ship, reducing the communication pressure of the follower. In the inclusion control, only multiple leaders communicate with the surface mother ship, receive instructions and plan the task trajectory, while the follower only communicates with some leaders and neighboring followers. In this process, all followers will act together with the convex hull formed by multiple leaders, and then reach the task area to start operations

[0004] In practical engineering problems, the multi-AUV system is a vulnerable network system, and in some cases, it also faces the risk of network-level attacks. The DoS attack is a main and common attack form in the field of network attacks, which can affect or even destroy the normal services and communications of the target network. Considering the complex real situation, actuator attacks can often pose a threat that cannot be ignored to the multi-agent system. The design of security control is an important issue in the multi-agent system. However, there are few research results on considering the multi-AUV system suffering from network attacks. In addition, the current research on actuator attacks generally assumes that the attack signal is bounded, which cannot fully show the maximum malice of the attacker because the attack signal may be unbounded Summary of the Invention

[0005] The present invention aims to solve the problem that the existing inclusion control method for multi-underwater vehicles cannot cope with DoS attacks

[0006] A virtual network elastic offset containment control method for multi-underwater vehicles considering composite attacks, for a multi-AUV system with M leader AUVs and N follower AUVs, let L = (1, 2, …, M), F = (M + 1, M + 2, …, M + N), and use a distributed elastic offset controller for control. The distributed elastic offset controller is as follows:

[0007]

[0008] where u i is the follower control input, i ∈ F; B 0i represents the thrust allocation matrix, is the corresponding transpose; M ηi represents the mass matrix of the AUV, is the transpose of the corresponding inverse matrix; C ηi represents the Coriolis force matrix, D ηi represents the fluid damping force matrix, g ηi represents the restoring force provided by gravity and buoyancy; η i (t) represents the position and angle vector of the follower AUV, is the first derivative of η i (t); is the first derivative of the virtual control quantity α i (t); z i (t) is the error between i and the virtual control quantity α is the transpose of z i (t); H i is a positive constant matrix, represents the compensation signal, is the adaptive parameter, is the first derivative of, θ is a positive constant; is a constant; ε i (t) is the state tracking error between the true state quantity η i (t) of follower i at the real physical network level and the observed quantity ξ i (t) at the virtual network level.

[0009] Furthermore, the virtual control quantity is where P i is a positive constant matrix, is the first derivative of the virtual observed quantity ξ i (t) of follower i at the virtual network level.

[0010] Furthermore, the observed quantity ξ i(t) is determined by the distributed resilient offset observer as shown below:

[0011]

[0012] where \(t\geq t_0\), \(t_0\) is a certain moment during the DoS attack process, and \(\xi\) i (t) represents the observed quantity of follower \(i\) at the virtual network layer. is the first derivative of \(\xi\) i (t); \(U\) is the designed gain matrix, and \(\xi\) ci (t) represents the local error of follower \(i\) related to neighbor information, and \(\chi\) i (t) represents the reference offset compensation signal; \(r\) i (t) represents the known reference offset, is the first derivative of \(r\) i (t).

[0013] Furthermore, the local error \(\xi\) of follower \(i\) related to neighbor information ci (t) is as follows:

[0014]

[0015] where \(\Upsilon\) N (t0,t), \(\Upsilon\) A (t0,t) are the non-attacked time interval and the attacked interval from \(t_0\) to \(t\); \(r\) j (t) is the known reference offset of follower \(j\); \(\xi\) j (t) is the observed quantity of follower \(j\) at the virtual network layer; \(\xi\) l (t) is the observed quantity of leader \(l\) at the virtual network layer; \(a\) ij represents the communication state between follower nodes. When follower node \(i\) can communicate with follower node \(j\), \(a\) ij = 1; otherwise \(a\) ij = 0.

[0016] Furthermore, the observed quantity \(\xi\) of leader \(l\) at the virtual network layer l (t)=\(\eta\) l (t), and \(\eta\) l (t) represents the position and angle vector of leader \(l\).

[0017] Furthermore, the total non-attacked time \(\Upsilon\) N (t0,t)=[t0,t] / \(\Upsilon\) A (t0,t).

[0018] Furthermore, the total duration \(\Upsilon\) of the DoS attack A (t0,t) satisfies where \(T_0>0\), \(\tau\)a > 1, and both are constants.

[0019] Furthermore, the kinematic models of the AUVs corresponding to the leader and the follower are as follows:

[0020]

[0021] where η a (t) represents the position and angle vector of the AUV, and the dot above the parameter represents the first derivative of the corresponding parameter. is the first derivative of η a (t), F a is a constant matrix, and a ∈ L ∪ F.

[0022] Beneficial effects:

[0023] For the multi-AUV system based on the leader-follower mode, this invention considers the situation where the multi-AUV system is subjected to DoS attacks and unbounded actuator attacks, and takes into account the offset in the inclusion error. By designing a distributed elastic offset observer and a distributed elastic offset inclusion control algorithm, the inclusion control consistency of the system is achieved. Therefore, this invention can effectively cope with the combined attacks of DoS attacks and unbounded actuator attacks. In particular, the elastic offset controller designed in this invention can not only successfully resist external attacks, but also has a wider application range. The effectiveness of the control law of this invention can be seen through numerical simulation. Description of the drawings

[0024] Figure 1 is a schematic diagram of the moving coordinate system;

[0025] Figure 2 is a model diagram of the AUV;

[0026] Figure 3 is a directed communication topology diagram of the leader-follower multi-AUV system;

[0027] Figure 4 is a diagram of the three-dimensional spatial position change of each AUV in the leader-follower multi-AUV system;

[0028] Figure 5 is a diagram of the position change of each AUV in the leader-follower multi-AUV system;

[0029] Figure 6 is a diagram of the angle change of each AUV in the leader-follower multi-AUV system;

[0030] Figure 7 is the virtual offset observation inclusion error of the follower AUV in the leader-follower multi-AUV system change diagram;

[0031] Figure 8For the follower AUV state tracking error ||ε i (t)|| 2 Variation diagram;

[0032] Figure 9 For the follower AUV design error ||z i (t)|| 2 Variation diagram;

[0033] Figure 10 For the follower AUV offset inclusion error ||e i (t)|| 2 Variation diagram;

[0034] Figure 11 For the 3D spatial position variation diagram of each AUV in the leader-follower multi-AUV system;

[0035] Figure 12 For the position variation diagram of each AUV in the leader-follower multi-AUV system;

[0036] Figure 13 For the angle variation diagram of each AUV in the leader-follower multi-AUV system;

[0037] Figure 14 For the virtual offset observation inclusion error of the follower AUV in the leader-follower multi-AUV system Variation diagram;

[0038] Figure 15 For the follower AUV state tracking error ||ε i (t)|| 2 Variation diagram;

[0039] Figure 16 For the follower AUV design error ||z i (t)|| 2 Variation diagram;

[0040] Figure 17 For the follower AUV offset inclusion error ||e i (t)|| 2 Variation diagram. Detailed implementation method

[0041] In view of the problems existing in the background technology, the present invention is directed to a leader-follower multi-AUV system, considering the situation where the multi-AUV system is subjected to DoS attacks and unbounded actuator attacks, and taking into account the offset in the inclusion error, and realizing the inclusion control consistency of the system by designing a distributed resilient offset observer and a distributed resilient offset inclusion control algorithm. The present invention will be described in detail below in conjunction with specific embodiments. Specific Embodiment 1:

[0043] This embodiment is a virtual network resilient offset inclusion control method for multi-underwater vehicles considering composite attacks. First, this embodiment focuses on the design process of the distributed resilient offset controller, which includes the following steps:

[0044] I. Establish a coordinate system:

[0045] (1) Inertial coordinate system E-xyz:

[0046] The inertial coordinate can be simply understood as a "fixed" coordinate, and its relative position with the earth remains unchanged. Usually, the reference plane is selected as the tangent plane of the earth's surface, and the coordinate plane Exy coincides with the tangent plane of the sea area where the AUV operates. Here, E is the origin of the inertial coordinate system; the positive direction of the Ex axis points to the north pole; the positive direction of the Ey axis is to the east geographically and is perpendicular to the Ex axis at the same time; the Ez axis is perpendicular to the Exy plane, and its positive direction points to the center of the earth.

[0047] (2) Moving coordinate system O-ξσζ:

[0048] The moving coordinate system is fixed on the object under study and is commonly called the body-fixed coordinate system. Its origin O is usually selected at the center of gravity of the AUV; the positive direction of the ξ axis is selected as the direction of the front end of the AUV; the σ axis is perpendicular to the ξ axis, and its positive direction points to the starboard side of the AUV; the ζ axis is perpendicular to the plane Oξσ and is located on the mid-longitudinal section of the AUV. The selection of the positive direction is different from that of the unmanned aerial vehicle and points to the bottom of the AUV. Its schematic diagram is as Figure 1 shown.

[0049] II. Construct the kinematic and dynamic equations of multi-AUV:

[0050] After determining the coordinate system used, the next step is carried out: expressing the motion equation of the AUV in space.

[0051] Assume that there are M leader AUVs and N follower AUVs in the leader-follower multi-AUV system. Set the following sets, let L = (1, 2,..., M), F = (M + 1, M + 2,..., M + N).

[0052] The kinematic model of the AUV can be expressed as

[0053]

[0054] Among them, represents the position and angle vector of the AUV, and the dot on the parameter represents the first derivative of the corresponding parameter. is the first derivative of η a (t), is a constant matrix, a ∈ L ∪ F.

[0055] The dynamic model of the follower AUV can be expressed as

[0056]

[0057] Among them, represents the position and angle vector of the follower AUV, is the first derivative of η i (t), and the double dot on the parameter represents the second derivative of the corresponding parameter. is the second derivative of η i (t), i ∈ F; is the control input, represents the mass matrix of the AUV, represents the Coriolis force matrix, represents the fluid damping force matrix, represents the restoring force provided by gravity and buoyancy, represents the thrust distribution matrix.

[0058] III. Design a distributed elastic offset observer:

[0059] In the virtual network layer, the following distributed elastic offset observer is proposed:

[0060]

[0061] Among them, t ≥ t0, t0 is a certain moment during the DoS attack, and the observed quantity ξ of the leader l in the virtual network layer l (t) = η l (t), ξ i (t) represents the observed quantity of the follower i in the virtual network layer, U is the designed gain matrix, and ξ ci (t) represents the local error of the follower i related to the neighbor information; is a constant matrix; r i (t) represents the known reference offset of the follower i, is the first derivative of r i (t), and χ i (t) represents the reference offset compensation signal.

[0062]

[0063] Among them, Υ N (t0, t), Υ A (t0, t) is the time interval during which there is no attack and the attacked interval from t0 to t; r j (t) represents the known reference offset of follower i; ξ j (t) represents the observed quantity of follower j in the virtual network layer; b il is the communication state between follower i and leader l, that is, the communication weight between the follower and the leader; a ij is the communication state between followers, the communication weight between followers.

[0064] When the multi - AUV system is under a DoS attack, the communication network in the real physical network layer and the virtual network layer is damaged, and ξ ci (t) = 0.

[0065] Define the observation offset of the virtual observed quantity ξ i (t) of follower i on the virtual network layer contains an error of

[0066]

[0067] Among them, is a constant, and there is i ∈ F, l ∈ L; ξ l (t) is the observed quantity of leader l in the virtual network layer.

[0068] IV. Design a distributed elastic offset controller:

[0069] Define the real - state quantity η i (t) of follower i on the real physical network layer and the state tracking error between the observed quantity ξ i (t) on the virtual network layer

[0070] ε i (t) = η i (t) - ξ i (t) (9)

[0071] On the real physical network layer, define the virtual control quantity

[0072]

[0073] Among them, P i is a positive constant matrix.

[0074] Define the error

[0075]

[0076] Furthermore, design the following distributed elastic offset controller:

[0077]

[0078] Among them, H i is a positive constant matrix, represents the compensation signal, is the adaptive parameter, and θ is a positive constant; is a known constant, and θ is a positive constant.

[0079] After the designed distributed elastic offset controller is obtained, the multi-underwater vehicle virtual network elastic offset containment control can be realized by using the designed distributed elastic offset controller.

[0080] The design proof process of the following distributed elastic offset controller is as follows:

[0081] In the present invention, directed communication is used between multi-AUV systems instead of undirected communication, which can save certain communication resources. The communication network of the multi-AUV system is described by using algebraic graph theory knowledge. Denote as a directed graph. Assume that there are M leaders and N followers in the multi-AUV system. Let L = {1, 2,..., M} represent the set of leader AUVs, and F = {M + 1, M + 2,..., M + N} represent the set of follower AUVs. Each AUV is regarded as a communication node. Let be the set of all communication nodes. In the text, represents the AUV numbered i'. The communication connection relationship between each communication node is regarded as an edge. Let be the set composed of all edges. The edge means that the AUV numbered j' can obtain the information of the AUV numbered i'. It is said that is 's child node, is 's parent node. represents the adjacency matrix.

[0082] Define as the subgraph about the follower AUV, where

[0083] For the adjacency matrix When and i ≠ j, a ij = 1 > 0; otherwise a ij = 0 (for an undirected graph, When , a ij = a ji = 1; otherwise a ij = 0). In the directed graph In , if there is a directed path from the leader l to the follower i, it indicates that the follower i can obtain information from the leader l, denoted as b il = 1 > 0; otherwise b il = 0.

[0084] Directed graph The Laplacian matrix of is defined as

[0085]

[0086] where

[0087]

[0088] Node to node The path in the directed graph can be represented as an ordered edge sequence m = 1, 2, ..., n. All satisfying the relationship nodes are the neighbors of the node, denoted by If in the directed graph all follower AUVs have at least one leader AUV as the root node, then the leader set L is said to be globally reachable.

[0089] Proof of the stability of the observation error:

[0090] Assume that there are M leader AUVs and N follower AUVs in the leader-follower multi-AUV system. Define the following sets, let L = (1, 2, …, M), F = (M + 1, M + 2, …, M + N).

[0091] Assumption 1: The leader AUVs are not disturbed and can move along the pre-set navigation trajectories.

[0092] Assumption 2: The communication between AUVs is directed communication, and each follower AUV has at least one leader AUV as the root node.

[0093] Assumption 3: The real part of the eigenvalues of the matrix F l is non-negative.

[0094] According to Assumption 1, the kinematic model of the leader AUV can be expressed as

[0095]

[0096] where represents the position and angle vector of the leader AUV, is a constant matrix, l ∈ L.

[0097] The dynamic model of the follower AUV can be expressed as

[0098]

[0099] where represents the position and attitude vector of the follower AUV, is the control input, represents the mass matrix of the AUV, represents the Coriolis force matrix, represents the hydrodynamic damping force, represents the restoring force provided by gravity and buoyancy, represents the thrust allocation matrix, i ∈ F.

[0100] In the present invention, it is assumed that during the operation of the multi-AUV system, it is simultaneously subjected to a DoS attack from an external attacker and an unbounded actuator attack. A DoS attack refers to an attack behavior in which the attacker occupies the signal bandwidth by sending a large amount of useless data, thereby cutting off the communication transmission between multi-AUV systems, making the AUV unable to obtain information about other AUVs, and the communication topology network falling into paralysis. An actuator attack can penetrate the actuator input of the AUV, tamper with the input signal of the AUV, and thus have an adverse impact on the normal movement of the AUV.

[0101] Definition 1: For a DoS attack, the nth attack interval is and represent the start time and end time of the attack respectively. At t ≥ t0, the total DoS attack time can be expressed as

[0102]

[0103] Correspondingly, the total un-attacked time can be obtained

[0104] Υ N (t0,t) = [t0,t] / Υ A (t0,t)(22)

[0105] Assumption 4: For the time interval [t0,t), Υ A (t0,t) represents the total duration of the DoS attack, and it is assumed that the following relationship holds

[0106]

[0107] where, T0 > 0, τ a > 1, and both are constants.

[0108] Remark 1: Assume that 4 indicates that the total duration of the DoS attack is limited. Considering the real mission scenario and the limited attack resources of the attacker and the self-healing mechanism of the multi-AUV system, the assumption of the total duration of the DoS attack is reasonable in the actual situation.

[0109] Definition 2: When the follower AUV is under an unbounded actuator attack, the control input for each follower AUV is defined as

[0110]

[0111] where, Δ i (t) represents the unknown unbounded actuator attack signal, and Δ i (t) is unbounded, but its derivative is bounded and satisfies which is a known constant.

[0112] Remark 2: When the derivative of the unbounded actuator attack signal is unbounded, the change amplitude of Δ i (t) will show a large order of magnitude and is easily detected and recognized, so it will be rejected or filtered by the actuator and cannot achieve the attack effect. Therefore, Definition 2 is reasonable.

[0113] Lemma 1: Consider a multi-agent system. Let L = {1, 2,..., M} denote the set of leaders and F = {M + 1, M + 2,..., M + N} denote the set of followers. The containment control error can be defined as

[0114]

[0115] where, x i and x l represent the follower and leader states respectively, and α il ≥0 is a positive constant. When , it indicates that the system achieves containment control consensus.

[0116] Combined with Lemma 1, the offset containment error of the follower AUV can be expressed as

[0117]

[0118] where, is a constant and there is r i (t) represents the known reference offset, which is continuously differentiable, and both ||r i (t)|| and are bounded, i ∈ F, l ∈ L. represents the centroid of the leader's encirclement area of follower i The offset between them is r i (t). When e i (t) is finally uniformly bounded, the follower i can converge to a position near a certain point at a distance r i (t) from the centroid position of the leader's surrounding circle, indicating that the offset inclusion control consistency is achieved when the multi-AUV system under the directed communication topology is subject to a composite attack.

[0119] Theorem 1: Consider the multi-AUV systems (21) and (22) in the leader-follower form. When the following conditions are satisfied

[0120]

[0121] where, is a positive constant, is a positive constant, and U is a positive design constant matrix, λ max (·) and λ min (·) represent the maximum and minimum values of the eigenvalues of the matrix respectively. For the case where the multi-AUV system is under a DoS attack, the offset observation error on the virtual network layer can finally converge to 0 under the action of the distributed resilient offset observer (3). Proof:

[0122] For the distributed resilient offset observer (3) when the communication network is normal

[0123]

[0124] where, for [ξ di (t) - ξ ri (t)], since all leaders have direct communication with only 1 follower, therefore, there exists such that [ξ di (t) - ξ ri (t)] = 0. Therefore, when the communication network is normal

[0125]

[0126] Consider the case where the multi-AUV system is under a DoS attack

[0127]

[0128] Therefore, in the case where the multi-AUV system is under a DoS attack and the communication network is damaged

[0129]

[0130] To sum up, when t ≥ t0 ≥ 0, the following equation holds on the virtual network layer

[0131]

[0132] For the offset observation error on the virtual network layer Design the following Lyapunov function

[0133]

[0134] When the communication network is normal

[0135]

[0136] It can be seen from (27) that

[0137]

[0138] In the case where the multi-AUV system is under a DoS attack and the communication network is damaged

[0139]

[0140] It can be seen from (28) that

[0141]

[0142] To sum up, when t≥t0≥0, the following formula holds

[0143]

[0144] At the multi-AUV system can communicate normally. Integrate

[0145]

[0146] At the communication network of the multi-AUV system is damaged and it cannot communicate normally. Integrate

[0147]

[0148] To sum up, the following formula holds

[0149]

[0150] According to (43), at

[0151]

[0152] At

[0153] ​​​​

[0154] Therefore, combining (44) and (45), when \(t\geq t_0\geq0\),

[0155]

[0156] Given \(\Upsilon\) N \((t_0,t)=t - t_0-\Upsilon\) A \((t_0,t)\), then we have

[0157]

[0158] where From (29), we know that \(\delta>0\), then is a bounded constant. Therefore, when \(t\rightarrow\infty\), \(V\) 1i \((t)\) converges to 0, and the offset observation error on the virtual network layer can finally converge to 0. The proof of Theorem 1 is completed.

[0159] Proof of consistency:

[0160] Define the state tracking error i between the true state quantity \(\eta\) i \((t)\) of follower \(i\) on the real physical network layer and the observed quantity \(\xi\)

[0161] \(\varepsilon\) i \((t)=\eta\) i \((t)-\xi\) i \((t)(48)\)

[0162] On the real physical network layer, define the virtual control quantity

[0163]

[0164] where \(P\) i is a positive constant matrix.

[0165] Define the error

[0166]

[0167] Design the following distributed elastic offset controller

[0168]

[0169] where \(H\) i is a positive constant matrix, represents the compensation signal, is the adaptive parameter, and \(\theta\) is a positive constant.

[0170] Theorem 2: Considering the multi-AUV system (19) and (20) in the leader-follower form, when suffering from a composite attack composed of DoS attacks and unbounded actuator attacks, the communication network between AUVs is damaged, and the control input signals of the actuators are also interfered. Under the action of the distributed resilient offset observer (3) and the distributed resilient offset controller (51), the system can achieve offset-inclusive control consensus.

[0171] Proof: Take the first derivative of ε i (t) with respect to time t

[0172]

[0173] Design the Lyapunov function for the state tracking error ε i (t)

[0174]

[0175] Combine (54) to find the first derivative of V 2i (t) with respect to time

[0176]

[0177] Furthermore, design the Lyapunov function as shown below

[0178]

[0179] Combine (50) to find the first derivative of V 3i (t) with respect to time

[0180]

[0181] Combining (20), (24) and (51), (58) can be further calculated

[0182]

[0183] where For Further calculation

[0184]

[0185] Because Combining (53), at the following equation holds

[0186]

[0187] Therefore, there exists t c > 0, for t ≥ t c , the following equation holds

[0188]

[0189] There is

[0190] Remark 3: Only discuss and do not discuss other cases because in other cases within an acceptable boundary, it is unstable within the boundary. Once it exceeds the boundary, the controller will pull ||z i (t)|| back into the boundary.

[0191] For t ≥ t c , combining (59) and (62), for continue to process

[0192]

[0193] where, because P i and H i are both positive constant matrices, so c1 = min[2λ min (P i ), 2λ min (H i )] > 0. According to (63), find the integral on the time interval [t c , t]

[0194]

[0195] where is a bounded constant. Therefore, according to (64), it can be known that when t ≥ t c > 0, V 3i (t) will eventually approach 0, and there is ε i (t) and z i (t) approach 0. In addition, the state tracking error is bounded, and it also satisfies

[0196] In summary, the follower AUV offset including error (26) can be expressed as

[0197]

[0198] From the previous proof, it can be seen that ε i (t) and will eventually converge to 0. Therefore, the offset including error e i (t) is ultimately uniformly bounded, and the boundary conditions satisfy

[0199] According to the above analysis, considering the leader-follower multi-AUV systems (19) and (20), when adopting a directed communication topology network and suffering from a composite attack composed of DoS network attacks and unbounded actuator attack signals, by constructing a virtual network layer and designing a distributed resilient offset observer (3) and a distributed resilient offset controller (51), under the combined action of the two, the follower AUV can finally converge to a point near a certain offset from the centroid position of the leader AUV's encirclement, and the offset inclusion control of the multi-AUV system is realized. The proof of Theorem 2 is completed.

[0200] Embodiment

[0201] To verify the effectiveness of the proposed distributed resilient offset inclusion control method under DoS attacks, a leader-follower multi-AUV system is selected as the verification object in the form of multiple leaders and multiple followers. In the simulation experiment, it is designed that there are 5 leader AUVs (denoted by numbers 1, 2, 3, 4, 5) and 7 follower AUVs (denoted by numbers 6, 7,..., 12) in the leader-follower multi-AUV system.

[0202] In the leader-follower multi-AUV system, the dynamic model of the leader AUV can be expressed by the following formula:

[0203]

[0204] For the actual model data of the AUV, refer to the reference, and the model of the AUV is as Figure 2 shown, and Table 1 and Table 2 respectively represent the inertia coefficients and hydrodynamic parameters of the AUV.

[0205] Table 1 Various inertia coefficients of the AUV

[0206]

[0207] Table 2 Hydrodynamic parameters of the AUV

[0208]

[0209] In the leader-follower multi-AUV system, the dynamic model of the follower AUV can be expressed by the following formula:

[0210]

[0211] where \(i = 6, 7,..., 12\). Considering that the system is under an unbounded actuator attack, \(u\) i (t) is redefined as

[0212]

[0213] where the unbounded actuator attack signal \(\Delta\) is selectedi $(t) = 10t×[2.1, 1.5, 0.8, 0.1, 0.15, 0.07]$ T , It is assumed that at $t = 11s$, all follower AUVs suffer from unbounded actuator attacks.

[0214] Tables 3, 4, and 5 represent the preset trajectories of the leader AUV and the initial states of the follower AUVs, respectively.

[0215] Table 3 Preset Trajectory of the Leader AUV

[0216]

[0217] Among them, $f1 = 0.02$, $f2 = 0.015$, $f3 = 0.01$, $f4 = 0$, $f5 = 0$, $f6 = 0$, and the constant matrix $F$ is selected l $= diag\{f1,f2,f3,f4,f5,f6\}$.

[0218] Table 4 Initial Positions and Angles of the Follower AUVs

[0219]

[0220]

[0221] Table 5 Initial Linear and Angular Velocities of the Follower AUVs

[0222]

[0223] The topology of the follower - leader multi - AUV system is as Figure 3 shown.

[0224] The trajectory of the leader in the follower - leader multi - AUV system is pre - designed, and 5 leader AUVs move according to the designed trajectory.

[0225] Scenario 1: Small offset (not exceeding the leader's encirclement)

[0226] For the selection of the parameters of the distributed elastic offset observer, $U = 4I6$, and the offset $r$ i $(t)$ is designed as follows

[0227]

[0228] For the selection of the parameters of the distributed elastic offset controller, $P$ i $= I6$, $H$ i $= I6$, $\theta = 1$, $i\in F$. The DoS attack time period is selected as $[3k, 3k + 1)s$,

[0229] Through simulation experiments, the following Figures 4 to 10 results are obtained.

[0230] Analysis Figure 4 shows that at 0 s, the followers are randomly distributed outside the convex hull formed by the leaders. Under the action of the distributed elastic containment control algorithm considering compound attacks designed in the present invention, as time goes by, at t = 10 s, all the follower AUVs can enter the convex hull, and all the follower AUVs move in a circular motion centered on the centroid position of the convex hull of the leader AUV. Moreover, from the images at t = 20 s and t = 30 s, it can be seen that the follower AUVs always move in coordination with the leader AUV and move with the movement of the convex hull formed by the leaders, indicating that the distributed offset containment control consistency of the leader-follower multi-AUV system can be achieved.

[0231] Analysis Figure 5 and Figure 6 show that the light red area represents the time period when the leader-follower multi-AUV system is under DoS attack. The red dotted line in the figure indicates that the follower AUV starts to be under unbounded actuator attack at t = 11 s. It can be seen that in the case of suffering from DoS attack and unbounded actuator attack, the state variables in each dimension of the follower AUV can converge to the vicinity of the convex hull range formed by the leader state variables at t = 5 s. There are fluctuations at t = 11 s, but it will continue to remain stable after a short time, indicating that all the follower AUVs can converge from outside the convex hull to the vicinity of the convex hull area, move in coordination with the leader AUV, and achieve offset containment control.

[0232] Analysis Figure 7 shows that from the information in the figure, the offset observation error on the virtual network layer converges to near 0 at t = 8 s, indicating that the observed quantity ξ i (t) on the virtual network layer can converge to the state near the centroid of the convex hull formed by the leader state variables with a distance of r i (t). The distributed elastic offset observer can play a good observation effect in the case of the system suffering from DoS attack, and the unbounded actuator attack has little effect on the observer.

[0233] Analysis Figure 8 and Figure 9 show that the state tracking error ||ε i (t)|| and the design error ||z i (t)|| both converge to near 0 at t = 7 s. Although there are fluctuations after t = 11 s due to the unbounded actuator attack, they will quickly stabilize near 0, indicating that the true state variables of the followers on the real physical network layer can accurately track the observed quantities on the virtual network layer.

[0234] Analysis Figure 10 shows that the offset containment error ||ei (t) will converge to near 0 at t = 8.5 s. Although there are fluctuations after t = 11 s due to unbounded actuator attacks, it will quickly stabilize near 0, indicating that the designed distributed resilient offset inclusion controller is effective under the combined attack composed of DoS attacks and unbounded actuator attacks. For the leader-follower multi-AUV system (19) and (20), the communication network adopts a directed communication topology. Considering that the system is under the combined attack composed of DoS network attacks and unbounded actuator attack signals, under the combined action of the distributed resilient offset observer (3) and the distributed resilient offset controller (51), the offset inclusion error of the follower AUVs is ultimately uniformly bounded, and all follower AUVs can finally converge to a point near a certain offset from the centroid position of the leader AUV's surrounding circle, realizing the consistency of the distributed resilient offset inclusion control of the multi-AUV system.

[0235] Scenario 2: The offset is large (beyond the "encirclement" of the leader's surrounding circle)

[0236] Selection of distributed resilient offset observer parameters, U = 4I6, offset r i (t) is designed as follows

[0237]

[0238] Selection of distributed resilient offset controller parameters, P i = I6, H i = I6, θ = 1, i ∈ F. The selected DoS attack time period is [3k, 3k + 1) s,

[0239] Through simulation experiments, the results as Figures 11 to 17 shown are obtained.

[0240] Analysis Figure 11 It can be seen that at 0 s, the followers are randomly distributed outside the convex hull formed by the leader. Under the action of the distributed resilient offset inclusion control algorithm considering combined attacks designed in the present invention, as time goes by, at t = 10 s, the follower AUVs have all been evenly distributed on the periphery of the convex hull formed by the leader AUV ("encirclement control"), and all follower AUVs move in a circular motion centered on the centroid position of the convex hull of the leader AUV. And from the images at t = 20 s and t = 30 s, it can be seen that the follower AUVs have been moving in coordination with the leader AUV and moving as the convex hull formed by the leader moves, indicating that the consistency of the distributed offset inclusion control of the leader-follower multi-AUV system can be realized.

[0241] Analysis Figure 12 and Figure 13, it can be seen that under the DoS attack and unbounded actuator attack, the state variables of the follower AUV in each dimension can converge to the vicinity of the convex hull formed by the leader state variables at t = 6 s. There are fluctuations at t = 11 s, but it will remain stable again after a short time, indicating that all follower AUVs can converge from outside the convex hull to the vicinity of the convex hull area, cooperate with the leader AUV in motion, and achieve offset containment control.

[0242] Analysis Figure 14 , it can be seen from the information in the figure that the offset observation error on the virtual network layer converges to around 0 at t = 7.5 s, indicating that the observed quantity ξ i (t) on the virtual network layer can converge to the state near the centroid of the convex hull formed by the leader state variables with a distance of r i (t). The distributed elastic offset observer can achieve good observation effects when the system is under DoS attack, and the unbounded actuator attack has almost no impact on the observer.

[0243] Analysis Figure 15 and Figure 16 , the state tracking error ||ε i (t)|| and the design error ||z i (t)|| both converge to around 0 at t = 7 s. Although there are fluctuations after t = 11 s due to the unbounded actuator attack, they will quickly stabilize around 0, indicating that the true state variables of the follower on the real physical network layer can accurately track the observed quantity on the virtual network layer.

[0244] Analysis Figure 17 , the offset containment error ||e i (t)|| will converge to around 0 at t = 8 s. Although there are fluctuations after t = 11 s due to the unbounded actuator attack, they will quickly stabilize around 0, indicating that the designed distributed elastic offset containment controller is effective under the combined attack considering DoS attack and unbounded actuator attack. For the leader-follower multi-AUV system (19) and (20), the communication network adopts a directed communication topology. When considering the combined attack composed of DoS network attack and unbounded actuator attack signal on the system, under the joint action of the distributed elastic offset observer (3) and the distributed elastic offset controller (51), the offset containment error of the follower AUV is ultimately uniformly bounded, and all follower AUVs can finally converge to a point near a certain offset from the centroid position of the leader AUV's encirclement. The follower AUVs complete the "encirclement" of the leader AUV, and the consistency of the distributed elastic offset containment control of the multi-AUV system is achieved.

[0245] The above numerical examples of the present invention are only for illustrating in detail the calculation model and calculation process of the present invention, rather than limiting the implementation manners of the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation manners here. Any obvious changes or modifications derived from the technical solutions of the present invention still fall within the protection scope of the present invention.

Claims

1. A virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks, characterized in that For a multi-AUV system with M leader AUVs and N follower AUVs, let L = (1, 2, …, M) and F = (M + 1, M + 2, …, M + N). A distributed elastic offset controller is used for control, and the distributed elastic offset controller is as follows: where, u i is the follower control input, i ∈ F; B 0i represents the thrust allocation matrix, is the corresponding transpose; M ηi represents the mass matrix of the AUV, is the transpose of the corresponding inverse matrix; C ηi represents the Coriolis force matrix, D ηi represents the fluid damping force matrix, g ηi represents the restoring force provided by gravity and buoyancy; η i (t) represents the position and angle vector of the follower AUV, is the first derivative of η i (t); is the first derivative of the virtual control quantity α i (t); z i (t) is the error between i and the virtual control quantity α is the transpose of z i (t); H i is a positive constant matrix, represents the compensation signal, is the adaptive parameter, is 's first derivative, θ is a positive constant; is a constant; ε i (t) is the state tracking error between the true state quantity η i (t) of the follower i at the real physical network level and the observed quantity ξ i (t) at the virtual network level.

2. The virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks according to claim 1, characterized in that, The virtual control quantity is where P i is a positive constant matrix, is the first derivative of the virtual observable ξ i (t) of follower i on the virtual network layer.

3. The virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks according to claim 2, characterized in that, The observable quantity ξ i (t) is determined by the distributed elastic offset observer shown as follows: where \(t\geq t_0\), \(t_0\) is a certain moment during the DoS attack, \(\xi\) i (t) represents the observable quantity of follower \(i\) in the virtual network layer, being the first derivative of \(\xi\) i (t); \(U\) is the designed gain matrix, \(\xi\) ci (t) represents the local error of follower \(i\) related to neighbor information, \(\chi\) i (t) represents the reference offset compensation signal; \(r\) i (t) represents the known reference offset, being the first derivative of \(r\) i (t).

4. The virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks according to claim 3, characterized in that The local error ξ ci (t) related to the follower i's neighbor information is as follows; Among them, Υ N (t0,t), Υ A (t0,t) represents the time interval without being attacked and the attacked interval from t0 to t; r j (t) is the known reference offset of follower j; ξ j (t) is the observed quantity of follower j in the virtual network layer; ξ l (t) is the observed quantity of leader l in the virtual network layer; a ij represents the communication state between follower nodes. When follower node i and follower node j can communicate, a ij = 1; otherwise a ij = 0.

5. The virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks according to claim 4, characterized in that The observed quantity ξ of the leader l in the virtual network layer l (t) = η l (t), η l (t) represents the position and angle vectors of the leader l.

6. A virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks according to any one of claims 1 to 5, characterized in that Total time of not being attacked Υ N (t0,t) = [t0,t] / Υ A (t0,t).

7. The virtual network elastic offset inclusion control method for multiple underwater vehicles considering composite attacks according to claim 6, characterized in that Total duration Υ of the DoS attack A (t0, t) satisfies where T0 > 0, τ a > 1, and both are constants.

8. A control method for virtual network elastic offset of multiple underwater vehicles considering composite attacks according to claim 7, characterized in that, The kinematic models of the AUVs corresponding to the leaders and followers are as follows: Among them, η a (t) represents the position and angle vector of the AUV. The dot on the parameter represents the first derivative of the corresponding parameter, is the first derivative of η a (t), F a is a constant matrix, and a ∈ L ∪ F.

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