Multi-AUV formation system distributed inclusion control method based on composite observer in consideration of external interference and time-varying topology

Abstract

The invention discloses a multi-AUV formation system distributed inclusion control method based on a composite observer in consideration of external interference and time-varying topology, and belongs to the technical field of control. According to the method, dynamic modeling is carried out on the AUV by considering the condition that the system has unknown external interference, topology switching signals of the multi-AUV system are described as conforming to a Markov random process, communication weights representing followers and a navigator and between the followers are determined, a distributed switching topology composite observer is designed based on the topology switching condition and the external interference, and the reliability of the distributed switching topology composite observer is improved. And a distributed switching topology composite observer and a distributed output inclusion control algorithm are provided in combination with the interference observer, so that cooperative control of the multi-AUV system is realized.

Description

Technical Field

[0001] The present invention belongs to the field of control technology, and particularly relates to a distributed containment control method for a multi-AUV formation system. Background Art

[0002] With the accelerated construction of a maritime power, marine equipment has attracted much attention. Among them, an autonomous underwater vehicle (AUV) is a robot that can operate autonomously underwater. Due to its advantages such as certain intelligence, large activity range, strong mobility, and strong concealment, it has a wide range of applications in civilian resource exploration and seabed survey, as well as military applications such as mine drainage and rapid underwater environment assessment. With the in-depth development of the ocean, it is increasingly difficult for a single AUV to meet the increasingly demanding operation requirements. The advantages of good redundancy, high efficiency, and robustness of a multi-AUV system have great potential application value in the face of large-scale and high-timeliness ocean exploration tasks. Compared with a centralized control system, a distributed control system has stronger robustness, flexibility, and scalability. In the leader-follower mode of a 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, which reduces the communication pressure on the follower.

[0003] In view of the complex and changeable characteristics of the actual ocean environment, in the face of unknown ocean current interference, the interference situation of AUVs with relatively small mass cannot be ignored. Moreover, the highly nonlinear and uncertain characteristics of the AUV's dynamic model itself increase the difficulty of AUV system trajectory tracking.

[0004] In practice, the working environment of a multi-AUV system is relatively complex and the communication resources are relatively limited. The network topology structure for information exchange between AUVs in the formation may change due to link failures or instability. Different from a fixed communication topology, a switched communication topology can flexibly switch the communication network according to the network communication situation, improving the robustness of the multi-AUV formation system. Although there have been studies on AUV formation control considering switched topologies, most of them do not consider the external interference on AUVs at the same time. Therefore, it not only affects the cooperative control effect of the multi-AUV system, but also limits its practicality. Summary of the Invention

[0005] The present invention aims to solve the problem that the existing cooperative control of multi-AUV systems does not consider the communication network topology structure and external interference at the same time, thus affecting the cooperative control effect of multi-AUV systems.

[0006] A distributed containment control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology performs distributed containment control on the multi-AUV formation system based on an output feedback containment controller;

[0007] The distributed output feedback includes the following controller:

[0008] τ i (t) = K 1i z i (t) + K 2i Φ i (t)

[0009]

[0010] Where τ i (t) is the control input, i ∈ F, and F = (M + 1, M + 2, …, M + N) represents the set of N followers; z i (t) represents the intermediate variable, and its specific form is reflected by the form of its first derivative K 1i , K 2i and U i are all gain matrices; Φ i (t) represents the distributed switching topology composite observer corresponding to the case where the topology switching signal {σ(t), t ≥ 0} of the multi - AUV system communication network conforms to the Markov random process and the system is subject to external interference; M ηi (t) is the mass matrix, C ηi (t) is the Coriolis force matrix, D ηi (t) is the fluid damping force matrix, I6, 0 6×6 are the identity matrix and the all - zero matrix respectively, W and B represent the gravity and buoyancy of the AUV, (x G , y G , z G ), (x B , y B , z B ) represent the centroid coordinates and buoyancy center coordinates of the AUV in the moving coordinate system; represents the disturbance observation value; is the output of the follower AUV, is a constant matrix.

[0011] Furthermore, the distributed switching topology composite observer Φ i (t) is as follows:

[0012]

[0013] Where δ i (t) represents the state observation value, represents the disturbance observation value, and P i is a design matrix.

[0014] Furthermore, the state observation value δ i (t) is determined by its corresponding first derivative in the form of a state observer as follows:

[0015]

[0016] where I6, 0 6×6 are the identity matrix and the all-zero matrix respectively, is a constant matrix, l ∈ L, and L = (1, 2, …, M) represents the set of M leaders; δ i (t) represents the state observation value, represents the state observer, is the first derivative of δ i (t); is a positive constant; σ(t) represents the topological state at time t, and t represents time; represents the communication weight between follower i and follower j, i, j ∈ F; represents the communication weight between follower i and leader l; is a matrix formed according to the position and angle vector η l (t) of the leader AUV and its corresponding first derivative, that is η l (t) represents the position and angle vector of the leader AUV, represents the first derivative of η l (t).

[0017] Furthermore, the is determined by the dynamic model of the leader; the dynamic model of the leader is

[0018] Furthermore, the disturbance observation value is determined by its corresponding first derivative in the form of a state observer as follows:

[0019]

[0020] where represents the disturbance observation value, represents the disturbance observer; is a matrix formed according to the position and angle vector η i (t) of the follower AUV and its corresponding first derivative, that is η i (t) represents the position and angle vector of the follower AUV, is ηi The first derivative of (t).

[0021] Furthermore, the is determined by the dynamic model of the follower; the dynamic model of the follower is

[0022] Furthermore, the output of the follower AUV

[0023] Furthermore, the topology switching signal {σ(t), t≥0} of the multi-AUV system communication network conforms to a Markov random process. The topology state is randomly selected only from the set S = {1, 2, …, s}, and the switching is based only on the current state. s represents the s-th topology state; the transition rate matrix is W = [ω kr s×s , and satisfies the following formula:

[0024]

[0025] where, represents the probability of the topology structure changing into other structures and maintaining the original structure; Δt represents the time change amount, and o(Δt) represents a higher-order infinitesimal function with respect to Δt; ω kr ≥0 represents the rate of transition from state k to state r, k, r ∈ S, and

[0026] Furthermore, the topology switching signal {σ(t), t≥0} of the multi-AUV system communication network conforms to a Markov random process. For the communication weight between follower i and follower j, if follower i can receive information from follower j, then otherwise When i = j, For the communication weight between follower i and leader l, if follower i can receive information from leader l, then otherwise

[0027] Furthermore, the Laplacian matrix corresponding to the communication network topology of the multi-AUV system is as follows:

[0028]

[0029] where,

[0030] Beneficial effects:

[0031] ​The research value and practical significance in the collaborative control design of multi-AUV systems. For the distributed containment control of the leader-follower multi-AUV system in the field of underwater robot intelligent control, this invention considers the situation where the system has unknown external disturbances, describes it using Markov random processes, and proposes a distributed switched topology composite observer and a distributed output containment control algorithm in combination with a disturbance observer. This invention simultaneously considers topology switching and external disturbances, not only achieving the collaborative control of the multi-AUV system, but also improving the collaborative control effect of the multi-AUV system. Description of the Drawings

[0032] Figure 1 It is a schematic diagram of the moving coordinate system;

[0033] Figure 2 It is the Markov switching topology structure diagram of the leader-follower multi-AUV system;

[0034] Figure 3 It is the topology switching signal change diagram of the leader-follower multi-AUV system;

[0035] Figure 4 It is the three-dimensional space position change diagram of each AUV in the leader-follower multi-AUV system;

[0036] Figure 5 It is the position change diagram of each AUV in the leader-follower multi-AUV system;

[0037] Figure 6 It is the angle change diagram of each AUV in the leader-follower multi-AUV system;

[0038] Figure 7 It is the follower AUV state observation containment error ||ξi(t)|| 2 change diagram;

[0039] Figure 8 It is the follower AUV disturbance observer error change diagram;

[0040] Figure 9 It is the follower AUV output containment error ||ei(t)|| 2 change diagram. Detailed Implementation Manner

[0041] In view of the problems in the background art, it has high research value and practical significance to consider topological switching and external disturbances in the collaborative control design of multi-AUV systems. For the distributed containment control of the leader-follower multi-AUV system in the field of underwater robot intelligent control, the present invention considers the situation where the system has unknown external disturbances, describes it in combination with the Markov random process, and proposes a distributed switched-topology composite observer and a distributed output containment control algorithm in combination with a disturbance observer to achieve the collaborative control of the multi-AUV system. The following is a detailed description in combination with specific embodiments. Specific Embodiment 1:

[0043] This embodiment is a distributed containment control method for a multi-AUV formation system based on a composite observer considering external disturbances and time-varying topologies. Before specific description, first, some parameter forms are paraphrased: Denotes the set of natural numbers, Denotes the set of real numbers, Denotes the set of n-dimensional real column vectors, Denotes the set of m×n real matrices, 0 n Denotes the n-dimensional all-zero column vector, 0 m×n Denotes the m×n all-zero matrix, 1 n Denotes the n-dimensional all-one column vector, I n Denotes the n-order identity matrix, A T Denotes the transpose of matrix A, A -1 Denotes the inverse matrix of matrix A, λ(A) denotes the set of eigenvalues of matrix A, diag[x1,...,x n denotes the diagonal matrix with diagonal elements x1,…,x n of, [·] m×n Denotes an m×n matrix, blockdiag{A1,...,A n} denotes the diagonal matrix with diagonal elements being matrices A1,…,A n of, Denotes the Kronecker product, ||x|| denotes the 2-norm of vector x, |x| denotes taking the absolute value of each element of vector x, M / N denotes the set that belongs to set M but does not belong to set N.

[0044] The distributed containment control method for a multi-AUV formation system based on a composite observer considering external disturbances and time-varying topologies described in this embodiment includes:

[0045] S1. Establish an inertial coordinate system and a motion coordinate system, then establish the kinematic and dynamic models of the multi-AUV, and finally give the design of the distributed switched-topology composite observer and the design of the distributed output containment controller.

[0046] S101. Establish a coordinate system:

[0047] Before constructing the mathematical model of the AUV, it is crucial to determine the spatial coordinate system in which the model is located. According to the recommendation of the International Towing Tank Conference (ITTC), two coordinate systems are adopted in the present invention. One is the most commonly used inertial coordinate system, and the other is the moving coordinate system commonly used for analyzing moving objects:

[0048] Inertial coordinate system E-ABC: The inertial coordinate system is the "fixed" coordinate system used in general research, and is also popularly known as the geodetic coordinate system, which is fixedly connected to the earth. Under normal circumstances, the tangent plane of the earth's surface is used as the reference plane, and the coordinate plane EAB is made to coincide with the tangent plane of the sea area where the AUV operates. E is the tangent point and also serves as the origin of the coordinate system. The positive direction of the EA axis points to the north pole, the EB axis is perpendicular to the EA axis and its positive direction is the geographical east, and the EC axis is perpendicular to the EAB plane, and its positive direction points to the center of the earth.

[0049] Moving coordinate system O-xyz: The moving coordinate system refers to the coordinate system that moves together with the object under study, and is also popularly known as the body-fixed coordinate system, that is, the "dynamic coordinate system" that moves simultaneously with the AUV body. Generally, its origin O is set at the center of gravity of the AUV, and the positive direction of the x-axis is the direction pointing to the front end of the AUV; the y-axis is perpendicular to the x-axis, and its positive direction points to the starboard side of the AUV; the z-axis is located on the longitudinal midsection plane of the AUV, perpendicular to the plane Oxy, and its positive direction is pointing to the bottom of the AUV.

[0050] The schematic diagram of the moving coordinate system of the present invention is as Figure 1 shown.

[0051] S102. Construct the kinematic and dynamic equations of multiple AUVs:

[0052] After determining the coordinate system used, the next step is to express the motion equation of the AUV in space. In the method of obtaining the kinematic equation of the AUV, it is necessary to explore the differences and connections between the inertial coordinate system and the moving coordinate system. The essence of the kinematic equation is the interpretation of the mutual transformation between the two. By analyzing the position, attitude, linear velocity and angular velocity of the AUV, the kinematic equation of the AUV can be given. Analyzing the situation in the moving coordinate system, when the velocity of the AUV changes at a certain moment in the moving coordinate system, in the inertial coordinate system, its corresponding position and attitude will change correspondingly. Therefore, it is necessary to use the coordinate transformation matrix to connect different coordinate systems.

[0053] 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).

[0054] The dynamic model of the leader AUV can be expressed as

[0055]

[0056] Among them, represents the position and angle vector of the leader AUV, represents the first and second derivatives of η l (t); is a constant matrix, l ∈ L.

[0057] Design the output form of the leader AUV

[0058]

[0059] Among them, represents the output of the leader, is a constant matrix.

[0060] Generally, considering the influence of external disturbances on the follower AUV, its dynamic model can be expressed as

[0061]

[0062] Among them, represents the position and angle vector of the follower AUV, is the control input, represents the unknown external disturbance, i ∈ F, is the mass matrix, is the Coriolis force matrix, is the fluid damping force matrix; is the restoring force of the heavy buoyancy, W and B represent the gravity and buoyancy of the AUV, (x G , y G , z G ), (x B , y B , z B ) represent the center of gravity coordinates and center of buoyancy coordinates of the AUV in the moving coordinate system.

[0063] Design the output form of the follower AUV

[0064]

[0065] Among them, is the output of the follower, is a constant matrix.

[0066] S103. Design communication topology switching:

[0067] Define that the topology switching signal {σ(t), t≥0} of the multi-AUV system conforms to a Markov random process. The topology state is randomly selected only from the set S = {1, 2,..., s}, and the switching is only based on the current state, where s represents the s-th topology state. The transition rate matrix is W = [ω kr s×s , and it satisfies the following formula

[0068]

[0069] where represents the probabilities of the topology structure transforming into other structures and remaining the original structure; t represents time, Δt represents the time variation, and o(Δt) represents a higher-order infinitesimal function with respect to Δt, whose meaning is that it approaches zero faster than Δt. It can be Δt 2 , Δt 3 …… and so on for such functions; ω kr ≥0 represents the rate of transition from state k to state r, k, r ∈ S, and Obviously, the transition rate matrix is W = [ω kr s×s and the sum of the elements in each row is 0.

[0070] Considering that the communication network topology switching signal of the multi-AUV system conforms to a Markov random process, let represent the communication weight between follower i and follower j, and σ(t) represents the topology state at the current moment, where i, j ∈ F. When i ≠ j, if follower i can receive information from follower j, then otherwise When i = j, represent the communication weight between follower i and leader l, and σ(t) represents the topology state at the current moment, where i ∈ F, l ∈ L. If follower i can receive information from leader l, then otherwise

[0071] Let σ(t) represents the topology state at the current moment, where i ∈ F. When and only when the switching signal σ(t) = k, k ∈ S, The label indicates definition.

[0072] Define the Laplacian matrix

[0073]

[0074] Define the communication network topology graph of the multi-AUV system as When and only when the switching signal σ(t) = k, ​​$k\in S$. Define the joint topological graph of the communication network of the multi-AUV system under the Markov switching topology as

[0075] S104. Design a distributed switched-topology composite observer:

[0076] Considering the case where the communication network of the multi-AUV system conforms to the Markov switching topology and the system is subject to external disturbances, design a distributed switched-topology composite observer

[0077]

[0078] where, is a positive constant, represents the composite observation value, $\delta$ i (t), represents the state observation value and the state observer, is actually the first derivative of $\delta$ i (t); represents the disturbance observation value and the disturbance observer, is actually the first derivative of; is a design matrix, $i\in F$.

[0079] Make the following definitions

[0080]

[0081]

[0082] It should be noted that $\varPhi$, $\delta(t)$, and $\varPhi$ i 、$\delta$ i (t), have subscript differences. $\varPhi$, $\delta(t)$, are the $\varPhi$ i 、$\delta$ i (t), sets corresponding to the followers; are the sets corresponding to the leader.

[0083] S105. Design a distributed output-feedback inclusion controller:

[0084] Based on the above-designed distributed switched-topology composite observer (9), for each follower $i\in F$, design the following distributed output-feedback inclusion controller

[0085] $\tau$ i (t)=K 1i $z$ i (t)+K 2i $\varPhi$ i(t) (16)

[0086]

[0087] Among them, represents an intermediate variable, and its specific form is reflected by its first derivative in the form of; K 1i , K 2i and U i are all designed gain matrices.

[0088] The design process of the present invention is as follows:

[0089] 1. Algebraic graph theory:

[0090] In the present invention, directed communication is used between multi-AUV formation systems, which can save some communication resources. Using the knowledge of algebraic graph theory to describe the communication network of the multi-AUV formation system, denote as a directed graph. Assume that there are N follower AUVs and M leader AUVs in the multi-AUV formation system. Let L = {1, 2,..., M} represent the set of leaders, and F = {M + 1, M + 2,..., M + N} represent the set of followers. Each AUV can be regarded as a node, and let be the set of all nodes. In the text, represents the i-th AUV. Regarding the communication connection relationship between each communication node as an edge, let be the set of all edges. represents that the j-th AUV can obtain the information of the i-th AUV. Define as 's parent node, and define as 's child node. represents the adjacency matrix.

[0091] Define as the subgraph regarding the follower AUV, where

[0092] For the adjacency matrix When and i ≠ j, a ij = 1 > 0; otherwise a ij = 0 (for an undirected graph, that is, when , a ij = a ji = 1; otherwise a ij = 0). In the directed graph if there is a directed path from leader l to follower i, it indicates that follower i can obtain information from leader l, denoted as bil = 1 > 0; Otherwise b il = 0.

[0093] Directed graph The Laplacian matrix of is defined as

[0094]

[0095] where

[0096]

[0097] to The path in the directed graph is described as an ordered sequence of edges m = 1, 2,..., n. Satisfying the relationship All The nodes are The neighbors of the nodes, denoted by If in the directed graph All followers have at least one leader as the root node, then the leader set L is said to be globally reachable.

[0098] Remark 1: This invention studies the leader-follower multi-AUV system, adopts a directed communication topology structure. An AUV can only obtain information from its own neighboring AUVs and cannot obtain information of other individuals. It is a distributed multi-AUV formation system, and the research is on the distributed cooperative control of the multi-AUV formation system.

[0099] 2. Multi-AUV formation system model:

[0100] Assumption 1: The leader AUV is not disturbed and can move along a pre-set navigation trajectory.

[0101] 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).

[0102] According to Assumption 1, the dynamic model of the leader AUV can be expressed as

[0103]

[0104] where Represents the position and angle vector of the leader AUV, Represents the first and second derivatives of η l (t); Is a constant matrix, l ∈ L. Write the above equation in matrix form

[0105]

[0106] Further simplification of it gives the following form

[0107]

[0108] wherein,

[0109] the output form of the Design Leader AUV

[0110]

[0111] wherein, represents the output of the leader, is a constant matrix.

[0112] Generally, considering the influence of external disturbances on the follower AUV, its dynamic model can be expressed as

[0113]

[0114] wherein, represents the position and angle vector of the follower AUV, is the control input, represents the unknown external disturbance, i ∈ F, is the mass matrix, is the Coriolis force matrix, is the restoring force of the heavy buoyancy.

[0115] Assumption 2: In water, for the restoring force of the AUV, the gravity W and the buoyancy B are equal, x G = x B , y G = y B , z G ≠ z B , and θ are both small angle quantities. In the moving coordinate system, the center of gravity coordinates of the AUV are (x G , y G , z G ), and the center of buoyancy coordinates are (x B , y B , z B ), θ represents the roll angle and pitch angle in the geodetic system. Then, according to the small angle assumption, g ηi can be rewritten in the following form

[0116]

[0117] wherein,

[0118] According to Hypothesis 2, (26) can be rewritten as

[0119]

[0120] Further simplifying it gives the following form

[0121]

[0122] where

[0123] Hypothesis 3: Assume that is bounded, and ||R|| c is a positive constant. In addition, R i (t) acts on the AUV body and is not affected by the topological network structure.

[0124] Design the output form of the follower AUV

[0125]

[0126] where is the output of the follower, is a constant matrix.

[0127] 3. Topological switching:

[0128] Define that the topological switching signal {σ(t), t ≥ 0} of the multi - AUV system conforms to a Markov random process. The topological state is randomly selected only from the set S = {1, 2,..., s}, and the switching is only based on the current state. The transition rate matrix is W = [ω kr s×s , satisfying the following formula

[0129]

[0130] where ω kr ≥ 0 represents the rate of transition from state k to state r, k, r ∈ S, and Obviously, the transition rate matrix is W = [ω kr s×s and the sum of the elements in each row is 0.

[0131] Hypothesis 4: Assume that the Markov process has a unique expected stationary distribution, denoted as γ = (γ1, γ2,..., γ s ) T ,

[0132] ​​Considering that the communication network topology switching signal of the multi-AUV system conforms to the Markov random process, let denote the communication weight between follower i and follower j, and σ(t) denote the topology state at the current moment, where i, j ∈ F. When i ≠ j, if follower i can receive information from follower j, then Otherwise When i = j, denote the communication weight between follower i and leader l, and σ(t) denote the topology state at the current moment, where i ∈ F and l ∈ L. If follower i can receive information from leader l, then Otherwise

[0133] Let σ(t) denote the topology state at the current moment, where i ∈ F. If and only if the switching signal σ(t) = k, k ∈ S.

[0134] Define the Laplacian matrix

[0135]

[0136] Define the communication network topology graph of the multi-AUV system as If and only if the switching signal σ(t) = k, k ∈ S. Define the joint topology graph of the communication network of the multi-AUV system under Markov switching topology as

[0137] Assumption 5: The joint topology graph is directed. In , each follower AUV has at least one leader AUV as the root node.

[0138] According to Assumption 3, define

[0139]

[0140]

[0141] where, if and only if the switching signal σ(t) = k, k ∈ S. According to the above analysis, L1 and L2 are constant matrices. It should be noted that the expected graph has the same structure as the joint topology graph . The Laplacian matrix of the expected graph can be expressed in the following form

[0142]

[0143] According to Hypothesis 4, compared with studying each topological subgraph k ∈ S, focusing on the joint topological graph can also achieve the goal of studying the cooperative control of multi-AUV systems. After the above analysis, it is not necessary for each follower in to have at least one leader as the root node. As long as each follower in the joint topological graph has at least one leader as the root node, distributed cooperative control consistency research can be carried out, which means relaxing the research constraints and being applicable to more diverse research scenarios.

[0144] Lemma 1: When Hypothesis 4 holds, the following equation holds

[0145]

[0146] where all elements in are non-negative.

[0147] According to Lemma 1, define

[0148]

[0149] where α il ≥ 0, and

[0150] Hypothesis 6: Matrices and are both known or measurable, i ∈ F, l ∈ L.

[0151] Hypothesis 7: For is non-negative, and holds, l ∈ L.

[0152] Hypothesis 8: For the following equation holds

[0153]

[0154] Hypothesis 9: For each follower AUV, the following matrix equation

[0155]

[0156] has a solution (Π i , Ξ i ), i ∈ F, l ∈ L.

[0157] When the following equation holds, the multi-AUV system can achieve output containing control consistency,

[0158]

[0159] where α il ≥ 0,

[0160] 4. Proof of stability:

[0161] According to Lemma 1 and (39), define the state observation inclusion error as

[0162]

[0163] where α il ≥ 0, i ∈ F.

[0164] Find the first derivative of δ(t) and with respect to time t

[0165]

[0166] where i ∈ F, j ∈ F; i ∈ F, l ∈ L.

[0167] Define the following error

[0168]

[0169] Find and the first derivative with respect to time t

[0170]

[0171] where

[0172] Define the following error

[0173]

[0174] Find the first derivative of ζ(t) with respect to time t

[0175]

[0176] where

[0177]

[0178] According to (54), the following equation holds

[0179]

[0180] where if and only if σ(t) = k, k ∈ S.

[0181] For the convenience of subsequent calculations, rewrite (53) in the following form

[0182]

[0183] where \(i\in F\).

[0184] Theorem 1: Considering the leader AUV and the state observer, when the following conditions are satisfied, finally holds,

[0185]

[0186] where, and are both positive constants, if and only if \(\sigma(t)=k\), \(k\in S\).

[0187] Proof:

[0188] Considering the case of Markov switching topology, for each case of \(\sigma(t)=k\), \(k\in S\), consider the following Lyapunov function

[0189]

[0190] where

[0191]

[0192] Find the first derivative with respect to time \(t\)

[0193]

[0194] For the complete topological state set of the multi-AUV system, the following definitions are given

[0195]

[0196] Then the derivative of can be expressed as

[0197]

[0198] According to Assumption 4, the topological switching signal \(\sigma(t)\) has a unique stationary distribution \(\gamma = (\gamma_1,\gamma_2,...,\gamma s ) T , then can be written in the following form

[0199]

[0200] Simplify (68)

[0201]

[0202] Among them,

[0203] According to the Cauchy - Schwarz inequality, the following equation holds

[0204]

[0205] Substituting (70) into (69) gives

[0206]

[0207] From (58) and (59), it can be seen that

[0208]

[0209] Find The integral with respect to time t

[0210]

[0211] According to the definition of, it can be known that ||ζ(τ)|| c represents the upper bound of ||ζ(τ)||, At the same time, Therefore, (74) can satisfy the following equation

[0212]

[0213] Among them, is a non - negative constant, then holds.

[0214] Combining the definitions of ξ(t) and ζ(t), the following relationship is obtained

[0215]

[0216] For any there exists such that the following equation holds

[0217]

[0218] Therefore

[0219]

[0220] Among them, is a non - negative constant, is a positive constant.

[0221] From (78), it can be seen that

[0222]

[0223] Under the action of the state observer (10), the multi-AUV system finally makes (81) hold. The proof of Theorem 1 is completed.

[0224] Theorem 2: For the multi-AUV system, under the action of the disturbance observer (11), the disturbance received by the AUV can be effectively estimated, and the estimation error can be ultimately uniformly bounded.

[0225] Proof:

[0226] The selected Lyapunov function is in the following form

[0227]

[0228] where

[0229] Find The first derivative with respect to time t

[0230]

[0231] According to Young's inequality, it can be known that

[0232]

[0233] Further calculate (83)

[0234]

[0235] where, a1 = 1 > 0,

[0236] Find The integral with respect to time t

[0237]

[0238] From (86), it can be seen that when t → ∞, It will finally converge to the following set

[0239]

[0240] According to (87) and the definition of, under the action of the disturbance observer (11), the disturbance received by the AUV can be effectively estimated, and the estimation error can be ultimately uniformly bounded. The proof of Theorem 2 is completed.

[0241] Theorem 3: Considering the leader-follower multi-AUV systems (22) and (26), in the context of Markov switching topology, let and Satisfy the Hurwitz matrix, and K 2i = Ξ i -K 1i Π i , i ∈ F. When Theorem 1 is satisfied, under the action of the distributed switched topology composite observer (9) and the distributed output feedback inclusion controller (16), the distributed output inclusion control consistency of the multi-AUV system can be achieved.

[0242] Proof:

[0243] First, make the following definitions

[0244]

[0245] The output inclusion error of the follower AUV is defined as follows

[0246]

[0247] Let U = blockdiag{U M+1 , U M+2 ,..., U M+N}, K1 = blockdiag{K 1(M+1) , K 1(M+2) ,..., K 1(M+N)}, K2 = blockdiag{K 2(M+1) , K 2(M+2) ,..., K 2(M+N)}, P = blockdiag{P M+1 , P M+2 ,..., P M+N}. Among them,

[0248]

[0249] According to (41), we can get

[0250]

[0251] where Π = blockdiag{Π M+1 , Π M+2 ,..., Π M+N}, and Ξ = K1Π + K2.

[0252] Define the error

[0253]

[0254] Find the first derivative of ε(t) and with respect to time t

[0255]

[0256] Combining (95), (99) and (100), we can obtain

[0257] e(t)=Cε(t) (104)

[0258] Analyzing (95)(103), because satisfies the Hurwitz matrix, and is ultimately uniformly bounded, i∈F, so is ultimately uniformly bounded. According to Theorem 1, there is established. Analyzing (102), because satisfies the Hurwitz matrix, i∈F, and there is and is ultimately uniformly bounded, and there is then is ultimately uniformly bounded. According to (104), it can be known that is ultimately uniformly bounded.

[0259] According to (103), because and i∈F, l∈L, then is ultimately uniformly bounded.

[0260] Therefore, for the leader-follower multi-AUV systems (22) and (26), adopting a directed communication network and considering the case of Markov switching topology, according to the joint topology graph, when the system is subject to external unknown disturbances, under the action of the distributed switching topology composite observer (9) and the distributed output feedback inclusion controller (16), it can make the outputs of all follower AUVs converge to the vicinity of a certain output state formed by the output of the leader AUV, realizing the distributed output inclusion control consistency of the multi-AUV system. The proof of Theorem 3 is completed.

[0261] The main innovation of the present invention lies in: using Markov switching topology to simulate the situation where the communication topology of the multi-AUV system changes with time, introducing the joint topology graph to relax the research limitations of the switching topology problem, and designing a composite observer and an output feedback controller while considering external disturbances, providing a new algorithmic solution for solving the distributed inclusion control problem of the multi-AUV system under switching topology.

[0262] Embodiment

[0263] To verify the effectiveness of the proposed distributed output containment control method under Markov switching topology, a leader-follower multi-AUV system is selected as the verification object, adopting the form of multiple leaders and multiple followers. In the simulation experiment, there are 5 leader AUVs (denoted as numbers 1, 2, 3, 4, 5) and 7 follower AUVs (denoted as numbers 6, 7, …, 12) designed in the leader-follower multi-AUV system.

[0264] In the leader-follower multi-AUV system, the dynamic model of the leader AUV can be expressed by the following formula:

[0265]

[0266] The output form of the leader AUV is

[0267]

[0268] Select

[0269] Referring to the relevant projects of the National Key Laboratory of Underwater Vehicle Technology and the hydrodynamic test results, the following AUV actual model data will be used as the reference basis. Tables 1 and 2 represent the inertial coefficients and hydrodynamic parameters of the AUV, respectively.

[0270] Table 1 Various inertial coefficients of the AUV

[0271]

[0272] Table 2 Hydrodynamic parameters of the AUV

[0273]

[0274] The designed motion trajectory of the leader AUV is:

[0275] Table 3 Preset trajectory of the leader AUV

[0276]

[0277] Among them, Select the constant matrix

[0278] Remark 2: When At this time,

[0279]

[0280] Because The magnitude of is negligible compared to η l (t), so it can be considered that Holds.

[0281] In the leader-follower multi-AUV system, the dynamic model of the follower AUV can be expressed by the following formula:

[0282]

[0283] where i = 6, 7,..., 12.

[0284] Set the initial position and angle of the follower AUV:

[0285] Table 4 Initial position and angle of the follower AUV

[0286]

[0287]

[0288] Set the initial linear velocity and angular velocity of the follower AUV:

[0289] Table 5 Initial linear velocity and angular velocity of the follower AUV

[0290]

[0291] The output form of the follower AUV is

[0292]

[0293] Select According to Note 1, holds. The leader-follower multi-AUV system adopts a directed communication topology network and is accompanied by switching topology behavior. The switching topology conforms to a Markov random process. The value set S of the switching topology signal {σ(t), t ≥ 0} is {1, 2, 3}, and the transition rate matrix is selected as

[0294]

[0295] The unique expected stationary distribution is Design the network communication topology subgraph and the joint topology graph as Figure 2 shown.

[0296] According to Figure 2 it can be seen that in each topology subgraph, there are followers without any leader as the root node, and only in the joint topology graph does each follower have at least one leader as the root node.

[0297] In the leader-follower multi-AUV system, the initial positions and velocities of the leader and followers are randomly selected to ensure that no two AUVs coincide at the initial time. The trajectory of the leader in the leader-follower multi-AUV system is pre-designed, and 5 leader AUVs move according to the designed trajectory.

[0298] Selection of parameters of the distributed switched topology composite observer, P i Designed by (97), Other parameter selections Satisfy (58), i ∈ F. The selection of the distributed output inclusion controller parameters, K 1i = -5000[I6I6], K 2i Calculated from, U i = [0 6×6 5000I6], i ∈ F.

[0299] Figure 3 Shows the change of the topology state of the leader-follower multi-AUV system over time. The topology structure switches among the topology subgraphs and The topology state at each moment can be intuitively observed, which conforms to the Markov random process.

[0300] The three-dimensional spatial position changes of each AUV in the leader-follower multi-AUV system are as Figure 4 shown. 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 designed switched topology distributed output inclusion control algorithm, as time progresses, at t = 10 s, all the follower AUVs have been able to enter the convex hull, 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 AUVs and moving with the movement of the convex hull formed by the leaders, indicating that the distributed output inclusion control consistency of the leader-follower multi-AUV system can be achieved.

[0301] The position changes of each AUV in the leader-follower multi-AUV system are as Figure 5 shown. The angle changes of each AUV in the leader-follower multi-AUV system are as Figure 6 shown. Analysis Figure 5 and Figure 6 shows that the state variables of each dimension of the follower AUVs can converge to the convex hull formed by the state variables of the leaders at t = 10.2 s, indicating that all the follower AUVs can enter the convex hull from outside and move in coordination with the leader AUVs, achieving the inclusion control.

[0302] The state observation of the follower AUVs in the leader-follower multi-AUV system includes the error ||ξ i(t)|| 2 The change is as Figure 7 shown, the error of the follower AUV disturbance observer in the leader-follower multi-AUV system changes as Figure 8 shown. Analyzing Figure 7 , it can be seen from the information in the figure that the state observation contains the error ||ξ i (t)|| 2 gradually converges to around 0 at t = 26 s, indicating that the state observation quantity can converge to the convex hull composed of the leader state quantities, and the state observer can play a good observation effect when the system considers the Markov switching topology. From Figure 8 it can be seen that the error of the disturbance observer converges to around 0 at t = 3 s, and the disturbance observer can compensate for the unknown external disturbance more accurately, reflecting good estimation performance.

[0303] The output of the follower AUV in the leader-follower multi-AUV system contains the error ||e i (t)|| 2 changes as Figure 9 shown. Analyzing Figure 9 , the output contains the error ||e i (t)|| 2 will gradually converge to around 0 at t = 27 s with time, indicating that the distributed output inclusion controller considering the switching topology is effective. For the leader-follower multi-AUV systems (22) and (26), the communication network adopts a directed communication topology form. When considering the situation of Markov switching topology and unknown external disturbances in the system, under the action of the distributed switching topology composite observer (9) and the distributed output feedback inclusion controller (16), the output inclusion error of the follower AUV is ultimately uniformly bounded, and the output quantities of all follower AUVs converge to the vicinity of an output state formed by the output quantities of the leader AUV, and the distributed output inclusion control consistency of the multi-AUV system is realized.

[0304] The above calculation examples of the present invention only illustrate the calculation model and calculation process of the present invention in detail, 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 distributed inclusion control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology, characterized in that: Distributed inclusive control of multi-AUV formation system based on output feedback inclusive controller; The distributed output feedback controller includes the following: τ i (t)=K 1i z i (t)+K 2i Φ i (t) Among them, τ i (t) is the control input, i∈F, F=(M+1,M+2,…,M+N) represents the set of N followers; z i (t) represents the intermediate variable, and its specific form is obtained by taking its first derivative The form of K 1i , K 2i and U i are all gain matrices; Φ i (t) represents the corresponding distributed switching topology composite observer when the topology switching signal {σ(t), t≥0} of the multi-AUV system communication network conforms to the Markov random process and the system is subject to external interference; M ηi (t) is the mass matrix, C ηi (t) is the Coriolis force matrix, D ηi (t) is the fluid damping force matrix, I6, 0 6×6 are the unit matrix and the matrix of all zeros respectively, W and B represent the gravity and buoyancy of the AUV, (x G ,y G ,z G )、(x B ,y B ,z B ) represents the coordinates of the center of gravity and the center of buoyancy of the AUV in the motion coordinate system; represents the interference observation value; is the output of the follower AUV, is a constant matrix.

2. According to claim 1, a distributed inclusion control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology is characterized in that: Distributed Switching Topology Composite Observer Φ i (t) are as follows: Among them, δ i (t) represents the state observation value, represents the interference observation value, P i is a design matrix.

3. According to claim 2, a distributed inclusion control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology is characterized in that: State observation value δ i (t) through its corresponding first derivative The state observer expressed in the form is determined as follows: in, I6,0 6×6 are the unit matrix and the matrix of all zeros respectively, is a constant matrix, l∈L, L=(1,2,…,M) represents the set of M pilots; δ i (t) represents the state observation value, represents the state observer, is δ i The first derivative of (t); is a positive constant; σ(t) represents the topological state at time t, and t represents time; represents the communication weight between follower i and follower j, i,j∈F; represents the communication weight between follower i and leader l; is the position and angle vector η of the leader AUV l (t) and the corresponding first derivative matrix, that is, η l (t) represents the position and angle vector of the leader AUV, Represents η l (t) is a first derivative.

4. According to claim 3, a distributed inclusion control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology is characterized in that: Said Determined by the navigator's dynamic model; the navigator's dynamic model is 5. According to claim 2, a distributed inclusion control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology is characterized in that: Interference Observation Through its corresponding first derivative The state observer expressed in the form is determined as follows: in, represents the interference observation value, represents the disturbance observer; is the position and angle vector η of the follower AUV i (t) and the corresponding first derivative matrix, that is, η i (t) represents the position and angle vector of the follower AUV, is η i (t) is a first derivative.

6. According to claim 1, a distributed inclusion control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology is characterized in that: Said Determined by the follower's dynamic model; the follower's dynamic model is 7. The distributed inclusive control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology according to claim 6 is characterized in that: Output of the follower AUV 8. A distributed inclusive control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology according to any one of claims 1 to 7, characterized in that: The topology switching signal {σ(t), t≥0} of the multi-AUV system communication network conforms to the Markov random process. The topological state is randomly selected only from the set S = {1, 2, ..., s}, and the switching is only based on the current state, s represents the sth topological state; the transfer rate matrix is ​​W = [ω kr ] s×s , satisfying the following formula: in, Represents the probability of a topological structure transforming into another structure and maintaining the original structure; Δt represents the time change, and o(Δt) represents a high-order infinitesimal function about Δt; ω kr ≥0 indicates the rate of transition from k state to r state, and 9. The distributed inclusive control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology according to claim 8 is characterized in that: The topology switching signal {σ(t), t≥0} of the multi-AUV system communication network conforms to the Markov random process. For the communication weight between follower i and follower j, If follower i can receive information from follower j, then otherwise When i=j, The communication weight between follower i and leader l If follower i can receive information from leader l, then otherwise 10. The distributed inclusive control method for a multi-AUV formation system based on a composite observer considering external interference and time-varying topology according to claim 9 is characterized in that: The Laplacian matrix corresponding to the communication network topology of the multi-AUV system is as follows: in,

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