Adaptive inclusion formation control method for heterogeneous singular multi-agent systems

Abstract

The invention discloses an adaptive inclusive formation control method for a heterogeneous singular multi-agent system, comprising: first constructing a heterogeneous singular multi-agent system and describing its inclusive formation control; then designing an adaptive state feedback formation controller and an adaptive dynamic output formation controller; then converting the inclusive formation control problem into an infinite norm constraint problem by designing the state transformation of a closed-loop system under the action of the adaptive state feedback formation controller; then combining pulse-free analysis and stability analysis to obtain the inclusive formation control method for the heterogeneous singular multi-agent system under the action of the adaptive state feedback formation controller; finally, according to the state transformation of the closed-loop system under the action of the adaptive dynamic output formation controller, the inclusive formation control method for the heterogeneous singular multi-agent system is obtained; the present invention realizes the inclusive formation control of the heterogeneous singular multi-agent system and can quickly estimate the virtual leader matrix information.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-agent system formation control, and specifically relates to an adaptive inclusive formation control method for a heterogeneous singular multi-agent system. Background Art

[0002] In recent years, with the rapid development of computing and communication technologies, the distributed cooperative control of multi-agent systems has been widely studied. Among them, consensus control is a basic research direction of distributed cooperative control of multi-agents, which includes formation control, cooperative output control, swarming control, etc. As a branch of consensus control, formation control usually involves multiple leader agents. The research on multi-leader formation control problems is based on many practical application scenarios. For example: a group of robots equipped with sensors can detect dangerous obstacles and maintain other faulty robots in a safe area until the fault is discovered and repaired. At present, the research on multi-leader formation control usually assumes that the leader agents do not communicate with each other and do not have collective behavior, which is not true in many complex practical applications.

[0003] In some practical applications, a leader agent is required to achieve control objectives through information exchange. Therefore, the problem of inclusive formation control has been proposed, where all leader agents form a formation control, with follower agents surrounded by the leader agent, achieving inclusive formation control. In inclusive formation control of multi-agent systems, obtaining information about the virtual leader system matrix is ​​a strict information transmission condition and increases communication costs between agents. Therefore, designing an adaptive controller to estimate the leader matrix information to reduce communication costs between agents is a challenge.

[0004] Currently, all results on inclusive formation control for multi-agent systems are based on normal multi-agent systems, where the dynamics model of the multi-agent system only contains differential constraints. This dynamics model cannot describe complex systems with algebraic constraints on state variables. Therefore, singular multi-agent systems, whose dynamics models contain both differential and algebraic constraints, have been proposed as an extension of normal multi-agent systems. To more accurately describe the diverse dynamics of singular multi-agent systems in practical applications, it is necessary to analyze more general heterogeneous singular multi-agent systems. Due to the algebraic constraints in singular multi-agent systems, inclusive formation control methods for normal multi-agent systems are ineffective. Furthermore, when the virtual leader agent system matrix is ​​unknown and the relative state variables are difficult to obtain, designing distributed adaptive controllers to implement inclusive formation control for heterogeneous singular multi-agent systems becomes even more challenging.

[0005] Currently, research on control involving formations has mostly focused on normal multi-agent systems, but has not addressed heterogeneous singular multi-agent systems whose dynamic models contain algebraic constraints. In practical applications, the system matrix of the virtual leader agent is often unknown, and the relative state variables are difficult to obtain. Therefore, control involving formations in heterogeneous singular multi-agent systems with adaptive leader matrix information estimation is an urgent problem to be solved.

[0006] Xuxi Zhang, Xianping Liu, Zhiguang Feng et al. disclosed Distributed containment control of singular heterogeneous multi-agent systems. (Xuxi Zhang, Xianping Liu, Zhiguang Feng, Distributed containment control of singular heterogeneous multi-agent systems[J], Journal of the Franklin Institute, 2020, 57(3): 1378-1399.) The sufficient conditions for containment control of heterogeneous singular multi-agent systems are given. However, since the distributed containment controller designed in this document relies on the state transmission of the heterogeneous singular multi-agent system, it has the disadvantage of high measurement and transmission costs during the containment control process.

[0007] Shan Zuo et al. published Time-varying output formation containment of general linear homogeneous (Shan Zuo, Yongduan Song, Frank L. Lewis, Ali Davoudi. Time-varying output formation containment of general linear homogeneous and heterogeneous multiagent systems[J],IEEE Transactions on Control of Network Systems,2019,6(2):537-548。), which studied the containment formation control problem of homogeneous normal multi-agent systems. However, this paper did not consider the differences between the matrices of different agent systems and the singular characteristics of multi-agent systems, and could not describe complex heterogeneous singular multi-agent systems with differences between agents. Summary of the Invention

[0008] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide an adaptive inclusive formation control method for a heterogeneous singular multi-agent system, which realizes the formation control of a heterogeneous singular multi-agent system by designing an adaptive state output formation controller independent of the system state information transmission. It has the characteristics of low measurement and transmission costs and can accurately describe the differences between agents.

[0009] In order to achieve the above object, the technical solution adopted by the present invention is:

[0010] The adaptive formation control method for heterogeneous singular multi-agent systems includes the following steps:

[0011] Step 1: Construct a heterogeneous singular multi-agent system, including follower agents, actual leader agents, and virtual leader agents; then describe the formation control problem of the heterogeneous singular multi-agent system;

[0012] Step 2: Based on the heterogeneous singular multi-agent system constructed in step 1, an adaptive state feedback formation controller is designed to obtain a closed-loop coupling system under the action of the adaptive state feedback formation controller; at the same time, an adaptive dynamic output formation controller is designed that is completely independent of the state information transmission of the heterogeneous singular multi-agent system constructed in step 1, and a closed-loop coupling system under the action of the adaptive dynamic output formation controller is obtained;

[0013] Step 3: By designing the state transformation of the closed-loop coupled system under the adaptive state feedback formation controller in step 2, a nonlinear singular system under the adaptive state feedback formation controller is obtained; the nonlinear singular system under the adaptive state feedback formation controller is analyzed in combination with the stability theory of nonlinear singular systems, including converting the formation control problem into an infinite norm constraint problem of the nonlinear singular system under the adaptive state feedback formation controller;

[0014] Step 4: By performing pulseless analysis and stability analysis on the nonlinear singular system under the adaptive state feedback formation controller obtained in step 3, the heterogeneous singular multi-agent system including formation control results are obtained;

[0015] At the same time, by designing the state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller in step 2, a nonlinear singular system under the action of the adaptive dynamic output formation controller is obtained; based on the pulse-free analysis and stability analysis, the heterogeneous singular multi-agent system including the formation control result is obtained.

[0016] The specific method for constructing a heterogeneous singular multi-agent system in step 1 is:

[0017] Build a heterogeneous singular multi-agent system consisting of follower agents and a de facto leader agent:

[0018]

[0019] Among them, when i=1,2,...,N, it represents the follower agent, marked as When i=N+1,N+2,...,N+M, it indicates that the leader agent is marked as variable is the state of the singular multi-agent system, the variable is the measured output of the singular multi-agent system, and the control input of the i-th agent is Matrix E i Satisfy the matrix constraint rank(E i )=r i <n i , that is, E i is a singular matrix; where and is the given system parameter matrix;

[0020] Building a heterogeneous singular multi-agent system with a virtual leader agent:

[0021]

[0022] Among them, the variable is the state of the virtual leader agent, is the measured output of the virtual leader agent.

[0023] The description method of the formation control problem included in step 1 is:

[0024] When there is a controller u i (t), so that the controller u i (t) The closed-loop system under the action of is pulse-free and satisfies:

[0025]

[0026] Where dist(x,F)=inf y∈F ||xy||2 represents the distance from x to the set F, lim represents the limit, inf represents the lower bound of the set, ||·|| represents the norm of the matrix, Then the formation control problem of heterogeneous singular multi-agent systems is solved.

[0027] In step 2, an adaptive state feedback formation controller is designed based on the heterogeneous singular multi-agent system, and a specific method for obtaining a closed-loop coupling system under the action of the adaptive state feedback formation controller is as follows:

[0028] Step 2.1: Based on the heterogeneous singular multi-agent system, design an adaptive state feedback formation controller:

[0029]

[0030] Among them, υ1<0, is the estimate of the virtual leader agent system matrix A0, is the state of the distributed controller, K 1i is the gain matrix to be designed, K 2i (t) is the time-varying gain matrix to be designed;

[0031] In step 2.2, combining the heterogeneous singular multi-agent system and the adaptive state feedback formation controller designed in step 2.1, the closed-loop coupling system under the action of the adaptive state feedback formation controller is obtained as follows:

[0032]

[0033] in,

[0034] In step 2, the adaptive dynamic output formation controller is designed to be completely independent of the state information transmission of the heterogeneous singular multi-agent system, and the specific method for obtaining the closed-loop coupling system under the action of the adaptive dynamic output formation controller is as follows:

[0035] Step 2.3, design an adaptive dynamic output formation controller that is completely independent of the system state information transmission:

[0036]

[0037] Among them, υ1<0, is the estimate of the virtual leader agent system matrix A0, is the state of the distributed controller, K is the state estimate of the follower agent and the actual leader agent, 1i is the gain matrix to be designed, K 2i (t) is the time-varying gain matrix to be designed;

[0038] In step 2.4, combining the heterogeneous singular multi-agent system and the adaptive dynamic output formation controller designed in step 2.3, the closed-loop coupling system under the action of the adaptive dynamic output formation controller is obtained as follows:

[0039]

[0040] in,

[0041] The specific method for designing the state transformation of the closed-loop coupled system under the action of the adaptive state feedback formation controller in step 3 to obtain the nonlinear singular system under the action of the adaptive state feedback formation controller is:

[0042] Step 3.1, design the state transformation according to the closed-loop coupled system under the action of the adaptive state feedback formation controller:

[0043] Feedback control matrix K 2i (t) satisfies K 2i (t) = U i (t)-K 1i Γ i (t), and in, is the solution of formula (9):

[0044]

[0045] When the formation function is f i (t) and satisfies The state transformation of the closed-loop coupling system under the action of the adaptive state feedback formation controller is given as:

[0046]

[0047] in,

[0048] is the Laplacian matrix corresponding to the follower agent, is the Laplacian matrix corresponding to the actual follower agent, col{·,…,·} represents the column vector spanned by each component, and diag{·,…,·} represents the diagonal matrix composed of each component;

[0049] Define the following matrix:

[0050]

[0051] In step 3.2, the state transformation of the closed-loop coupled system under the action of the adaptive state feedback formation controller and the heterogeneous singular multi-agent system given in step 3.1 is combined to obtain the nonlinear singular system under the action of the adaptive state feedback formation controller:

[0052]

[0053] in,

[0054]

[0055] The specific method for converting the formation control problem in step 3 into an infinite norm constraint problem of a nonlinear singular system under the action of an adaptive state feedback controller is as follows:

[0056] Step 3.3, if equation (12) holds:

[0057]

[0058] Then the formation control problem is solved under the action of the adaptive state feedback formation controller, wherein,

[0059] ||·|| ∞ Represents the infinity norm of a matrix

[0060] On this basis, a linear singular system is constructed And design the Lyapunov function Then the Lyapunov function along the linear singular system The derivative of is:

[0061]

[0062] in,

[0063] when When Established;

[0064] Furthermore, the matrix satisfy

[0065]

[0066] Then, the matrix pair is pulse-free, therefore, φ is stable;

[0067] In addition, the measurement output of the follower agent satisfies

[0068]

[0069] in, and are the Laplacian matrices corresponding to the actual leader agent and the follower agent, respectively, and therefore contain the control constraints be satisfied;

[0070] At the same time, the measured output of the actual leader agent satisfies

[0071]

[0072] in, Therefore, the constraint lim for formation control is t→∞ ||y i (t)-y0(t)-C0f i (t)||=0 is satisfied;

[0073] further, The proof is as follows

[0074]

[0075] Among them, ρ(·) represents the spectral radius of the matrix, max i represents the maximum value among all i, so the formation control problem is transformed into an infinite norm constraint problem of a nonlinear singular system under the action of an adaptive state feedback formation controller.

[0076] In step 4, the non-linear singular system under the adaptive state feedback formation controller is subjected to pulseless analysis and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result. The specific steps are:

[0077] Step 4.1: If the communication topology between heterogeneous singular multi-agents can be normalized and contains a spanning tree with a virtual leader as the root node, then Established;

[0078] because Established, obtained by calculation Right now Established, among which is the Laplacian matrix, is the connection weight matrix, Indicates that the i-th agent can receive the information of the virtual leader, otherwise d i =0, in addition, the matrix The following inequality is satisfied:

[0079]

[0080] so Established;

[0081] Step 4.2, given and If the system matrix (A0, C0) is detectable, And there exists a matrix satisfy:

[0082]

[0083] in, but, and All are established, ||·|| ∞ Represents the infinite norm of a matrix;

[0084] First, according to the bounded real lemma of the system, when When it was established, be satisfied;

[0085] Furthermore, for any We get formula (21):

[0086]

[0087] this means in addition, Therefore, we get

[0088] Step 4.3, when in step 4.1 Established, step 4.2 Established and parameters If there is an invertible matrix and any matrix satisfy

[0089]

[0090] in, Then, the inclusive formation control in heterogeneous singular multi-agent systems is realized under the action of the adaptive state feedback formation controller;

[0091] First, by Combining Schur's complement theorem and formula (22), we can know that Established;

[0092] In addition, according to step 4.1 and step 4.2 get:

[0093]

[0094] Therefore, the heterogeneous singular multi-agent system realizes inclusive formation control under the action of the adaptive state feedback formation controller.

[0095] The specific steps for designing the state transformation of the closed-loop coupling system under the action of the adaptive dynamic output formation controller in step 4 are as follows:

[0096] According to the closed-loop coupled system under the action of the adaptive dynamic output formation controller, the state transformation is designed:

[0097] Feedback control matrix K 2i (t) satisfies K 2i (t) = U i(t)-K 1i Γ i (t), and in, Yes The solution is when the formation function is fi(t) and satisfies The state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller is given as:

[0098]

[0099] in,

[0100] is the Laplacian matrix corresponding to the follower agent, is the Laplacian matrix corresponding to the actual follower agent, col{·,…,·} represents the column vector spanned by each component, and diag{·,…,·} represents the diagonal matrix composed of each component;

[0101] According to the state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller, the nonlinear singular system under the action of the adaptive dynamic output formation controller is obtained as follows:

[0102]

[0103] in,

[0104] In step 4, the nonlinear singular system under the adaptive dynamic output formation controller is subjected to pulseless and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result. The specific steps are:

[0105] For the nonlinear singular system under the action of adaptive dynamic output formation controller, the parameters If the matrix pair (E i ,A i +M 2i C i ) has no pulse, and the inequality If it holds true, then the formation control problem is solved under the action of the adaptive dynamic output formation controller;

[0106] First, given a singular matrix And the reversible model transformation matrix:

[0107]

[0108] For any matrix X i

[0109] The nonlinear singular system under the action of the adaptive state feedback formation controller satisfies Depend on Established, derived If (E i ,A i +M 2i C i ) allows, then in, represents the left negative half plane, so, Established;

[0110] Furthermore, the matrix and (E i ,A i +M 2i C i ) No pulse, based on model transformation get:

[0111]

[0112] Therefore, the matrix No pulse;

[0113] At the same time, the measurement output of the follower singular multi-agent satisfies:

[0114]

[0115] in, therefore, The validity of the proposed method is established, that is, the inclusive control of the follower singular multi-agents under the action of the adaptive state feedback formation controller is realized;

[0116] And the measured output of the actual leader singular multi-agent system satisfies:

[0117]

[0118] in,

[0119] therefore, The formation control of the actual leader singular multi-agent under the action of the adaptive state feedback formation controller is realized.

[0120] Compared with the prior art, the present invention has the following beneficial effects:

[0121] 1. The present invention adopts a heterogeneous singular multi-agent system to achieve an accurate description of an actual heterogeneous singular multi-agent system with different singular systems, which includes a normal multi-agent system as its special case. Therefore, it has a wider range of applications.

[0122] 2. The present invention designs an adaptive dynamic output formation controller that is completely independent of the state information transmission of the heterogeneous singular multi-agent system, realizing inclusive formation control. The controller does not need to measure and transmit the state variables of the singular multi-agent system, and therefore has stronger feasibility.

[0123] 3. The adaptive dynamic output formation controller designed in the present invention realizes the adaptive estimation of the system matrix of the virtual leader intelligent agent. The controller does not need to directly measure the virtual leader intelligent agent matrix information, which effectively reduces the information measurement cost and transmission cost involved in formation control.

[0124] In summary, compared with the existing technology, the present invention adopts a heterogeneous singular multi-agent system, which can more accurately describe the complex actual singular multi-agent model; at the same time, the adaptive dynamic output formation controller designed by the present invention does not need to measure and transmit the state variables of the singular multi-agent system, and is easier to implement than the traditional state-dependent controller; further, the formation controller designed by the present invention realizes inclusive formation control when the virtual leader matrix information is unknown, effectively reducing the communication cost between the virtual leader agent and the actual leader agent. BRIEF DESCRIPTION OF THE DRAWINGS

[0125] Figure 1 This is a system control flow chart of the method of the present invention.

[0126] Figure 2 This is the error state trajectory diagram of the simulation experiment of the present invention under the action of the adaptive state feedback formation controller.

[0127] Figure 3(a) is a schematic diagram of the output state trajectory of all singular agent systems under the action of the adaptive state feedback formation controller in the simulation experiment from 0s to 6s; Figure 3(b) is a schematic diagram of the output state trajectory of all singular agent systems under the action of the adaptive state feedback formation controller in the simulation experiment from 30s to 50s;

[0128] Figure 4 3 is the error state trajectory diagram of the simulation experiment of the present invention under the action of the adaptive dynamic output formation controller.

[0129] Figure 5(a) is a schematic diagram of the output state trajectory of all singular agent systems under the action of the adaptive dynamic output formation controller in the simulation experiment from 0s to 6s; Figure 5(b) is a schematic diagram of the output state trajectory of all singular agent systems under the action of the adaptive dynamic output formation controller in the simulation experiment from 30s to 50s; DETAILED DESCRIPTION

[0130] The present invention will be described in detail below with reference to the accompanying drawings.

[0131] like Figure 1 As shown, the adaptive formation control method of a heterogeneous singular multi-agent system includes the following steps:

[0132] Step 1: Construct a heterogeneous singular multi-agent system, including follower agents, actual leader agents, and virtual leader agents; then describe the formation control problem of the heterogeneous singular multi-agent system;

[0133] Step 2: Based on the heterogeneous singular multi-agent system constructed in step 1, an adaptive state feedback formation controller is designed to obtain a closed-loop coupling system under the action of the adaptive state feedback formation controller; at the same time, an adaptive dynamic output formation controller is designed that is completely independent of the state information transmission of the heterogeneous singular multi-agent system constructed in step 1, and a closed-loop coupling system under the action of the adaptive dynamic output formation controller is obtained;

[0134] Step 3: By designing the state transformation of the closed-loop coupled system under the adaptive state feedback formation controller in step 2, a nonlinear singular system under the adaptive state feedback formation controller is obtained; the nonlinear singular system under the adaptive state feedback formation controller is analyzed in combination with the stability theory of nonlinear singular systems, including converting the formation control problem into an infinite norm constraint problem of the nonlinear singular system under the adaptive state feedback formation controller;

[0135] Step 4: By performing pulseless analysis and stability analysis on the nonlinear singular system under the adaptive state feedback formation controller obtained in step 3, the heterogeneous singular multi-agent system including formation control results are obtained;

[0136] At the same time, by designing the state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller in step 2, a nonlinear singular system under the action of the adaptive dynamic output formation controller is obtained; based on the pulse-free analysis and stability analysis, the heterogeneous singular multi-agent system including the formation control result is obtained.

[0137] The specific method for constructing a heterogeneous singular multi-agent system in step 1 is:

[0138] Build a heterogeneous singular multi-agent system consisting of follower agents and a de facto leader agent:

[0139]

[0140] Among them, when i=1,2,...,N, it represents the follower agent, marked as When i=N+1,N+2,...,N+M, it indicates that the leader agent is marked as variable is the state of the singular multi-agent system, the variable is the measured output of the singular multi-agent system, and the control input of the i-th agent is Matrix E i Satisfy the matrix constraint rank(E i )=r i <n i , that is, E i is a singular matrix; where and is the given system parameter matrix;

[0141] Building a heterogeneous singular multi-agent system with a virtual leader agent:

[0142]

[0143] Among them, the variable is the state of the virtual leader agent, is the measured output of the virtual leader agent.

[0144] The description method of the formation control problem included in step 1 is:

[0145] When there is a controller u i (t), so that the controller u i (t) The closed-loop system under the action of is pulse-free and satisfies:

[0146]

[0147] Where dist(x,F)=inf y∈F ||xy||2 represents the distance from x to the set F, lim represents the limit, inf represents the lower bound of the set, ||·|| represents the norm of the matrix, Then the formation control problem of heterogeneous singular multi-agent systems is solved.

[0148] In step 2, an adaptive state feedback formation controller is designed based on the heterogeneous singular multi-agent system, and a specific method for obtaining a closed-loop coupling system under the action of the adaptive state feedback formation controller is as follows:

[0149] Step 2.1: Based on the heterogeneous singular multi-agent system, design an adaptive state feedback formation controller:

[0150]

[0151] Among them, υ1<0, is the estimate of the virtual leader agent system matrix A0, is the state of the distributed controller, K 1i is the gain matrix to be designed, K 2i(t) is the time-varying gain matrix to be designed;

[0152] In step 2.2, combining the heterogeneous singular multi-agent system and the adaptive state feedback formation controller designed in step 2.1, the closed-loop coupling system under the action of the adaptive state feedback formation controller is obtained as follows:

[0153]

[0154] in,

[0155] In step 2, the adaptive dynamic output formation controller is designed to be completely independent of the state information transmission of the heterogeneous singular multi-agent system, and the specific method for obtaining the closed-loop coupling system under the action of the adaptive dynamic output formation controller is as follows:

[0156] Step 2.3, design an adaptive dynamic output formation controller that is completely independent of the system state information transmission:

[0157]

[0158] in, is the estimate of the virtual leader agent system matrix A0, is the state of the distributed controller, K is the state estimate of the follower agent and the actual leader agent, 1i is the gain matrix to be designed, K 2i (t) is the time-varying gain matrix to be designed;

[0159] In step 2.4, combining the heterogeneous singular multi-agent system and the adaptive dynamic output formation controller designed in step 2.3, the closed-loop coupling system under the action of the adaptive dynamic output formation controller is obtained as follows:

[0160]

[0161] in,

[0162] The specific method for designing the state transformation of the closed-loop coupled system under the action of the adaptive state feedback formation controller in step 3 to obtain the nonlinear singular system under the action of the adaptive state feedback formation controller is:

[0163] Step 3.1, design the state transformation according to the closed-loop coupled system under the action of the adaptive state feedback formation controller:

[0164] Feedback control matrix K 2i (t) satisfies K 2i (t) = U i (t)-K1i Γ i (t), and in, is the solution of formula (9):

[0165]

[0166] When the formation function is f i (t) and satisfies The state transformation of the closed-loop coupling system under the action of the adaptive state feedback formation controller is given as:

[0167]

[0168] in,

[0169] is the Laplacian matrix corresponding to the follower agent, is the Laplacian matrix corresponding to the actual follower agent, col{·,…,·} represents the column vector spanned by each component, and diag{·,…,·} represents the diagonal matrix composed of each component;

[0170] Define the following matrix:

[0171]

[0172] In step 3.2, the state transformation of the closed-loop coupled system under the action of the adaptive state feedback formation controller and the heterogeneous singular multi-agent system given in step 3.1 is combined to obtain the nonlinear singular system under the action of the adaptive state feedback formation controller:

[0173]

[0174] in,

[0175]

[0176] The specific method for converting the formation control problem in step 3 into an infinite norm constraint problem of a nonlinear singular system under the action of an adaptive state feedback controller is as follows:

[0177] Step 3.3, if equation (12) holds:

[0178]

[0179] Then the formation control problem mentioned above is solved under the action of the adaptive state feedback formation controller, where

[0180]

[0181] ||·|| ∞ Represents the infinity norm of a matrix

[0182] On this basis, a linear singular system is constructed And design the Lyapunov function Then the Lyapunov function along the linear singular system The derivative of is:

[0183]

[0184] in,

[0185] when When Established;

[0186] Furthermore, the matrix satisfy

[0187]

[0188] Then, the matrix pair is pulse-free, therefore, φ is stable;

[0189] In addition, the measurement output of the follower agent satisfies

[0190]

[0191] in, and are the Laplacian matrices corresponding to the actual leader agent and the follower agent, respectively, and therefore contain the control constraints be satisfied;

[0192] At the same time, the measured output of the actual leader agent satisfies

[0193]

[0194] in, Therefore, the constraint lim for formation control is t→∞ ||y i (t)-y0(t)-C0f i (t)||=0 is satisfied;

[0195] further, The proof is as follows

[0196]

[0197] Among them, ρ(·) represents the spectral radius of the matrix, max i represents the maximum value among all i, so the formation control problem is transformed into an infinite norm constraint problem of a nonlinear singular system under the action of an adaptive state feedback formation controller.

[0198] In step 4, the non-linear singular system under the adaptive state feedback formation controller is subjected to pulseless analysis and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result. The specific steps are:

[0199] Step 4.1: If the communication topology between heterogeneous singular multi-agents can be normalized and contains a spanning tree with a virtual leader as the root node, then Established;

[0200] because Established, obtained by calculation Right now Established, among which is the Laplacian matrix, is the connection weight matrix, d i > 0 means that the i-th agent can receive the information of the virtual leader, otherwise d i =0, in addition, the matrix The following inequality is satisfied:

[0201]

[0202] so Established;

[0203] Step 4.2, given and If the system matrix (A0, C0) is detectable, And there exists a matrix satisfy:

[0204]

[0205] in, but, and All are established, ||·|| ∞ Represents the infinite norm of a matrix;

[0206] First, according to the bounded real lemma of the system, when When it was established, be satisfied;

[0207] Furthermore, for any We get formula (21):

[0208]

[0209] this means in addition, Therefore, we get

[0210] Step 4.3, when in step 4.1 Established, step 4.2 Established and parameters If there is an invertible matrix and any matrix satisfy

[0211]

[0212] in, Then, the inclusive formation control in heterogeneous singular multi-agent systems is realized under the action of the adaptive state feedback formation controller;

[0213] First, by Combining Schur's complement theorem and formula (22), we can know that Established;

[0214] In addition, according to step 4.1 and step 4.2 get:

[0215]

[0216] Therefore, the heterogeneous singular multi-agent system realizes inclusive formation control under the action of the adaptive state feedback formation controller.

[0217] The specific steps for designing the state transformation of the closed-loop coupling system under the action of the adaptive dynamic output formation controller in step 4 are as follows:

[0218] According to the closed-loop coupled system under the action of the adaptive dynamic output formation controller, the state transformation is designed:

[0219] Feedback control matrix K 2i (t) satisfies K 2i (t) = U i (t)-K 1i Γ i (t), and in, Yes The solution is when the formation function is f i (t) and satisfies The state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller is given as:

[0220]

[0221] in,

[0222]

[0223] is the Laplacian matrix corresponding to the follower agent, is the Laplacian matrix corresponding to the actual follower agent, col{·,…,·} represents the column vector spanned by each component, and diag{·,…,·} represents the diagonal matrix composed of each component;

[0224] According to the state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller, the nonlinear singular system under the action of the adaptive dynamic output formation controller is obtained as follows:

[0225]

[0226] in,

[0227]

[0228] In step 4, the nonlinear singular system under the adaptive dynamic output formation controller is subjected to pulseless and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result. The specific steps are:

[0229] For the nonlinear singular system under the action of adaptive dynamic output formation controller, the parameters If the matrix pair (E i ,A i +M 2i C i ) has no pulse, and the inequality If it holds true, then the formation control problem is solved under the action of the adaptive dynamic output formation controller;

[0230] First, given a singular matrix And the reversible model transformation matrix:

[0231]

[0232] The nonlinear singular system under the action of the adaptive state feedback formation controller satisfies Depend on Established, derived If (E i ,A i +M 2i C i ) allows, then in, represents the left negative half plane, so, Established;

[0233] Furthermore, the matrix and (E i ,A i +M 2i C i ) No pulse, based on model transformation get:

[0234]

[0235] Therefore, the matrix No pulse;

[0236] At the same time, the measurement output of the follower singular multi-agent satisfies:

[0237]

[0238] in,

[0239]

[0240] therefore, The validity of the proposed method is established, that is, the inclusive control of the follower singular multi-agents under the action of the adaptive state feedback formation controller is realized;

[0241] And the measured output of the actual leader singular multi-agent system satisfies:

[0242]

[0243] in,

[0244] therefore, The formation control of the actual leader singular multi-agent under the action of the adaptive state feedback formation controller is realized.

[0245] Simulation experiment

[0246] In order to verify the formation control effect, the present invention uses matlab to perform simulation verification. The matrix parameters of the heterogeneous singular multi-agent system are:

[0247]

[0248] The set of follower agents and the actual leader agent are represented as and The connection weight matrix between heterogeneous singular multi-agents is

[0249] For all actual leader agents Assume that the corresponding formation function is if If holds, then the inclusive control of the singular follower multi-agent is established, that is, Therefore, the error vector is defined as The error vector e stabilizes at the same time as the formation control of the heterogeneous singular multi-agent system. The state trajectory of Figure 2 Based on Figure 2 The simulation results show that the error vectors e1, e2, and e3 are stable. Therefore, under the action of the adaptive state feedback formation controller, the follower singular multi-agents are driven into a convex combination of the measured outputs of the actual leader singular multi-agent. The trajectories of the measured outputs of all heterogeneous singular multi-agent systems from 0 to 6 seconds are shown in Figure 3(a); the trajectories of the measured outputs of all heterogeneous singular multi-agent systems from 30 to 50 seconds are shown in Figure 3(b). As shown in Figure 3(b), using the designed adaptive state feedback formation controller, all actual leader agents form a square formation around the virtual leader agent, and the follower agents are driven into a convex combination of the actual leader agents.

[0250] Adopting adaptive dynamic output formation controller, error vector The state trajectory of Figure 4 was displayed in . Figure 4 Simulation results show that the error vectors e1, e2, and e3 reach stability around 30 seconds. Under the adaptive dynamic output formation controller, the measured output trajectories of all heterogeneous singular multi-agent systems from 0 to 6 seconds are shown in Figure 5(a); the measured output trajectories of all heterogeneous singular multi-agent systems from 30 to 50 seconds are shown in Figure 5(b). Figure 5(b) shows a detailed demonstration of formation control under the adaptive dynamic output formation controller. Using the designed adaptive dynamic output formation controller, all actual leader agents form a square formation around the virtual leader agent, and the follower agents are driven into a convex combination of the actual leader agents.

[0251] From the above simulation experiments, it can be seen that compared with the prior art, the present invention considers the formation control of heterogeneous singular multi-agent systems, that is, the parameter matrices of all agent systems in formula (30) are inconsistent. This heterogeneous singular multi-agent system can more accurately describe the complex actual system than the homogeneous singular multi-agent system. At the same time, the present invention considers the singular multi-agent system, that is, the system matrix E i ,i=1,…,7 is not full rank, when E i=I, the system of the present invention degenerates into a normal multi-agent system, that is, the singular multi-agent system considered by the present invention includes the normal multi-agent system as a special case; in addition, the adaptive controller designed by the present invention realizes the formation control when the virtual leader matrix information is unknown, as shown in Figures 3(a), 3(b), 5(a) and 5(b), which effectively reduces the communication cost between agents; further, the dynamic output formation controller designed by the present invention only relies on the transmission of measured outputs without the need for measurement and transmission of the system state, which is easier to implement than the controller based on state transmission.

Claims

1. An adaptive formation control method for heterogeneous singular multi-agent systems, characterized by: The following steps are involved: Step 1: construct a heterogeneous singular multi-agent system, wherein the heterogeneous singular multi-agent system includes a follower agent, an actual leader agent, and a virtual leader agent; and then describe the formation control problem of the heterogeneous singular multi-agent system; Step 2: Based on the heterogeneous singular multi-agent system constructed in step 1, an adaptive state feedback formation controller is designed to obtain a closed-loop coupling system under the action of the adaptive state feedback formation controller; at the same time, an adaptive dynamic output formation controller is designed that is completely independent of the state information transmission of the heterogeneous singular multi-agent system constructed in step 1, and a closed-loop coupling system under the action of the adaptive dynamic output formation controller is obtained; Step 3: By designing the state transformation of the closed-loop coupled system under the adaptive state feedback formation controller in step 2, a nonlinear singular system under the adaptive state feedback formation controller is obtained; the nonlinear singular system under the adaptive state feedback formation controller is analyzed in combination with the stability theory of nonlinear singular systems, and the formation control problem described in step 1 is converted into an infinite norm constraint problem of the nonlinear singular system under the adaptive state feedback formation controller; Step 4: By performing pulseless analysis and stability analysis on the nonlinear singular system under the adaptive state feedback formation controller obtained in step 3, the heterogeneous singular multi-agent system including formation control results are obtained; At the same time, by designing the state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller in step 2, a nonlinear singular system under the action of the adaptive dynamic output formation controller is obtained; based on the pulse-free analysis and stability analysis, the heterogeneous singular multi-agent system including the formation control result is obtained.

2. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: The specific method for constructing a heterogeneous singular multi-agent system in step 1 is: Build a heterogeneous singular multi-agent system consisting of follower agents and a de facto leader agent: Among them, when i=1,2,...,N, it represents the follower agent, marked as When i=N+1,N+2,...,N+M, it indicates that the leader agent is marked as variable is the state of the singular multi-agent system, the variable is the measured output of the singular multi-agent system, and the control input of the i-th agent is Matrix E i Satisfy the matrix constraint rank(E i )=r i <n i , that is, E i is a singular matrix; where and is the given system parameter matrix; Building a heterogeneous singular multi-agent system with a virtual leader agent: Among them, the variable is the state of the virtual leader agent, is the measured output of the virtual leader agent.

3. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: The description method of the formation control problem included in step 1 is: When there is a controller u i (t), so that the controller u i (t) The closed-loop system under the action of is pulse-free and satisfies: and Where dist(x,F)=inf y∈F ||xy||2 represents the distance from x to the set F, lim represents the limit, inf represents the lower bound of the set, ||·|| represents the norm of the matrix, Then the formation control problem of heterogeneous singular multi-agent systems is solved.

4. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: In step 2, an adaptive state feedback formation controller is designed based on the heterogeneous singular multi-agent system, and a specific method for obtaining a closed-loop coupling system under the action of the adaptive state feedback formation controller is as follows: Step 2.1: Based on the heterogeneous singular multi-agent system, design an adaptive state feedback formation controller: Among them, υ1<0, is the estimate of the virtual leader agent system matrix A0, is the state of the distributed controller, K 1i is the gain matrix to be designed, K 2i (t) is the time-varying gain matrix to be designed; In step 2.2, combining the heterogeneous singular multi-agent system and the adaptive state feedback formation controller designed in step 2.1, the closed-loop coupling system under the action of the adaptive state feedback formation controller is obtained as follows: in, 5. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: In step 2, the adaptive dynamic output formation controller is designed to be completely independent of the state information transmission of the heterogeneous singular multi-agent system, and the specific method for obtaining the closed-loop coupling system under the action of the adaptive dynamic output formation controller is as follows: Step 2.3, design an adaptive dynamic output formation controller that is completely independent of the system state information transmission: Among them, υ1<0, is the estimate of the virtual leader agent system matrix A0, is the state of the distributed controller, K is the state estimate of the follower agent and the actual leader agent, 1i is the gain matrix to be designed, K 2i (t) is the time-varying gain matrix to be designed; In step 2.4, combining the heterogeneous singular multi-agent system and the adaptive dynamic output formation controller designed in step 2.3, the closed-loop coupling system under the action of the adaptive dynamic output formation controller is obtained as follows: in, 6. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: The specific method for designing the state transformation of the closed-loop coupled system under the action of the adaptive state feedback formation controller in step 3 to obtain the nonlinear singular system under the action of the adaptive state feedback formation controller is: Step 3.1, design the state transformation according to the closed-loop coupled system under the action of the adaptive state feedback formation controller: Feedback control matrix K 2i (t) Satisfaction and in, is the solution of formula (9): When the formation function is f i (t) and satisfies The state transformation of the closed-loop coupling system under the action of the adaptive state feedback formation controller is given as: in, is the Laplacian matrix corresponding to the follower agent, is the Laplacian matrix corresponding to the actual follower agent, col{·,…,·} represents the column vector spanned by each component, and diag{·,…,·} represents the diagonal matrix composed of each component; Define the following matrix: In step 3.2, the state transformation of the closed-loop coupled system under the action of the adaptive state feedback formation controller and the heterogeneous singular multi-agent system given in step 3.1 is combined to obtain the nonlinear singular system under the action of the adaptive state feedback formation controller: in, 7. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: The specific method for converting the formation control problem in step 3 into an infinite norm constraint problem of a nonlinear singular system under the action of an adaptive state feedback controller is as follows: Step 3.3, if equation (12) holds: Then the formation control problem mentioned above is solved under the action of the adaptive state feedback formation controller, where ||·|| ∞ Represents the infinity norm of a matrix On this basis, a linear singular system is constructed And design the Lyapunov function Then the Lyapunov function along the linear singular system The derivative of is: in, For any matrix X i when When Established; Furthermore, the matrix satisfy Then, the matrix pair is pulse-free, therefore, φ is stable; In addition, the measurement output of the follower agent satisfies in, and are the Laplacian matrices corresponding to the actual leader agent and the follower agent, respectively, and therefore contain the control constraints be satisfied; At the same time, the measured output of the actual leader agent satisfies in, Therefore, the constraint lim for formation control is t→∞ ||y i (t)-y0(t)-C0f i (t)||=0 is satisfied; further, The proof is as follows Among them, ρ(·) represents the spectral radius of the matrix, max i represents the maximum value among all i, so the formation control problem is transformed into an infinite norm constraint problem of a nonlinear singular system under the action of an adaptive state feedback formation controller.

8. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: In step 4, the non-linear singular system under the adaptive state feedback formation controller is subjected to pulseless analysis and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result. The specific steps are: Step 4.1: If the communication topology between heterogeneous singular multi-agents can be normalized and contains a spanning tree with a virtual leader as the root node, then Established; because Established, obtained by calculation Right now Established, among which is the Laplacian matrix, is the connection weight matrix, d i > 0 means that the i-th agent can receive the information of the virtual leader, otherwise d i =0, in addition, the matrix The following inequality is satisfied: so Established; Step 4.2, given and If the system matrix (A0, C0) is detectable, And there exists a matrix satisfy: in, but, and All are established, ||·|| ∞ Represents the infinite norm of a matrix; First, according to the bounded real lemma of the system, when When it was established, be satisfied; Furthermore, for any We get formula (21): this means in addition, Therefore, we get Step 4.3, when in step 4.1 Established, step 4.2 Established and parameters If there is an invertible matrix and any matrix satisfy in, Then, the inclusive formation control in heterogeneous singular multi-agent systems is realized under the action of the adaptive state feedback formation controller; First, by Combining Schur's complement theorem and formula (22), we can know that Established; In addition, according to step 4.1 and step 4.2 get: Therefore, the heterogeneous singular multi-agent system realizes inclusive formation control under the action of the adaptive state feedback formation controller.

9. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: The specific steps for designing the state transformation of the closed-loop coupling system under the action of the adaptive dynamic output formation controller in step 4 are as follows: According to the closed-loop coupled system under the action of the adaptive dynamic output formation controller, the state transformation is designed: Feedback control matrix K 2i (t) satisfies K 2i (t) = U i (t)-K 1i Γ i (t), and in, Yes The solution is when the formation function is f i (t) and satisfies The state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller is given as: in, is the Laplacian matrix corresponding to the follower agent, is the Laplacian matrix corresponding to the actual follower agent, col{·,…,·} represents the column vector spanned by each component, and diag{·,…,·} represents the diagonal matrix composed of each component; According to the state transformation of the closed-loop coupled system under the action of the adaptive dynamic output formation controller, the nonlinear singular system under the action of the adaptive dynamic output formation controller is obtained as follows: in, 10. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: In step 4, the nonlinear singular system under the adaptive dynamic output formation controller is subjected to pulseless and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result. The specific steps are: For the nonlinear singular system under the action of adaptive dynamic output formation controller, the parameters If the matrix pair (E i ,A i +M 2i C i ) has no pulse, and the inequality If it holds true, then the formation control problem is solved under the action of the adaptive dynamic output formation controller; First, given a singular matrix And the reversible model transformation matrix: The nonlinear singular system under the action of the adaptive state feedback formation controller satisfies Depend on Established, derived If (E i ,A i +M 2i C i ) allows, then in, represents the left negative half plane, so, Established; Furthermore, the matrix and (E i ,A i +M 2i C i ) No pulse, based on model transformation get: and Therefore, the matrix No pulse; At the same time, the measurement output of the follower singular multi-agent satisfies: in, therefore, The validity of the proposed method is established, that is, the inclusive control of the follower singular multi-agents under the action of the adaptive state feedback formation controller is realized; And the measured output of the actual leader singular multi-agent system satisfies: in, therefore, The formation control of the actual leader singular multi-agent under the action of the adaptive state feedback formation controller is realized.

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