Heterogeneous singular multi-agent system adaptive inclusion formation control method

Abstract

The self-adaptive inclusion formation control method for the heterogeneous singular multi-agent system comprises the following steps: firstly, constructing the heterogeneous singular multi-agent system, and describing inclusion formation control of the heterogeneous singular multi-agent system; designing an adaptive state feedback formation controller and an adaptive dynamic output formation controller; secondly, converting a formation control problem into an infinite norm constraint problem by designing state transformation of a closed-loop system under the action of an adaptive state feedback formation controller; then, in combination with pulse-free analysis and stability analysis, a heterogeneous singular multi-agent system inclusion formation control method under the action of the adaptive state feedback formation controller is obtained; and finally, obtaining the heterogeneous singular multi-agent system including formation control method according to the state transformation of the closed-loop system under the action of the self-adaptive dynamic output formation controller. According to the method, the formation control of the heterogeneous singular multi-agent system is realized, and the virtual leader matrix information can be quickly estimated.

Description

Technical Field

[0001] This invention belongs to the field of formation control technology for multi-agent systems, specifically relating to an adaptive inclusion formation control method for heterogeneous singular multi-agent systems. Background Technology

[0002] In recent years, with the rapid development of computing and communication technologies, distributed cooperative control of multi-agent systems has been extensively studied. Among these, consensus control is a fundamental research direction in multi-agent distributed cooperative control, encompassing formation control, cooperative output control, and swarm control. As a branch of consensus control, formation control typically involves multiple leader agents. Research on multi-leader formation control is based on many practical application scenarios. For example, a group of robots equipped with sensors can detect dangerous obstacles and maintain other malfunctioning robots within a safe area until the fault is discovered and repaired. Currently, research on multi-leader formation control often assumes that the leader agents do not communicate with each other and do not possess collective behavior, which is not valid in many complex practical applications.

[0003] In some practical applications, leader agents need to achieve control objectives through information exchange. Therefore, the inclusion formation control problem has been proposed, where all leader agents form a formation, and follower agents are surrounded by leader agents, thus achieving inclusion formation control. In inclusion formation control of multi-agent systems, obtaining the virtual leader system matrix information is a strict information transmission condition and increases the communication cost between agents. Therefore, how to design an adaptive controller to estimate the leader matrix information to reduce the communication cost between agents is a problem that needs to be solved.

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

[0005] Currently, research on formation control in heterogeneous singular multi-agent systems (MAS) largely focuses on normal MAS, neglecting MAS with algebraic constraints in their dynamic models. In practical applications, the system matrix of the virtual leader agent is often unknown, and the relative state variables are difficult to obtain. Therefore, formation control in MAS with adaptive leader matrix information estimation is a pressing problem that needs to be solved.

[0006] Xuxi Zhang, Xianping Liu, Zhiguang Feng et al. published "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.) They provided sufficient conditions for containment control of heterogeneous singular multi-agent systems. However, because the distributed containment controller designed in this paper relies on the state transmission of the heterogeneous singular multi-agent system, it has the disadvantage of high measurement and transmission costs in 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 multiagent systems. However, this paper did not consider the differences between the matrices of different agent systems and the singular characteristics of multiagent systems, and could not describe complex heterogeneous singular multiagent systems with differences between agents. Summary of the Invention

[0008] To overcome the shortcomings of the prior art, the present invention aims to provide an adaptive grouping control method for heterogeneous singular multi-agent systems. This method achieves grouping control of heterogeneous singular multi-agent systems by designing an adaptive state output grouping controller that is independent of system state information transmission. It features low measurement and transmission costs and can accurately describe the differences between agents.

[0009] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[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, design an adaptive state feedback formation controller to obtain a closed-loop coupled system under the action of the adaptive state feedback formation controller; at the same time, design an adaptive dynamic output formation controller that is completely independent of the state information transmission of the heterogeneous singular multi-agent system constructed in Step 1 to obtain a closed-loop coupled system under the action of the adaptive dynamic output formation controller.

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

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

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

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

[0017] Establish a heterogeneous singular multi-agent system comprising follower agents and actual leader agents:

[0018]

[0019] Where i = 1, 2, ..., N represents the follower agent, denoted as When i = N+1, N+2, ..., N+M, it indicates that the leader agent is labeled as... variable It refers to the state and variables of a singular multi-agent system. It is the measurement output of a singular multi-agent system, and the control input of the i-th agent is... Matrix E i Satisfying matrix constraints rank(E) i ) = r i <n i E i It is a singular matrix; where, and It is a given system parameter matrix;

[0020] Establish a heterogeneous singular multi-agent system that includes a virtual leader agent:

[0021]

[0022] Among them, variables This refers to the state of the virtual leader intelligent agent. It is the measurement output of the virtual leader intelligent agent.

[0023] The method for describing the formation control problem in step 1 is as follows:

[0024] When controller u exists i (t), causing the controller u i The closed-loop system under the action of (t) has no pulse and satisfies:

[0025]

[0026] Where, dist(x,F)=inf y∈F ||xy||2 represents the distance from x to set F, lim denotes the limit, inf denotes the infimum of the set, and ||·|| denotes the norm of the matrix. This solves the problem of formation control in heterogeneous singular multi-agent systems.

[0027] In step 2, based on the heterogeneous singular multi-agent system, the specific method for designing an adaptive state feedback formation controller to obtain the closed-loop coupled system under the action of the adaptive state feedback formation controller is as follows:

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

[0029]

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

[0031] Step 2.2, combining the heterogeneous singular multi-agent system described above and the adaptive state feedback formation controller designed in Step 2.1, yields the closed-loop coupled system under the action of the adaptive state feedback formation controller as follows:

[0032]

[0033] in,

[0034] The specific method for designing an adaptive dynamic output formation controller that is completely independent of the state information transmission of the heterogeneous singular multi-agent system in step 2, and obtaining the closed-loop coupled 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 system state information transmission.

[0036]

[0037] Where, υ1<0, It is an estimate of the matrix A0 of the virtual leader intelligent agent system. The state of the distributed controller. For state estimation of both follower and actual leader agents, K 1i K is the gain matrix to be designed. 2i (t) is the time-varying gain matrix to be designed;

[0038] Step 2.4, combining the heterogeneous singular multi-agent system described above and the adaptive dynamic output formation controller designed in Step 2.3, yields the closed-loop coupled system under the action of the adaptive dynamic output formation controller 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, and obtaining the nonlinear singular system under the action of the adaptive state feedback formation controller, is as follows:

[0042] Step 3.1, based on the closed-loop coupled system under the action of the adaptive state feedback formation controller, design the state transition:

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

[0044]

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

[0046]

[0047] in,

[0048] It is the Laplacian matrix corresponding to the follower agent. It is the Laplacian matrix corresponding to the actual follower agent, where 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] Step 3.2, combining the state transformations of the heterogeneous singular multi-agent system and the closed-loop coupled system under the action of the adaptive state feedback formation controller given in Step 3.1, the nonlinear singular system under the action of the adaptive state feedback formation controller is obtained as follows:

[0052]

[0053] in,

[0054]

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

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

[0057]

[0058] The formation control problem described above is then solved under the action of the adaptive state feedback formation controller, wherein,

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

[0060] Based on this, construct linear singular systems. And design Lyapunov functions. Then the Lyapunov function along the linear singular system The derivative is:

[0061]

[0062] in,

[0063] when When, the derivative Established;

[0064] Furthermore, matrix pairs satisfy

[0065]

[0066] Then, matrix pair Since there is no pulse, φ is stable;

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

[0068]

[0069] in, and These are the Laplacian matrices corresponding to the actual leader agent and the follower agents, respectively; therefore, they contain control constraints. Satisfaction;

[0070] Meanwhile, the measured output of the actual leader agent satisfies

[0071]

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

[0073] further, The proof is as follows

[0074]

[0075] Where ρ(·) represents the spectral radius of the matrix, max i This means taking the maximum value among all i. Therefore, 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] The specific steps in step 4 of performing impulse-free analysis and stability analysis on the nonlinear singular system under the action of the adaptive state feedback formation controller to obtain the heterogeneous singular multi-agent system including the formation control results are as follows:

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

[0078] because It is true, obtained through calculation. Right now Established, among which, It is a Laplacian matrix. For the connection weight matrix, This indicates that the i-th intelligent agent can receive information from the virtual leader; otherwise, d i =0, in addition, matrix The following inequalities must be 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 Both are true, ||·|| ∞ The infinity norm of a matrix;

[0084] First, based on the bounded real lemma of the system, when At the time of its establishment, Satisfaction;

[0085] Furthermore, for any Equation (21) is obtained:

[0086]

[0087] this means in addition, It is always true, therefore, we get

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

[0089]

[0090] in, Therefore, formation control in heterogeneous singular multi-agent systems is realized under the action of an adaptive state feedback formation controller.

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

[0092] Additionally, according to step 4.1 and in step 4.2 get:

[0093]

[0094] Therefore, heterogeneous singular multi-agent systems can achieve formation control under the action of an adaptive state feedback formation controller.

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

[0096] Based on the closed-loop coupled system under the action of the adaptive dynamic output formation controller, design the state transition:

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

[0098]

[0099] in,

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

[0101] Based on 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] The specific steps in step 4 of performing impulse-free and stability analysis on the nonlinear singular system under the action of the adaptive dynamic output formation controller to obtain the heterogeneous singular multi-agent system including the formation control results are as follows:

[0105] For nonlinear singular systems under adaptive dynamic output formation controllers, parameters If the matrix pair (E) i A i +M 2i C i No pulse, and inequalities If this is established, then the formation control problem is solved under the action of the adaptive dynamic output formation controller;

[0106] First, given a singular matrix Transformation matrix of invertible model:

[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, deduced If (E) i A i +M 2i C i If permitted, then in, Representing the left negative half-plane, therefore, Established;

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

[0111]

[0112] Therefore, matrix pairs No pulse;

[0113] Meanwhile, the measurement outputs of the follower singular multi-agent satisfy:

[0114]

[0115] in, therefore, The establishment of this means that the inclusion control of the follower singular multi-agent 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 singular multi-agent leader under the action of the adaptive state feedback formation controller is realized.

[0120] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0121] 1. This invention employs a heterogeneous singular multi-agent system, which enables an accurate description of actual heterogeneous singular multi-agent systems with different singular systems. It includes normal multi-agent systems as a special case, and therefore has a wider range of applications.

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

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

[0124] In summary, compared with existing technologies, this invention employs a heterogeneous singular multi-agent system, which can more accurately describe complex real-world singular multi-agent models. Furthermore, the adaptive dynamic output formation controller designed in this invention does not require measuring and transmitting the state variables of the singular multi-agent system, making it easier to implement than traditional state-dependent controllers. Moreover, the formation controller designed in this invention achieves formation control even when the virtual leader matrix information is unknown, effectively reducing the communication cost between the virtual leader agent and the actual leader agent. Attached Figure Description

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

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

[0127] Figure 3(a) shows 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) shows 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 The figure in the middle is the error state trajectory diagram of the simulation experiment of this invention under the action of the adaptive dynamic output formation controller.

[0129] Figure 5(a) shows the output state trajectory of all singular intelligent agent systems under the action of the adaptive dynamic output formation controller in the simulation experiment from 0s to 6s; Figure 5(b) shows the output state trajectory of all singular intelligent agent systems under the action of the adaptive dynamic output formation controller in the simulation experiment from 30s to 50s. Detailed Implementation

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

[0131] like Figure 1As shown, the adaptive formation control method for heterogeneous singular multi-agent systems 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, design an adaptive state feedback formation controller to obtain a closed-loop coupled system under the action of the adaptive state feedback formation controller; at the same time, design an adaptive dynamic output formation controller that is completely independent of the state information transmission of the heterogeneous singular multi-agent system constructed in Step 1 to obtain a closed-loop coupled system under the action of the adaptive dynamic output formation controller.

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

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

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

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

[0138] Establish a heterogeneous singular multi-agent system comprising follower agents and actual leader agents:

[0139]

[0140] Where i = 1, 2, ..., N represents the follower agent, denoted as When i = N+1, N+2, ..., N+M, it indicates that the leader agent is labeled as... variable It refers to the state and variables of a singular multi-agent system. It is the measurement output of a singular multi-agent system, and the control input of the i-th agent is... Matrix E i Satisfying matrix constraints rank(E) i ) = r i <n i E i It is a singular matrix; where, and It is a given system parameter matrix;

[0141] Establish a heterogeneous singular multi-agent system that includes a virtual leader agent:

[0142]

[0143] Among them, variables This refers to the state of the virtual leader intelligent agent. It is the measurement output of the virtual leader intelligent agent.

[0144] The method for describing the formation control problem in step 1 is as follows:

[0145] When controller u exists i (t), causing the controller u i The closed-loop system under the action of (t) has no pulse and satisfies:

[0146]

[0147] Where, dist(x,F)=inf y∈F ||xy||2 represents the distance from x to set F, lim denotes the limit, inf denotes the infimum of the set, and ||·|| denotes the norm of the matrix. This solves the problem of formation control in heterogeneous singular multi-agent systems.

[0148] In step 2, based on the heterogeneous singular multi-agent system, the specific method for designing an adaptive state feedback formation controller to obtain the closed-loop coupled system under the action of the adaptive state feedback formation controller is as follows:

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

[0150]

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

[0152] Step 2.2, combining the heterogeneous singular multi-agent system described above and the adaptive state feedback formation controller designed in Step 2.1, yields the closed-loop coupled system under the action of the adaptive state feedback formation controller as follows:

[0153]

[0154] in,

[0155] The specific method for designing an adaptive dynamic output formation controller that is completely independent of the state information transmission of the heterogeneous singular multi-agent system in step 2, and obtaining the closed-loop coupled 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 system state information transmission.

[0157]

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

[0159] Step 2.4, combining the heterogeneous singular multi-agent system described above and the adaptive dynamic output formation controller designed in Step 2.3, yields the closed-loop coupled system under the action of the adaptive dynamic output formation controller 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, and obtaining the nonlinear singular system under the action of the adaptive state feedback formation controller, is as follows:

[0163] Step 3.1, based on the closed-loop coupled system under the action of the adaptive state feedback formation controller, design the state transition:

[0164] Feedback control matrix K 2i (t) satisfies K 2i (t)=Ui (t)-K 1i Γ i (t), and in, The solution to equation (9) is:

[0165]

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

[0167]

[0168] in,

[0169] It is the Laplacian matrix corresponding to the follower agent. It is the Laplacian matrix corresponding to the actual follower agent, where 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] Step 3.2, combining the state transformations of the heterogeneous singular multi-agent system and the closed-loop coupled system under the action of the adaptive state feedback formation controller given in Step 3.1, the nonlinear singular system under the action of the adaptive state feedback formation controller is obtained as follows:

[0173]

[0174] in,

[0175]

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

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

[0178]

[0179] The formation control problem described above is then solved under the action of the adaptive state feedback formation controller, wherein...

[0180]

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

[0182] Based on this, construct linear singular systems. And design Lyapunov functions. Then the Lyapunov function along the linear singular system The derivative is:

[0183]

[0184] in,

[0185] when When, the derivative Established;

[0186] Furthermore, matrix pairs satisfy

[0187]

[0188] Then, matrix pair Since there is no pulse, φ is stable;

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

[0190]

[0191] in, and These are the Laplacian matrices corresponding to the actual leader agent and the follower agents, respectively; therefore, they contain control constraints. Satisfaction;

[0192] Meanwhile, the measured output of the actual leader agent satisfies

[0193]

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

[0195] further, The proof is as follows

[0196]

[0197] Where ρ(·) represents the spectral radius of the matrix, max i This means taking the maximum value among all i. Therefore, 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] The specific steps in step 4 of performing impulse-free analysis and stability analysis on the nonlinear singular system under the action of the adaptive state feedback formation controller to obtain the heterogeneous singular multi-agent system including the formation control results are as follows:

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

[0200] because It is true, obtained through calculation. Right now Established, among which, It is a Laplacian matrix. For the connection weight matrix, d i >0 indicates that the i-th intelligent agent can receive information from the virtual leader; otherwise, d i =0, in addition, matrix The following inequalities must be 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 Both are true, ||·|| ∞ The infinity norm of a matrix;

[0206] First, based on the bounded real lemma of the system, when At the time of its establishment, Satisfaction;

[0207] Furthermore, for any Equation (21) is obtained:

[0208]

[0209] this means in addition, It is always true, therefore, we get

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

[0211]

[0212] in, Therefore, formation control in heterogeneous singular multi-agent systems is realized under the action of an adaptive state feedback formation controller.

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

[0214] Additionally, according to step 4.1 and in step 4.2 get:

[0215]

[0216] Therefore, heterogeneous singular multi-agent systems can achieve formation control under the action of an adaptive state feedback formation controller.

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

[0218] Based on the closed-loop coupled system under the action of the adaptive dynamic output formation controller, design the state transition:

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

[0220]

[0221] in,

[0222]

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

[0224] Based on 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] The specific steps in step 4 of performing impulse-free and stability analysis on the nonlinear singular system under the action of the adaptive dynamic output formation controller to obtain the heterogeneous singular multi-agent system including the formation control results are as follows:

[0229] For nonlinear singular systems under adaptive dynamic output formation controllers, parameters If the matrix pair (E) i A i +M 2i C i No pulse, and inequalities If this is established, then the formation control problem is solved under the action of the adaptive dynamic output formation controller;

[0230] First, given a singular matrix Transformation matrix of invertible model:

[0231]

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

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

[0234]

[0235] Therefore, matrix pairs No pulse;

[0236] Meanwhile, the measurement outputs of the follower singular multi-agent satisfy:

[0237]

[0238] in,

[0239]

[0240] therefore, The establishment of this means that the inclusion control of the follower singular multi-agent 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 singular multi-agent leader under the action of the adaptive state feedback formation controller is realized.

[0245] Simulation Experiment

[0246] To verify the effects of formation control, this invention utilizes MATLAB for simulation verification. The matrix parameters of the heterogeneous singular multi-agent system are:

[0247]

[0248] The sets of follower agents and actual leader agents are respectively represented as follows: and The connection weight matrix between heterogeneous singular multi-agents is For all actual leader intelligent agents Assume its corresponding formation function is if If this holds true, then the singular follower multi-agent containment control is established, that is... Therefore, an error vector is defined. The settling time of the error vector e is consistent with the time at which the heterogeneous singular multi-agent system, including formation control, is formed. An adaptive state feedback formation controller is employed, and the error vector... The state trajectory in Figure 2 It was displayed in the middle. Based on... Figure 2 Simulation results show that 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 the convex combination of the measurement outputs of the actual leader singular multi-agents. The trajectories of the measurement outputs of all heterogeneous singular multi-agent systems from 0 to 6 s are shown in Figure 3(a); the trajectories of the measurement outputs of all heterogeneous singular multi-agent systems from 30 s to 50 s are shown in Figure 3(b). From Figure 3(b), it can be seen that with 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 the convex combination of the actual leader agents.

[0250] An adaptive dynamic output formation controller is used, with an error vector... The state trajectory in Figure 4 It was displayed in the middle. By Figure 4 Simulation results show that error vectors e1, e2, and e3 stabilize at approximately 30 seconds. Under the action of the adaptive dynamic output formation controller, the trajectories of the measurement outputs of all heterogeneous singular multi-agent systems from 0 to 6 seconds are shown in Figure 5(a); the trajectories of the measurement outputs of all heterogeneous singular multi-agent systems from 30 to 50 seconds are shown in Figure 5(b). Figure 5(b) provides a specific demonstration of formation control under the action of 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 follower agents are driven into the convex combination of the actual leader agents.

[0251] As can be seen from the above simulation experiments, compared with the prior art, the present invention considers the inclusion of formation control in heterogeneous singular multi-agent systems, that is, the parameter matrices of all agent systems in equation (30) are inconsistent. This heterogeneous singular multi-agent system can more accurately describe complex real systems than homogeneous singular multi-agent systems. At the same time, the present invention considers singular multi-agent systems, that is, the system matrix E i If i = 1, ..., 7 is not full rank, then when E iWhen I = 1, the system of the present invention degenerates into a normal multi-agent system, that is, the singular multi-agent system considered in the present invention includes a normal multi-agent system as a special case; in addition, the adaptive controller designed in the present invention implements 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; furthermore, the dynamic output formation controller designed in the present invention only relies on the transmission of measured output without the measurement and transmission of the system state, which is easier to implement than the controller based on state transmission.

Claims

1. A method for adaptive formation control of heterogeneous singular multi-agent systems, characterized in that: The following steps are involved: Step 1, constructing 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 describing 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 that is completely independent of the state information transmission of the heterogeneous singular multi-agent system constructed in step 1 is designed to obtain a closed-loop coupling system under the action of the adaptive dynamic output formation controller; Step 3, by designing the state transformation of the closed-loop coupling 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, a heterogeneous singular multi-agent system including formation control results is obtained; At the same time, by designing the state transformation of the closed-loop coupling system under the action of the adaptive dynamic output formation controller in step 2, the 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, and the variable is the measured output of the singular multi-agent system, and the control input of the ith 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 infimum 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; 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 of obtaining the closed-loop coupling system under the action of the adaptive dynamic output formation controller is: 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, is the state estimate of the follower agent and the actual leader agent, K 1i is the gain matrix to be designed, K 2i (t) is the time-varying gain matrix to be designed; 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 coupling 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 coupling 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: Step 3.2, combining the heterogeneous singular multi-agent system and the state transformation of the closed-loop coupling system under the adaptive state feedback formation controller given in step 3.1, the nonlinear singular system under the adaptive state feedback formation controller is obtained as follows: in, 7. The adaptive formation control method for heterogeneous singular multi-agent systems according to claim 1, characterized in that: The specific method of converting the formation control problem into the infinite norm constraint problem of the nonlinear singular system under the action of the adaptive state feedback controller in step 3 is: 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 infinite 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 of 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 It means taking the maximum value among all i. Therefore, 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 nonlinear singular system under the action of 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, and 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 established, be satisfied; Furthermore, for any We get formula (21): this means in addition, Always holds true, so 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 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 of 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 coupling 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 coupling 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 coupling 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 action of the adaptive dynamic output formation controller is subjected to pulse-free and stability analysis to obtain the heterogeneous singular multi-agent system including the formation control result, and 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 is established, 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 adaptive state feedback formation controller satisfies Depend on Established, derived If (E i ,A i +M 2i C i ) allows, then in, represents the negative left 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, It holds true, that is, the inclusive control of the follower singular multi-agent 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, It holds true, that is, 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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