Method for realizing time-varying formation trajectory tracking of high-order all-drive discrete multi-agent system based on LQR

Abstract

The invention discloses a method for realizing distributed time-varying formation trajectory tracking by a high-order all-drive discrete multi-agent system based on an LQR, and belongs to the field of multi-agent system formation control. The method is technically characterized by comprising the steps that a high-order all-drive mathematical model of a discrete multi-agent system is established, a feedback controller is designed, and a high-order all-drive model of a closed-loop multi-agent system is obtained; constructing a high-order all-drive model of a time-varying formation trajectory tracking error system, and proposing a linear quadratic performance index; proposing a design method of the suboptimal distributed controller; and according to the suboptimal distributed controller, a control vector expression introduced by a follower is obtained, and sufficient conditions for realizing time-varying formation trajectory tracking of the high-order all-drive multi-agent system are given. According to the method, the influence of nonlinearity on formation control of the multi-agent system is overcome, and the formation control target can be realized on the premise of minimum energy consumption.

Description

[0001] The present invention relates to the field of high-order all-wheel drive multi-agent system and formation control, and in particular to a time-varying formation trajectory tracking control method using distributed LQR. Background Art

[0002] As the most popular research direction in the field of multi-agent systems, the formation control problem has received increasing attention in recent years due to its prospects for use in various application fields and the challenging theoretical problems involved. Its control goal is generally to make the state of each agent satisfy certain specified constraints.

[0003] The advantage of the all-wheel drive model is that its all-wheel drive characteristics facilitate the elimination of known nonlinear terms in the system, thereby converting nonlinear problems into linear problems to a certain extent. In addition, considering the energy loss problem of the multi-agent system in the control process, the linear quadratic regulator (LQR) technology can enable the multi-agent system to achieve the required control objectives with minimal energy consumption, avoiding a large amount of resource waste. Therefore, the study of using the all-wheel drive system model to describe the nonlinear multi-agent system, minimize the LQR performance index of the formation trajectory tracking error, and achieve the time-varying formation trajectory tracking control target has important theoretical significance and practical application value. Summary of the invention

[0004] The present invention proposes a method for realizing time-varying formation trajectory tracking of a high-order all-wheel drive discrete multi-agent system based on LQR. By establishing a mathematical model of the high-order all-wheel drive multi-agent system and designing a suboptimal distributed controller, the time-varying formation trajectory tracking problem of the high-order all-wheel drive discrete multi-agent system based on LQR is solved.

[0005] The present invention proposes a method for realizing time-varying formation trajectory tracking by a high-order all-wheel drive discrete multi-agent system based on LQR, and the method comprises the following steps:

[0006] Step 1: Establish a high-order all-wheel drive mathematical model of the discrete multi-agent system, design a feedback controller, and obtain a high-order all-wheel drive model of the closed-loop multi-agent system;

[0007] Step 2: Construct a high-order all-wheel drive model of the time-varying formation trajectory tracking error system and propose a linear quadratic performance index;

[0008] Step 3: Propose a design method for a suboptimal distributed controller;

[0009] Step 4: Based on the suboptimal distributed controller, the control vector expression introduced by the follower is obtained, and sufficient conditions for realizing time-varying formation trajectory tracking of a high-order all-wheel drive multi-agent system are given.

[0010] Furthermore, the high-order full-drive mathematical model of the discrete multi-agent system in step 1 is:

[0011] According to the all-wheel drive system method, the all-wheel drive model of N followers is:

[0012]

[0013] in, and is the state vector and control input vector of the ith follower, and are respectively a known continuous nonlinear vector function and a known continuous matrix function, They are m-dimensional vector space and m×m-dimensional Euclidean space respectively;

[0014] The leader's all-wheel drive models are:

[0015]

[0016] in, and are the state vector and control input vector of the leader, and are respectively a known continuous nonlinear vector function and a known continuous matrix function;

[0017] The feedback controller is expressed as follows:

[0018]

[0019] Among them, A 0～n-1 =[A0 A1…A n-1 ], A i (i=0,1,...,n-1) is the parameter matrix, v i (k) is the control vector introduced by the follower, v r (k) is the control vector known to the leader, i=1,2,...,N is the formation offset vector, and They are matrix functions and The inverse matrix function of

[0020] Therefore, the closed-loop high-order all-wheel drive model of the follower is:

[0021]

[0022] in,

[0023]

[0024] The leader’s closed-loop high-level all-wheel drive model is:

[0025]

[0026] Furthermore, the high-order all-wheel drive model and linear quadratic performance index of the time-varying formation trajectory tracking error system described in step 2 are:

[0027] make is the time-varying formation trajectory tracking error of the ith follower, then the high-order all-wheel drive model of the time-varying formation trajectory tracking error system is:

[0028]

[0029] in,

[0030]

[0031]

[0032]

[0033] For the time-varying formation trajectory control problem of high-order all-wheel drive multi-agent systems, the following linear quadratic performance indicators are proposed:

[0034]

[0035] Among them, the positive array Zhengding array And the matrices Q1 and R are known positive definite matrices, and the matrix Q2 is a known semi-positive definite matrix, It is the communication topology of the multi-agent system The Laplace matrix of .

[0036] Furthermore, the suboptimal distributed controller in step 3 is:

[0037] According to optimal control theory, for a given positive definite matrix and positive definite matrices Suboptimal Distributed Controller Minimizing Quadratic Performance Index J The expression is:

[0038]

[0039]

[0040] in, is the algebraic Riccati equation The symmetric positive definite solution of is the algebraic Riccati equation The correct solution of The off-diagonal elements of Communication topology diagram The maximum in-degree of Has the following structure:

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047] θ is the parameter to be designed, Communication topology diagram The adjacency matrix of .

[0048] Further, step 4 is based on the suboptimal distributed controller described in step 3 The expression of the control vector introduced by the follower is:

[0049]

[0050] When θ satisfies the following conditions

[0051]

[0052]

[0053]

[0054]

[0055]

[0056] hour, The characteristic roots of are all within the unit circle, where σ min (·) and σ max (·) are the minimum and maximum eigenvalues ​​of the matrix, respectively. Therefore, the time-varying formation trajectory tracking error system is in the suboptimal distributed controller Therefore, the control input v(k) is applied to the high-order all-drive discrete multi-agent system to achieve The specified time-varying formation trajectory is tracked while minimizing the quadratic performance indicator described in step 2.

[0057] The beneficial technical effects of the present invention are:

[0058] The present invention studies the time-varying formation trajectory tracking control problem of a high-order all-wheel drive discrete multi-agent system based on LQR. On the one hand, the all-wheel drive model is used to eliminate the known nonlinear terms in the system, thereby converting the nonlinear optimal control problem into a linear optimal control problem to a certain extent, thereby simplifying the difficulty of solving. On the other hand, the use of LQR technology can enable the multi-agent system to achieve the desired control objectives with minimal energy consumption, reducing the energy loss in the control process. With the help of mathematical tools such as graph theory and matrix theory, a suboptimal distributed controller is constructed based on the local algebraic Riccati equation, so that the high-order all-wheel drive discrete multi-agent system can achieve the desired time-varying formation trajectory tracking. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] The above and other objects, features and advantages of the exemplary embodiments of the present invention will become readily understood by reading the detailed description below with reference to the accompanying drawings. In the accompanying drawings, several embodiments of the present invention are shown in an exemplary and non-limiting manner, in which:

[0060] Figure 1 A flow chart of a method for realizing time-varying formation trajectory tracking by a high-order all-wheel drive discrete multi-agent system based on LQR according to the present invention;

[0061] Figure 2 It is a communication topology diagram of a high-order all-drive discrete multi-agent system in an embodiment of the present invention;

[0062] Figure 3 This is an example diagram of the state trajectory of a leader in an embodiment of the present invention;

[0063] Figure 4 This is an example diagram of realizing time-varying formation trajectory tracking control of a high-order all-wheel drive discrete multi-agent system under the action of LQR in an embodiment of the present invention;

[0064] Figure 5 This is an example diagram of the formation trajectory tracking error of the intelligent agent in the x direction in an embodiment of the present invention;

[0065] Figure 6 This is an example diagram of the formation trajectory tracking error of the intelligent agent in the y direction in an embodiment of the present invention; DETAILED DESCRIPTION

[0066] The principles and spirit of the present invention will be described below with reference to several exemplary embodiments. It should be understood that these embodiments are provided only to enable those skilled in the art to better understand and implement the present invention, and are not intended to limit the scope of the present invention in any way. On the contrary, these embodiments are provided to make the present disclosure more thorough and complete, and to fully convey the scope of the present disclosure to those skilled in the art.

[0067] Those skilled in the art know that the embodiments of the present invention can be implemented as a system, device, equipment, method or computer program product. Therefore, the present disclosure can be specifically implemented in the following forms, namely: complete hardware, complete software (including firmware, resident software, microcode, etc.), or a combination of hardware and software. In this article, it is to be understood that any number of elements in the drawings is for illustration and not limitation, and any naming is only for distinction and does not have any limiting meaning.

[0068] In order to overcome the influence of nonlinear terms on optimal control and minimize the energy loss of the multi-agent system in the process of time-varying formation trajectory tracking, this paper proposes a method for high-order all-wheel drive discrete multi-agent system to realize time-varying formation trajectory tracking based on LQR. Figure 1 As shown, the method specifically comprises the following steps:

[0069] Step 1: Establish a high-order all-wheel drive mathematical model of the discrete multi-agent system, design a feedback controller, and obtain a high-order all-wheel drive model of the closed-loop multi-agent system;

[0070] Step 2: Construct a high-order all-wheel drive model of the time-varying formation trajectory tracking error system and propose a linear quadratic performance index;

[0071] Step 3: Propose a design method for a suboptimal distributed controller;

[0072] Step 4: Based on the suboptimal distributed controller, the control vector expression introduced by the follower is obtained, and sufficient conditions for realizing time-varying formation trajectory tracking of a high-order all-wheel drive multi-agent system are given.

[0073] The present invention is described in detail below.

[0074] In step one, a high-order all-wheel drive mathematical model of a discrete multi-agent system is established, and a feedback controller is designed to obtain a high-order all-wheel drive model of a closed-loop multi-agent system.

[0075] According to the embodiment of the present invention, the intelligent agents are regarded as the vertices of the graph and a communication topology graph is defined. To describe the communication relationship between N agents, is the vertex set of the agent, For edge sets, is a non-negative weighted adjacency matrix. The directed edge (i, j)∈ε from node i to node j indicates that the jth agent can receive information from the ith agent through the network. The adjacent element a associated with the edge (i, j) ji is positive, that is For any node i, if (i, j) ∈ ε, then node j is called a neighbor node of node i; the set of all neighbor nodes of node i is denoted as Laplacian matrix of weighted graph Defined as

[0076] According to the all-wheel drive system method, the all-wheel drive model of N followers is:

[0077]

[0078] in, and is the state vector and control input vector of the ith follower, and are respectively a known continuous nonlinear vector function and a known continuous matrix function, They are m-dimensional vector space and m×m-dimensional Euclidean space respectively;

[0079] The leader's all-wheel drive models are:

[0080]

[0081] in, and are the state vector and control input vector of the leader, and are respectively a known continuous nonlinear vector function and a known continuous matrix function;

[0082] The feedback controller is expressed as follows:

[0083]

[0084] Among them, A 0～n-1 =[A0 A1…A n-1 ], A i (i=0,1,...,n-1) is the parameter matrix, v i (k) is the control vector introduced by the follower, v r (k) is the control vector known to the leader, i=1,2,...,N is the formation offset vector, and They are matrix functions and The inverse matrix function of

[0085] Therefore, the closed-loop high-order all-wheel drive model of the follower is:

[0086]

[0087] in,

[0088]

[0089] The leader’s closed-loop high-level all-wheel drive model is:

[0090]

[0091] In step 2, a high-order all-wheel drive model of the time-varying formation trajectory tracking error system is constructed, and a linear quadratic performance indicator is proposed.

[0092] For a high-order all-wheel drive discrete multi-agent system, if the following conditions are met, the high-order all-wheel drive discrete multi-agent system can be realized by Specified time-varying formation trajectory tracking control.

[0093]

[0094] make is the time-varying formation trajectory tracking error of the ith follower, then the high-order all-wheel drive model of the time-varying formation trajectory tracking error system is:

[0095]

[0096] in,

[0097]

[0098]

[0099]

[0100] For the time-varying formation trajectory control problem of high-order all-wheel drive multi-agent systems, the following linear quadratic performance indicators are proposed:

[0101]

[0102] Among them, the positive array Zhengding array And the matrices Q1 and R are known positive definite matrices, and the matrix Q2 is a known semi-positive definite matrix, It is the communication topology of the multi-agent system The Laplace matrix of .

[0103] In step three, a design method for a suboptimal distributed controller is proposed.

[0104] According to optimal control theory, for a given positive definite matrix and positive definite matrices Suboptimal Distributed Controller Minimizing Quadratic Performance Index J The expression is:

[0105]

[0106]

[0107] in, is the algebraic Riccati equation The symmetric positive definite solution of is the algebraic Riccati equation The correct solution of The off-diagonal elements of Communication topology diagram The maximum in-degree of Has the following structure:

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114] θ is the parameter to be designed, Communication topology diagram The adjacency matrix of .

[0115] Furthermore, the control vector expression introduced by the follower in step 4 is:

[0116]

[0117] When θ satisfies the following conditions

[0118]

[0119]

[0120]

[0121]

[0122]

[0123] hour, The characteristic roots of are all within the unit circle, where σ min (·) and σ max (·) are the minimum and maximum eigenvalues ​​of the matrix, respectively. Therefore, the time-varying formation trajectory tracking error system is in the suboptimal distributed controller Therefore, the control input v(k) is applied to the high-order all-drive discrete multi-agent system to achieve The specified time-varying formation trajectory is tracked while minimizing the quadratic performance indicator described in step 2.

[0124] The technical effect of the present invention is further verified through simulation experiments.

[0125] Consider that four followers (N=4) track the reference trajectory of the leader and form the desired time-varying formation and n=2. At this time, the simplified system matrices of the leader and followers are as follows:

[0126]

[0127] like Figure 2 Communication topology diagram The corresponding adjacency matrix and Laplace matrix are:

[0128]

[0129] The initial states of the four followers are:

[0130]

[0131] The time-varying formation vector is designed as follows:

[0132]

[0133] The control inputs to the leader are:

[0134]

[0135] The matrix R = I2 and Has the following form:

[0136]

[0137] in,

[0138]

[0139] and is calculated based on two low-dimensional algebraic Riccati equations, and Available

[0140]

[0141] The distributed controller gain is:

[0142]

[0143] in, And set Therefore, the designed LQR-based distributed control method can realize the time-varying formation trajectory tracking control of high-order all-wheel drive discrete multi-agent systems.

[0144] The experimental results are as follows Figure 3-Figure 6 As shown, Figure 3 Figure 4 are the position trajectory of the leader and the position trajectory of all agents, which illustrates that the high-order all-wheel drive discrete multi-agent system can realize time-varying formation trajectory tracking control under the designed distributed LQR control. Figure 5 and Figure 6 is the state error trajectory of the agent, which shows that the state error trajectory of the agent will eventually tend to zero over time.

Claims

1. A method for tracking time-varying formation trajectories by a high-order all-wheel-drive discrete multi-agent system based on LQR, characterized in that: The following steps are involved: Step 1: Establish a high-order all-wheel drive mathematical model of the discrete multi-agent system, design a feedback controller, and obtain a high-order all-wheel drive model of the closed-loop multi-agent system; Step 2: Construct a high-order all-wheel drive model of the time-varying formation trajectory tracking error system and propose a linear quadratic performance index; Step 3: Propose a design method for a suboptimal distributed controller; Step 4: Based on the suboptimal distributed controller, the control vector expression introduced by the follower is obtained, and sufficient conditions for realizing time-varying formation trajectory tracking of a high-order all-wheel drive multi-agent system are given.

2. According to the method of establishing a high-order all-wheel drive discrete mathematical model of a multi-agent system in claim 1, a feedback controller is designed to obtain a high-order all-wheel drive model of a closed-loop multi-agent system, characterized in that: The step one is: According to the all-wheel drive system method, the all-wheel drive model of N followers is: in, and is the state vector and control input vector of the ith follower, and are respectively a known continuous nonlinear vector function and a known continuous matrix function, They are m-dimensional vector space and m×m-dimensional Euclidean space respectively; The all-wheel drive models of the leaders are: in, and are the state vector and control input vector of the leader, and are respectively a known continuous nonlinear vector function and a known continuous matrix function; The feedback controller is expressed as follows: Among them, A 0～n-1 =[A0 A1…A n-1 ], A i (i=0,1,...,n-1) is the parameter matrix, v i (k) is the control vector introduced by the follower, v r (k) is the control vector known to the leader, is the formation offset vector, and They are matrix functions and The inverse matrix function of Therefore, the closed-loop high-order all-wheel drive model of the follower is: in, The leader’s closed-loop high-level all-wheel drive model is:

3. According to the high-order all-wheel drive model of the time-varying formation trajectory tracking error system of claim 2, a linear quadratic performance index is proposed, characterized in that: The second step is: make is the time-varying formation trajectory tracking error of the ith follower, then the high-order all-wheel drive model of the time-varying formation trajectory tracking error system is: in, For the time-varying formation trajectory control problem of high-order all-wheel drive multi-agent systems, the following linear quadratic performance indicators are proposed: Among them, the positive array Zhengding array And the matrices Q1 and R are known positive definite matrices, and the matrix Q2 is a known semi-positive definite matrix, It is the communication topology of the multi-agent system The Laplace matrix of .

4. The method for designing a suboptimal distributed controller according to claim 3, characterized in that: The step three is: According to optimal control theory, for a given positive definite matrix and positive definite matrices Suboptimal Distributed Controller Minimizing Quadratic Performance Index J The expression is: in, is the algebraic Riccati equation The symmetric positive definite solution of is the algebraic Riccati equation The correct solution of The off-diagonal elements of Communication topology diagram The maximum in-degree of Has the following structure: θ is the parameter to be designed, Communication topology diagram The adjacency matrix of .

5. According to claim 4, the control vector expression introduced by the follower obtained according to the suboptimal distributed controller provides sufficient conditions for realizing the time-varying formation trajectory tracking of the high-order all-wheel drive multi-agent system, characterized in that: The step four is: When θ satisfies the following conditions hour, The characteristic roots of are all inside the unit circle, where σ min (·) and σ max (·) are the minimum and maximum eigenvalues ​​of the matrix, respectively. Therefore, the time-varying formation trajectory tracking error system is in the suboptimal distributed controller Therefore, the control input v(k) is applied to the high-order all-drive discrete multi-agent system to achieve The specified time-varying formation trajectory is tracked while minimizing the quadratic performance indicator described in step 2.

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