Method and system for finite-time time-varying formation tracking control of multi-agent system

Abstract

The application discloses a finite time time-varying formation tracking control method and system of a multi-agent system, relates to the time-varying formation tracking control field of an agent system, and comprises the following steps: a dynamic model of a leader and a follower of a heterogeneous nonlinear multi-agent system is established; a distributed finite time state observer of the leader with unknown input is constructed; an adaptive algorithm is used to solve an output control adjustment equation of the follower, so that a feedback control law of the follower is obtained; an additional variable is introduced under the premise that the heterogeneous nonlinear multi-agent system realizes finite time output time-varying formation control under the influence of unknown input of the leader; and a time-varying formation control protocol of the follower is designed based on the state observer, the feedback control law of the follower and the additional variable. The scheme of the application realizes finite time consistency or formation tracking control for the heterogeneous nonlinear multi-agent system.

Description

Technical Field

[0001] This invention belongs to the field of formation tracking control technology for multi-agent systems, specifically involving finite-time time-varying formation tracking control for heterogeneous nonlinear multi-agent systems. Based on a finite-time observer and an adaptive output adjustment method, a novel distributed output time-varying formation tracking controller is proposed. Background Technology

[0002] Formation control of multi-agent systems is currently a research hotspot in the field of control, with significant application prospects in numerous areas. This technology integrates multi-sensor, communication, and cooperative control techniques, offering unique advantages compared to single-agent systems. In practical scenarios such as autonomous vehicles, satellite formation control, marine exploration, and UAV formations, multi-agent formation and tracking control technology is essential for achieving typical tasks. Therefore, in-depth research and widespread application of multi-agent system technology are crucial for promoting development in these fields.

[0003] The main goal of formation control is to design suitable controllers that enable intelligent agents with information interaction capabilities to achieve the required formation according to task requirements. Several classic formation control methods have been proposed and validated in the field of robotics, including behavior-based, virtual structure, and leader-follower formation control. However, these methods all have their own drawbacks, affecting the efficiency of multi-agent systems. For example, with the leader-follower control method, if the leader becomes unable to function due to various factors, the entire formation configuration cannot be maintained.

[0004] With advancements in consensus control theory, researchers have utilized distributed consensus theory to study multi-agent formation control or formation tracking problems due to its robustness, low computational cost, and good stability. Many consensus-based formation control methods exist, each with numerous advantages. Notably, multi-agents not only need to form the desired formation shape but also need to ensure that the formation tracks the leader's state trajectory within a finite timeframe under real-world conditions. However, much current research focuses on fixed-form multi-agent consensus tracking or formation tracking problems, with few studies exploring how to achieve formation control within a finite timeframe. For practical applications such as formation penetration and cooperative detection, the formation of multi-agents may need to change over time. For example, unmanned surface vessels need to rapidly change formation shapes to pass through narrow waterways. Furthermore, many previous studies were based on homogeneous systems, while in the real world, multi-agent systems may consist of various types of intelligent robots. Therefore, exploring formation tracking control in heterogeneous nonlinear multi-agent systems is also essential. Thus, designing suitable time-varying formation tracking controllers for heterogeneous multi-agent systems is of great significance. Summary of the Invention

[0005] The purpose of this invention is to provide a finite-time time-varying formation tracking control method and system for multi-agent systems, aiming to solve the need for formation to change over time in multi-agent systems containing different types of agents in practical applications, namely, finite-time consistency or formation tracking control of heterogeneous multi-agent systems.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] A finite-time time-varying formation tracking control method for a multi-agent system, the method comprising:

[0008] Establish dynamic models of leaders and followers in heterogeneous nonlinear multi-agent systems;

[0009] Based on the dynamic model, a distributed finite-time state observer for a leader with unknown inputs is constructed, and the leader state and leader system matrix are estimated using the state observer.

[0010] Based on the observation results of the state observer, the output control regulation equation of the follower is solved using an adaptive algorithm to obtain the feedback control law of the follower;

[0011] An additional variable is introduced; this additional variable is proposed under the premise of ensuring that the heterogeneous nonlinear multi-agent system can achieve time-varying formation control with finite-time output under the influence of unknown leader input;

[0012] Based on the observation results of the state observer, the feedback control law of the follower, and the additional variables, an adaptive finite-time output time-varying formation control protocol for the follower is designed; the observation results include the leader state and the leader system matrix estimated by the state observer.

[0013] Optionally, the dynamics model includes a follower dynamics model and a leader dynamics model;

[0014] The follower dynamics model includes:

[0015]

[0016] y i (t)=C i x i (t),

[0017] Where i = 1, 2, ..., N; N is the number of followers; x i (t),y i (t),u i (t) represent the state, output, and control input vectors of follower i, respectively; Describe the intrinsic nonlinear dynamic equation of the known i-th follower; A i B is the dynamic matrix of follower i; i C is the control input matrix for follower i; i Let i be the output matrix of the follower. For x i The first derivative of (t).

[0018] The leader dynamics model includes:

[0019]

[0020] y0(t)=Fx0(t)

[0021] Where x0(t) and y0(t) represent the leader's state and output, respectively; D is the leader's dynamic matrix; E is the leader's external control input matrix; and F is the leader's output matrix. It is the first derivative of x0(t).

[0022] Optionally, the expression for the state observer is:

[0023]

[0024] in,

[0025] In the formula, η i (t) represents the estimate of the leader's state x0(t) by the i-th follower at time t; For η i The first derivative of (t); vec -1 This represents the inverse transformation from a vector to a matrix; sign() is the sign function; T represents transpose; γ i >0; ξ i (t) represents the formation tracking neighborhood error; 0 < β < 1; κ1, κ2, κ3, κ4 represent positive constants; 0 < β1 < 1, 0 < β2 < 1, 0 < β3 < 1, 0 < β4 < 1; D0=vec(D), E0=vec(E), F0=vec(F); w ij Let w represent the communication weight between the j-th leader and the i-th follower. If a communication path exists, then w... ij =1, otherwise w ij =0; w i0 D represents the weight between the i-th follower and the leader; D0, E0, and F0 represent the vector forms of the leader's state matrices D, E, and F, respectively; D i (t), Ei (t), F i (t), d i Let D, E, F and the bias d be the estimated values ​​of the state matrix D, E, F and the bias d of the i-th follower, respectively.

[0026] Optionally, the follower's output control adjustment equation includes a first adjustment equation, a second adjustment equation, and an equation for restoring the true parameters of the solution;

[0027] The first adjustment equation is:

[0028]

[0029] Among them, (X) i U i To satisfy the solution of the first regulating equation, X i U is the state feedback gain matrix of follower i. i It is the input feedback gain matrix of follower i;

[0030] The second adjustment equation is:

[0031] 0 = B i R i -X i E

[0032] R i To satisfy the solution of the second regulating equation, R i The feedback gain matrix is ​​the reference signal required for the tracking of follower i;

[0033] The equation that restores the true parameters is:

[0034]

[0035] Among them, 0<κ; 0<φ<1;

[0036]

[0037] Optionally, the expression for the additional variable is:

[0038]

[0039] in, For the extra variable ζ i The first derivative of (t); Indicates coordinate transformation; φ i (t) satisfies h i (t)=C i φ i (t); h i (t) is the output vector obtained by linear transformation of the follower's state vector; K0i This represents the gain matrix to be designed; 0 < β 1i <ν i ;v i (t) represents formation tracking compensation; Q 1i and Q 2i These represent positive definite matrices.

[0040] Optionally, the expression for the adaptive finite-time output time-varying formation control protocol of the follower is:

[0041] u i (t)=K 2i (t)η i (t)+K 1i (x i (t)-φ i (t)-ζ i (t))--d i (t)R i (t)g i (t)+v i (t)+K 0i ζ i (t)-f i (x i (t))

[0042] Among them, K 1i ,K 2i (t) represent the two gain matrices to be designed, K 2i (t)=U i (t)-K 1i X i (t);

[0043] H i (t)=Q 1i B i R i (t).

[0044] The present invention also provides a finite-time time-varying formation tracking control system for a multi-agent system, the system comprising:

[0045] The dynamics model building module is used to build dynamics models of leaders and followers in heterogeneous nonlinear multi-agent systems;

[0046] A state observer construction module is used to construct a distributed finite-time state observer for a leader with unknown inputs based on the dynamic model, and to estimate the leader state and the leader system matrix using the state observer;

[0047] An adaptive solver module is used to solve the output control regulation equation of the follower based on the observation results of the state observer using an adaptive algorithm, so as to obtain the feedback control law of the follower.

[0048] An additional variable definition module is used to introduce an additional variable; the additional variable is proposed under the premise of ensuring that the heterogeneous nonlinear multi-agent system can achieve time-varying formation control with finite-time output under the influence of unknown leader input;

[0049] A time-varying formation control protocol construction module is used to design an adaptive finite-time output time-varying formation control protocol for the follower based on the observation results of the state observer, the feedback control law of the follower, and the additional variables; the observation results include the leader state estimated by the state observer and the leader system matrix.

[0050] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0051] This invention provides a finite-time time-varying formation tracking control method and system for multi-agent systems. A distributed finite-time observer is designed to estimate the unknown inputs of the leader, including the leader's state and system matrix. Compared to existing finite-time observers, the assumption that the leader's system matrix is ​​available to all followers is removed. A novel distributed finite-time output time-varying formation control protocol for heterogeneous nonlinear multi-agent systems is developed, overcoming the challenge of non-full row rank in the follower input matrix and further expanding the application scope of finite-time formation tracking control. An adaptive algorithm is proposed to solve the agent output regulation equation, guaranteeing a solution within a finite time, extending the exponential convergence of existing results to finite-time convergence. Clearly, based on the distributed finite-time observer, the adaptive algorithm for solving the agent output regulation equation, and the time-varying formation control protocol, finite-time consensus or formation tracking control for heterogeneous nonlinear multi-agent systems is achieved. Attached Figure Description

[0052] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 This is the structure of a multi-agent, observer-based, finite-time time-varying formation control system for unknown inputs provided in Embodiment 1 of the present invention.

[0054] Figure 2Here is a flowchart of a finite-time time-varying formation tracking control method for a multi-agent system provided in Embodiment 1 of the present invention;

[0055] Figure 3 This is the communication topology between intelligent agents provided in Embodiment 1 of the present invention;

[0056] Figure 4 This is a snapshot of the heterogeneous nonlinear control system provided in Embodiment 1 of the present invention within 15 seconds under the action of a finite-time time-varying formation controller.

[0057] Figure 5 The formation tracking error curve provided in Embodiment 1 of the present invention;

[0058] Figure 6 This is the formation tracking neighborhood error curve provided in Embodiment 1 of the present invention;

[0059] Figure 7 This is the follower state estimation deviation curve provided in Embodiment 1 of the present invention;

[0060] Figure 8 A snapshot of the output of the multi-agent system provided in Embodiment 1 of the present invention within 7 seconds of the method proposed in the present invention and the classical method;

[0061] Figure 9 Error curves of output formation tracking under two different control protocols provided in Embodiment 1 of the present invention. Detailed Implementation

[0062] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0063] This invention provides a finite-time time-varying formation tracking control method and system for multi-agent systems, aiming to solve the need for formation to change over time in multi-agent systems containing different types of agents in practical applications, namely, finite-time consistency or formation tracking control of heterogeneous multi-agent systems.

[0064] To achieve the above objectives, this invention provides a distributed finite-time formation tracking control protocol based on proximity interaction. The ultimate goal is for followers to track a leader and achieve the desired formation within a finite time. To eliminate the follower's need for leader system matrix information and its unknown control inputs, a finite-time observer is constructed to estimate the leader's state and system matrix, compensating for the influence of unknown inputs. A novel finite-time distributed output time-varying formation controller is proposed by introducing coordinate transformation techniques with additional variables, eliminating the generalized inverse assumption that the follower's input matrix needs to be found in existing results. Using Lyapunov theory, it is proved that the expected finite-time output time-varying formation controller can be implemented in a finite time using the considered heterogeneous nonlinear multi-agent system. Finally, the stability of the proposed control protocol is demonstrated.

[0065] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0066] This embodiment provides a finite-time time-varying formation tracking control method for a multi-agent system. It is a distributed finite-time formation tracking control method based on proximity interaction. The control system design schematic is shown below. Figure 1 As shown, the system designs two novel distributed observers for each follower to estimate the dynamic matrix and state of the leader, whose input is unknown. A novel finite-time time-varying formation controller based on the constructed observers is proposed, utilizing coordinate transformation techniques that introduce additional variables. To determine the parameters in the constructed controller, an adaptive algorithm for solving the agent's output regulation equations is presented.

[0067] like Figure 2 As shown, the method mainly consists of the following parts:

[0068] S1: Establish dynamic models of leaders and followers in heterogeneous nonlinear multi-agent systems.

[0069] For a general heterogeneous nonlinear multi-agent system, the dynamics of the i-th follower are described as follows:

[0070]

[0071] y i (t)=C i x i (t),(1)

[0072] Where i = 1, 2, ..., N; N is the number of followers; These represent the state, output, and control input vectors of follower i, respectively. And rank(B) i ) = mi ; (A i B i ) is stable, and (C) i A i It is detectable; Let A represent the intrinsic nonlinear dynamic equation of the known i-th follower. i The dynamic behavior of the system is defined by the dynamic matrix of follower i; B i For the control input matrix of follower i, we define how the control input affects the dynamic behavior of the system; C i Let i be the output matrix of the follower, defining the relationship between the system's output and its state; For x i The first derivative of (t).

[0073] The dynamic model of the leader is given by the following equation:

[0074]

[0075] in, Let x0(t) represent the leader's state and output respectively, where x0(t) is bounded; D is the leader's dynamic matrix; E is the leader's external control input matrix; and F is the leader's output matrix. Let x0(t) be the first derivative. (D, E) is detectable, and the information of matrices D, E, F can only be derived from followers that include the leader as a neighbor; that is, the information of matrices D, E, F can only be inferred from followers that include the leader as a neighbor. In a multi-agent system, there is usually one leader and some followers. Followers can obtain the leader's state information through communication with the leader, but they cannot directly observe the leader. Therefore, followers can only infer the values ​​of matrices D, E, F through their own state information and the information obtained from the leader's neighbors. Specifically, matrices D, E, F are usually used to describe the topology of the system, including the connections between agents, communication methods, etc. Since followers cannot directly observe the leader, they can only infer the topology of the entire system through their own neighbor information, and thus derive the information of matrices D, E, F.

[0076] This represents the leader's external control inputs, which are continuous and bounded, i.e.

[0077] ||u0(t)|| ∞ <d(3)

[0078] Here, d is a positive integer that can only be received by a few followers whose neighbors are the leader.

[0079] As can be seen from (1) and (2), the above dynamic equations describe a heterogeneous nonlinear multi-agent system model with unknown leader input, where the dynamics of each agent are heterogeneous in terms of nonlinear dynamics, system matrix and state dimension.

[0080] Once the dynamic model is established, all subsequent calculations are performed based on this model.

[0081] S2: Based on the dynamic model, construct a distributed finite-time state observer for the leader with unknown input, and use the state observer to estimate the leader state and the leader system matrix.

[0082] Since the leader's state cannot be applied to all followers, it is necessary to design a state observer by using relative information from neighboring agents to obtain x0(t). However, the difficulty in constructing a distributed state observer lies in the fact that the upper state observer boundary of the leader's system matrix and the unknown control inputs (i.e., D, E, F, d) can only be used for a subset of the followers, which is a common problem in many practical systems. d is the unknown control input of the leader, and u0 is the control input we give to the leader, which is known. The design of the observer aims to solve the problems of unobservable leader state and incompletely known system matrix in order to achieve TVFT.

[0083] To ensure that the heterogeneous nonlinear multi-agent system can achieve time-varying formation control with finite-time output, the following observer is constructed to estimate the state of the leader and the leader system matrix:

[0084]

[0085]

[0086]

[0087]

[0088]

[0089] In the formula, η i (t) represents the estimate of the leader's state x0(t) by the i-th follower at time t; For η i The first derivative of (t); vec -1 This represents the inverse transformation from a vector to a matrix; the underlined parts of Di(t) and Ei(t) indicate that the matrices Di(t) and Ei(t) are converted into vector form for easier calculation and processing; sign() is the sign function; T represents transpose; γ i>0; 0 < β < 1; β is a constant between 0 and 1, which is the growth exponent used to adjust the growth rate of the function to achieve faster convergence and smaller observation error; κ1, κ2, κ3, κ4 represent positive constants; 0 < β1 < 1, 0 < β2 < 1, 0 < β3 < 1, 0 < β4 < 1; D0=vec(D), E0=vec(E), F0=vec(F); w ij Let w represent the communication weight between the j-th leader and the i-th follower. If a communication path exists, then w... ij =1, otherwise w ij =0; w i0 D represents the weight between the i-th follower and the leader; D0, E0, and F0 represent the vector forms of the leader's state matrices D, E, and F, respectively; D i (t), E i (t), F i (t), d i Let ξ represent the estimated values ​​of the state matrices D, E, F, and bias d of the i-th follower, respectively. i (t) represents the error in forming the tracking neighborhood, which is a vector; ξ i The formal description of (t) is as follows:

[0090]

[0091] D i (t),E i (t),F i (t),d i (t) is constructed to estimate D0, E0, F0, d; (the parameter with index i represents the state variable in the i-th observer, and the parameter with index 0 is the unknown parameter in the system, that is, the system matrix of the leader to be estimated, which needs to be estimated through the constructed observer (the parameter with index i).

[0092] The following are the estimated errors for each observer, used to evaluate the effectiveness of the observations.

[0093] make Then, according to formulas (2) and (4), we have

[0094]

[0095] in

[0096] make Then system (10) can be obtained as follows:

[0097]

[0098] in γ = diag{γ1,γ2,…,γ N}

[0099] Formula (4) is the observer expression, Formula (10) is the estimation error of each observer, used to evaluate the effect of the observation, and Formula (11) fuses the observation results of each observer to make the optimal estimate.

[0100] Topology diagram of a multi-agent system If there exists a spanning tree rooted at the leader, then the topology among the followers is undirected. Therefore, for a graph... We can derive the following Laplace matrix L:

[0101]

[0102] Based on the above provisions, D can be verified. i (t),E i (t),F i (t) and d i (t) can estimate D0, E0, F0, d at a fixed time T1, and the distributed observer (4) can estimate D0, E0, F0, d at a finite time t. * Converging to the leader state, i.e. This verifies D. i (t),E i (t),F i (t) and d i (t) can estimate D0, E0, F0, d within a fixed time T1, and η i (t) can converge to a finite time t. * (0<t * The state of the leader (i.e., x0(t)) with unknown control inputs.

[0103] S3: Based on the observation results of the state observer, the output control regulation equation of the follower is solved using an adaptive algorithm to obtain the feedback control law of the follower.

[0104] To overcome the difficulty of coupling between the state dimension and the system matrix, the following criteria are proposed for the system under consideration.

[0105] For follower i, we can find a solution (X) that satisfies the following regulator equation (the first regulator equation). i U i ):

[0106]

[0107] For follower i, we can always find a solution R.i The following regulator equation (second regulation equation) is satisfied:

[0108] 0 = B i R i -X i E,(13)

[0109] Among them, 0<κ; 0<φ<1;

[0110] Equations (12) and (13) are based on the assumptions made in the case of output regulation control of a multi-agent system. In this case, the goal is to design a control strategy that enables the follower's output to asymptotically track the desired reference signal under unknown disturbances and system uncertainties.

[0111] Regulation equations (12) and (13) are the equations that need to be solved to achieve output regulation control. Solution (X) i U i ) and R i Let X represent the feedback control law of the i-th follower, where X i It is the state feedback gain matrix, U i It is the input feedback gain matrix, R i It is the reference signal feedback gain matrix.

[0112] The solvability condition of the regulating equation is based on the rank of the matrix being equal to n. i +p, this matrix depends on the system matrix A i B i and C i And the unknown matrix D. However, since the follower does not know matrix D, it cannot directly check the solvability condition.

[0113] To address this issue, an adaptive method is employed, whereby the online estimator uses available measurement data to estimate the unknown matrix D. Based on these estimates, the follower can adaptively solve the regulation equations to obtain the desired feedback control law. Therefore, an adaptive algorithm is used to estimate the unknown matrix D in real time and solve the regulation equations (12) and (13), enabling the follower to achieve output regulation control under system uncertainty.

[0114] The above provisions are two necessary conditions for the output regulation and control of the system under consideration. If If all hold true, then regulator equations (12) and (13) are respectively Solvable in Let D represent the spectrum of D. However, matrices D, E, and F are not available for all followers. That is, the regulator equation (12) cannot be directly examined. Inspired by adaptive methods and previous work, matrix D is estimated. i(t),E i (t),F i (t) can be adaptively used to solve equations (12) and (13).

[0115] Therefore, under the premises of (12) and (13), for any initial value The following equation (the equation that restores the true parameters) holds:

[0116]

[0117] in,

[0118] These two matrices are used to construct the state estimation error of follower i. of.

[0119] In the first matrix, is n i ×n i The identity matrix represents the state matrix X of follower i. i The first n i The first component can be accurately estimated. The last two components of the matrix are both zero, representing the state matrix X of follower i. i The components following the equation cannot be accurately estimated. The purpose of this matrix is ​​to improve the accuracy of state estimation by only considering the components that can be accurately estimated and ignoring the components that cannot be accurately estimated when estimating the state of follower i.

[0120] In the second matrix, A i B i C i It is the state matrix of the i-th follower, n i m i Let X represent the state dimension and input dimension of the i-th follower, respectively. This matrix is ​​used to estimate the state matrix X of follower i. i U i R i The estimation error is linked to the estimation errors of the leader's state matrices E and F, thus achieving distributed state estimation. Specifically, this matrix links the estimation errors of the follower i's state matrix X... i The estimation error is related to the estimation error of the leader's state matrix E, and the input matrix U of the follower i is used. i The estimation error is related to the estimation error of the leader's state matrix F, and the perturbation matrix R of the follower i is also considered. i The estimation error of the leader is related to the estimation error of the leader's state matrix B. Thus, the state estimation of the followers can be achieved through the leader's state estimation error.

[0121] It is the transpose of the state matrix D of the leader node; This represents the Kronecker product.

[0122] There is a unique bounded solution so In a finite time t * It meets the system standards.

[0123] → indicates that the variable or expression on the left is transformed into the variable or expression on the right. In this formula, It is the estimated value of the state vector of the i-th follower. This represents the state vector X of follower i. i Input vector U i and perturbation vector R i Arrange them in column vector form, and then transform them into a single column vector. Therefore, The meaning is to estimate the state vector of the i-th follower. This is transformed into a column vector, where the elements of the column vector are the state vector X of follower i. i Input vector U i and perturbation vector R i Element.

[0124] In addition, if in Then it is assumed that in a finite time t * Inside

[0125] and Let X and Y represent the state vectors of follower i respectively. i Input vector U i and perturbation vector R i The estimated value.

[0126] This represents the estimated value of the state vector of follower i. Transform into the real state vector X i .

[0127] This represents the estimated value of the input vector of follower i. Transform into the real input vector U i .

[0128] This represents the estimated value of the perturbation vector of follower i. Transform into the real perturbation vector R iThe purpose of these transformations is to convert the estimation errors of the follower's state vector, input vector, and perturbation vector into the true state vector, input vector, and perturbation vector in distributed state estimation, thereby improving the accuracy of state estimation.

[0129] Formula (14) is derived based on the assumptions of formulas (12) and (13). The proof steps will be introduced later. In general, it utilizes vectors. The structure is decomposed into matrix X. i U i and R i The vectorized representation of these matrices. Therefore, by applying the inverse vectorization operation vec^{-1}, the values ​​of these matrices can be recovered. Ultimately, Equation 14 provides a method for retrieving values ​​from vectors. Recovering the true parameter X i U i and R i The method.

[0130] Proof of Formula 14: Regulator equation (12) can be expressed as:

[0131]

[0132] Transform the equation above:

[0133] Π i θ i =b i ,

[0134] in,

[0135]

[0136]

[0137] According to formulas (12) and (13), for follower i, the first equation has a solution pair (X). i U i ,R i ).therefore, Based on previous work (J. Huang, “Nonlinear Output Regulation: Theory and Applications,” Philadelphia, PA, USA: SIAM, 2004), equation (14) has a unique bounded solution. satisfy In a finite time t * Inside.

[0138] S4: Introduce an additional variable; the additional variable is proposed under the premise of ensuring that the heterogeneous nonlinear multi-agent system can achieve time-varying formation control with finite-time output under the influence of unknown input from the leader.

[0139] To ensure that the system under consideration can achieve time-varying formation control with finite-time output under the influence of unknown leader input, an additional variable ζ is introduced. i (t)(i=1,2,…,N) is as follows:

[0140]

[0141] in, For the extra variable ζ i The first derivative of (t); Indicates coordinate transformation, Satisfy h i (t)=C i φ i (t) and K 0i Q represents the gain matrix to be designed; 1i and Q 2i Let β and β represent positive definite matrices, respectively, where 0 < β 1i <ν i .

[0142] h i (t) is the output vector obtained by linearly transforming the state vector of the follower, C i It is a coefficient matrix, φ i (t) is a vector to be solved for, used to satisfy h i (t)=C i φ i (t).

[0143] ν i It is a control parameter in formation tracking compensation, used to control the convergence speed of state error. When ν i The larger the value, the faster the convergence speed of the state error.

[0144] S5: Based on the observation results of the state observer, the feedback control law of the follower, and the additional variables, design the adaptive finite-time output time-varying formation control protocol of the follower; the observation results include the leader state estimated by the state observer and the leader system matrix.

[0145] Based on the distributed adaptive observer (4) and additional variables (15), an adaptive finite-time output time-varying formation control protocol is designed as follows:

[0146]

[0147] in, The formation tracking compensation is determined by the desired time-varying formation information, K. 1i ,K 2i (t) represent the two gain matrices to be designed, K 2i (t)=U i (t)-K 1i X i (t); g i (t) The design is as follows to suppress the influence of unknown input from the leader:

[0148]

[0149] and H i (t)=Q 1i B i R i (t).

[0150] definition According to formulas (1)(2) and (15)(16), we have

[0151]

[0152] For the i-th follower, the finite-time output time-varying formation control protocol (16) can be determined by the following steps:

[0153] (1): Consider the vector h i (t) specifies the time-varying formation. For a given h i (t)=C i φ i φ of (t) i (t), check the following finite-time output time-varying formation feasibility condition (19):

[0154]

[0155] If the feasibility condition is met, continue; otherwise, the required φ cannot be found under the time-varying formation control protocol (16) with finite-time output. i (t) and stop.

[0156] (2): v represents the formation tracking compensation. i (t) is given by the following formula:

[0157]

[0158] Where v i (t) is defined in the finite-time output control protocol (16).

[0159] (3): K 1i K 0i Designed as A respectivelyi +B i K 1i and A i +B i K 0i Solve for the positive definite matrix under the following conditions.

[0160]

[0161] This ensures that the time-varying formation with finite-time output can be realized by the system under consideration (1) under the constructed adaptive controller (16).

[0162] The following is a simulation example of the method provided in this implementation.

[0163] Consider a heterogeneous nonlinear multi-agent system consisting of five agents. The communication topology between the agents is as follows: Figure 3 As shown. According to Figure 3 Based on the interactive topology diagram shown and the definition in equation (9), we can obtain w ik (k = 0, 1, ..., 4). The dynamics of followers and leaders are described by multi-agent systems (1) and (2), where

[0164] The leader's control input is u0(t) = [0.1cos(4t), 0.1sin(4t)] T Furthermore, the follower's output is expected to enable time-varying formation tracking, given by the following formula:

[0165]

[0166] Where i = 1, 2, 3, 4.

[0167] Based on the design steps of a finite-time output time-varying formation controller, there exists φ i (t)(i=1,2,3,4) satisfies the feasible condition (19), and φ i (t) can be expressed by φ1(t)=[3cos(2t),3sin(2t)] T φ2(t)=[3cos(2t+π / 2),-6sin(2t+π / 2),3sin(2t+π / 2),6cos(2t+π / 2)] T ,φ3(t)=[3cos(2t+π),-6sin(2t+π),3sin(2t+π),6cos(2t+π)] T,φ4(t)=[3cos(2t+3π / 2),-6sin(2t+3π / 2),-12cos(2t+3π / 2),3sin(2t+3π / 2),6cos(2t+3π / 2),-12sin(2t+3π / 2)] T We obtain v1(t) = [-6sin(2t), 6cos(2t)] based on the formation tracking compensation (20). T ,v2(t)=[-12cos(2t+π / 2),-12sin(2t+π / 2)] T ,v3(t)=[-6sin(2t+π)-9cos(2t+π),-9sin(2t+π)+6cos(2t+π / 2)] T ,v4(t)=[0,12sin(2t+3π / 2)+6cos(2t+3π / 2),0,6sin(2t+3π / 2)-12cos(2t+3π / 2)] T .

[0168] The gain matrix K can be obtained using MATLAB. 1i (i = 1, 2, 3, 4), where K 01 =K 11 =[-3 -6], Q 11 =Q 21 =2I2,

[0169] In the time-varying formation control protocol (16) with finite-time output. i (t),x0(t),η i (t),ζ i The initial value of (t) is generated by a random number t between -2 and 2.

[0170] Case 1: The nonlinear system is described as follows:

[0171]

[0172]

[0173]

[0174]

[0175] According to step S5, the time-varying formation of the finite-time output is implemented by the considered system under the constructed adaptive controller (16). Figure 4-7 The simulation results are shown.

[0176] Case 2: If f in the multi-agent system (1) i (x i If (t))=0, then the method proposed in this invention can also be applied to study finite-time output time-varying formation tracking of heterogeneous linear multi-agent systems. To better demonstrate the advantages of this method, a comparative simulation is performed with the classic observer-based time-varying formation control method, where the time-varying formation controller is described in the following form:

[0177] u i (t)=K 2i η i (t)+K 1i (x i (t)-φ i (t))+v i (t)

[0178] To ensure that factors other than the control algorithm remain the same, and disregarding the effects of nonlinear functions and unknown leader inputs, our finite-time time-varying formation tracking controller is reconstructed into the following form:

[0179] u i (t)=K 2i (t)η i (t)+K 1i (x i (t)-φ i (t)-ζ i (t))+v i (t)+K 0i ζ i (t)

[0180] Figure 8 The agent's output snapshots over 7 seconds are represented using both the method proposed in this invention and the classical method, where the four followers are represented by squares and the leader by rhombuses. The tracking error of the two output formations is ||e yi The process of change of (t)||(i=1,2,3,4) is as follows Figure 9 As shown.

[0181] The advantages of this invention are:

[0182] (1) A distributed finite-time observer is designed to estimate the unknown inputs of the leader and the system matrix under influence. Compared with existing finite-time observers, the assumption that the leader's system matrix is ​​available to all followers is removed.

[0183] (2) A new distributed finite-time output time-varying formation control protocol for heterogeneous nonlinear multi-agent systems was developed, which overcomes the challenge of the follower input matrix not being full row rank and further expands the application scope of finite-time formation tracking control.

[0184] (3) An adaptive algorithm is proposed to solve the regulation equation of the agent output. The proposed algorithm guarantees that the solution of the equation can be obtained in a finite time, and extends the exponential convergence of the existing results to finite time convergence.

[0185] Example 2

[0186] This embodiment provides a finite-time time-varying formation tracking control system for a multi-agent system, the system comprising:

[0187] The dynamics model building module is used to build dynamics models of leaders and followers in heterogeneous nonlinear multi-agent systems.

[0188] A state observer construction module is used to construct a distributed finite-time state observer for a leader with unknown inputs based on the dynamic model, and to estimate the leader state and the leader system matrix using the state observer.

[0189] An adaptive solver module is used to solve the output control regulation equation of the follower based on the observation results of the state observer, and to obtain the feedback control law of the follower.

[0190] An additional variable definition module is used to introduce an additional variable; the additional variable is proposed under the premise of ensuring that the heterogeneous nonlinear multi-agent system can achieve time-varying formation control with finite-time output under the influence of unknown input from the leader.

[0191] A time-varying formation control protocol construction module is used to design an adaptive finite-time output time-varying formation control protocol for the follower based on the observation results of the state observer, the feedback control law of the follower, and the additional variables; the observation results include the leader state estimated by the state observer and the leader system matrix.

[0192] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple; relevant parts can be referred to the method section.

[0193] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for finite-time time-varying formation tracking control of multi-agent systems, characterized in that, The method comprises: establishing a dynamic model of a leader and a follower of a heterogeneous nonlinear multi-agent system; based on the dynamic model, constructing a distributed finite-time state observer of the leader with unknown input, and estimating the leader state and the leader system matrix by using the state observer; based on the observation result of the state observer, solving the output control adjustment equation of the follower by using an adaptive algorithm to obtain a feedback control law of the follower; an additional variable is introduced; the additional variable is proposed under the premise of guaranteeing that the heterogeneous nonlinear multi-agent system realizes finite-time output time-varying formation control under the influence of unknown input of the leader; based on the observation result of the state observer, the feedback control law of the follower and the additional variable, an adaptive finite-time output time-varying formation control protocol of the follower is designed; the observation result includes the leader state and the leader system matrix estimated by the state observer.

2. The method of claim 1, wherein, The dynamic model comprises a follower dynamic model and a leader dynamic model; The follower dynamic model comprises: y i (t) = C i x i (t), where i = 1, 2,..., N; N is the number of followers; x i (t), y i (t), u i (t) are the state, output and control input vectors of the i-th follower, respectively; represents the known intrinsic nonlinear dynamics of the i-th follower; A i is the dynamic matrix of the i-th follower; B i is the control input matrix of the i-th follower; C i is the output matrix of the i-th follower; is the first order derivative of x i (t). The leader dynamic model comprises: y0(t)=Fx0(t) wherein x0(t), y0(t) represent the state and output of the leader respectively; D is the dynamic matrix of the leader; E is the external control input matrix of the leader; F is the output matrix of the leader; is the first order derivative of x0(t).

3. The method of claim 2, wherein, The expression of the state observer is: wherein where η i (t) represents the estimation of the leader state x0(t) by the i-th follower at time t; is the first order derivative of η i (t); vec -1 represents the inverse transformation from vector to matrix; sign() is the sign function; T represents the transpose; γ i > 0; ξ i (t) represents the formation tracking neighborhood error; 0 < β < 1; κ1, κ2, κ3, κ4 represent normal numbers respectively; 0 < β1< 1, 0 < β2< 1, 0 < β3< 1, 0 < β4< 1; D0 = vec(D), E0 = vec(E), F0 = vec(F); w ij represents the communication weight between the j-th leader and the i-th follower, w ij = 1 if there is a communication path, otherwise w ij = 0; w i0 represents the weight between the i-th follower and the leader; D0, E0, F0 represent the vector form of the state matrix D, E, F of the leader respectively; D i (t), E i (t), F i (t), d i represent the estimation of the state matrix D, E, F and the bias d of the i-th follower respectively.

4. The method of claim 3, wherein, The output control adjustment equation of the follower comprises a first adjustment equation, a second adjustment equation and an equation for restoring true parameters of a solution; The first adjustment equation is: where (X i , U i ) is the solution to the first regulation equation, Xi is the state feedback gain matrix of the follower i, U i is the input feedback gain matrix of the follower i; The second adjustment equation is: 0 = B i R i -X i E R i To satisfy the solution of the second regulation equation, R i Feedback gain matrix for the reference signal required for tracking of the follower i; The equation for restoring true parameters of a solution is: wherein 0 < K; 0 < φ < 1; 5. The method of claim 4, wherein, The expression of the additional variable is: wherein is an additional variable ζ i the first derivative of h denotes a coordinate transformation; φ i h i (t) = C i φ i (t); h i (t) is an output vector obtained by linear transformation of the state vector of the follower; K 0i denotes a gain matrix to be designed; 0 < β 1i < v i ; v i (t) denotes a platoon tracking compensation; Q 1i and Q 2i denote positive definite matrices, respectively.

6. The method of claim 5, wherein, The expression of the adaptive finite-time output time-varying formation control protocol of the follower is: u i (t) = K 2i (t) η i (t) + K 1i (x i (t) - φ i (t) - ζ i (t) - d i (t) R i (t) g i (t) + v i (t) + K 0i ζ i (t) - f i (x i (t) where K 1i , 2i (t) are two gain matrices to be designed, respectively, 2i (t) = U i (t) - K 1i X i (t); H i (t) = Q 1i B i R i (t).

7. A finite-time time-varying formation tracking control system for a multi-agent system, characterized in that, The system comprises: a dynamic model construction module, configured to establish a dynamic model of a leader and a follower of a heterogeneous nonlinear multi-agent system; a state observer construction module, configured to, based on the dynamic model, construct a distributed finite-time state observer of the leader with unknown input, and estimate the leader state and the leader system matrix by using the state observer; an adaptive solving module, configured to, based on an observation result of the state observer, solve an output control adjustment equation of the follower by using an adaptive algorithm to obtain a feedback control law of the follower; an additional variable definition module, configured to introduce an additional variable; the additional variable is proposed under the premise of guaranteeing that the heterogeneous nonlinear multi-agent system realizes finite-time output time-varying formation control under the influence of unknown input of the leader; a time-varying formation control protocol construction module, configured to, based on the observation result of the state observer, the feedback control law of the follower and the additional variable, design an adaptive finite-time output time-varying formation control protocol of the follower; the observation result includes the leader state and the leader system matrix estimated by the state observer.

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