A fixed-time consensus control method under matching and mismatching disturbances

Abstract

The application relates to the technical field of multi-agent control, and particularly discloses a fixed-time consensus control method under matched and unmatched disturbances, which comprises the following steps: constructing a multi-agent system dynamics model, establishing a communication network topology structure, designing a fixed-time convergent non-recursive disturbance observer, estimating the matched disturbance and the unmatched disturbance; constructing a fixed-time controller based on the output result of the disturbance observer; and driving the follower to track the leader state within a fixed time based on the fixed-time controller, wherein the fixed time is irrelevant to the initial state of the system. Through the design of the fixed-time observer and the controller, a control method based on distributed error is established, the proposed method gives a unified control framework, can be applied to a first-order, second-order and high-order disturbed multi-agent system, and enables the system to reach convergence within a convergence time independent of the initial value.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of multi-agent control, and particularly relates to a fixed-time consensus control method under matched and unmatched disturbances. BACKGROUND

[0002] Consensus control of multi-agent systems (MAS) is one of the core problems in the field of distributed cooperative control. The consensus control of high-order multi-agent systems has become a research hotspot due to its closer proximity to actual engineering needs (such as robot formation and unmanned aerial vehicle cluster). Existing research mainly focuses on the following directions:

[0003] For the basic theory of consensus control of high-order systems, early research has mainly focused on linear high-order systems, and consensus is achieved by designing a distributed protocol based on neighbor state error. For example, the method based on sliding mode control constructs a nonlinear sliding mode surface to ensure that the system state converges to the preset trajectory in a finite time. For nonlinear high-order systems, backstepping and dynamic surface control (DSC) are widely used, but these methods usually rely on the accurate known system model and have insufficient robustness to disturbances.

[0004] For disturbance processing strategies, in practical applications, multi-agent systems are often affected by external disturbances (such as environmental noise and communication delay) and internal disturbances (such as model uncertainty). For matched disturbances (which can be directly counteracted by control input), sliding mode control and disturbance observer (DO) are the mainstream methods; for unmatched disturbances (which cannot be directly counteracted by control input), adaptive control and neural network approximation techniques are used to compensate for unknown disturbances. However, existing methods often only handle a single type of disturbance, and there is little research on simultaneously dealing with matched and unmatched disturbances.

[0005] For the development of fixed-time consensus control, fixed-time control, as an improvement of finite-time control, has a convergence time that only depends on the design parameters and not on the initial state, and has stronger robustness. In recent years, the application of fixed-time control in multi-agent systems has gradually increased, such as event-triggered fixed-time control protocol and distributed fixed-time observer. However, the fixed-time control of high-order systems still faces challenges, especially when there are both matched and unmatched disturbances. The existing methods have high conservatism, and the convergence time estimation depends on the complex construction of Lyapunov functions.

[0006] For communication constraints and robustness enhancement, there are often time delays, packet loss and bandwidth limitations and other non-ideal factors in actual communication networks. Event-triggered mechanism, quantized communication and prediction compensation strategy are introduced to improve communication efficiency, but high-order systems are more sensitive to delay, and quantization error may affect the convergence accuracy. For example, although the event-triggered method proposed by Ni Junkang team can reduce the communication frequency, the selection of the trigger threshold needs to balance the convergence speed and stability.

[0007] Most of the existing consensus control methods for multi-agent systems with disturbance only consider matched disturbance, and ignore the influence of mismatched disturbance on the control system. Although some methods focus on control design with mismatched disturbance, there are problems such as simple system, strong disturbance assumption, and convergence time dependent on initial value of the system. SUMMARY

[0008] The purpose of the present application is to provide a fixed-time consensus control method under matched and mismatched disturbance to solve the problems raised in the background art.

[0009] To achieve the above purpose, the present application provides the following technical scheme: a fixed-time consensus control method under matched and mismatched disturbance, the method comprising:

[0010] Obtaining state information of a leader and a plurality of followers, and constructing a multi-agent system dynamics model based on the state information of the leader and the plurality of followers;

[0011] Establishing a communication network topology by taking the leader and the plurality of followers in the multi-agent system as communication nodes;

[0012] Designing a fixed-time convergence non-recursive disturbance observer to estimate the matched disturbance and the mismatched disturbance;

[0013] Constructing a fixed-time controller based on the output results of the disturbance observer;

[0014] Driving the followers to track the state of the leader within a fixed time based on the fixed-time controller, and the fixed time is independent of the initial state of the system.

[0015] As a further scheme of the present application, an n-order multi-agent system is composed of N followers and a leader, and the leader dynamics model in the multi-agent system dynamics model is:

[0016] ,

[0017] ,

[0018]

[0019] ;

[0020] in, Let u0 be the state vector, and u0 be the control input of the leader. and These represent mismatch interference and matching interference from leaders, respectively.

[0021] The dynamics model of the i-th follower is:

[0022] ,

[0023] ,

[0024]

[0025] ;

[0026] in, i = 1, 2, ..., N, X i Let be the state vector of the i-th node. This is the control input for the i-th node. and These represent the mismatched interference and matched interference of the system, respectively, and it is assumed that... , , , i = 1, 2, ..., N , and It is a known positive constant.

[0027] As a further embodiment of the present invention, the interference observer includes:

[0028] For an observer with mismatched interference, it is represented as:

[0029] ;

[0030] For an observer with matched interference, it is represented as:

[0031] ;

[0032] in, , ,..., , , They are respectively for , ,..., , , The observed estimated variables, For the state variables of the agent, 、 、 and are parameters of the observer, and satisfy the following selection conditions respectively:

[0033] satisfy the recursive relation , and for a sufficiently small ;

[0034] satisfy the recursive relation , for a sufficiently small ;

[0035] ;

[0036] and , .

[0037] As a further scheme of the present application, there is a fixed time T1>0, such that for any t>T1, it is made that , ,..., , , , another observer can be constructed in the same way, and the observation estimation variable in the observer converges within a fixed time T2, when t>T1+T2, the n-order dynamic system of the leader and the follower is:

[0038] ;

[0039] ;

[0040] wherein, ;

[0041] ;

[0042] , is an agent state variable, is an agent state variable after being transformed by the observer, and for the leader state and and the transformation of the leader state variable and is the same as above, , , i=1, 2,..., N.

[0043] As a further scheme of the present application, the construction process of the fixed time controller is as follows:

[0044] ;

[0045] ;

[0046] ;

[0047] ;

[0048] ;

[0049] wherein, is the control input of the agent, is a partial component variable of the agent control input, is a distributed observer state variable error based on an observer construction, is an integral sliding mode, is a partial component variable of the integral sliding mode, 0 < p < 1, q > 0, normal number can simultaneously make the polynomials and are both Hurwitz polynomials with respect to the Laplace operator s, i = 1, 2,..., N, constant , parameters and satisfy the following definitions: , .

[0050] As a further scheme of the present application, it further includes stability analysis of the fixed time convergence based on Lyapunov function, and determines the fixed time.

[0051] As a further scheme of the present application, the augmented graph of the network topology contains a spanning tree with the leader as the root node.

[0052] Compared with the prior art, the beneficial effects of the present application are: through the design of the fixed time observer and the controller, a control method based on distributed error is established, the proposed method gives a unified control framework, which can be applied to the first-order, second-order and high-order disturbed multi-agent system, so that the system can reach convergence within the convergence time without relying on the initial value. BRIEF DESCRIPTION OF DRAWINGS

[0053] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description are only some embodiments of the present application.

[0054] Figure 1 A flow chart of a fixed time consensus control method under matching and non-matching interference provided by the embodiments of the present application is shown in the figure.

[0055] Figure 2 A fixed-time consensus control method flow architecture diagram under matching and mismatching interference is provided for an embodiment of the present application.

[0056] Figure 3 A network topology structure of a multi-agent system is provided for an embodiment of the present application.

[0057] Figure 4 A multi-agent system position trajectory diagram is provided for an embodiment of the present application.

[0058] Figure 5 A multi-agent system error curve diagram is provided for an embodiment of the present application.

[0059] Figure 6 A multi-agent system control input diagram is provided for an embodiment of the present application. DETAILED DESCRIPTION

[0060] In order to make the technical problems to be solved by the present application, technical solutions and beneficial effects clearer, the present application will be further described in detail below in combination with the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and do not limit the present application.

[0061] As shown in Figure 1 and Figure 2 , in an embodiment of the present application, a fixed-time consensus control method under matching and mismatching interference, the method comprises:

[0062] Step S100, obtaining state information of a leader and a plurality of followers, and constructing a multi-agent system dynamics model based on the state information of the leader and the plurality of followers;

[0063] Step S200, establishing a communication network topology structure by taking the leader and the plurality of followers in the multi-agent system as communication nodes;

[0064] Step S300, designing a fixed-time convergent non-recursive disturbance observer to estimate the matching disturbance and the mismatching disturbance;

[0065] Step S400, constructing a fixed-time controller based on the output result of the disturbance observer;

[0066] Step S500, driving the followers to track the leader state within a fixed time based on the fixed-time controller, and the fixed time is irrelevant to the initial state of the system.

[0067] As a preferred embodiment of the present application, an n-order multi-agent system is composed of N followers and a leader, and the leader dynamics model in the multi-agent system dynamics model is:

[0068] ,

[0069] ,

[0070]

[0071] ;

[0072] wherein, is the state vector of the i-th follower, u0is the control input of the leader, u0is known, and denote the mismatched disturbance and the matched disturbance of the leader, respectively;

[0073] The i-th follower is described by the n-th order dynamics:

[0074] ,

[0075] ,

[0076]

[0077] ;

[0078] wherein, is the state vector of the i-th node, i is the control input of the i-th node, is the state vector of the i-th node, and denote the mismatched disturbance and the matched disturbance of the system, respectively, and satisfy the assumption , , , , i = 1, 2,..., N, , and are known positive constants.

[0079] Embodiments of the present application aim to propose a control so that the multi-agent system can reach state consensus within a fixed time, i.e., for any agent satisfies the formula:

[0080] ;

[0081] wherein is the output state of the agent, is the output state of the leader, and T is the upper bound of the convergence time that can be obtained.

[0082] As a preferred embodiment of the present application, the augmented graph of the network topology contains a spanning tree rooted at the leader.

[0083] In the present embodiment, there is a graph used to represent the network topology among followers in the multi-agent system. The vertex set represents each node of the follower, represents the signal transmission relationship between each follower, means that there is an arc from node i to node j, and the neighbor of node i is defined as For the topology of the weighted graph, it is usually represented by the adjacency matrix , where a ij > 0 when ij (i.e., follower i can receive signals from follower j), otherwise a = 0.

[0084] Assume that there is no self-loop in the network topology of the system, and the network topology does not change over time (i.e., a ij = 0 and A is a constant matrix). Define as the weighted in-degree of node i and define the diagonal matrix . Define the Laplacian matrix of the graph .

[0085] Define the augmented graph of the graph , where the vertex set and the edge set . The directed graph is used to represent the network topology of the entire multi-agent system containing the leader. Define the matrix , where b i > 0 if the i-th node can receive signals from the leader (i.e., ).

[0086] The augmented graph of the network topology of the entire multi-agent system contains a spanning tree rooted at the leader, i.e., the signal of the leader can traverse the entire multi-agent system.

[0087] Symbolically represented as: , where a > 0, x ∈ R, is the standard symbol function.

[0088] As a preferred embodiment of the present application, a method is proposed for estimating the unmatched disturbance and the matched disturbance fixed-time convergent non-recursive observer, and is designed as:

[0089] For the observer of mismatched disturbance, it is expressed as:

[0090] ;

[0091] For the observer of matched disturbance, it is expressed as:

[0092] ;

[0093] wherein, , ,..., , , are observation estimation variables of , ,..., , , , is a state variable of the agent, , , and are parameters of the observer, and respectively satisfy the following selection conditions:

[0094] satisfy the recursive relationship , and for a sufficiently small ;

[0095] satisfy the recursive relationship , for a sufficiently small ;

[0096] ;

[0097] and , wherein , are selected to make the matrices and both Hurwitz.

[0098] As a preferred embodiment of the present application, since , ,..., are estimations of , ,..., , the observer error variable , Thus, the error system of the observer is

[0099] ;

[0100] ;

[0101]

[0102] ;

[0103] According to Finite-and fixed-time differentiators utilising HOSM techniques. IET Control Theory & Applications, 2017, the error system of the observer is fixed-time stable, that is, there exists a positive number T1>0, such that for any t>T1, the observer error is 0, that is , ,..., , , .

[0104] Let be the observation estimation variable of .

[0105] Similarly, another observer can be constructed, and the observation estimation variable in the observer can converge within a fixed time T2.

[0106] Therefore, the following transformation is made:

[0107] ;

[0108] ;

[0109] where , is the state variable of the agent, is the agent state variable after the observer transformation, and for the leader state and , the similar transformation can also be made to obtain the leader state variable and after the observer transformation.

[0110] Therefore, when t>T1+T2, the n-order dynamic system of the leader and the follower is

[0111] ;

[0112] ;

[0113] wherein, , , i = 1, 2,..., N.

[0114] As a preferred embodiment of the present application, the construction process of the fixed-time controller is as follows:

[0115] ;

[0116] ;

[0117] ;

[0118] ;

[0119] ;

[0120] wherein, is the control input of the agent, is a partial component variable of the agent control input, is a distributed observer state variable error constructed based on an observer, is an integral sliding mode, is a partial component variable of the integral sliding mode, 0 < p < 1, q > 0, normal number can simultaneously make the polynomials and are both Hurwitz polynomials with respect to the Laplace operator s, i = 1, 2,..., N, constant , the parameters and satisfy the following definitions: , .

[0121] As a preferred embodiment of the present application, it further includes stability analysis of fixed-time convergence based on Lyapunov function, and determination of fixed-time.

[0122] In the present embodiment, the following explains the proposal of the control can realize the fixed-time consensus of the multi-agent system.

[0123] First, define the consensus error as , then get the n-order system of the consensus error at t > T1 + T2 as:

[0124] ;

[0125] ;

[0126] Let ;

[0127] Let ;

[0128] Then the consistency error system can be rewritten as: , .

[0129] The n-order dynamics system formula of leader and follower is only valid when t > T1 + T2, when time The state equation is:

[0130] ;

[0131] ;

[0132] where, ; , ;

[0133] Let , then the following formula is obtained: ;

[0134] Since the observer error system is fixed-time stable, it is bounded in . Without loss of generality, it is assumed that , for i = 1, 2,..., N, where C i is a constant.

[0135] For the integral sliding mode , the Lyapunov function is constructed, and the concept of Filippov solution and set-valued derivative can be defined as:

[0136] ;

[0137] Therefore, the derivative of V s with respect to time t is:

[0138]

[0139] where is a constant. Therefore, is bounded in the time interval , and according to the definition of integral , it can be known that the consistency error system and the initial multi-agent system are bounded in the given control in the time interval . When t > T1 + T2, since , the following transformation is obtained:

[0140] ;

[0141] Introduce the fixed-time convergence analysis, if a positive definite function satisfies ;

[0142] where, where , are positive integers and satisfy , then , and

[0143] The upper bound of time can be estimated by the following formula: .

[0144] Based on the above convergence analysis results, it can be concluded that there is a positive number T3, such that V s and the integral can converge to in a fixed time. Therefore, when , the consistency error system can be rewritten as: , .

[0145] According to the fixed-time convergence analysis, consider an n-order system (A(s) = s^n + a_(n-1)s^(n-1) +... + a_0, B(s) = b_m s^m + b_(m-1)s^(m-1) +... + b_0): ) :

[0146] ;

[0147] ;

[0148]

[0149] ;

[0150] ;

[0151] where, is the state vector of the system, is the control input of the system. Take a positive number , which can make the polynomials and both be Hurwitz polynomials with respect to the Laplace operator s, then there is a constant , such that for any , the system can be stabilized to the origin in a fixed time through the following feedback control.

[0152] ;

[0153] The parameters and satisfy the following definitions:

[0154] ;

[0155] ;

[0156] Therefore, there is a positive constant T4 such that the consensus error system at this time can be stabilized at the origin within a fixed time T3.

[0157] In summary, define , the consensus error system can be stabilized at the origin within a fixed time T. In other words, this means that the original multi-agent system can reach consensus with the leader within a fixed time under the given control u.

[0158] Based on this, specific embodiments are proposed, such as Figures 3 to 6 As shown in the figure, a 3-order multi-agent system is composed of a leader with node 0 and followers with nodes 1-5. The dynamic of the leader is:

[0159] ;

[0160] ;

[0161] ;

[0162] The dynamic of the follower is:

[0163] ;

[0164] ;

[0165] , i = 1, 2, 3, 4, 5.

[0166] Where , respectively, represent the position, velocity, acceleration and control input of the leader and follower, is the unmatched disturbance in the system, is the matched disturbance.

[0167] Let the control input of the leader be , the disturbance terms be , and , the initial state of the leader be , the initial position of the follower be randomly generated within the interval , the initial velocity be randomly generated within the interval , and the initial acceleration be randomly generated within the interval . The network topology of the multi-agent system is shown in Figure 3 , so its corresponding adjacency matrix is:

[0168] ;

[0169] matrix , parameters of the observer are selected , , .

[0170] parameters of the controller p = q = 0.1, c1 = 2, c2 = 3, c3 = 5, = 0.75. A fixed time step of 10 -4 is used for simulation, and simulation results are as shown in Figure 4 and Figure 5 . It can be seen that the position, velocity, and acceleration errors between each follower and the leader converge to 0 quickly, which means that the multi-agent system reaches consensus within a fixed time.

[0171] The above merely describes preferred embodiments of the present application and is not used to limit the present application, and any modification, equivalent replacement, and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A fixed-time consensus control method under matched and unmatched disturbances, characterized in that, The method comprises: acquiring state information of a leader and a plurality of followers, and constructing a multi-agent system dynamics model based on the state information of the leader and the plurality of followers; establishing a communication network topology by taking the leader and the plurality of followers in the multi-agent system as communication nodes; designing a fixed-time convergent non-recursive disturbance observer to estimate matched disturbance and unmatched disturbance; constructing a fixed-time controller based on an output result of the disturbance observer; driving the followers to track the state of the leader within a fixed time based on the fixed-time controller, wherein the fixed time is irrelevant to an initial state of the system; the construction process of the fixed-time controller is as follows: ； ； ； ； ； wherein is a control input for the agent, is a partial constituent variable of the control input for the agent, is a distributed observer state variable error constructed based on an observer, is an integral sliding mode, is a partial constituent variable of the integral sliding mode, 0 < p < 1, q > 0, normal number can simultaneously make the polynomials and both about Laplace operator s are Hurwitz polynomials, i = 1, 2,..., N, constant , parameter and satisfy the following definitions: , .

2. The fixed-time consensus control method in the presence of matching and mismatching disturbances according to claim 1, wherein an n-order multi-agent system is composed of N followers and a leader, and a leader dynamics model in the multi-agent system dynamics model is: ； where is the state vector, u0is the leader's control input, and denote the leader's unmatched and matched disturbances, respectively. an i-th follower dynamics model is: ； where , i = 1, 2,..., N, X i is the state vector of the ith node, is the control input of the ith node, and represent the unmatched disturbance and matched disturbance of the system, respectively, and it is assumed that , , , , i = 1, 2,..., N, , and are known positive constants.

3. The fixed-time consensus control method in the presence of matching and mismatching disturbances according to claim 2, characterized in that, the disturbance observer comprises: for an observer of the unmatched disturbance, represented as: ； for an observer of the matched disturbance, represented as: ； wherein , , , , are observation estimation variables of , , , , , is a state variable of the agent, , , and are parameters of the observer, and satisfy the following selection conditions, respectively: , satisfies the recurrence relation , and for a sufficiently small ; , satisfying the recurrence relation , , for a sufficiently small ; ； and , .

4. The fixed-time consensus control method in the presence of matching and mismatching disturbances according to claim 3, wherein There exists a fixed time T1>0 such that for any t>T1, we have , ,..., , , , Similarly, another observer can be constructed, and the observation estimation variable in the observer converges within a fixed time T2, when t>T1+T2, the n-order dynamic system of the leader and the follower is: ； ； wherein ; ； , is the agent state variable, is the agent state variable transformed by the observer, and for the leader state and is transformed as above for the leader state variable and is transformed as above for the leader state variable , , i = 1, 2,..., N.

5. The fixed-time consensus control method in the presence of matching and mismatching disturbances according to claim 1, wherein, stability analysis on the fixed-time convergence is further included based on a Lyapunov function, and the fixed time is determined.

6. The fixed-time consensus control method in the presence of matching and mismatching disturbances according to claim 1, wherein, an augmented graph of the network topology contains a spanning tree with the leader as a root node.

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