CN110083179B - Consistency tracking control method for multi-agent system in preset time - Google Patents

Abstract

The invention relates to a consistency tracking control method for a preset time multi-agent system, which comprises the following steps: designing a preset time distributed observer to estimate a tracking error in preset time; determining parameters of the distributed observer according to preset time; designing a non-singular terminal sliding mode consistency control protocol in preset time to enable a follower multi-agent system to track the track of an upper-leading intelligent system in the preset time; determining controller parameters which can ensure the continuity and the differentiability of the sliding mode surface and eliminate the control singularity; the proposed distributed observer and consistency control protocol is deployed on each follower agent to achieve scheduled time consistency tracking. Compared with the existing fixed-time consistency scheme, the method can obtain the convergence time estimation with lower conservatism, and the method also helps to reduce the conservatism of the controller design. Therefore, the method of the invention has wide application prospect in the application of the multi-agent system with higher requirements on control precision and convergence time.

Description

Consistency tracking control method for multi-agent system in preset time

Technical Field

The invention relates to the technical field of multi-agent system cooperative control, in particular to a consistency tracking control method for a multi-agent system in preset time.

Background

In recent years, distributed cooperative control of a multi-agent system is widely applied in various fields, including networked robot operation hand coordination control, robot tracking and deployment, unmanned system formation and microgrid energy management. One fundamental problem with distributed cooperative control is that the coherence protocol is designed such that all agents synchronize through local communication, a problem known as coherence. The consistency problem can be further divided into the following steps according to the leadership in the system: no leadership consistency problem and consistency tracking problem. Leaderless consistency refers to the convergence of all agents to an average of the initial states or a weighted average of the initial states. Consistent following refers to all following agents tracking the track of the leader, and the targets of the group can be specified through the state of the leader. Therefore, compared with leaderless consistency, consistency tracking improves the flexibility of group target realization and is widely concerned by academia and industry.

The key to solving the consistency tracking problem is to design a consistency control protocol. An important performance index for evaluating a designed conformance control protocol is convergence speed. The relevant literature shows that the improvement of algebraic connectivity helps to improve the convergence speed of the consistency control protocol. This motivates researchers to speed up convergence by designing optimal weights or selecting better communication topologies. However, these methods can only achieve gradual consistency, which means that the intelligent system cannot achieve precise consistency in a limited time. The limited time consistency can realize the accurate consistency of the multi-agent system in a limited time. Compared with gradual consistency, the limited time consistency has higher consistency precision, faster convergence speed and stronger robustness to interference and uncertainty. For some applications where control accuracy and convergence time are highly desirable, it is more desirable to achieve time-limited consistency. Therefore, researchers spend a lot of effort designing a time-limited consistency control protocol, the homogeneous method, the power-up integrator method, and the terminal sliding mode method for designing a time-limited consistency control protocol. However, the convergence time of the existing time-limited coherence control protocol depends on the initial value of the agent. However, the initial value of the actual intelligent system is difficult to obtain, which brings difficulty in estimating the convergence time. Furthermore, as the initial values approach infinity, the consistency times approach infinity, which prevents the limited time consistency from being applied to multi-agent systems with very large initial values.

To improve the performance of existing finite time consistency protocols, fixed time stability is introduced into the consistency control protocol design. Fixed-time consistency has an attractive property-the upper bound on convergence time is a constant independent of initial state, which helps controller design and settling time estimation, making fixed-time control suitable for applications with a tight upper bound on convergence time, such as power system control and remote operating system control. Fixed-time control protocols have been used to achieve consistency in multi-agent systems with undirected communication topologies, directed communication topologies, and switched communication topologies. However, there are two major problems with the existing results. First, the upper bound on convergence time derived from stability analysis is very conservative. Secondly, the upper bound of the convergence time of the fixed-time consistency protocol is a complex function of the control parameters, and there is no explicit relationship between the adjustment parameters and the upper bound of the stabilization time. In order to overcome the above problems, it is necessary to develop a predetermined time stabilization control. Compared with the fixed-time steady control, the predetermined-time steady control has the following two advantages: firstly, the convergence time upper bound of the stable control of the preset time is equal to the adjustment parameter of the controller, which is convenient for the design of the controller to meet the requirement of the stable time; second, the tuning parameter of the controller is the minimum upper bound of the settling time, thereby avoiding overestimation of the convergence time boundary. Extending the predetermined time stability to solve the consistency tracking problem requires solving two challenging scientific problems — the predetermined time observation problem and the control singularity problem. To date, no document has proposed a predetermined time consistency control protocol. The implementation of the predetermined time leading-following consistency helps to determine the control protocol gain and reduce the conservatism of the upper bound estimation of the consistency time, and helps to apply the predetermined time consistency to the application occasions with strict requirements on convergence time. Therefore, the research on the problem of the consistency tracking of the preset time has important guiding function and application prospect for the application of the practical system.

Disclosure of Invention

Technical problem to be solved

In order to avoid the defects of the prior art, the invention provides a consistency tracking control method of a preset time multi-agent system, which is used for facilitating the estimation of the upper bound of consistency time and the design of a controller and reducing the conservatism of the consistency time estimation.

Technical scheme

A consistency tracking control method for a multi-agent system at preset time is characterized in that the multi-agent system consists of a leader and N followers, and an interaction network formed by the N followers is an undirected graph G_sEach follower is connected with the leader through a path, and the motion of the ith follower is as follows:

wherein x is_1iIs in a position type state, x_2iIn a velocity-type state, u_iIs a control input;

the leader's dynamics are:

wherein x is₁₀Is in a position type state, x₂₀In a velocity-type state, u₀Is a bounded control input, i.e. there is a known normal number λ, such that | u₀Lambda is less than or equal to, all followers can not obtain control input information of the leader, but the constant lambda is known to all followers;

the consistency tracking control scheme comprises the following steps:

step 1: designing a distributed observer for each follower:

xi in the formula_iAnd η_iIs the ith follower distributed observer state, α, β, λ are observer gains, p and q are such that p is satisfied<Positive odd number of q, sig (.)^α＝|·|^αsign(·)；

Step 2: root of herbaceous plantAccording to the required observation time 2T_bDetermining the gain of the distributed observer:

in the formula of_min(H) Is the minimum eigenvalue of the matrix H, H ═ L_s+B，L_sIs shown as a drawing G_sLaplacian matrix of (B ═ diag { a) }₁₀,...,a_N0}，a_i0>0 means that the follower is communicatively connected to the leader, otherwise a_i00; enabling observer states at predetermined times 2T_bInternally converging to a tracking error state;

and step 3: defining the ith following agent tracking error e_1i＝x_1i-x₁₀，e_2i＝x_2i-x₂₀(ii) a After the observer obtains the leading state information, to enable the ith following agent to be able to do so at a pre-specified time T_c+T_dInternally tracking the track of the upper leader, and designing a nonsingular terminal sliding mode consistency control protocol for the ith following agent:

wherein m and n are normal numbers, T_cSlip form surface s for a pre-specified settling time_iIs defined as:

in the formula

k₁，k₂Is a positive real number, T_dEpsilon is a small positive constant for a pre-specified stabilization time;

and 4, step 4: controlling parameter k to make sliding mode surface and its derivative continuous₁，k₂Is selected as k₁＝1+m，k₂-m; to eliminate the control singularity, the control parameters m, n need to satisfy 0<m<1/2，0<n<1/2；

And 5: deploying a controller (5) and a distributed observer (3) into the ith follower such that the states of all followers are able to be at a pre-specified time 2T_b+T_c+T_dInternally tracking the trajectory of the leader, i.e. for any time T ≧ 2T_b+T_c+T_dAll have x_1i(t)＝x₁₀(t)，x_2i(t)＝x₂₀(t) is true.

Advantageous effects

In view of the high requirements of many practical multi-agent systems on control precision and consistency time, the invention can realize the accurate consistency tracking of the multi-agent systems in a specified time, and the gain of the distributed observer/consistency tracking control protocol can be directly determined by the specified consistency time, thereby improving the flexibility of convergence time adjustment and controller design and facilitating the controller design. Furthermore, the proposed method reduces the conservatism of the convergence time upper bound estimate, and thus the conservatism of the controller design. Therefore, the proposed consistency tracking control method can show obvious advantages when applied to a system with high requirements on control accuracy and convergence time. The innovation of the present invention with respect to the prior art is achieved in three aspects:

(a) the method provides a novel preset time consistency tracking control scheme to solve the problem of preset time tracking of a multi-agent system, the control scheme comprises a distributed observer and a nonsingular terminal sliding mode control protocol, and the problems of preset time distributed observation and singular control are solved.

(b) The consistency time may be pre-specified according to the convergence time requirement, and the gain of the proposed distributed observer/consistency tracking control protocol may be directly determined by the specified consistency time.

(c) Compared with the existing fixed-time consistency scheme, the method provided by the invention can obtain the convergence time estimation with lower conservatism, which also helps to reduce the conservatism of the controller design.

Drawings

FIG. 1 is a block diagram of the design steps of a predetermined time consistency tracking control scheme

FIG. 2 Single link manipulator Multi-agent System interaction topology

Observer state xi in fig. 3_iAnd tracking error state e_1iTime response of

State η of observer of FIG. 4_iAnd tracking error state e_2iTime response of

FIG. 5 agent state x_1iTime response of

FIG. 6 agent state x_2iTime response of

Detailed Description

The invention will now be further described with reference to the following examples and drawings:

referring to FIGS. 1-6, a multi-agent system contemplated by the present invention is comprised of a leader and N followers, the interaction network of which is an undirected graph G_sEach follower is connected with the leader through a path, and the motion of the ith follower is as follows:

wherein x is_1iIs in a position type state, x_2iIn a velocity-type state, u_iIs a control input.

The leader's dynamics are:

wherein x is₁₀Is in a position type state, x₂₀In a velocity-type state, u₀Is a bounded control input, i.e. there is a known normal number λ, such that | u₀Lambda ≦ lambda, all followers cannot get the leader's control input information, but the constant lambda is known to all followers.

The invention aims to provide a consistency tracking control method of a preset time multi-agent system, which is convenient for estimating the upper bound of consistency time and designing a controller and reduces the conservatism of the consistency time estimation.

With reference to fig. 1, the design process of the consistency tracking control scheme proposed by the present invention includes the following steps:

the method comprises the following steps: designing a distributed observer for each follower:

xi in the formula_iAnd η_iIs the ith follower distributed observer state, α, β, λ are observer gains, p and q are such that p is satisfied<Positive odd number of q, sig (.)^α＝|·|^αsign(·)。

Step two: according to the required observation time 2T_bDetermining gain of a distributed observer

In the formula of_min(H) Is the minimum eigenvalue of the matrix H, H ═ L_s+B，L_sIs shown as a drawing G_sLaplacian matrix of (B ═ diag { a) }₁₀,...,a_N0}，a_i0>0 means that the follower is communicatively connected to the leader, otherwise a_i00. Enabling observer states at predetermined times 2T_bInternally converging to a tracking error state.

Step three: defining the ith following agent tracking error e_1i＝x_1i-x₁₀，e_2i＝x_2i-x₂₀. After the observer obtains the leading state information, to enable the ith following agent to be able to do so at a pre-specified time T_c+T_dInternally tracking the track of the upper leader, and designing a nonsingular terminal sliding mode consistency control protocol for the ith following agent:

wherein m and n are normal numbers, T_cSlip form surface s for a pre-specified settling time_iIs defined as:

in the formula

k₁，k₂Is a positive real number, T_dFor a pre-specified settling time, ε is a small positive constant (e.g., ε may be equal to 0.1).

Step four: controlling parameter k to make sliding mode surface and its derivative continuous₁，k₂Is selected as k₁＝1+m，k₂To eliminate control singularity, the control parameters m, n need to satisfy 0<m<1/2,0<n<1/2。

Step five: deploying a controller (5) and a distributed observer (3) into the ith follower such that the states of all followers are able to be at a pre-specified time 2T_b+T_c+T_dInternally tracking the trajectory of the leader, i.e. for any time T ≧ 2T_b+T_c+T_dAll have x_1i(t)＝x₁₀(t)，x_2i(t)＝x₂₀(t) is true.

Example (b): single-connecting-rod mechanical arm multi-agent system preset time consistency tracking control

The single-link manipulator system is an important system in the field of industrial automation, the accuracy and the rapidity of coordination control of the single-link manipulator system are of great significance for improving the efficiency and the quality of finishing action instructions, and the development of a rapid and accurate single-link manipulator coordination control algorithm is of great significance for improving the industrial automation level of China. The effectiveness of the above-mentioned predetermined time consistency tracking control method is illustrated by taking a single-link manipulator multi-agent system as an example. The single-link manipulator multi-agent system is composed of 3 follower agents and 1 leader agent, and the interaction topology is shown in fig. 2. The dynamics of the follower agent can be described as:

wherein the content of the first and second substances,

q_irespectively representing angular acceleration, angular velocity and angular position, J, of the ith connection_iRepresenting the total moment of inertia of the i-th connection and the motor, B_iRepresenting the damping coefficient, M_iDenotes the total mass of the ith connection, g denotes the acceleration of gravity, l_iRepresenting the distance between the ith connecting joint axis and the center of gravity. If the control input is designed as

Order to

The dynamics of the follower agent may be expressed as:

the same expression is used for the system (1).

The dynamics of the leader agent can be described as:

wherein the content of the first and second substances,

and u₂₀Is the angular acceleration of the reference trajectory and the control input of the leader. Let x₁₀＝q₀,

Then (9) has the same form as (2).

The leader agent's initial value is selected to be (x)₁₀(0),x₂₀(0) (0,0.1), the initial value of the follower agent is chosen to be (x)₁₁(0),x₂₁(0),x₁₂(0),x₂₂(0),x₁₃(0),x₂₃(0) (0.4,0.35,0.2,0.3,0.5,0.25) and the leader's control input is selected to be u₂₀-0.1sin (T), the distributed observer and controller parameters are selected as p-5, q-9, T_b＝1,α＝β＝14.9054,λ＝0.1，m＝0.3,n＝0.4,k₁＝1.3,k₂＝-0.3,ε＝0.1,T_c＝T_dThe state and lead state estimated by the distributed observer are shown in fig. 3-4, from which it can be seen that the proposed distributed observer can give an accurate estimate of the tracking error state within 0.2 s. The multi-agent system state under the action of the proposed controller is shown in fig. 5-6, and as can be seen from fig. 5-6, the proposed control protocol can achieve accurate consistency tracking of the multi-agent system within 1.5 s. The simulation results confirm that the proposed control scheme can achieve a predetermined time consistency tracking.

Claims (1)

1. A consistency tracking control method for a multi-agent system at preset time is characterized in that the multi-agent system consists of a leader and N followers, and an interaction network formed by the N followers is an undirected graph G_sEach follower is connected with the leader through a path, and the motion of the ith follower is as follows:

wherein x is_1iIs in a position type state, x_2iIn a velocity-type state, u_iIs a control input;

the leader's dynamics are:

wherein x is₁₀Is in a position type state, x₂₀In a velocity-type state, u₀Is a bounded control input, i.e. there is a known normal lambda', such that | u₀λ 'or less, all followers cannot get the control input information of the leader, but the constant λ' is known to all followers;

the consistency tracking control scheme comprises the following steps:

step 1: designing a distributed observer for each follower:

xi in the formula_iAnd η_iIs the ith follower distributed observer state, α, β, λ are observer gains, p and q are positive odd numbers satisfying p < q, sig (·)^α＝|·|^αsign(·)；

Step 2: according to the required observation time 2T_bDetermining the gain of the distributed observer:

in the formula of_min(H) Is the minimum eigenvalue of the matrix H, H ═ L_s+B，L_sIs shown as a drawing G_sLaplacian matrix of (B ═ diag { a) }₁₀,...,a_N0}，a_i0> 0 means that the follower is communicatively connected to the leader, otherwise a_i00; enabling observer states at predetermined times 2T_bInternally converging to a tracking error state;

and step 3: defining the ith following agent tracking error e_1i＝x_1i-x₁₀，e_2i＝x_2i-x₂₀(ii) a After the observer obtains the state information of the leader, the ith following intelligent agent can be pointed in advanceFixed time T_c+T_dInternally tracking the track of the upper leader, and designing a nonsingular terminal sliding mode consistency control protocol for the ith following agent:

wherein m and n are normal numbers, T_cSlip form surface s for a pre-specified settling time_iIs defined as:

in the formula

k₁，k₂Is a positive real number, T_dEpsilon is a small positive constant for a pre-specified stabilization time;

and 4, step 4: controlling parameter k to make sliding mode surface and its derivative continuous₁，k₂Is selected as k₁＝1+m，k₂-m; in order to eliminate control singularity, the control parameter m and n need to satisfy 0-m-1/2 and 0-n-1/2;

and 5: deploying a controller (5) and a distributed observer (3) into the ith follower such that the states of all followers are able to be at a pre-specified time 2T_b+T_c+T_dInternally tracking the trajectory of the leader, i.e. for any time T ≧ 2T_b+T_c+T_dAll have x_1i(t)＝x₁₀(t)，x_2i(t)＝x₂₀(t) is true.

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