Distributed event-triggered control for multi-agent formation stabilization and tracking

09/19/2017 ∙ by Christophe Viel, et al. ∙ 0

This paper addresses the problem of formation control and tracking a of desired trajectory by an Euler-Lagrange multi-agent systems. It is inspired by recent results by Qingkai et al. and adopts an event-triggered control strategy to reduce the number of communications between agents. For that purpose, to evaluate its control input, each agent maintains estimators of the states of the other agents. Communication is triggered when the discrepancy between the actual state of an agent and the corresponding estimate reaches some threshold. The impact of additive state perturbations on the formation control is studied. A condition for the convergence of the multi-agent system to a stable formation is studied. Simulations show the effectiveness of the proposed approach.

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1 Introduction

Distributed cooperative control of a multi-agent system (MAS) usually requires significant exchange of information between agents. In early contributions, see, e.g., [Olfati2007, Ren2008], communication was considered permanent. Recently, more practical approaches have been proposed. For example, in [wen2012, Wen2012_2, Wen2013], communication is intermittent, alternating phases of permanent communication and of absence of communication. Alternatively, communication may only occur at discrete time instants, either periodically as in [Garcia2014J2], or triggered by some event, as in [dimarogonas2012, fan2013, zhang2015, viel2016].

This paper proposes a strategy to reduce the number of communications for displacement-based formation control while following a desired reference trajectory. Agent dynamics are described by Euler-Lagrange models and include perturbations. This work extends results presented in [Qingkai2015] by introducing an event-triggered strategy, and results of [Qingchen2015, sun2015, tang2011] by addressing systems with more complex dynamics than a simple integrator. To obtain efficient distributed control laws, each agent uses an estimator of the state of the other agents. The proposed distributed communication triggering condition (CTC) involves the inter-agent displacements and the relative discrepancy between actual and estimated agent states. A single a priori trajectory has to be evaluated to follow the desired path. Effect of state perturbations on the formation and on the communications are analyzed. Conditions for the Lyapunov stability of the MAS have been introduced. The time interval between two consecutive communications by the same agent is shown to be strictly positive.

This paper is organized as follows. Related work is detailed in Section 2. Some assumptions are introduced in Section 3 and the formation parametrization is described in Section 4. As the problem considered here is to drive a formation of agents along a desired reference trajectory, the designed distributed control law consists of two parts. The first part (see Section 4) drives the agents to some target formation and maintains the formation, despite the presence of perturbations. It is based on estimates of the states of the agents described in Section 5.1. The second part (see Section 5) is dedicated to the tracking of the desired trajectory. Communication instants are chosen locally by each agent using an event-triggered approach introduced in Section 6. A simulation example is considered in Section 7 to illustrate the reduction of the communications obtained by the proposed approach. Finally, conclusions are drawn in Section 8.

2 Related work

Event-triggered communication is a promising approach to save energy. It is well-suited to applications where communications should be minimized, e.g., to improve furtivity, reduce energy consumption, or limit collisions between transmitted data packets. Application examples with such constraints are exposed in [linsenmayer2014, linsenmayer2015] for the case of a fleet of vehicles, or in [aragues2011] where agents aim at merging local feature-based maps. The main difficulty consists in determining the CTC that will ensure the completion of the task assigned to the MAS, e.g., reaching some consensus, maintaining a formation, etc. In a distributed strategy, the states of the other agents are not permanently available, thus each agent usually maintains estimators of the state of its neighbours to evaluate their control laws. Nevertheless, without permanent communication, the quality of the state estimates is difficult to evaluate. To address this issue, each agent maintains an estimate of its own state using only the information it has shared with its neighbours. When the discrepancy between this own state estimate and its actual state reaches some threshold, the agent triggers a communication. This is the approach considered, e.g., in [zhu2014, garcia2014J, seyboth2013, garcia2015, viel2016, dimarogonas2012, viel2017]. These works differ by the complexity of the agents’ dynamics [zhu2014, garcia2014J, seyboth2013], the structure of the state estimator [dimarogonas2012, garcia2015, viel2016, viel2017], and the determination of the threshold for the CTC [seyboth2013, viel2017].

Most of the event-triggered approaches have been applied in the context of consensus in MAS [dimarogonas2012, seyboth2013, garcia2015]. This paper focuses on distributed formation control, which has been considered in [Qingchen2015, sun2015, tang2011]. Formation control consists in driving and maintaining all agents of a MAS to some reference, possibly time-varying configuration, defining, , their relative positions, orientations, and speeds. Various approaches have been considered, such as behavior-based flocking [Reynolds1999, Vicsek1995, olfati2006, rochefort2011, moreno2014], or formation tracking [do2008, chao2012, arboleda2013, mei2011, ren2004].

Behavior-based flocking [Reynolds1999, Vicsek1995, olfati2006, rochefort2011, moreno2014] imposes several behavior rules (attraction, repulsion, imitation) to each agent. Their combination leads the MAS to follow some desired behavior. Such approach requires the availability to each agent of observations of the state of its neighbours. These observations may be deduced from measurements provided by sensors embedded in each agent or from information communicated by its neighbours. In all cases, these observations are assumed permanently available. In addition, if a satisfying global behavior may be obtained by the MAS, behavior-based flocking cannot impose a precise configuration between agents.

Different formation-tracking methods have been considered. In leader-follower techniques [do2008, chao2012, arboleda2013, mei2011], based on mission goals, a trajectory is designed only for some leader agent. The other follower agents, aim at tracking the leader as well as maintaining some target formation defined with respect to the leader. A virtual leader has been considered in [chao2001, chao2012, Yohan] to gain robustness to leader failure. This requires a good synchronization among agents of the state of the virtual leader. Virtual structures have been introduced in [ren2004, wang2014]

, where the agent control is designed to satisfy constraints between neighbours. Such approaches also address the problem of leader failure. In distance-based control, the constraints are distances between agents. In displacement-based control, relative coordinate or speed vectors between agents are imposed. In tensegrity structures

[Qingkai2015, nabet2009] additional flexibility in the structure is considered by considering attraction and repulsion terms between agents, as formalized by [alfakih2013]. In addition to constraints on the structure of the MAS, [sun2009] imposes some reference trajectory to each agent. In most of these works, permanent communication between agents is assumed.

Some recent works combine event-triggered approaches with distance-based or displacement-based formation control [Qingchen2015, sun2015, tang2011]. In these works, the dynamics of the agents are described by a simple integrator, with control input considered constant between two communications. The proposed CTCs consider different threshold formulations and require each agent to have access to the state of all other agents. A constant threshold is considered in [sun2015]. A time-varying threshold is introduced in [Qingchen2015, tang2011]. The CTC depends then on the relative positions between agents and the relative discrepancy between actual and estimated agent states. These CTCs reduce the number of triggered communications when the system converges to the desired formation. A minimal time between two communications, named inter-event time, is also defined. Finally, in all these works, no perturbations are considered.

LBC techniques have been introduced in [rego2013, xu2006, aguiar2007, yook2002] to reduce the number of communications in trajectory tracking problems. MAS with decoupled nonlinear agent dynamics are considered in [rego2013, aguiar2007]. Agents have to follow parametrized paths, designed in a centralized way. CTCs introduced by LBC lead all agents to follow the paths in a synchronized way to set up a desired formation. Communication delays, as well as packet losses are considered. Nevertheless, if input-to-state stability conditions are established, absence of Zeno behavior is not analyzed.

3 Notations and hypotheses

Table 1 summarizes the main notations used in this paper.

Consider a MAS consisting of a network of agents which topology is described by an undirected graph . is the set of nodes and the set of edges of the network. The set of neighbours of Agent  is . is the cardinal number of . For some vector , we define where is the absolute value of the -th component of . Similarly, the notation will be used to indicate that each component of is non negative, , . A continuous function is said to belong to class if for each fixed , the function is strictly increasing and , and for each fixed , the function is decreasing and . A continuous function is said to belong class if it is strictly increasing and . A continuous function is said to belong class if it belongs to class , with and .

vector of coordinates of Agent  in some global fixed reference frame
vector , configuration of the MAS
state vector of Agent 
estimate of performed by Agent .
estimate of performed by Agent .
estimate of performed by Agent .
estimation error between and .
relative coordinate vector between agents and .
desired value for .
reference trajectory
reference trajectory for Agent ,
trajectory error for Agent ,
time at which the -th message is sent by Agent .
time at which the -th message sent by Agent  is received by Agent .
Table 1: Main notations

Let be the vector of coordinates of Agent  in some global fixed reference frame and let be the configuration of the MAS. The dynamics of each agent is described by the Euler-Lagrange model

(1)

where is some control input described in Section 4.2, is the inertia matrix of Agent , is the matrix of the Coriolis and centripetal term on Agent , accounts for gravitational acceleration supposed to be known and constant, and is a time-varying state perturbation satisfying . The state vector of Agent  is . Assume that the dynamics satisfy the following assumptions, where Assumptions A1, A2 and A3 have been previously considered, e.g., in [mei2011, liu2016, makkar2007]:

A1)

is symmetric positive and there exists satisfying , .

A2)

is skew symmetric or negative definite and there exists

satisfying , and .

A3)

The left-hand side of (1) can be linearly parametrized as

(2)

for all vectors , where is a regressor matrix with known structure and is a vector of unknown but constant parameters associated with the -th agent.

Moreover, one assumes that

A4)

For each is such that , with known and .

A5)

Each Agent  is able to measure without error its own state ,

A6)

There is no packet losses or communication delay between agents.

In what follows, the notations and are used to replace and .

4 Formation control problem

This section aims at designing a decentralized control strategy to drive a MAS to a desired target formation in some global reference frame , while reducing as much as possible the communications between agents. The target formation is first described in Section 4.1. The potential energy of a MAS with respect to the target formation is introduced to quantify the discrepancy between the target and current formations. The proposed distributed control, introduced in Section 4.2, tries to minimize the potential energy. To evaluate the control input of each agent despite the communications at discrete time instants only, estimators of the coordinate vectors of all agents are managed by each agent, as presented in Section 5.1. The presence of perturbations increases the discrepancy between the state vector and their estimates. A CTC is designed to limit this discrepancy by updating the estimators as described in Section 6.

4.1 Formation parametrization

Consider the relative coordinate vector between two agents and and the target relative coordinate vector for all . A target formation is defined by the set . Consider, without loss of generality, the first agent as a reference agent and introduce the target relative configuration vector . Any target relative configuration vector can be expressed as .

The potential energy of the formation, introduced for tensegrety formations in [nabet2009, Qingkai2015], represents the disagreement between and

(3)

where the are some spring coefficients, which can be be positive or null. The values of the s that make a given an equilibrium formation may be chosen using the method developed in [Qingkai2015]. Moreover, we take and if , i.e., if and are not neighbors. Since is connected, the minimum number of non-zero coefficients to properly define a target formation is . A number of non-zero larger than introduces robustness in the formation, in particular with respect to the loss of an agent.

Definition 1.

[Qingkai2015] The MAS asymptotically converges to the target formation with a bounded error iff there exists some such as

(4)

A control law designed to reduce the potential energy allows a bounded convergence of the MAS. To describe the evolution of , one introduces as in [Qingkai2015]

(5)
(6)
(7)

where and characterize the evolution of the discrepancy between the current and target formations and is a positive scalar design parameter.

Note that since if , one has . Consequently, Agent  can evalutate and using only information from its neighbors.

4.2 Distributed control

The control law proposed in [Qingkai2015] is defined as and aims at reducing , thus making the MAS converge to the target formation in case of permanent communication. In this approach, each agent evaluates its control input using the state vectors of its neighbours obtained via permanent communication. Here, in a distributed context with limited communications between agents, agents cannot have permanent access to . Thus, one introduces the estimate of performed by Agent  to replace the missing information in the control law. The MAS configuration estimated by Agent  is denoted as . The way is evaluated is described in Section 5.1.

In a distributed context with limited communications, with the help of , Agent  is able to evaluate

(8)
(9)

with and . Using and , Agent  is able to evaluate the following adaptive distributed control input to be used in (1)

(10)
(11)

where , are design parameters and is an arbitrary symmetric positive definite matrix.

Section 5.1 details the estimator of needed in (10).

5 Time-varying formation and tracking

In this section, the MAS has to follow some reference trajectory , while remaining in a desired formation. Agent , taken as the reference agent, aims at following . It is assumed that all agents have access to . Moreover, assume that the target formation can be time-varying and is represented by the relative configuration vector . Therefore the reference trajectory of each agent can be expressed as .

Definition 2.

The MAS reaches its tracking objective iff there exists and such that (4) is satisfied and

(12)

i.e., iff the reference agent asymptotically converges to the reference trajectory, and the MAS asymptotically converges to the target formation with bounded errors.

A distributed control law is designed to satisfy this target. Introduce the trajectory error terms

The terms , , , and introduced in Sections 4 are now redefined as follows to address the trajectory tracking problem

(13)
(14)
(15)
(16)
(17)
(18)

where is a positive design parameter which may be used to control the tracking error with respect to the reference trajectory. When no reference trajectory is considered, .

From these terms, a new distributed control input to be used in (1) is defined for Agent  as

(19)
(20)

where and .

5.1 Communication protocol and estimator dynamics

5.1.1 Communication protocol

In what follows, the time instant at which the -th message is sent by Agent is denoted . Let be the time at which the -th message sent by Agent  is received by Agent . According to Assumption A6, for all . When a communication is triggered at by Agent , it transmits a message containing , , and . Upon reception of this message, the neighbours of Agent  update their estimate of the state of Agent  using this information.

Figure 1: Example of transmission times by Agent  of -th message and reception times of -th message by Agent .

5.1.2 Estimator dynamics

Agent  evaluates the estimate of for all its neighbors as

(21)
(22)
(23)

where and are estimates of and computed from and using

(24)

The estimator (21) managed by Agent  requires an estimate of the control input evaluated by Agent . This estimate, used by Agent , is evaluated as

(25)
(26)
(27)

where is the estimate of , , and if , , in the case of a reference trajectory to be tracked and else. Note that if , . The estimator (21)-(23) only requires that Agent  receives messages from Agent  to evaluate and (25)-(27).

Errors appear between and its estimate obtained by any other Agent  due to the presence of state perturbations, the non-permanent communication, and the mismatch between , , and . The errors for the estimates performed by Agent  are expressed as

(28)
(29)

These errors are used in Section 6 to trigger communications when and become too large. Figure 2 summarizes the overall structure of the estimator and controller.

Figure 2: Formation control system architecture

Using Assumption A6 and considering the structure of the estimator (21)-(23), one has for all and . This simplifies the stability analysis in Appendix 9.2.

6 Event-triggered communications

Theorem 1 introduces a CTC used to trigger communications to ensure a bounded asymptotic convergence of the MAS to the target formation. Each agent knows the initial values of the state of its neighbors. In practice, this condition can be satisfied by triggering a communication at time .

Let and , , and . Using Assumption A4, define also for and

(30)

and .

Theorem 1.

Consider a MAS with agent dynamics given by (1) and the control law (19). Consider some design parameters , , ,

and . In absence of communication delays, the system (1) is input-to-state practically stable (ISpS),see [jiang1996] or Appendix 9.1, and the agents can be driven to some target formation such that

(31)

with

(32)

where , if the communications are triggered when one of the following conditions is satisfied

(33)
(34)

with , and .

Moreover, consecutive communication triggering time instants satisfy .

The proof of Theorem 1 is given in Appendix 9.2 and the proof of in Appendix 9.3.

The CTCs proposed in Theorem 1 are analyzed assuming that the estimators of the state of the agents and the communication protocol is such that ,

(35)
(36)

These properties are actually satisfied if the communication protocol described in Section and the state estimator are employed. Theorem 1 is valid independently of the way the estimate of is evaluated provided that (35) and (36) are satisfied.

From (31) and (33), one sees that can be used to adjust the trade-off between the bound on the formation and tracking errors and the amount of triggered communications. If , there is no perturbation and is perfectly known, the system converges asymptotically.

The CTC (34) is related to the discrepancy between and . Choosing a small value of may lead to frequent communications. On the contrary, when is large, (33) is more likely to be satisfied. A value of that corresponds to a trade-off between the two CTCs (33) and (34) has thus to be found to minimize the amount of communications.

The CTCs (33) and (34) mainly depend on and . A communication is triggered by Agent  when the state estimate of its own state vector is not satisfying, , when and becomes large. To reduce the number of triggered communications, one has to keep and as small as possible. This may be achieved by increasing the accuracy of the estimator, as proposed in [viel2017], but possibly at the price of a more complex structure for the estimator or the number of connection in the communication graph.

The perturbations have a direct impact on and , and, as a consequence, on the frequency of communications. (32) shows the impact of and on the formation and tracking errors: in presence of perturbations, the formation and tracking errors cannot reach a value below a minimum value due to the perturbations. At the cost of a larger formation and tracking errors, can reduce the number of triggered communications and so can reduce the influence of perturbations on the CTC (33).

The discrepancy between the actual values of and and of their estimates and determines the accuracy of , so , and the estimation errors. Even in absence of state perturbations, due to the linear parametrization, it is likely that , and , which leads to the satisfaction of the CTCs at some time instants. Thus, the CTC (33) leads to more communications when the model of the agent dynamics is not accurate, requiring thus more frequent updates of the estimate of the states of agents.

The choice of the parameters , , and also determines the number of broadcast messages. Choosing the spring coefficients such that is small leads to a reduction in the number of communication triggered due to the satisfaction of (33).

7 Simulation results

The performance of the proposed algorithm is evaluated considering a set of agents. Two models will be considered to describe the dynamics of the agents.

7.1 Models of the agent dynamics and estimator

7.1.1 Double integrator with Coriolis term (DI)

The first model consists in the dynamical system

with and where

(37)

Then the vectors , are obtained using (2). In place of the estimator in Section 5.1 a first less accurate estimate of made by Agent , is evaluated as

(38)
(39)

This estimator allows one to better observe the tradeoff between the potential energy of the formation and the communication requirements.

For this dynamical model, the parameters of the control law (19) and the CTC (33) have been selected as: , , , , , , and .

7.1.2 Surface ship (SS)

The second model considers surface ships with coordinate vectors , , in a local earth-fixed frame. For Agent , represents its position and its heading angle. The dynamics of the agents is described by the surface ship dynamical model taken from [Kyrkjeb2007], assumed identical for all agents, and expressed in the body frame as

(40)

where is the velocity vector in the body frame, is the control input, is the perturbation, and