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Decentralised event‐triggered consensus of double integrator multi‐agent systems with packet losses and communication delays

2016; Institution of Engineering and Technology; Volume: 10; Issue: 15 Linguagem: Inglês

10.1049/iet-cta.2016.0107

1751-8652

Eloy García, Yongcan Cao, David W. Casbeer,

Stability and Control of Uncertain Systems

IET Control Theory & ApplicationsVolume 10, Issue 15 p. 1835-1843 Research ArticleFree Access Decentralised event-triggered consensus of double integrator multi-agent systems with packet losses and communication delays Eloy Garcia, Corresponding Author Eloy Garcia elgarcia@infoscitex.com Infoscitex Corp., Dayton, OH, 45431 USA Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433 USASearch for more papers by this authorYongcan Cao, Yongcan Cao Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, TX, 78249 USASearch for more papers by this authorDavid W. Casbeer, David W. Casbeer Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433 USASearch for more papers by this author Eloy Garcia, Corresponding Author Eloy Garcia elgarcia@infoscitex.com Infoscitex Corp., Dayton, OH, 45431 USA Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433 USASearch for more papers by this authorYongcan Cao, Yongcan Cao Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, TX, 78249 USASearch for more papers by this authorDavid W. Casbeer, David W. Casbeer Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433 USASearch for more papers by this author First published: 01 October 2016 https://doi.org/10.1049/iet-cta.2016.0107Citations: 21AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The event-triggered consensus problem with agents described by double integrator dynamics is addressed in this study. The authors consider the problem of non-consistent packet losses where the broadcast channel from one agent to its neighbours can drop the event-triggered packets of information, where the transmitting agent is unaware that the packet was not received and the receiving agents have no knowledge of the transmitted packet. They also consider the constraints associated with communication delays. In this study, they consider directed graphs, and they also relax the consistency on the packet dropouts and the delays. By relaxing the consistency they allow the dropouts and delays for a packet broadcast by one agent to be different for each receiving node. Under these constraints, an event-triggered consensus protocol is designed for the agents to achieve consensus asymptotically while reducing transmissions of measurements. In addition, positive inter-event times are obtained which guarantee that Zeno behaviour does not occur. 1 Introduction In recent years, consensus problems over reliable and infinite bandwidth communication networks have been well studied due to their applications in sensor networks and multi-agent systems coordination [1–4]. Many consensus protocols rely on the assumption that continuous exchange of information among agents is possible. Since continuous communication is not possible in many applications, it becomes important to discern how frequently agents should communicate to preserve the system properties that stemmed from continuous information exchange. The sampled-data approach has been commonly used to estimate the sampling periods [5–7]. An important drawback of periodic transmission is that it requires synchronisation between the agents, that is, all agents need to transmit their information at the same time instants and, in some cases, it requires a conservative sampling period for worst case situations. More recently, the event-triggered control paradigm has been used to design consensus protocols that account for limited bandwidth communication channels, by reducing the number of transmitted measurements by each agent in the network [8–10]. Different from periodic (or time-triggered) implementations, in event-triggered control information or measurements are not transmitted periodically in time, rather they are triggered by the occurrence of certain events. In event-triggered broadcasting [11–15], a subsystem sends its local state to the network only when it is necessary, that is, only when a measure of the local subsystem state error is above a specified threshold. Event-triggered control strategies have also been applied to stabilise multiple coupled subsystems as in [16–20]. Guinaldo et al. [21] considered delays and packet losses in the stabilisation problem of coupled subsystems. Two event-triggered communication protocols were proposed in [21]. The first protocol preserves state consistency in the sense that all neighbours of a given node use the same version of the transmitted state by that node. In the presence of packet losses multiple retransmissions of the same measurement may be needed until it is guaranteed that all neighbours receive the update. At that point the transmitting node sends a permission message that allows neighbours to start using the new transmitted measurement. Due to disadvantages in retransmissions and extra delays a second protocol is presented where the state consistency is relaxed and the neighbours of a given node are allowed to use different versions of the state of that node. However, acknowledgment (ACK) messages are required. In this case, retransmissions are used for neighbours whose ACK message is not received by the transmitting node. Meanwhile, the neighbours that successfully received the measurement update can use it to recalculate their control inputs regardless of any remaining neighbour nodes that have not received the update. Event-triggered control provides a more robust and efficient use of network bandwidth. Its implementation in multi-agent systems also provides a highly decentralised way to schedule transmission instants, which eliminates the need for the synchronisation required by periodic sampled-data approaches. Decentralised event-triggered consensus protocols give each agent the ability to broadcast information. These decisions are made locally by each agent based only on current, local information. Different authors have extended the event-triggered consensus approach, for instance, Chen and Hao [22] studied event-triggered consensus for discrete time integrators. The authors of [23] used event-triggered techniques for consensus problems involving a combination of discrete time single and double integrators. The authors of [24] studied event-triggered consensus of single integrator systems using non-linear consensus protocols. The authors of [25] investigated the event-triggered consensus of second order systems. The event-triggered consensus problem with general linear dynamics has been addressed in [26–30]. In this paper, we consider the event-triggered consensus problem of agents described by double integrator dynamics. In addition, we address the problems of non-consistent packet dropouts and non-consistent communication delays which to the best of our knowledge, have not been addressed before using the event-triggered control paradigm. Previous event-triggered consensus approaches such as [8–10] assumed that every event-based packet transmission, containing measurement updates, will arrive at its corresponding destination. In many practical scenarios however, this is not the case and packet dropouts occur. We consider the general case of packet dropouts in multi-agent systems where the same transmitted packet may arrive at some destinations and may be lost by other intended receiving agents. This type of unreliable communication is not consistent with respect to packet dropouts. Similarly, communication delays (when packets are successfully received) are not consistent if a particular broadcast packet could arrive at each receiving node at different time instants. The design of event-triggered consensus protocols in the presence of delays and packet losses represents an important challenge. Several event-triggered control methods [21, 31] demand the broadcasting and reception of long sequences of messages that include inquiries and ACKs for every single event that is generated. This type of controllers are undesired since each message is subject to delay or it may never be received at the destination nodes. The consensus protocol presented in this paper is only based on broadcasting (only one broadcasting per single event) and it does not require inquiry or acknowledgment messages. This paper extends the results in [32] where agents with single integrator dynamics were considered. The extension to address systems with double integrator dynamics is not straightforward. In particular, we do not implement a zero-order hold (ZOH) approach. In order to provide better performance in terms of reduction of communication and in the presence of packet losses and communication delays, we implement a model-based approach for multi-agent consensus [26]. The remainder of this paper is organised as follows. Section 2 provides a brief background on graph theory and describes the problem and the consensus protocol. Section 3 analyses the problem of packet dropouts using event-triggered control. Asymptotic consensus for agents with double integrator dynamics and subject to packet losses and communication delays is shown in Section 4. Examples are given in Section 5 and Section 6 concludes the paper. 2 Preliminaries Notation. The notations and represent column vectors of all ones and all zeros, respectively. and denote the set of real numbers and the set of complex numbers, respectively. For any , represents the real part of . represents a Jordan block of size corresponding to eigenvalue and denotes the Kronecker product. The boldface represents the exponential of the scalar and represents the matrix exponential of matrix . 2.1 Graph theory For a team of agents, the communication among them can be described by a directed graph , where denotes the agent set and denotes the edge set. An edge in the set denotes that agent can obtain information from agent , but not necessarily vice versa. For an edge , agent is a neighbour of agent . The set is called the set of neighbours of agent , and is its cardinality. We also define . A directed path from agent to agent is a sequence of edges in a directed graph of the form , where . A directed graph has a directed spanning tree if there exists at least one agent with directed paths to all other agents. The adjacency matrix of a directed graph is defined by if and otherwise. The Laplacian matrix of is defined as , where represents the degree matrix which is a diagonal matrix with entries . If a directed graph has a directed spanning tree, then the corresponding Laplacian matrix has only one eigenvalue equal to zero, , and the following holds for the remaining eigenvalues [33]: , for . 2.2 Problem statement Consider a group of agents with double integrator dynamics which are interconnected by means of a directed communication graph. Each agent can be described by the following equation: (1)for , where The local control inputs are given by (2)for . The variables and represent decoupled models implemented by the local agent ; their dynamics can be described by (3)and (4)The meaning of (3) and (4) is as follows. The variable represents a model of the local agent dynamics and it is updated at the event time instants denoted by . The variables represent models of the states of agents implemented by agent , such that , and they are not updated at every event time because packet dropouts occur in the communication channels. The models are updated only when a measurement from agent is successfully received by agent . The event time instant associated with a successful arrival from agent to agent is denoted as . The update rule in (4) is defined as , where represents the communication delay from agent to agent associated to the received packet . The implementation of dynamical models (3) and (4) represents a sharp difference with respect to [32]. In that paper, the ZOH approach is used where measurements received from neighbours are kept constant until new updates are received again. The use of models provides estimates of the real states of neighbours and reduce communication instants. However, the analysis and design of event-triggered controllers are more complicated due to the additional dynamics pertaining to (3) and (4). We consider the event-triggered consensus problem of agents (1) in the presence of packet losses. This means that when any agent generates its own events and broadcasts its state measurement, there is no guarantee that all destination agents will receive the transmitted state measurement. In this paper, we consider a general or non-consistent type of packet dropouts [20]. In [20] a packet of information broadcasted by a subsystem is lost in some communication links, but it is not lost in other links, i.e. it may successfully arrive to a subset of nodes. Therefore, some agents may receive different sets of measurements from the same subsystem. This means that a broadcasted measurement may be successfully received by all, some, or none of the receiving (or destination) agents , for . In addition, we consider non-consistent communication delays. In multi-agent systems consistent delays refer to the case where the delay associated with the transmitted state is the same for every receiving agent. By non-consistent delays we refer to the more general case where the delay associated to a transmitted state can be different to every receiving agent. In this case, we define as the time it takes the measurement which is released at time to arrive to agent , for every such that . For instance, if agents 2 and 3 receive information from agent 1, then the measurement released at time will arrive (if it is not dropped by either agent 2 or 3) to agent 2 at time and to agent 3 at time , where, in general, . Also, the delay in the communication channel from to is time-varying, i.e. might not be equal to for and for . 3 Packet dropouts analysis The main consequence of dealing with non-consistent packet dropouts and non-consistent communication delays is that agents , for , will generate different estimates of the state of agent since each agent will successfully receive different sets of updates from agent . In this section we characterise the state error only in the presence of packet losses. In Section 4, we consider the joint effect of packet losses and communication delays. Define the errors (5) (6)The error represents agent's local state error and it can be continuously measured in order for agent to decide when to broadcast its state. On the other hand, the error represents the state error corresponding to agent as seen by agent , which cannot be measured by agent , since agent does not know the current variable . Furthermore, error cannot be measured by agent , since agent does not have access to the state . Lastly, due to packet losses and delays, the error cannot immediately be reset to . How to manage and impose bounds on the error represents an important challenge. This problem is addressed in the remaining of this paper. Let us assume that there exist a uniform Maximum Allowable Number of Successive Dropouts (MANSD) [19, 34], denoted as , where is an integer. This means that if a measurement transmitted by agent at time is successfully received by agent , then, at most consecutive dropouts are allowed from to and, in the worst case, the measurement transmitted at time will be successfully received by agent . Due to presence of packet losses each agent will impose a maximum time between events , i.e. an event is generated if . The need for this maximum inter-event time is explained at the end of Section 4. Proposition 1.Assume that the MANSD is , for . If agent 's events, for , are generated according to the following condition: (7)where , then, the error due to packet losses is bounded by for where (8) Proof.Let us consider the state error due to packet losses. Assume without loss of generality that the last update transmitted by agent and successfully received by agent takes place at time . Hence, we have that . Agent will generate the next event at time (and broadcast its current state ). This event is generated because the condition (7) is satisfied. This means that either or the time is such that (and ). In general, we have that the following holds , where represents the value of just before the event at time takes place. Note that we consider the state errors evaluated just before the event time instants (which can be denoted as ); however, to simplify notation, we refer to the error evaluated at time instants simply as .Assume that the update at time is dropped, so . Similarly, agent will generate the following event at time (and broadcast its current state ) and we have that is satisfied. If the number of successive dropouts after the last successful update is the MANSD, , then we have that the error at time , just before the update is successfully received by agent , satisfies the following: (9)From (3), we have that for any and for any . For instance, the last term inside the norm brackets in (9) can be written as follows: Then we have that (10)Similarly, from (4) we have that and the first two terms inside the norm brackets of (10) cancel out. Then, we can write the following: Thus in the worst case, the error when the MANSD occurs is bounded by Let , for some , where . Thus, the time instant can be represented by . Similarly, and so on. We can write the following: Since, by definition, , we have that , for , and we can write In addition, let us consider and if some packet is received at then we would have that since . Then, in general, we have that for , where is given by (8). □ Hereafter, we will refer to as simply with the understanding that depends on the design parameter . 4 Decentralised event-triggered consensus protocol Let us write (1) as follows: (11)Define the vector . Then, we can write the overall system as follows: (12)where , , , and (13)Due to the pair is controllable [35], we have that for there exists a (independent of the communication graph) symmetric and positive definite solution to (14)Let (15) (16)Also, there exists a similarity transformation such that is in Jordan canonical form. Define and . Thus, we can obtain the transformed system dynamics (17)Since we have where the matrix contains Jordan blocks corresponding to the eigenvalues for . Then, the transformed system dynamics can be expressed as (18)where , represents the first two rows of and contains the remaining rows of . If the communication graph has a spanning tree then, by selection of the controller gains (15) and (16) we have that the matrix is a Hurwitz matrix. Thus, there exist such that the relation holds. Define . Let us also define as This means that Due to communication delays, the update will not be received by agent until time , in the worst-case delay. Hence, we aim at finding a bound on the error within the extended time interval . Define the following: (19)and define the error due to delays as follows: (20)for . Note that . Let us write the following: (21)We can also write . Hence, we have that (22)and the following expression is obtained: (23)These relationships are illustrated in Figs. 1 and 2 for each element of the state of agent . Note that because of the model dynamics, the element of the state representing the position is modelled as a first-order hold while the velocity component is modeled as ZOH. These cases are captured in the two-dimensional states and state errors in (19)–(23). Let . We now use the fact that and the relation to obtain for . Fig. 1Open in figure viewerPowerPoint State errors: positions Fig. 2Open in figure viewerPowerPoint State errors: velocities In addition, given a we can always guarantee that there exist some such that for . The previous relation holds because , , and is continuous in the interval . However, in Theorem 1, an estimate of the largest admissible delay will be obtained as a function of the design parameter . We have obtained the following bound on the error : (24)for , where . Theorem 1.Assume that the communication graph has a spanning tree and MANSD is , for . Then, for some , the equation (25)has only one solution and this solution is positive, i.e. . Also, agents (1) with decentralised control inputs (2) achieve consensus asymptotically in the presence of packet losses and communication delays if the event time instants, for , are generated according to condition (7) where , , (26)and Proof.To prove Theorem 1, the following observation is required. Note that because of threshold (7), the error is reset to zero at the event instants , i.e. . Thus, the error satisfies and we have that .Let us define and write , where (27)Then we have that (28)where . We can write the following: (29)The response of can be bounded as follows: As goes to infinity we have that (30)In order to show asymptotic consensus we first note that , and then, we use the similarity transformation to obtain the original state from . Note that the first column of contains the right eigenvector of associated with . (31)and the agents achieve consensus asymptotically. Note that , as it is expected, since the agents have double integrator dynamics.We will now determine admissible delay . Here, we want to determine the largest possible value of such that holds, for and for selected design parameter . The dynamics of the error can be written using (11), (19), and (20) as follows: (32)for . We have the following: where . Note that Then, we have Then, the error response during the time interval can be bounded as follows: (33)where (34)We realise that the time that it takes for the last expression in (33) to grow from zero, at time , to reach the threshold is less or equal than the time it takes the error to grow from zero, at time , to reach the same threshold. Hence, the admissible delay upper-bound needs to satisfy (35)which can also be written as (36)In order to show monotonicity of the previous expression and, ultimately, that the corresponding equation has only one root, it is convenient to write the left-hand side of (36) in terms of the coefficients and ; doing so we obtain (37)We first note that both and are positive for any . Thus, the term for any and it is also monotonically increasing. Similarly, the term is monotonically increasing. This can be shown simply by obtaining its derivative with respect to which is given by . The term is negative for and it is positive for . Hence, we conclude that the expression is monotonically increasing and (25) has exactly one root. Due to the monotonicity property, we also conclude that the largest possible value of such that (37) holds is obtained when the expression is satisfied with equality.In the previous paragraph, it is important to note that we considered the case only to show that (25) has only one root. This is practical in order to solve this equation numerically since it is guaranteed that the numerical solution represents the correct value of . However, in terms of the admissible communication delays, negative values of are meaningless. Finally, we can see that the solution of (25) is positive since the solution of is given by and the parameter . □ Finally, as in every event-triggered control approach, it is necessary to avoid the presence of Zeno behaviour. The following theorem establishes a positive lower-bound on the inter-event time intervals of every agent. The existence of some guarantees that events are never triggered infinitely fast and, therefore, guarantees that Zeno behaviour does not occur. Theorem 2.Given and communication delays where is given by the solution of (25), agents (1) implementing control inputs (2) do not exhibit Zeno behaviour if the event time instants, for , are generated according to condition (7) where , . Furthermore, the inter-event times for every agent are bounded below by the positive time , i.e. (38)where (39)and (40) Proof.In order to establish a positive lower bound on the inter-event time intervals (and avoid Zeno behaviour) we obtain the following expression for the dynamics of the error (41)for with . We have (42)Then, we obtain the following (43)Further, the error response during the time interval can be bounded as follows: (44)where .Thus, the time that it takes for the last expression in (44) to grow from zero, at time , to reach the threshold is less or equal than the time it takes the error to grow from zero, at time , to reach the same threshold and generate the following event at time , i.e. . Thus, we wish to find a lower-bound such that the following holds: (45)equivalently (46)An explicit solution for can be obtained as follows. For a given we first obtain the admissible delay and determine the values of and in (34). Note that and depend on . Then, since , the following relation holds for any Thus, we solve for in the following inequality: and the explicit solution for the lower bound on the inter-event time intervals is given by (39). By the selection , we have that for any , ensuring that remains bounded and that , i.e. the minimum inter-event time is positive. □ Remark 1.The time event given by the design parameter guarantees that the sequence . In the presence of packet dropouts and using a pure event-triggered scheme we could have the case that an agent's local state error satisfies , for , i.e. after some update at time , agent 's local control input is , for , and no further events are triggered; however, since packet dropouts are possible, some or all agents that receive information from agent may never receive the final update. The addition of a time-event ensures that all agents are finally updated with the correct state values even if the error remains equal to zero after some update. Note that the time event is not needed if packet dropouts do not exist and every measurement update always arrives at all of their destinations. Finally, and no less important, is the fact that each agent can independently select its own local parameter . The results in this paper hold in the same way by defining 5 Example Consider five agents connected using a directed graph. The entries of the adjacency matrix are given by and the remaining entries of are equal to zero. The communication channel is subject to packet dropouts and the MANSD is , where . We choose the parameters , , , . The admissible delays and the minimum inter event times are shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Admissible delays and lower-bound on inter-event time intervals Fig. 4 shows the response of the five agents. Measurement updates generated by agent based on its local events may be lost and may not be received by some or by all of the intended agents , such that , and for . The maximum number of successive dropped packets is . It can be seen from Fig. 4 that the states of all agents converge in their corresponding dimension, i.e. consensus is reached. The time intervals between events are shown for each one of the agents in Fig. 5; however, some of these updates do not reach their destinations. Fig. 6 shows the receiving time intervals from agent 2 to agents 1 and 3; it also shows the receiving time intervals from agent 3 to agents 2 and 4. It can be clearly seen, for instance, that only a fraction of the measurement updates generated by agent 2 are able to reach agents 1 or 3. Thus, the corresponding receiving time intervals are much greater, in general, than the broadcasting time intervals. A similar situation occurs to every agent, but only two agents were selected to show this scenario due to space constraints. Fig. 4Open in figure viewerPowerPoint Positions (top) and velocities (bottom) of five agents Fig. 5Open in figure viewerPowerPoint Broadcasting time intervals for each agent Fig. 6Open in figure viewerPowerPoint Receiving time intervals for agents such that (top) and (bottom) An alternative to the event-triggered consensus protocol presented in this paper is the periodic or time-driven implementation (assuming synchronisation of sampling periods and update time instants is possible). The range of values of the update period for which the agents with double integrator dynamics will achieve consensus under a time-driven or periodic model-based implementation can be obtained from [36]. Necessary and sufficient conditions for consensus of double integrators using periodic updates are provide in [36] (although delays and dropouts are not considered in that reference). Two cases can be studied: the ZOH implementation and the model-based implementation. In any case, one can use these results to conclude that reduction of communication in the time-driven strategy is limited by the range of possible values one can choose for the update period (outside this range the overall system is unstable and the agents' states do not converge), while the back-up time event used in this paper can take any finite value. For the example in this section a periodic implementation free of delays and packet losses will require an update period in the range . In the presence of non-consistent communication delays and packet losses, it is expected that the upper limit on the update periods will be smaller and more communication will be needed. However, the periodic model-based consensus of multi-agent systems in the presence of non-consistent communication delays and packet losses seems to remain as an open problem. Remark 2.The double integrator model is a simple model yet it captures some important dynamical aspects of multi-agent systems. An example with real world applications is given by the recovery of autonomous aerial vehicles. Consider a large UAV (or mother-ship) that needs to recover a group of smaller or micro-UAVs after they were deployed and they have performed their tasks. Only acceleration commands can be given to each one of the vehicles. Also, the large UAV cannot stop, but it needs to travel at a certain speed range. Hence, the single integrator model is not suitable whereas the double integrator model can capture these requirements. The consideration of directed graphs in this paper also makes this scenario feasible since the mother-ship speed profile is usually not affected by any of the micro-UAVs. 6 Conclusions The consensus problem of agents with double integrator dynamics and with packet losses and communication delays was studied in this paper. A decentralised event-triggered consensus protocol was proposed and it was shown that the group of agents achieves consensus asymptotically when they are connected using a directed graph and that Zeno behaviour is avoided. We provided methods to obtain admissible delays and to determine the lower-bounds on the inter-event time intervals. The use of event-triggered control and communication techniques allows for further reduction of communication compared to periodic implementations. More importantly, the event-triggered consensus protocol presented in this paper provides a higher level of decentralisation since it is not necessary for agents to know a global sampling period and global communication time instants as in sampled-data approaches. 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