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Necessary and Sufficient Conditions for Consensus Tracking of Multi-Agent Time-Delay Systems
Proceedings, 14th IFAC Workshop on Time Delay Systems Proceedings, 14th IFAC Workshop on Time Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Proceedings, 14th IFAC IFAC Workshop Workshop on Time Time Delay Systems Proceedings, 14th on Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Available online at www.sciencedirect.com Pesti Hungary, 28-30, 2018 Pesti Vigadó, Vigadó, Budapest, Budapest, Hungary, June June 28-30, 2018 Proceedings, 14th IFAC Workshop on Time Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018
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IFAC PapersOnLine 51-14 (2018) 37–42
Necessary and Sufficient Conditions for Necessary and Sufficient Conditions for Necessary and Sufficient Conditions Necessary and Sufficient Conditions for for Consensus Tracking of Multi-Agent Consensus Tracking of Multi-Agent Necessary and Sufficient Conditions for Consensus Tracking of Multi-Agent Consensus TrackingSystems of Multi-Agent Time-Delay Time-Delay Systems Consensus Tracking of Multi-Agent Time-Delay Systems Systems Time-Delay ∗ ∗∗ ∗∗∗ Time-Delay Systems Souradip Souradip De De ∗ Soumya Soumya Ranjan Ranjan Sahoo Sahoo ∗∗ Pankaj Pankaj Wahi Wahi ∗∗∗
∗ ∗∗ ∗∗∗ Souradip Souradip De De ∗ Soumya Soumya Ranjan Ranjan Sahoo Sahoo ∗∗ Pankaj Pankaj Wahi Wahi ∗∗∗ ∗ ∗∗ ∗∗∗ Department of IIT Kanpur, Souradip De SoumyaEngineering, Ranjan Sahoo Pankaj Wahi India Department of Electrical Electrical Engineering, IIT Kanpur, Kanpur, Kanpur, India Department of of Electrical Electrical Engineering, IIT Kanpur, Kanpur, Kanpur, Kanpur, India India (e-mail: [email protected]). Department IIT (e-mail:Engineering, [email protected]). ∗∗ ∗ Department of Electrical (e-mail: [email protected]). Engineering, IIT Kanpur, Kanpur, India ∗∗Department of Electrical Engineering, IIT Kanpur, Kanpur, (e-mail: [email protected]). Department of Electrical Engineering, IIT Kanpur, Kanpur, India India ∗∗ ∗∗ Department of Electrical Engineering, IIT Kanpur, Kanpur, India (e-mail: [email protected]) (e-mail: [email protected]). Department of Electrical Engineering, IIT Kanpur, Kanpur, India (e-mail: [email protected]) ∗∗∗ ∗∗∗∗∗ Department of Mechanical (e-mail: [email protected]) Engineering, IIT Kanpur, Kanpur, Department of Electrical Engineering, IIT Kanpur, Kanpur, India (e-mail: [email protected]) Department of Mechanical Engineering, IIT Kanpur, Kanpur, ∗∗∗ ∗∗∗ Department of ofIndia Mechanical Engineering, IIT Kanpur, Kanpur, Kanpur, Kanpur, (e-mail: [email protected]) (e-mail: [email protected]) Department Mechanical Engineering, IIT India (e-mail: [email protected]) ∗∗∗ (e-mail: [email protected]) Department ofIndia Mechanical IIT Kanpur, Kanpur, India (e-mail:Engineering, [email protected]) India (e-mail: [email protected]) Abstract: Abstract: In In this this paper, paper, a a distributed distributed protocol protocol is is proposed proposed to to solve solve the the consensus consensus tracking tracking Abstract: In this paper, a distributed protocol is proposed to solve the consensus tracking problem under heterogeneous input and communication delays. In contrast Abstract: In this paper, a distributed protocol is proposed to solve the consensus tracking problem under heterogeneous input and communication delays. In contrast to to consensus consensus problem under heterogeneous input communication delays. In contrast to consensus which be achieved without knowledge the delay, aa Abstract: In paper,even a distributed protocol is of proposed to solve consensus tracking problem heterogeneous input and and communication delays. In the contrast consensus which can canunder be this achieved even without knowledge of the communication communication delay,to tracking tracking which can be achieved even without knowledge of the communication delay, tracking a general trajectory requires precise information about the individual delays. The proposed problem under heterogeneous input and communication delays. In contrast to consensus which be achieved even without knowledge of the delay, a generalcan trajectory requires precise information about the communication individual delays. Thetracking proposed general trajectory requires precise information about the individual delays. The proposed protocol is as tracking controller in with estimator which be achieved without knowledge of the delay, a general trajectory requires precise information about the communication individual delays. Thetracking proposed protocolcan is customized customized as a aeven tracking controller in conjunction conjunction with aa consensus-based consensus-based estimator protocol is customized as a tracking controller in conjunction with aa consensus-based estimator for the desired trajectory. The tracking controllers accommodate the input delays whereas general trajectory requires precise information about the individual delays. The proposed protocol is customized as a tracking controller in conjunction with consensus-based estimator for the desired trajectory. The tracking controllers accommodate the input delays whereas for the trajectory. The tracking controllers accommodate the delays whereas communication delays govern the stability of the estimators. The problem of singleprotocol is customized as a tracking controller with consensus-based for the desired desired trajectory. The controllers accommodate the input input delaysestimator communication delays govern thetracking stability of in theconjunction estimators. The atracking tracking problem ofwhereas singlecommunication delays govern the stability of the estimators. The tracking problem of singleintegrator agents is first addressed and later adapted for double-integrator agents. The choice for the desired trajectory. The tracking controllers accommodate the input delays whereas communication the stability the estimators. The tracking agents. problemThe of singleintegrator agentsdelays is firstgovern addressed and laterofadapted for double-integrator choice integrator agents is first addressed and later adapted for double-integrator agents. The choice of coupled single-integrator estimators for the double-integrator agents eases gain tuning. communication delays govern the stability of the estimators. The tracking problem of singleintegrator agents is first addressed and later adapted for double-integrator agents. The choice of coupled single-integrator estimators for the double-integrator agents eases gain tuning. of coupled single-integrator estimators for the double-integrator agents gain Simulations are to demonstrate effectiveness of technique. integrator is firstout addressed and later adapted for double-integrator agents. Thetuning. choice of coupledagents single-integrator forthe the double-integrator agents eases eases gain tuning. Simulations are carried carried out toestimators demonstrate the effectiveness of the the proposed proposed technique. Simulations are carried out to demonstrate the effectiveness of the proposed technique. of coupled single-integrator estimators for the double-integrator agents eases gain tuning. Simulations are carried out to demonstrate the effectiveness of the proposed technique. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. SimulationsTrajectory are carried out to demonstrate the effectiveness of the proposed technique. Keywords: Keywords: Trajectory Tracking, Tracking, Delay, Delay, Root Root Tendency, Tendency, Multi-Agent Multi-Agent System. System. Keywords: Trajectory Tracking, Delay, Root Tendency, Multi-Agent Keywords: Trajectory Tracking, Delay, Root Tendency, Multi-Agent System. System. Keywords: Trajectory Tracking, Delay, Root Tendency, Multi-Agent System. 1. cerned 1. INTRODUCTION INTRODUCTION cerned agent agent and and its its neighbouring neighbouring agent agent (Olfati-Saber (Olfati-Saber and and 1. INTRODUCTION cerned agent and its neighbouring agent (Olfati-Saber and Murray, 2004; Zhang et al., 2017). In our 1. INTRODUCTION cerned agent and its neighbouring agent (Olfati-Saber and Murray, 2004; Zhang et al., 2017). In our work, work, we we conconMurray, 2004; Zhang et In our work, we sider presence of communication and follow the 1. INTRODUCTION cerned agent and its (Olfati-Saber and Murray, 2004; Zhang et al., al., 2017). 2017).agent Indelay our work, we conconCooperative sider the the presence ofneighbouring communication delay and follow the Cooperative control control of of multi-agent multi-agent system system has has attracted attracted aa second sider the presence of communication delay and follow the approach. Consensus of multi-agent system with 2004; Zhang et al., 2017). Indelay our and work, we with conCooperative control of system has aa Murray, sider presence of communication follow the lot in past The of secondtheapproach. Consensus of multi-agent system Cooperative control of multi-agent multi-agent system has attracted attracted lot of of attention attention in the the past decade. decade. The study study of multimultisecond approach. of system with communication and delays has been studied sider presence Consensus of communication delay follow the lot of attention in the past decade. The of second approach. Consensus of multi-agent multi-agent system with agent include synchronization, trackCooperative control of multi-agent system has attracted a both both the communication and input input delays hasand been studied lot of cooperative attention in control the past decade. The study study of multimultiagent cooperative control include synchronization, trackboth communication and input delays has been studied in Tian and Liu (2008). The authors show that consensus second approach. Consensus of multi-agent system with agent cooperative control include synchronization, trackboth communication and input delays has been studied ing, formation, flocking. Several consensus algorithms have lot attentionflocking. in control the past decade. The algorithms study of multiin Tian and Liu (2008). The authors show that consensus agent cooperative include synchronization, tracking,of formation, Several consensus have conditions in Tian and Liu (2008). The authors show consensus not depend on communication delay. Howboth communication and input delays hasthat been studied ing, formation, flocking. Several consensus algorithms have in Tian anddo The authors show that consensus been used to these problems and be in agent cooperative control include synchronization, trackconditions doLiu not(2008). depend on communication delay. Howing, flocking. Several consensus algorithms have beenformation, used to solve solve these problems and can can be found found in ever, conditions do not depend on communication delay. Howfor the tracking problem, consensus conditions may in Tian The show conditions that consensus been used to solve these problems and can be found in conditions doLiu not(2008). depend on authors communication delay. HowPanteley (2015); Hu (2012); et (2013); ing, formation, flocking. Several algorithms have ever, forand the tracking problem, consensus may been usedand to Loria solve these problems andCao can in ever, Panteley and Loria (2015); Hu consensus (2012); Cao etbeal. al.found (2013); for the tracking problem, consensus conditions may depend both on the communication and input delay. Tian conditions do not depend on communication delay. HowPanteley and Loria (2015); Hu (2012); Cao et al. (2013); ever, forboth the tracking problem, consensus conditions may Tanner et al. (2007) and references therein. Consensus albeen used to solve these problems and can be found in depend on the communication and input delay. Tian Panteley and Loria (2015); Hu (2012); Cao et al. (2013); Tanner et al. (2007) and references therein. Consensus al- and depend on communication and delay. Tian (2009) consider diverse input delays and ever, forboth the tracking problem, consensus conditions may Tanner etare al. (2007) and therein. aldepend both on the the communication and input input delay. Tian gorithms in nature and consider exchange Panteley Loria (2015); (2012); Cao Consensus et al. (2013); and Liu Liu (2009) consider diverse input delays and solve solve Tanner al. distributive (2007) and references references therein. Consensus al- and gorithmsetand are distributive in Hu nature and consider exchange Liu (2009) consider diverse input delays and solve the tracking problem for double-integrator agents when depend both on the communication and input delay. Tian gorithms are distributive in nature and consider exchange and Liu (2009) consider diverse input delays and solve of information among neighbouring agents such that all Tanner et al. (2007) and references therein. Consensus althe tracking problem for double-integrator agents when gorithms are distributive in nature and consider of information among neighbouring agents suchexchange that all reference the tracking problem for double-integrator agents when trajectory has aa constant velocity. Xie and and Liu (2009) consider diverse input delays andCheng solve of information among neighbouring agents such that all the tracking problem double-integrator agents when agents in aa common value (Ren et gorithms are network distributive inon nature and consider exchange reference trajectory hasfor constant velocity. Xie and Cheng of information amongagree neighbouring agents such that all reference agents in the the network agree on common value (Ren et al., al., trajectory has a constant velocity. Xie and Cheng (2014) derive conditions for trajectory tracking for doublethe tracking problem for double-integrator agents when agents in the network agree on a common value (Ren et al., reference trajectory has a constant velocity. Xie and Cheng 2007; Chen al., 2011; Qin al., In the present of information among neighbouring agents that all (2014) derive conditions for trajectory tracking for doubleagents in theet agree onet a common value (Ren et al., 2007; Chen etnetwork al., 2011; Qin et al., 2012). 2012). Insuch the present (2014) derive conditions for trajectory tracking doubleintegrator agents in the of homogeneous reference trajectory constant Xie for andcommuCheng 2007; Chen et al., 2011; Qin et al., 2012). In the present (2014) derive conditions for trajectory tracking for doublework, we mainly focus on consensus tracking which means agents in the network agree on a common value (Ren et al., integrator agents in has the apresence presence ofvelocity. homogeneous commu2007; Chen et al., 2011; Qin et al., 2012). In the present work, we mainly focus on consensus tracking which means nication integrator agents in the presence of homogeneous commudelays. The work in Meng et al. (2011) addresses (2014) derive conditions for trajectory tracking for doublework, we mainly focus on consensus tracking which means integrator agents in the presence of homogeneous commuthe whole group follows autonomously a time-varying ref2007; Chen et al., 2011; Qin et al., 2012). In the present nication delays. The work in Meng et al. (2011) addresses work, we mainly focus onautonomously consensus tracking which means the whole group follows a time-varying ref- tracking nication delays. The work in Meng et al. addresses issues when both type of are and integrator agents in the presence homogeneous commuthe whole group aa time-varying refnication The work in Meng et al. (2011) (2011) addresses erence trajectory. In we address the of work, we mainly focus onautonomously consensus which means tracking delays. issues when both type ofofdelays delays are present present and the whole group follows follows autonomously time-varying reference trajectory. In particular, particular, we tracking address the effect effect of find tracking issues when both type of delays are present and conditions for uniform ultimate boundedness of the nication delays. The work in Meng et al. (2011) addresses erence trajectory. In particular, we address the effect of tracking issues when both type of delays are present and both communication and input delays on the stability the whole group follows autonomously a time-varying reffind conditions for uniform ultimate boundedness of the erence trajectory. In particular, we address the effect of both communication and input delays on the stability of tracking find conditions for uniform ultimate boundedness of the errors. To the best of our knowledge, asymptotic issues when both type of delays are present and both communication and input delays on the stability of find conditions for uniform ultimate boundedness of the our proposed consensus tracking protocol. erence trajectory. In and particular, we address effect of tracking errors. To the best of our knowledge, asymptotic both communication input delays on thethe stability our proposed consensus tracking protocol. tracking errors. To best our knowledge, asymptotic of tracking error zero in presence find conditions for uniform ultimate boundedness of the our consensus protocol. tracking errors. To the the best of of our to knowledge, asymptotic both communication andtracking input delays on theaddressed stability in of convergence convergence of the the tracking error to zero in the the presence our proposed proposed consensus tracking protocol. The problem of consensus tracking has been convergence of the tracking error to zero in the presence of heterogeneous communication and input delays has not tracking errors. To the best of our knowledge, asymptotic of the tracking errorand to zero indelays the presence The proposed problem of consensus tracking has been addressed in convergence our consensus tracking protocol. of heterogeneous communication input has not The problem of tracking has been addressed in Hu Peng et Hu al. (2015), where of heterogeneous communication and input delays has not reported yet. In De et al. (2017), a consensus-based The problem of consensus consensus tracking addressed in been convergence of the tracking error to zero in the presence of heterogeneous communication and input delays has not Hu (2012); (2012); Peng et al. al. (2013); (2013); Hu et ethas al. been (2015), where the the been reported yet. In De et al. (2017), a consensus-based Hu (2012); Peng et (2013); Hu et al. (2015), where effect of is not considered. In multi-agent system, inbeen reported yet. In al. aabeen consensus-based The problem of tracking has addressed in estimator with aa communication tracking controller designed to Hu (2012); Peng et al. al. (2013); Hu al. been (2015), where the the of heterogeneous andhas input delays has not been reported In De De et et al. (2017), (2017), consensus-based effect of delay delay is consensus not considered. In et multi-agent system, inestimator with yet. tracking controller has been designed to effect of delay is not considered. In multi-agent system, instantaneous exchange of information among neighbouring estimator with a tracking controller has been designed to Hu (2012); Peng et al. (2013); Hu et al. (2015), where the track a time-varying trajectory in the presence of heterogeeffect of delay is not considered. In multi-agent system, inbeen reported Intrajectory De et al.in(2017), abeen consensus-based with yet. a tracking controller has designed to stantaneous exchange of information among neighbouring estimator track a time-varying the presence of heterogestantaneous exchange of information among neighbouring agents may not be possible leading to communication detrack a time-varying trajectory in presence of heterogeeffect delay is be notpossible considered. In multi-agent system, dein- neous delays with no communication In stantaneous exchange of information neighbouring estimator with a tracking beendelay. designed to track time-varying trajectory in the the has presence of heterogeagentsofmay not leading toamong communication neousainput input delays with nocontroller communication delay. In this this agents may not be possible leading to communication delay. the actuation of an (dos Junior neous input delays with no communication delay. In this stantaneous exchange of information neighbouring we extend the control architecture reported in De agents mayin possible leading toamong communication de- work, track a time-varying trajectory in the presence of heterogeneous input delays with no communication delay. In this lay. Delays Delays innot thebe actuation of an agent, agent, (dos Santos Santos Junior work, we extend the control architecture reported in De lay. Delays the actuation of an (dos Santos Junior et 2015) are referred as input delays. the work, we extend the control architecture reported De agents mayin possible leading to communication de- et to heterogeneous communicalay. Delays innot thebe actuation an agent, agent, (dosIn Junior neous input delays no communication delay. Inin work, we extend thewith control architecture reported inthis De et al., al., 2015) are referred asof input delays. InSantos the existing existing et al. al. (2017) (2017) to accommodate accommodate heterogeneous communicaet al., 2015) are referred as input delays. In the existing literature, two types of consensus algorithms can be found et al. (2017) to accommodate heterogeneous communicalay. Delays in the actuation of an agent, (dos Santos Junior tion delays. The major contributions of this paper can be et al., 2015) are referred as input delays. In the existing we extend the control architecture reported in De et al.delays. (2017) to accommodate heterogeneous communicaliterature, two types of consensus algorithms can be found work, tion The major contributions of this paper can be literature, two types of consensus algorithms can be found for accommodating communication delay. The first one tion delays. The major contributions of this paper can et al., 2015) are referred as input delays. In the existing summarized as literature, two types of consensus algorithms can be found et al. (2017) to accommodate heterogeneous communicadelays. The major contributions of this paper can be be for accommodating communication delay. The first one tion summarized as for accommodating communication delay. The first one deals with the relative difference of current state of the as literature, of consensus can be found for communication first one tion The major contributions of this paper can be summarized as and dealsaccommodating with two the types relative differencealgorithms of delay. currentThe state of the summarized •• delays. Necessary deals with the relative difference of current state of the concerned agent and the delayed state of its neighbouring for accommodating communication delay. The first one Necessary and sufficient sufficient conditions conditions are are derived derived for for deals with the relative difference of current state of the summarized as concerned agent and the delayed state of its neighbouring •• tracking Necessary sufficient conditions are derived for aa and time-varying presence concerned agent and the delayed state of its neighbouring Necessary and sufficient trajectory conditions in arethe derived for agent (Seuret et al., 2008). The second approach considers deals with the relative difference of current state of the tracking time-varying trajectory in the presence concerned agent and the delayed state of its neighbouring agent (Seuret et al., 2008). The second approach considers tracking aa and time-varying trajectory in presence communication and input delays • of Necessary sufficient conditions arethe derived for agent (Seuret et The approach considers tracking time-varying trajectory in the presence the difference of delayed state the concerned agent and2008). the state ofof neighbouring of heterogeneous heterogeneous communication and input delays agent (Seuret et al., al., 2008). The second second considers the relative relative difference ofdelayed delayed stateapproach ofitsboth both the conconof heterogeneous communication and input delays tracking a time-varying trajectory in the presence the relative difference of delayed state of both the conof heterogeneous communication and input delays agent (Seuret et al., 2008). The second approach considers the relative difference of delayed state of both the conof heterogeneous communication and input delays the relative difference of delayedFederation state ofofboth the Control) con- 37 Hosting Copyright © IFAC 2405-8963 © 2018 2018, IFAC (International Automatic by Elsevier Ltd. All rights reserved. ∗ ∗ ∗ ∗
Copyright © 2018 IFAC 37 Peer review responsibility of International Federation of Automatic Copyright © 2018 37 Copyright © under 2018 IFAC IFAC 37 Control. 10.1016/j.ifacol.2018.07.195 Copyright © 2018 IFAC 37
2018 IFAC TDS 38 Budapest, Hungary, June 28-30, 2018
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
for the tracking controller coupled with a consensusbased estimator. • It is found that controller and estimator gains are inversely proportional to the input delays and communication delays respectively. Arbitrarily large bounded communication and input delays can be tolerated by suitably adjusting the gains. • Modelling the estimator as coupled single-integrator agents helps gain tuning for double-integrator agents. This benefit can be extended to higher order integrators.
lim �xd (t) − xi (t)� = 0,
t→∞
lim �x˙ d (t) − vi (t)� = 0,
t→∞
(4)
in the presence of heterogeneous communication delays τicom and input delays τiin . 3. CONSENSUS TRACKING WITH TIME-DELAYS To track a general desired trajectory we present a consensus-based estimator along with a tracking controller for each agent. The purpose of the estimator is to estimate the desired trajectory whereas the tracking controller is responsible to reduce the error between estimates and actual states of the agent. When one agent has information on the desired trajectory it updates its estimator based on the available information. In the absence of desired trajectory information, the ith agent updates its estimate using the information received from agent (i + 1).
2. PROBLEM FORMULATION The objective of a group of autonomous agents is to track a desired trajectory in spite of heterogeneous communication and input delays. We first design a protocol for a group of single-integrator agents, and then, extend this idea to double-integrator agents. In our design agent i pursues agent i + 1 if it has no information on the desired trajectory, xd (t). However, when the information on desired trajectory is available to ith agent, it pursues the desired trajectory.
3.1 Single-integrator model The control law for single-integrator agents is designed as ˆi (t)) , i = 1, 2, . . . , N, ui (t) = x˙ d t + τiin − Ki (xi (t) − x (5) where x ˆi (t) represents estimate of desired trajectory made th by i agent and Ki ∈ R is the respective controller gain. The estimator has the dynamics xi+1 (t − τ com ) xˆ˙ i (t) =x˙ d (t) + ci (1 − aid ) (ˆ
Consider a group of N agents with single-integrator dynamics. The dynamics of ith agent, i = 1, 2, . . . , N is x˙ i (t) =ui t − τiin (1) where xi ∈ R is the position of ith agent, and ui ∈ R is the control input. The control input for ith agent is delayed by τiin . The input delay τiin is heterogeneous through out the network. We also consider the presence of heterogeneous communication delay τicom for i = 1, 2, . . . , N when information flows from (i + 1)th agent to ith agent over communication channel. Assumption 1. Both the communication delays τicom and input delays τiin , ∀i are constant and bounded.
i
ˆi (t)) , (6) −ˆ xi (t − τicom )) + γi aid (xd (t) − x where ci , γi ∈ R. The weight aid is 1 if ith agent has the knowledge of xd (t), and 0 otherwise. Before deriving the conditions on the design parameters, we present a lemma which is essential to prove the necessary and sufficient conditions for tracking.
This lemma extends the condition for the roots of the quasipolynomial ρ1 (s) = s + ae−sτ to be in the open lefthalf complex plane presented in De et al. (2017). Lemma 1. The characteristic quasipolynomial ρ1 (s) = s+ ae−sτ has all the roots in the open left-half complex plane π . if and only if 0 < a < 2τ
We aim to design a protocol such that for any initial position xi (0), all agents converge to the desired trajectory, that is, lim �xd (t) − xi (t)� = 0 (2) t→∞
in the presence of heterogeneous communication delays τicom and input delays τiin . We make the following assumption along with Assumption 1: Assumption 2. For single-integrator agents, only x˙ d (t) is available to all agents for all time. Next, we attempt to solve the tracking problem when the agents have double-integrator dynamics. The dynamics of ith agent, i = 1, 2, . . . , N is given by x˙ i (t) = vi (t), (3) v˙ i (t) =ui t − τiin , where xi ∈ R and vi ∈ R are the position and velocity of ith agent, and ui ∈ R is the control input. We make the following assumption along with Assumption 1: Assumption 3. For double-integrator agents, only x ¨d (t) is available to all agents for all time. In this scenario, we aim to design a protocol such that for any initial position xi (0) and velocity vi (0), all agents converge to the desired trajectory, that is ∀i, 38
Proof. We will prove this by the method of contradiction. If a = 0, ρ1 (s) will always has a root at the origin. Now let a be negative. When τ = 0, the root of ρ1 (s) is at s = −a > 0. Therefore, the root is in the open right-half complex plane. Let us analyze the behavior of the roots as τ increases. For all the roots of ρ1 (s) to have negative real parts for some τ (> 0), roots in the open right-half plane have to cross the imaginary axis from right-half to left-half complex plane. As s = 0 cannot be a root for a < 0, the roots will have to cross the imaginary axis at jω, ω > 0. ds as We get the root sensitivity dτ s=jω ∂ρ1 (s) ase−sτ ds ∂τ = = − ∂ρ1 (s) dτ s=jω 1 − aτ e−sτ s=jω ∂s s=jω ω 2 (1 − jωτ ) ω2 −s2 = = . = 1 + sτ s=jω 1 + jωτ 1 + ω2τ 2
2018 IFAC TDS Budapest, Hungary, June 28-30, 2018
Hence, the root tendency is �
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
Proof. We define two error variables as ǫi1 (t) = xd (t) − x ˆi (t), and ǫi2 (t) = xi (t) − x ˆi (t). The tracking problem can be seen as the convergence of the error variables ǫ1 (t) and ǫ2 (t) to zero with ǫ1 = [ǫ11 ǫ21 . . . ǫN 1 ]⊤ , and ǫ2 = [ǫ12 ǫ22 . . . ǫN 2 ]⊤ . Using (5) and (6), time derivative of ǫi1 (t) and ǫi2 (t) can be found as ˆ˙ i (t) ǫ˙i1 (t) =x˙ d (t) − x
�� � ds �� RT|s=jω = sgn Re dτ �s=jω � � ω2 = +1. (7) = sgn 1 + ω2τ 2 This signifies that if at least one root is in the open righthalf complex plane, then that particular root cannot cross the imaginary axis from right to left, and hence, for a < 0 there always will be at least one root in the open righthalf complex plane for any delay 0 ≤ τ < ∞. When the parameter a > 0, then following De et al. (2017) the delay π margin can be found as τ = 2a . Therefore, all the roots of ρ1 (s) will be in the open left-half complex plane if and π only if 0 < a < . 2τ �
= − ci (1 − aid ) (ǫi1 (t − τicom ) � −ǫ(i+1)1 (t − τicom ) − γi aid ǫi1 (t),
−γ1 a1d
Γ=
..
..
. −γN aN d
sIN − Γ −
∆(s) = det
. −γi aid
N �
−sτ com i
0 ... Ci = 0 .. . 0
, 0
Ci e
i=1
−Γ −
N �
−sτ com i Ci e
sIN −
i=1
N � i=1
0 0 ··· 0 . . . . . . . . . 0 −ci (1 − aid ) ci (1 − aid ) · · · . . . . . . . . . 0 0 ··· 0
= det in −sτ
Ai e
i
The characteristic quasipolynomial of (10) is conferred in (12) and can be interpreted as ∆(s) = ρa (s).ρb (s),
π , ∀i. 2τicom
sIN − Γ −
N �
0 . . . 0 , . . . 0
−sτ com i Ci e
i=1
0
Ai = �
× det
π
, 2τicom γi > 0,
..
. −Ki
..
.
.
(11)
0
�
sIN −
N �
−sτ in i Ai e
i=1
if aid = 0, if aid = 1
∀i.
�
(12)
(15)
The error ǫ(t) decays to zero if and only if the design parameters satisfy (14) and (15). As ǫ(t) → 0, all agents track the desired trajectory xd (t). Remark 1. Controller gains Ki and estimator gains ci can be designed independently. Controller and estimator gains are inversely proportional to τiin and τicom respectively.
�
= i=1 s +�γi aid + ci (1 − aid ) e , ρb (s) with ρa (s) �N � −sτiin . The error ǫ(t) decays to zero if = i=1 s + Ki e and only if all the roots of (13) are in the open lefthalf complex plane. Following Lemma 1, the necessary and sufficient conditions for the roots of ρb (s) to be in the open left-half complex plane can be obtained as 0 < Ki <
�
0 < ci <
(13) −sτicom
�N �
(8)
ǫ˙i2 (t) =x˙ i (t) − x ˆ˙ i (t) � � = − Ki ǫi2 t − τiin − ci (1 − aid ) � � × ǫi1 (t − τicom ) − ǫ(i+1)1 (t − τicom ) − γi aid ǫi1 (t). (9) The dynamics of the errors can be represented as � � � N �� � Ci 0 Γ0 ǫ(t) ˙ = ǫ(t) + Ci 0 Γ0 i=1 � � � N �� � � � 0 0 ǫ t − τiin , (10) × ǫ (t − τicom ) + 0 Ai i=1 � ⊤ � ⊤ where ǫ(t) = ǫ1 (t) ǫ⊤ . The matrices Γ, Ci , and Ai 2 (t) are defined in (11).
Lemma 1 is useful for designing the parameters Ki and ci . The necessary and sufficient conditions for tracking a desired trajectory are presented in Theorem 1. Theorem 1. Consider a group of Single-integrator agents given by (1). The control law (5) along with the estimator (6) solves the tracking problem if and only if π 0 < Ki < in , ∀i, 2τi π 0 < ci < com , if aid = 0, 2τi , ∀i. γi > 0, if aid = 1
39
Next, the objective is to develop a tracking algorithm for double-integrator case. In view of this, the tracking controller has been designed along with an estimator that captures the structure of the estimator used in singleintegrator case. The estimator dynamics in this case resembles two coupled single-integrator estimators.
(14)
� com � is For aid = 1, the root of s + γi aid + ci (1 − aid ) e−sτi at s = −γi . This establishes that for stability when � γi > 0, com � aid = 1. When aid = 0, we require roots of s + ci e−sτi to have negative real parts. With the aid of Lemma 1, we get condition on ci as 0 < ci < 2τ πcom . Therefore, roots of i ρa (s) are in the open left-half complex plane if and only if
3.2 Double-integrator model The control law for the double-integrator agents is designed as 39
2018 IFAC TDS 40 Budapest, Hungary, June 28-30, 2018
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
� � ui (t) = x ¨d t + τiin − Kip (xi (t) − x ˆi (t)) − Kiv (vi (t) − vˆi (t)) , i = 1, 2, . . . , N,
1 τ< � tan−1 √ 2 b + b4 + 4c2 2
(16)
where x ˆi (t) and vˆi (t) represent estimated position and velocity of ith agent. The gains Kip , Kiv ∈ R are the design parameters. The estimator has the dynamics x ˆ˙ i (t) =ˆ vi (t) + cip (1 − aid ) (ˆ xi+1 (t − τicom ) −ˆ xi (t − τicom )) + γip aid (xd (t) − xˆi (t)) (17) xd (t) + civ (1 − aid ) (ˆ vi+1 (t − τicom ) vˆ˙ i (t) =¨ −ˆ vi (t − τicom )) + γiv aid (x˙ d (t) − vˆi (t)) ,
Proof. Firstly for c = 0, ρ2 (s) will have one root at origin for any τ . Similar to Lemma 1, we will prove by contradiction. When τ = 0, the roots are at s = √ −b± b2 −4c . It can be noted that there will be at least one 2 root in the closed right-half complex plane when at least one of the coefficients of ρ2 (s) is nonpositive. The root sensitivity at jω crossing can be found as � � � bs2 e−sτ + cse−sτ ds �� � = � � −sτ −sτ −sτ dτ 2s + b (−τ se + e ) − cτ e
δ˙i2 (t) = − civ (1 − aid ) (δi2 (t − τicom ) � −δ(i+1)2 (t − τicom ) − γiv aid δi2 (t),
s=jω
Hence, the root tendency is �� � � � ds �� RT|s=jω = sgn Re dτ �s=jω � � � � ω 2 2c2 + ω 2 b2 = +1. = sgn (2c − ω 2 bτ )2 + ω 2 (b + cτ )2 (18) Presence of at least one root in the closed right-half complex plane at τ = 0 and the root tendency given by (18) reveals that the ρ2 (s) has at least one root in the closed right-half complex plane always. This contradicts what we assume. Therefore, the necessary condition for stability is that b, c > 0. Under this scenario, following De et al. (2017) the condition for all roots to be in the open left-half complex plane can be found as Γp =
.
−γip aid
..
. −γN p aN d
,
Cip
0 .. . = 0 .. . 0
(20)
δ˙i3 (t) =δi4 (t) − cip (1 − aid ) (δi1 (t − τicom ) � (21) −δ(i+1)1 (t − τicom ) − γip aid δi1 (t), � � � � δ˙i4 (t) = − Kip δi3 t − τiin − Kiv δi4 t − τiin + δ˙i2 (t). (22) �⊤ � ⊤ ⊤ ⊤ ⊤ Let us define δ(t) = δ1 (t) δ2 (t) δ3 (t) δ4 (t) and hence errors (19), (20), (21), and (22) can be expressed in a more compact form as Γp IN 0 0 C 0 0 0 N ip � 0 Γv 0 0 0 Civ 0 0 ˙ = δ(t) Γ 0 0 I δ(t) + C ip 0 0 0 p N i=1 0 Civ 0 0 0 Γv 0 0 � 0 0 0 0 N � 0 0 0 0 � � in . × δ (t − τicom ) + 0 0 0 0 δ t − τi i=1 0 0 Aip Aiv (23) Here, all the matrices, Γp , Γv , Cip , Civ , Aip , Aiv are defined in (25). The characteristic quasipolynomial of (23) is conferred in (26).
ω 2 (c + jωb) −ω 2 bτ + 2c + jω(b + cτ ) � � ω 2 (c + jωb) (2c − ω 2 bτ ) − jω(b + cτ ) = (2c − ω 2 bτ )2 + ω 2 (b + cτ )2 =
..
√ b4 + 4c2 b. 2c2
Proof. We define error variables as δi1 (t) = xd (t) − x ˆi (t), δi2 (t) = x˙ d (t) − vˆi (t), δi3 (t) = xi (t) − x ˆi (t), ⊤ vi (t). Also, we define δ1 = [δ11 δ21 . . . δN 1 ] , δi4 (t) = vi (t)−ˆ ⊤ ⊤ δ2 = [δ12 δ22 . . . δN 2 ] , δ3 = [δ13 δ23 . . . δN 3 ] and δ4 = ⊤ [δ14 δ24 . . . δN 4 ] to express the error dynamics in a compact form. Using (16) and (17), the time derivative of δi1 (t), δi2 (t), δi3 (t) and δi4 (t) can be found as δ˙i1 (t) =δi2 (t) − cip (1 − aid ) (δi1 (t − τicom ) � (19) −δ(i+1)1 (t − τicom ) − γip aid δi1 (t),
2
−γ1p a1d
b2 +
Lemma 1 and 2 are useful to derive the necessary and sufficient conditions for asymptotic convergence of our control protocol for double-integrator case. The conditions on the design parameters are presented in Theorem 2. Theorem 2. Consider a group of double-integrator agents given by (3). The control law (16) along with the estimator (17) solves the tracking problem if and only if � � � √ 1+ 1+βi2 2 −1 tan βi 2 � √ , βi > 0, ∀i, 0 < Kiv < 1+ 1+βi2 in τi 2 π 0 < cip , civ < com , if aid = 0, 2τi , ∀i. γip , γiv > 0, if aid = 1
where cip , civ , γip , γiv ∈ R. Without loss of generality, we 2 can take Kip = β2i Kiv with βi ∈ R. Here, we present a lemma that analyzes the characteristics of roots of ρ2 (s) = s2 + bse−sτ + ce−sτ . This is required to prove the necessity and sufficiency of the obtained conditions. Lemma 2. The characteristic quasipolynomial ρ2 (s) = s2 + bse−sτ + ce−sτ has all the roots in the open left half complex plane if and if b, c > 0, and τ < � only √ 2 4 + 4c2 1 b + b � tan−1 b. √ 2c2 b2 + b4 + 4c2
s=jω
�
0 0 ··· 0 . . . . . . . . . 0 −cip (1 − aid ) cip (1 − aid ) · · · . . . . . . . . . 0 0 ··· 0
40
0 . . . 0 , . . . 0
0
Aip =
..
.
−Kip
..
.
,
0
(24)
2018 IFAC TDS Budapest, Hungary, June 28-30, 2018
−γ1v a1d
Γv
=
..
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
. −γiv aid
..
. −γN v aN d
∆(s) = det
sIN − Γp −
N �
,
Civ
0 ... = 0 .. . 0
−sτ com i
0 . . . 0 , . . . 0
Aiv
..
=
0
−IN
. −Kiv
sIN − Γv −
�
−sτ com i
0
i=1
−Γp −
N �
−sτ com i Cip e
0
�
× det sIN − Γv − sIN
sIN
−IN
i=1
0
−Γv −
N �
−sτ com i
Civ e
i=1
i=1 N �
Civ e
i=1
N in × det � Aip e−sτi − i=1
−sτicom
−
N �
−sτ in i
Aip e
i=1
sIN −
.
0
0
Civ e
..
.
(25)
0
N
0
0
i=1
N �
in −sτ
Aiv e
i=1
(26)
i
4. SIMULATION RESULTS
We simplify the characteristic quasipolynomial as � � N � −sτicom Cip e ∆(s) = det sIN − Γp −
0 0 ··· 0 . . . . . . . . . 0 −civ (1 − aid ) civ (1 − aid ) · · · . . . . . . . . . 0 0 ··· 0
Cip e
41
In this section, we show trajectory tracking for a group of 4 agents for both the single-integrator and double-integrator case. The desired trajectory in both the cases are same and given by xd (t) = 1 + 10 sin(t). Only agent 1 has the knowledge of the desired trajectory xd (t).
�
−IN N � in Aiv e−sτi sIN −
4.1 Illustration 1: Single-Integrator Case The communication delays and input delays are heterogeneous. The initial position, input delays and communication delays for each agent are tabulated in Table 1. Without loss of generality, we can take initial estimates of the desired trajectory as zero.
i=1
xi − xd
ρb (s).¯ ρc (s), (27) = ρ¯a (s).¯ � �N � −sτicom with ρ¯a (s) = , )e i=1 s + γip aid + cip (1 − aid� � �N −sτicom ρ¯b (s) = �i=1 s + γiv aid + civ (1 − aid )�e , and ρ¯c (s) Agents State xi (t0 ) Input delay τiin (in seconds) Communication delay τicom (in �N 2 −sτiin −sτiin seconds) s . Our control al+ K se + K e = iv ip i=1 Agent 1 2 0.3 1 Agent 2 5 0.6 0.5 gorithm will converge if and only if all the roots of the Agent 3 12 0.9 0.7 quasipolynomials ρ¯a (s), ρ¯b (s), ρ¯c (s) reside in the open leftAgent 4 15 1.2 0.1 half complex plane. According to Lemma 2, the conditions Table 1. Initial positions, input delays and on controller gains has been found as Kip , Kiv > 0, and communication delays in a group of 4 agents � � � 2 4 + 4K 2 � Kiv + Kiv ip 1 −1 � tan Kiv . According to Theorem 1, we select controller gains as τiin < � � 2 2Kip 2 + 4 +4K 2 Kiv Kiv K1 = 3.6652, K2 = 1.8326, K3 = 1.2217, K4 = 0.9163 and ip 2 estimator gains as c2 = 2.1991, c3 = 1.5708, c4 = 10.9956. 2 Therefore, with Kip = β2i Kiv , ρc (s) has roots in the open As xd (t) is known to only agent 1, we have to ensure γ1 > 0. We choose γ1 = 1. The error between the position left-half complex plane if and only if � � � √ 1+ 1+βi2 15 x1 − xd tan−1 β2i 2 x2 − xd x3 − xd 10 � √ 0 < Kiv < , βi > 0, ∀i. (28) 2 x4 − xd 1+ 1+βi τiin 2 5 The necessary and sufficient conditions for the roots of 0 ρ¯a (s), and ρ¯b (s) to have negative real parts can be obtained following the proof of Theorem 1. We get the conditions −5 on cip and civ as 0 < cip , civ <
π
, 2τicom
if aid
−10 0
= 0,
, ∀i. (29) γip , γiv > 0, if aid = 1 Therefore, error δ(t) converge to zero if and only if (28) and (29) are satified. This means position and velocity of each agent converges to that of desired trajectory.
10
20
30
Time(seconds)
40
50
60
Fig. 1. Error between position of agents and xd (t). of agents and the desired trajectory is shown in Fig. 1. It shows that the proposed control methodology ensures trajectory tracking. 41
2018 IFAC TDS 42 Budapest, Hungary, June 28-30, 2018
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4.2 Illustration 2: Double-Integrator Case
De, S., Sahoo, S.R., and Wahi, P. (2017). Trajectory tracking control with heterogeneous input delay in multiagent system. Journal of Intelligent & Robotic Systems, 1–24. dos Santos Junior, C.R., Souza, F.O., and Savino, H.J. (2015). Consensus analysis in multi-agent systems subject to delays and switching topology. IFACPapersOnLine, 48(12), 147–152. Hu, G. (2012). Robust consensus tracking of a class of second-order multi-agent dynamic systems. Systems & Control Letters, 61(1), 134–142. Hu, J., Geng, J., and Zhu, H. (2015). An observerbased consensus tracking control and application to event-triggered tracking. Communications in Nonlinear Science and Numerical Simulation, 20(2), 559–570. Meng, Z., Ren, W., Cao, Y., and You, Z. (2011). Leaderless and leader-following consensus with communication and input delays under a directed network topology. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 41(1), 75–88. Olfati-Saber, R. and Murray, R.M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Panteley, E. and Loria, A. (2015). On practical synchronisation and collective behaviour of networked heterogeneous oscillators. IFAC-PapersOnLine, 48(18), 25–30. Peng, Z., Wang, D., Li, T., and Wu, Z. (2013). Leaderless and leader-follower cooperative control of multiple marine surface vehicles with unknown dynamics. Nonlinear Dynamics, 74(1-2), 95–106. Qin, J., Zheng, W.X., and Gao, H. (2012). Coordination of multiple agents with double-integrator dynamics under generalized interaction topologies. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 42(1), 44–57. Ren, W., Beard, R.W., and Atkins, E.M. (2007). Information consensus in multivehicle cooperative control. IEEE Control Systems, 27(2), 71–82. Seuret, A., Dimarogonas, D.V., and Johansson, K.H. (2008). Consensus under communication delays. In Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, 4922–4927. IEEE. Tanner, H.G., Jadbabaie, A., and Pappas, G.J. (2007). Flocking in fixed and switching networks. IEEE Transactions on Automatic control, 52(5), 863–868. Tian, Y.P. and Liu, C.L. (2008). Consensus of multi-agent systems with diverse input and communication delays. IEEE Transactions on Automatic Control, 53(9), 2122– 2128. Tian, Y.P. and Liu, C.L. (2009). Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations. Automatica, 45(5), 1347–1353. Xie, D. and Cheng, Y. (2014). Consensus tracking control of multi-agent systems with an active virtual leader: time delay case. IET Control Theory & Applications, 8(17), 1815–1823. Zhang, Y., Li, R., Zhao, W., and Huo, X. (2017). Stochastic leader-following consensus of multi-agent systems with measurement noises and communication timedelays. Neurocomputing.
The initial position, input delays and communication delays are same as given in Illustration 4.1. The initial velocity for the 4-agent system is given by v1 (t0 ) = 3, v2 (t0 ) = 5, v3 (t0 ) = 6, v4 (t0 ) = 9. Initially, all the estimates are zero. According to Theorem 2, we design controller parameters as K1v = 2, K2v = 1.5, K3v = 1, K4v = 0.5, and βi = 1, ∀i. The estimator gains are c2p = 1.5708, c3p = 1.1220, c4p = 7.8540, c2v = 2.1991, c3v = 1.5708, c4v = 10.9956. As only agent 1 has the knowledge of xd (t), we need to design γ1p and γ1v . We set these two parameters as 1. The error between the states of each agent and the desired trajectory are shown in Figs. 2 and 3. It can be that the errors decay to zero and all agents track the desired trajectory.
xi − x d
20
x1 − xd x2 − xd x3 − xd x4 − xd
0 −20 −40 0
10
20
30
Time(seconds)
40
50
60
Fig. 2. Error between position of agents and xd (t). 40
v1 − x˙ d v2 − x˙ d v3 − x˙ d v4 − x˙ d
x˙ i − x˙ d
20 0 −20 −40 0
10
20
30
Time(seconds)
40
50
60
Fig. 3. Error between velocity of agents and x˙ d (t). 5. CONCLUSION The effect of heterogeneous input delays and communication delays in the consensus tracking problem is investigated. Necessary and sufficient conditions are derived for the proposed protocol. Controller and estimator gains are designed independently and they have inverse relationship with input and communication delays respectively. In future, the effect of time-varying delays and information loss can be investigated REFERENCES Cao, Y., Yu, W., Ren, W., and Chen, G. (2013). An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics, 9(1), 427–438. Chen, Y., L¨ u, J., Han, F., and Yu, X. (2011). On the cluster consensus of discrete-time multi-agent systems. Systems & Control Letters, 60(7), 517–523. 42
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IFAC PapersOnLine 51-14 (2018) 37–42
Necessary and Sufficient Conditions for Necessary and Sufficient Conditions for Necessary and Sufficient Conditions Necessary and Sufficient Conditions for for Consensus Tracking of Multi-Agent Consensus Tracking of Multi-Agent Necessary and Sufficient Conditions for Consensus Tracking of Multi-Agent Consensus TrackingSystems of Multi-Agent Time-Delay Time-Delay Systems Consensus Tracking of Multi-Agent Time-Delay Systems Systems Time-Delay ∗ ∗∗ ∗∗∗ Time-Delay Systems Souradip Souradip De De ∗ Soumya Soumya Ranjan Ranjan Sahoo Sahoo ∗∗ Pankaj Pankaj Wahi Wahi ∗∗∗
∗ ∗∗ ∗∗∗ Souradip Souradip De De ∗ Soumya Soumya Ranjan Ranjan Sahoo Sahoo ∗∗ Pankaj Pankaj Wahi Wahi ∗∗∗ ∗ ∗∗ ∗∗∗ Department of IIT Kanpur, Souradip De SoumyaEngineering, Ranjan Sahoo Pankaj Wahi India Department of Electrical Electrical Engineering, IIT Kanpur, Kanpur, Kanpur, India Department of of Electrical Electrical Engineering, IIT Kanpur, Kanpur, Kanpur, Kanpur, India India (e-mail: [email protected]). Department IIT (e-mail:Engineering, [email protected]). ∗∗ ∗ Department of Electrical (e-mail: [email protected]). Engineering, IIT Kanpur, Kanpur, India ∗∗Department of Electrical Engineering, IIT Kanpur, Kanpur, (e-mail: [email protected]). Department of Electrical Engineering, IIT Kanpur, Kanpur, India India ∗∗ ∗∗ Department of Electrical Engineering, IIT Kanpur, Kanpur, India (e-mail: [email protected]) (e-mail: [email protected]). Department of Electrical Engineering, IIT Kanpur, Kanpur, India (e-mail: [email protected]) ∗∗∗ ∗∗∗∗∗ Department of Mechanical (e-mail: [email protected]) Engineering, IIT Kanpur, Kanpur, Department of Electrical Engineering, IIT Kanpur, Kanpur, India (e-mail: [email protected]) Department of Mechanical Engineering, IIT Kanpur, Kanpur, ∗∗∗ ∗∗∗ Department of ofIndia Mechanical Engineering, IIT Kanpur, Kanpur, Kanpur, Kanpur, (e-mail: [email protected]) (e-mail: [email protected]) Department Mechanical Engineering, IIT India (e-mail: [email protected]) ∗∗∗ (e-mail: [email protected]) Department ofIndia Mechanical IIT Kanpur, Kanpur, India (e-mail:Engineering, [email protected]) India (e-mail: [email protected]) Abstract: Abstract: In In this this paper, paper, a a distributed distributed protocol protocol is is proposed proposed to to solve solve the the consensus consensus tracking tracking Abstract: In this paper, a distributed protocol is proposed to solve the consensus tracking problem under heterogeneous input and communication delays. In contrast Abstract: In this paper, a distributed protocol is proposed to solve the consensus tracking problem under heterogeneous input and communication delays. In contrast to to consensus consensus problem under heterogeneous input communication delays. In contrast to consensus which be achieved without knowledge the delay, aa Abstract: In paper,even a distributed protocol is of proposed to solve consensus tracking problem heterogeneous input and and communication delays. In the contrast consensus which can canunder be this achieved even without knowledge of the communication communication delay,to tracking tracking which can be achieved even without knowledge of the communication delay, tracking a general trajectory requires precise information about the individual delays. The proposed problem under heterogeneous input and communication delays. In contrast to consensus which be achieved even without knowledge of the delay, a generalcan trajectory requires precise information about the communication individual delays. Thetracking proposed general trajectory requires precise information about the individual delays. The proposed protocol is as tracking controller in with estimator which be achieved without knowledge of the delay, a general trajectory requires precise information about the communication individual delays. Thetracking proposed protocolcan is customized customized as a aeven tracking controller in conjunction conjunction with aa consensus-based consensus-based estimator protocol is customized as a tracking controller in conjunction with aa consensus-based estimator for the desired trajectory. The tracking controllers accommodate the input delays whereas general trajectory requires precise information about the individual delays. The proposed protocol is customized as a tracking controller in conjunction with consensus-based estimator for the desired trajectory. The tracking controllers accommodate the input delays whereas for the trajectory. The tracking controllers accommodate the delays whereas communication delays govern the stability of the estimators. The problem of singleprotocol is customized as a tracking controller with consensus-based for the desired desired trajectory. The controllers accommodate the input input delaysestimator communication delays govern thetracking stability of in theconjunction estimators. The atracking tracking problem ofwhereas singlecommunication delays govern the stability of the estimators. The tracking problem of singleintegrator agents is first addressed and later adapted for double-integrator agents. The choice for the desired trajectory. The tracking controllers accommodate the input delays whereas communication the stability the estimators. The tracking agents. problemThe of singleintegrator agentsdelays is firstgovern addressed and laterofadapted for double-integrator choice integrator agents is first addressed and later adapted for double-integrator agents. The choice of coupled single-integrator estimators for the double-integrator agents eases gain tuning. communication delays govern the stability of the estimators. The tracking problem of singleintegrator agents is first addressed and later adapted for double-integrator agents. The choice of coupled single-integrator estimators for the double-integrator agents eases gain tuning. of coupled single-integrator estimators for the double-integrator agents gain Simulations are to demonstrate effectiveness of technique. integrator is firstout addressed and later adapted for double-integrator agents. Thetuning. choice of coupledagents single-integrator forthe the double-integrator agents eases eases gain tuning. Simulations are carried carried out toestimators demonstrate the effectiveness of the the proposed proposed technique. Simulations are carried out to demonstrate the effectiveness of the proposed technique. of coupled single-integrator estimators for the double-integrator agents eases gain tuning. Simulations are carried out to demonstrate the effectiveness of the proposed technique. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. SimulationsTrajectory are carried out to demonstrate the effectiveness of the proposed technique. Keywords: Keywords: Trajectory Tracking, Tracking, Delay, Delay, Root Root Tendency, Tendency, Multi-Agent Multi-Agent System. System. Keywords: Trajectory Tracking, Delay, Root Tendency, Multi-Agent Keywords: Trajectory Tracking, Delay, Root Tendency, Multi-Agent System. System. Keywords: Trajectory Tracking, Delay, Root Tendency, Multi-Agent System. 1. cerned 1. INTRODUCTION INTRODUCTION cerned agent agent and and its its neighbouring neighbouring agent agent (Olfati-Saber (Olfati-Saber and and 1. INTRODUCTION cerned agent and its neighbouring agent (Olfati-Saber and Murray, 2004; Zhang et al., 2017). In our 1. INTRODUCTION cerned agent and its neighbouring agent (Olfati-Saber and Murray, 2004; Zhang et al., 2017). In our work, work, we we conconMurray, 2004; Zhang et In our work, we sider presence of communication and follow the 1. INTRODUCTION cerned agent and its (Olfati-Saber and Murray, 2004; Zhang et al., al., 2017). 2017).agent Indelay our work, we conconCooperative sider the the presence ofneighbouring communication delay and follow the Cooperative control control of of multi-agent multi-agent system system has has attracted attracted aa second sider the presence of communication delay and follow the approach. Consensus of multi-agent system with 2004; Zhang et al., 2017). Indelay our and work, we with conCooperative control of system has aa Murray, sider presence of communication follow the lot in past The of secondtheapproach. Consensus of multi-agent system Cooperative control of multi-agent multi-agent system has attracted attracted lot of of attention attention in the the past decade. decade. The study study of multimultisecond approach. of system with communication and delays has been studied sider presence Consensus of communication delay follow the lot of attention in the past decade. The of second approach. Consensus of multi-agent multi-agent system with agent include synchronization, trackCooperative control of multi-agent system has attracted a both both the communication and input input delays hasand been studied lot of cooperative attention in control the past decade. The study study of multimultiagent cooperative control include synchronization, trackboth communication and input delays has been studied in Tian and Liu (2008). The authors show that consensus second approach. Consensus of multi-agent system with agent cooperative control include synchronization, trackboth communication and input delays has been studied ing, formation, flocking. Several consensus algorithms have lot attentionflocking. in control the past decade. The algorithms study of multiin Tian and Liu (2008). The authors show that consensus agent cooperative include synchronization, tracking,of formation, Several consensus have conditions in Tian and Liu (2008). The authors show consensus not depend on communication delay. Howboth communication and input delays hasthat been studied ing, formation, flocking. Several consensus algorithms have in Tian anddo The authors show that consensus been used to these problems and be in agent cooperative control include synchronization, trackconditions doLiu not(2008). depend on communication delay. Howing, flocking. Several consensus algorithms have beenformation, used to solve solve these problems and can can be found found in ever, conditions do not depend on communication delay. Howfor the tracking problem, consensus conditions may in Tian The show conditions that consensus been used to solve these problems and can be found in conditions doLiu not(2008). depend on authors communication delay. HowPanteley (2015); Hu (2012); et (2013); ing, formation, flocking. Several algorithms have ever, forand the tracking problem, consensus may been usedand to Loria solve these problems andCao can in ever, Panteley and Loria (2015); Hu consensus (2012); Cao etbeal. al.found (2013); for the tracking problem, consensus conditions may depend both on the communication and input delay. Tian conditions do not depend on communication delay. HowPanteley and Loria (2015); Hu (2012); Cao et al. (2013); ever, forboth the tracking problem, consensus conditions may Tanner et al. (2007) and references therein. Consensus albeen used to solve these problems and can be found in depend on the communication and input delay. Tian Panteley and Loria (2015); Hu (2012); Cao et al. (2013); Tanner et al. (2007) and references therein. Consensus al- and depend on communication and delay. Tian (2009) consider diverse input delays and ever, forboth the tracking problem, consensus conditions may Tanner etare al. (2007) and therein. aldepend both on the the communication and input input delay. Tian gorithms in nature and consider exchange Panteley Loria (2015); (2012); Cao Consensus et al. (2013); and Liu Liu (2009) consider diverse input delays and solve solve Tanner al. distributive (2007) and references references therein. Consensus al- and gorithmsetand are distributive in Hu nature and consider exchange Liu (2009) consider diverse input delays and solve the tracking problem for double-integrator agents when depend both on the communication and input delay. Tian gorithms are distributive in nature and consider exchange and Liu (2009) consider diverse input delays and solve of information among neighbouring agents such that all Tanner et al. (2007) and references therein. Consensus althe tracking problem for double-integrator agents when gorithms are distributive in nature and consider of information among neighbouring agents suchexchange that all reference the tracking problem for double-integrator agents when trajectory has aa constant velocity. Xie and and Liu (2009) consider diverse input delays andCheng solve of information among neighbouring agents such that all the tracking problem double-integrator agents when agents in aa common value (Ren et gorithms are network distributive inon nature and consider exchange reference trajectory hasfor constant velocity. Xie and Cheng of information amongagree neighbouring agents such that all reference agents in the the network agree on common value (Ren et al., al., trajectory has a constant velocity. Xie and Cheng (2014) derive conditions for trajectory tracking for doublethe tracking problem for double-integrator agents when agents in the network agree on a common value (Ren et al., reference trajectory has a constant velocity. Xie and Cheng 2007; Chen al., 2011; Qin al., In the present of information among neighbouring agents that all (2014) derive conditions for trajectory tracking for doubleagents in theet agree onet a common value (Ren et al., 2007; Chen etnetwork al., 2011; Qin et al., 2012). 2012). Insuch the present (2014) derive conditions for trajectory tracking doubleintegrator agents in the of homogeneous reference trajectory constant Xie for andcommuCheng 2007; Chen et al., 2011; Qin et al., 2012). In the present (2014) derive conditions for trajectory tracking for doublework, we mainly focus on consensus tracking which means agents in the network agree on a common value (Ren et al., integrator agents in has the apresence presence ofvelocity. homogeneous commu2007; Chen et al., 2011; Qin et al., 2012). In the present work, we mainly focus on consensus tracking which means nication integrator agents in the presence of homogeneous commudelays. The work in Meng et al. (2011) addresses (2014) derive conditions for trajectory tracking for doublework, we mainly focus on consensus tracking which means integrator agents in the presence of homogeneous commuthe whole group follows autonomously a time-varying ref2007; Chen et al., 2011; Qin et al., 2012). In the present nication delays. The work in Meng et al. (2011) addresses work, we mainly focus onautonomously consensus tracking which means the whole group follows a time-varying ref- tracking nication delays. The work in Meng et al. addresses issues when both type of are and integrator agents in the presence homogeneous commuthe whole group aa time-varying refnication The work in Meng et al. (2011) (2011) addresses erence trajectory. In we address the of work, we mainly focus onautonomously consensus which means tracking delays. issues when both type ofofdelays delays are present present and the whole group follows follows autonomously time-varying reference trajectory. In particular, particular, we tracking address the effect effect of find tracking issues when both type of delays are present and conditions for uniform ultimate boundedness of the nication delays. The work in Meng et al. (2011) addresses erence trajectory. In particular, we address the effect of tracking issues when both type of delays are present and both communication and input delays on the stability the whole group follows autonomously a time-varying reffind conditions for uniform ultimate boundedness of the erence trajectory. In particular, we address the effect of both communication and input delays on the stability of tracking find conditions for uniform ultimate boundedness of the errors. To the best of our knowledge, asymptotic issues when both type of delays are present and both communication and input delays on the stability of find conditions for uniform ultimate boundedness of the our proposed consensus tracking protocol. erence trajectory. In and particular, we address effect of tracking errors. To the best of our knowledge, asymptotic both communication input delays on thethe stability our proposed consensus tracking protocol. tracking errors. To best our knowledge, asymptotic of tracking error zero in presence find conditions for uniform ultimate boundedness of the our consensus protocol. tracking errors. To the the best of of our to knowledge, asymptotic both communication andtracking input delays on theaddressed stability in of convergence convergence of the the tracking error to zero in the the presence our proposed proposed consensus tracking protocol. The problem of consensus tracking has been convergence of the tracking error to zero in the presence of heterogeneous communication and input delays has not tracking errors. To the best of our knowledge, asymptotic of the tracking errorand to zero indelays the presence The proposed problem of consensus tracking has been addressed in convergence our consensus tracking protocol. of heterogeneous communication input has not The problem of tracking has been addressed in Hu Peng et Hu al. (2015), where of heterogeneous communication and input delays has not reported yet. In De et al. (2017), a consensus-based The problem of consensus consensus tracking addressed in been convergence of the tracking error to zero in the presence of heterogeneous communication and input delays has not Hu (2012); (2012); Peng et al. al. (2013); (2013); Hu et ethas al. been (2015), where the the been reported yet. In De et al. (2017), a consensus-based Hu (2012); Peng et (2013); Hu et al. (2015), where effect of is not considered. In multi-agent system, inbeen reported yet. In al. aabeen consensus-based The problem of tracking has addressed in estimator with aa communication tracking controller designed to Hu (2012); Peng et al. al. (2013); Hu al. been (2015), where the the of heterogeneous andhas input delays has not been reported In De De et et al. (2017), (2017), consensus-based effect of delay delay is consensus not considered. In et multi-agent system, inestimator with yet. tracking controller has been designed to effect of delay is not considered. In multi-agent system, instantaneous exchange of information among neighbouring estimator with a tracking controller has been designed to Hu (2012); Peng et al. (2013); Hu et al. (2015), where the track a time-varying trajectory in the presence of heterogeeffect of delay is not considered. In multi-agent system, inbeen reported Intrajectory De et al.in(2017), abeen consensus-based with yet. a tracking controller has designed to stantaneous exchange of information among neighbouring estimator track a time-varying the presence of heterogestantaneous exchange of information among neighbouring agents may not be possible leading to communication detrack a time-varying trajectory in presence of heterogeeffect delay is be notpossible considered. In multi-agent system, dein- neous delays with no communication In stantaneous exchange of information neighbouring estimator with a tracking beendelay. designed to track time-varying trajectory in the the has presence of heterogeagentsofmay not leading toamong communication neousainput input delays with nocontroller communication delay. In this this agents may not be possible leading to communication delay. the actuation of an (dos Junior neous input delays with no communication delay. In this stantaneous exchange of information neighbouring we extend the control architecture reported in De agents mayin possible leading toamong communication de- work, track a time-varying trajectory in the presence of heterogeneous input delays with no communication delay. In this lay. Delays Delays innot thebe actuation of an agent, agent, (dos Santos Santos Junior work, we extend the control architecture reported in De lay. Delays the actuation of an (dos Santos Junior et 2015) are referred as input delays. the work, we extend the control architecture reported De agents mayin possible leading to communication de- et to heterogeneous communicalay. Delays innot thebe actuation an agent, agent, (dosIn Junior neous input delays no communication delay. Inin work, we extend thewith control architecture reported inthis De et al., al., 2015) are referred asof input delays. InSantos the existing existing et al. al. (2017) (2017) to accommodate accommodate heterogeneous communicaet al., 2015) are referred as input delays. In the existing literature, two types of consensus algorithms can be found et al. (2017) to accommodate heterogeneous communicalay. Delays in the actuation of an agent, (dos Santos Junior tion delays. The major contributions of this paper can be et al., 2015) are referred as input delays. In the existing we extend the control architecture reported in De et al.delays. (2017) to accommodate heterogeneous communicaliterature, two types of consensus algorithms can be found work, tion The major contributions of this paper can be literature, two types of consensus algorithms can be found for accommodating communication delay. The first one tion delays. The major contributions of this paper can et al., 2015) are referred as input delays. In the existing summarized as literature, two types of consensus algorithms can be found et al. (2017) to accommodate heterogeneous communicadelays. The major contributions of this paper can be be for accommodating communication delay. The first one tion summarized as for accommodating communication delay. The first one deals with the relative difference of current state of the as literature, of consensus can be found for communication first one tion The major contributions of this paper can be summarized as and dealsaccommodating with two the types relative differencealgorithms of delay. currentThe state of the summarized •• delays. Necessary deals with the relative difference of current state of the concerned agent and the delayed state of its neighbouring for accommodating communication delay. The first one Necessary and sufficient sufficient conditions conditions are are derived derived for for deals with the relative difference of current state of the summarized as concerned agent and the delayed state of its neighbouring •• tracking Necessary sufficient conditions are derived for aa and time-varying presence concerned agent and the delayed state of its neighbouring Necessary and sufficient trajectory conditions in arethe derived for agent (Seuret et al., 2008). The second approach considers deals with the relative difference of current state of the tracking time-varying trajectory in the presence concerned agent and the delayed state of its neighbouring agent (Seuret et al., 2008). The second approach considers tracking aa and time-varying trajectory in presence communication and input delays • of Necessary sufficient conditions arethe derived for agent (Seuret et The approach considers tracking time-varying trajectory in the presence the difference of delayed state the concerned agent and2008). the state ofof neighbouring of heterogeneous heterogeneous communication and input delays agent (Seuret et al., al., 2008). The second second considers the relative relative difference ofdelayed delayed stateapproach ofitsboth both the conconof heterogeneous communication and input delays tracking a time-varying trajectory in the presence the relative difference of delayed state of both the conof heterogeneous communication and input delays agent (Seuret et al., 2008). The second approach considers the relative difference of delayed state of both the conof heterogeneous communication and input delays the relative difference of delayedFederation state ofofboth the Control) con- 37 Hosting Copyright © IFAC 2405-8963 © 2018 2018, IFAC (International Automatic by Elsevier Ltd. All rights reserved. ∗ ∗ ∗ ∗
Copyright © 2018 IFAC 37 Peer review responsibility of International Federation of Automatic Copyright © 2018 37 Copyright © under 2018 IFAC IFAC 37 Control. 10.1016/j.ifacol.2018.07.195 Copyright © 2018 IFAC 37
2018 IFAC TDS 38 Budapest, Hungary, June 28-30, 2018
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
for the tracking controller coupled with a consensusbased estimator. • It is found that controller and estimator gains are inversely proportional to the input delays and communication delays respectively. Arbitrarily large bounded communication and input delays can be tolerated by suitably adjusting the gains. • Modelling the estimator as coupled single-integrator agents helps gain tuning for double-integrator agents. This benefit can be extended to higher order integrators.
lim �xd (t) − xi (t)� = 0,
t→∞
lim �x˙ d (t) − vi (t)� = 0,
t→∞
(4)
in the presence of heterogeneous communication delays τicom and input delays τiin . 3. CONSENSUS TRACKING WITH TIME-DELAYS To track a general desired trajectory we present a consensus-based estimator along with a tracking controller for each agent. The purpose of the estimator is to estimate the desired trajectory whereas the tracking controller is responsible to reduce the error between estimates and actual states of the agent. When one agent has information on the desired trajectory it updates its estimator based on the available information. In the absence of desired trajectory information, the ith agent updates its estimate using the information received from agent (i + 1).
2. PROBLEM FORMULATION The objective of a group of autonomous agents is to track a desired trajectory in spite of heterogeneous communication and input delays. We first design a protocol for a group of single-integrator agents, and then, extend this idea to double-integrator agents. In our design agent i pursues agent i + 1 if it has no information on the desired trajectory, xd (t). However, when the information on desired trajectory is available to ith agent, it pursues the desired trajectory.
3.1 Single-integrator model The control law for single-integrator agents is designed as ˆi (t)) , i = 1, 2, . . . , N, ui (t) = x˙ d t + τiin − Ki (xi (t) − x (5) where x ˆi (t) represents estimate of desired trajectory made th by i agent and Ki ∈ R is the respective controller gain. The estimator has the dynamics xi+1 (t − τ com ) xˆ˙ i (t) =x˙ d (t) + ci (1 − aid ) (ˆ
Consider a group of N agents with single-integrator dynamics. The dynamics of ith agent, i = 1, 2, . . . , N is x˙ i (t) =ui t − τiin (1) where xi ∈ R is the position of ith agent, and ui ∈ R is the control input. The control input for ith agent is delayed by τiin . The input delay τiin is heterogeneous through out the network. We also consider the presence of heterogeneous communication delay τicom for i = 1, 2, . . . , N when information flows from (i + 1)th agent to ith agent over communication channel. Assumption 1. Both the communication delays τicom and input delays τiin , ∀i are constant and bounded.
i
ˆi (t)) , (6) −ˆ xi (t − τicom )) + γi aid (xd (t) − x where ci , γi ∈ R. The weight aid is 1 if ith agent has the knowledge of xd (t), and 0 otherwise. Before deriving the conditions on the design parameters, we present a lemma which is essential to prove the necessary and sufficient conditions for tracking.
This lemma extends the condition for the roots of the quasipolynomial ρ1 (s) = s + ae−sτ to be in the open lefthalf complex plane presented in De et al. (2017). Lemma 1. The characteristic quasipolynomial ρ1 (s) = s+ ae−sτ has all the roots in the open left-half complex plane π . if and only if 0 < a < 2τ
We aim to design a protocol such that for any initial position xi (0), all agents converge to the desired trajectory, that is, lim �xd (t) − xi (t)� = 0 (2) t→∞
in the presence of heterogeneous communication delays τicom and input delays τiin . We make the following assumption along with Assumption 1: Assumption 2. For single-integrator agents, only x˙ d (t) is available to all agents for all time. Next, we attempt to solve the tracking problem when the agents have double-integrator dynamics. The dynamics of ith agent, i = 1, 2, . . . , N is given by x˙ i (t) = vi (t), (3) v˙ i (t) =ui t − τiin , where xi ∈ R and vi ∈ R are the position and velocity of ith agent, and ui ∈ R is the control input. We make the following assumption along with Assumption 1: Assumption 3. For double-integrator agents, only x ¨d (t) is available to all agents for all time. In this scenario, we aim to design a protocol such that for any initial position xi (0) and velocity vi (0), all agents converge to the desired trajectory, that is ∀i, 38
Proof. We will prove this by the method of contradiction. If a = 0, ρ1 (s) will always has a root at the origin. Now let a be negative. When τ = 0, the root of ρ1 (s) is at s = −a > 0. Therefore, the root is in the open right-half complex plane. Let us analyze the behavior of the roots as τ increases. For all the roots of ρ1 (s) to have negative real parts for some τ (> 0), roots in the open right-half plane have to cross the imaginary axis from right-half to left-half complex plane. As s = 0 cannot be a root for a < 0, the roots will have to cross the imaginary axis at jω, ω > 0. ds as We get the root sensitivity dτ s=jω ∂ρ1 (s) ase−sτ ds ∂τ = = − ∂ρ1 (s) dτ s=jω 1 − aτ e−sτ s=jω ∂s s=jω ω 2 (1 − jωτ ) ω2 −s2 = = . = 1 + sτ s=jω 1 + jωτ 1 + ω2τ 2
2018 IFAC TDS Budapest, Hungary, June 28-30, 2018
Hence, the root tendency is �
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
Proof. We define two error variables as ǫi1 (t) = xd (t) − x ˆi (t), and ǫi2 (t) = xi (t) − x ˆi (t). The tracking problem can be seen as the convergence of the error variables ǫ1 (t) and ǫ2 (t) to zero with ǫ1 = [ǫ11 ǫ21 . . . ǫN 1 ]⊤ , and ǫ2 = [ǫ12 ǫ22 . . . ǫN 2 ]⊤ . Using (5) and (6), time derivative of ǫi1 (t) and ǫi2 (t) can be found as ˆ˙ i (t) ǫ˙i1 (t) =x˙ d (t) − x
�� � ds �� RT|s=jω = sgn Re dτ �s=jω � � ω2 = +1. (7) = sgn 1 + ω2τ 2 This signifies that if at least one root is in the open righthalf complex plane, then that particular root cannot cross the imaginary axis from right to left, and hence, for a < 0 there always will be at least one root in the open righthalf complex plane for any delay 0 ≤ τ < ∞. When the parameter a > 0, then following De et al. (2017) the delay π margin can be found as τ = 2a . Therefore, all the roots of ρ1 (s) will be in the open left-half complex plane if and π only if 0 < a < . 2τ �
= − ci (1 − aid ) (ǫi1 (t − τicom ) � −ǫ(i+1)1 (t − τicom ) − γi aid ǫi1 (t),
−γ1 a1d
Γ=
..
..
. −γN aN d
sIN − Γ −
∆(s) = det
. −γi aid
N �
−sτ com i
0 ... Ci = 0 .. . 0
, 0
Ci e
i=1
−Γ −
N �
−sτ com i Ci e
sIN −
i=1
N � i=1
0 0 ··· 0 . . . . . . . . . 0 −ci (1 − aid ) ci (1 − aid ) · · · . . . . . . . . . 0 0 ··· 0
= det in −sτ
Ai e
i
The characteristic quasipolynomial of (10) is conferred in (12) and can be interpreted as ∆(s) = ρa (s).ρb (s),
π , ∀i. 2τicom
sIN − Γ −
N �
0 . . . 0 , . . . 0
−sτ com i Ci e
i=1
0
Ai = �
× det
π
, 2τicom γi > 0,
..
. −Ki
..
.
.
(11)
0
�
sIN −
N �
−sτ in i Ai e
i=1
if aid = 0, if aid = 1
∀i.
�
(12)
(15)
The error ǫ(t) decays to zero if and only if the design parameters satisfy (14) and (15). As ǫ(t) → 0, all agents track the desired trajectory xd (t). Remark 1. Controller gains Ki and estimator gains ci can be designed independently. Controller and estimator gains are inversely proportional to τiin and τicom respectively.
�
= i=1 s +�γi aid + ci (1 − aid ) e , ρb (s) with ρa (s) �N � −sτiin . The error ǫ(t) decays to zero if = i=1 s + Ki e and only if all the roots of (13) are in the open lefthalf complex plane. Following Lemma 1, the necessary and sufficient conditions for the roots of ρb (s) to be in the open left-half complex plane can be obtained as 0 < Ki <
�
0 < ci <
(13) −sτicom
�N �
(8)
ǫ˙i2 (t) =x˙ i (t) − x ˆ˙ i (t) � � = − Ki ǫi2 t − τiin − ci (1 − aid ) � � × ǫi1 (t − τicom ) − ǫ(i+1)1 (t − τicom ) − γi aid ǫi1 (t). (9) The dynamics of the errors can be represented as � � � N �� � Ci 0 Γ0 ǫ(t) ˙ = ǫ(t) + Ci 0 Γ0 i=1 � � � N �� � � � 0 0 ǫ t − τiin , (10) × ǫ (t − τicom ) + 0 Ai i=1 � ⊤ � ⊤ where ǫ(t) = ǫ1 (t) ǫ⊤ . The matrices Γ, Ci , and Ai 2 (t) are defined in (11).
Lemma 1 is useful for designing the parameters Ki and ci . The necessary and sufficient conditions for tracking a desired trajectory are presented in Theorem 1. Theorem 1. Consider a group of Single-integrator agents given by (1). The control law (5) along with the estimator (6) solves the tracking problem if and only if π 0 < Ki < in , ∀i, 2τi π 0 < ci < com , if aid = 0, 2τi , ∀i. γi > 0, if aid = 1
39
Next, the objective is to develop a tracking algorithm for double-integrator case. In view of this, the tracking controller has been designed along with an estimator that captures the structure of the estimator used in singleintegrator case. The estimator dynamics in this case resembles two coupled single-integrator estimators.
(14)
� com � is For aid = 1, the root of s + γi aid + ci (1 − aid ) e−sτi at s = −γi . This establishes that for stability when � γi > 0, com � aid = 1. When aid = 0, we require roots of s + ci e−sτi to have negative real parts. With the aid of Lemma 1, we get condition on ci as 0 < ci < 2τ πcom . Therefore, roots of i ρa (s) are in the open left-half complex plane if and only if
3.2 Double-integrator model The control law for the double-integrator agents is designed as 39
2018 IFAC TDS 40 Budapest, Hungary, June 28-30, 2018
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
� � ui (t) = x ¨d t + τiin − Kip (xi (t) − x ˆi (t)) − Kiv (vi (t) − vˆi (t)) , i = 1, 2, . . . , N,
1 τ< � tan−1 √ 2 b + b4 + 4c2 2
(16)
where x ˆi (t) and vˆi (t) represent estimated position and velocity of ith agent. The gains Kip , Kiv ∈ R are the design parameters. The estimator has the dynamics x ˆ˙ i (t) =ˆ vi (t) + cip (1 − aid ) (ˆ xi+1 (t − τicom ) −ˆ xi (t − τicom )) + γip aid (xd (t) − xˆi (t)) (17) xd (t) + civ (1 − aid ) (ˆ vi+1 (t − τicom ) vˆ˙ i (t) =¨ −ˆ vi (t − τicom )) + γiv aid (x˙ d (t) − vˆi (t)) ,
Proof. Firstly for c = 0, ρ2 (s) will have one root at origin for any τ . Similar to Lemma 1, we will prove by contradiction. When τ = 0, the roots are at s = √ −b± b2 −4c . It can be noted that there will be at least one 2 root in the closed right-half complex plane when at least one of the coefficients of ρ2 (s) is nonpositive. The root sensitivity at jω crossing can be found as � � � bs2 e−sτ + cse−sτ ds �� � = � � −sτ −sτ −sτ dτ 2s + b (−τ se + e ) − cτ e
δ˙i2 (t) = − civ (1 − aid ) (δi2 (t − τicom ) � −δ(i+1)2 (t − τicom ) − γiv aid δi2 (t),
s=jω
Hence, the root tendency is �� � � � ds �� RT|s=jω = sgn Re dτ �s=jω � � � � ω 2 2c2 + ω 2 b2 = +1. = sgn (2c − ω 2 bτ )2 + ω 2 (b + cτ )2 (18) Presence of at least one root in the closed right-half complex plane at τ = 0 and the root tendency given by (18) reveals that the ρ2 (s) has at least one root in the closed right-half complex plane always. This contradicts what we assume. Therefore, the necessary condition for stability is that b, c > 0. Under this scenario, following De et al. (2017) the condition for all roots to be in the open left-half complex plane can be found as Γp =
.
−γip aid
..
. −γN p aN d
,
Cip
0 .. . = 0 .. . 0
(20)
δ˙i3 (t) =δi4 (t) − cip (1 − aid ) (δi1 (t − τicom ) � (21) −δ(i+1)1 (t − τicom ) − γip aid δi1 (t), � � � � δ˙i4 (t) = − Kip δi3 t − τiin − Kiv δi4 t − τiin + δ˙i2 (t). (22) �⊤ � ⊤ ⊤ ⊤ ⊤ Let us define δ(t) = δ1 (t) δ2 (t) δ3 (t) δ4 (t) and hence errors (19), (20), (21), and (22) can be expressed in a more compact form as Γp IN 0 0 C 0 0 0 N ip � 0 Γv 0 0 0 Civ 0 0 ˙ = δ(t) Γ 0 0 I δ(t) + C ip 0 0 0 p N i=1 0 Civ 0 0 0 Γv 0 0 � 0 0 0 0 N � 0 0 0 0 � � in . × δ (t − τicom ) + 0 0 0 0 δ t − τi i=1 0 0 Aip Aiv (23) Here, all the matrices, Γp , Γv , Cip , Civ , Aip , Aiv are defined in (25). The characteristic quasipolynomial of (23) is conferred in (26).
ω 2 (c + jωb) −ω 2 bτ + 2c + jω(b + cτ ) � � ω 2 (c + jωb) (2c − ω 2 bτ ) − jω(b + cτ ) = (2c − ω 2 bτ )2 + ω 2 (b + cτ )2 =
..
√ b4 + 4c2 b. 2c2
Proof. We define error variables as δi1 (t) = xd (t) − x ˆi (t), δi2 (t) = x˙ d (t) − vˆi (t), δi3 (t) = xi (t) − x ˆi (t), ⊤ vi (t). Also, we define δ1 = [δ11 δ21 . . . δN 1 ] , δi4 (t) = vi (t)−ˆ ⊤ ⊤ δ2 = [δ12 δ22 . . . δN 2 ] , δ3 = [δ13 δ23 . . . δN 3 ] and δ4 = ⊤ [δ14 δ24 . . . δN 4 ] to express the error dynamics in a compact form. Using (16) and (17), the time derivative of δi1 (t), δi2 (t), δi3 (t) and δi4 (t) can be found as δ˙i1 (t) =δi2 (t) − cip (1 − aid ) (δi1 (t − τicom ) � (19) −δ(i+1)1 (t − τicom ) − γip aid δi1 (t),
2
−γ1p a1d
b2 +
Lemma 1 and 2 are useful to derive the necessary and sufficient conditions for asymptotic convergence of our control protocol for double-integrator case. The conditions on the design parameters are presented in Theorem 2. Theorem 2. Consider a group of double-integrator agents given by (3). The control law (16) along with the estimator (17) solves the tracking problem if and only if � � � √ 1+ 1+βi2 2 −1 tan βi 2 � √ , βi > 0, ∀i, 0 < Kiv < 1+ 1+βi2 in τi 2 π 0 < cip , civ < com , if aid = 0, 2τi , ∀i. γip , γiv > 0, if aid = 1
where cip , civ , γip , γiv ∈ R. Without loss of generality, we 2 can take Kip = β2i Kiv with βi ∈ R. Here, we present a lemma that analyzes the characteristics of roots of ρ2 (s) = s2 + bse−sτ + ce−sτ . This is required to prove the necessity and sufficiency of the obtained conditions. Lemma 2. The characteristic quasipolynomial ρ2 (s) = s2 + bse−sτ + ce−sτ has all the roots in the open left half complex plane if and if b, c > 0, and τ < � only √ 2 4 + 4c2 1 b + b � tan−1 b. √ 2c2 b2 + b4 + 4c2
s=jω
�
0 0 ··· 0 . . . . . . . . . 0 −cip (1 − aid ) cip (1 − aid ) · · · . . . . . . . . . 0 0 ··· 0
40
0 . . . 0 , . . . 0
0
Aip =
..
.
−Kip
..
.
,
0
(24)
2018 IFAC TDS Budapest, Hungary, June 28-30, 2018
−γ1v a1d
Γv
=
..
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
. −γiv aid
..
. −γN v aN d
∆(s) = det
sIN − Γp −
N �
,
Civ
0 ... = 0 .. . 0
−sτ com i
0 . . . 0 , . . . 0
Aiv
..
=
0
−IN
. −Kiv
sIN − Γv −
�
−sτ com i
0
i=1
−Γp −
N �
−sτ com i Cip e
0
�
× det sIN − Γv − sIN
sIN
−IN
i=1
0
−Γv −
N �
−sτ com i
Civ e
i=1
i=1 N �
Civ e
i=1
N in × det � Aip e−sτi − i=1
−sτicom
−
N �
−sτ in i
Aip e
i=1
sIN −
.
0
0
Civ e
..
.
(25)
0
N
0
0
i=1
N �
in −sτ
Aiv e
i=1
(26)
i
4. SIMULATION RESULTS
We simplify the characteristic quasipolynomial as � � N � −sτicom Cip e ∆(s) = det sIN − Γp −
0 0 ··· 0 . . . . . . . . . 0 −civ (1 − aid ) civ (1 − aid ) · · · . . . . . . . . . 0 0 ··· 0
Cip e
41
In this section, we show trajectory tracking for a group of 4 agents for both the single-integrator and double-integrator case. The desired trajectory in both the cases are same and given by xd (t) = 1 + 10 sin(t). Only agent 1 has the knowledge of the desired trajectory xd (t).
�
−IN N � in Aiv e−sτi sIN −
4.1 Illustration 1: Single-Integrator Case The communication delays and input delays are heterogeneous. The initial position, input delays and communication delays for each agent are tabulated in Table 1. Without loss of generality, we can take initial estimates of the desired trajectory as zero.
i=1
xi − xd
ρb (s).¯ ρc (s), (27) = ρ¯a (s).¯ � �N � −sτicom with ρ¯a (s) = , )e i=1 s + γip aid + cip (1 − aid� � �N −sτicom ρ¯b (s) = �i=1 s + γiv aid + civ (1 − aid )�e , and ρ¯c (s) Agents State xi (t0 ) Input delay τiin (in seconds) Communication delay τicom (in �N 2 −sτiin −sτiin seconds) s . Our control al+ K se + K e = iv ip i=1 Agent 1 2 0.3 1 Agent 2 5 0.6 0.5 gorithm will converge if and only if all the roots of the Agent 3 12 0.9 0.7 quasipolynomials ρ¯a (s), ρ¯b (s), ρ¯c (s) reside in the open leftAgent 4 15 1.2 0.1 half complex plane. According to Lemma 2, the conditions Table 1. Initial positions, input delays and on controller gains has been found as Kip , Kiv > 0, and communication delays in a group of 4 agents � � � 2 4 + 4K 2 � Kiv + Kiv ip 1 −1 � tan Kiv . According to Theorem 1, we select controller gains as τiin < � � 2 2Kip 2 + 4 +4K 2 Kiv Kiv K1 = 3.6652, K2 = 1.8326, K3 = 1.2217, K4 = 0.9163 and ip 2 estimator gains as c2 = 2.1991, c3 = 1.5708, c4 = 10.9956. 2 Therefore, with Kip = β2i Kiv , ρc (s) has roots in the open As xd (t) is known to only agent 1, we have to ensure γ1 > 0. We choose γ1 = 1. The error between the position left-half complex plane if and only if � � � √ 1+ 1+βi2 15 x1 − xd tan−1 β2i 2 x2 − xd x3 − xd 10 � √ 0 < Kiv < , βi > 0, ∀i. (28) 2 x4 − xd 1+ 1+βi τiin 2 5 The necessary and sufficient conditions for the roots of 0 ρ¯a (s), and ρ¯b (s) to have negative real parts can be obtained following the proof of Theorem 1. We get the conditions −5 on cip and civ as 0 < cip , civ <
π
, 2τicom
if aid
−10 0
= 0,
, ∀i. (29) γip , γiv > 0, if aid = 1 Therefore, error δ(t) converge to zero if and only if (28) and (29) are satified. This means position and velocity of each agent converges to that of desired trajectory.
10
20
30
Time(seconds)
40
50
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Fig. 1. Error between position of agents and xd (t). of agents and the desired trajectory is shown in Fig. 1. It shows that the proposed control methodology ensures trajectory tracking. 41
2018 IFAC TDS 42 Budapest, Hungary, June 28-30, 2018
Souradip De et al. / IFAC PapersOnLine 51-14 (2018) 37–42
4.2 Illustration 2: Double-Integrator Case
De, S., Sahoo, S.R., and Wahi, P. (2017). Trajectory tracking control with heterogeneous input delay in multiagent system. Journal of Intelligent & Robotic Systems, 1–24. dos Santos Junior, C.R., Souza, F.O., and Savino, H.J. (2015). Consensus analysis in multi-agent systems subject to delays and switching topology. IFACPapersOnLine, 48(12), 147–152. Hu, G. (2012). Robust consensus tracking of a class of second-order multi-agent dynamic systems. Systems & Control Letters, 61(1), 134–142. Hu, J., Geng, J., and Zhu, H. (2015). An observerbased consensus tracking control and application to event-triggered tracking. Communications in Nonlinear Science and Numerical Simulation, 20(2), 559–570. Meng, Z., Ren, W., Cao, Y., and You, Z. (2011). Leaderless and leader-following consensus with communication and input delays under a directed network topology. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 41(1), 75–88. Olfati-Saber, R. and Murray, R.M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Panteley, E. and Loria, A. (2015). On practical synchronisation and collective behaviour of networked heterogeneous oscillators. IFAC-PapersOnLine, 48(18), 25–30. Peng, Z., Wang, D., Li, T., and Wu, Z. (2013). Leaderless and leader-follower cooperative control of multiple marine surface vehicles with unknown dynamics. Nonlinear Dynamics, 74(1-2), 95–106. Qin, J., Zheng, W.X., and Gao, H. (2012). Coordination of multiple agents with double-integrator dynamics under generalized interaction topologies. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 42(1), 44–57. Ren, W., Beard, R.W., and Atkins, E.M. (2007). Information consensus in multivehicle cooperative control. IEEE Control Systems, 27(2), 71–82. Seuret, A., Dimarogonas, D.V., and Johansson, K.H. (2008). Consensus under communication delays. In Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, 4922–4927. IEEE. Tanner, H.G., Jadbabaie, A., and Pappas, G.J. (2007). Flocking in fixed and switching networks. IEEE Transactions on Automatic control, 52(5), 863–868. Tian, Y.P. and Liu, C.L. (2008). Consensus of multi-agent systems with diverse input and communication delays. IEEE Transactions on Automatic Control, 53(9), 2122– 2128. Tian, Y.P. and Liu, C.L. (2009). Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations. Automatica, 45(5), 1347–1353. Xie, D. and Cheng, Y. (2014). Consensus tracking control of multi-agent systems with an active virtual leader: time delay case. IET Control Theory & Applications, 8(17), 1815–1823. Zhang, Y., Li, R., Zhao, W., and Huo, X. (2017). Stochastic leader-following consensus of multi-agent systems with measurement noises and communication timedelays. Neurocomputing.
The initial position, input delays and communication delays are same as given in Illustration 4.1. The initial velocity for the 4-agent system is given by v1 (t0 ) = 3, v2 (t0 ) = 5, v3 (t0 ) = 6, v4 (t0 ) = 9. Initially, all the estimates are zero. According to Theorem 2, we design controller parameters as K1v = 2, K2v = 1.5, K3v = 1, K4v = 0.5, and βi = 1, ∀i. The estimator gains are c2p = 1.5708, c3p = 1.1220, c4p = 7.8540, c2v = 2.1991, c3v = 1.5708, c4v = 10.9956. As only agent 1 has the knowledge of xd (t), we need to design γ1p and γ1v . We set these two parameters as 1. The error between the states of each agent and the desired trajectory are shown in Figs. 2 and 3. It can be that the errors decay to zero and all agents track the desired trajectory.
xi − x d
20
x1 − xd x2 − xd x3 − xd x4 − xd
0 −20 −40 0
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Time(seconds)
40
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60
Fig. 2. Error between position of agents and xd (t). 40
v1 − x˙ d v2 − x˙ d v3 − x˙ d v4 − x˙ d
x˙ i − x˙ d
20 0 −20 −40 0
10
20
30
Time(seconds)
40
50
60
Fig. 3. Error between velocity of agents and x˙ d (t). 5. CONCLUSION The effect of heterogeneous input delays and communication delays in the consensus tracking problem is investigated. Necessary and sufficient conditions are derived for the proposed protocol. Controller and estimator gains are designed independently and they have inverse relationship with input and communication delays respectively. In future, the effect of time-varying delays and information loss can be investigated REFERENCES Cao, Y., Yu, W., Ren, W., and Chen, G. (2013). An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics, 9(1), 427–438. Chen, Y., L¨ u, J., Han, F., and Yu, X. (2011). On the cluster consensus of discrete-time multi-agent systems. Systems & Control Letters, 60(7), 517–523. 42