Synchronization of Lagrangian Systems With Irregular Communication Delays

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY Synchronization of Lagrangian Systems With Irregular Communication Delays Abdelkader Abdessameud, Ilia G Polushin, and Abdelhamid Tayebi Abstract The synchronization problem of networked Euler-Lagrange systems with unknown parameters is addressed The information flow in the network is represented by a directed communication graph and is subject to unknown and possibly discontinuous time-varying communication delays with unknown upper bounds We propose a control scheme that achieves position synchronization, ie, all the positions of the systems converge to a common final position, provided that the directed communication graph contains a spanning tree The convergence analysis of the proposed scheme is based on the multidimensional small-gain framework Simulation results on a network of ten robot manipulators are given to illustrate the effectiveness of the proposed control scheme Index Terms Adaptive control, communication delays, cooperative control, small-gain theorem, synchronization I INTRODUCTION With the recent advances in the cooperative control of linear multiagent systems [1], the motion synchronization of multiple nonlinear mechanical systems has recently received an increased interest in the control community In particular, the synchronization problem of the wide class of systems modeled by Euler-Lagrange equations has been the focus of several research work with applications to robot networks [2] [5] and spacecraft formations [6] For networked dynamical systems, synchronization schemes are developed using local information exchange such that all systems reach an agreement on their states, or a common objective Although a similar problem is well studied for linear multi-agent systems, the synchronization of nonlinear systems is more challenging, especially in practical environments imposing restrictions on the information exchange in the network with the presence of communication delays The effects of communication delays in networked Euler-Lagrange systems have been studied in several recent works In [2], it has been shown that output synchronization for passive nonlinear systems can be achieved despite the communication delays provided that the interconnection between the systems in the network is represented by a balanced and strongly connected directed communication graph A similar formalism has been used in [3] and [4], where the work of [7] has been extended to solve the cooperative trajectory tracking problem of multiple Euler-Lagrange systems More recently, the authors in [8] have proposed synchronization schemes for uncertain Euler-Lagrange systems with and without reference trajectory assignment Using Lyapunov-Krasovskii functionals and the final value theorem, the authors Manuscript received December 22, 2011; revised July 26, 2012 and December 19, 2012; accepted March 29, 2013 Date of publication June 19, 2013; date of current version December 19, 2013 This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada under Discovery Grants RGPIN1510 and RGPIN/ Recommended by Associate Editor Y Hong A Abdessameud and I G Polushin are with the Department of Electrical and Computer Engineering, University of Western Ontario, N6A 5B9 London, ON, Canada ( aabdessa@ieeeorg; ipolushi@uwoca) A Tayebi is with the Department of Electrical and Computer Engineering, University of Western Ontario, N6A 5B9 London, ON, Canada and the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada ( tayebi@ieeeorg) Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAC in [8] have shown that position synchronization is achieved under a directed communication topology in the presence of communication delays In [9], a different approach has been proposed for the synchronization of nonlinear relative degree two multi-agent systems with delayed communication This approach can be applied to the class of Euler-Lagrange systems interconnected according to an undirected communication graph With the same assumption on the communication graph, a synchronization scheme that accounts for input saturations for networked Euler-Lagrange systems has been proposed in [10] However, all the results cited above rely on the basic assumption that the communication delays are constant Despite the interesting results cited above, the synchronization of networked Euler-Lagrange systems still poses several challenges in more realistic applications Indeed, communication over networks imposes communication constraints that include unknown and discontinuous time-varying communication delays To study the effects of time-varying communication delays, Lyapunov-Krasovskii functionals are generally used to derive sufficient conditions on the control parameters and the upper bounds of the communication delays so that synchronization is achieved Using this analysis method, some results have been reported in the literature related to the synchronization of nonlinear systems, see for example [11], [12] in the case of spacecraft formations and [13] in the case of unmanned aerial vehicles As witnessed in these papers, an undirected communication topology is always imposed to the interconnected systems, in addition to the differentiability and/or known upper bounds of the time-varying communication delays In this technical note, we present a solution to the synchronization problem of networked Euler-Lagrange systems in the presence of timevarying (possibly discontinuous) communication delays with unknown upper bounds The systems in the network are subject to parameter uncertainties and are interconnected according to a directed communication graph The stability and convergence analysis presented in this work is based on a new multidimensional ISS/IOS small-gain approach for systems with communication constraints similar to the one presented in [14] It is shown that synchronization is achieved under sufficient conditions that can always be satisfied by an appropriate choice of the control gains In addition, our approach relies on weak assumptions on the communication delays that can be discontinuous with unknown upper bounds Moreover, the directed communication graph is only assumed to contain a spanning tree A preliminary version of the results in this technical note has been published in [15] Simulation results that confirm the validity of the proposed approach are presented II PROBLEM STATEMENT Consider a network of not necessarily identical systems governed by the Euler-Lagrange equations of the form 1 for, with is the vector of generalized configuration coordinates, is the positive-definite inertia matrix, is the vector of coriolis/centrifugal forces, is the vector of gravitational force, and is the vector of torques associated with the system We consider the following common properties of Euler-Lagrange systems; P1 Each system in (1) admits a linear parametrization of the form, where, is a known 1 Throughout the technical note, is used to denote the Euclidean norm of a vector, and is used to denote the induced norm of a matrix (1) IEEE

2 188 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY 2014 regressor matrix, and is the vector of the system s parameters P2 The coriolis matrix is defined such that P3 For all, there exists such that In addition, and are bounded independently from To achieve synchronization, the Euler-Lagrange systems exchange information over a network described by the directed communication graph Theset is the set of nodes or vertices, describing the set of systems in the network, is the set of ordered pairs of nodes, called edges, and is the weighted adjacency matrix An edge indicates that system can receive information from system, but not necessarily vice versa The weighted adjacency matrix is defined such that, if,and if A directed path is a sequence of edges in a directed graph of the form,where A directed graph is said to contain a spanning tree if there exists at least one node having a directed path to all of the other nodes The Laplacian matrix associated to the directed graph is defined such that:,and for We assume that the model parameters of the systems in the network are not exactly known, each system can sense its state vector with no delay, and for any pair of nodes, the information of the -th system is received by the -th system with the communication delay The following assumption is imposed on the communication delays Assumption 1: For each, the communication delay can be decomposed into the sum of two terms, where the components and have the following properties: i) There exists a function such that for all,,and holds for all ii) The function satisfies: as iii) There exists such that the inequality: holds for almost all,, with iv) There exists such that: holds for almost all The subscripts and indicate that and are the smooth and the irregular components of the communication delay, respectively In particular, part i) implies the existence of an upper bound of the smooth part of the communication delays, given by, which is possibly a time-varying unbounded function that does not grow faster than the time itself Also, part iii) implies that the time derivative is well-defined for almost all and satisfies, where defined Our objective is to design a control scheme that achieves synchronization such that all systems synchronize their positions, ie,,for,with,for III PRELIMINARY RESULTS In this section, we present some definitions and technical results that will be used in the subsequent analysis Consider an affine nonlinear system of the form (2) where, for, for,and,,for,and,for, are locally Lipschitz functions of the corresponding dimensions,, We assume that for any initial condition and any inputs that are uniformly essentially bounded on, the corresponding solution is well defined for all Definition 1: A system of the form (3) is said to be weakly input-tooutput stable (WIOS) if there exist 2 and,with and, such that the following inequalities hold along the trajectories of the system for any measurable uniformly essentially bounded inputs and for all : i) uniform boundedness:,wehave ii) asymptotic gain: The notion of WIOS is a version of the input-to-output stability (IOS) property [16] (See Definition 3 in Appendix A) In the definition above, the function,where,,iscalledtheios (input-to-output stability) gain from the input to the output Inthe subsequent analysis, we will mostly deal with the case where the IOS gains are linear functions of the form, for each, where In this case, we will simply say that the system has linear IOS gains It can be checked directly that for system (3), the input-to-state stability (ISS) [16] (see Definition 2 in Appendix A) implies the WIOS property defined above Also, the input-to-output stability (IOS) property [16] implies WIOS, but the converse does not hold as demonstrated in [16, p 194] The convergence analysis in our technical note is based on the ISS/IOS small-gain approach The ISS/IOS small-gain theorems were introduced in [17] [19]; since then, a significant number of important contributions has been made in this area (see, for instance, the special issue [20] and references therein) The following WIOS small-gain theorem for systems with communication constraints is the key technical tool used in our work and is proved in Appendix B Theorem 1: Consider a system of the form (3) Suppose the system is WIOS with linear IOS gains Suppose also that each input,, is a Lebesgue measurable function satisfying and for almost all,where,and satisfy Assumption 1 Let,where,,, If,where is the spectral radius of the matrix, then the trajectories of the system (3) with input-output constraints (4), (5) are well defined for all and such that all the outputs,, and all the inputs,, are uniformly bounded and satisfy, as (4) (5) (3) 2 A continuous function is said to belong to class ( ) if it is strictly increasing and satisfies A function belongs to class if as [21] Also,,where is zero function, for all

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY IV MAIN RESULT We propose the following control scheme: where and the vector can take arbitrary initial values The matrix is symmetric positive definite and is a strictly positive scalar gain The vector is an additional input that will be designed, using the position of the -th system and the received positions of its neighbors, such that the closed loop dynamics of each system is WIOS and the small-gain result of Theorem 1 can be applied to guarantee the boundedness and convergence of the systems trajectories The main difficulty here is to design these inputs in a way that allows to fulfill the requirements of Theorem 1 through an appropriate tuning of the control gains For this purpose, we propose the following dynamic scheme to generate the variable : where, can be selected arbitrarily,, is the -th element of the adjacency matrix of the directed communication graph, and, are strictly positive scalar gains Before stating our main result, we define the following parameters: for,where, are the roots of It is clear that for strictly positive,,and Theorem 2: Consider the network of -systems described by (1), where the interconnection between the systems is described by the directed communication graph Let the controller be defined by (6) with (7)-(8) and suppose Assumption 1 holds If the control gains of each -th system with satisfy 3 (6) (7) (8) (9) (10) then and,for, are uniformly bounded and converge asymptotically to zero Furthermore, if the directed communication graph contains a spanning tree, and in Assumption 1, point i), satisfies,then all systems synchronize their positions to the same final position, ie,,forall One interesting feature of the above result is that Assumption 1 imposes very mild conditions on the communication delays In particular, the delays can be time-varying and discontinuous, and the upper bound of the delay is not required Note that the assumption that and are available is not restrictive and can be satisfied in practical situations Also, the result of Theorem 2 achieves synchronization provided that the directed communication graph contains a spanning tree, which is a weak assumption on the interconnection topology Remark 1: The small-gain condition (10) does not impose additional constraints on the communication delays, and can always be satisfied for any given values of and One possible tuning method is to set to obtain In this case, condition (10) is satisfied if (11) 3 Note that indicates that the -th system does not receive information from any other system in the network Therefore, condition (10) is imposed only on systems that receive information from at least one other neighbor in the team with It is clear that (11) can always be satisfied by choosing sufficiently large or choosing sufficiently small Note that these gains can be selected arbitrarily, however, additional consideration may need to be taken into account especially in the case of a highly irregular communication process In fact, large values of and/or large will impose large ( ) as per (11) In this case, small values of require the use of small weights of the interconnection links in the network, which may affect the system performance in terms of convergence rate Also, since and are injected in (6), large values of may lead to high input torques, at least in the transient, which may make the system more sensitive to certain types of noise A compromise is therefore needed in this case, and can always be achieved through the choice of the control gains in any practical situation V PROOF OF THEOREM 2 Let us begin by introducing the following variables: (12) (13) (14) with given in Assumption 1 The closed loop dynamics of the -th system with are obtained, using (1), (6), (7)-(8), and (12) (14), as (15) (16) (17) (18) (19) where the arguments of, and have been omitted for simplicity, the vectors,,,,and are the states, and are the inputs, and the output is given by (20) Proposition 1: The system (15) (19) with the inputs, and the output (20) is weakly IOS Moreover, the IOS gains with respect to the inputs, are and respectively Proof: Applying Lemma 1, given in Appendix C, to the trajectories of the system (19), we see that the following estimate: (21) holds for any Inequality (21) indicates, in particular, that the system (19) is ISS with respect to inputs,, with the corresponding ISS gains both equal to one Similarly, applying Lemma 1 to (17) (18) yields us to write (22) where is defined in (9) Inequality (22) implies that the system (17)-(18) is ISS with respect to inputs,, In particular, the ISS gains of the system (17)-(18) with respect to the

4 190 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY 2014 inputs, are both equal to Taking into account the fact that a cascade connection of two ISS subsystems is ISS [16], one concludes that the system (17) (19) is ISS with respect to inputs,, On the other hand, the uniform boundedness of, (with the upper bound independent on other variables) can be shown using the following Lyapunov function candidate, whose time-derivative along the trajectories of (15)-(16) is Combination of these two facts proves the uniform boundedness property given in Definition 1 To prove the asymptotic gain property, note that the input-to-state stability of (17) (19) implies that and are uniformly bounded for any uniformly bounded inputs,,since wasshowntobe uniformly bounded Consequently, we can see, using properties P1 and P3, that the right-hand side of (15) is uniformly bounded, and hence is uniformly bounded This implies that is uniformly bounded, and therefore the uniform continuity of Now, applying Barbălat lemma [21, Lemma 82], we conclude that as As a result, we can conclude that as, for any uniformly bounded inputs, This, with the ISS property of (17) (19), leads us to conclude that the system (15) (19), (20) is weakly IOS The estimates of IOS gains can be verified directly by combining (21), (22), and (20) The proof is complete Now, let us consider the -th system with In this case,, and the corresponding closed loop dynamics become (23) (24) (25) (26) with output Using the same Lyapunov function in the proof of Proposition 1, and by noting that system (25)-(26) is exponentially stable, we can conclude that (23) (26) has uniformly bounded state trajectories, and as Therefore, one can formally consider (23) (26) to be similar to (15) (19), however, with zero IOS gains with respect to inputs and From the above analysis, the networked systems (1) with input (6) and (7)-(8) can be considered as a system that consists of subsystems of the form (15) (19), where Each -th subsystem (15) (19) has two inputs,,, and one output Therefore, the overall system has inputs and outputs It is convenient to order them as follows: for,and, for Proposition 1 indicates that thus defined system is weakly IOS Moreover, based on the above described order of the inputs and outputs, the elements of the IOS gain matrix are as follows otherwise On the other hand, using Assumption 1, the following estimates of the inputs and can be derived (27) (28) where, in the latter inequality, we use notation and Therefore, the elements of the interconnection matrix are as follows: if From the above expressions, we conclude that the elements of the closed-loop gain matrix are as follows otherwise Moreover, notice that for all, which implies that for all, ie, the diagonal elements of are all zeros Taking into account that the elements of are nonnegative, one can apply the Geršgorin disc theorem [22] to conclude that if which is satisfied by (10) Now, applying the small-gain theorem (Theorem 1), we conclude that all, and,for,are uniformly bounded and,, as This, with Proposition 1, leads to the conclusion that,,, are uniformly bounded and,,, as Then, using (12) (14), we have, which implies, using (12), that is uniformly bounded and converges asymptotically to zero, for To prove the last point in the theorem, note that we can verify from parts i) and iv) of Assumption 1, with and as,that as for all Then, exploiting the above results with the relation,we conclude that as,for,which is equivalent to,where is the Laplacian matrix of the communication graph, is the identity matrix, is the vector containing all for,and is the Kronecker product Finally, since implies that if the directed communication graph contains a spanning tree [23], we conclude that as,forall Remark 2: Following the proof of Theorem 1, it can be verified that the effects of noise in the communication channels which is typically the most significant source of noise in networked systems on the system can be attenuated by an appropriate choice of the control gains To demonstrate this, assume that the delayed position is received with some additive communication error,andlet In this case, the estimate (28) becomes This, with the result of Proposition 1, indicates that the IOS gain of the system (15) (20) with respect to the disturbances in the communication channel is proportional to Using the tuning method described in Remark 1, it can be noted that increasing the gain would decrease the sensitivity of the system to the noise in the communication channels VI SIMULATION RESULTS In this section, we provide simulation results of a network of ten planar two degrees of freedom rigid manipulator arms (with revolute joints) For any generalized coordinates,the inertia matrices, Coriolis and centrifugal matrices, and gravity vectors are given, respectively, by:,, and with,,,,,

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY Fig 1 Joint angles, with Fig 2 Joint angles derivatives, with,,,and Intheseexpressions,, the variables,, are given as:,,,, and, with,,, and The following parametrization is considered for the control law (6):, with,,,,,,,,and The vector of estimated parameters is We implement the control scheme in Theorem 2, with,,for,andthe communication topology between the systems is represented by the directed graph, with, and the adjacency matrix, with,for,and zero otherwise It can be easily verified that contains a spanning tree We consider the following communication delays for each :, with, is a uniform random function, and the constants,,,,,,,,,,for It should be noted that the delay functions used here are bounded and satisfy Assumption 1 with,forall Also, the control gains are selected as:,,,and,whichleadsto,where can be computed from the adjacency matrix of Note that this choice of the gains satisfies the small-gain conditions (10) for every system in the team Figs 1 and 2 show the obtained results, where it can be seen that all systems synchronize their positions despite the presence of irregular communication delays VII CONCLUSIONS We addressed the synchronization problem of uncertain Euler-Lagrange systems in the presence of time-varying discontinuous communication delays Using the multidimensional IOS small-gain approach, we proposed a control scheme that achieves adaptive synchronization under a directed communication graph topology containing a spanning tree The sufficient conditions for synchronization derived in this technical note can be satisfied by an appropriate choice of the control gains To the best of our knowledge, the synchronization problem of Euler-Lagrange systems has been addressed in the literature only for the case of constant communication delays, which is a particular case of this work In fact, our approach handles time-varying (possibly discontinuous) communication delays with unknown upper bounds Possible directions for future research include extension of these results to the case of unbounded irregular communication delays and significant information losses, as well as synchronization with prescribed nonzero velocities in the case of directed and switching communication topologies APPENDIX A Input-to-State Stability and Input-to-Output Stability The notions of input-to-state stability and input-to-output stability [16] for MIMO (multiple inputs multiple outputs) systems can be defined as follows Definition 2: A system of the form:,issaidtobeinput-to-state stable (ISS) if there exist and,, such that the following inequalities hold along the trajectories of the system for any Lebesgue measurable uniformly essentially bounded inputs : i) uniform boundedness:,wehave ii) asymptotic gain: Definition 3: A system of the form (3) is said to be input-to-output stable (IOS) if for any initial condition and any measurable uniformly essentially bounded inputs,, the corresponding solution is well-defined for all, and there exist 4 and,,, such that the following inequality holds for all and 4 A continuous function is said to belong to class ( ) if, for each fixed, as a function of and, for each fixed, is strictly decreasing in and as [21]

6 192 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY 2014 B Proof of Theorem 1 Let be the positive orthant in, ie, Denote, and, where are the functions from the definition of WIOS (Definition 1) Also, denote,where and all are from Assumption 1 First of all, note that (4) implies that is well-defined for all Also, (4) together with the WIOS property imply that where the supremum of a vector argument is understood in the element-wise sense The inequality (5) together with local Lipschitzness of imply that the solution of the interconnected system (3), (5) exists at least for all where Furthermore, under the assumptions of the theorem, we have and therefore (31) becomes, or (32) Since has full rank, (32) implies This proves the convergence,for The convergence of,, now follows from (5) and Assumption 1, parts ii) and iv) The proof of Theorem 1 is complete C Lemma The following lemma can be found for example in [21] and [25] Lemma 1: Consider an LTI system,where is the state vector, is the input vector, and is a Hurwitz matrix such that for some Then for any initial condition, the solution of the above system satisfies: REFERENCES These inequalities imply that (29) where is the unit matrix of an appropriate dimension It is well-known that, for a matrix with all nonnegative elements, the condition is equivalent to the fact that is invertible and the inverse has all elements nonnegative (see, for example [24, page 109]) Therefore, (29) implies that (30) Now, it is easy to see that Indeed, assume the converse, ie, From (30) and (5), it follows that all inputs,, are uniformly essentially bounded for all, and therefore the solution is well-defined for all Again, the inequality (5) together with local Lipschitzness of imply that a solution of the interconnected system (3), (5) exists at least for all where This contradiction implies that ; in particular, (30) becomes The uniform boundedness of,, therefore, is proven The uniform boundedness of,, now follows directly from (4), (5) To prove convergence, note that Due to Assumption 1, we have (31) [1] WRen,RWBeard,andEMAtkins, Informationconsensusin multivehicle cooperative control: Collective group behavior through local interaction, IEEE Control Syst Mag, vol 27, no 2, pp 71 82, 2007 [2] N Chopra and M W Spong, Output synchronization of nonlinear systems with time delay in communication, in Proc IEEE Conf Decision Control, 2006, pp [3] M W Spong and N Chopra, Synchronization of networked Lagrangian systems, in Lagrangian and Hamiltonian Methods for Nonlinear Control, F Bullo and K Fujimoto, Eds Berlin, Germany: Springer Verlag, 2006, vol 366, Lecture Notes in Control and Information Sciences, pp [4] S J Chung and J J Slotine, Cooperative robot control and concurrent synchronization of Lagrangian systems, IEEE Trans Robot, vol 25, no 3, pp , 2009 [5] W Ren, Distributed leaderless consensus algorithms for networked Euler-Lagrange systems, Int J Control, vol 82, no 11, pp , 2009 [6] S J Chung, U Ahsun, and J J E Slotine, Application of synchronization to formation flying spacecraft: Lagrangian approach, J Guid, Control and Dynam, vol 32, no 2, pp , 2009 [7] J-J E Slotine and W Li, On the adaptive control of robot manipulators, Int J Robot Res, vol 6, no 3, pp 49 59, 1987 [8] E Nuño, R Ortega, L Basañez, and D Hill, Synchronization of networks of nonidentical Euler-Lagrange systems with uncertain parameters and communication delays, IEEE Trans Autom Control, vol 56, no 4, pp , Apr 2011 [9] U Münz, A A Papachristodoulou, and F Allgöwer, Robust consensus controller design for nonlinear relative degree two multi-agent systems with communication constraints, IEEE Trans Autom Control, vol 56, no 1, pp , Jan 2011 [10] A Abdessameud and A Tayebi, Synchronization of networked Lagrangian systems with input constraints, in Proc 18th IFAC World Congr, Milan, Italy, 2011, pp [11] J Erdong, J Xiaoleib, and S Zhaoweia, Robust decentralized attitude coordination control of spacecraft formation, Syst and Control Lett, vol 57, pp , 2008 [12] A Abdessameud, A Tayebi, and I G Polushin, Attitude synchronization of multiple rigid bodies with communication delays, IEEE Trans Autom Control, vol 57, no 9, pp , 2012 [13] A Abdessameud and A Tayebi, Formation control of VTOL Unmanned Aerial Vehicles with communication delays, Autom, vol 47, pp , 2011 [14] I G Polushin and S N Dashkovskiy, A small gain framework for networked cooperative teleoperation, in Proc 8th IFAC Symp Nonlinear Control Syst (NOLCOS), Bologna, Italy, 2010, pp [15] A Abdessameud, I G Polushin, and A Tayebi, Adaptive synchronization of networked Lagrangian systems with irregular communication delays, in Proc IEEE Conf Decision Control, 2012, pp

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 59, NO 1, JANUARY [16] E D Sontag, Input-to-state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory,PNistriandGStefani,Eds Berlin, Germany: Springer Verlag, 2006, pp [17] Z-P Jiang, A R Teel, and L Praly, Small-gain theorem for ISS systems and applications, Mathem of Control, Signals, and Syst, vol7, pp , 1994 [18] A R Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Trans Autom Control, vol AC-41, no 9, pp , Sep 1996 [19] Z-P Jiang and I M Y Mareels, A small-gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE Trans Autom Control, vol 42, no 3, pp , Mar 1997 [20] S Dashkovskiy, Z-P Jiang, and B Rüffer, Special issue on robust stability and control of large-scale nonlinear systems, Mathem of Control, Signals, and Syst, vol 24, no 1, pp 1 2, 2012 [21] H K Khalil, Nonlinear Systems, third ed Upper Saddle River, NJ: Prentice Hall, 2002 [22] R A Horn and C R Johnson, Matrix Analysis Cambridge, UK: Cambridge University Press, 1985 [23] W Ren and R W Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans Autom Control, vol 50, no 5, pp , May 2005 [24] M Vidyasagar, Input-Output Analysis of Large-Scale Interconnected Systems: Decomposition, Well-Posedness and Stability Berlin/ Heidelberg/New York: Springer Verlag, 1981 [25] M J Corless and A E Frazho, Linear Systems and Control: An Operator Perspective New York/Basel: Marcel Dekker, 2003 Networked Decision Making for Poisson Processes With Applications to Nuclear Detection Chetan D Pahlajani, Ioannis Poulakakis, and Herbert G Tanner Abstract This paper addresses a detection problem where a network of radiation sensors has to decide, at the end of a fixed time interval, if a moving target is a carrier of nuclear material The problem entails determining whether or not a time-inhomogeneous Poisson process due to the moving target is buried in the recorded background radiation In the proposed method, each of the sensors transmits once to a fusion center a locally processed summary of its information in the form of a likelihood ratio The fusion center then combines these messages to arrive at an optimal decision in the Neyman-Pearson framework The approach offers a pathway toward the development of novel fixed-interval detection algorithms that combine decentralized processing with optimal centralized decision making Index Terms Decision making, inhomogeneous Poisson processes, nuclear detection, sensor networks I INTRODUCTION The physical quantities of interest in many scientific problems can be captured by random processes characterized by discrete events that are highly localized in time Such phenomena can be mathematically modeled and analyzed within the framework of point processes [1] [4] Manuscript received July 26, 2012; revised February 22, 2013; accepted June 01, 2013 Date of publication June 10, 2013; date of current version December 19, 2013 Recommended by Associate Editor L Zaccarian C D Pahlajani is with the Department of Mathematical Sciences, University of Delaware, Newark, DE USA ( chetan@mathudeledu) I Poulakakis and H G Tanner are with the Department of Mechanical Engineering, University of Delaware, Newark, DE USA ( {poulakas, btanner}@udeledu) Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAC Applications include nuclear detection [5] [7], queueing networks [1], [8], optical communications [9], neuroscience [10], and others Our primary focus in this work is the detection of illicit radioactive (nuclear) materials in transit Of special interest in this regard are Poisson processes, which provide the natural models describing the emission and measurement of radiation The problem of detecting (moving and stationary) radioactive sources using networks of sensors has received a fair bit of attention in the literature In situations where the parameters (location, trajectory, activity) of the source are unknown, Bayesian methods are frequently used [6], [11] [13], embedding the issue of detection in a parameter estimation problem While powerful, Bayesian methods for source parameter estimation exhibit computational complexity exponential in the number of parameters estimated, posing challenges for their implementation in real time for networks with more than ten nodes [6], [11] An important insight and one that serves as the starting point for our analysis is that in many cases of interest, the problem of source localization can be decoupled from the problem of source detection Indeed, there are improved methods [14] [16] for tracking the carrier of a potential radioactive source using sensor modalities other than a Geiger counter Armed with this observation, source detection reduces to the problem of deciding whether the counts observed by a spatially distributed network of radiation sensors correspond solely to background radiation, or whether they also include emission from a radioactive source with known parameters In this setting, [6] explores the Signal-to-Noise Ratio (SNR) resulting from the combination of data from a network of radiation sensors, allowing for spatially varying background rates The analysis is restricted, however, to uniform linear source motion and does not provide a decision test The costs and benefits of using networked sensors for moving sources, together with a threshold test (based on the total number of recorded counts) are addressed in [17], assuming uniform background and constant geometry between source and sensor 1 For the case of a stationary source and correlated sensor measurements, a distributed detection scheme is developed in [18] using the theory of copulas The work in [11] studies detection (via Bayesian estimation) for a moving source, but the motion is required to be linear with constant velocity Detection and parameter estimation for an unknown number of static radioactive point sources are treated in [12], [13] Evidently, the networked detection problem for general source motion with spatially varying background intensity has yet to be studied Motivated by the above, we pose the following binary hypothesis testing problem: a spatially distributed network of radiation sensors records impinging photons over a fixed time interval, and has to decide at the end of the interval whether the registered counts correspond solely to background radiation, or whether the counts are the superposition of background radiation with the emission from a moving suspected radioactive source with known intensity If a source is in fact present, the relative motion between the source and the sensors leads to a time-inhomogeneous Poisson arrival process at the sensors Under the assumption of conditionally independent sensor observations, we identify an optimal Neyman-Pearson decision scheme that combines decentralized processing (local processing at each individual sensor) with centralized decision making via a fusion center The method relies on the sensors communicating processed information in the form of locally-computed likelihood ratios to the fusion center The fusion center then combines these messages to arrive at a decision, without the need for any additional information such as the location or the raw data of individual sensors This approach combines the significantly lower 1 Our analysis indicates that the optimal test involves comparing the likelihood ratio against a threshold, rather than the total number of counts IEEE