Coupled Distributed Estimation and Control for Mobile Sensor Networks

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 10, OCTOBER 2012 2609

and � �� , �� , � � ��. Now, bythe two-step optimization algorithm in Remark 2 starting with ��

��, we obtain the values of �, �� and � in Table I with different �

and �. Table I shows that for the present example, when � and �

increase, the value of �� decreases, which implies the smaller admis-sible initial domain. Now, consider the case with � � ��, � � ��,and � � �� , �� , � � �� in Table I. In this case, wecan calculate the corresponding state feedback controller

��

Then, the simulation results of state trajectories of the closed-loop sat-urated system, and the corresponding domains �� , ��, and�� are given in Fig. 1.

V. CONCLUSION

In this technical note, we have developed a method to designstate feedback controllers for a class of uncertain discrete time-delaysystems subject to control input saturation and bounded externaldisturbances. The designed controllers guarantee that there exists anadmissible initial condition domain such that the closed-loop system isuniformly exponentially convergent to a ball with certain exponentialdecay rate for every initial condition from the admissible domain.

REFERENCES

[1] E.-K. Boukas, “Discrete-time systems with time-varying time delay:Stability and stabilizability,” Math. Problems Eng., vol. 2006, pp. 1–10,2006.

[2] Y.-Y. Cao, Z. Lin, and T. Hu, “Stability analysis of linear time-delaysystems subject to input saturation,” IEEE Trans. Circuits Syst. I, vol.49, pp. 233–240, 2002.

[3] W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependent guaranteedcost control for uncertain discrete time systems with delay,” Proc. Inst.Elect. Eng., vol. 150, pp. 412–416, 2003.

[4] H. Fang and Z. Lin, “A further result on global stabilization of oscil-lators with bounded delayed input,” IEEE Trans. Autom. Control, vol.51, pp. 121–128, 2006.

[5] E. Fridman and U. Shaked, “Stability and guaranteed cost control ofuncertain discrete delay system,” Int. J. Control, vol. 78, pp. 235–246,2005.

[6] J. M. Gomes da Silva, Jr., F. Lescher, and D. Eckhard, “Design oftime-varying controllers for discrete-time linear systems with input sat-uration,” IET Control Theory Appl., vol. 1, pp. 155–162, 2007.

[7] J. M. Gomes da Silva, Jr. and S. Tarbouriech, “Local stabilization ofdiscrete-time linear systems with saturating controls: An LMI-basedapproach,” IEEE Trans. Autom. Control, vol. 46, pp. 119–125, 2001.

[8] A. Hmamed, A. Benzaouia, and H. Bensalah, “Regulator problem forlinear continuous time-delay systems with nonsymmetrical constrainedcontrol,” IEEE Trans. Autom. Control, vol. 40, pp. 1615–1619, 1995.

[9] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysisand Design. Boston, MA: Birkhäuser, 2001.

[10] T. Hu, Z. Lin, and B. M. Chen, “Analysis and design for discrete-timelinear systems subject to actuator saturation,” Syst. Control Lett., vol.45, pp. 97–112, 2002.

[11] W. Lan and J. Huang, “Semiglobal stabilization and output regulationof singular linear systems with input saturation,” IEEE Trans. Autom.Control, vol. 48, pp. 1274–1280, 2003.

[12] V. J. S. Leite and M. F. Miranda, “Robust stabilization of discrete-timesystems with time-varying delay: An LMI approach,” Math. ProblemsEng., vol. 2008, pp. 1–15, 2008.

[13] Z. Lin and A. Saberi, “Semi-global exponential stabilization of lineardiscretetime systems subject to input saturation via linear feedbacks,”Syst. Control Lett., vol. 24, pp. 125–132, 2001.

[14] S. Oucheriah, “Robust exponential convergence of a class of linear de-layed systems with bounded controllers and disturbances,” Automatica,vol. 42, pp. 1863–1867, 2006.

[15] I. R. Petersen, “A stabilization algorithm for a class of uncertain linearsystems,” Syst. Control Lett., vol. 8, pp. 351–357, 1987.

[16] S. Tarbouriech, G. Garcia, P. L. D. Peres, and I. Queinnec, “Stabi-lization of linear discrete time delay systems with additive disturbanceand saturating actuators,” in Proc. IFAC Symp. Robust Control Design,Prague, Czech Republic, Jun. 21–23, 2000, pp. 261–266.

[17] S. Tarbouriech and J. M. Gomes da Silva, Jr., “Synthesis of controllersfor continuous-time delay systems with saturating controls via LMI’s,”IEEE Trans. Autom. Control, vol. 45, pp. 105–111, 2000.

[18] S. Tarbouriech, P. L. D. Peres, G. Garcia, and I. Queinnec, “Delay-de-pendent stabilisation and disturbance tolerance for time-delay systemssubject to actuator saturation,” Proc. Inst. Elect. Eng., vol. 149, pp.387–393, 2002.

[19] K. Yakoubi and Y. Chitour, “Linear systems subject to input saturationand time delay: Global asymptotic stabilization,” IEEE Trans. Autom.Control, vol. 52, pp. 874–879, 2007.

[20] B. Zhou, Z. Lin, and G. Duan, “Stabilization of linear systems withinput delay and saturation-a parametric Lyapunov equation approach,”Int. J. Robust Nonlin. Control, vol. 20, pp. 1502–1519, 2010.

[21] Z. Zuo, D. W. C. Ho, and Y. Wang, “Fault tolerant control for sin-gular systems with actuator saturation and nonlinear perturbation,” Au-tomatica, vol. 46, pp. 569–576, 2010.

Coupled Distributed Estimation andControl for Mobile Sensor Networks

Reza Olfati-Saber and Parisa Jalalkamali

Abstract—In this paper, we introduce a theoretical framework for cou-pled distributed estimation and motion control of mobile sensor networksfor collaborative target tracking. We use a Fisher Information theoreticmetric for quality of sensed data. The mobile sensing agents seek to improvethe information value of their sensed data while maintaining a safe-distancefrom other neighboring agents (i.e., perform information-driven flocking).We provide a formal stability analysis of continuous Kalman-Consensus fil-tering (KCF) algorithm on a mobile sensor network with a flocking-basedmobility control model. The discrete-time counterpart of this coupled es-timation and control algorithm is successfully applied to tracking of twotypes of targets with stochastic linear and nonlinear dynamics.

Index Terms—Collaborative target tracking, distributed Kalman fil-tering, flocking, information-driven control, mobile sensor networks.

I. INTRODUCTION

Collaborative tracking of multiple targets (or events) in an environ-ment arise in a variety of surveillance and security applications andintelligent transportation. Most of the past research on target trackinghas been focused on the use of centralized algorithms that run on staticmultisensor platforms [1]. Centralized Kalman filtering plays a crucialrole in such target tracking algorithms.

Distributed estimation for static sensor networks has attracted manyresearchers in recent years [2]–[7]. The existing distributed algorithms

Manuscript received May 25, 2011; revised October 16, 2011 and October 16,2011; accepted February 04, 2012. Date of publication March 08, 2012; date ofcurrent version September 21, 2012. This work was presented in part at ACC 11.This work was supported in part by an NSF CAREER award. Recommended byAssociate Editor H. Zhang.

The authors are with the Thayer School of Engineering, DartmouthCollege, Hanover, NH 03755 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2012.2190184

0018-9286/$31.00 © 2012 IEEE

2610 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 10, OCTOBER 2012

for target tracking using mobile sensor networks are extremely limitedto a few instances [8], [9]. In [10] the KCF algorithm of the first authoris successfully applied to multitarget tracking using camera networks.

In this paper, we present a systematic analysis framework for mobilesensor networks with a flocking-based mobility control model that runa novel distributed Kalman filtering algorithm [11] for collaborativetracking of a single target.

The sensors in our framework have an information value function�� where �� denotes the target range and defined as the dis-tance between the agent and the predicted position of target �. In ad-dition, �� is a decreasing function of the target range. According tothis model of quality of sensed data, the information value of a sensorincreases as the sensor comes closer to the target. This notion of theinformation value that was also used in [8] is the same as the trace ofthe Fisher Information Matrix (FIM) of sensed data for target trackingapplications [12], [13].

We propose a solution to the problem of collision-free tracking of amobile target via mobile sensor networks using a combination of theflocking and Kalman-Consensus Filtering algorithms of [2] and [11].

The major challenge in analysis of the resulting coupled estimationand control algorithm for mobile sensor networks that we call informa-tion-driven flocking is that each sensing agent �� has its own dedicated�-agent called �� (See [14] for the definition of �- and �-agent). Thestate of �� is the estimate of the state of target � by agent � and the� dif-ferent estimates �� of the target are distinct. In the flocking algorithmspresented in [14], all � �-agents are the same. This change results in aperturbed structural dynamics of the flock where the perturbation termsdepend on the estimation errors.

Our main result is to establish that the coupled distributed estima-tion and control algorithm for a mobile sensor network has a combinedcost (Lyapunov function) that is monotonically decreasing in time andguarantees reaching a consensus on estimates of the state of the targetby all mobile sensors. We also introduce a cascade nonlinear normalform and stability analysis for structural dynamics of mobile sensornetworks performing information-driven flocking.

The outline of the paper is as follows. Some basic notations andproblem setup are discussed in Section II. Our main theoretical resultson distributed target tracking algorithms for mobile sensor networksare provided in Section III. Our experimental results are presented inSection V. Finally, concluding remarks are made in Section VI.

II. PRELIMINARIES: NOTATIONS AND PROBLEM SETUP

Consider � mobile sensors �� with the dynamics��

(1)

where �� and the goal to track the state of a mobile target� with dynamics

�� (2)

The sensing agents make the following partial-state noisy measure-ments of the state of �

�� (3)

where the matrices , �, and �� are generally time-varying and ofappropriate dimensions and � and �� are zero-mean Gaussian noise.

Let � � �� be the proximity graph (network) of the mobile sen-sors. The set of vertices of � is � � �� . Let � � � bethe interaction range of every sensor. Then, the set of edges of � is atime-varying set defined as

�� (4)

and the set of neighbors �� of sensor � on this proximity network is

��

The main problem of interest is to design distributed motion controland estimation algorithms that achieve two objectives: i) the group ofsensing agents improve their collective information value

�� and ii)

avoid collisions during tracking of target �. We refer to this problem as“information-driven flocking.” We propose a solution to this problemusing a combination of flocking and Kalman-Consensus Filtering al-gorithms [11].

III. DISTRIBUTED TRACKING WITH MOBILE SENSORS

The Kalman-Consensus filtering algorithm (or Algorithm 1) relieson reaching a consensus on estimates obtained by local Kalman filtersrather than distributed averaging-based Kalman filtering. Algorithm 1is the discrete-time analog of the continuous-time Kalman-Consensusfilter described in the following.

Theorem 1: (Kalman-Consensus Filter [2]) Consider a sensor net-work with a continuous-time linear sensing model in (3). Suppose eachnode applies the following distributed estimation algorithm

��

��

��

��

��

�� (5)

with a Kalman-Consensus estimator and initial conditions �� and �� . Then, the collective dynamics of the estima-tion errors !� � � � �� (without noise) is a stable linear systemwith a Lyapunov function � �!� � �

�� !��

�� !�. Moreover, ��

�� !� � � where

�� "��

��

��

and �" � " � �� is the #-dimensional Laplacian of the network (�denotes the Kronocker product). Moreover, all estimators asymptoti-cally reach a consensus, i.e., �� .

The following flocking algorithm is a modified form of [14, Alg. 2].Algorithm 2: (flocking with � distinct �-agents) Let ��

�� be the estimate of the state of target � by mobile sensor� obtained via Kalman-Consensus filtering. Then, each sensing agent�� with dynamics in (1) applies the following distributed control tointeract with its neighboring sensors on ��:

� ��

$ ��

%�� (6)

where �� is a linear feedback for tracking particle �� with state ��:

�� &�� &�� &�� &� � � (7)

where �� ' � � (�� is a subnormal vector con-necting agent � to agent �. Please, refer to [14] for the definitions of $ ,the )-norm � �� , and smooth adjacency elements %��.

Algorithm 1 Kalman-Consensus Filter [11] (one cycle)

Given ��, ��, and messages #� � �� *� � �� +� �� ,

1:Obtain measurement �� with covariance ��.2:Compute information vector and matrix of node �

��

��

*� ��

��

3:Broadcast message#� � �� ,�� to neighbors.4:Receive messages from all neighbors.5:Fuse information matrices and vectors

-� ��

�� .� ��

*� �


6:Compute the Kalman-Consensus state estimate

��

�

��

��

��

� �

��

��

��

� � ��

7:Update the state of the information filter (�� is the updated �)

��

� � ��

��

Remark 1: According to the flocking framework in [14], there existsa smooth potential function in explicit form

��

��

��

� (8)

with � � �� such that �� can be stated as a distributed

gradient-based control:

��

�� (9)

�� denotes the partial derivative with respect to ��.Note that the state estimates generated by Algorithm 1 is directly

used in (7) of Algorithm 2 for distributed mobility-control of the sen-sors. We refer to the combined Algorithms 1 and 2 as the cascade dis-tributed estimation and control algorithm for collision-free distributedtracking of a mobile target �. The analysis of the this discrete-time cou-pled estimation and control algorithm is tremendously challenging andis one of our future research objectives.

In this paper, we seek to provide the stability analysis of the con-tinuous-time version of this coupled distributed estimation and controlalgorithm.

IV. STABILITY ANALYSIS: COUPLED DISTRIBUTED

ESTIMATION AND CONTROL ALGORITHMS

The formulation of our main analytical result as well as the followingassumptions are inspired by our experimental observations and consis-tent collective behavior of a group of mobile sensors tracking two typesof mobile targets: 1) a linear target; and 2) a maneuverable nonlineartarget called particle-in-the-box. Both models of the motion of targetswill be discussed in detail in Section V. The notions of flocks, structuralstability, and cohesion of flocks are used in the following propositionand defined in [14].

A flock is a connected network of dynamic agents. Flocking is thecollective behavior of a network of dynamic agents with the objectiveto self-assemble and maintain a connected network in a collision-freemanner. A flock is called cohesive if all the agents can be contained ina ball of finite radius.

Assumption 1: Assume there exists a finite time �� such thatthe proximity graph �� becomes connected for all � � ��.

The following definition clarifies that the Laplacian and algebraicconnectivity of the networks used in flocking and KCF algorithms arenot the same.

Definition 1: (Laplacian and �� of the proximity networks inflocking vs. KCF) Let �� be the smooth adjacency elementsof the proximity network of mobile agents with configuration� � � �� . We represent the adjacency matrix of flockingwith �� and its Laplacian and algebraic connectivitywith � and � � � �� , respectively. The adjacency matrix

� � �� of networked filters in KCF has 0–1 elements, i.e.,�� if �� and �� , otherwise. Similarly, we denotethe Laplacian and algebraic connectivity of the networked filters with�� and �� , respectively.

Assumption 2: Assume there exist constant thresholds �� such that the algebraic connectivity functions � �� and �� along the trajectoryof mobile agents cross the levels � and �, respectively, at time�� and remain above those threshold valuesthereafter, i.e., � �� for all � � ��.

Assumption 3: The parameters �� in the tracking feedback�� of the flocking algorithm satisfy � ! � ! � and � � � � �where � is defined in Assumption 2.

Here is our main theoretical result:Proposition 1: Consider a network of � mobile sensing agents with

dynamics (1), the sensing model in (3), and the proximity graph ��with the set of edges (4). Suppose that the agents apply the Kalman-Consensus filter in (5) to obtain � estimates �� of the state of a mobiletarget � with dynamics (2). These state estimates of the target determinethe states of � �-agents ��. Suppose that every sensing agent " tracksits associated �-agent �� by applying the flocking algorithm in (6). Let�� and � be the collective dynamics of the � networked estimatorsand mobility-controlled agents, respectively, and denote their cascadewith �. Then, the following statements hold:

(i) � can be separated into three subsystems that consist of thestructural and translational dynamics of the group of mobile sen-sors in cascade with the error dynamics of the Kalman-Con-sensus filter.

(ii) Given Assumption 1, the agents form a cohesive flock in finitetime.

(iii) Suppose that Assumptions 1 through 3 hold. Then, the solutionsof the structural dynamics of the flock of mobile sensors areasymptotically stable.

(iv) Given the assumptions in part (iii), all estimators asymptoticallyreach a consensus on the state estimates of the target �� (for the error dynamics of KCF with zero noise).

The proof of proposition 1 is relatively lengthy; therefore, we presentthe proof in separate parts.

Proof of Part (i)

Let us first determine the error dynamics of the Kalman-Consensusfilter in (5). The estimation error of sensor " is defined as #� � �� ,thus error dynamics of (5) (without noise) is in the form:

�#� � ��#� � ��

��

�#� � #��

with �� $�%�. Defining block diagonal matrices � � ��and � � �� and # � � ��#��, one can rewrite the last equation

�# � �# � �� # � ��# (10)

where �� . According to Theorem 1, the error dynamics�# � ��# is stable and has a quadratic Lyapunov function & �#� �#��# � � #

�� #�.

The flocking dynamics of the agents can be written as

��

��

��

� ��

(11)

or

��

��

��

� ��

�


After defining the block matrix � � �� , one can express thelast equation in a form with an input ��:

��

��

��

with a linear tracking feedback

��

This enables us to express the dynamics of � as the cascade of its esti-mation and control subsystems �� and �:

� ��

�� (12)

where �� is a positive definite damping matrix, � �� , and � � �� is a constant matrix. System (12) is thecascade normal form of estimation and control subsystems of a mobilesensor network in which its sensing agents apply the flocking algorithmfor mobility control and the Kalman-Consensus filter for distributedtracking.

According to [14], since �� is a linear feedback, the flocking dy-namics � can be further decomposed as the cascade of structural andtranslational dynamics of particles. The position and velocity of thecenter of mass (CM) of the particles is given by

� ��

��

��

��

��

Consider a moving frame centered at �. Then, the position and ve-locity of agent � can be written as �� and �� .We refer to the dynamics of the motion of the group of agents in themoving frame coordinates as structural dynamics. The structural andtranslational dynamics of � can be written as

��

��

with �� representing the column vector of ones and

� ��

where the perturbation terms � � �� and �� depend on the targetestimation errors by the sensors and are defined as

��

��

��

��

��

��

��

The normal form of � can be written as follows

��

��

� ��

��

Proof of Parts (ii) to (iv)

The solutions of the structural dynamics in cascade with �� is calledcohesive for all � � � if the position of all agents remains in a ball ofradius �� for � � �. Note that this cascade nonlinear system is glob-ally Lipschitz and all of its solutions are bounded for arbitrary initial

Fig. 1. Experimental results for the linear target: (a) MSE for distributed targettracking. (b) Average information value. (c) � plots for flocking (smoothblue curve) and Kalman-Consensus filtering (piecewise constant red curve).(d) A target trajectory and fused estimates of 20 sensors.

conditions. The global Lipschitz property is a byproduct of the designof the smooth potential function �� which has a globally boundedgradient. This implies that over the interval �� the solutions of thecascade system and therefore the position of all agent remain bounded.For all � � � , the proximity graph �� is connected and thus hasa finite diameter �� at any time �. Define the diameter ofthe flock as

��

��

Then, �� and by setting �� the position of the agents remain cohesive for all � � � inside a ballof radius ��.

To establish stability of the flock, we need to construct an en-ergy-type Lyapunov function � for the cascade of �� and ��. Let�� be the Hamiltonian of the unper-turbed structural dynamics �� and � �� be the Lyapunovfunction of ��. We propose the following Lyapunov function for thecascade nonlinear system ��:

��

� � �� (13)

Before computing ��, let us state a simple inequality. For an � ! matrix " and two vectors � � � and # � �, the followinginequality holds:

��"#� ��

�� "#��

�

�� $��"�#��

In the special case of " � � � �� , we have

��#� ��

�� #

��

where �� . By direct differentiation, we obtain

��

� ��


From Theorem 1 and Assumptions 1 and 2, for all � � ��, one gets

��

where �� always exists based on Assumption2.

Now, let us compute �� . We have

��

�

��

Note that ��

�� thus

�

��

��

��

In addition, �� . Hence

��

��

��

��

��

� ��

� ��

�

�

�

��

��

��

Based on the above upper bounds, we get

��

��

Given the fact that

��

and setting �� , one concludes

��

� � ��

if the following two conditions hold:

��

��

� (14)

Given the definition of ��

and �� and Assumption 2, we have ��

�� and �� . By choosing � � �� and � � � � �� (as inAssumption 3) both conditions will be satisfied. Thus

��

Based on LaSalle’s invariance principle, for any set of initial condi-tions, the solutions of the cascade system �� asymptotically con-verge to the largest invariant set in

� � ��

where � is the equilibria of the unperturbed structural dynamics. Fromthe equilibria in �, only the local minima of �� are asymptoticallystable.

The proof of part (iv) is a byproduct of the above stability analysis:the estimation errors �� asymptotically vanish for all sensors and there-fore all state estimates become the same.

Remark 2: If in addition to Assumptions 1 through 3, Conjectures1 and 2 in [14] hold, then almost every solution of the structural dy-namics of the flock asymptotically converges to a quasi �-lattice. Inall of our experimental results, we have observed finite-time self-as-sembly of quasi-�-lattices.

Fig. 2. Experimental results for the nonlinear target: (a) MSE for distributedtarget tracking. (b) Average information value. (c) � plots for flocking (smoothblue curve) and Kalman-Consensus filtering (piecewise constant red curve).(d) A target trajectory and fused estimates of 30 sensors.

Fig. 3. Snapshots of a mobile sensor network tracking a maneuvering targetwith a flocking-based motion control algorithm. The target is marked by a redcircle and the estimates are marked by green dots.

V. EXPERIMENTAL RESULTS

In this section, we apply our coupled distributed estimation and con-trol algorithm—namely, KCF plus flocking—to two types of targets: 1)a target with a linear model which is a particle moving in ��; and 2) amaneuvering target with nonlinear dynamics. The later target remainsin a rectangular region (box) for all time � � �.

A. Linear Target

Consider a particle in � with a linear dynamics

��

with

� ��

� � ��

��

��

where � � �� is the discretization step-size. The sensor makes noisymeasurements of the position of the target, i.e.,

��


The noise statistics for zero-mean Gaussian signals �� and �� are

��

where �� if � � � and �� , otherwise. According to the modelof information value in [8], the measurement error covariance matrixof sensor � is � � �� where �� is the information valuefunction

��

� ��

��

(15)

where �� , �� , and � � � � �. In our experi-ment, we use a mobile sensor network with � � �� agents. The param-eters of � are � � ��, � � �, and � � ��. The interaction range ofthe agents in the flock is � � �� and their desired interagent distanceis � � �. For the KCF algorithm, �� , �� with� � �, and � � ��.

Fig. 1 shows the MSE of tracking error over 10 random runs, theaverage information value, and the algebraic connectivity plots duringtracking. From Fig. 1(c), one can readily verify that Assumptions 1through 2 hold.

B. Maneuvering Nonlinear Target: Particle-in-the-Box

We also consider a maneuvering target with the following nonlineardynamics:

�� (16)

where �� denotes the state of the

target at time �. The target moves inside and outside of a square field�� . Matrix �� is defined as

��

��

� ��

�

� !� �� !��

�� "��

� "��

where �� and �� determine the dynamics of the target inside and out-side of the region, respectively, and "�#� is a switching function taking0–1 values defined by

"�#� �� #� �� #�

�

��#� �� # � �;�� # $ �

In addition, matrix � is given by

� � �� %� % ��

�

��

where � �� is the step-size, �� , � � ��, � � ��, !� � ��,and !� � �� are the parameters of a PD controller, and the elementsof �� are normal zero-mean Gaussian noise with � � ��. Theinitial condition of the target is �� with � � � and�� . The parameters of the information value function in (15)are � � � ��, � � ��, � � �, � � ��, and � � �. We consider amobile sensor network with � � �� nodes with a linear sensing model

and

��

� � � �

Fig. 2 illustrates the tracking estimation error, average informationvalue, and the algebraic connectivity plots for the nonlinear target withsnapshots shown in Fig. 3. Similarly, Assumptions 1 and 2 hold basedon Fig. 2(c).

VI. CONCLUSION

We introduced a theoretical framework for coupled distributed esti-mation and flocking-based control of mobile sensor networks for col-laborative target tracking. The mobile sensing agents seek to improvethe information value of their sensed data while avoiding interagentcollisions. We demonstrated that the coupled dynamics of the com-bined distributed estimation and control algorithm has a separable cas-cade nonlinear normal form. Then, we provided the stability analysisof the structural dynamics of a flock with � dedicated &-agents in cas-cade with the error dynamics of the continuous-time KCF. Based onour experimental results, the discrete-time counterpart of the informa-tion-driven flocking algorithm is effectively applicable to tracking botha linear and a nonlinear maneuverable target.

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Coupled Distributed Estimation and Control for Mobile Sensor Networks