\u003ctitle\u003eCooperative controls with intermittent communication\u003c/title\u003e

$Page 1: \u003ctitle\u003eCooperative controls with intermittent communication\u003c/title\u003e$
Cooperative Controls with Intermittent Communication

Dan Shena, Genshe Chen∗a, Jose B. Cruz, Jr.b, Khanh Phamc, Erik Blaschd, Robert Lynche

aDCM Research Resources, LLC, 14163 Furlong Way, Germantown, MD, USA 20874; bDept. of ECE, 2015 Neil Ave, the Ohio State University, Columbus, OH USA 43210;

cAir Force Research Laboratory, Space Vehicles Directorate, Kirtland AFB, NM, USA 87117; dAir Force Research Laboratory, Sensors Directorate, Wright-Patterson AFB, OH, USA 45433

eNaval Undersea Warfare Center, Newport, RI, USA 02841

ABSTRACT

In this paper, we propose a solution to the cooperative path planning with limited communication problem in two phases. In the first (offline) phase, a Pareto-optimal path problem is formulated to find a reference path and the graph cuts minimization method is used to speedily calculate the optimal solution. In the second (online) phase, a foraging algorithm is used to dynamically refine the reference path to meet the dynamic constraints of unmanned aerial vehicle (UAVs), during which an open-loop feedback optimal (OLFO) controller is used to estimate the states which may be unavailable due to infrequent battlefield information updates. Furthermore, an adaptive Markov decision process is proposed to deal with intermittent asynchronous information flow. The method is demonstrated in a simulation for a swarm of Unmanned Air Vehicle (UAV) teams with various communication ranges.

Keywords: Path planning, Pareto-optimal, open-loop feedback, Markov decision process, UAV, swarms

1. INTRODUCTION Much effort has been made to study cooperative control for autonomous, unmanned air vehicle (UAV) teams, as these teams are envisioned in near future to be capable of performing many key battlefield roles. UAV teams (or swarms) could perform (1) cooperative target search, acquisition, and tracking, (2) persistent Intelligence, Surveillance, and Reconnaissance (ISR), (3) persistent area denial, and (4) coordinated attack. In some applications, UAVs in the team may have to autonomously act in a coordinated fashion while only receiving information updates from other team members, ground operators, or remote sensing/command platforms intermittently. Current techniques for coordinated control often assume continuous and uninterrupted data transmission among the team members. However, UAV team members’ communication are often limited for variable and uncertain intervals, whether due to terrain masking, urban terrain, jamming, stealth requirements, antenna placement and vehicle orientation, bandwidth limitations, or other reasons.

Applications and scenarios of cooperative control with communication constraints have been receiving increasing attention during the past several years [1, 2]. A driving reason behind this attention is the fact that due to limited or intermittent communications, UAV team members may have varying degrees of information available at different times, or may only receive occasional updates from a remote platform or operator. The vehicles still need to be able to act cooperatively without full knowledge of the world state, or of one another’s actions. Innovative approaches are requested to deal with cooperative control issues arising from intermittent and asynchronous communications. Applications requiring UAV cooperative control include information observations related to time critical targets, moving targets, pop-up threats or targets, new team member information, or other factors.

The goal of this paper is to construct a mathematically rigorous framework for evaluation and refinement of path planning of UAV teams in the presence of intermittent and asynchronous communication. The initial phase of the research will involve the design of a framework inside which interactions among system tasks, feedback control, and dynamic communication algorithms can be analyzed and synthesized to achieve desired performance. ∗ [email protected]. This work was supported in part by the US Air Force under contracts FA9453-09-C-0175.

Space Missions and Technologies, edited by Joseph Lee Cox, Manfred G. Bester, Wolfgang Fink, Proc. of SPIE Vol. 7691, 76910D · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.849786

Proc. of SPIE Vol. 7691 76910D-1

Downloaded From: http://spiedigitallibrary.org/ on 10/22/2013 Terms of Use: http://spiedl.org/terms

Our proposed Pareto Foraging Algorithm with open-loop feedback optimal (OLFO) controller is illustrated in Figure 1. We tackle cooperative path planning with limited communication problem in two phases: in the first (offline) phase, a Pareto-optimal path problem is formulated to find a reference path and the graph cuts minimization method is used to speedily calculate the optimal solution; in the second (online) phase, a foraging algorithm is used to dynamically refine the reference path to meet the dynamic constraints of UAVs, during which an OLFO controller is used to estimate the states which may be unavailable due to infrequent battlefield information update. Furthermore, an adaptive Markov decision process is proposed to deal with intermittent asynchronous information flow.

Figure 1. Flowchart of Pareto Foraging algorithm with OLFO controller.

The rest of the paper is organized as follows. In section 2, we present a foraging model of path planning with Pareto-optimal and OLFO. Section 3 describes the adaptive fuzzy Markov-chain approach for intermittent communication in the path planning foraging model. Numerical examples are simulated in Section 4. Finally we draw conclusions in Section 5.

2. PATH PLANNING WITH PARETO-OPTIMAL FORAGING AND OLFO 2.1 Pareto Optimal Algorithm for Offline Path Planning

The concept of a Pareto-optimal strategy is useful when we consider cooperative strategies in a team of M UAVs. A set of decision choices is said to be Pareto-optimal if for any other set of choices of decisions, it is impossible to make one team member better off without necessarily making someone else worse off. When we use the minimal-risk as the goal of path planning, the Pareto-optimal path means that the survival probability of any single UAV cannot be improved without decreasing the survivability of any other UAV in the same group. So the Pareto-optimal paths can be obtained by

*

1

arg min [ ]M

j jj

Jρ ρρ λ ρ

=

= ∑ (1)

where path ρ is a mapping from time to space [ ]: 0, nTρ →ℜ , [ ]jJ ρ is the objective function of the jth UAV, and

0 1jλ< < are the Pareto cooperation coefficients.

2.2 A Foraging Model of Online Path Planning with OLFO

The reference path obtained offline from Pareto-optimal algorithm may very likely contain line segments with turn angles exceeding the dynamic constraints of the UAVs. The reference paths that are not physically available can be eliminated by the foraging algorithms which dynamically refine the reference path to be within the bounds on the turning radius of the UAVs.



Consider a swarm of M vehicles (UAVs) in an n-dimensional Euclidian space. We model the individuals as points and ignore their dimensions. Let i nx R∈ be the position of the ith UAV in the swarm, i nv R∈ be the velocity, and

i nu R∈ be the control input, then the dynamic model for the ith agent is i ix v=& (2)

( )( ) ( )( )

2

21,

1 ˆ ˆ2ˆ ˆ ˆ êxp i

i jM p p

i i i i i i i i j i i i ip v v v r p p f fxij j i s

e ev u k e k e k e e k J x d

r= ≠

⎛ ⎞− −⎜ ⎟= = − + + − − ∇ −⎜ ⎟

⎜ ⎟⎝ ⎠

∑& (3)

where i ipe x x= − is the position error,

1

1 Mi

ix x

M =

= ∑ ,

i ive v v= − is the velocity error,

1

1 Mi

i

v vM =

= ∑ ,

( )i iJ x is the cost function for the ith UAV,

( ), ( ), ( )i i if p vd t d t d t are the sensing errors of ( ), ,i

i i i ip vx

J x e e∇ respectively,

î i ip p pe e d= − is the position error with sensing error,

î i iv v ve e d= − is the velocity error with sensing error,

0, 0i ip vk k> > are the contraction gains,

the gain 0ifk > is proportional to that UAV’s desire to follow its profile,

0irk > is the repulsion gain, 0i

sr > is its repulsion range. The foraging algorithm with perfect communication has been discussed in [1], in which the interaction between UAVs is formulated as an “attraction-repel” method (as in potential fields analysis) in which each and every UAV seeks a “comfortable” position related to its neighboring platforms. In the case of limited communication capacity, the latest battlefield information may not be always available to calculate the cost functions of the UAVs. Limited communication makes it difficult to apply the “attraction-repel” algorithm to determine the refined path. We thus propose to use open-loop feedback optimal (OLFO) concept to estimate the states of UAVs based on a Dubin’s car model [5], in which UAVs are supposed to move only forward with bounds on the turning radius, as well as solving the foraging algorithm. The OLFO controller is indeed an adaptive model predictive controller, which uses the optimal open-loop initial decision at each stage, and re-computes the open-loop optimal deterministic control after re-evaluating the uncertainty system state at each and every time step [3, 4]. The basic structure of OLFO is shown in Figure 2, and the basic principle, the “moving horizon” strategy, is illustrated in Figure 3. We can see from Figure 2 that at the present time, the control behavior during the prediction horizon is considered. In more detail, if the present time is indexed by k , we assume that the control sequence 0, 1 , , … , has been applied to the system and that the observation sequence , 0, , 0, 1, … , is available. We would like to find a “future” control sequence as to optimize the cost-to-go over the prediction horizon conditioned on the total available information at the present time. Suppose an optimal control sequence , , , … , is calculated. Using only the first element of this optimal control sequence is feed forward control. At the next time step, for the prediction horizon which has now moved one time step forward (“moving horizon”) and whose length may have changed, a new optimal control sequence is calculated. The control loop is therefore closed in cycles (optimal feedback control).



The cost-to-go function of the foraging problem at time step k , ikJ is now determined by the state estimate ˆ i

kx which

can be generated by a standard Kalman filter algorithm over the prediction horizon kt . The length of the prediction horizon will be predicted by a Markov-Chain model as we will explain in the next section. By defining the estimated

state sequence { }1ˆ ˆ ˆ ˆ( , ) , ,...,kk k k k tx k k t x x x+ ++ = , where 1ˆ ˆ ˆ ˆ{ , ,..., }i i M

k k k kx x x x+= , and the control sequence

{ }1( , ) , ,...,k

i i i ik k k k tU k k t u u u+ ++ , the cost-to-go function can be expressed as

( )ˆ( , ), ( , )i ik k kJ f x k k t U k k t= + + (4)

We would like to find a set of waypoints ˆ( , )kx k k t+ to minimize the cost-to-go function, i.e., , arg min , , , which may be solved by standard optimization methods as in gradient descent or the greedy method. At the next point of time, the prediction horizon 1kt + moves one time interval forward and the length will be predicted by the Markov-Chain

model. The OLFO algorithm will carry out state estimation over 1kt + again. The procedure will repeat until the destination is reached.

3. ADAPTIVE FUZZY MARKOV-CHAIN MODEL Due to various reasons such as terrain masking, urban terrain, jamming, stealth requirements, antenna placement and vehicle orientation and bandwidth limitations, UAV team members’ communication is limited for variable, uncertain intervals. Innovative approaches are requested to deal with cooperative control issues arising from intermittent and asynchronous communications. Meanwhile, as we have seen from the OLFO model, the length of prediction horizon plays an important role in determining the algorithm results. Especially in the presence of limited communication, the length of horizon determines the efficiency of the OLFO controller. In order to select a horizon which reflects the current battlefield environment at each and every time step, we present an adaptive fuzzy Markov-Chain model to estimate the arrival time of the next information update.

Figure 2. Basic structure of the OLFO controller.

Figure 3. "Moving horizon" strategy of the OLFO controller.



Let the present time be indexed by k . Let t be the communication interval between the time steps of the current information update and the next information arrival. Since [0, )t∈ +∞ , we propose to define a random sequence

{ , 0, {1, 2,..., }}k QD d i k i Q N= = ≥ ∈ = to represent the communication intervals by the following fuzzy rules:

1

1

max

1 02 0

j j

Q

tt l

ij l t l

N t l

−

=⎧⎪ < ≤⎪⎪⎪= ⎨ < ≤⎪⎪⎪

≥⎪⎩

M M

M M

(5)

where 1 max{ ,... }l l is a set of typical interval values which may be determined arbitrarily or experimentally.

3.1 First-order Markov-Chain Model

If the next communication interval depends only on the previous update intervals, we model the predicted next communication intervals as a first-order Markov-chain such that, for any k , 1kd + is independent of 0 1 1, ,... kd d d − . Mathematically,

1 1 0 1Pr( | , ,..., ) Pr( | )k k k k kd j d i d m d s d j d i+ − += = = = = = = , ,i j Q∈ (6)

We also assume that the one-step transition probabilities ( ){ 1Pr |k kd j d i+ = = , },i j Q∈ depend only on the states i

and j and are independent of the index k , i.e.,

( )1Pr | k k ijd j d i p+ = = = (7)

for all ,i j Q∈ and for any k. The transition probabilities are stationary or homogeneous. We then may construct a one-step transition probability matrix of the first-order Markov-Chain,

11 12 1

21 22 2(1)

1 2

Q

Q

Q Q Q Q

N

N

N N N N

p p p

p p pP

p p p

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

L

L L L L

L (8)

where for any ,i j Q∈ , 0ijp ≥ and for each i Q∈ , 1ijj Q

p∈

=∑ .

3.2 Higher-order Markov Chain Model

The dependence structure of order one may not be sufficient to model some situations. For a more realistic model, it is assumed that future communication intervals might depend on more than one previous state, or possibly out-of order measurements that are delayed For example, the next information arrival could depend on the previous n communication intervals of information updates, which can be described as follows:

( )( )

1 1 1 2 1 0 1

1 1 1 2 1

Pr | , , , , ,

Pr | , , ,k k k k n n k

k k k k n n

d j d i d i d i d i

d j d i d i d i+ − − + +

+ − − +

= = = = =

= = = = =

L L

L (9)

where 1 1, , , ; 1ki i j Q k n+ ∈ ≥ −L . We explain a second-order Markov chain model and the higher order models can be developed similarly. In terms of the previous notations, the second-order dependence is characterized as



( ) ( )1 1 0 0 1 1Pr | , , , Pr | , , , ,k k k k k kd l d j d i d i d l d j d i i j l Q+ − + −= = = = = = = = ∈L (10) The one-step transition probability matrix for the Markov chain of order two is given by

111 112 11

121 122 12(2)

1 2

Q

Q

Q Q Q Q Q Q Q

N

N

N N N N N N N

p p p

p p pP

p p p

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

L

L L L L

L

(11)

where for all , ,i j l Q∈ , we assume

( )1 1Pr , 0ijl k k kp d l d j d i+ −= = = = > (12)

and for each pair ( , )i j Q Q∈ × , 1ijll Q

p∈

=∑ .

The prediction accuracy of the next information update time critically depends on the transition probabilities, which in turn can be precise enough only when the order of the Markov-Chain model is appropriately chosen with respect to the available historical data. A hypothesis test algorithm proposed by Liu in [2] can be used to estimate the order of Markov chains with stationary transition probability matrices.

3.3 Estimates of Stationary Transition Probabilities

An adaptive approach to estimate the stationary transition probability matrix for the first-order Markov chain model was developed in [8-9]. We explain how to apply this adaptive algorithm for the first-order and second-order model, and the extension to a higher-order Markov chain model is straightforward. Let ( )ijn k represent the status of transition from the

communication interval state being i Q∈ at time 1k − to state being j Q∈ time k . That is,

11 if ,( )

0 otherwisek k

ij

d i d jn k − = =⎧

= ⎨⎩

For the first-order Markov chain, the updated stationary transition probability ˆ ( )ijp k of the next update interval being

categorized by kd j= given the previous communication interval being categorized by 1kd i− = for all ,i j Q∈ , at

discrete event index k can be estimated by

1

1 1

( )ˆ ( )

( )Q

k

ijh

ij N k

iss h

n hp k

n h

=

= =

=∑

∑∑ (13)

This update procedure increases the estimated stationary transition probability from being state i to state j and reduces it for the other states from state i . For a second-order Markov chain model, the transition probabilities ˆ ( )ijlp k can be estimated by

1

1 1

( )ˆ ( )

( )Q

k

ijlh

ijl N k

ijss h

n hp k

n h

=

= =

=∑

∑∑ (14)



where ( )ijln k also stands for the transition status which is defined as

2 11 if , , ( )

0 otherwisek k k

ijl

d i d j d ln k − −= = =⎧

= ⎨⎩

(15)

3.4 Estimate of next information arrival

After estimating the transition probability matrices, we propose to consider the maximum likelihood strategy to determine the next communication interval. Given the previous communication interval state being 1

ikd − , an estimated

next communication interval state is selected as j

k kd d= with ârg max{ ( )}ijj Qj p k

∈= (16)

By examining the fuzzy rules, we then know that the corresponding predicted communication interval is 1j jl t l− < ≤ , and the worst case is that jt l= , which is to be used by the OLFO controller to estimate the current states.

4. NUMERICAL SIMULATIONS 4.1 Scenario

In the scenario shown in Figure 4, suppose there are some starting points and a final destination area in which there are some destination points. Between the starting area and final destination area there are some dangerous areas which are known to be hostile. Besides these, there are also some possible pop-up areas which are also dangerous to UAVs. The farther away from the center point of a pop-up area, the less possibility that a pop-up will appear there. For UAVs to get to the final destination points as safely as possible, usually UAVs should try to avoid the dangerous areas and the possible pop-up areas in their paths. This requires us to model the situation with some comprehensive risk-benefit functions. The simplest way to identify a path is to assign some way points for it. Moreover, there is a pop-up threat in the middle. It is supposed to be detected during the path execution stage rather than the static path planning stage. In this scenario, we assume that the communication range of each UAV is limited so that each UAV can only share information with other UAVs within the range. It is more realistic than our original centralized algorithm, whose disadvantages are dramatic: There has to be some super-node acting as the processing center which requires much higher energy, storage and computing capabilities. This will largely reduce the lifetime and survivability of the whole sensor network, especially in the case of autonomous sensor networks. The first step is to conduct static path planning to find the approximate and discrete path composed of way points. This part is based on Pareto-optimal and Voronio concept. The result is shown in Figure 5. We can see the whole x-y space is sampled according to the “importance” of each spot. (The spot, where the change objective function is larger, is deemed as a more important point.) Since the additional pop-up threat will be detected during the path refined stage (foraging algorithm), the result of static path planning is same as before. Note that the dense cost contours represent both the known and the estimated hostile areas. The blue dots are the path-planning solution that minimizes the distance from the

Figure 4. A scenario in a virtual battlefield



start to the dused as refer

We carried ounlimited coalso investiga

estination; avoence points in

out different exommunication rated the effects

oiding the hostnext stage --- d

Fig

xperiments incrange (same as of the initial p

ile areas. Thosdynamic path e

gure 5. A scenar

luding (a) largs centralized cpositions of the

se blue dots (wexecution.

rio in a virtual ba

ge communicatcase in our orie UAVs.

waypoints of ap

attlefield

tion range, (b) iginal path plan

pproximate and

small communnning algorith

d discrete path

nication rangem). For each c

) will be

, and (c) case, we



4.2 Larg

Figure 6: L

For UAVs wUAVs, the teThe UAVs wcommunicatidestination acloser duringteams mergecommunicati

4.3 Smal

Figure 8: S

For small codistances botsome UAVs communicati

ge Communica

arger communicposi

with close initiaeam of UAVs

will fly togetheion range, thealong the referg the journey to and become oion range (dTh

ll Communica

Small communicposi

ommunication th in Fig.8 andwill fly close

ion range.

ation Case

cation range and itions

al positions (Fcan share info

er to the destine 10 UAVs foence waypointo the destinatioone team. In bohreshold = 3.5)

ation Case

ation range and itions

range (dThresd Fig. 9, all Uenough to shar

close initial

Fig. 6), since thormation from ation from theorm three subts. Since all su

on. When the doth experimentcan avoid the

close initial

shold=1.0) expUAVs can onlyre information

Figure 7: La

he communicathe start, so a start position.

b-teams. Each ub-teams use tdistances are smts (Fig. 6 and Fpop-up threats

Figure 9: Sm

periments, becay share inform, but these kin

arger communicposit

ation range is lll the effects o. In Fig. 7, sinc

sub-team canthe same referemaller and withFig. 7), our revs detected along

mall communicaposit

ause the commation at the beds of sub-team

ation range and tions

larger than all of repels and ace some distann share informence points, th

hin the communvised algorithmg the reference

ation range and ations

munication raneginning. Duri

ms are not stabl

apart initial

the distances attraction are avnces are larger mation and flyhey will fly clonication range,

m with large bue way-points.

apart initial

nge is smaller ing the enroutele because of t

between vailable. than the

y to the oser and , all sub-

ut limited

than the e flights, the small



It is not diffiformation in additional prfounded in F Both experim

Figure 10: S

In Fig. 11, thcollisions. In

We proposeddynamic, asy(OLFO) apporientated obbenefit of fodynamic condecision procachieve a coconducted towhich means

[1] Chen, Gplanning

[2] Liu, Y., on Aeros

[3] Dreyfus,120-134

[4] Tse, E. Control,

cult to find thaFig. 8 with clo

rocedure of flyig. 8.

ments show tha

Small communicposi

he highlighted an real applicatio

d a frameworkynchronous co

proach has sevbjective functioraging is neanstraints; 3) OLcess is used to onvergent, fasto demonstrate s less energy, s

G., Shen, D. Crg for multiple aCruz, Jr., J. B

space and Elec, S. E., “Some

4, 1962. and Athans, M

, vol. AC-17, p

at the formationose initial posiying apart, whi

at the revised al

cation range and itions

area in Fig. 10 on, this is dang

k of cooperativommunication

veral advantageons can guara

ar optimal but LFO acts as anpredict the next, risky minimour revised n

storage and com

ruz Jr., J.B., Kaerial platforms

B., and Schumactronic Systemse types of optim

M., “Adaptive pp. 38-51, no. 1

n of Fig. 9, wititions. This is bich causes mor

lgorithm can a

apart initial

was zoomed. Wgerous, so a rela

5. COve path plannin. Our Pareto-es of our coopantee goal arri

very efficientn identifier of sxt arrival times

mum and efficinon-centralizedmputing power

REF

Kwan, C., Ridds”, Infotech@Aacher, C. J., “Ps, vol. 43, no.2mal control of

stochastic con1, 1972.

th further apartbecause in the re of an overs

avoid pop-up th

Figure 11: Z

We can see thaatively larger c

ONCLUSIOng problem fo-optimal foragperative path pival and threatt, robust, convsystem model as of intermittenient trajectory

d path planningr.

FERENCES

dle, S. P, CoxAerospoace; APop-Up Threat 2 April 2007.

stochastic sys

ntrol for a cla

t initial positiofirst part of the

shot as found i

hreats during th

oomed view of t

at some UAVs communication

ONS r a team of UA

ging algorithmplanning approt/obstacle avoivergent, smootand an estimatont information

planning algog algorithms w

, S. and Matthrlington, VA; UModels for Pe

tems,” SIAM J

ass of linear sy

ons, is better ane journey in Fiin Figure 8. Th

he refinement o

the area highligh

fly very closeln range is recom

AVs in the prm with open-looaches, includiidance with mth and easy toor for system sinflows; and 5orithm. Numerwith limited c

hews, C., “A nUSA; 26-29 Seersistent Area

Journal of Con

ystems,” IEEE

nd more stable ig.8 the UAVshen more over

of the path.

hted in Fig. 9

ly but have nommended.

resence of interoop feedback ing: 1) Pareto

minimum costs;o consider thestates; 4) The ) The combinarical simulatio

communication

novel cooperatept. 2005. pp. 1Denial,” IEEE

ntrol, vol. 2, n

E Trans. on Au

than the have an rshot are

rmittent, optimal

-optimal ; 2) The

e vehicle Markov

ation can ons were n ranges,

tive path 1-10. E Trans.

no. 1, pp.

utomatic



[5] Dubins, L. E., “On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents,” American J. of Mathematics, vol 79, pp 497-516, 1957.

[6] Shen, D., G. Chen, J Cruz, Jr., and E. Blasch, “A Game theoretic Data Fusion Aided Path planning Approach for Cooperative UAV ISR”, IEEE Aerospace Conference, 2008.

[7] Ahmadzadeh, A., B. Sayyar-Roudsari, and A. Homaifar, “A Hybrid projected gradient-evolutionary search algorithm for capacitated multi-source multi-UAVs scheduling with time windows,” Ch 1 in Recent Development s in Cooperative Control and Optimization, (eds) S. Butenko, R. Murphey, and P Pardalos, Kluwer, 2004.

[8] DeLima, P and D. Pack, “Maximizing Search Coverage Using Future Path Projection for Cooperative Multiple UAVs with Limited Communication Ranges,” Ch 6 in Optimization and Cooperative Control Strategies – Proc. Cooperative Control and Optimization,” (eds) M. J. Hirsch, C. W. Commander, P. M. Pardalos, and R. Murphey, Springer, 2009.

[9] K. Pham, “New Results in Stochastic Cooperative Games: Strategic Coordination for Multi-resolution Performance Robustness,” Ch 16 in Optimization and Cooperative Control Strategies – Proc. Cooperative Control and Optimization,” (eds) M. J. Hirsch, C. W. Commander, P. M. Pardalos, and R. Murphey, Springer, 2009.

[10] Chen, H, G. Chen, E. Blasch, and T. M. Schuck, “Robust Track Association and Fusion with Extended Feature Matching,” Ch 19 in Optimization and Cooperative Control Strategies – Proc. Cooperative Control and Optimization,” (eds) M. J. Hirsch, C. W. Commander, P. M. Pardalos, and R. Murphey, Springer, 2009.



Documents

\u003ctitle\u003eCooperative controls with intermittent communication\u003c/title\u003e