Formation Control Protocols for Nonlinear Dynamical ...or dissipative dynamical systems. In Sec. 4, we specialize the results of Sec. 3 to linear dynamical systems. We then turn our

Wassim M. HaddadSchool of Aerospace Engineering,

Georgia Institute of Technology,

Atlanta, GA 30332-0150

e-mail: [email protected]

Sergey G. NersesovDepartment of Mechanical Engineering,

Villanova University,

Villanova, PA 19085-1681


Qing HuiDepartment of Mechanical Engineering,

Texas Tech University,

Lubbock, TX 79409-1021


Masood GhasemiDepartment of Mechanical Engineering,

Villanova University,

Villanova, PA 19085-1681


Formation Control Protocols forNonlinear Dynamical SystemsVia Hybrid Stabilization of SetsIn this paper, we develop a hybrid control framework for addressing multiagent forma-tion control protocols for general nonlinear dynamical systems using hybrid stabilizationof sets. The proposed framework develops a novel class of fixed-order, energy-basedhybrid controllers as a means for achieving cooperative control formations, which caninclude flocking, cyclic pursuit, rendezvous, and consensus control of multiagent systems.These dynamic controllers combine a logical switching architecture with the continuoussystem dynamics to guarantee that a system generalized energy function whose zero levelset characterizes a specified system formation is strictly decreasing across switchings.The proposed approach addresses general nonlinear dynamical systems and is not lim-ited to systems involving single and double integrator dynamics for consensus and forma-tion control or unicycle models for cyclic pursuit. Finally, several numerical examplesinvolving flocking, rendezvous, consensus, and circular formation protocols for standardsystem formation models are provided to demonstrate the efficacy of the proposedapproach. [DOI: 10.1115/1.4027501]

Keywords: energy-based set stabilization, formation control, hybrid control, impulsivedynamical systems, consensus protocols, multiagent systems

1 Introduction

Using system-theoretic thermodynamic concepts, an energy-and entropy-based hybrid controller architecture was proposed inRefs. [1] and [2] as a means for achieving enhanced energy dissi-pation in lossless and dissipative dynamical systems. Thesedynamic controllers combined a logical switching architecturewith continuous dynamics to guarantee that the system plantenergy is strictly decreasing across switchings. The general frame-work developed in Ref. [1] leads to closed-loop systems describedby impulsive differential equations [2]. In particular, the authorsin Refs. [1] and [2] construct hybrid dynamic controllers thatguarantee that the closed-loop system is consistent with basicthermodynamic principles. Specifically, the existence of anentropy function for the closed-loop system is established thatsatisfies a hybrid Clausius-type inequality. Special cases ofenergy- and entropy-based hybrid controllers involving state-dependent switching were also developed to show the efficacy ofthe approach.

Recent technological advances in communications and compu-tation have spurred a broad interest in control of networks andcontrol over networks [3]. Network systems involve distributeddecision making for coordination of networks of dynamic agentsand address a broad area of applications including cooperativecontrol of unmanned air vehicles, microsatellite clusters, mobilerobotics, and congestion control in communication networks. Inmany applications involving multiagent systems, groups of agentsare required to agree on certain quantities of interest. For example,in a group of autonomous vehicles, this property might be a com-mon heading angle or a shared communication frequency. More-over, it is important to develop information consensus protocolsfor networks of dynamic agents, wherein a unique feature of theclosed-loop dynamics under any control algorithm that achievesconsensus is the existence of a continuum of equilibria

representing a state of equipartitioning or consensus [4–6]. Undersuch dynamics, the limiting consensus state achieved is not deter-mined completely by the dynamics, but depends on the initial sys-tem state as well. For such systems possessing a continuum ofequilibria, semistability [7–9], and not asymptotic stability, is therelevant notion of stability. In addition, system-theoretic thermo-dynamic concepts [4–6,10] have proved invaluable in addressingLyapunov stability and convergence for nonlinear dynamicalnetworks.

Convergence and state equipartitioning also arise in numerouscomplex large-scale dynamical networks that demonstrate adegree of synchronization. System synchronization typicallyinvolves coordination of events that allows a dynamical system tooperate in unison resulting in system self-organization. The onsetof synchronization in populations of coupled dynamical networkshave been studied for various complex networks including net-work models for mathematical biology, statistical physics, kinetictheory, bifurcation theory, as well as plasma physics [11]. Syn-chronization of firing neural oscillator populations also appears inthe neuroscience literature [12,13].

Alternatively, in other applications of multiagent systems,groups of agents are required to achieve and maintain a prescribedgeometric shape. This formation control problem includes flock-ing [14,15] and cyclic pursuit [16], wherein parallel and circularformations of vehicles are sought. For formation control of multi-ple vehicles, cohesion, separation, and alignment constraints aretypically required for individual agent steering which describehow a given vehicle maneuvers based on the positions and veloc-ities of nearby agents. Specifically, cohesion refers to a steeringrule, wherein a given vehicle attempts to move toward the averageposition of local vehicles, separation refers to collision avoidancewith nearby vehicles, whereas alignment refers to velocity match-ing with nearby vehicles.

Since a specified formation of multiagent systems, which caninclude flocking, cyclic pursuit, rendezvous, or consensus, can becharacterized by a hyperplane or manifold in the state space; inthis paper, we extend the results of Refs. [1] and [2] to develop astate-dependent hybrid control framework for addressing

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedApril 23, 2013; final manuscript received April 22, 2014; published online July 9,2014. Assoc. Editor: Shankar Coimbatore Subramanian.

Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2014, Vol. 136 / 051020-1Copyright VC 2014 by ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2014 Terms of Use: http://asme.org/terms

multiagent formation control protocols for general nonlinear dy-namical systems using hybrid stabilization of sets. The proposedframework involves a novel class of fixed-order, energy-basedhybrid controllers as a means for achieving cooperative controlformations. These dynamic controllers combine a logical switch-ing architecture with continuous dynamics to guarantee that asystem generalized energy function, whose zero level set charac-terizes a specified system formation, is strictly decreasing acrossswitchings. The general framework leads to hybrid closed-loopsystems described by impulsive differential equations andaddresses general nonlinear dynamical systems without limitingconsensus and formation control protocols to single and double in-tegrator models.

The contents of the paper are as follows: In Sec. 2, we establishdefinitions, notation, and review some basic results on impulsivedifferential equations which provide the mathematical foundationfor designing formation control protocols for nonlinear dynamicalsystems using logic-based hybrid controllers. In Sec. 3, we presenta general state-dependent hybrid control framework for stabiliza-tion of sets. The main result in this section extends the results ofRef. [1] to hybrid stabilization of sets and is not limited to losslessor dissipative dynamical systems. In Sec. 4, we specialize theresults of Sec. 3 to linear dynamical systems. We then turn ourattention to hybrid control design for parallel and rendezvousformations, and consensus control of multiagent systems forstandard single and double integrator formation models in Secs. 5and 6, respectively. In Sec. 7, we use the results of Secs. 2 and 3to design hybrid controllers for cyclic pursuit. Finally, we drawconclusions in Sec. 8.

2 Hybrid Control and Impulsive Dynamical Systems

In this section, we establish definitions, notation, and reviewsome basic results on impulsive dynamical systems [2]. Let R

denotes the set of real numbers, Rþ denotes the set of nonnega-tive real numbers, Rn denotes the set of n� 1 real column vectors,

Zþ denotes the set of nonnegative integers, Zþ denotes the set of

positive integers, ð�ÞT denotes transpose, In denotes the n� n iden-tity matrix, and 0n�n denotes the n� n zero matrix. Furthermore,

let @S;S�, and S denote the boundary, the interior, and the closure

of the subset S � Rn, respectively. We write �k k for the Euclid-ean vector norm, �k kF for the Frobenius matrix norm,BeðaÞ; a 2 Rn, e> 0, for the open ball centered at a with radius e,and V0ðxÞ for the Fr�echet derivative of V at x. Finally, we writexðtÞ !M as t ! 1 to denote that x(t) approaches the set M,that is, for every e > 0, there exists T> 0 such that distðxðtÞ;MÞ< e for all t> T, where distðp;MÞ ¼D infx2M p� xk k.

In this paper, we consider continuous-time nonlinear dynamicalsystems of the form

_xpðtÞ ¼ fpðxpðtÞ; uðtÞÞ; xpð0Þ ¼ xp0; t � 0 (1)

yðtÞ ¼ hpðxpðtÞÞ (2)

where t� 0, xpðtÞ 2 Dp � Rnp ;Dp is an open set, uðtÞ 2 Rm;fp : Dp �Rm ! Rnp is smooth (i.e., infinitely differentiable) on

Dp �Rm, and hp : Dp ! Rl is smooth. Furthermore, we considerhybrid (i.e., resetting) dynamic controllers of the form

_xcðtÞ ¼ fccðxcðtÞ; yðtÞÞ; xcð0Þ ¼ xc0; ðxcðtÞ; yðtÞÞ 62 Zc (3)

DxcðtÞ ¼ fdcðxcðtÞ; yðtÞÞ; ðxcðtÞ; yðtÞÞ 2 Zc (4)

uðtÞ ¼ hccðxcðtÞ; yðtÞÞ (5)

where t� 0, xcðtÞ 2 Dc � Rnc ;Dc is an open set, DxcðtÞ¼D xcðtþÞ � xcðtÞ, where xcðtþÞ ¼

DxcðtÞ þ fdcðxcðtÞ; yðtÞÞ ¼ lime!0þ

xcðtþ eÞ, ðxcðtÞ; yðtÞÞ 2 Zc; fcc : Dc �Rl !Rnc is smooth on

Dc �Rl; hcc : Dc �Rl !Rm is smooth, fdc : Dc �Rl !Rnc is

continuous, and Zc � Dc �Rl is the resetting set. Note that, forgenerality, we allow the hybrid dynamic controller to be of fixeddimension nc, which may be less than the plant order np.

The equations of motion for the closed-loop dynamical systems(1)–(5) have the form

_xðtÞ ¼ fcðxðtÞÞ; xð0Þ ¼ x0; xðtÞ 62 Z (6)

DxðtÞ ¼ fdðxðtÞÞ; xðtÞ 2 Z (7)

where

x ¼Dxp

xc

� �2 Rn; fcðxÞ ¼D

fpðxp; hccðxc; hpðxpÞÞÞfccðxc; hpðxpÞÞ

� �;

fdðxÞ ¼D0

fdcðxc; hpðxpÞÞ

� �(8)

and Z ¼D fx 2 D : ðxc; hpðxpÞÞ 2 Zcg, with n ¼D np þ nc andD ¼D Dp �Dc. We refer to the differential equation (6) as thecontinuous-time dynamics, and we refer to the difference equation(7) as the resetting law. Note that although the closed-loop statevector consists of plant states and controller states, it is clear fromEq. (8) that only those states associated with the controller arereset. To ensure well posedness of the solutions to Eqs. (6) and(7), we make the following additional assumptions [2].

ASSUMPTION 1. If x 2 ZnZ, then there exists e> 0 such that, forall 0< d< e, wðd; xÞ 62 Z, where w(�,�) denotes the solution to thecontinuous-time dynamics (6).

ASSUMPTION 2. If x 2 Z, then xþ fdðxÞ 62 Z.Assumption 1 ensures that if a trajectory reaches the closure of

Z at a point that does not belong to Z, then the trajectory must bedirected away from Z; that is, a trajectory cannot enter Z througha point that belongs to the closure of Z but not to Z. Furthermore,Assumption 2 ensures that when a trajectory intersects the reset-ting set Z, it instantaneously exits Z. Finally, we note that ifx0 2 Z, then the system initially resets to xþ0 ¼ x0 þ fdðx0Þ 62 Z,which serves as the initial condition for the continuous-timedynamics (6).

A function x : I x0! D is a solution to the impulsive dynamical

systems (6) and (7) on the interval I x0� R with initial condition

x(0)¼ x0, where I x0denotes the maximal interval of existence of

a solution to Eqs. (6) and (7), if x(�) is left continuous and x(t)satisfies Eqs. (6) and (7) for all t 2 I x0

. For further discussion onsolutions to impulsive differential equations, see Refs. [2] and[17–22]. For convenience, we use the notation s(t, x0) to denotethe solution x(t) of Eqs. (6) and (7) at time t� 0 with initial condi-tion x(0)¼ x0.

For a particular closed-loop trajectory x(t), we let tk ¼D skðx0Þdenote the kth instant of time at which x(t) intersects Z, and wecall the times tk the resetting times. Thus, the trajectory of theclosed-loop systems (6) and (7) from the initial conditionx(0)¼ x0 is given by w(t, x0) for 0< t t1. If and when the trajec-tory reaches a state x1 ¼D xðt1Þ satisfying x1 2 Z, then the state isinstantaneously transferred to xþ1 ¼

Dx1 þ fdðx1Þ according to the

resetting law (7). The trajectory x(t), t1< t t2, is then given bywðt� t1; x

þ1 Þ, and so on. Our convention here is that the solution

x(t) of Eqs. (6) and (7) is left continuous, that is, it is continuouseverywhere except at the resetting times tk, and xk ¼D xðtkÞ¼ lime!0þ xðtk � eÞ and xþk ¼

DxðtkÞ þ fdðxðtkÞÞ ¼ lime!0þ xðtk þ eÞ

for k¼1, 2,….It follows from Assumptions 1 and 2 that for a particular initial

condition, the resetting times tk¼ sk(x0) are distinct and welldefined [2]. Since the resetting set Z is a subset of the state spaceand is independent of time, impulsive dynamical systems of theform Eqs. (6) and (7) are time-invariant systems. These systemsare called state-dependent impulsive dynamical systems [2]. Since

051020-2 / Vol. 136, SEPTEMBER 2014 Transactions of the ASME


the resetting times are well defined and distinct and since the solu-tion to Eq. (6) exists and is unique, it follows that the solution ofthe impulsive dynamical systems (6) and (7) also exists and isunique over a forward time interval. For details on the existenceand uniqueness of solutions of impulsive dynamical systems inforward time, see Refs. [17–20].

Remark 2.1. Let x 2 D satisfy fd(x*)¼ 0. Then x 62 Z. To seethis, suppose x 2 Z. Then x þ fdðxÞ ¼ x 2 Z, which contra-dicts the assumption that if x 2 Z, then xþ fdðxÞ 62 Z. Hence, ifx¼ x* is an equilibrium point of Eqs. (6) and (7), then x 62 Z.

For the statement of the next result, the following key assump-tion is needed.

ASSUMPTION 3. Consider the impulsive dynamical systems (6)and (7), and let s(t, x0), t� 0, denote the solution to Eqs. (6) and(7) with initial condition x0. Then, for every x0 62 Z and everye> 0 and t 6¼ tk, there exists d(e, x0, t)> 0 such that if

x0 � zk k < dðe; x0; tÞ; z 2 D, then sðt; x0Þ � sðt; zÞk k < e.Assumption 3 is a weakened version of the quasi-continuous

dependence assumption given in Refs. [2] and [21] and is a gener-alization of the standard continuous dependence property fordynamical systems with continuous flows to dynamical systemswith left-continuous flows. Specifically, by letting t � [0, 1),Assumption 3 specializes to the classical continuous dependenceof solutions of a given dynamical system with respect to the sys-tem’s initial conditions x0 2 D for every time instant. It should benoted that the standard continuous dependence property fordynamical systems with continuous flows is defined uniformly intime on compact intervals. Since solutions of impulsive dynamicalsystems are not continuous in time and solutions are not continu-ous functions of the system initial conditions, Assumption 3involving pointwise continuous dependence is needed to apply thehybrid invariance principle developed in Refs. [2] and [21] tohybrid closed-loop systems. Sufficient conditions that guaranteethat the impulsive dynamical systems (6) and (7) satisfy a strongerversion of Assumption 3 are given in Refs. [2] and [21] (see alsoRef. [23]). The following proposition provides a generalization ofProposition 4.1 in Ref. [21] for establishing sufficient conditionsfor guaranteeing that the impulsive dynamical systems (6) and (7)satisfy Assumption 3.

PROPOSITION 2.1 [1]. Consider the impulsive dynamical system Ggiven by Eqs. (6) and (7). Assume that Assumptions 1 and 2 hold,s1(�) is continuous at every x 62 Z such that 0< s1(x)<1, and ifx 2 Z, then xþ fdðxÞ 2 ZnZ. Furthermore, for every x 2 ZnZsuch that 0< s1(x)<1, assume that the following statements hold:

(i) If a sequence fxig1i¼1 2 D is such that limi!1 xi ¼ x andlimi!1 s1ðxiÞ exists, then either both fd(x)¼ 0 andlimi!1 s1ðxiÞ ¼ 0 or limi!1 s1ðxiÞ ¼ s1ðxÞ.

(ii) If a sequence fxig1i¼1 2 ZnZ is such that limi!1 xi ¼ xand limi!1 s1ðxiÞ exists, then limi!1 s1ðxiÞ ¼ s1ðxÞ.

then G satisfies Assumption 3.The following result provides sufficient conditions for estab-

lishing continuity of s1(�) at x0 62 Z and sequential continuity of

s1(�) at x0 2 ZnZ, that is, limi!1 s1ðxiÞ ¼ s1ðx0Þ for fxig1i¼1 62 Zand limi!1 xi ¼ x0. For this result, the following definition isneeded. First, however, recall that the Lie derivative of a smoothfunction X : D ! R along the vector field of the continuous-time

dynamics fc(x) is given by LfcXðxÞ ¼D ðd=dtÞXðwðt; xÞÞjt¼0

¼ ð@XðxÞ=@xÞfcðxÞ, and the zeroth and higher order Lie

derivatives are, respectively, defined by L0fcXðxÞ ¼D XðxÞ and

LkfcXðxÞ ¼D Lfc ðLk�1

fcXðxÞÞ, where k� 1.

Definition 2.1 [1]. Let Q ¼D fx 2 D : XðxÞ ¼ 0g, whereX : D ! R is an infinitely differentiable function. A point x 2 Qsuch that fc(x) 6¼ 0 is k-transversal to Eq. (6) if there exists k � {1,2, …} such that

LrfcXðxÞ ¼ 0; r ¼ 0;…; 2k � 2; L2k�1

fcXðxÞ 6¼ 0: (9)

PROPOSITION 2.2 [1]. Consider the impulsive dynamical systems(6) and (7). Let X : D ! R be an infinitely differentiable functionsuch that Z ¼ fx 2 D : XðxÞ ¼ 0g and assume that every x 2 Zis k-transversal to Eq. (6). Then at every x0 62 Z such that0< s1(x0)<1, s1(�) is continuous. Furthermore, if x0 2 ZnZ issuch that s1(x0) � (0, 1) and either (i) fxig1i¼1 2 ZnZ or (ii)limi!1 s1ðxiÞ > 0, where fxig1i¼1 62 Z is such that limi!1 xi ¼ x0

and limi!1 s1ðxiÞ exists, then limi!1 s1ðxiÞ ¼ s1ðx0Þ.Remark 2.2. Let x0 62 Z be such that limi!1 s1ðxiÞ 6¼ s1ðx0Þ for

some unbounded sequence fxig1i¼1 62 Z with limi!1 xi ¼ x0.Then it follows from Proposition 2.2 that limi!1 s1ðxiÞ ¼ 0.

Remark 2.3. The notion of k-transversality introduced in Defini-tion 2.1 differs from the well-known notion of transversality[24,25] involving an orthogonality condition between a vectorfield and a differentiable submanifold. In the case where k¼ 1,Definition 2.1 coincides with the standard notion of transversalityand guarantees that the solution of the closed-loop systems (6)and (7) is not tangent to the closure of the resetting set Z at theintersection with Z [2]. In general, however, k-transversality guar-antees that the sign of XðxðtÞÞ changes as the closed-loop systemtrajectory x(t) transverses the closure of the resetting set Z at theintersection with Z.

Remark 2.4. Proposition 2.2 is a nontrivial generalization ofProposition 4.2 of Ref. [21] and Lemma 3 of Ref. [23]. Specifi-cally, Proposition 2.2 establishes the continuity of s1(�) in the casewhere the resetting set Z is not a closed set. In addition, thek-transversality condition given in Definition 2.1 is also a general-ization of the transversality conditions given in Refs. [21] and[23] by considering higher order derivatives of the function Xð�Þrather than simply considering the first-order derivative as in Refs.[21] and [23].

The next result characterizes impulsive dynamical systemlimit sets in terms of continuously differentiable functions. Inparticular, we show that the system trajectories of a state-dependent impulsive dynamical system converge to an invariantset contained in a union of level surfaces characterized by thecontinuous-time system dynamics and the resetting system dy-namics. Note that for addressing the stability of sets of an impul-sive dynamical system the usual set stability definitions are valid[9]. Specifically, for a positively invariant set D0 � D;D0 isLyapunov stable with respect to Eqs. (6) and (7) if and only if,for every open neighborhood O1 � D of D0, there exists an openneighborhood O2 � O1 of D0 such that xðtÞ 2 O1, t� 0, for allx0 2 O2. Equivalently, D0 is Lyapunov stable with respect toEqs. (6) and (7) if and only if, for all e> 0, there existsd¼ d(e)> 0 such that if distðx0;D0Þ < d, then distðsðt; x0Þ;D0Þ< e; t � 0. D0 is attractive with respect to Eqs. (6) and (7) if andonly if there exists an open neighborhood O3 � D of D0 suchthat the omega limit set x(x0) of Eqs. (6) and (7) is contained inD0 for all x0 2 O3. D0 is asymptotically stable with respectto Eqs. (6) and (7) if and only if D0 is Lyapunov stable andattractive. Equivalently, D0 is asymptotically stable withrespect to Eqs. (6) and (7) if and only if D0 is Lyapunov stableand there exits e> 0 such that if distðx0;D0Þ < e, thendistðsðt; x0Þ;D0Þ ! 0 as t!1. Asymptotic stability is global ifthe previous statement holds for all x0 2 Rn.

It is important to note here that since state-dependent impulsivedynamical systems are time invariant [2], the notions of asymp-totic stability and uniform asymptotic stability with respect to ini-tial times are equivalent. However, unlike continuous-time anddiscrete-time dynamical systems, wherein asymptotic set stabilityof autonomous systems is equivalent to the existence of classK and L functions a(�) and b(�), respectively, such that ifdistðx0;D0Þ < d, d> 0, then distðxðtÞ;D0Þ aðdistðx0;D0ÞÞbðtÞ,t� 0, this is not generally true for state-dependent impulsivedynamical systems. That is, asymptotic stability might not be uni-form with respect to compact sets of initial conditions. If, how-ever, for every compact set the first time-to-impact function s1(x0)is uniformly bounded with respect to the system initial conditions,then it can be shown that asymptotic stability is uniform with

Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2014, Vol. 136 / 051020-3


respect to compact sets of initial conditions. For further details onthis subtle point, see Ref. [26].

THEOREM 2.1. Consider the impulsive dynamical systems (6)and (7) and assume Assumptions 1–3 hold. Assume Dci � D is apositively invariant set with respect to Eqs. (6) and (7), assumethat if x0 2 Z, then x0 þ fdðx0Þ 2 ZnZ, and assume that thereexists a continuously differentiable function V : Dci ! R suchthat

V0ðxÞfcðxÞ 0; x 2 Dci; x 62 Z (10)

Vðxþ fdðxÞÞ VðxÞ; x 2 Dci; x 2 Z (11)

Let R ¼D fx 2 Dci : x 62 Z; V0ðxÞfcðxÞ ¼ 0g [ fx 2 Dci : x 2 Z;Vðxþ fdðxÞÞ ¼ VðxÞg and let M denote the largest invariant setcontained in R. If x0 2 Dci, then xðtÞ !M as t ! 1. Further-

more, if D0 � D�

ci;VðxÞ ¼ 0; x 2 D0, VðxÞ > 0; x0 2 DcinD0, andthe set R contains no invariant set other than the set D0, then theset D0 is asymptotically stable with respect to Eqs. (6) and (7),and Dci is a subset of the domain of attraction of Eqs. (6) and (7).

Proof. The proof of this result is similar to the proof ofCorollary 5.1 given in Ref. [21] and, hence, is omitted. �

Remark 2.5. Setting D ¼ Rn and requiring V(x) ! 1 asxk k ! 1 in Theorem 2.1, it follows that the set D0 is globally

asymptotically stable. A similar remark holds for Theorem 2.2below.

THEOREM 2.2. Consider the impulsive dynamical systems (6)and (7) and assume Assumptions 1–3 hold. Assume Dci � D is apositively invariant set with respect to Eqs. (6) and (7) such that

D0 � D�

ci, assume that if x0 2 Z, then x0 þ fdðx0Þ 2 ZnZ, andassume that for every x0 2 DcinD0, there exists s� 0 such thatxðsÞ 2 Z, where x(t), t� 0, denotes the solution to Eqs. (6) and(7) with the initial condition x0. Furthermore, assume there existsa continuously differentiable function V : Dci ! R such thatVðxÞ ¼ 0; x 2 D0;VðxÞ > 0, x0 2 DcinD0

Vðxþ fdðxÞÞ < VðxÞ; x 2 Dci; x 2 Z (12)

and Eq. (10) is satisfied. Then the set D0 � Dci is asymptoticallystable with respect to Eqs. (6) and (7) and Dci is a subset of thedomain of attraction.

Proof. It follows from Eq. (12) that R ¼ fx 2 Dci : x 62 Z;V0ðxÞfcðxÞ ¼ 0g. Since for every x0 2 DcinD0, there exists s� 0such that xðsÞ 2 Z, it follows that the largest invariant setcontained in R is D0. Now, the result is a direct consequence ofTheorem 2.1. �

3 Hybrid Stabilization of Sets

In this section, we present a hybrid controller design frameworkfor stabilization of sets. Specifically, we consider nonlineardynamical systems Gp of the form given by Eqs. (1) and (2). Fur-thermore, we consider hybrid resetting dynamic controllers Gc ofthe form

_xcðtÞ ¼ fccðxcðtÞ; yðtÞÞ; xcð0Þ ¼ xc0; ðxcðtÞ; yðtÞÞ 62 Zc (13)

DxcðtÞ ¼ gðyðtÞÞ � xcðtÞ; ðxcðtÞ; yðtÞÞ 2 Zc (14)

ycðtÞ ¼ hccðxcðtÞ; yðtÞÞ (15)

where xcðtÞ 2 Dc � Rnc ;Dc is an open set, yðtÞ 2 Rl,ycðtÞ 2 Rm, fcc : Dc �Rl ! Rnc is smooth on Dc �Rl,g : Rl ! Dc is continuous, and hcc : Dc �Rl ! Rm is smooth.

Consider the negative feedback interconnection of Gp and Gc

given by y¼ uc and u¼�yc. In this case, the closed-loop systemG is given by

_xðtÞ ¼ fcðxðtÞÞ; xð0Þ ¼ x0; xðtÞ 62 Z; t � 0 (16)

DxðtÞ ¼ fdðxðtÞÞ; xðtÞ 2 Z (17)

where t� 0, xðtÞ ¼D ½xTp ðtÞ; xT

c ðtÞ�T, Z ¼D fx 2 D : ðxc; hpðxpÞÞ

2 Zcg,

fcðxÞ ¼fpðxp;�hccðxc; hpðxpÞÞÞ

fccðxc; hpðxpÞÞ

� �; fdðxÞ ¼

0

gðhpðxpÞÞ � xc

� �(18)

The objective is to design the hybrid resetting controller (13)–(15)in such a way that the set D0 ¼ fðxp; xcÞ 2 Dp �Dc : xp 2 Dp0g,whereDp0 � Dp, is asymptotically stable with respect to the closed-loop systems (16) and (17). In order to do this, we associate with theplant a generalized energy function Vp : Dp ! Rþ such thatVp(xp)¼ 0, xp 2 Dp0, and VpðxpÞ > 0; xp 2 DpnDp0. Furthermore,we associate with the controller a generalized energy functionVc : Dc �Rl ! Rþ such that Vcðxc; yÞ � 0; xc 2 Dc; y 2 Rl, andVcðxc; yÞ ¼ 0 if and only if xc¼ g(y). Finally, we associate with theclosed-loop system the generalized energy functionVðxÞ ¼D VpðxpÞ þ Vcðxc; hpðxpÞÞ.

Next, we construct the resetting set for the closed-loop systemG in the following way:

Z¼fðxp;xcÞ2Dp�Dc :Lfc Vcðxc;hpðxpÞÞ¼0 and Vcðxc;hpðxpÞÞ>0g(19)

The resetting set Z is thus defined to be the set of all points in theclosed-loop state space that correspond to the instant when thecontroller is at the verge of decreasing its generalized energyfunction Vc(�). By resetting the controller states, the generalizedenergy function Vp(�) can never increase after the first resettingevent. Furthermore, if the closed-loop system generalized energyfunction V(�) is conserved between resetting events, then adecrease in Vp(�) is accompanied by a corresponding increase inVc(�). Hence, this approach allows the generalized plant energy toflow to the controller, where it increases the emulated generalizedcontroller energy but does not allow the emulated generalizedcontroller energy to flow back to the plant after the first resettingevent.

This energy dissipating hybrid controller effectively enforces aone-way generalized energy transfer between the plant and thecontroller after the first resetting event. For practical implementa-tion, knowledge of xc and y is sufficient to determine whether ornot the closed-loop state vector is in the set Z. That is, the fullstate xp need not be known in order to determine whether or notthe closed-loop state vector is in the set Z, neither is it needed forfeedback control between resettings determined by Eq. (15).

The next theorem gives sufficient conditions for asymptoticstability of the set D0 � Dp �Dc with respect to the closed-loopsystem G using state-dependent hybrid controllers.

THEOREM 3.1. Consider the closed-loop impulsive dynamicalsystem G given by Eqs. (16) and (17) and assume that Dci � D is

a positively invariant set with respect to G such that D0 � D�

ci,where D0 ¼ fðxp; xcÞ 2 Dp �Dc : xp 2 Dp0g and Dp0 � Dp.Assume that there exists a continuously differentiable function

Vp : Dp ! Rþ such that VpðxpÞ ¼ 0; xp 2 Dp0, and VpðxpÞ > 0;xp 2 DpnDp0, and assume there exists a smooth (i.e., infinitely

differentiable) function Vc : Dc �Rl ! Rþ such that Vcðxc; yÞ� 0; xc 2 Dc; y 2 Rl, and Vcðxc; yÞ ¼ 0 if and only if xc¼ g(y).Furthermore, assume that every x0 2 Z is k-transversal toEq. (16) and

_VpðxpðtÞÞ þ _VcðxcðtÞ; yðtÞÞ ¼ 0; xðtÞ 62 Z (20)

where y¼ uc¼ hp(xp) and Z is given by Eq. (19). Then, the setD0 � Dci is asymptotically stable with respect to the closed-loopsystem G. Finally, if Dp ¼ Rnp ;Dc ¼ Rnc and V(�) is radially



unbounded, then the set D0 � Dci is globally asymptotically sta-ble with respect to G.

Proof. First, note that since Vc(xc, y)� 0, xc 2 Dc; y 2 Rl, itfollows that

Z¼fðxp;xcÞ2Dp�Dc :Lfc Vcðxc;hpðxpÞÞ¼0 and Vcðxc;hpðxpÞÞ�0g¼fðxp;xcÞ2Dp�Dc :XðxÞ¼0g (21)

where XðxÞ ¼ Lfc Vcðxc; hpðxpÞÞ. Next, we show that if the k-trans-versality condition (9) holds, then Assumptions 1–3 hold and, forevery x0 2 Dci, there exists s� 0 such that xðsÞ 2 Z. Note that ifx0 2 ZnZ, that is, Vcðxcð0Þ; hpðxpð0ÞÞÞ ¼ 0, and Lfc Vcðxcð0Þ;hpðxpð0ÞÞÞ ¼ 0, it follows from the k-transversality condition thatthere exists d> 0 such that for all t � (0, d], Lfc VcðxcðtÞ;hpðxpðtÞÞÞ 6¼ 0. Hence, since VcðxcðtÞ; hpðxpðtÞÞÞ ¼ Vcðxcð0Þ;hpðxpð0ÞÞÞ þ tLfc VcðxcðsÞ; hpðxpðsÞÞÞ for some s � (0, t] andVc(xc,y)� 0, xc 2 Dc; y 2 Rl, it follows that VcðxcðtÞ; hpðxpðtÞÞÞ> 0, t � (0, d], which implies that Assumption 1 is satisfied. Fur-thermore, if x 2 Z then, since Vc(xc,y)¼ 0 if and only if xc¼ g(y),it follows from Eq. (17) that xþ fdðxÞ 2 ZnZ. Hence, Assump-tion 2 holds.

Next, consider the set Mc ¼D fx 2 Dci : Vcðxc; hpðxpÞÞ ¼ cg,

where c� 0. It follows from the k-transversality condition that forevery c� 0, Mc does not contain any nontrivial trajectory ofG. To see this, suppose, ad absurdum, there exists a nontrivialtrajectory xðtÞ 2 Mc, t� 0, for some c� 0. In this case, itfollows that ðdk=dtkÞVcðxcðtÞ; hpðxpðtÞÞÞ ¼ Lk

fcVcðxcðtÞ; hpðxpðtÞÞÞ

� 0, k¼ 1, 2,…, which contradicts the k-transversality condition.Next, we show that for every x0 62 Z; x0 62 D0, there exists

s> 0 such that xðsÞ 2 Z. To see this, suppose, ad absurdum,xðtÞ 62 Z, t� 0, which implies that

d

dtVcðxcðtÞ; hpðxpðtÞÞÞ 6¼ 0; t � 0 (22)

orVcðxcðtÞ; hpðxpðtÞÞÞ ¼ 0; t � 0 (23)

If Eq. (22) holds, then it follows that VcðxcðtÞ; hpðxpðtÞÞÞ is a(decreasing or increasing) monotonic function of time. Hence, itfollows from the monotone convergence theorem [Ref. [9], p. 37]that VcðxcðtÞ; hpðxpðtÞÞÞ ! c as t!1, where c� 0 is a constant,which implies that the positive limit set of the closed-loop systemis contained in Mc for some c� 0, and hence, is a contradiction.Similarly, if Eq. (23) holds, thenM0 contains a nontrivial trajec-tory of G also leading to a contradiction. Hence, for every x0 62 Z,there exists s> 0 such that xðsÞ 2 Z. Thus, it follows that for ev-ery x0 62 Z, 0< s1(x0)<1. Now, it follows from Proposition 2.2

that s1(�) is continuous at x0 62 Z. Furthermore, for all x0 2 ZnZand for every unbounded sequence fxig1i¼1 2 ZnZ converging to

x0 2 ZnZ, it follows from the k-transversality condition and Prop-

osition 2.2 that limi!1 s1ðxiÞ ¼ s1ðx0Þ. Next, let x0 2 ZnZ andlet fxig1i¼1 2 Dci be such that limi!1 xi ¼ x0 and limi!1 s1ðxiÞexists. In this case, it follows from Proposition 2.2 that eitherlimi!1 s1ðxiÞ ¼ 0 or limi!1 s1ðxiÞ ¼ s1ðx0Þ. Furthermore, since

x0 2 ZnZ corresponds to the case where Vcðxc0; hpðxp0ÞÞ ¼ 0, itfollows that xc0 ¼ gðhpðxp0ÞÞ, and hence, fd(x0)¼ 0. Now, it fol-lows from Proposition 2.1 that Assumption 3 holds.

Next, note that if x0 2 Z and x0 þ fdðx0Þ 62 D0, then it followsfrom the above analysis that there exists s> 0 such that xðsÞ 2 Z.Alternatively, if x0 2 Z and x0 þ fdðx0Þ 2 D0, then the solution ofthe closed-loop system reaches D0 in finite time, which is a stron-ger condition than reaching D0 as t!1.

To show that the set D0 � Dci is asymptotically stable, considerthe Lyapunov function candidate VðxÞ ¼ VpðxpÞ þ Vcðxc; hpðxpÞÞcorresponding to the total generalized energy function. It followsfrom Eq. (20) that

_VðxðtÞÞ ¼ 0; xðtÞ 62 Z (24)

Furthermore, it follows from Eqs. (18) and (19) that

DVðxðtkÞÞ ¼ Vcðxcðtþk Þ; hpðxpðtþk ÞÞÞ � VcðxcðtkÞ; hpðxpðtkÞÞÞ¼ VcðgðhpðxpðtkÞÞÞ; hpðxpðtkÞÞÞ � VcðxcðtkÞ; hpðxpðtkÞÞÞ¼ �VcðxcðtkÞ; hpðxpðtkÞÞÞ< 0; xðtkÞ 2 Z; k 2 Zþ (25)

Thus, it follows from Theorem 2.2 that the set D0 � Dci is asymp-totically stable. Finally, if Dp ¼ Rnp ;Dc ¼ Rnc , and V(�) isradially unbounded, then global asymptotic stability is immediateusing standard arguments. �

To demonstrate the utility of Theorem 3.1, let the set Dp0 begiven by the zero level set of the function Qp : Dp ! Rsp and letVp : Dp ! Rþ be given by

VpðxpÞ ¼ QTðxpÞPQðxpÞ; xp 2 Dp (26)

where P 2 Rsp�sp and P> 0. Furthermore, let Vc : Dc

�Rl ! Rþ be given by

Vcðxc;hpðxpÞÞ¼ðxc�gðhpðxpÞÞÞTPcðxc�gðhpðxpÞÞÞ;ðxp;xcÞ2Dp�Dc

(27)

where Pc 2 Rnc�nc and Pc> 0. In this case, the functions fcc(�,�),hcc(�,�), and g(�) can be selected using Eq. (20) in Theorem 3.1.These constructions are shown for the specific problems of con-sensus and formation control for multiagent systems in Secs. 4–6.

4 Specialization to Linear Dynamical Systems

In this section, we specialize the results of Sec. 3 to the class oflinear dynamical systems given by

_xpðtÞ ¼ AxpðtÞ þ BuðtÞ; xpð0Þ ¼ xp0; t � 0 (28)

yðtÞ ¼ CxpðtÞ (29)

where xpðtÞ 2 Rn;A 2 Rn�n;B 2 Rn�m, and C 2 Rl�n. Here, forsimplicity of exposition, we assume np¼ nc¼ n and C¼ In. Thecase where C 6¼ In can be addressed using an identical analysis asshown below with F2, H2, and M in Eqs. (32)–(34) replaced byF2C, H2C, and MC, respectively. For the systems (28) and (29),we construct a hybrid feedback controller of the form (13)–(15)that asymptotically stabilizes the set D0 given by

D0 ¼ fðxp; xcÞ 2 Dp �Dc : xp 2 Dp0g (30)

where

Dp0 ¼ fxp 2 Dp : Txp ¼ 0g (31)

and T 2 Rsp�n. Specifically, we set

fccðxc; xpÞ ¼ F1xc þ F2xp (32)

hccðxc; xpÞ ¼ �H1xc � H2xp (33)

gðxpÞ ¼ Mxp (34)

where F1 2 Rn�n;F2 2 Rn�n, H1 2 Rm�n;H2 2 Rm�n, andM 2 Rn�n. Thus, the closed-loop system (28), (29), and (13)–(15)with the negative feedback interconnection u¼�yc is given by

_xpðtÞ ¼ ðAþ BH2ÞxpðtÞ þ BH1xcðtÞ; ðxpðtÞ; xcðtÞÞ 62 Z (35)



_xcðtÞ ¼ F1xcðtÞ þ F2xpðtÞ; ðxpðtÞ; xcðtÞÞ 62 Z (36)

DxcðtÞ ¼ MxpðtÞ � xcðtÞ; ðxpðtÞ; xcðtÞÞ 2 Z (37)

where Z is given by Eq. (19).Next, define the generalized energy functions

VpðxpÞ ¼1

2xT

p TTTxp; xp 2 Dp (38)

Vcðxc; xpÞ ¼1

2ðxc �MxpÞTPcðxc �MxpÞ; ðxp; xcÞ 2 Dp �Dc

(39)

where Pc 2 Rn�n and Pc> 0. Note that Vp(xp)¼ 0, xp 2 Dp0, andVp(xp)> 0, xp 2 DpnDp0. Furthermore, note that Vc(xc, xp)� 0,ðxp; xcÞ 2 Dp �Dc, and Vc(xc, xp)¼ 0 if and only if xc¼ g(xp). Forthe closed-loop system (35)–(37), condition (20) in Theorem 3.1gives

_VpðxpðtÞÞ þ _VcðxcðtÞ; xpðtÞÞ¼ xT

p ðtÞðTTTBH1 þ FT2 Pc � ATMTPc � HT

2 BTMTPc

�MTPcF1 þMTPcMBH1ÞxcðtÞþ xT

p ðtÞðTTTAþ TTTBH2 �MTPcF2

þMTPcMAþMTPcMBH2ÞxpðtÞþ xT

c ðtÞðPcF1 � PcMBH1ÞxcðtÞ ¼ 0; ðxpðtÞ; xcðtÞÞ 62 Z(40)

Since xp and xc are independent state variables, Eq. (40) holds ifand only if there exist skew-symmetric matrices Ap 2 Rn�n andAc 2 Rn�n such that

TTTBH1 þ FT2 Pc � ATMTPc � HT

2 BTMTPc �MTPcF1

þMTPcMBH1 ¼ 0 (41)

TTTAþ TTTBH2 �MTPcF2 þMTPcMAþMTPcMBH2 ¼ Ap

(42)

PcF1 � PcMBH1 ¼ Ac (43)

The skew-symmetric matrices Ap 2 Rn�n and Ac 2 Rn�n arefree design parameters. Furthermore, if the matrices H1 2 Rm�n

and H2 2 Rm�n are fixed, then it follows from Eqs. (41)–(43) that

F1 ¼ P�1c Ac þMBH1 (44)

F2 ¼ MAþMBH2 � P�1c AcM � P�1

c HT1 BTTTT (45)

where M 2 Rn�n satisfies

TTTAþ TTTBH2 þMTAcM þMTHT1 BTTTT ¼ Ap (46)

Note that if Ac is skew symmetric, then MTAcM is also skew sym-metric. In this case, we can set Ap ¼ ~Ap þMTAcM, where~Ap 2 Rn�n is an arbitrary skew-symmetric matrix, so that

NM ¼ L (47)

where

N ¼D TTTBH1 (48)

L ¼D � ~Ap � ATTTT � HT2 BTTTT (49)

Recall that a solution M to the matrix equation (47) exists ifand only if [27, Fact 6.4.43, p. 421]

NN†L ¼ L (50)

where N† 2 Rn�n is the Moore–Penrose generalized inverse ofN 2 Rn�n. If Eq. (50) is satisfied, then every solution to Eq. (47)is given by

M ¼ N†Lþ Y � N†NY (51)

where Y 2 Rn�n is an arbitrary matrix; and if Y¼ 0, then tr MTMis minimized. Thus, the existence of a hybrid controller thatasymptotically stabilizes the set D0 given by Eqs. (30) and (31) ischaracterized by a matrix condition (50). Finally, if

TTTBH1 6¼ 0 (52)

then the k-transversality condition (9) is satisfied. To see this, notethat Eq. (52) implies that _VpðxpÞ �= 0, which, using Eq. (20),implies that _Vcðxc; xpÞ �= 0. This shows that k-transversality condi-tion, with k¼ 1, holds for the closed-loop system (35)–(37). Notethat Eq. (52) is guaranteed by Eq. (50).

5 Hybrid Control Design for Parallel and

Rendezvous Formations

In this section, we apply the hybrid control framework devel-oped in Sec. 4 to multiagent systems composed of double integra-tor agents executing various coordinated tasks. First, we considerthe parallel formation problem for multiagent systems. Specifi-cally, let q denote the number of mobile agents so thatxp ¼ ½xT

p1; xTp2�

T 2 R2dq, where xp1 2 Rdq represents a vector ofpositions, xp2 ¼

D_xp1 2 Rdq represents a vector of velocities, d rep-

resents the number of degrees-of-freedom of each agent, and Aand B in Eq. (28) are given by

A ¼ 0dq�dq Idq

0dq�dq 0dq�dq

� �; B ¼ 0dq�dq

Idq

� �(53)

The control aim is to design a hybrid feedback control law sothat a parallel formation is achieved, wherein the agents arecollectively required to maintain a prescribed geometric shapewith constant velocities, and the relative position between anytwo mobile agents is asymptotically stabilized to a constant value.For this task, we set d¼ 2 and let q¼ 5 so that xp1¼ [x1, y1,…, x5,y5]T, where xi, yi, i¼ 1,…,5, are, respectively, horizontal and ver-tical coordinates of the ith agent. The individual agent dynamicsare thus given by

€xiðtÞ ¼ uxiðtÞ; xið0Þ ¼ xi0; t � 0 (54)

€yiðtÞ ¼ uyiðtÞ; yið0Þ ¼ yi0 (55)

where i¼ 1,…, 5 and uxi and uyi are individual control inputs inthe horizontal and vertical directions, respectively.

For our hybrid controller design, we set H1¼ [I10, I10],H2¼ [010�10, H22], Pc¼ 2I20, ~Ap ¼ 0, where

H22 ¼

�2 0 1 0 0 0 0 0 0 0

0 �2 0 0 0 0 0 0 0 0

1 0 �2 0 1 0 0 0 0 0

0 0 0 �2 0 0 0 0 0 0

0 0 1 0 �2 0 1 0 0 0

0 0 0 0 0 �2 0 0 0 0

0 0 0 0 1 0 �2 0 1 0

0 0 0 0 0 0 0 �2 0 0

1 0 0 0 0 0 1 0 �2 0

0 0 0 0 0 0 0 0 0 �2

2666666666666664

3777777777777775

(56)

and we choose Ac 2 R20�20 to be a random skew-symmetricmatrix. The specifications of the parallel formation along the x



axis with equal distances along the y axis and equal velocities canbe characterized by Eq. (31) with

T ¼T1 07�10

09�10 T2

" #(57)

where

T1 ¼

1 0 �1 0 0 0 0 0 0 0

0 0 1 0 �1 0 0 0 0 0

0 0 0 0 1 0 �1 0 0 0

0 0 0 0 0 0 1 0 �1 0

0 1 0 �2 0 1 0 0 0 0

0 0 0 1 0 �2 0 1 0 0

0 0 0 0 0 1 0 �2 0 1

26666666666664

37777777777775

(58)

T2 ¼

0 1 0 0 0 0 0 0 0 0

0 1 0 �1 0 0 0 0 0 0

0 0 0 1 0 �1 0 0 0 0

0 0 0 0 0 1 0 �1 0 0

0 0 0 0 0 0 0 1 0 �1

1 0 �1 0 0 0 0 0 0 0

0 0 1 0 �1 0 0 0 0 0

0 0 0 0 1 0 �1 0 0 0

0 0 0 0 0 0 1 0 �1 0

26666666666664

37777777777775

(59)

In this case, condition (50) is verified and M 2 R20�20 is obtainedfrom Eq. (51) with Y¼ 0. Consequently, the matrices F1 2 R20�20

and F2 2 R20�20 are computed using Eqs. (44) and (45).For our simulation, we set xpð0Þ ¼ ½0:4; �1; �0:3; 1:3;

�1:3; �0:8; �0:1; �0:7; 1:1; 1:9; �0:34; �0:26; 0:2; �0:12; 0:47;0:47; 0:15; 0:36; �0:1; 0:13�T and xc(0) ¼Mxp(0). Figure 1 showsthe positions of the agents in the plane, whereas Figs. 2 and 3show the control forces in x and y directions, respectively, actingon each agent. Figures 4 and 5 show the agent velocities in the xand y directions, respectively. Finally, Fig. 6 shows the time his-tory of the generalized energy functions Vp(xp(t)) and Vc(xc(t),xp(t)), t� 0.

For the next task, we design a hybrid controller (13)–(15) forthe rendezvous problem of planar double integrator agents. Spe-cifically, a cooperative rendezvous task requires that each agentdetermines the rendezvous time and location through team nego-tiation. In the following simulation, we consider four agents com-ing to a square formation with zero terminal velocities. In this

case, T ¼ T1 06�8

08�8 T2

� �, where

T1 ¼

1 0 �1 0 1 0 �1 0

0 1 0 �1 0 1 0 �1

1 0 1 0 �1 0 �1 0

0 1 0 �1 0 �1 0 1

1 �1 �1 0 0 0 0 1

0 0 0 1 1 �1 �1 0

26666664

37777775

(60)

Fig. 1 Agent positions in the plane

Fig. 2 Control forces in x direction

Fig. 3 Control forces in y direction

Fig. 4 Velocities in x direction



T2 ¼

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

1 0 �1 0 0 0 0 0

0 1 0 �1 0 0 0 0

0 0 1 0 �1 0 0 0

0 0 0 1 0 �1 0 0

0 0 0 0 1 0 �1 0

0 0 0 0 0 1 0 �1

266666666664

377777777775

(61)

For our simulation, we set H1¼ [I8, I8], H2¼ [08�8, �2I8],Ac ¼ ~Ap ¼ 0, and Pc¼ 0.7I16. As in our previous example, condi-tion (50) is verified and M 2 R16�16 is obtained from Eq. (51)with Y¼ 0. For the initial conditions xpð0Þ ¼ ½10; 10; 1; 5;2; 4; 9; 1; 0; 0; 0; 0; 0; 0; 0; 0�T and xc(0)¼Mxp(0), Fig. 7 shows thepositions of the agents in the plane, whereas Figs. 8 and 9 showthe control forces in x and y directions, respectively, acting oneach agent. Finally, Fig. 10 shows the time history of the general-ized energy functions Vp(xp(t)) and Vc(xc(t), xp(t)), t� 0.

6 Hybrid Control Design for Consensus in

Multiagent Networks

In this section, we specialize the results of Sec. 4 to designhybrid consensus controllers for multiagent networks of single in-tegrator systems. Specifically, the consensus problem involves thedesign of a dynamic protocol algorithm that guarantees systemstate equipartition [4,6], that is, limt!1 xpiðtÞ ¼ a 2 R for

i¼ 1,…, q, where xpi(t) denotes the ith component of the systemstate vector xp(t). In particular, consider q continuous-time inte-grator agents with dynamics

_xpiðtÞ ¼ uiðtÞ; xið0Þ ¼ xi0; t � 0; i ¼ 1;…; q (62)

yiðtÞ ¼ xpiðtÞ (63)

where, for each i � {1,…, q}, xpiðtÞ 2 R denotes the informationstate and uiðtÞ 2 R denotes information control input for all t� 0.In this case, the set Dp0 ¼ fxp 2 Rq : xp1 ¼ � � � ¼ xpqg, where

xp ¼D ½xp1;…; xpq�T, characterizes the state of consensus in themultiagent network.

In the following analysis, we construct a hybrid feedback con-troller (13)–(15) that asymptotically stabilizes a more generalform of the classical consensus steady state for the multiagent net-work characterized by

D0 ¼ fðxp; xcÞ 2 Dp �Dc : xp 2 Dp0g (64)

whereDp0 ¼ fxp 2 Dp : Txp ¼ 0g (65)

and T 2 Rsp�q. Clearly, Dp0 given by Eq. (65) characterizes theequipartitioned consensus state of a multiagent network with

Fig. 5 Velocities in y direction

Fig. 6 Generalized energy functions Vp(xp(t)) and Vc(xc(t),xp(t)) versus time


Fig. 8 Control forces in x direction



T ¼

1 �1 0 0 � � � 0

0 1 �1 0 � � � 0

0 0 1 �1 � � � 0

..

. . .. ..

.

0 0 0 0 � � � �1

266664

377775 2 Rðq�1Þ�q (66)

In order to stabilize D0 given by Eq. (64), consider the hybridfeedback controller (13)–(15) with

fccðxc; xpÞ ¼ Fðxc � xpÞ (67)

hccðxc; xpÞ ¼ �Hðxc � xpÞ (68)

gðxpÞ ¼ Mxp (69)

where F 2 Rq�q;H 2 Rq�q, and M 2 Rq�q. In this case,Eqs. (41)–(43) developed for a general class of linear dynamicalsystems specialize to

PcF� PcMH ¼ Ac (70)

� TTTH þMTAc ¼ Ap (71)

Ap ¼ Ac (72)

for the systems (62) and (63). Note that Eqs. (70)–(72) can be fur-ther simplified to give

F�MH ¼ P�1c Ac (73)

� TTTH þMTAc ¼ Ac (74)

Note that if q is even, then we can always choose a skew-symmetric matrix Ac 2 Rq�q such that A�1

c exists. In this case, itfollows from Eqs. (73) and (74) that

M ¼ A�1c ðAc � HTTTTÞ (75)

F ¼ P�1c Ac þ A�1

c ðAc � HTTTTÞH (76)

For the following numerical example, we consider four agentswith the dynamics given by Eqs. (62) and (63) and the objectivebeing to stabilize the equipartitioned consensus state withT 2 R3�4 given by Eq. (66). For our design, we set

Ac ¼

0 1 �1 0

�1 0 1 �1

1 �1 0 0:50 1 �0:5 0

2664

3775 (77)

H¼ 1.5I4, and Pc¼ 0.75I4 so that M 2 R4�4 and F 2 R4�4 arecomputed using Eqs. (75) and (76). Note that with theabove choice of T 2 R3�4 and H 2 R4�4, condition (52) is satis-fied. For the initial conditions, xp(0)¼ [0.5, 1, 0.7, 1.2]T andxc(0)¼Mxp(0), Fig. 11 shows the system states history versustime, whereas Fig. 12 shows the control input history versus time.Finally, Fig. 13 shows the time history of the generalized energyfunctions Vp(xp(t)) and Vc(xc(t), xp(t)) versus time. It can be seenfrom Fig. 12 that the control inputs ui, i¼ 1,…, 4, are discontinu-ous functions of time.

7 Hybrid Control Design for Cyclic Pursuit

In this section, we use the results of Sec. 2 to develop a hybridresetting controller of the form (3)–(5) to achieve circular forma-tions [28] involving cyclic pursuit [16,29,30]. The proposed con-troller has a leaderless, dynamic distributed architecture, which is

Fig. 9 Control forces in y direction

Fig. 10 Generalized energy functions Vp(xp(t)) and Vc(xc(t),xp(t)) versus time

Fig. 11 Plant states xp versus time

Fig. 12 Control inputs versus time



more robust and exhibits faster convergence than static, leader-based, or partially leaderless control designs [16,28–30]. Considerq mobile autonomous agents in a plane described by the unicyclemodel given by

_xiðtÞ ¼ viðtÞ cos hiðtÞ; xið0Þ ¼ xi0; t � 0 (78)

_yiðtÞ ¼ viðtÞ sin hiðtÞ; yið0Þ ¼ yi0 (79)

_hiðtÞ ¼ xiðtÞ; hið0Þ ¼ hi0 (80)

where, for each i � {1,…, q}, ½xi; yi�T 2 R2 denotes the positionvector of the ith agent, hi 2 R denotes the orientation of the ithagent, vi 2 R denotes the velocity of the ith agent, xi 2 Rdenotes the angular velocity of the ith agent, and ui ¼ ½vi;xi�T isthe control input of the ith agent.

For our result, we assume that the graph G of the communica-tion topology for the mobile agents is undirected and stronglyconnected [3]. The control aim is to design ui by means of neigh-boring information so that a circular formation is achieved; that is,the system state asymptotically converges to an invariant manifoldcharacterized by the following constraints [29]:

vi ¼ c1; xi ¼ c2;Xq

i¼1

sin hi ¼ 0;Xq

i¼1

cos hi ¼ 0 (81)

Xj2N i

½ðxi � xjÞ cos hi þ ðyi � yjÞ sin hi� ¼ 0; i ¼ 1;…; q (82)

where c1 2 R; c2 2 R, and N i denote the set of all neighborswhich can communicate with the ith agent. Note that ifhiðtÞ � hjðtÞ ¼ ð2ðj� iÞ=qÞ for all i, j¼ 1,…, q, then

Pqi¼1

sin hi ¼ 0 andPq

i¼1 cos hi ¼ 0.Define hiðxpÞ ¼

D Pj2N i½ðxi � xjÞ cos hi þ ðyi � yjÞ sin hi�; ~hi

¼D hi þ ð2i=qÞp, i¼ 1,…, q, hðxpÞ ¼D ½h1ðxpÞ;…; hqðxpÞ�T, and

~h ¼D ½~h1;…; ~hq�T, where xp ¼D ½~xT; ~yT; hT�T, ~x ¼D ½x1;…; xq�T; ~y¼D ½y1;…; yq�T, and h ¼D ½h1;…; hq�T, and consider the hybridresetting controller given by

_xc1ðtÞ ¼ �Lxc1ðtÞ þ hðxpðtÞÞ; xc1ð0Þ ¼ xc10; t � 0; xðtÞ 62 Z(83)

_xc2ðtÞ ¼ �Lxc2ðtÞ þ L~hðtÞ; xc2ð0Þ ¼ xc20; xðtÞ 62 Z (84)

Dxc1ðtÞ ¼ �L†Lxc1ðtÞ; xðtÞ 2 Z (85)

Dxc2ðtÞ ¼ �L†Lxc2ðtÞ; xðtÞ 2 Z (86)

vðtÞ ¼ �xc1ðtÞ � hðxpðtÞÞ (87)

xðtÞ ¼ �xc2ðtÞ (88)

where L 2 Rq�q denotes the Laplacian of the graphG; x ¼D ½xT

p ; xTc �

T, xc ¼D ½xT

c1; xTc2�

T; xc1 2 Rq, and xc2 2 Rq. Since,by assumption, G is undirected and strongly connected, it followsthat L¼LT� 0 and the rank of L is q� 1.

Remark 7.1. It follows from Eqs. (87) and (88) that the systemvelocities and angular velocities are reset due to the resetting ofxc1(t) and xc2(t) when xðtÞ 2 Z. This resetting involves an impulsevelocity as discussed in Ref. [31]. Using Eq. (4.17) of Ref. [31],p. 125, it follows from Eqs. (85)–(88) that

vðtþÞ ¼ vðtÞ þðtþ

t

L†Lxc1ðsÞdðs� tÞds (89)

xðtþÞ ¼ xðtÞ þðtþ

t

L†Lxc2ðsÞdðs� tÞds (90)

where L†Lxc1(t) and L†Lxc2(t) can be viewed as input momentsand d(t) is the Dirac delta function. These equations are a restate-ment of Eq. (4.17) of Ref. [31], p. 125. Hence, in this case, thecontrol input is an impulse. The notion of impulse velocities hasbeen used in engineering dynamics to model dynamical processeswhen a moment acts over a very short time interval (see Ref. [31]for further details).

Next, let

VpðxpÞ ¼1

2~xTL~xþ 1

2~yTL~yþ 1

2~hTL2 ~h (91)

and

Vcðxc; xpÞ ¼1

2xT

c1xc1 þ1

2xT

c2Lxc2 (92)

and define the resetting set Z by

Z¼fx2R5q :xTc1½�Lxc1þhðxpÞ�þxT

c2Lð�Lxc2þL~hÞ¼0; xTc2Lxc2>0g

(93)

Furthermore, note that XðxÞ ¼ xTc1½�Lxc1 þ hðxpÞ� þ xT

c2Lð�Lxc2

þ L~hÞ and

fcðxÞ ¼

diag½cos h1;…; cos hq�½�xc1 � hðxpÞ�diag½sin h1;…; sin hq�½�xc1 � hðxpÞ�

�xc2

�Lxc1 þ h

�Lxc2 þ L~h

2666664

3777775 (94)

where “diag” denotes a diagonal matrix. In addition, note that

@h

@~x¼

Lð1;1Þ cos h1 Lð1;2Þ cos h1 … Lð1;qÞ cos h1

Lð2;1Þ cos h2 Lð2;2Þ cos h2 … Lð2;qÞ cos h2

..

. ... . .

. ...

Lðq;1Þ cos hq Lðq;2Þ cos hq … Lðq;qÞ cos hq

2666664

3777775 (95)

@h

@~y¼

Lð1;1Þ sin h1 Lð1;2Þ sin h1 … Lð1;qÞ sin h1

Lð2;1Þ sin h2 Lð2;2Þ sin h2 … Lð2;qÞ sin h2

..

. ... . .

. ...

Lðq;1Þ sin hq Lðq;2Þ sin hq … Lðq;qÞ sin hq

2666664

3777775 (96)

@h

@h¼ diag½�h1ðxpÞ;…; �hqðxpÞ� (97)

where L(i,j) denotes the (i, j)th entry of L, i, j¼ 1,…, q, and

�hiðxpÞ ¼DP

j2N i½�ðxi � xjÞ sin hi þ ðyi � yjÞ cos hi�, i¼ 1,…, q.

Fig. 13 Generalized energy functions Vp and Vc versus time



Then, it follows that

LfcXðxÞ ¼ xTc1

@h

@~xdiag½cos h1;…; cos hq�ð�xc1 � hÞ

þ xTc1

@h

@~ydiag½sin h1;…; sin hq�ð�xc1 � hÞ � xT

c1

@h

@hxc2

� ðLxc2ÞTLxc2 þ ð�Lxc1 þ hÞTð�Lxc1 þ hÞþ ð�Lxc1ÞTð�Lxc1 þ hÞ � 2ð�Lxc2 þ L~hÞTð�LÞ� ð�Lxc2 þ L~hÞ þ ðL~hÞTð�LÞð�Lxc2 þ L~hÞ¼ xT

c1CðhÞð�xc1 � hÞ þ xTc1SðhÞð�xc1 � hÞ

� xTc1diag½�h1;…; �hq�xc2 � ðLxc2ÞTLxc2

þ ð�Lxc1 þ hÞTð�Lxc1 þ hÞ þ ð�Lxc1ÞTð�Lxc1 þ hÞ� 2ð�Lxc2 þ L~hÞTð�LÞð�Lxc2 þ L~hÞþ ðL~hÞTð�LÞð�Lxc2 þ L~hÞ (98)

where h¼ h(xp), �hi ¼ �hiðxpÞ, i¼ 1,…, q, and

CðhÞ

¼

Lð1;1Þ cos2 h1 Lð1;2Þ cosh1 cosh2 … Lð1;qÞ cosh1 coshq

Lð2;1Þ cosh2 cosh1 Lð2;2Þ cos2 h2 … Lð2;qÞ cosh2 coshq

..

. ... . .

. ...

Lðq;1Þ coshq cosh1 Lðq;2Þ coshq cosh2 … Lðq;qÞ cos2 hq

2666664

3777775

(99)

SðhÞ

¼

Lð1;1Þ sin2 h1 Lð1;2Þ sinh1 sinh2 … Lð1;qÞ sinh1 sinhq

Lð2;1Þ sinh2 sinh1 Lð2;2Þ sin2 h2 … Lð2;qÞ sinh2 sinhq

..

. ... . .

. ...

Lðq;1Þ sinhq sinh1 Lðq;2Þ sinhq sinh2 … Lðq;qÞ sin2 hq

2666664

3777775

(100)

Hence, if LfcXðxÞ 6¼ 0 for all x 2 R5q satisfying

xc1 Lxc2½ � �Lxc1 þ h�Lxc2 þ L~h

� �¼ 0 (101)

anddiag½cos h1;…; cos hq�ð�xc1 � hÞdiag½sin h1;…; sin hq�ð�xc1 � hÞ

�xc2

�Lxc1 þ h

�Lxc2 þ L~h

2666664

3777775 6¼ 0 (102)

then the k-transversality condition (9) holds with k¼ 1.Now, it follows from Eqs. (91), (92), (78)–(80), (83), (84), (87),

and (88) that

_VpðxpÞ þ _Vcðxc; xpÞ¼ ~xTL _~xþ ~yTL _~yþ ~hLTL _hþ xT

c1 _xc1 þ xTc2L _xc2

¼ ~xTL _~xþ ~yTL _~y� ~hLTLxc1 þ xTc1ð�Lxc1 þ hÞ

þ xTc2Lð�Lxc2 þ L~hÞ

¼ ~xTL _~xþ ~yTL _~yþ xTc1h� xT

c1Lxc1 � xTc2L2xc2

¼ ~xTL _~xþ ~yTL _~yþ ð�h� vÞTh� xTc1Lxc1 � xT

c2L2xc2

¼ ~xTL _~xþ ~yTL _~y� vTh� hTh� xTc1Lxc1 � xT

c2L2xc2; x 62 Z(103)

Note that ð1=2Þ~xTL~xþ ð1=2Þ~yTL~y ¼ ð1=4ÞPq

i¼1

Pj2N iðxi � xjÞ2

þð1=4ÞPq

i¼1

Pj2N iðyi � yjÞ2, and hence,

~xTL _~xþ ~yTL _~y ¼ 1

2

Xq

i¼1

Xj2N i

ðxi � xjÞð _xi � _xjÞ

þ 1

2

Xq

i¼1

Xj2N i

ðyi � yjÞð _yi � _yjÞ

¼ 1

2

Xq

i¼1

Xj2N i

ðxi � xjÞðvi cos hi � vj cos hjÞ

þ 1

2

Xq

i¼1

Xj2N i

ðyi � yjÞðvi sin hi � vj sin hjÞ

¼ 1

2

Xq

i¼1

Xj2N i

vi½ðxi � xjÞ cos hi þ ðyi � yjÞ sin hi�

þ 1

2

Xq

i¼1

Xj2N i

vj½ðxj � xiÞ cos hj þ ðyj � yiÞ sin hj�

¼Xq

i¼1

Xj2N i

vi½ðxi � xjÞ cos hi þ ðyi � yjÞ sin hi�

¼Xq

i¼1

vihi

¼ vTh; x 62 Z (104)

Hence, combining Eqs. (103) and (104) yields that

_VpðxpÞ þ _Vcðxc; xpÞ ¼ �hTh� xTc1Lxc1 � xT

c2L2xc2 0; x 62 Z(105)

Alternatively, noting that L(Iq� L†L)¼ 0 and Iq � L†L��

F 1, it

follows from Eqs. (91), (92), (85), and (86) that

DVpðxpÞ þ DVcðxc; xpÞ ¼1

2xT

c1ðIq � L†LÞTðIq � L†LÞxc1

� 1

2xT

c1xc1 �1

2xT

c2Lxc2

� 1

2xT

c2Lxc2

< 0; x 2 Z

(106)

Next, let R ¼D fx 2 D : _VpðxpÞ þ _Vcðxc; xpÞ ¼ 0g, where

D � R5q is positively invariant with respect to Eqs. (78)–(80) and(83)–(88). Then it follows that R ¼ fx 2 D : Lxc1 ¼ 0;Lxc2 ¼ 0; h ¼ 0g. Let M denote the largest invariant set con-

tained in R and note that Le ¼ LTe ¼ 0 and rank L¼ q� 1, where

e ¼D ½1;…; 1�T 2 Rq. Then it follows from Eq. (84) that

eT _xc2 ¼ 0. Since on M;Lxc2 ¼ 0, it follows that xc2 ¼ c1e,and hence, x¼�c1 e, where c1 2 R. Now, it follows from

Eq. (84) that L~h ¼ 0. Thus, it follows that ~hi ¼ ~hj, and hence,hi � hj ¼ ð2ðj� iÞ=qÞp for every i, j¼ 1,…, q, which further

implies thatPq

i¼1 sin hi ¼ 0 andPq

i¼1 cos hi ¼ 0. Next, sinceLxc1¼ 0 and h¼ 0 on M, it follows from Eq. (83) that _xc1 ¼ 0,and together with Lxc1¼ 0, we thus have xc1¼ c2e, where c2 2 R.Hence, it follows from Eq. (87) that on M; v ¼ �c2e. Finally, itfollows from Theorem 2.1 that there exists D0 � D such that forevery system initial condition in D0; ðxðtÞ; yðtÞ; hðtÞ; xcðtÞ; zcðtÞÞ!M as t !1, which implies convergence to a circular forma-tion characterized by the manifold (81) and (82).



To show the efficacy of our framework, let q¼ 10 and let

L ¼

2 �1 0 0 0 0 0 0 0 �1

�1 2 �1 0 0 0 0 0 0 0

0 �1 2 �1 0 0 0 0 0 0

0 0 �1 2 �1 0 0 0 0 0

0 0 0 �1 2 �1 0 0 0 0

0 0 0 0 �1 2 �1 0 0 0

0 0 0 0 0 �1 2 �1 0 0

0 0 0 0 0 0 �1 2 �1 0

0 0 0 0 0 0 0 �1 2 �1

�1 0 0 0 0 0 0 0 �1 2

2666666666666664

3777777777777775

(107)

For this system, the transversality condition was verified numeri-cally for k¼ 1. A group of 10 agents is initialized with randominitial positions ð~x; ~yÞ in the range of [�10, 10]� [�10, 10]. Theinitial states of the hybrid resetting controller (83)–(88) arechosen randomly within [�10, 10]. Figure 14 shows circular for-mation is achieved using the hybrid resetting controller (83)–(88).Figures 15–17 show the time histories of the velocities and orien-tations for circular formation design.

8 Conclusion

In this paper, we have developed a general energy-based hybridcontrol framework for formation control protocols of generaldynamical systems using hybrid stabilization of sets. The pro-posed framework is used to develop a novel class of fixed-order,

energy-based hybrid controllers as a means for achievingcooperative control formations which include flocking, cyclic pur-suit, rendezvous, and consensus control of multiagent systems.Specifically, a specified formation is characterized by a hyper-plane or manifold in the state space and a hybrid feedback archi-tecture is designed that achieves set stabilization for the desiredformation thereby addressing formation control protocols for gen-eral nonlinear dynamical models.

Acknowledgment

This research was supported in part by the Air Force Office ofScientific Research under Grant No. FA9550-12-1-0192, theOffice of Naval Research under Grant No. N00014-09-1-1195,and the Defense Threat Reduction Agency under Grant Nos.HDTRA1-10-1-0090 and HDTRA1-13-1-0048.

References[1] Haddad, W. M., Chellaboina, V., Hui, Q., and Nersesov, S. G., 2007, “Energy-

and Entropy-Based Stabilization for Lossless Dynamical Systems Via HybridControllers,” IEEE Trans. Autom. Control, 52(9), pp. 1604–1614.

[2] Haddad, W. M., Chellaboina, V., and Nersesov, S. G., 2006, Impulsive andHybrid Dynamical Systems. Stability, Dissipativity, and Control, Princeton Uni-versity Press, Princeton, NJ.

[3] Mesbahi, M., and Egerstedt, M., 2010, Graph Theoretic Methods in MultiagentNetworks, Princeton University Press, Princeton, NJ.

[4] Hui, Q., and Haddad, W. M., 2008, “Distributed Nonlinear Control Algorithmsfor Network Consensus,” Automatica, 44(9), pp. 2375–2381.

[5] Chellaboina, V., Haddad, W. M., Hui, Q., and Ramakrishnan, J., 2008, “OnSystem State Equipartitioning and Semistability in Network Dynamical Sys-tems With Arbitrary Time-Delays,” Syst. Control Lett., 57(8), pp. 670–679.

[6] Hui, Q., Haddad, W. M., and Bhat, S. P., 2008, “Finite-Time Semistability andConsensus for Nonlinear Dynamical Networks,” IEEE Trans. Autom. Control,53(8), pp. 1887–1900.

[7] Bhat, S. P., and Bernstein, D. S., 2003, “Nontangency-Based Lyapunov Testsfor Convergence and Stability in Systems Having a Continuum of Equilibria,”SIAM J. Control Optim., 42(5), pp. 1745–1775.


Fig. 15 Agent velocities versus time

Fig. 16 Agent angular velocities versus time

Fig. 17 Agent orientations versus time



http://dx.doi.org/10.1109/TAC.2007.904452

http://dx.doi.org/10.1016/j.automatica.2008.01.011

http://dx.doi.org/10.1016/j.sysconle.2008.01.008


http://dx.doi.org/10.1137/S0363012902407119

[8] Bhat, S. P., and Bernstein, D. S., 2010, “Arc-Length-Based Lyapunov Tests forConvergence and Stability With Applications to Systems Having a Continuumof Equilibria,” Math. Control Signals Syst., 22(2), pp. 155–184.

[9] Haddad, W. M., and Chellaboina, V., 2008, Nonlinear Dynamical Systemsand Control: A Lyapunov-Based Approach, Princeton University Press, Prince-ton, NJ.

[10] Haddad, W. M., Chellaboina, V., and Nersesov, S. G., 2005, Thermodynamics:A Dynamical Systems Approach, Princeton University Press, Princeton, NJ.

[11] Strogatz, S. H., 2000, “From Kuramoto to Crawford: Exploring the Onset ofSynchronization in Populations of Coupled Oscillators,” Physica D, 143(1–4),pp. 1–20.

[12] Brown, E., Moehlis, J., and Holmes, P., 2004, “On the Phase Reduction andResponse Dynamics of Neural Oscillator Populations,” Neural Comput., 16(4),pp. 673–715.

[13] Hui, Q., Haddad, W. M., and Bailey, J. M., 2011, “Multistability, Bifurcations,and Biological Neural Networks: A Synaptic Drive Firing Model for CerebralCortex Transition in the Induction of General Anesthesia,” Nonlinear Anal.Hybrid Syst., 5(3), pp. 554–573.

[14] Tanner, H. G., Jadbabaie, A., and Pappas, G. J., 2003, “Stable Flocking ofMobile Agents, Part I: Fixed Topology,” 42nd IEEE Conference on Decisionand Control, Maui, HI, December 9–12, pp. 2010–2015.

[15] Olfati-Saber, R., 2006, “Flocking for Multi-Agent Dynamic Systems:Algorithms and Theory,” IEEE Trans. Autom. Control, 51(3), pp. 401–420.

[16] Marshall, J. A., Broucke, M. E., and Francis, B. A., 2004, “Formations ofVehicles in Cyclic Pursuit,” IEEE Trans. Autom. Control, 49(11), pp.1963–1974.

[17] Lakshmikantham, V., Bainov, D. D., and Simeonov, P. S., 1989, Theory ofImpulsive Differential Equations, World Scientific, Singapore.

[18] Bainov, D. D., and Simeonov, P. S., 1989, Systems With Impulse Effect: Stabil-ity, Theory and Applications, Ellis Horwood Limited, UK.

[19] Bainov, D. D., and Simeonov, P. S., 1995, Impulsive Differential Equations:Asymptotic Properties of the Solutions, World Scientific, Singapore.

[20] Samoilenko, A. M., and Perestyuk, N. A., 1995, Impulsive Differential Equa-tions, World Scientific, Singapore.

[21] Chellaboina, V., Bhat, S. P., and Haddad, W. M., 2003, “An Invariance Princi-ple for Nonlinear Hybrid and Impulsive Dynamical Systems,” Nonlinear Anal.Theory Methods Appl., 53(3–4), pp. 527–550.

[22] Michel, A. N., Wang, K., and Hu, B., 2001, Qualitative Theory of DynamicalSystems: The Role of Stability Preserving Mappings, Marcel Dekker, Inc., NewYork.

[23] Grizzle, J. W., Abba, G., and Plestan, F., 2001, “Asymptotically Stable Walkingfor Biped Robots: Analysis Via Systems With Impulse Effects,” IEEE Trans.Autom. Control, 46(1), pp. 51–64.

[24] Guillemin, V., and Pollack, A., 1974, Differential Topology, Prentice-Hall,Englewood Cliffs, NJ.

[25] Dubrovin, B. A., Fomenko, A. T., and Novikov, S. P., 1985, ModernGeometry—Methods and Applications: Part II: The Geometry and Topology ofManifolds, Springer, New York.

[26] Goebel, R., and Teel, A. R., 2006, “Solutions to Hybrid Inclusions Via Set andGraphical Convergence With Stability Theory Applications,” Automatica,42(4), pp. 573–587.

[27] Bernstein, D. S., 2005, Matrix Mathematics, Princeton University Press, Prince-ton, NJ.

[28] Justh, E. W., and Krishnaprasad, P. S., 2004, “Equilibria and Steering Laws forPlanar Formations,” Syst. Control Lett., 52(1), pp. 25–38.

[29] Hui, Q., and Haddad, W. M., 2007, “Continuous and Hybrid Distributed Con-trol for Multiagent Coordination: Consensus, Flocking, and Cyclic Pursuit,”American Control Conference (ACC ’07), New York, July 9–13, pp.2576–2581.

[30] El-Hawwary, M. I., and Maggiore, M., 2013, “Distributed Circular FormationStabilization for Dynamic Unicycles,” IEEE Trans. Autom. Control, 58(1), pp.149–162.

[31] Kasdin, N. J., and Paley, D. A., 2011, Engineering Dynamics, PrincetonUniversity Press, Princeton, NJ.



http://dx.doi.org/10.1007/s00498-010-0054-3

http://dx.doi.org/10.1016/S0167-2789(00)00094-4

http://dx.doi.org/10.1162/089976604322860668

http://dx.doi.org/10.1016/j.nahs.2010.12.002

http://dx.doi.org/10.1016/j.nahs.2010.12.002

http://dx.doi.org/10.1109/CDC.2003.1272910

http://dx.doi.org/10.1109/CDC.2003.1272910



http://dx.doi.org/10.1016/S0362-546X(02)00316-4

http://dx.doi.org/10.1016/S0362-546X(02)00316-4

http://dx.doi.org/10.1109/9.898695

http://dx.doi.org/10.1109/9.898695

http://dx.doi.org/10.1016/j.automatica.2005.12.019

http://dx.doi.org/10.1016/j.sysconle.2003.10.004

http://dx.doi.org/10.1109/ACC.2007.4282465


Documents

Formation Control Protocols for Nonlinear Dynamical ...or dissipative dynamical systems. In Sec. 4, we specialize the results of Sec. 3 to linear dynamical systems. We then turn our