Passivity as a Design Tool for Group Coordination - uni-stuttgart

1380 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007 

Passivity as a Design Tool for Group Coordination 

Murat Arcak, Senior Member, IEEE 

Abstract—We pursue a group coordination problem where the 

objective is to steer the differences between output variables of the 

group members to a prescribed compact set. To stabilize this set 

we study a class of feedback rules that are implementable with 

local information available to each member. When the information 

flow between neighboring members is bidirectional, we show that 

the closed-loop system exhibits a special interconnection structure 

which inherits the passivity properties of its components. By exploiting 

this structure we develop a passivity-based design framework, 

which results in a broad class of feedback rules that encompass 

as special cases some of the existing formation stabilization 

and group agreement designs in the literature. The passivity approach 

offers additional design flexibility compared to these special 

cases, and systematically constructs a Lurie-type Lyapunov function 

for the closed-loop system. We further study the robustness of 

these feedback laws in the presence of a time-varying communication 

topology, and present a persistency of excitation condition 

which allows the interconnection graph to lose connectivity pointwise 

in time as long as it is established in an integral sense. 

Index Terms—Cooperative systems, distributed control, Lyapunov 

methods, passivity. 

I. INTRODUCTION 

THE increasing number of feedback applications in sensor 

networks, cooperative robotics, and vehicle formations 

have brought group coordination problems to the forefront of 

control theory. The main challenge in these applications is to 

achieve a prescribed group behavior with local feedback rules, 

rather than with centralized controllers. A major focus of research 

in group coordination is formation stability, where such 

local rules have been proposed in the form of artificial attraction 

and repulsion forces with neighboring vehicles [1]–[3]. A 

related topic of wide interest is the agreement problem, where 

the objective is to steer variables of interest (position, heading, 

phase of oscillations, etc.) to a common value across the network 

[4]–[8]. The application areas for the agreement problem 

range from the study of flocking and schooling behaviors in 

groups of natural organisms [9], to distributed computing [10], 

and to forest fire surveillance [11]. 

Manuscript received September 26, 2005; revised July 11, 2006. Recommended 

by Associate Editor D. Nesic. This work was supported in part by the 

Air Force Office of Scientific Research under award No. FA9550–07–1–0308, 

and by the Center for Automation Technologies and Systems (CATS) under a 

block grant from the New York State Office of Science, Technology, and Academic 

Research (NYSTAR). 

The author is with the Department of Electrical, Computer, and Systems Engineering, 

Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail: arcakm@rpi.edu). 

Color versions of one or more of the figures in this paper are available online 

at http://ieeexplore.ieee.org. 

Digital Object Identifier 10.1109/TAC.2007.902733 

In this paper, we formulate a coordination problem that is applicable 

to formation stabilization and group agreement as special 

cases, and present a class of feedback laws that solve this 

problem with local information. These feedback laws assume 

that, if a communication link exists between two members of 

the group, then both members gain access to each other’s information. 

A key observation in the paper is that such bidirectional 

communication gives rise to a special interconnection structure 

which guarantees that the resulting feedback loop will inherit 

the passivity properties of its components. By exploiting this 

structure, we develop a design method which results in a broad 

class of feedback laws that achieve passivity and, thus, stability 

of the interconnected system. The passivity approach also leads 

to a systematic construction of a Lyapunov function in the form 

of a sum of storage functions for the subsystems. As detailed 

in the paper, several existing feedback rules for formation stability 

and group agreement become special cases in our passivity 

framework. 

The coordination task studied in this paper is to steer the differences 

between the output variables of group members to a 

prescribed compact set. This compact set is a sphere when the 

outputs are positions of vehicles that must maintain a given distance 

in a formation, or the origin if the outputs are variables 

that must reach an agreement across the group. We thus formulate 

this task as a set stability problem [12], [13] as defined in 

Section I-A, and use passivity as a tool for constructing a stabilizing 

feedback law and a Lyapunov function with respect to 

this set. We prove global asymptotic stability with additional 

assumptions that guarantee appropriate detectability properties 

for trajectories away from the set. 

The subsequent sections are organized as follows: Section I-A 

introduces the notation and definitions used in the paper, and 

Section I-B presents the problem formulation. Section II develops 

a passivity-based design procedure by exploiting the 

structure of the interconnected system. It then gives the main 

stability result (Theorem I) and illustrates it with examples 

from formation control and from synchronous operation in 

power systems. Section III focuses on the agreement problem 

as a special case of Theorem I. It first presents a corollary to 

Theorem I that yields a class of continuous-time agreement 

protocols, and next develops a discrete-time analogue of this 

result (Theorem 2). It finally addresses time-varying communication 

topologies and employs the persistency of excitation 

concept from adaptive control to prove that group agreement is 

achievable when graph connectivity is established over a period 

of time (Theorem 3). This result is significant because it does 

not require pointwise connectivity of the graph, and is applicable 

to a class of nonlinear agreement protocols. The proofs 

of the theorems are presented in Section IV. The conclusion is 

given in Section V. 

0018-9286/$25.00 © 2007 IEEE 

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ARCAK: PASSIVITY AS A DESIGN TOOL 1381 

A. Notation and Definitions 

• A function is said to be , if its partial derivatives exist 

and are continuous up to order . 

• Given a function we denote by its 

gradient vector, and by its Hessian matrix. 

• For a closed set , denotes the distance from the point 

to ,defined as 

• A closed invariant set is uniformly asymptotically stable 

with region of attraction if for each there exists 

such that 

and, if for each and , there exists 

such that for every initial condition the resulting 

trajectory satisfies 

Several results on set stability and, in particular, converse 

Lyapunov theorems are presented in [12] and [13]. 

• Following [13], we use the notation to indicate 

a sequence of points in converging to a point on the 

boundary of ,orif is unbounded, having the property 

. 

• The dynamic system 

is said to be passive if there exists a 

such that 

(1) 

(2) 

(3) 

(4) 

storage function 

for some positive semidefinite function . We say that 

(4) is strictly passive if is positive definite. A static 

nonlinearity is passive if, for all , 

and strictly passive if (6) holds with strict inequality 

. 

• The Kronecker product of matrices 

and 

is defined as 

. 

(5) 

(6) 

. .. . . (7) 

where 

and 

is an identity matrix of arbitrary dimension , and 

are assumed to be compatible for multiplication. 

B. Problem Statement 

We consider a group where each member 

is 

represented by a vector that consists of variables to 

be coordinated with the rest of the group. The topology of information 

exchange between these members is described by a 

graph. We say that the th and th members are “neighbors” if 

they have access to the relative information , in which 

case we let the th and th nodes of the graph be connected by a 

link. To simplify our derivations we assign an orientation to the 

graph by considering one of the nodes to be the positive end of 

the link. The choice of orientation does not change the results 

because in this paper the information flow between neighbors is 

assumed to be bidirectional. Denoting by the total number 

of links, we recall that the incidence matrix of the 

graph is defined as [14] 

if th node is the positive end of the th link 

if th node is the negative end of the th link 

otherwise. 

(10) 

Because the sum of its rows is zero, the rank of is at most 

. Indeed, the rank is when the graph is connected, 

that is when a path exists between every two distinct nodes, 

and less than otherwise. The columns of are linearly 

independent when no cycles exist in the graph. 

Our objective is to develop coordination laws that are implementable 

with local information (the th member can use the 

information if the th member is a neighbor) and that 

guarantee the following two group behaviors: 

B1) Each member achieves in the limit a common velocity 

vector 

prescribed for the group; that is 

, . 

B2) If th and th members are neighbors connected by link 

, then the difference variable 

if 

if 

is the positive end 

is the positive end 

(11) 

converges to a prescribed compact set , 

. 

Examples of such target sets include the origin if s are 

variables that must reach an agreement within the group, or a 

sphere in if s are positions of vehicles that must maintain 

a prescribed distance. The formulation B1-B2 can thus be employed 

to design and stabilize a formation of vehicles, or to synchronize 

variables in a distributed network of computers, satellites, 

power generators, etc. 

Note that if the columns of are linearly dependent, that is 

if the graph contains cycles, then s are mutually dependent. 

Indeed, from (11), the concatenated vectors 

and satisfies the properties 

(8) 

(9) 

satisfy 

(12) 

(13) 



Fig. 1 The task of the internal feedback design is to render the plant passive 

from the external feedback signal u to the velocity error y . The resulting passive 

block is denoted by H . 

where is the identity matrix and “ ” represents the 

Kronecker product, which means that is constrained to lie in 

the range space 

. Thus, for B2 above to be feasible, 

the target sets must be consistent in the sense that 

II. THE PASSIVITY APPROACH 

(14) 

A. Design Procedure 

We now present our passivity-based design to achieve objectives 

B1-B2 above. We assume that an internal feedback loop 

has been designed for each node 

to render its 

dynamics passive from an external feedback signal to the velocity 

error 

(15) 

A complete characterization of the class of systems which can 

be rendered passive from to is beyond the scope of this 

paper. However, this class is broad, and encompasses mechanical 

systems modeled by Euler-Lagrange equations [15, Section 

4.1]. Upon application of this internal feedback the node dynamics 

take the form of Fig. 1, where the properties that must 

be satisfied by the passive block are further specified below. 

We design an external feedback law of the form 

(16) 

where s are the relative distance variables as in (11), and the 

multivariable nonlinearities 

are to be designed 

to render the target sets invariant and asymptotically stable. 

The external feedback law (16) is indeed implementable with 

available signals because only for links that are connected 

to node . 

To specify the properties s in Fig. 1 and s in (16) must 

satisfy we note that their interconnection is as in Fig. 2, where 

denotes the -vector of ones 

and (16) is rewritten in the compact form 

(17) 

(18) 

Fig. 2 exhibits a “symmetric” interconnection structure in which 

the coupling of the block-diagonal feedforward and feedback 

subsystems is due to multiplication by the matrices and 

. Because post-multiplication by a matrix and pre-multiplication 

by its transpose preserve passivity and, thus, stability 

properties, this structure allows us to proceed with a passivitybased 

design of , , and , . 

If is a static block, we restrict it to be of the form 

where 

(19) 

is a locally Lipschitz function satisfying 

(20) 

The only exception to (20) is when one of the members, say the 

first one, is the “leader” of the group in the sense that does 

not contain feedback terms from other members. We encompass 

this situation by letting 

If 

is a dynamic block of the form 

(21) 

(22) 

we assume and are locally Lipschitz functions 

such that 

, and 

(23) 

Our main restriction on (22) is that it be strictly passive with 

a , positive definite, radially unbounded storage function 

satisfying 

for some positive definite function . 

Likewise, we restrict the feedforward nonlinearities 

to be of the form 

where is a nonnegative function 

(24) 

(25) 

(26) 

defined on an open set in which is allowed to 

evolve. To steer s to their target sets while keeping 

them within , we design the function to grow unbounded 

as approaches (see Section I-A for this notation), 

and let and its gradient vanish on the set 

: 

(27) 

(28) 

(29) 

When , (27) means that is radially unbounded. 

As shown in [13, Remark 2], a continuous function satisfying 

(27) and (28) exists for any given open set and compact 



Fig. 2. A block diagram representation for the interconnection of the subsystems in Fig. 1 via the external feedback (18). The vectors x,y,z,u and are as defined 

in (12) and (17), I is the p 2 p identity matrix, 1 is the N -dimensional column vector with all entries equal to 1, and “” represents the Kronecker product. 

subset . In this paper we assume that the sets and 

are chosen such that smoothness of and (29) are 

also achievable. 

B. Stability Analysis 

In Theorem I below, we prove passivity for the feedforward 

and feedback subsystems of Fig. 2 by using the storage functions 

(30) 

i) The feedforward path in Fig. 2 is passive from to , 

and from to . 

ii) The feedback path is passive from input to output . 

iii) If Assumption 1 holds then the set 

(32) 

is uniformly asymptotically stable with region of attraction 

respectively, where denotes the subset of indices 

which correspond to dynamic blocks . We then prove asymptotic 

stability of the set of points where and by 

taking as a Lyapunov function the sum of the two storage functions 

in (30). This construction results in a Lurie-type Lyapunov 

function because its key ingredient is the integral of the 

feedback nonlinearity . 

When the columns of are linearly dependent, that is when 

the graph contains cycles, the constraint (13) implies that s are 

mutually dependent, because the vector lies in the range space 

. In this case in (30), restricted to the subspace 

, may acquire new critical points outside the sets 

, thus giving rise to undesired equilibria. Indeed, from Fig. 2, 

the set of equilibria is 

(31) 

which means that the following assumption must hold true to 

ensure that no equilibria arises outside the sets . 

Assumption 1: 

and 

imply . 

When the columns of are linearly independent Assumption 

I holds because, then, the null space of is trivial and, thus, 

implies , and follows from 

(29). When the columns of are linearly dependent, whether 

Assumption 1 holds or not depends on the sets . It holds in 

agreement problems where is the origin (see Section III), 

and fails in Example 1 below where s are spheres. 

Theorem 1: Consider the block diagram in Fig. 2 where 

is bounded and piecewise continuous, and , and 

, are designed as in (19)–(24) and (25)–(29) 

for given open sets and compact subsets , 

and assume the sets are consistent as in (14). Then: 

(33) 

If Assumption 1 fails, all trajectories 

starting in 

converge to the set of equilibria in (31). 

The proof is given in Section IV. When Assumption 1 fails 

Theorem 1 proves that the trajectories convergence to the set 

of equilibria in (31). In this case one can conclude “generic 

convergence” to from almost all initial conditions by showing 

that the equilibria outside are unstable (see Example 1 below). 

Convergence to means that the trajectories converge to 

the set of points where the difference variables belong to their 

target sets . It also implies that and, thus, in (15) tend to 

zero, which means that objectives B1 and B2 in Section I-B are 

indeed achieved. 

C. Examples 

Example 1(Formation Control): When applied to the 

problem of formation stabilization, Theorem 1 recovers the stability 

results of [1] and [2], and gives a passivity interpretation. 

These references study the coordination of vehicles, modeled 

as fully actuated point masses 

(34) 

where is the position of each mass and 

is the force input. They then design as a sum of artificial 

interaction forces with the neighboring vehicles. To show that 

our formulation encompasses this application, we note that the 

internal feedback 

(35) 



Fig. 3. (a) Desired formation where jz j =1, k =1; 2;3. (b) An undesired 

equilibrium where the masses are aligned and the repulsion forces due to jz j < 

1 and jz j < 1 counterbalance the attraction force due to jz j > 1. 

and the change of variables 

form 

bring (34) into the 

(36) 

(37) 

where the -subsystem with input and output plays 

the role of in Fig. 1. This subsystem is indeed passive with 

the storage function 

and satisfies (23). The 

design of s in [1], [2] is as in (16), with s representing 

artificial interaction forces between neighboring vehicles. 

To stabilize the formation in Fig. 3(a) while avoiding collisions 

we let be the unit circle, , and let the 

potential functions be of the form 

(38) 

where is a , strictly increasing function such 

that 

and such that, as , in (38). Then, 

satisfies (27)-(29), and the interaction forces 

(39) 

(40) 

guarantee local asymptotic stability of the desired formation 

in Fig. 3(a) from Theorem 1. In particular plays the 

role of a “spring force” which creates an attraction force when 

and a repulsion force when . Assumption 1 

fails in this example because additional equilibria arise when the 

point masses are aligned as in Fig. 3(b), and the attraction force 

between the two distant masses counterbalances the repulsion 

force due to the intermediate mass (cf. [16, Fig. 3.1(c)]). Such 

equilibria cannot be eliminated with the choice of the function 

because 

(41) 

(42) 

have a unique solution since are increasing 

and onto and, thus, the set of points where , 

and 

constitute new equilibria as 

in Fig. 3(b). Because these equilibria are unstable, however, 

we conclude generic convergence to the desired formation in 

Fig. 3(a) from all initial conditions except for those that lie on 

the stable manifolds of the unstable equilibria. 

Example 2 (Synchronous Operation in Power Networks): In 

Example 1 interaction forces were designed to coordinate the 

motion of decoupled subsystems. In several other applications 

such interactions arise due to existing physical coupling, and 

exhibit the passive feedback structure studied in this paper. An 

excellent example is a multi-machine power system where the 

th generator is governed by (see [17]): 

(43) 

(44) 

in which is the angle of the rotor shaft, is the synchronous 

speed for the network, is the deviation of the rotor speed from 

, and and are the inertia and damping constants, respectively. 

and are the internal voltages of the th and th 

machines, and is the transfer susceptance between them. 

The equilibrium point for is , which is determined 

by solving equations that match the mechanical input power to 

the electrical output. From two solutions for in ,we 

study the one that satisfies 

because the 

other one results in an unstable equilibrium. 

To show that this example fits into our framework, we note 

that the model (43) and (44) is as in Fig. 1 where 

(45) 

and the -subsystem with input and output is passive 

with storage function 

. Furthermore (45) is 

of the form (16) where 

is the gradient of the potential function 

(46) 

(47) 

which satisfies (28) and (29) locally around 

.Itis 

also positive definite because its Hessian at is positive due to 

the assumption 

. The Lyapunov function 

of Theorem 1 thus proves local asymptotic stability for the equilibrium 

comprising of s. Our Lurie-type construction consists 

of the components in (30), and coincides with the Lyapunov 

function derived in [18]. 

III. PASSIVE PROTOCOLS FOR GROUP AGREEMENT 

A. The Agreement Problem as a Special Case of Theorem 1 

In several cooperative control applications, it is of interest to 

achieve agreement of group variables such as position, heading, 

phase of oscillations, etc. To apply Theorem 1 to this problem 



we let denote a vector of variables to be synchronized 

with those of other members, and select the target sets for the 

difference variables in (11) to be . With this choice 

of s, (14) is satisfied. Likewise, (26)–(29) and Assumption 

1 hold if s are selected to be positive definite, radially 

unbounded functions on such that 

(48) 

In particular, Assumption 1 holds because 

and imply that and are orthogonal 

to each other which, in view of (48), is possible only when 

. Thus, from Theorem 1, the feedback law (49) with 

achieves global asymptotic stability for the 

equilibrium , which guarantees that the difference 

variables converge to zero. 

Corollary 1: Consider members 

, interconnected 

as described by the graph representation (10), and let 

, denote the differences between the variables 

of neighboring members as in (11). Let be positive 

definite, radially unbounded functions satisfying (48), and let 

. Then the agreement protocol 

(49) 

where denotes the output at time of a static or dynamic 

block satisfying (19)–(23), guarantees 

B. Extension to Discrete-Time Protocols 

We now study discrete-time agreement protocols of the form 

(52) 

where is the time index, and is a discrete-time 

dynamic block or a static nonlinearity. The block diagram representation 

of this discrete-time model is as in Fig. 2, with integral 

blocks replaced by summation blocks, and and replaced by, 

respectively 

(53) 

Unlike the continuous-time design in Section II, passivity of 

the feedforward path cannot be achieved in discrete-time because 

the phase lag of the summation block exceeds 90 .To 

overcome this obstacle, we trade the shortage of passivity in the 

feedforward path with the excess of passivity in the feedback 

path. To guarantee such excess in the feedback path, we restrict 

static blocks by 

(54) 

where is a positive definite function and the constant 

quantifies the excess of passivity. Likewise, if is a dynamic 

block of the form 

(50) 

for every pair of nodes which are connected by a path. 

If the graph is connected then Corollary 1 implies that all variables 

are synchronized in the limit. When , that 

is when s and s are scalars, (48) means that 

is a sector nonlinearity which lies in the first and 

third quadrants. Corollary 1 thus encompasses the result of [19], 

which proposed agreement protocols of the form 

(51) 

where is the set of neighbors for the th member, and 

plays the role of our . However, both 

[19] and a related result in [20] assume that the nonlinearities 

satisfy an incremental sector assumption which is more 

restrictive than the sector condition (48) of Corollary 1. An 

independent study in [21] takes a similar approach to synchronization 

as [20]; however, it further restricts the coupling 

terms to be linear. Our feedback law (49) in Corollary 

1 generalizes (51) by applying to its right-hand side (RHS) 

the additional operation , which may either represent a 

passive filter or another sector nonlinearity as specified in 

Section II. Because in (49) can be dynamic, Corollary 1 is 

applicable, unlike other agreement results surveyed in [22], to 

plants with higher-order dynamics than an integrator. 

(55) 

we assume that (23) holds and that there exists a positive definite 

and radially unbounded storage function satisfying 

(56) 

for some positive definite function . 

For the feedforward subsystem we design the nonlinearity 

to be the gradient of a , positive definite, radially unbounded 

function that satisfies (48). We further restrict 

the Hessian of this function to be upper bounded by 

(57) 

where the constant is to be specified. With this condition 

we show in Theorem 2 below that the storage function in 

(30) satisfies 

(58) 

where quantifies the shortage of passivity from input to 

output . We then prove that this shortage is compensated for 

by the excess of passivity in (54) and (56) if 

(59) 



where denotes the largest eigenvalue of the graph Laplacian 

matrix . 

Theorem 2: Consider members 

, interconnected 

as described by the graph representation (10), and let , 

denote the differences between the variables of 

neighboring members as in (11). Suppose that s are either dynamic 

blocks (55) satisfying (56), or static blocks 

satisfying (54), or possibly a combination of such static and 

dynamic blocks. Under these conditions, if we design 

to be , positive definite, radially unbounded functions satisfying 

(48), (57), and (59), then the feedback law (52) with 

achieves global asymptotic stability of the 

origin . In particular, if a path exists between the th 

and th nodes in the graph representation (10), then 

(60) 

Denoting by the set of neighbors of the th member, and 

by the number of elements in this set, the author of [23] 

showed that an upper bound on is 

rewritten in vector form (12) and (17) as 

where the multivariable nonlinearity 

has the property 

(64) 

(65) 

(66) 

is piecewise continuous because its entries exhibit step 

changes when a change occurs in the communication topology. 

The minimum eigenvalue for the graph Laplacian matrix 

is 

(67) 

with eigenvector , the -vector of ones. A standard result in 

algebraic graph theory states that the graph is connected if and 

only if the second smallest eigenvalue of its Laplacian is strictly 

positive (see e.g. [14, Item 4e and Corollary 6.5]). If the graph 

remains connected for all , that is, if 

(68) 

A more conservative bound for 

topology of the graph is 

(61) 

that does not depend on the 

(62) 

An important conclusion from (59) is that less communication 

between members allows a broader class of s and s for the 

protocol (52) because, a smaller value for in (59) translates 

to larger in (57) and smaller in (54) and (56). 

When s are scalars, condition (57) of Theorem 2 means 

that the slope of the nonlinearity 

is to be 

bounded above by . Although this condition was not used 

in the continuous-time result of Theorem 1, it plays a crucial role 

in the proof of Theorem 2 (see Section IV) for establishing the 

discrete-time property (58). The assumption of an upper bound 

on the slope of the nonlinearity was introduced independently 

in [24]–[26], to extend the continuous-time Popov Criterion to 

discrete-time systems. Indeed, the Popov Criterion makes use of 

a Lurie-type Lyapunov function that incorporates the integral of 

the system nonlinearity, which is similar to the use of s 

in our Lyapunov construction. 

C. Time-Varying Communication Topology 

Thus far we have assumed that the communication topology 

between the members does not change with time. To study convergence 

properties for a time-varying incidence matrix , 

we restrict our attention to the class of agreement protocols 

(63) 

for some constant that does not depend on time, then it 

is not difficult to show that s in (64), (65) reach an agreement 

despite the time-varying . We now prove agreement under 

a less restrictive persistency of excitation condition which stipulates 

that graph connectivity be established over a period of 

time, rather than pointwise in time: 

Theorem 3: Consider the system (64), (65) where 

comprises of the components , concatenated 

as in (12), is a locally Lipschitz nonlinearity satisfying 

(66), and is piecewise continuous incidence matrix. 

Let be an matrix with orthonormal rows that 

are each orthogonal to ; that is 

(69) 

If there exist constants and such that, for all , 

(70) 

then the protocol (64) and (65) achieves (50) for all 

. 

The “persistency of excitation” condition (70) means that 

is nonsingular when integrated over a period 

of time, and not necessarily pointwise in time. Since, by construction 

of in (69), inherits all eigenvalues 

of 

except the one at zero, its smallest eigenvalue is 

(71) 

which means that nonsingularity of the matrix 

is equivalent to connectivity of the graph. Because Theorem 3 

does not require nonsingularity of 

pointwise 

in time, it allows the graph to lose pointwise connectivity as 



long as it is established in the integral sense of (70). The pointwise 

connectivity situation (68) is a special case of Theorem 3 

because, then, (70) readily holds with . 

A discrete-time extension of Theorem 3 is possible, but is 

omitted due to space limitations. In [5, Theorem 9] the authors 

study linear agreement protocols with time-varying topology 

and prove convergence when the graph is connected pointwise 

in time. The authors of [4] consider a class of linear discretetime 

protocols, and show that it is sufficient to assume joint connectivity, 

that is, connectivity for a union of the graphs collected 

over an interval of time [4, Proposition 1]. Our persistency of 

excitation condition in Theorem 3 is similar to this joint connectivity 

assumption, and is further applicable to nonlinear protocols 

of the form (63). 

A. Proof of Theorem 1 

IV. PROOFS 

i) To prove passivity from to we use in (30) as 

a storage function, and obtain from (25), (13), (18) and 

: 

(72) 

To show passivity from to we substitute 

in (72) and use the property 

which follows because the sum of the rows of 

is zero, thus obtaining 

(73) 

ii) To establish passivity of the feedback path, we let denote 

the subset of indices for which is 

a dynamic block as in (22), and employ the storage function 

in (30), which yields 

(74) 

(27). From (73), (74), and (75), this Lyapunov function 

yields the negative semidefinite derivative 

(78) 

which implies that the trajectories 

are 

bounded on , for any within the maximal 

interval of definition for the differential equations 

(49) and (22). Because this bound does not depend on 

, and because is bounded, from (49) we can find a 

bound on that grows linearly in . This proves that 

there is no finite escape time because, if were finite, 

by letting we would conclude that exists, 

which is a contradiction. 

Having proven the existence of solutions for all ,we 

conclude from (78) stability of the set . However, because the 

RHS of (78) vanishes on a superset of , to prove attractivity 

of we appeal to the Invariance Principle. 1 To investigate the 

largest invariant set where 

we note from (23) that 

if holds identically then . Likewise, the static 

blocks satisfy (20) or (21), which means that the RHS of (78) 

vanishes when , . Indeed, if the first member 

satisfies (20), then follows directly. If it satisfies 

(21) instead of (20), still holds because the sum of the 

rows of being zero implies, from (18), that 

(79) 

We thus conclude that , which means from (18) that 

lies in the null space 

. Using the Invariance Principle 

[27, Lemma 4.1] which states that all bounded solutions 

approach their positive limit set, which is invariant, we conclude 

that the trajectories converge to the set in (31). When 

Assumption 1 holds, coincides with , which proves asymptotic 

stability of with region of attraction , while uniformity 

of asymptotic stability follows from the time-invariance of the 

-dynamics. 

Adding to the RHS of (74) 

(75) 

B. Proof of Theorem 2 

We first study passivity properties of the feedforward and 

feedback subsystems using the storage functions in (30). For 

the feedforward subsystem we note that 

which is nonnegative because the static blocks satisfy 

(20) or (21), we get 

(76) 

and, thus, conclude passivity with input and output . 

iii) To prove asymptotic stability of the set we use the Lyapunov 

function 

(77) 

which is zero on the set due to property (28), and grows 

unbounded as approaches due to property 

(80) 

and obtain from Taylor’s Theorem (see, e.g., [28, Sec. A.6]) 

(81) 

where is as defined in (53), and 

for some . Substitution of (25), (57), and (81) into (80) 

1 The Invariance Principle is indeed applicable because the dynamics of (z; ) 

are autonomous: Although v(t) appears in the block diagram in Fig. 2, it is 

canceled in the _z equation because (D I )(1 v(t)) = 0. 



yields (58) which, when combined with 

(16), results in 

and 

We then denote 

(90) 

(82) 

Because the spectral radius of is , and because 

has the same eigenvalues as 

(with the multiplicity of each multiplied by ), we conclude 

(83) 

Likewise, for the feedback subsystem, we obtain from (54) and 

(56) 

and conclude global uniform asymptotic stability for (87) from 

Lemma 1 below. 

Lemma 1: Consider the time-varying system 

(91) 

where , is an piecewise continuous matrix 

of , and 

is a locally Lipschitz nonlinearity 

satisfying (66). If satisfies and 

(92) 

for some constants , and that do not depend on , then the 

origin is globally uniformly asymptotically stable. 

Proof of Lemma 1: We let 

(84) 

Stability of the origin follows because, from (83), 

(84) and (59), the Lyapunov function (77) satisfies 

(85) 

and note that the Lyapunov function 

satisfies 

(93) 

(94) 

(95) 

To conclude asymptotic stability we note that the summation of 

the RHS of (85) from to converges, which means 

that as , if and if . Moreover, 

because of assumption (23), implies and, hence, 

. It then follows from the arguments in part (iv) of the 

proof of Theorem 1 that, because of assumption (48), 

implies 

, which proves 

that and concludes the proof of asymptotic stability. 

C. Proof of Theorem 3 

We define the new variable 

(86) 

where is as in (69) and, thus, (50) is equivalent to .To 

prove asymptotic stability of , we note from (64) and (65) 

that 

where we obtained the second equation by substituting 

(87) 

(88) 

and by using (86). To see that (88) holds, note that 

is an orthogonal projection matrix onto the span of , and that 

is in the null space of for all which, together, 

imply 

(89) 

from which we conclude global uniform stability. To prove 

global uniform asymptotic stability we employ the Nested 

Matrosov Theorem [29, Theorem 1]. To this end we introduce 

the auxiliary function 

where 

from 

, and 

from (92). Furthermore, 

we obtain 

Next, substituting (97), (98) and 

(91), we get 

(96) 

(97) 

(98) 

, from which 

(99) 

obtained from 

(100) 

When in (95), it follows from (66) that and, 

thus, 

. Furthermore, 

and together imply , which means that all 



conditions of [29, Theorem 1] are satisfied and, hence, 

globally uniformly asymptotically stable. 

V. CONCLUSION 

The notion of passivity evolved from a network theory concept 

to a versatile feedback design tool. It has been instrumental 

in the development of absolute stability criteria in the 1960’s, 

in adaptive control and large-scale system studies in the 1970’s 

and 1980’s, as well as in the nonlinear control design procedures 

developed in the 1990’s [30]. In this paper we showed that passivity 

is a suitable design approach for a class of group coordination 

problems in which the communication structure is bidirectional. 

In particular, we exploited the symmetry (post-multiplication 

by and pre-multiplication by its transpose 

) in the interconnection structure of Fig. 2, which is due 

to such bidirectional communication. For unidirectional communication 

topologies significant results have been obtained 

using a number of different approaches, such as the use of Laplacian 

properties for directed graphs in [5] and [31], input-to-state 

stability in [32], eigenvalue structure of circulant matrices in 

[33], and set-valued Lyapunov theory in [6]. Although unidirectional 

interconnection topologies do not preserve passivity 

properties in general, the design methodology of this paper can 

be extended to assign strict forms of passivity to compensate for 

the loss of passivity in the interconnection. Such an extension is 

currently being pursued by the author. 

The continuous-time feedback laws obtained within the passivity 

framework are “scalable” in the sense that their design parameters 

do not depend on the number of members in the group. 

In the discrete-time extension, however, the number of members 

restricted the growth of nonlinear terms [see (59)–(62)] due to 

the shortage of passivity resulting from the sampling process. 

In addition to stabilizing feedback rules, our passivity approach 

constructed a Lurie-type Lyapunov function which can be used 

as a starting point for several robust redesigns. Indeed, robustness 

against measurement and quantization noise, and against 

time-delays that occur in the communication links are important 

problems that will be addressed in future work. Another 

promising research direction is to relax the assumption that the 

group reference velocity is available to each member. We 

are presently developing an adaptive design where is available 

only to a leader, while other members estimate this information 

from their relative distance measurements. 

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Murat Arcak (S’97–M’00–SM’05) received the 

B.S. degree in electrical and electronics engineering 

from the Bogazici University, Istanbul, Turkey, in 

1996, and the M.S. and Ph.D. degrees in electrical 

and computer engineering from the University of 

California, Santa Barbara, in 1997 and 2000, under 

the direction of P. Kokotović. 

He is an Associate Professor of electrical, computer 

and systems engineering with the Rensselaer 

Polytechnic Institute, Troy, NY. He joined the faculty 

at Rensselaer in 2001. He has also held visiting 

faculty positions at the University of Melbourne, Australia, and with the Massachusetts 

Institute of Technology, Cambridge. His research is in nonlinear control 

theory and its applications, with particular interest in robust and observer-based 

feedback designs, and in distributed control of networks of dynamic systems. In 

these areas, he has published more than 90 journal and conference papers and 

organized several technical workshops. 

Dr. Arcak is a member of SIAM and an Associate Editor for the IFAC Journal 

Automatica. He received a CAREER Award from the National Science Foundation 

in 2003, the Donald P. Eckman Award from the American Automatic 

Control Council in 2006, and the SIAM Control and System Theory Prize in 

2007.

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