Accelerated linear iterations for distributed averaging

Annual Reviews in Control 35 (2011) 160–165

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Accelerated linear iterations for distributed averaging q Ji Liu, A. Stephen Morse ⇑ Department of Electrical Engineering, Yale University, New Haven, Connecticut, United States

a r t i c l e

i n f o

Article history: Received 1 August 2011 Accepted 17 September 2011 This paper is dedicated to David Q. Mayne on the occasion of his 80th birthday.

Keywords: Consensus Distributed algorithms Linear iterations Convergence rates Matrix analysis

a b s t r a c t Distributed averaging deals with a network of n > 1 agents and the constraint that each agent is able to communicate only with its neighbors. The purpose of the distributed averaging problem is to devise a protocol which will enable all n agents to asymptotically determine in a decentralized manner, the average of the initial values of their scalar agreement variables. Most distributed averaging protocols involve a linear iteration which depends only on the current estimates of the average. Building on the idea proposed in Muthukrishnan, Ghosh, and Schultz (1998), this paper investigates an augmented linear iteration for fast distributed averaging in which local memory is exploited. A thorough characterization of the behavior of the augmented system is obtained under appropriate assumptions. It is shown that the augmented linear iteration can solve the distributed averaging problem faster than the original linear iteration, but the adjustable parameter must be chosen carefully. The optimal choice of the parameter and the corresponding fastest rate of convergence are also provided in closed form. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction There has been considerable interest recently in developing algorithms for distributing information among the members of a group of sensors or mobile autonomous agents via local interactions. Notable among these are those algorithms intended to cause such a group to reach a consensus in a distributed manner (Jadbabaie, Lin, & Morse, 2003; Blondel, Hendrichkx, Olshevsky, & Tsitsiklis, 2005; Moreau, 2005; Olfati-Saber & Murray, 2004; Ren & Beard, 2005). Consensus problems have a long history in statistics (DeGroot, 1974) and computer science (Pease, Shostak, & Lamport, 1980) and have a tradition in systems and control theory starting with the early work of Tsitsiklis (1984), Tsitsiklis, Bertsekas, and Athans (1986). Many different problems in various disciplines of science and engineering are closely related to consensus problems. They include subjects such as flocking (Jadbabaie et al., 2003), rendezvousing (Lin, Morse, & Anderson, 2003), formation (Fax & Murray, 2004), and synchronization (Freris, Graham, & Kumar, 2011). For a survey on the most recent works in this area see (Olfati-Saber, Fax, & Murray, 2007). One particular type of consensus process which has received much attention lately is called distributed averaging (Xiao & Boyd, 2004). In the development of algorithms to perform distributed

q An earlier version of this paper was presented at the IFAC Workshop on 50 Years of Nonlinear Control and Optimization (London, UK, September 30-October 1, 2010), dedicated to David Q. Mayne on the occasion of his 80th birthday. ⇑ Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (A.S. Morse).

1367-5788/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.arcontrol.2011.10.005

averaging, one typically encounters linear update equations of the form

xi ðt þ 1Þ ¼

X

aij xj ðtÞ;

i 2 f1; 2; . . . ; ng;

t P 1

ð1Þ

j2N i

where n > 1 is an integer, xi(t) is a real scalar, N i is a subset of the set {1, 2, . . . , n} containing i, and the aij are constants satisfying aij ¼ 0; j R N i ; i 2 f1; 2; . . . ; ng, n X

aij ¼ 1;

j 2 f1; 2; . . . ; ng and

i¼1

n X

aij ¼ 1;

i 2 f1; 2; . . . ; ng

j¼1

Typically these are the update equations for a group of n autonomous agents with labels 1 to n; in this case N i is the set of labels of agent i’s neighbors including itself. By introducing an n-vector x whose ith entry is xi, (1) can be written as

xðt þ 1Þ ¼ AxðtÞ;

t P 1

ð2Þ

where A is a real-valued n  n matrix whose row and column sums all equal 1. It is clear that any such matrix must have an eigenvalue at value 1 and moreover that x(t) will converge to a finite limit just in case all remaining eigenvalues lie strictly inside of the unit circle in the complex plane.1 In such cases, the rate at which x(t) converges {in the worst case} is determined by A’s ‘‘sub-spectral radius’’ where by the sub-spectral radius we mean the second largest among the magnitudes of the n eigenvalues of A. In the sequel we call any real 1 As a consequence of the constraints on the aij, it is easy to see that if x(t) P converges, in the limit each of its entries must equal the average 1n ni¼1 xð1Þ. We will not make use of this fact in this paper.

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n  n matrix A an averaging matrix if its row and column sums all equal 1 and its sub-spectral radius is less than 1; we use the symbol r2 to denote the sub-spectral radius of A. In distributed averaging applications, A typically has only nonnegative entries in which case it is doubly stochastic.2 Over the past fifteen years several clever ideas have emerged for speeding up the convergence of x(t) by augmenting (1) with delayed values of xi(t) (Aysal, Oreshkin, & Coates, 2009; Johansson & Johansson, 2008; Muthukrishnan et al., 1998; Oreshkin, Coates, & Rabbat, 2010; Xiong & Kishore, 2010). In a pioneering paper (Muthukrishnan et al., 1998), Muthukrishnan et al. propose augmented update equations of the form X xi ðt þ 1Þ ¼ ðg þ 1Þ aij xj ðtÞ  gxi ðt  1Þ; i 2 f1; 2; . . . ; ng; t P 1 j2N i

ð3Þ where g is a real, constant design parameter not depending on i. In matrix terms one thus has the augmented system

xðt þ 1Þ ¼ ð1 þ gÞAxðtÞ  gxðt  1Þ;

t P 1

ð4Þ

Under the assumption that A is a symmetric averaging matrix with nonnegative entries the authors of Muthukrishnan et al. (1998) show that there exists a value of g 2 (0, 1) dependent on A with which a faster rate at which x(t) converges is obtained compared with the original system (2). This result is inspired by the work of Golub and Varga (1961) and derived using matrix iterative analysis (Varga, 1962). Recently the more sophisticated iteration

xðt þ 1Þ ¼ ðg þ 1ÞAxðtÞ  gðh3 AxðtÞ þ h2 xðtÞ þ h1 xðt  1ÞÞ;

t P 1 ð5Þ

is proposed in Aysal et al. (2009) to further speed up convergence, again for the case when A is symmetric; here h1, h2, and h3 are prespecified real-valued constants. It is clear that in the case when h1 = 1 and h2 = h3 = 0, the system (5) is the same as the system (4). The system (5) is analyzed in Oreshkin et al. (2010) under special spectrum assumptions; the analysis also assumes that h2 P 0 and h3 P 1 so the result in Oreshkin et al. (2010) does not apply to the system (4). Also related to this paper is the work of Cao, Spielman, and Yeh (2006) in which Cao et al. propose a modified gossiping3 algorithm intended to speed up convergence without proof of correctness, but with convincing experimental results. The algorithm has recently been analyzed in Liu, Anderson, Cao, and Morse (2009). Although exploiting the same over-relaxation technique in Muthukrishnan et al. (1998), the algorithm in Cao et al. (2006) admits a time-varying system in which the update matrix depends on time; by way of contrast the system (4) is time-invariant. Therefore there is a fundamental difference between the work of Muthukrishnan et al. (1998) and Cao et al. (2006). In this paper we revisit the original acceleration rule proposed in Muthukrishnan et al. (1998) but without the assumption of nonnegativity. We do not assume that A is symmetric but we do retain one of symmetry’s implications, namely that A has a real spectrum. Thus a larger class of averaging matrices is considered compared with the work of Muthukrishnan et al. (1998). The main contribution of this paper is to thoroughly characterize the behavior of the system (4). The technique of analysis in this paper is different from that in Muthukrishnan et al. (1998) and somewhat more general results are obtained. In addition to recalculating by new means, the optimal value of g and corresponding fastest convergence rate 2 By a doubly stochastic matrix is meant a nonnegative n  n matrix whose row sums and column sums all equal 1. 3 Gossiping is an alternative approach to distributed averaging (Boyd, Ghosh, Prabhakar, & Shah, 2006).

which were originally calculated in Muthukrishnan et al. (1998), the present paper also derives a closed-form characterization of the range of values of g which accelerate convergence. 2. Main results Distributed averaging, as considered here, deals with a group of n > 1 agents labeled 1 to n. Each agent i has control over a real-valued scalar quantity xi called an agreement variable which the agent is able to update from time to time. Agents are able to communicate only with their neighbors. Neighbor relations are described by a given simple, undirected, n-vertex graph N called a neighbor graph in which vertices correspond to agents and edges indicate neighbor relations. Agent j is a neighbor of agent i if (i, j) is an edge in N. Thus the neighbors of an agent i have the same labels as the vertices in N which are adjacent to vertex i; we use the symbol N i to denote the set of labels of the neighbors of agent i. Initially, each agent i has or acquires a real number yi which might be a measured temperature or something similar. The distributed averaging problem is to devise a protocol which enables the n agents to reach a consensus in the sense that all n agreement variables ultimately reach the same value at the average n 1X y n i¼1 i

yavg ¼

in the limit as t ? 1 using only information acquired from its neighbors. For this to be possible, no matter what the initial values of the agreement variables are, it is clearly necessary that N be a connected graph. We assume that this is so. There are many variants of this problem. For example, instead of real numbers, the yi may be integer-valued (Kashyap, Basar, & Srikant, 2007). Another variant assumes that the edges of N change over time (Xiao, Boyd, & Lall, 2005). This paper considers the case when the yi are real numbers and N does not depend on time. In the sequel we will prove the following result. Theorem 1. Let A be an averaging matrix with a real spectrum. P P Suppose that xið1Þ = yi, i 2 {1, 2, . . . , n}, and ni¼1 xi ð0Þ ¼ ni¼1 yi . Then the state vector x(t) of the augmented system (4) converges to yavg1 1. 2. 3. 4. 5.

if and only if 1 < g < 1 as fast as the original system (2) if g = 0 or g ¼ r22 slower than the original system (2) if 1 < g < 0 or faster than the original system (2) if 0 < g < r22 fastest when

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  r22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g¼ 1 þ 1  r22 1

r22 < g < 1

ð6Þ

3. Analysis The augmented system (4) can be written in a state space form. Toward this end, let

uðtÞ ¼ xðt  1Þ and zðtÞ ¼



xðtÞ



uðtÞ

Then z(t) evolves of the form

zðt þ 1Þ ¼ BzðtÞ

ð7Þ

where

B¼



ðg þ 1ÞA gI I



0

It is clear that the system (7), as well as the augmented system (4), converges if and only if q2(B) < 1 and the convergence rate {in the

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worst case} is determined by q2(B), where q2(B) is the sub-spectral radius of B. It is of interest to look at two special cases corresponding to g = 1 and g = 0. First, with g = 1, it is obvious that B has eigenvalues 1 and 1 with multiplicity n. In this case q2(B) = 1, which implies that the system (7) does not converge. Second, with g = 0, there holds

B¼



A 0 I



ðg þ 1ÞA gI I



0

 g is a real constant. Suppose g – 1 or 0. Then B is nonsingluar. where z If 1 is an eigenvector of B with eigenvalue k, then z1 is an eigenvecz2 tor of A with eigenvalue

l¼

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jli j2 ðg þ 1Þ2  4g

ð13Þ

Without loss of generality, we assume that

ð14Þ

It is clear that l1 = 1 and jl2j = r2 < 1.

Lemma 1. Let A be an n  n real matrix and B be a 2n  2n matrix given by



1 2

qi ¼ jli jðg þ 1Þ þ

jl1 j P jl2 j P jl3 j P    P jln j

0

The block triangular structure of B implies that the eigenvalues of B comprise the eigenvalues of A together with n zeros. Thus q2(B) = r2 when g = 0, which implies that the system (7) converges if and only if the original system (2) converges and that their convergence rates are the same. This is accordance with the fact that no memory is actually used in the case when g = 0. Next we will consider how q2(B) varies as a function of g. Before doing this, we first show the existence of the desired convergence of the augmented system (4). Toward this end we need the following lemma.

B¼

From (11) and (12), with real li and 1 < g < 1, if li is positive, then jki1j P jki2j; if li is negative, then jki1j 6 jki2j. Let qi = max{jki1j, jki2j}. Then it is easy to see that

k g þ g þ 1 ðg þ 1Þk

ð8Þ

 l and k satisfies Conversely, if z1 is an eigenvector of A with eigenvalue z (8), then k is an eigenvalue of B with eigenvector 1 where z2 ¼ zk1 . z2 The simple proof is omitted.

Lemma 2. Let A be an n  n real matrix and B be a 2n  2n real matrix given by

 B¼

ðg þ 1ÞA gI I



0

where 1 < g < 1. Suppose that A has a real spectrum and the magnitudes of all of its eigenvalues, with the exception of a single eigenvalue of value 1, are strictly less than 1. Then B has eigenvalue 1 with multiplicity 1 and all the remaining 2n  1 eigenvalues are strictly less than 1 in magnitude. Proof of Lemma 2. We consider the complex eigenvalues and real eigenvalues of B separately. For any complex eigenvalue, from (10), pffiffiffi its magnitude equals g , which is strictly less than 1 in the case when 0 6 g < 1 {equation (9) does not have complex roots when 1 < g < 0}. Next we turn to the real eigenvalues. From (13) it can be checked that @ qi/@jlij > 0 and @ qi/@g < 0, which implies that qi increases as jlij increases or g decreases. Thus an upper bound of qi is given by (13) with jlij = 1 and g = 1. With these values, the right side of (13) evaluates as 1. This implies that qi < 1 for any g 2 (1, 1) and li, i 2 {2, 3, . . . , n}. Recall that in the case when li = 1, the two corresponding eigenvalues of B are 1 and g. It therefore can be concluded that B has an eigenvalue at 1 and all the remaining 2n  1 eigenvalues are strictly less than 1 in magnitude. h We are led to the following result:

Remark. Lemma 1 implies that in the cases when g – 1 or 0, the 2n eigenvalues of the matrix B are determined by the n eigenvalues of the matrix A through equation (8) which can be written in a quadratic form

Proposition 1. Suppose that A is an averaging matrix with a real spectrum. Then the system (7) converges if and only if 1 < g < 1.

k2  lðg þ 1Þk þ g ¼ 0

Proposition 2. Let A be an averaging matrix with a real spectrum. Suppose that 1 < g < 1, g – 0, and xi(1) = yi, i 2 {1, 2, . . . , n}. Then for the system (7) and each i 2 {1, 2, . . . , n}, xi(t) converges to yavg if and only if

ð9Þ

Note that from (9), if g = 1, then k = 1 or 1; if g = 0, then k = l or 0. That is to say, equation (9) also holds in the cases when g = 1 and g = 0. Thus for any g and each eigenvalue l of A, there are two corresponding eigenvalues of B which are the two roots of equation (9). Let li, i 2 {1, 2, . . . , n}, denote the eigenvalues of A and let ki1, ki2 be the two eigenvalues of B corresponding to li . Assume that A is an averaging matrix with a real spectrum. Then the li are all real and one of the li equals 1. If for some li, the roots of equation (9) are complex, then

pffiffiffi jki1 j ¼ jki2 j ¼ g

ð10Þ

If, on the other hand, li satisfies the condition that equation (9) has two real roots, then the corresponding two eigenvalues of B are

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ki1 ¼ li ðg þ 1Þ þ l2i ðg þ 1Þ2  4g 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 l2i ðg þ 1Þ2  4g ki2 ¼ li ðg þ 1Þ  2 2

Since we are interested in the average yavg, more can be said:

n X

xi ð0Þ ¼

i¼1

n X

yi

i¼1

The proof of Proposition 2 depends on the following lemma. Lemma 3. Suppose that M is a matrix in Rnn and c, d are two vectors in Rn such that c0 d – 0. Then the equation 0

lim M t ¼

t!1

dc c0 d

holds if and only if

ð11Þ ð12Þ

Note that when li = 1, both ki1 and ki2 are real and they equal 1 and g. Since the system (7) converges if and only if q2(B) < 1, it is clear that the system (7) does not converge if g 6 1 or g P 1. In the sequel we will consider the case when 1 < g < 1.

 0 dc 0 and f(1) 6 0; thus f(g) has a unique zero point between 1 and 1 which is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  l2i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gi ¼ 1 þ 1  l2i 1

ð15Þ

That is to say, for each li, if 1 < g 6 gi, then ki1 and ki2 are real; if gi < g < 1, then ki1 and ki2 are complex. In addition, from (15), it is easy to see that gi is an increasing function of jlij and in particular, g1 = 1. These facts and (14) imply that

I



0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  r22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and g ¼ 1 þ 1  r22

Then when g⁄ < g < 1, q2 ðBÞ ¼



1

pffiffiffi g.

Combining Lemmas 4 and 5, more can be said: Proposition 3. Suppose that A is an averaging matrix with a real spectrum. Let

B¼

2

ðg þ 1ÞA gI



ðg þ 1ÞA gI I



0

where 1 < g < 1. Then the minimum of q2(B) is unique. The optimal value of g at this minimum is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  r22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 1 þ 1  r22 1

and the minimum of q2(B) is

q2 ¼

r2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1  r22

ð17Þ

1 < g n 6 g n1 6    6 g 2 < g 1 ¼ 1 Therefore, if 1 < g 6 gn, B has 2n real eigenvalues; if gk+1 < g 6 gk, k 2 {2, 3, . . . , n  1}, B has 2k real eigenvalues and 2n  2k complex eigenvalues; if g2 < g < 1, B has 2 real eigenvalues and 2n  2 complex eigenvalues. Lemma 4. Suppose that A is an averaging matrix with a real spectrum. Let



ðg þ 1ÞA gI B¼ I 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  r22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and g  ¼ 1 þ 1  r22



We are now in a position to prove Theorem 1.

1

Then when 1 < g 6 g⁄,

1 2

q2 ðBÞ ¼ r2 ðg þ 1Þ þ

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r22 ðg þ 1Þ2  4g

Proof of Proposition 3. From Lemmas 4 and 5, it is clear that q2(B) is a continuous function of g on the interval (1, 1). When 1 < g 6 g⁄, by Lemma 4 and the fact that @ q2/@g < 0, q2(B) is a decreasing function of g. When g⁄ < g < 1, q2(B) is an increasing function of g by Lemma 5. Since q2(B) is continuous at g⁄, g⁄ is the unique point in the interval (1, 1) which minimizes q2(B). h

ð16Þ

Proof of Lemma 4. Since 0 6 r2 < 1, it is clear that g⁄ P 0. We consider the case when 1 < g < 0 first. In this case all 2n eigenvalues of B are real. Recall that from (13) qi increases as jlij increases, so q2 P qk for all k 2 {3, 4, . . . , n}. Since for l1 = 1 the two corresponding eigenvalues of B, k11 and k12, are 1 and g respectively, q2(B) = max {q2, g}. Note that q2 equals the right side of (16). Also it can be verified that q2 P g. Thus q2(B) = q2 when 1 < g < 0.

Proof of Theorem 1. The assertion 1 and assertion 5 are respectively direct consequences of Propositions 2 and 3. In addition, from the proof of Proposition 3, q2(B) is a decreasing function of g when 1 < g 6 g⁄ and an increasing function of g when g⁄ < g < 1. By Lemmas 4 and 5, it can be verified that q2(B) equals r2 when g = 0 or g ¼ r22 . Moreover 0 2 (1, g⁄] and r22 2 ½g  ; 1Þ. Thus the rest assertions of Theorem 1 follow at once. h Here we define the acceleration ratio as

g¼

log q2 log r2

to measure the maximum speed-up of the augmented system. It can be easily verified that

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15

Acknowledgments

acceleration ratio η

The research of J. Liu and A. S. Morse is supported by the US Air Force Office of Scientific Research and the National Science Foundation. 10

References

5

0

0

0.2

0.4

0.6

0.8

1

0 < σ2 < 1 Fig. 1. Acceleration ratio g as a function of r2.

g¼1

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 log 1 þ 1  r22 log r2

and that g > 1 if 0 < r2 < 1. We plot g as a function of r2 in Fig. 1. It is shown that the acceleration ratio g increases monotonically as r2 increases from 0 to 1. When 0 < r2 < 0.6, the convergence rate of the augmented system is slightly faster than the original system. When the value of r2 is close to 1, there is a substantial acceleration of the augmented system.

4. Concluding remarks It is worth emphasizing that if an averaging matrix A has a complex spectrum, then the analysis of the spectrum of the matrix B, as well as q2(B), will become more complicated. In addition, the following example shows that equation (6) does not hold for such A. Consider an asymmetric doubly stochastic matrix

2

:125 :5 :25 :125

3

6 :125 :5 :25 :125 7 6 7 A¼6 7 4 :25 0 :5 :25 5 :5

0

0

:5

which has a complex spectrum. In this case, r2 = 0.3536. According to (6), the optimal g should be 0.0334. But the numerical search shows that q2(B) is minimized when g 2 (0.015, 0.025). Analysis of the cases in which A has a complex spectrum is a subject for future research. From (17) it is clear that faster convergence of distributed averaging can be achieved using the augmented system (4) instead of the original system (2) if r2 > 0. The range of g, as well as the optimal value of g, that accelerates the convergence depends on r2 which is a global information. Thus all n agents are required to acquire this global value. In Oreshkin et al. (2010) a decentralized estimation of r2 is proposed. One of the problems with the idea of distributed averaging with local memory, which apparently is not widely appreciated, is that it is difficult to come up with provably correct algorithms for accelerated convergence without making restrictive assumptions. For example, we are unaware of any accelerated linear iteration for distributed averaging only using local information. Whether or not such algorithms can be devised remains to be seen.

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J. Liu, A.S. Morse / Annual Reviews in Control 35 (2011) 160–165 A. Stephen Morse received the B.S.E.E. degree from Cornell University, the M.S. degree from the University of Arizona, and the Ph.D. degree from Purdue University. Following three years at the Office of Control Theory and Application (OCTA), NASA Electronics Research Center, Cambridge, MA, he joined the Yale University faculty, where he is currently the Dudley Professor of Engineering. His main interest is in network synthesis, optimal control, multivariable control, adaptive control, urban transportation, vision-based control, hybrid and nonlinear systems, and, most recently, coordination and control of large grouping of mobile autonomous agents.

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He received the George S. Axelby Outstanding Paper Award from the IEEE Control Systems Society in 1993 and 2005, the American Automatic Control Council’s Best Paper Award (twice), and the IEEE Technical Field Award for Control Systems in 1999. He is a life fellow of the IEEE, and a member of the National Academy of Engineering and the Connecticut Academy of Science and Engineering.