University of Groningen Coordination networks under noisy measurements and sensor biases Shi, Mingming

Coordination networks under noisy measurements and sensor biases

Shi, Mingming

DOI:

10.33612/diss.99968844

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Shi, M. (2019). Coordination networks under noisy measurements and sensor biases. Rijksuniversiteit Groningen. https://doi.org/10.33612/diss.99968844

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Coordination networks under

noisy measurements and

sensor biases

of Science and Engineering, University of Groningen, the Netherlands.

This research was supported by Chinese Scholarship Council (CSC), the Chinese Ministry of Education.

The research reported in this dissertation is part of the research program of the Dutch Institute of Systems and Control (DISC). The author has successfully completed the educational program of DISC.

The cover picture is downloaded from Unsplash uploaded by Meagan Carsi-ence.

The thesis is printed by IPSKAMP printing. ISBN 978-94-034-2015-8 (printed version) ISBN 978-94-034-2014-1 (electronic version)

Coordination networks under noisy

measurements and sensor biases

PhD Thesis

to obtain the degree of PhD at the University of Groningen

on the authority of the Rector Magnificus Prof. dr. E. Sterken

and in accordance with the decision by the College of Deans This thesis will be defended in public on

Friday 25 October 2019 at 12:45 hours

by

Mingming Shi born on 18 May 1989

Prof. C. De Persis Co-Supervisor Dr. P. Tesi Assessment committee Prof. C. Altafini Prof. D. Bauso Prof. J. Hendrickx

Xia Man,

Shi Junchang, Zhao Xiaoge, Shi Kaiyu, Shi Panpan

Acknowledgements

”one time, one meeting.”

–Sen no Rikyū

How time flies! The four years PhD time is going to draw an end. I still feel that my first arrival at the central station of Groningen happened not long ago. The four years’ staying in this small, quiet and beautiful city undoubtedly leaves a unique and deep impression in my mind and will still shine in my later life. Even though I am not a person who socialize much, there are still a lot of memories emerging now when I am sitting in the corner of my office. In fear of losing these moments many years later, I write this chapter to record and thank the people who flash into my mind.

My sincere gratitude goes to my Supervisors Claudio De Persis, who is know-ledgeable, sharp and witty. I would like to thank him gratefully for his pa-tience when I got stuck, his rigor when I was sloppy, and his encouragement when I was unconfident. He always encourages me to try my best in pursuing simple and interesting problems, and criticizes me for the messy communic-ation skills. Looking back to the first year when I was here, these criticisms and encouragement make me improve greatly in being as a Phd. I would also like to thank my co-promoter Pietro Tesi for his guidance in the problem for-mulation and paper writing. He is easygoing, smart and good at listening. With not many words, he could easily convey sharp thought. All in all, pur-suing the Phd under the supervision of my supervisors is very comfortable and more like to learning and fighting with two experienced friends. I am also grateful to Nima Monshizadeh for the nice discussion and help in working on the problem in the fifth chapter. His flexibility and quick thinking really impress me.

I would like to thank the assessing committee for their dedication in reading my thesis and provide me with their instructive suggestions. I should also thank the group support team, especially Frederika for her effort in the confer-ences, events and group outings. Great thanks to Qin Yuzhen and Ji Chengtao

for accepting to be my paranymphs. They are also two of my best friends in Groningen. I spend a lot of time with you in having food, playing Mahjongg and card games, watching movies, travelling and discussing.

My gratitude also goes to my former office-mates and -neighbors, Erik, Danial, Shuai, Tobias, Sebastian and Tjardo. Much gratitude goes to Shuai, Erik and Tobias for their daily help, both academically and non-academically. I am grateful to Tobias for the translation of the summary of the thesis. I enjoyed the discussion about the research on the noisy consensus with Danial. Talking with the current members of our group, Mehran, Meichen, Guopin, Monique and Alexandro during the lunch also brings me a lot fun in the last year of the Phd.

I would also like to express my gratitude to the (former) Chinese member in DTPA and JBI, which are all kind and easygoing, especially for Qin Yuzhen and his wife Ge Shanshan, Yang Qingkang, Jiao Junjie, Chen Liangming, Xue Dong, Guo Miao and his wife Tian Bei, Zhou Ning, Zhang Hongyu, Cheng Xiaodong, Huang Jie, Bai Xiaoshan. I would miss the time of dinners and card games with you. My foreigner colleagues are always humorous and bringing fun to the chat. I feel pleased to talk with them, particularly with Alain, Yu, Hadi, Nelson, Martijn, Matthijs.

Many thanks to my (former) neighbors, Yan Xingchen, He Wei, Guo Hao, Fu Haigen, Xu Liang, Wu huala, Tao Yehan, Xu Qi, Ge Luo. We shared a lot of fun and silly times in the weekends, which often started with a hotpot or barbe-cue and are followed by board games. Sometime, these times were also shared with my (our) friends, Gu Xianming, Guo Yafei and Li Jiuling, Tian Yuchen, Zhang Yanxi and Liu Bin. I also miss the walking with some of you in the cit-ies and the mountains of the eastern Europe, which was exhaustive but full of laughing. I want to thank Deng Chenglong, Zhang Xuewen and Xiao Chengy-ong for reserving the ground for the badminton and helping me to improve my skill. I would also like to thank other friends I made in Groningen, Liu Yuru, Gao Kai, Cao Huangtang, Tang Xiaoying, etc. who have helped me directly and indirectly, and shared fun with me in the last four years.

Apart from my friends in Groningen, I feel grateful to Ye Dong, which guided me in the master research and convinced me to apply for the Phd position abroad. Without his help, there is no chance for me to end up with the Phd degree in Groningen. I would also like to thank the friends in my master who still share their life with me: Li Haiqin, Hou Zhili, Jiang Bingqiang, Huang Simeng, Fang Xiang, Sun Rui, Wan Neng, Han Weixin, Shao Qi, Shen Donghao

and Ren Bin. Chatting with you really release my pressure in pursuing the Phd. Hope we will keep in touch.

At last, the deepest gratitude goes to my parents for their insistence on the higher education of my sisters and mine, which was not easy for a family with three children in a small village of China in the early of the last decade. I should also thank my sisters who accompanied and took care of my parents in the last four years. Endless love to my wife. I feel guilty that I can not spend more time with you in the last four years.

Mingming Shi Groningen 17th_{of September, 2019}

Contents

1 introduction

1

1.1 Network coordination under communication noise . . . 2

1.2 Self triggered coordination . . . 4

1.3 Sensor bias estimation . . . 4

1.4 Outline of this thesis . . . 5

2 preliminaries

7

2.2 Graph-theoretic notions . . . 7

2.3 Bipartite graphs . . . 8

2.4 Compressed sensing . . . 9

3 self-triggered network coordination over noisy

communica-tion channels

13

3.2 Framework and Outline of the Main Results . . . 15

3.3 Self-triggered Coordination with Adaptive Consensus Thresholds 17 3.4 Noiseless Case . . . 19

3.5 Noisy Case . . . 22

3.6 Adaptive Thresholds, Sign Function and Node-to-node Error . . 32

3.7 Numerical Examples . . . 38

3.8 Conclusions . . . 44

3.9 Appendix A: Communication Delays . . . 45

4 benefits of saturating information in coordination networks

with noise

55

4.2 Problem formulation – Self-triggered interval consensus . . . . 57

4.3 Main Result . . . 59

4.4 Asymptotic consensus . . . 67

4.5 Numerical examples . . . 68

4.6 Conclusion . . . 72

4.7 Appendix B: Proofs of the results of the asymptotic consensus . 75

5 bias estimation in sensor networks

83

5.1 Introduction . . . 84

5.2 Problem formulation – biases estimation in sensor networks . . 87

5.3 Non-bipartite graphs . . . 90

5.4 Bipartite graphs . . . 94

5.5 Numerical Simulations . . . 105

5.6 Conclusion . . . 110

6 conclusions and future research

111

6.2 Future Research . . . 112

bibliography

115

summary

127

1

Introduction

In the last two decades there has been a strong drive towards a systematic un-derstanding of the modelling and evolution of complex large-scale systems, ranging from biology and sociology to engineering. The investigated subjects include, but are not limited to, flocking in animals, opinion formation in so-cial networks, power sharing in energy systems, formation of mobile vehicles, distributed estimation and computation in sensor networks. To achieve con-trol objectives of these complex networks, it is vital to efficiently monitor and control the systems conditions, especially for large scale artificial systems, like power networks or sensor networks since they play a major role in modern life. In these systems, sensors and actuators are deployed over the system and ex-change measurements and control signals with computing units over a com-munication medium, which introduces constraints on the scheduling and the duration of the transmission. To take into account of the constraints imposed by these sensors and communication devices, a large research activity has been developed in three directions. The first one has systematically investigated so-called event-based control methods in which measurements are sampled and control inputs are scheduled according to a state-dependent law Heemels et al. (2012), Tabuada (2007), Anta and Tabuada (2010). The second direction has methodically exploited the graph structure underlying large-scale systems to design distributed control laws that use information available only locally, while achieving a global coordination task, such as consensus or synchroniz-ation Cao et al. (2013); Panteley and Loría (2017); Kadowaki and Ishii (2014); Akashi et al. (2018). The third research direction has investigated the combined problem in which each local controller collects information from its neigh-bours and schedules new control values in an event-based fashion Nowzari et al. (2019); Yu and Dimarogonas (2018); Liu et al. (2017). These lines of re-search have provided methods to efficiently utilize the sensors and commu-nications for designing and controlling the network systems.

Since data transmission is often carried out through digital (i.e., packet-based) channels, the communication medium can introduce issues such as packet loss, transmission delay and error. Furthermore, sensors may be installed

in a hostile environment and operates without any protection. This makes it possible for the malicious agents to introduce errors in the measurements. The existence of errors in the measurements exchanged among the agents in a network system thus inspired this thesis, which focuses on the analysis and design of coordination algorithms under noisy measurements and faulty sensors.

1.1

network coordination under communication

noise

In the network coordination problem, the agents perform actions cooperatively to achieve global objectives or optimize performances of the whole systems. Typical examples are consensus Cao et al. (2013), synchronization Scardovi and Sepulchre (2009), formation de Marina et al. (2015) and resource alloca-tion Nedić and Liu (2018). Achieving coordinaalloca-tion usually requires the agents to exchange information with others. Hence the presence of communication noise is one of the major issues that arise in problems of network coordination involving data exchange. In this thesis, we focus on consensus algorithms Cao et al. (2013), by which the agents in the network gradually agree on a value for the variables of interest based on the information available locally. Since consensus is the prototypical problem in distributed coordination, consider-ing consensus under communication noise lays the foundation for investigat-ing the influence of communication noise on other distributed coordination problems.

Even if one neglects network-related issues such as finite transmission rate, dropouts and delay, developing noise-robust consensus algorithms is a very challenging task. The intuitive reason is that the consensus dynamics is de-scribed by means of the Laplacian matrix. Since the Laplacian matrix is only marginally stable (has an eigenvalue at zero), the communication noise can cause the states of the nodes to diverge. This means that even if consensus is achieved the consensus value needs not to be bounded, in which case conver-gence may become useless.

Most of the works in this area assume that the noise has specific statistical properties, for example that it is white Li and Zhang (2009); Cheng et al. (2011),

Brownian-like Li et al. (2014) or martingale Li and Zhang (2010); Huang et al.

(2010). In contrast, only few works have approached the problem where, due to uncertain channel characteristics, one can only regard noise as a bounded signal (unknown-but-bounded). Arguably, the lack of the statistical properties

of the noise makes it much more difficult to ensure state boundedness since one cannot rely on features such as zero-mean or stationarity. In Kingston et al. (2005), the authors consider a Kalman-based coordination scheme, and show that consensus (the disagreement among the nodes) satisfies input-to-state sta-bility properties, but no results are given regarding boundedness of the state trajectories. In Shi and Johansson (2013), the authors study robust and integral robust consensus with respect to L_∞and L1norms of the noise function, but

again the analysis only involves the disagreement variable and no results are given regarding state boundedness. This is also the case in Garulli and Gianni-trapani (2011) where the authors consider discrete consensus under bounded measurement noise, and in Franceschelli et al. (2013) where the authors pro-pose discontinuous interaction rules to mitigate the effect of disturbances on the node disagreement. In the slightly different context of leader-following consensus, Arabi et al. (2017) considers the issue of sensor noise, but assumes that the noise is a smooth signal. While effective to account for sensor bias, this hypothesis is hardly met with communication noise. A framework closer to the problem considered in this thesis is in Bauso et al. (2009). There, the au-thors propose a coordination scheme that guarantees approximate consensus along with boundedness of the state trajectories, but an upper bound on the magnitude of the noise is required to be known, which is also assumed in a re-cent paper Hendrickx et al. (2020). State boundedness and exact consensus are achieved in Zhou et al. (2013), but the result requires the restrictive assumption on the integral of the absolute value of the noise to be finite.

A problem related to consensus with communication noise is network coordin-ation with misbehaving agents LeBlanc et al. (2013); Vaidya et al. (2012); Dibaji and Ishii (2017); Senejohnny et al. (2019); Kikuya et al. (2017). These two prob-lems are similar in the sense that the nodes in both settings may receive in-correct state information from its neighbours. However, in the problem con-sidered in this thesis, all the transmission channels can be corrupted, while in the scenario considering misbehaving nodes there is usually a bound on the number of the malicious nodes, which implies that the communications among normal nodes are not subject to noise. On the other hand, in consensus with misbehaving nodes the error added to the state that is transmitted to the neighbours of abnormal nodes can be arbitrarily large. Instead, in the scenario we consider the communication noise is an unknown but bounded signal.

1.2

self triggered coordination

As mentioned before, data transmission is often carried out through digital devices, hence it can only occur at discrete times. In most of the coordin-ation algorithms this implies that the nodes can not communicate continu-ously. Due to this reason, sampled control of network coordination has been studied for a long time Dimarogonas and Johansson (2009); Xiao and Chen (2012); Liu et al. (2010); Cao and Ren (2010); Wu et al. (2016). In the recent years, with the purpose of reducing the communication channel occupation and saving the energy consumption during data transmission, the attention has turned into the event- or self-triggered implementation of the coordina-tion algorithms, by which the nodes can exchange data sporadically Dimaro-gonas and Johansson (2009); Kadowaki and Ishii (2014); Seyboth et al. (2013); Nowzari et al. (2019); Yu and Dimarogonas (2018); Liu et al. (2017); De Persis and Frasca (2013). In this thesis, we proposed two new self-triggered con-sensus algorithms, which relax the requirement of continuously monitoring the neighbours’ states presented in the event-triggered consensus method. The algorithms are based on the approach in De Persis and Frasca (2013). In this approach, the nodes take sampling and update their control inputs asynchron-ously based on the last state sampling of their neighbours. Apart from the sampling times, the access of the nodes to the communication channel is not needed. Hence the dynamics of each node is decoupled from those of the neighbours over two consecutive sampling instants. This provides more flex-ibility in the design of the coordination systems as it is more easy for the nodes to implement heterogeneous tasks and achieve network coordination simul-taneously.

1.3

sensor bias estimation

In this thesis, we also investigate the problem of eliminating sensor measure-ment errors. Specifically, we focus on the scenario that each sensor takes meas-urements of the relative states of its neighbours and the reading of the sensor may be affected by a constant bias. Relative state measurements are used widely in sensor localization Zhao and Zelazo (2016); Barooah and Hespanha (2007); Carron et al. (2014); Safavi et al. (2018); Shames et al. (2013); Wang and Tian (2018); Bof et al. (2016); Ravazzi et al. (2013, 2018). Sensor biases can lead to inaccuracy in the estimation capabilities of the sensors. When sensors are used for formation control, relative state measurement biases can result in the

distortion of the formation shape and excursion of the vehicle position from the prescribed one Meng et al. (2016a); de Marina et al. (2015); Liu et al. (2016); Sukumar et al. (2018). Finally, relative state measurements are also used in the statistical ranking, where the order of a group of the same kind of products, like movies or websites, needs to be determined by the pairwise assessment Jiang et al. (2011); Osting et al. (2014); Ravazzi et al. (2018). As a good ranking may bring commercial benefit for the product owner, a bias in the pairwise assessment may lead to incorrect ranking and mislead the consumers. Hence, in many applications, it is important to remove the biases from relative state measurements.

Extensive literature in distributed localization with erroneous relative state measurement deals with obtaining an approximate estimation of the node po-sition from the relative measurements Zhao and Zelazo (2016); Barooah and Hespanha (2007); Carron et al. (2014); Safavi et al. (2018); Shames et al. (2013); Wang and Tian (2018); Bof et al. (2016); Ravazzi et al. (2013, 2018). Some pa-pers also discuss how to attenuate the measurement biases in formation con-trol de Marina et al. (2015); Liu et al. (2016); Sukumar et al. (2018) under dif-ferent assumptions or scenarios. However, the problem of estimating the bi-ases and identifying the biased sensors has received less attention Bolognani et al. (2010); Shames et al. (2012); Carlone et al. (2014), and no systematic result has been presented characterizing the number of biased sensors and graph-theoretic conditions that allow for all the biases to be estimated.

The bias estimation problem considered in this thesis is also related to the sparse recovery problem, which aims at reconstructing the unknowns from an undetermined linear equation Rauhut (2010). Methods from sparse recovery have been applied to system identification Hayden et al. (2016), fault detection Hashimoto et al. (2018), and secure state estimation under attacks Chang et al. (2015); Hu et al. (2018); Shoukry and Tabuada (2016).

1.4 outline of this thesis

The main contribution of this thesis is given in three chapters. In Chapter 3 and 4, we address the network coordination in the presence of unknown but bounded communication noise. In Chapter 5, we focus on the problem of es-timating and removing biases in sensor measurements. We summarize the content of the chapters below.

and a few basic results in compressive sensing, which will be used in the rest of the thesis.

In Chapter 3 to achieve practical consensus and guarantee that the trajectories of the nodes in the consensus network are bounded in the presence of commu-nication noise, we propose a state-dependent self-triggered network consensus algorithm, which sets the control input of each node to zero when the abso-lute value of the node disagreement is less than a threshold. The threshold is adjusted adaptively according to the node state, which is proven to make the system robust to the communication noise.

In Chapter 4 we propose a new self-triggered consensus algorithm following the same framework of Chapter 3. To improve the bounds on the system state and the node disagreement, the algorithm assigns each node an interval which represents the expected region of evolution, and saturates the noisy transmis-sions received from the neighbours based on this interval.

In Chapter 5 we investigate how to estimate the biases of the sensors from the relative state measurements. Our major objective is to answer the question on how the topology of the network and the number of biased sensors affect our ability to identify all the biased sensors and estimate the corresponding biases. In Chapter 6 we provide some conclusions and suggestions for future research.

2

Preliminaries

2.1 notation

Given p scalar-valued variables z1,z2, . . . ,zp, we define z := col(z1,z2, . . . ,zp)

and let diag{z} represent the diagonal matrix with the ith diagonal entry equal to the ith element of z. We denote bySzthe support of z, which is the set of

indices that correspond to the nonzero entries of z, and by∥z∥0the 0-norm of

z, which is the number of elements inSz. We also let|z| denote its Euclidean

norm and|z|∞its infinity norm. When z is a scalar,|z| represents its absolute

value. The vector 1p and 0p represent the p-dimensional vectors with all

ele-ments equal to 1 and 0, respectively. Given a matrix A, Airepresents its ith row

and aijrepresents its element in the ith row and jth column. The cardinality of

a set S is denoted by|S|. For two sets S and M, we let S \ M = {x ∈ S | x /∈ M} represent the complement of M in S. Given a signal s mapping R≥0to Rn, we

define|s|∞:= sup_t∈R≥0|s(t)|_∞and say that s is bounded if|s|∞is finite.

2.2 graph-theoretic notions

For a network with n nodes, let its topology be represented by an undirected and connected graph G ={V, E}, with V = {1, 2, ..., n} being the set of nodes and E ⊆ V × V the set of edges, where {i, j} ∈ E, or equivalently, node i is a neighbour of node j, means that node i can receive information from node j and vice versa. We denote the set of neighbors of node i byNiand let di=|Ni|.

The adjacency matrix A of G is defined as aij = 1 if node j is the neighbor

of node i and aij = 0 otherwise. For an undirected graph G, we can assign

arbitrary orientations to the edges such that each edge{i, j} ∈ E has a head and a tail. The edge-node incidence matrix B ∈ Rm×nof G, with m = |E|, is defined as bij=1 if j is the head node of the edge i∈ E and bij=−1 if j is the

tail node. The Laplacian matrix L of G is an n× n matrix given by lij = −aij

for j ̸= i and lii =

∑

j∈Niaij =di. Since G is undirected, it is well-known that

L = B⊤B. The incidence matrix can be decomposed as the head incidence

matrix B+ ∈ Rm×nand the tail incidence matrix B− ∈ Rm×n, which are given

by

b+,ij =

{

1, if node j is the head 0, otherwise

b_−,ij =

{

−1, if node j is the tail

0, otherwise

We also let R denote the signless edge-node incidence matrix with rij=|bij|. It

is easy to verify that B = B++B−and R = B+− B−. Let d = [d1d2...dn]⊤and D = diag{d}. The matrix A + D is called the signless Laplacian matrix. When

G is undirected, A + D = R⊤R. Hence, A + D is positive semi-definite and all

its eigenvalues μ1≤ μ2≤ · · · ≤ μnare real and nonnegative.

A pathPijfrom node i to node j is a sequence of nodes and edges such that

each successive pair of nodes in the sequence is adjacent. The length of a path is the number of edges in the path. The distance between node i and j is the length of the shortest path from i to j. We denote by DGthe diameter of G,

which is the maximum distance between any two nodes.

2.3

bipartite graphs

In Chapter 5 we show that the ability to recover the biases from relative meas-urements depends on whether the measurement graph is bipartite or not. Hence, in this subsection we give a short introduction on bipartite graphs and their matrix properties. A graph G is bipartite if the vertex set V can be partitioned into two sets V+ and V− in such a way that no two vertices from the same

set are adjacent. The sets V+ and V− are called the colour classes of G and

(V+,V−)is a bipartition of G. For a bipartite graph, the following result holds:

Theorem 2.1. Asratian et al. (1998) A graph G is bipartite if and only if G has no

cycle of odd length.

An algebraic characterization of bipartite graphs is provided next.

Lemma 2.1. An undirected and connected graph G is bipartite if and only if the

sign-less incidence matrix R does not have full column rank. Moreover, if G is bipartite,

then any n− 1 columns of R are linearly independent.

Proof. To prove the first part, suppose that Rv = 0 for some nonzero vector v ∈ Rn_{. It is easy to see that|v}

fact, for every pair (r, s) of adjacent nodes, we must have vr =−vsotherwise

Rv̸= 0. Since the graph is connected and since v must be nonzero, we obtain

the claim. Note that this also shows that any two nodes i, j ∈ V with vi = vj

should not be adjacent. Let V+/V− contain the nodes corresponding to the

entries of v with value a/− a, then any node in V+/V₋should not be adjacent

to other nodes in V+/V₋. This implies that G is bipartite and (V+,V₋)is a

bipartition of G. Conversely if G is bipartite, there exists a bipartition (V+,V₋)

of G. By letting the elements of v corresponding to V+and V₋ be a and−a,

respectively, with a ̸= 0, we have Rv = 0, which shows that R does not have full column rank.

For the second part, we prove it by contradiction. Suppose there exist some dependent columns of R and let the index set of these columns be S ⊂ V,

with|S| ≤ n − 1, then there should exist a nonzero vector v ∈ R|S|such that

RSv = 0 where RSis the matrix whose columns are those indexed by S. The

latter implies the existence of a nonzero vector ˜v, whose nonzero entries are given by v, and satisfies R˜v = 0. However, from the proof of the first part, the absolute values of all the elements of ˜v should be equal to each other. Hence,

v must be the zero vector, which is a contradiction. ■

The if and only if part of the statement above is also provided in (Bapat, 2010, Lemma 2.17). We provide the proof here, since it is used in proving the second part of the statement as well as in other parts of the paper.

For later use, by the proof of Lemma 2.1, we note that

Rv = 0n⇐⇒ ∃a ∈ R s.t. vi=

{

a i∈ V+

−a i ∈ V− (2.1)

for a bipartite graph with bipartition (V+,V−).

Lemma 2.2. Cvetković et al. (2007) The smallest eigenvalue of the signless

Lapla-cian matrix A + D of an undirected and connected graph is equal to zero if and only if the graph is bipartite. In case the graph is bipartite, zero is a simple eigenvalue.

2.4 compressed sensing

In the field of compressed sensing or sparse signal recovery, one of the most important problems is how to find the sparsest solution from the

number-deficient measurements. Formally, consider the following linear equation

y = Fx (2.2)

where x∈ Rnis the vector of unknown variables, y∈ Rpis the vector of known values, and F ∈ Rp×n is a matrix defining the linear relation from x to y. It is assumed that p < n, thus equation (2.2) is under-determined. It is then of interest to find solutions x such that∥x∥0≪ n, and in particular to seek for the

sparsest solution of (2.2). Let us define the set of k-sparse vectors as

Wk:={x ∈ Rn| ∥x∥0≤ k}. (2.3)

The following result provides a sufficient condition under which the solution of (2.2) can be uniquely determined.

Lemma 2.3. Given an integer s≥ 0, let 2s ≤ p, and assume that any matrix made of 2s columns of F is full column rank. If x∈ Wsis a solution of (2.2), then there exists

no other solution of (2.2) inWs.

Remark 2.1. Under the assumptions of the lemma, the solution x∈ Wsof (2.2)

is also the solution to min

x∈Rn ∥x∥0

s.t. y = Fx, (2.4)

that is, the sparsest solution to (2.2). The proof of Lemma 2.3 descends from

(Hayden et al., 2016, Lemma 1). ■

However, solving x from (2.2) under the assumption that∥x∥0 ≤ s is

cumber-some when s is not small, as it requires to combinatorially search for s columns of F whose span contains y. A typical way to avoid this exhaustive search is to change the problem into the following ℓ1-norm optimization problem

min

x∈Rn ∥x∥1 (2.5)

s.t. y = Fx

where∥x∥1 =

∑

i=1,...,n|xi| denotes the 1 norm of x, the vector y is known from

(2.2) and the objective function and the constraint are both convex. Problem (2.5) can be solved by linear programming Rauhut (2010). The ℓ1-norm

following definition and result characterize the relation between the matrix F, the equation (2.2) and the ℓ1-norm minimization problem.

Definition 2.1 (Nullspace Property). A matrix F ∈ Rp×n _{is said to satisfy the}

nullspace property of order s, with s being a positive integer, if for any set

S⊂ V = {1, 2, ..., n} with |S| ≤ s and any nonzero vector v in the null space of

F, the condition below holds

∥vS∥1<∥vSc∥1, (2.6)

where vS∈ R|S|and vSc ∈ R|S

c|_{are subvectors of v whose elements are indexed}

by S and Sc, respectively, and Sc=V\ S.

The null space property is usually difficult to verify and a more restrictive but more conveniently checkable condition known as restricted isometry property is considered (Rauhut, 2010, p. 8). Yet, in the special cases that are of interest to us the null space property can be easily confirmed, and we will persist with it in the sequel.

Theorem 2.2. (Rauhut, 2010, Theorem 2.3) Every vector x ∈ Wsis the unique

solution of the ℓ1-norm minimization problem (2.5), with y = Fx, if and only if F

satisfies the null space property of order s.

We highlight the role of this theorem explictly in connection with the equation (2.2). For a given y∈ Rp_{, let x∈ R}n_{be a solution of (2.2). Assume that∥x∥}

0≤ s

and F satisfies the null space property of order s, with 0 < s < n. By Theorem 2.2, x is the unique solution of (2.5), with y = Fx. Stated directly, there exists a unique solution x∗of (2.5), with y = Fx, and it satisfies x∗ =x. Hence, under

the given condition of s-sparsity of the vector x solution of (2.2) and the null space property of order s of the matrix F, solving the optimization problem (2.5), with y = Fx, univocally returns x.

3

Self-Triggered Network

Coordination over Noisy

Communication Channels

abstract

This chapter investigates coordination problems over packet-based commu-nication channels. We consider the scenario in which the commucommu-nication between network nodes is corrupted by unknown-but-bounded noise. We introduce a novel coordination scheme, which ensures practical consensus in the noiseless case, while preserving bounds on the nodes disagreement in the noisy case. The proposed scheme does not require any global information about the net-work parameters and/or the operating environment (the noise characteristics). Moreover, network nodes can sample at independent rates and in an aperiodic manner. The analysis is substantiated by extensive numerical simulations.

Published as:

M. Shi, C. De Persis, and P. Tesi, “Self-triggered network coordination over noisy com-munication channels,” in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), Dec 2017, pp. 3942–3947.

M. Shi, P. Tesi, and C. De Persis, “Self-triggered network coordination over noisy com-munication channels,” IEEE Transactions on Automatic Control, pp. 1–1, 2019.

3.1 introduction

In this chapter, we consider a coordination algorithm that can handle

unknown-but-bounded noise without requiring the knowledge of a noise upper bound.

In order to prevent state divergence, we propose a state-dependent coordination scheme where each node dynamically adjusts its update rule depending on the magnitude of its state. This approach can be regarded as a coarse dynamic quantization strategy, which updates the quantization based on the state of the nodes Carli et al. (2010). We show that this approach prevents state diver-gence and guarantees, in the noiseless case, a maximum consensus error for the worst case over the initial vector of states, which is reminiscent of

normal-ized consensus metrics Boyd et al. (2006); Dimakis et al. (2010). As for the noisy

case, we show that this approach guarantees that both disagreement and state variables scale nicely (linearly) with the noise magnitude.

From a technical point of view, our approach employs a self-triggered control scheme De Persis and Frasca (2013). Each node uses a local clock to decide its update times. At each update time, the node polls its neighbors, collects the data and determines whether it is necessary to modify its controls along with its next update time. Similar to event-triggered control Heemels et al. (2012); Dimarogonas et al. (2012); Nowzari et al. (2019); Kadowaki and Ishii (2014),

self-triggered control features the remarkable property that the communication

among nodes occurs only at discrete time instants Anta and Tabuada (2010)-De Persis and Postoyan (2017). Moreover, the nodes can sample independ-ently and aperiodically. Thus, the proposed approach is appealing also from the perspective of finding coordination algorithms that are practically imple-mentable (as we will see, including the case where the data exchange encoun-ters delays).

The proposed self-triggered algorithm shares similarities with several pair-wise gossip or multi-gossip approaches with randomized Boyd et al. (2006) and deterministic Liu et al. (2011) protocols. There is however a major dif-ference, namely that while for gossiping algorithms the inter-node interaction times occur at multiples of discrete time-steps, in self-triggered consensus al-gorithms the update instants are established on the basis of current node meas-urements and can take any value on the continuous-time axis. Moreover, to the best of our knowledge, gossiping has not been considered in connection with unknown-but-bounded noise, even in the recent literature Shi et al. (2016)-Yu et al. (2017).

3.2 framework and outline of the main results

3.2.1 network dynamics

We consider a network of n dynamical systems that are interconnected over an undirected and connected graph G = (I, E), with I ={1, 2, ..., n} the set of nodes and E the set of edges. Each node is described by

˙xi=ui

zi=xi+wi (3.1)

where i ∈ I; xi ∈ R is the state; ui ∈ R is the control input, and zi ∈ R is the

output where wi∈ R is a bounded signal, which models communication noise.

Note that this model implies that all the neighbors of node i will receive the same corrupted information. As it will become clear in the sequel, it is possible to replace the second of (3.1) with zij=xi+wij, where i∈ I and j ∈ Ni, so that

each neighbor of node i receives a different corrupted information. We will not pursue this model in order to keep the notation as streamlined as possible. According to the usual notion of consensus Cao et al. (2013), the network nodes should converge, asymptotically or in a finite time, to an equilibrium point where all the nodes have the same value lying somewhere between the min-imum and maxmin-imum of their initial values. In the presence of noise, however, convergence to an exact common value is in general impossible to achieve. As outlined hereafter, the main contribution of this chapter is a new coordination scheme that ensures practical (approximate) consensus, namely convergence to a set whose radius depends on the noise amplitude.

3.2.2 outline of the main results

One way to define practical consensus is via the normalized error between the nodes. We consider a coordination scheme that, in the noiseless case, guaran-tees that all the network nodes remain between the minimum and the max-imum of their initial values, and converge in a finite time to a point belonging to the set

E := {x ∈ Rn_:|∑

j∈Ni

(xj− xi)| < max{ϵ, ϵχ₀}, ∀i ∈ I} (3.2)

where ϵ ∈ (0, 1) is a design parameter, and χ₀ := |xi(0)|∞. In words, when χ₀ > 1 the coordination scheme guarantees that, in a finite time, each node

reaches a local average that satisfies

|∑j∈Ni(xj− xi)|

χ₀ ≤ ϵ (3.3)

The parameter ϵ determines the desired accuracy level for the consensus final value, which is normalized to the magnitude of the initial data. In this way, a maximum error ϵ is guaranteed for the worst case over the initial vector of measurements. If instead χ₀ ≤ 1 then the tolerance reduces to ϵ. We will further comment on this point in Section 3.6.

As for the noisy case, the coordination scheme guarantees that the error scales nicely with respect to the noise magnitude. Specifically, let

r := max{ϵ, ϵχ0} +

(_ϵ

3 +3dmax )

|w|∞ (3.4)

where dmax := |d|_∞ denotes the maximum among the nodes degrees. The

scheme guarantees that, in a finite time, the network state enters the set

D := {x ∈ Rn_:|∑

j∈Ni

(xj− xi)| < r, ∀i ∈ I} (3.5)

and remains there forever with convergence in the event that w goes to zero. Moreover, the state remains confined in a set whose radius depends on ϵ and

|w|∞.

From an implementation point of view, the proposed scheme enjoys the fol-lowing features:

(i) No knowledge of χ0is required.

(ii) No knowledge of|w|∞is required.

(iii) The control action is fully distributed.

(iv) The communication between network nodes occurs only at discrete time instants. Moreover, the nodes can sample independently and in an aperi-odic manner.

These features indicate the implementation does not require any global in-formation about the network parameters and/or the operating environment (the noise). The last feature renders the proposed scheme applicable when coordination is through packet-based communication networks.

The main derivations will be carried out assuming that there are no commu-nication delays, which are tackled at last in Section 3.9. The analysis shows that, in practice, delays have the same effect as an additional noise source. For this reason, also numerical simulations will be restricted to the delay-free case.

3.3 self-triggered coordination with adaptive

consensus thresholds

3.3.1 adaptive consensus thresholds

As discussed in the previous section, we aim at considering a normalized error between network nodes. To this end, each node has a local variable

ϵi(t) :=

{

ϵ|xi(t)| if|xi(t)| ≥ 1

ϵ otherwise (3.6)

that specifies the threshold used to assess whether or not consensus is achieved. In contrast with previous self-triggered schemes De Persis and Frasca (2013); Senejohnny et al. (2018), this threshold is adaptive as it scales dynamically with the state magnitude. It is exactly this feature that ensures robustness against noise.

Notice that xi is used by node i to construct the threshold ϵi, which amounts

to assuming that each node has access to its own state without noise. This as-sumption can be relaxed and all the results continue to hold with the difference that the state bound (3.2) and the consensus set radius (r in equation (3.4)) will be enlarged. We neglect the details for this situation since it does not affect the general idea of the chapter.

3.3.2 control action and triggering times

For each i∈ I, let {ti

k}k∈N0with t

i

0=0 be the sequence of time instants at which

node i collects data from its neighbors. At these time instants, the node updates its control action and determines when the next update will be triggered.

For each i∈ I, let avew_i (t) :=∑

j∈Ni

(zj(t)− xi(t)) (3.7)

denote the local noisy average.

The control action makes use of a quantized sign function, The control signals take values in the set U := {−1, 0, +1}, and the specific quantizer of choice is

sign_α: R→ U, α > 0, which is given by

sign_α(z) :=

{

sign(z) if|z| ≥ α

0 otherwise (3.8)

The control action is given by

ui(t) = signϵi(ti_k) ( avew_i(tik) ) (3.9) for t∈ [ti k,tik+1[.

The triggering times are given by ti

k+1=tik+Δ i k, where Δik:=          | avew i (tik)| 4di if | avew i(tik)| ≥ ϵi(tik) ϵ 4di otherwise (3.10)

Note that by construction the first triggering event for all the nodes happens at time t = 0, and the inter-sampling times are bounded away from zero. The latter guarantees the existence of a unique Carathèodory solution for the state trajectories.

Remark 3.1. In the noise-free case, the control law (3.9) is an approximation of the pure (non-quantized) sign function which yields “max-min” consensus Cortés (2006), that is convergence to the centre of the interval containing the nodes initial values. Specifically, in the noise-free case, the scheme reduces to the one in Cortés (2006) when ϵi(·) ≡ 0 and the flow of information among

nodes is continuous. We refer the reader to Sections VII-B for further

Remark 3.2. Although the chapter focuses on networks of dynamical systems of the form (3.1), it is not hard to tackle synchronization problems involving lin-ear dynamics as in Scardovi and Sepulchre (2009), since synchronization can be reduced to a consensus problem by means of suitable coordinate transform-ations. For the noise-free case self-triggered algorithms for the synchroniza-tion of linear systems have been studied in De Persis (2013), and for the noise-free case with packet dropouts in Senejohnny et al. (2016). These algorithms can be modified in the spirit of (3.6)-(3.10) for the case of noisy measurements and the analysis carried out in the rest of the chapter can be extended to the

synchronization problem of linear systems. ■

3.4 noiseless case

We start by investigating the properties of this coordination scheme in the ab-sence of communication noise. For ease of notation, we let

avei(t) :=

∑

j∈Ni

(xj(t)− xi(t)) (3.11)

denote the noiseless average. Note that in the noiseless case avew

i(t) = avei(t)

for every t∈ R≥0.

Let

x := max

i∈I xi(0), x := mini∈I xi(0) (3.12)

We have the following result.

Theorem 3.1. Consider a network of n dynamical systems as in (3.1) with w ≡ 0,

which are interconnected over an undirected connected graph G = (I, E). Let each local control input be generated in accordance with (3.6)-(3.10). Then, for every initial

condition, the state x converges in a finite time to a point belonging to the setE in (3.2).

Moreover, maxi∈Ixi(t)≤ x and mini∈Ixi(t)≥ x for every t ∈ R≥0.

Proof. We start with showing the last property. We only show that maxi∈Ixi(t)≤

x for every t ∈ R_≥0since the other case is analogous. We prove the claim by

contradiction. Suppose there exists a time t_∗such that maxi∈Ixi(t∗) = x and ui(t∗) > 0, with i being the index of the node exceeding x for the first time

(clearly, more than one node could exceed x at the same time but this does not affect the analysis). Note that t∗cannot be a switching time for node i. In fact, if this were true, then we would have ui(t∗) > 0, which would require

avei(t∗)≥ ϵi(t∗) > 0, which is not possible because xs(t∗)≤ x = xi(t∗)for all

s ∈ I, by definition of t∗and i. Thus, we focus on the case where t∗ is not a

switching time. Let ti

kbe the last sampling instant smaller than t∗, which implies xs(tik)≤ x for

all s ∈ I. Notice that ti

kis well defined even if t∗occurs during the first

inter-sampling interval of node i because xs(0)≤ x for all s ∈ I. Since ui(t) = 1 for

all t∈ [ti

k,tik+1[, it holds that

xi(t) = xi(tik) + (t− tik) (3.13)

Evaluating the last identity at t = t∗, we get x− xi(tik) =t∗− tik<tik+1− tik=Δ

i

k (3.14)

Observe now that in order for xito grow we must also have| avei(tik)| = avei(tik)≥ ϵi(tik). This requires xi(tik) <x. In fact, if xi(tik) =x then node i could not grow

as xs(ti_k)≤ x for all s ∈ I. By (3.10), we have

Δik = 1 4di| avei (ti_k)| = 1 4di ∑ j∈Ni ( xj(tik)− xi(tik) ) ≤ 1 4(x− xi(t i k)) (3.15)

where the inequality comes again from the fact that xs(tik)≤ x for all s ∈ I. The

proof follows recalling the inequality (3.14). In fact, this implies

x− xi(tik) <Δ i k≤ 1 4(x− xi(t i k)) (3.16)

We now focus on the property of convergence. Consider the Lyapunov func-tion

V(x) := 1

2x

T_{Lx (3.17)}

where L is the Laplacian matrix related to the graph G. By letting ti

k= max{tih≤

t, h∈ N0}, the evolution of V along the solutions to (3.1) satisfies

˙ V(x(t)) = u⊤(t)Lx(t) = − n ∑ i=1 avei(t) sign_ϵi(ti k) (avei(tik)) = − ∑ i:| avei(tik)|≥ϵi(tik) avei(t) sign_ϵi(ti k) (avei(tik)) = − ∑ i:| avei(ti_k)|≥ϵi(ti_k) avei(t) sign(avei(tik)) (3.18)

where the last equality follows from the definition of the quantized sign func-tion. Observe now that if avei(ti_k)≥ ϵi(ti_k)then

avei(t) ≥ avei(tki)− 2di(t− tik) ≥ avei(tik)− 1 2| avei(t i k)| = avei(tik)− 1 2avei(t i k) = avei(t i k) 2 (3.19)

for all t∈ [ti_k,ti_k+1]. This implies that avei(t) preserves the sign during

continu-ous flow. Similarly, if avei(tik)≤ −ϵi(tik)then

avei(t) ≤ avei(tik) +2di(t− tik)≤

avei(ti_k)

2 (3.20)

Hence,

= | avei(t)| (3.21) This leads to ˙ V(x(t)) ≤ − ∑ i:| avei(ti_k)|≥ϵi(ti_k) | avei(t)| ≤ − ∑ i:| avei(ti_k)|≥ϵi(ti_k) | avei(tik)| 2 ≤ − ∑ i:| avei(ti_k)|≥ϵi(ti_k) ϵ 2 (3.22)

since ϵi(t) ≥ ϵ for all t ∈ R≥0. Thus, there exists a finite time T such that

each node satisfies| avei(tik)| ≤ ϵi(tik)for every k such that tik ≥ T, otherwise V would take on negative values. This shows that all the controls eventually

become zero, which implies that x(t) = x(T) for all t≥ T. Hence, we also have

ϵi(t) = ϵi(T) for all t ≥ T and for all i ∈ I. Since the network state remains

within the initial envelope, we have ϵi(t)≤ max{ϵ, ϵχ₀} for all t ∈ R≥0and for

all i∈ I, which yields the desired result. ■

3.5

noisy case

In this section, we study convergence and boundedness properties of the pro-posed scheme in the presence of noise. We first show that the propro-posed co-ordination method ensures boundedness of the state trajectories.

3.5.1

boundedness of the state trajectories

Let γ := ( 1 3 + 4 3 dmax ϵ ) |w|∞ (3.23)

We have the following result.

Theorem 3.2. Consider a network of n dynamical systems as in (3.1), which are

input be generated in accordance with (3.6)-(3.10). Then, for every initial condition, the state x satisfies

max i∈I xi(t)≤ { x if|x| ≥ γ γ otherwise (3.24) and min i∈I xi(t)≥ { x if|x| ≥ γ −γ otherwise (3.25) for every t∈ R≥0.

Proof. We will only prove the result regarding maxi∈Ixi(t) since the other can

be proved in an analogous manner. Notice that avew

i (t) = avei(t) + φi(t) for all t∈ R≥0and all i∈ I, where we defined

φ_i(t) :=∑

j∈Ni

wj(t) (3.26)

Clearly, we have

|φi(t)| ≤ dmax|w|∞ (3.27)

for all t∈ R≥0and all i∈ I.

Case 1:|x| ≥ γ. We show that there is no node that can exceed x. Suppose that

there exists a time t∗such that maxi∈Ixi(t∗) =x and ui(t∗) >0, with i the index

of the node exceeding x for the first time (clearly, more than one node could exceed x at the same time but this does not affect the analysis). In contrast with the proof of Theorem 3.1, here t∗may potentially be a switching time, since it

could happen that avew

i (t∗)≥ ϵi(t∗)even though xs(t∗)≤ x = xi(t∗)for all s∈ I due to the presence of the noise w. The case in which t∗is a switching instant

falls into the case studied in the next paragraph.

Let tikbe the last sampling instant not greater than t∗, which implies xs(tik)≤ x

for all s ∈ I. Notice that tikis well defined even if t∗ occurs in the first

Sub-case 1: xi(ti_k) >x−1₃|w|∞. The condition for xito grow is

avew_i (ti_k) = avei(tik) +φi(t i

k)≥ ϵi(tik) (3.28)

Since xs(ti_k)≤ x for all s ∈ I, we have

avei(tik) ≤ dix− dixi(tik) ≤ di(x− (x − 1 3|w|∞)) ≤ 1 3dmax|w|∞ (3.29)

By combining (3.27) and (3.29), in order for xito grow we must necessarily

have 4

3dmax|w|∞≥ ϵi(t

i

k) (3.30)

This leads to a contradiction. In fact, if |xi(ti_k)| ≥ 1 then ϵi(ti_k) = ϵ|xi(ti_k)|.

Moreover,|xi(tik)| > |x| −13|w|∞. Hence, we must necessarily have

4

3dmax|w|∞>ϵ(|x| − 1

3|w|∞) (3.31)

which implies|x| < γ, thus leading to a contradiction. If instead |xi(ti_k)| < 1

then ϵi(ti_k) =ϵ and we must have

4

3dmax|w|∞≥ ϵ (3.32)

This leads again to a contradiction since, by hypothesis, we must have γ≤ |x| and|x| < |xi(ti_k)| + ₃1|w|∞<1 + 13|w|∞. This would imply

4

3dmax|w|∞<ϵ.

Sub-case 2: xi(ti_k)≤ x − 1₃|w|∞. By construction, xican grow at most up to

xi(tik) + 1 4di (avei(tik) +φi(tik)) = 3 4xi(t i k) + 1 4di ∑ j∈Ni (xj(tik) +wj(tik))

≤3 4xi(t i k) + 1 4(x +|w|∞) (3.33)

where the inequality follows since xs(tik)≤ x for all s ∈ I. Since xi(tik) ≤ x −

1

3|w|∞we conclude that xican grow at most up to

3 4(x− 1 3|w|∞) + 1 4(x +|w|∞) =x (3.34)

which leads to a contradiction.

Case 2. |x| < γ. The proof of this case is exactly same as for the previous case

with x replaced by γ. ■

3.5.2 consensus properties under low-magnitude noise

We start with a simple result which shows that convergence is preserved under noise whenever|w|∞is sufficiently small compared to ϵ. Moreover, the state

remains within the initial envelope like in the noiseless case.

Theorem 3.3. Consider a network of n dynamical systems as in (3.1), which are

inter-connected over an undirected inter-connected graph G = (I, E). Let each local control input

be generated in accordance with (3.6)-(3.10). Suppose that ϵ > 2dmax|w|∞. Then, for

every initial condition, the state x converges in a finite time to a point belonging to the setD in (3.5). Moreover, maxi∈Ixi(t)≤ x and mini∈Ixi(t)≥ x for all t ∈ R≥0. Proof. We first show the last property. This can be done following the same

steps as in the noiseless case. Again, we only show that maxi∈Ixi(t) ≤ x for

all t∈ R≥0. Suppose that there exists a time t∗such that maxi∈Ixi(t∗) =x and ui(t∗) >0, with i the index of the first node exceeding x (clearly, more than one

node could exceed x at the same time but this does not affect the analysis). Let

ti

kbe the last sampling instant not greater than t∗, which implies xs(tik)≤ x for

all s ∈ I. Notice that ti

kis well defined even if t∗occurs during the first

inter-sampling interval of node i because xs(0) ≤ x for all s ∈ I. Clearly, we must

necessarily have| avew

i (tik)| = avewi(tik)≥ ϵi(tik). Moreover, xi(t)≤ xi(tik) + (t− t i k) (3.35) for all t∈ [ti k,tik+1].

By (3.10), we have Δik = 1 4di| ave w i(tik)| = 1 4di ∑ j∈Ni ( xj(tik)− xi(tik) +wj(tik) ) ≤ 1 4(x− xi(t i k) +|w|∞) (3.36)

where the inequality follows from the fact that xs(tik) ≤ x for all s ∈ I. By

hypothesis, ti

kis the last sampling instant not greater than t∗. Hence, since the

control input is constant over [ti

k,tik+1]and because ximust exceed x we must

have x < xi(ti_k+1). Hence, x− xi(tik) <Δ i k≤ 1 4(x− xi(t i k) +|w|∞) (3.37)

This inequality is possible only when

x− xi(tik) <

1

3|w|∞ (3.38)

However, this implies avew_i (ti k) = ∑ j∈Ni ( xj(tik)− xi(tik) +wj(tik) ) ≤ dmax(x− xi(tik) +|w|∞) < 4 3dmax|w|∞<ϵ (3.39)

where the last inequality follows since 2dmax|w|_∞ < ϵ by hypothesis. This

implies that avew

i(tik) <ϵi(tik), thus leading to a contradiction.

We now focus on convergence. Let V be defined as in (3.17), and consider the evolution of V along the solutions to (3.1). By letting ti

k= max{tih≤ t, h ∈ N0},

we have ˙

= −

n

∑

i=1

avei(t) signϵi(tik)(ave

w i(tik)) = − ∑ i:| avew i(t i k)|≥ϵi(tik) avei(t) sign_ϵi(ti k) (avew_i(ti_k)) = − ∑ i:| avew i(tik)|≥ϵi(t i k) avei(t) sign(avewi (tik)) (3.40)

where the last equality follows from the definition of the quantized sign func-tion. Observe now that if avew

i(tik)≥ ϵi(tki)then sign(avewi(tik)) =1. Moreover,

avei(t) ≥ avei(tki)− 2di(t− tik) ≥ avei(tik)− 1 2| ave w i(tik)| ≥ avei(tik)− 1 2ave w i(tik) = 1 2ave w i(tik)− φi(tik) ≥ 1 2ϵ− dmax|w|∞ (3.41) for all t ∈ [ti

k,tik+1]. Similarly, if avewi(tik) ≤ −ϵi(tik)then sign(avewi(tik)) = −1,

and avei(t) ≤ avei(tki) +2di(t− tik) ≤ avei(tik) + 1 2| ave w i(tik)| ≤ −1 2ϵ + dmax|w|∞ (3.42) This leads to ˙ V(x(t))≤ − ∑ i:| avew i(tik)|≥ϵi(t i k) ( 1 2ϵ− dmax|w|∞ ) (3.43)

for all t≥ 0. Since ϵ > 2dmax|w|∞,

1

for some α > 0, since all the quantities involved are constant. Hence, there exists a finite time T′ after which each node satisfies | avew

i(tik)| < ϵi(tik)for

every k such that ti

k ≥ T′, otherwise V would take on negative values. Since

x remains within the initial envelope then | avewi (t)| ≤ di(2χ0+|w|∞)for all

t∈ R≥0. Thus Δik≤ max{ϵ, (2χ0+|w|∞)}/4 := ¯Δ for every k ∈ N0. This shows

that all the controls eventually become zero not later than T := T′+ ¯Δ, which implies that xi(t) = xi(T) and avei(t) = avei(T) for all t ≥ T. Moreover, since

x remains within the initial envelope we also have ϵi(t)≤ max{ϵ, ϵχ0} for all

t∈ R≥0. Taking any tik≥ T we then have | avei(t)| = | avei(tik)|

≤ | avew

i(tik)| + dmax|w|∞

≤ max{ϵ, ϵχ0} + dmax|w|∞ (3.45)

The proof is concluded by noting that the right side of (3.45) is upper bounded

by r. ■

3.5.3 consensus properties under general noise

In general, condition ϵ > 2dmax|w|∞need not be satisfied if|w|∞is unknown.

Even if |w|∞ is known, enforcing this condition might lead to large errors

between network nodes. To this end, we study the properties of the proposed approach for the general case of noise which are unknown but bounded. We have the following result.

Theorem 3.4. Consider a network of n dynamical systems as in (3.1), which are

in-terconnected over an undirected connected graph G = (I, E). Let each local control input be generated in accordance with (3.6)-(3.10). Then, for every initial condition,

the network state x enters in a finite time the setD in (3.5) and remains there forever.

Moreover, x converges in a finite time to a point belonging to the setD in (3.5) when

the noise converge to zero.

We prove two technical results which are instrumental for the proof of The-orem 3.4.

Lemma 3.1. Consider the same assumptions and conditions as in Theorem 3.4. For

any i∈ I, it holds that

ϵi(tik)≤ r −

5

3dmax|w|∞ (3.46)

for every k∈ N0.

Proof. By Theorem 3.2, we have

|xi(tik)| ≤ max{|x|, |x|, γ} ≤ χ0+γ (3.47) Hence, ϵi(tik) = max{ϵ, ϵ|xi(tik)|} ≤ max{ϵ, ϵ(χ0+γ)} ≤ max{ϵ, ϵχ0} + ϵγ = r−5 3dmax|w|∞ (3.48)

where the last equality holds by the definitions (3.4) and (3.23) of r and γ

respectively. ■

The second result shows that the average preserves the sign as long as its ab-solute value remains large enough compared with the radius r.

Lemma 3.2. Consider the same assumptions and conditions as in Theorem 3.4.

Con-sider any index i ∈ I and any M ∈ N0. If| avei(ti_k+m)| ≥ r for m = 0, 1, . . . , M

then

sign(avei(tik+m)) = sign(avei(tik)),m = 1, 2, . . . , M + 1 (3.49)

Proof. Assume without loss of generality that avei(ti_k)≥ r, the other case being

analogous. From Lemma 3.1, we have avew_i(tik) ≥ avei(tik)− dmax|w|∞

≥ r − dmax|w|_∞

Hence, ui(ti_k) =1. Moreover, avei(t) ≥ avei(tki)− 2di(t− tik) ≥ avei(tik)− 1 2ave w i(tik) = 1 2avei(t i k)− 1 2φi(t i k) ≥ 1 2r− 1 2dmax|w|∞ ≥ 1 2max{ϵ, ϵχ0} (3.51) for all t∈ [ti k,tik+1].

We then conclude that avei(ti_k+1) >0. Thus aveipreserves its sign. ■

We can now proceed with the proof of Theorem 3.4.

Proof of Theorem 3.4. We only show the result for the case ϵ ≤ 2dmax|w|∞since

the other case can be derived from Theorem 3.3. To begin with, we introduce three sets into which we partition the set of switching times of each node i. For each i∈ I, let Si1:= { ti k: | avewi (tik)| ≥ ϵi(tik)∧ | avei(tik)| ≥ r } Si2:= { ti k: | avewi (tik)| ≥ ϵi(tik)∧ | avei(tik)| < r } Si3:= { ti k: | avewi (tki)| < ϵi(tik) } (3.52)

Clearly, ti_k∈ Si1∪ Si2∪ Si3for every k∈ N0.

Pick any i∈ I, and assume by contradiction that there exists a time t∗such that | avei(tik)| ≥ r for all tik≥ t∗. In view of Lemma 3.1, uiis never zero from t∗on

since the condition above yields| avew

i(tik)| ≥ r − dmax|w|∞≥ ϵi(tik). Moreover,

by Lemma 3.2, sign(avei(ti_k+m)) = sign(avei(ti_k))for every m. Hence, either

ui(t) = 1 for all ti_k ≥ t∗ or ui = −1 for all ti_k ≥ t∗. This would imply that xi

diverges, violating the state boundedness property of Theorem 3.2.

By the foregoing arguments, there exists a time instant tiksuch that| avei(tik)| <

r. This implies that ti

k∈ S/ i1, or, equivalently, that tik∈ Si2∪ Si3. Thus it remains

to show that transitions from Si2and Si3to Si1are not possible. We analyze the