University of Groningen
Coordination networks under noisy measurements and sensor biases
Shi, Mingming
DOI:
10.33612/diss.99968844
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date: 2019
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Shi, M. (2019). Coordination networks under noisy measurements and sensor biases. Rijksuniversiteit Groningen. https://doi.org/10.33612/diss.99968844
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum.
6
Conclusions and future
research
6.1 conclusions
In this thesis we proposed methods to mitigate the effects of corrupted com-munication and measurement on network systems.
In Chapter 3 and 4 we investigated network coordination policies in the pres-ence of communication noise. The noise is assumed to be unknown but bounded, and no knowledge on the noise bound is required. In Chapter 3 we proposed a self-triggered coordination framework. By introducing an adaptive threshold in the consensus algorithm, and relating the threshold to the state of the node, the proposed method achieves practical consensus and boundedness of the system state. It is also shown that the node disagreement scales linearly with the magnitude of the noise and the norm of the initial state. The analysis and the numerical results reveal that the adaptive threshold is a promising way to robustify network coordination systems against communication noise. Yet, the bound on the system state depends on the magnitude of the noise, which may be large for large noise. To improve the bound on the consensus error and make the state bound decoupled from the noise, in Chapter 4, we pro-posed another coordination policy under the same framework of Chapter 3, in which each node saturates the received information based on a so-called favorite interval. This interval models the system prescribed or safe evolving region. We showed that the proposed method can disentangle the node dis-agreement from the norm of the initial system condition and also make the state evolve within a set closely around the favorite interval, independent of the noise magnitude. Moreover, when the nodes share a global clock, the method can be modified to make the node disagreement decay to zero in the noiseless situation, a result which is not possible to achieve by the adaptive threshold method. Hence, saturating information is an effective tool to contain the effect of the noise on the state, as well as to refine the consensus accuracy. In Chapter 5 we considered the problem of cooperatively estimating biases
112 conclusions and future research
from relative state measurements. We characterized the conditions on the number of biased sensors such that the sensors biases can be exactly recovered. The analysis shows that the ability to estimate the biases depends on the type of measurement graphs. For non-bipartite graphs, all the sensors biases can be recovered under biased relative state measurements, while for bipartite graphs more than half of the nodes should be unbiased, a number that can be re-duced to 2 by assuming heterogeneity of the biases. We also proposed three algorithms to estimate the biases.
6.2 future research
The results presented in this thesis serve as a starting point for the solution of various other problems in network systems under corrupted measurements. We suggest a few possible research lines worth to be investigated.
6.2.1
network coordination over noisy communication
In the presence of unknown but bounded communication noise, one already sees that the adaptive threshold method proposed in Chapter 3 can bound the node disagreement and confine the system state. The final value to which the state converges may be predicated when the noise follows a specific statistic distribution. For example, as observed in the numerical results for random graphs and random geometric graphs (Subsection 3.7.2), when the noise is white with zero mean, the states of the nodes are fluctuating around the centre of the interval containing the nodes initial values Cortés (2006), which reminds us of the concept of mean square consensus Li and Zhang (2010). This leads to two questions : a) Can this observation be proved theoretically? b) Under what conditions on the graphs or on the noise statistic features, does this observation hold?
As for Chapter 4, a challenging direction is to extend the result to the situation in which each node adopts a private favorite interval which may be different from those of the others. Convergence of the system under heterogeneous fa-vorite intervals has been shown in Fontan et al. (2017). However, extending the analysis therein to the situation where communication noise appears may lead to too conservative bounds on the system convergence interval and the node disagreement.
The proposed schemes in Chapter 3 and 4 can guarantee a small node-to-node error without requiring to choose the consensus threshold to be too small. In
turn, this can be beneficial for moderating the node-to-node error and keep-ing the communication cost low. Investigatkeep-ing this point in more detail for different graphs certainly represents an interesting venue for future research. An implicit assumption in Chapter 3 and 4 is that each node is able to measure its absolute state, which plays an important role in the proposed algorithms. A harder problem is that each node can only measure the relative states of its neighbours, as assumed in Chapter 5. Following the framework in Chapter 3 and 4, a naive approach to address this problem would be to let the threshold of each node increase unless it is larger than the absolute value of the noisy average. In detail, each node i∈ V could update the threshold as follows,
ϵ(ti k+1) = { ϵ(ti k), if| avewi(tik)| < ε(tik) ρϵ(ti k), if| avewi(tik)| ≥ ε(tik)
with ε(ti0) >0 and ρ > 1, the value to be carefully chosen. However, the proof
that such mechanism would work is not available.
In this thesis, we only consider systems with single integrator dynamics. It is worth to extend the results in Chapter 3 and 4 to systems with more complex agent dynamics. Even though it has been mentioned in Chapter 3 that exten-sion to systems with general linear dynamics is possible using the state trans-formation in Scardovi and Sepulchre (2009), De Persis (2013), it is not trivial to consider network systems with nonlinear agents Panteley and Loría (2017), and second-order agent dynamics De Persis and Postoyan (2017), De Persis et al. (2013) under communication noise.
Finally, it is also interesting to study the problem of achieving specific goals for coordination networks under unknown but bounded communication noise. For example, in the max-consensus algorithm Zhang et al. (2016); Nakamura et al. (2018); Muniraju et al. (2019), the estimate of the maximum of the nodes’ initial states also drifts in the presence of noise. Hence it is natural to consider how to prevent this drift and obtain a practical estimate of the max-consensus under communication noise. Also, in distributed optimization Nedić and Liu (2018), an unbounded state could result from the action of an unknown but bounded communication noise. Additionally, as mentioned in Wang et al. (2019), tackling the influence of noise on distributed algorithms for solving linear equations is still terra incognita.
114 conclusions and future research
6.2.2
bias estimation in sensor networks
The problem considered in Chapter 5 can be further investigated along a few directions. First, if the sensors are affected by noise in addition to the biases, one could study how noise impacts the accuracy of the estimation of the biases Lee et al. (2015). Second, the result for bipartite graphs could also be used for problems where the distance and angle of arrival measurements are affected by biases Meng et al. (2016a); Bolognani et al. (2010); Lee and Ahn (2016); Oh and Ahn (2014); Giridhar and Kumar (2006); Hashimoto et al. (2018).
Finally, in Chapter 5 we assume that the measurements taken by a node of the states of its neighbours is affected by the same bias. It is of interest to relax this assumption. For each neighbour j of node i, we can consider the relative state measurement zij=xj−xi+wij, with wij∈ R denoting the bias, and assume that
wijis different from wikwhen wij ̸= 0, with k any other neighbour of node i.
Considering this scenario, Shames et al. (2012) has worked on identifying the measurements zijwith wij̸= 0. It assumes that the value of wijis independent
of that of wik, for k∈ Niand k̸= j. Thus if wij̸= 0, wikcould be zero. However,
this may not be reasonable if node i uses one sensor to measure the relative states of all the neighbours, since any nonzero wijimplies that the sensor taken
by node i is faulty, which will also give incorrect measurements of the states of other neighbours of node i. Hence if wij ̸= 0 for some j ∈ Ni, wikshould be
nonzero for all k∈ Ni. In other words, if the sensor taken by node i is faulty,
wik̸= 0 for all k ∈ Ni. How does this property affect the result in Shames et al.
