A Design Method for Distributed Luenberger Observers Francisco F. C. Rego 12 , A. Pedro Aguiar 3 , Ant ´ onio M. Pascoal 2 and Colin N. Jones 1* †‡ May 30, 2018 Abstract The paper addresses the problem of designing distributed ob- servers for discrete linear time-invariant (LTI) systems with distributed sensor nodes subjected to bounded measurement noise. A solution is proposed in terms of a distributed LTI Lu- enberger observer, thus departing from common linear time- varying solutions rooted in consensus-based distributed esti- mation techniques, and dispensing with the need for the ex- change of covariance matrices. It is shown, under the con- ditions of collective observability, strong connectivity of the sensor communication network, and invertibility of the state transition matrix, that the resulting observer yields ultimate boundedness of the estimation error. A design example is given where the asymptotic performance of the proposed ob- server is shown to be similar to that obtained using a time- varying distributed Kalman filtering approach. 1 Introduction 1.1 Motivation Spawned by recent advances in wireless sensor networks and distributed sensing, there has been a flurry of activity on the topic of distributed state estimation, see for example [1–3] and the references therein. Distributed state estimation and control have a wide range of applications, from network lo- calization to environmental monitoring and formation control of vehicles (see [4–6] for an introduction to these topics). One of the most studied families of distributed estima- tion algorithms in discrete-time are distributed Kalman fil- ters, which extend the theory of Kalman filtering to a dis- tributed setting. Among the many references on this sub- ject we cite [7–15], see Section 3. All methods developed so far for distributed estimation in a stochastic linear setting require that the estimation error covariances computed lo- cally be exchanged among nodes, which increases the amount of data to be communicated. The issue of bandwidth effi- *1 Francisco F. C. Rego and Colin N. Jones are with LA3, STI, EPFL, Lau- sanne, Switzerland, { francisco.fernandescastrorego, colin.jones } @epfl.ch †2 Francisco F. C. Rego and Ant´ onio M. Pascoal are with the In- stitute for Systems and Robotics (ISR), IST, Univ. Lisbon, Portugal, antonio@isr.ist.utl.pt ‡3 A. Pedro Aguiar is with the Research Center for Systems and Tech- nologies (SYSTEC) and the Faculty of Engineering of the University of Porto (FEUP), Portugal, pedro.aguiar@fe.up.pt ciency is of paramount importance in practical applications, since lower bandwidth translates into lower energy consump- tion and therefore increased operational autonomy. Moreover, since in these methods the estimates have time-varying dy- namics, it is difficult to obtain in hand convergence rates for the estimation errors. The above two issues, the need to exchange covariances and the difficulty in computing guaranteed error convergence rates, do not occur for distributed Luenberger observers, also named distributed linear time-invariant (LTI) observers, or distributed fixed gain observers, where the dynamics of the es- timation errors are linear time-invariant, unlike Kalman filters for which the observer gains are typically time-varying. Dis- tributed Luenberger observers have been the object of many recent studies [16–21], described in detail in Section 3. How- ever, the methods developed so far require strong assumptions on the kind of dynamical systems considered, or the number of times that data are exchanged among nodes at each filter iteration. Furthermore, convergence of the estimates can only be guaranteed if the norm of the state transition matrix is be- low a certain value. As of the writing of this paper, and to the best of the au- thors’ knowledge only references [22, 23] discuss LTI dis- tributed observers that guarantee ultimate boundedness of the estimation error for any LTI discrete-time system satisfying only a collective observability property, as defined in As- sumption A1. However, the methods proposed may suffer from shortcomings in terms of performance, as mentioned in Section 3 and clearly shown with the help of a simulation in Section 5. Borrowing from the theory in [10], in this paper we present an alternative design for a distributed Luenberger observer with guaranteed stability, which requires only collective ob- servability, with asymptotic behavior similar to [10], as will be seen in Section 5. However, in contrast with the method in [10], since the observer proposed in this paper is time- invariant, there is no need to exchange covariances, thus re- ducing the communications bandwidth requirements and al- lowing for the computation of a guaranteed convergence rate for the estimation errors. 1.2 Paper structure The paper is structured as follows. Section 2 formulates the problem of distributed observer design and describes the as- sumptions required. Section 3 provides a literature survey on