Design of a Distributed Quantized Luenberger Filter for Bounded Noise Francisco F. C. Rego 12 , Ye Pu 1 , Andrea Alessandretti 12 , A. Pedro Aguiar 3 , Antonio M. Pascoal 2 , Colin N. Jones 1 May 30, 2018 Abstract This paper addresses the problem of distributed state estimation for linear systems with process and measure- ment noise, in the case of limited communication data rate, where the data exchanged between agents is quan- tized. We propose a linear distributed Luenberger ob- server and we derive a set of conditions on the design pa- rameters of the quantizer to guarantee ultimate bound- edness of the estimation error. The latter is shown to depend on the L 2 norm of the disturbance signals and on the number of bits. A numerical example illustrates the performance of the proposed algorithm. 1 Introduction Motivated by advances in wireless sensor networks, there has been growing interest in the topic of distributed state estimation, see for example [1–4] and the refer- ences therein. However, there are important applica- tions where such methods can not be used directly due to bandwidth limitations e.g., localization of underwater vehicles [5]. In [2–4] the authors propose consensus based linear state estimators. Consensus can either be achieved on a linear transformation of the measurements (the mea- surement form), or on a linear transformation of the state estimate (the information form). Motivated by the fact that certain applications have stringent communication limitations, we propose a consensus based distributed Luenberger observer where the messages exchanged dur- ing the consensus step are quantized. For an overview of the theory of quantized consensus, see for example [6–9] and the references therein. We consider in this paper that the consensus algorithm is performed with progres- sive quantization as in [7] and propose a method to select *1 Francisco F. C. Rego, Ye Pu, Andrea Alessandretti and Colin N. Jones are with LA3, STI, EPFL, Lausanne, Switzerland, { francisco.fernandescastrorego, y.pu, andrea.alessandretti, colin.jones } @epfl.ch *2 Francisco F. C. Rego Andrea Alessandretti and Antonio M. Pascoal are with the Institute 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 Technologies (SYSTEC) and the Faculty of Engineering of the University of Porto (FEUP), Portugal, pedro.aguiar@fe.up.pt the quantization parameters to achieve ultimate bound- edness of the estimation errors. In particular, we show that the proposed method is suitable for distributed state estimation with limited data rate between nodes and known disturbance bounds. Only global observability is required. Moreover, we de- termine the ultimate bound of the estimation error, and we show that the maximum allowed convergence rate de- pends on the number of iterations of the consensus algo- rithm and can be made arbitrarily close to the perfor- mance of the centralized case by increasing the number of iterations. The implementation of the algorithm is straightforward, but must be done off-line with global information about the system and the network. The key contributions of the paper are the following: • We propose a distributed linear state estimation scheme that takes into account limited data-rate communications among agents. • We provide conditions on the design parameters of the algorithm to guarantee ultimate boundedness of the estimation error. • Given that the above mentioned conditions are sat- isfied, we derive explicit bounds on the estimation error norm. 1.1 Notation Throughout this paper we will use the symbol ⊗ for the Kronecker product. The symbol ‖·‖ represents the L 2 norm, and ‖·‖ ∞ represents the L ∞ norm. The nota- tion |·| represents the cardinality of a set. The notation ⌊·⌋ represents the floor operator, or the rounding down to the closest lower integer, the function sgn(·) is the sign function, and ρ(·) the spectral radius of a square matrix. I M represents an M × M identity matrix, and 1 represents a N × 1 vector with ones in every entry. When clear from the context the superscript of a vari- able, e.g. x i , refers to the node index of that variable where i ∈{1,...,N } := N . The operator row(·) repre- sents the operator defined by row(X i ) := [X 1 ,...,X N ], the operator col(·) represents the column operator, i.e. col(X i ) := row(X i T ) T and the operator diag(X i ) results in a block diagonal matrix whose diagonal elements are X 1 ,...,X N . 1