Automatica 49 (2013) 360–369 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Identification of ARMA models using intermittent and quantized output observations ✩ Damián Marelli a,1 , Keyou You b , Minyue Fu a,c a School of Electrical Engineering and Computer Science, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia b Department of Automation, Tsinghua University, Beijing, 100084, PR China c Department of Control Science and Engineering, Zhejiang University, 388 Yuhangtang Road, Hangzhou, Zhejiang Province, 310058, PR China article info Article history: Received 11 April 2011 Received in revised form 28 May 2012 Accepted 4 September 2012 Available online 8 December 2012 Keywords: Identification methods Network-based computing systems ARMA model Finite-level quantization Packet dropout abstract This paper studies system identification of ARMA models whose outputs are subject to finite-level quantization and random packet dropouts. Using the maximum likelihood criterion, we propose a recursive identification algorithm, which we show to be strongly consistent and asymptotically normal. We also propose a simple adaptive quantization scheme, which asymptotically achieves the minimum parameter estimation error covariance. The joint effect of finite-level quantization and random packet dropouts on identification accuracy are exactly quantified. The theoretical results are verified by simulations. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction System identification of plants with quantized observations is significant in understanding the modeling capacity for sys- tems with limited sensor information, and the trade off between communication resources and identification performance (Wang, Zhang, & Yin, 2003). This work is concerned with the identification of autoregressive moving average (ARMA) models whose quan- tized outputs must be communicated through a digital noisy chan- nel. A motivating example is given by a sensor and an estimator communicating over wireless channels with limited resources in terms of bandwidth and transmission power. By modeling the packet dropout process as an independent and identically dis- tributed (i.i.d.) Bernoulli process, this paper aims to quantify the joint effect of finite-level quantization and packet dropouts on the identification accuracy of ARMA models. The key difference of quantized identification from the classical identification problem ✩ The material in this paper was partially presented at the 16th IFAC Symposium on System Identification (SYSID 2012), July 11–13, 2012, Brussels, Belgium. This paper was recommended for publication in revised form by Associate Editor Johan Schoukens under the direction of Editor Torsten Söderström. E-mail addresses: Damian.Marelli@newcastle.edu.au (D. Marelli), youky@tsinghua.edu.cn (K. You), Minyue.Fu@newcastle.edu.au (M. Fu). 1 Tel.: +61 2 4921 7845; fax: +61 2 49216993. is that the estimator is no longer able to access the original analog amplitude (unquantized) observations. Especially under aggres- sive quantization, the discrete-valued observations supply limited information on system outputs, and hence introduce new chal- lenges in system modeling, identification and control. In addition, channel errors, e.g., packet dropouts, further induce information loss that influences identification accuracy. Recently, research on quantized identification/estimation con- stitutes a vast body of literature, see e.g., Wang, Yin, Zhang, and Zhao (2010); Xiao, Ribeiro, Luo, and Giannakis (2006) and refer- ences therein. In Xiao et al. (2006) and the references therein, various quantized estimation algorithms are developed in the con- text of wireless sensor networks. In Wang et al. (2010), a compre- hensive treatment on quantized system identification is presented for single-input-single-output linear discrete time-invariant sta- ble systems. Based on periodic inputs, they study the computa- tional complexity and the impact of disturbances and unmodeled dynamics on the identification accuracy. In the same spirit, various models such as rational models, Wiener systems and Hammerstein systems have been studied as well. Although their identification algorithms are shown to be optimal in the sense of asymptoti- cally achieving the Cramer–Rao lower bound (Wang et al., 2010), the assumption on periodic inputs makes the identification algo- rithm inappropriate for tracking control applications. Moreover, input design is of essential importance in system identification to 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.11.020