Optimal Kalman Filtering with Random Sensor Delays, Packet Dropouts and Missing Measurements Maryam Moayedi , Yeng Chai Soh, and Yung Kuan Foo, Member, IEEE Abstract— In this paper an optimal Kalman filter design problem is studied for networked stochastic linear discrete-time systems with random measurement delays, packet dropouts and missing measurements. Any of these three uncertainties in the measurement can occur in the network in the same run. Based on a Markov chain, we develop a unified/combined model to accommodate random delay, packet dropouts and missing measurements. Some simulation examples are presented to show the effectiveness of the proposed approach. I. INTRODUCTION S co the result of the increasing development in mmunication networks, control and state estimation over network has attracted great attention during the past few years (see e.g. [9]). The feedback control systems wherein the control loops are closed through a real-time network are called networked control systems (NCSs) (see e.g. [6]). In a NCS, data typically travel through the communication networks from sensors to the controller and from controller to the actuators. As a direct consequence of the finite bandwidth for data transmission over networks, time-delay is inevitable in networked systems where a common medium is used for data transfers. This delay, either constant, time varying, or random, can degrade the performance of a control system if the design is done without due consideration given to the delay. In many instances it can even destabilize the system. In addition, some packets not only suffer transmission delay but, even worse, can be lost during transmission. This phenomena is known as ‘packet dropout’; see [2], [15] for some further discussions. In practical applications, there may also be a nonzero probability that an observation consists of noise only, i.e. the measurements contain missing observations. The missing observations can arise for a variety of reasons, see [1], [7], [4] and [16] for more detailed discussions. Hence sensor delays, packet dropouts and missing measurements are some of the challenging problems faced by control practitioners in NCS, [2]. The filtering problem for systems with any of these uncertainties has received much attention during the past few years. See [1], [3], [4], [5], [8], [10], [11], [12], [13], [14] for example. M. Moayedi is with Nanyang Technological University, Singapore. (e-mail: mary0008@ntu.edu.sg) Y.C. Soh is with Nanyang Technological University, Singapore. (e-mail: eycsoh@ntu.edu.sg). Y.K. Foo is with LW Electrical and Mechanical Engineering Private Limited, Singapore (e-mail: fooyk@leunwah.com.sg). In most of the literature, the aforementioned uncertainties in data transmission networks are usually assumed to happen separately. Very few works have been reported regarding the filtering problem for NCSs with mixed uncertainties in the measurement transmission network. Recently in [15] the robust estimation for uncertain systems with signal transmission delay and data packet dropout has been considered. However, in their approach, the filter designed is essentially a continuous-time design involving an event-driven zero-order hold (ZOH). In [16], the H ∞ H ∞ filter design problem is studied for a class of networked systems where two kinds of incomplete measurements, namely measurements with random delay and measurements with stochastic missing phenomenon are simultaneously considered. To the best of our knowledge, the filtering problems for NCSs with three simultaneous mixed uncertainties, i.e. random sensor delay, packet dropout and uncertain observation (missing measurement), have not been investigated in the literature. This motivates our present work. In this paper, we consider the case where any of all three types of uncertain observations (sensor delay, packet dropout and missing measurement) may occur in a single run. To achieve this aim, we use a finite-state Markov chain to model the uncertain system whose state is to be estimated. The design of the optimal estimator is then obtained via minimizing the approximate expected estimation error covariance matrix. One advantage of this approach is that it allows us to handle precedence constraint. We also use Markov chain in simulation for the purpose of evaluating the performance of the filters designed. This permits us to impose precedence constraints and hence more realistically models the real situation. For example, if a measurement packet arrives at discrete time , then it could not be arriving again at time because a packet could not be arriving twice. k 1 k + The organization of the paper is as follows. In the next section, we model the complete uncertain system via Markov chain and we present the various state equations used to model the uncertain system with measurement delay, packet dropout and missing measurement. We also discuss how the approach proposed in this paper can be readily adapted to admit multiple-step sensor delays and packet dropouts. In section 3, we present our main result and we discuss how a linear time-invariant filter using the same approach may be found. In section 4, we give several A 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 ThC03.6 978-1-4244-4524-0/09/$25.00 ©2009 AACC 3405