State Estimation via a Serial Network with Different Transmission Periods Young Soo Suh, Vinh Hao Nguyen, Young Shick Ro, and Hee Jun Kang Department of Electrical Enginnering, University of Ulsan Namgu, Ulsan, 680-749, Korea e-mail: suh@ieee.org ABSTRACT This paper considers an estimation problem, where sensors are connected through a serial network and sensor data are sent to the estimation board periodically. A rate monotonic scheduling is used for sensor data packets scheduling. Given the bandwidth, the design parameter affecting estimation error is the transmission period of each sensor. A new optimization algorithm to determine transmission period of each sensor is proposed so that the overall estimation error covariance is small in the Kalman filter framework. The computed transmis- sion period is depending on system models and noise covariances. The proposed algorithm is verified through a numerical simulation. I. I NTRODUCTION Recently sensors and actuators in monitoring and control systems are increasingly connected through serial networks instead of traditional cables [1], [2]. Main mo- tivation behind this trend is networked monitoring and control systems are more flexible in physical connection and configuration. In this paper, we consider an estimation problem, where sensor data are transmitted through a serial net- work. Many problems, which do not exist in conven- tional estimation problems, arise due to use of the network: transmission delay, packet loss, and limited bandwidth of the network. Estimation with time delay was considered in [3] and packet loss problems are considered in [4], [5]. To overcome limited network bandwidth, transmission data size reduction using special encoder-decoder was considered in [6], [7]. Although theoretically appealing, its usefulness in real implementation is rather limited as pointed out in [8]. Most control networks employ packet based transmission and large reduction of data size does not result in large reduction of packet size due to large fixed packet overhead. Under limited network bandwidth, reasonable ap- proach is to transmit important sensor data more fre- quently and less important data less frequently. In [8], fast changing output data are given higher priority so that they are transmitted more frequently. In [9], estima- tors were used at each sensor node to reduce network traffic. When the estimate value deviates from the actual output by more than a prespecified tolerance, the actual sensor data are transmitted. In both cases, intelligent sensor nodes with computational capabilities are used and dynamic scheduling methods are employed. In this paper, we consider networked monitoring systems, where function of sensor nodes is limited to periodic transmission of sensor data without compu- tational power. In this situation, the design parameter is transmission period of each sensor and scheduling methods. Transmission period of each sensor is de- termined so that estimation error covariance is small in the Kalman filter framework. Since contribution of each sensor to estimation error covariance is different, transmission period of each sensor is generally different. As the scheduling method, the rate monotonic scheduling method [10], which is known to be an optimal fixed scheduling method, is used. II. PROBLEM FORMULATION Consider the following system ˙ x(t) = Ax(t)+ w(t) y(t) = Cx(t)+ v(t) (1) where x ∈ R n is the state we want to estimate and y ∈ R p is the measurement. Process noise w(t) and measurement noise v(t) are uncorrelated, zero mean white Gaussian random processes satisfying E{w(t)w(s) ′ } = Qδ(t − s) E{v(t)v(s) ′ } = Rδ(t − s) E{w i (t)v j (s)} = 0, 1 ≤ i ≤ n, 1 ≤ j ≤ p where w i and v j are i-th element of w and j -th element of v, respectively. Output y(t) is transmitted through a serial network to the estimator board, where state x(t) is estimated, as in Fig. 1. The following assumptions are made on the network data transmission. 1. Output i (1 ≤ i ≤ p) is transmitted periodically to the estimator board with the period T i . 2. Non-preemptive, priority-based transmission is used.