A polynomial approach for observer design in networked control systems with unknown packet dropout rate. I. Pe˜ narrocha, D. Dolz and R. Sanchis Abstract— In this work, the observer design for systems oper- ating over communication networks with previously unknown packet dropout rate (PDR) is addressed. It is assumed that the PDR is time-varying and that it can be estimated online by means of the acknowledgement on new data arrival. The observer gains depend on the estimated PDR and they are designed minimizing the H∞ norm from disturbances and measurement noises to estimation error over all the possible PDRs. The observer gains are rational (including as a particular case a polynomial) functions of the estimated PDR of the desired order. An optimization over polynomials has been carried out in order to find a solution for the proposed filter. I. INTRODUCTION In the last years many processes in industry are controlled or supervised through sensors, actuators and controllers connected to a shared network (wired or wireless). One of the interesting control problems that arise in those scenarios is the state estimation from measurements that are acquired through a network. The main difficulties are the problems of packet dropout, network induced delays, or intermittent partial observations. Most of the proposals in the literature can be classified in two generic groups: Kalman filter based algorithms (e.g. [12], [10], [11]), in which the estimator implements a modified Kalman filter to compute on line the gains of an estimator, and off-line computed gains strategies (e.g. [9], [13], [5]) in which the estimator gains are previously computed and stored. The use of a Kalman filter with irregular observations that follow a Bernoulli distribution was firstly studied in depth in [12], where the conditions for the existence of an estable estimator were addressed, demonstrating the existence of a critical value for the measurements arrival probability to get a bounded filter when dealing with transmission of a packet containing measurements from several sensors. The main drawback is that the online computation of the gains requires a high computer power, and, furthermore, the algorithm does not give as a result a value of the bound of the estimation error. On the other hand, the off-line computed gains approaches lead to a low computer cost algorithm and allow to obtain in advance a bound of the estimation error. Previous works (as [9], [13], [5], [6], [2]) propose a constant gain, or a set of constant gains, that are not a function of the packet dropout rate or the successful transmission probability. When these are not known in advance, or are time varying, the resulting observer is very conservative. I. Pe˜ narrocha, D. Dolz and R. Sanchis are with Department of Industrial System Engineering and Design, Universitat Jaume I of Castell´ o, Spain {ipenarro,ddolz,rsanchis}@uji.es In this work, the design of a rational gain-scheduled observer is addressed for networks with packet dropout whose successful transmission ratio is unknown in advance and time-varying. The implemented observer gain at each instant is a function of an estimation of the packet arrival rate on the observer node, as a difference with other works. This leads to a better estimator performance with a slight increase in the computational cost. The design is addressed assuring stochastic stability and H ∞ performance over the disturbance, noises and time-varying and uncertain packet dropout rate. Then, an LMI optimization problem is derived from an optimization over polynomial constraints that tries to minimize the state estimation error covariance for the overall packet successful transmission rate. The degree of the polynomials of the proposed Lyapunov function and the observer gains is a tuning parameter that can be selected as a compromise between the computational complexity of the optimization problem to be solved, and the achievable performance. In order to overcome the optimization over polynomials, the sum of squares (SOS) approach is used (see [1], [8], [3], [7] and [4] or [8] for a tool that allows to implement these methods). The paper has the following structure: in Section II the system is defined including the characteristics of the network and the proposed state estimation algorithm. In Section III the proposed solution for the polynomial observer design is presented, including the necessary existing results about SOS decomposition. Finally, in Section IV some examples show the validity of the approach. II. PROBLEM STATEMENT A. System description Let us assume a linear time invariant discrete time system defined by equations x k = Ax k-1 + Bu k-1 + B w w k-1 , (1) y k = Cx k + v k , (2) where x ∈ R n is the state, u ∈ R nu is the input, w ∈ R nw is the state disturbance, y ∈ R ny are the measured outputs and v k ∈ R ny the measurement noise. Let us assume that the output measurements are acquired through a network with packet dropout, and let us define the binary variable called availability factor of new data as α k =  0, if y k is not received, 1, if y k is received. (3) 52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy 978-1-4673-5716-6/13/$31.00 ©2013 IEEE 5933