Optimal Linear Filtering for Networked Control Systems With Random Matrices, Correlated Noises, and Packet Dropouts

Abstract

In this paper, the optimal linear filtering problem for linear discrete-time stochastic systems with random matrices, correlated noises and packet dropouts is studied where the random matrices are real and appear both in the the state and measurement equations. Using an equivalent transformation for random matrices and some results presented in this paper, an optimal linear filter for the system under consideration is developed. The developed optimal filter has a recursive pattern, and its computational complexity does not change with time. Number examples with simulation results are given to examine the performance of the developed optimal linear filter. INDEX TERMS Optimal linear filtering, discrete-time, networked control system, random matrix, packet dropout.

A Two-Phase Distributed Filtering Algorithm for Networked Uncertain Systems with Fading Measurements under Deception Attacks

“…For the simulation, let us consider that the sensor network has the same topological structure as that in [25], represented by a digraph G = (V, E , A), with set of nodes V = {1, 2, 3, 4}, set of edges E = { (1, 1), (1,2), (1,3), (2,2), (2,3), (2,4), (3,1), (3,3), (3,4), (4,1), (4,2), (4, 4) }, and binary adjacency matrix A = (a ij ) m×m , such that a ij = 1 if and only if (i, j) ∈ E and a ij = 0 otherwise.…”

“…Considering that θ = 0.75 and choosing the attack probabilities as λ (1) = λ (2) = 0.6 and λ (3) = λ (4) = 0.7, Figure 1 shows, for the first state component, the error variances of the local filter (obtained using only the measurements from the ith sensor itself) and those of the proposed intermediate and distributed filters at every sensor node i. From Figure 1, it is observed, on the one hand, that the error variance corresponding to the intermediate filter is significantly less than that of the local filter and, on the other, that the distributed filter outperforms all the intermediate filters in its neighborhood.…”

“…(I) Impact of the missing measurement phenomenon: Assuming, as in Figure 1, that the probabilities of attacks are λ (1) = λ (2) = 0.6 and λ (3) = λ (4) = 0.7, the effect of the missing measurement phenomenon is studied by analyzing how the distributed filtering error variances are influenced by the probability θ that the signal is present in the measured outputs of Sensors 3 and 4. First, Figure 3a shows the distributed error variances for the values θ = 0.1 to 0.9, at Sensor Node 4.…”

“…Probability θ Finally, we present a comparative analysis of the proposed distributed filter and the distributed filter [25] for networked systems with random parameter matrices and correlated noises. Considering the same probabilities (θ = 0.75, λ (1) = λ (2) = 0.6 and λ (3) = λ (4) = 0.7) as in Figure 1 and using one thousand independent simulations, both filtering estimates are compared on the basis of the empirical mean squared error values, which are calculated as: is the jth component of the filter calculated in the ith sensor node, at the sampling time k, in the sth simulation run. The results are displayed in Figure 5, which shows that, for both the first and second components, the proposed distributed filter outperforms the filter in [25].…”

“…It is well known that networked systems may often undergo random imperfections and disturbances (for example, missing and fading measurements, uncertainties of multiplicative noises, random delays, packet dropouts, and so on), which, if not addressed properly, are likely to impair the performance of the estimators. For this reason, considerable research efforts have been focused on the analysis of mathematical models involving these network-induced random phenomena and the design of estimation algorithms that do not neglect their effects (see, e.g., [1][2][3][4] and the references therein).…”