A simple closed-form solution to the discrete Lyapunov equation (DLE) is established for certain families of matrices. This solution is expressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all 2 × 2, 3 × 3 and certain families of matrices. For general matrices, the proposed ED based closed-form solution can be used as an efficient numerical solution when the ED can be computed. The result is then extended to give closed-form solutions for a generalization of the DLE, called the Stein equation. The proposed explicit solution's complexity is of the same order as iterative solutions and significantly smaller than known closed-form solutions. These solutions may prove convenient for analysis and synthesis problems related to these equations due to their compact form.
This letter considers the problem of estimating the state of a scalar dynamical system over a wireless channel that is lossy, capacity, and power limited. The limited power assumption, which is valid for most real problems, links the number of quantisation bits and packet error rate. As the number of quantisation bits increases, the quantisation noise decreases but the packet loss rate increases. It is shown that the estimation error variance (EEV) is minimised for a range of quantisation bits. Further, it is shown that operating beyond the optimal range sharply increases the EEV. Simulation results corroborating the analytical results are also presented.
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