Evaluation of variances in hybrid MTL systems with stochastic parameters via SDAE approach

The application of stochastic differential equation (SDE) approach find its usage in various areas of the engineering when stochastic changes in physical systems are to be considered [1]. In the electrical engineering this theory can very be useful at the solution of systems with distributed parameters, particulary containing multiconductor transmission lines (MTL), which are often exploited in high-speed circuits for the data transmission [2]. In the paper, an attention will be paid to hybrid (lumped-distributed) systems with the MTLs as their distributed parts, whereas parameters of the system can vary randomly. In case of the MTL itself, with rather simple terminating elements, just the SDE theory would be applicable when taking into account some proper numerical technique. Usually, evaluation of the dispersion of responses of the system is needed which can be resolved via confidence intervals determination. To do it, either the solution of the SDE and subsequent statistical processing is applied or the direct solution of variances via Lyapunov-like equations can be performed. In hybrid systems, however, due to lumped-parameter parts included, a non-differential (algebraic) part is generally present in respective mathematical model. Therefore, a stochastic differential-algebraic equation (SDAE) approach has to be considered. The SDAE solution is more complicated in general while a proper numerical technique has to be chosen [3]. Utilization of approach in [3, 4] leads to necessity to repeatedly solve the SDAEs and to process the results statistically. In this paper a way of direct solution of variances and confidence intervals will be elaborated to be able to avoid the SDAE solution itself, namely by formulating and numerically solving Lyapunov-like equations adapted for hybrid MTL systems. All the computations will be done by using the Matlab language, with some examples provided.