Application of Stochastic Differential-Algebraic Equations in Hybrid MTL Systems Analysis

The paper deals with the application of stochastic differential-algebraic equations (SDAE) in the field of the time-domain simulation of hybrid (lumped/distributed) systems with randomly varying parameters. A core of the method lies on the theory of stochastic differential equations (SDE) considering the system responses as stochastic processes. However, due to a hybrid nature of the system, namely its lumped parameter part, non-differential (algebraic) parts arise generally in the solution. Herein, multiconductor transmission lines (MTL) play a role of the distributed-parameter parts of the hybrid system. The MTL model is designed as a cascade connection of generalized RLCG T-networks, while the state-variable method is applied for its description. The MTL boundary conditions are incorporated through a modified nodal analysis (MNA) to cover arbitrarily complex circuits. System responses are formed by the sets of stochastic trajectories completed by corresponding sample means and respective confidence intervals. To get the results a weak stochastic backward Euler scheme is used, consistent with the Ito stochastic calculus. All the computer simulations have been performed in the Matlab language environment. DOI: http://dx.doi.org/10.5755/j01.eee.20.5.7098


I. INTRODUCTION
Stochastic differential equations (SDE) approach finds its place in many fields of the engineering when stochastic changes in physical systems should be taken into account [1], [2].In the electrical engineering it can cover a number of random processes arising in electrical circuits [3]- [5].Interesting possibility is to apply this approach for systems with distributed parameters, namely single or multiconductor transmission lines (MTL), including those implemented on semiconductor substrates, which are exploited in high-speed mixed (analog/digital) circuits to transmit data [6]- [8].In the paper, an attention is paid to hybrid (lumped/distributed) systems with multiconductor transmission lines (MTL) as their distributed-parameter parts, whereas MTLs' primary parameters can vary randomly.Due to the lumped-parameter parts of the system, however, a non-differential (algebraic) part in its description is generally present.That is why Manuscript received November 27, 2014; accepted February 16, 2014.This work was performed in laboratories supported by the SIX project CZ.1.05/2.1.00/03.0072 the operational program Research and Development for Innovation, and supported by the project no.FEKT-S-14-2281.stochastic differential-algebraic equations (SDAE) have to be utilized in this case.Some attention has been paid to basic RLCG networks, [9], [10], which can be used as building blocks for more complex physical models, as transmission lines (TL) [6], [7].An RLCG model of a simple TL is shown in [11], [12], where stochastic responses caused by random excitation and the model's parameters have been analysed.The multiconductor transmission lines (MTL) represented by generalized RLCG-based models were simulated in [13], with boundary conditions folded via generalized Thévenin equivalents of terminal circuits, and in [14], where modified nodal analysis (MNA) was used in this respect, enabling to consider an arbitrarily complex lumped-parameter part.A hybrid system with several MTLs in Fig. 1 will be used for further considerations, while the MTLs' models in Fig. 2 are employed.Such a system has also been simulated in [15], however, a numerical method applied in this paper differs, namely it is of a weak convergence utilizing two-point distributed random variables as an initial stochastic process, which is more convenient if sample moments are needed in the resultant stochastic process [2].In the paper, some MTLs per-unit-length (p.u.l.) parameters are regarded as randomly varying, which is marked by * in Fig. 2.

II. HYBRID SYSTEM BLOCK DIAGRAM
We will consider a hybrid system in Fig. 1 composed of a lumped parameter section and a number of MTLs, namely the MTLr, r = 1,2,...,P.
Based on the concept in [12] the lumped-parameter section is described by the MNA method    where CN and GN are N × N matrices with entries defined by lumped reactive and resistive components, respectively, vN(t) is an N × 1 vector of node voltages appended by currents of independent voltage sources and inductors, iN(t) is an N × 1 vector of the exciting source waveforms, iL (r) (t) and iR (r) (t) are nr × 1 vectors of currents entering the r-th MTLr from its left and right side, respectively, with nr as the number of its active wires, SL (r) and SR (r) are N × nr selector matrices, with entries sij  {0,1}, mapping current vectors into the node space of the hybrid system, respectively.These currents satisfy the MTLs boundary conditions, and are bound with respective voltages through vN(t) vector by: where r = 1,2,...,P, where mr is the number of sections of the r-th MTL model, and T denotes a transposition.
A conversion of DAE ( 17) into its stochastic counterpart consists in multiplying it by dt and then replacement of all the products ξij,k (r) (t)dt by dWij,k (r) (t), with Wij,k (r) (t) denoting Wiener processes [1].By this we get a linear stochastic DAE (SDAE) with multiplicative noises where δBW(t) arises from (18), after making the above stated operations and introducing respective Wiener processes.It is common to mark a stochastic solution by a capital letter, Z(t), to distinguish it from a deterministic solution, z(t).

IV. SDAE NUMERICAL SOLUTION
As is obvious from ( 15) the matrix A in ( 16), (17), and thus in (29), is singular in general due to a singularity of CN matrix in (1), which leads to discussed DAEs and SDAEs.There are various approaches for the solution of the SDAEs, see e.g.[4], [5], [16], depending mainly on their index.For example, it is first possible to decouple the SDAE into its differential and algebraic parts by using a canonical projectors method [5].The former can be regarded as the SDE describing Itô stochastic processes, the other as its algebraic constraints.However, this procedure is not recommended for the numerical solution due to possible numerical instabilities [5].Instead, in practice, common oneor multi-step backward differentiation formulae are usually applied for the index-1 SDAEs [16].
For numerical simulations we have to generate the random increments of the Wiener processes as independent Gauss random variables, with means E[ΔW (l) ] = 0 and E[(ΔW (l) ) 2 ] = h.Then, a stochastic backward Euler scheme can be written as which is consistent with the Itô calculus.The matrix A + Bh is now regular as CN + GN is now a regular matrix.Our problem, however, does not require a good pathwise approximation but only an approximation of the expectation μ = E[Z] and the standard deviation σ = √E[(Z-μ) 2 ] of the SDAE solutions.We can use the weak version of (30), where we replace the increments of the Wiener processes ΔW (l) by simplier two-point distributed random variables , where P( A numerical efficiency of the weak scheme is better than of the strong one.That is important in our problem of MTL systems, where we have a large system with many independent Wiener processes.
The stochastic responses at the MTL2 output, vout(t), including excitation voltage vin(t), and the MTL3 output, i2(t), are shown in Fig. 4. Sample statistics were stated through 100 realizations, 99 % confidence intervals (dash red) for mean values (solid blue), and individual samples (dash orange, for vicinities circa ±2.6 σ), were obtained via a student-t distribution [2].More detailed information about the statistical estimates is shown e.g. in [2], [9], [17].The sample means were also verified via deterministic solutions.

VI. CONCLUSIONS
The paper has dealt with a method for the simulation of stochastic responses in hybrid systems containing MTLs with randomly varying parameters based on SDEs theory.The method was developed for fluctuating series resistances and shunt conductances of respective MTLs' models.The work follows concepts in [14], [15], when by using the MNA, the MTLs can be integrated into arbitrarily complex hybrid systems.By this, however, a system of stochastic DAEs is generally obtained which is usually more difficult to solve.By contrast to the previous works, for numerical simulations we used a weak backward Euler scheme that is more effective to solve sample moments.Besides, the higher order stochastic numerical schemes, we plan to use in future, involve multiple Itô integrals of higher multiplicity which are difficult to generate.In weak schemes these multiple integrals can be replaced by simpler random variables and we can obtain quite efficient higher order weak schemes.
All simulations were performed in the Matlab language utilizing techniques of sparse matrices processings.The above stated approach could be an interesting alternative to other stochastic or probabilistic methods, like e.g.[19]- [21] often used to study random changes in interconnects of highspeed electronic systems.Our work is focused on improving the efficiency of the numerical simulations in this field.