سنتز اتوماتیک از مدل های نامشخص برای شبیه سازی مدار خطی : روش تئوری آشوب چند جمله ای

A generalized and automated process for the evaluation of system uncertainty using computer simulation is presented. Wiener–Askey polynomial chaos and generalized polynomial chaos expansions along with Galerkin projections, are used to project a resistive companion system representation onto a stochastic space. Modifications to the resistive companion modeling method that allow for individual models to be produced independently from one another are presented. The results of the polynomial chaos system simulation are compared to Monte Carlo simulation results from PSPICE and C++. The comparison of the simulation results from the various methods demonstrates that polynomial chaos circuit simulation is accurate and advantageous. The algorithms and processes presented in this paper are the basis for the creation of a computer-aided design (CAD) simulator for linear networks containing uncertain parameters.

Representation of uncertainty, when applied to circuit simulation, can be a powerful tool for producing and designing robust systems. Various methods have been proposed which can be used to quantify and propagate uncertainty. There are interesting examples of research in which Artificial Intelligence techniques are applied to provide a simplified or qualitative definition of the physics of systems when uncertainty is involved. Reviews of this topic can be found in [1], [2], [3], [4], [5] and [6]. Similar issues have been considered in the electrical engineering field to overcome specific design problems caused by uncertainty [7] and [8]. The numerical evaluation of the effects of uncertainty is traditionally achieved by using the Monte Carlo method, which is widely accepted as an “exact method” for determining uncertainty [9]. The Monte Carlo method can give the entire probability density function (PDF) of any system variable through reiterations of the system simulation. Another method used for the evaluation of uncertainty is “true worst-case circuit tolerance” analysis, which gives only the results concerning the upper and lower statistical bounds of the circuit response to uncertainty [10]. The polynomial chaos approach to the evaluation of uncertainty also yields the full PDF of the system’s variables using only one execution of the simulation. A method to automate the process of generating a circuit’s polynomial chaos representation for use in simulation is presented in the paper. It has been common practice in engineering to analyze systems based on deterministic mathematical models with precisely defined input data. However, since such ideal situations are rarely encountered in practice, the need to address uncertainties is now clearly recognized and there has been a growing interest in the application of probabilistic and other methods. Among the existing methods for uncertainty analysis, Ghanem and Spanos pioneered a polynomial chaos expansion method [11] and [12]. This method is based on the homogeneous chaos theory of Wiener [13], which uses a spectral expansion of random variables. The use of the term chaos in this context represents uncertainty and should not be confused with the chaos theory used frequently in theoretical physics. The homogeneous chaos expansion employs Hermite orthogonal polynomials in terms of Gaussian random variables. Cameron and Martin have proved that this expansion converges to any L2 functional in random space in the L2 sense [14]. Combined with Karhunen–Loeve decomposition [15] of the inputs, polynomial chaos results in computationally tractable algorithms for large engineering systems. Recently, a more general framework, called generalized polynomial chaos or Askey chaos, has been proposed [16]. This expansion technique utilizes more orthogonal polynomials from the Askey scheme than the original homogeneous chaos theory [17] and can represent general non-Gaussian processes. Applications to ODE, PDE, Navier–Stokes equations and flow-structure interactions have been reported and convergence has been demonstrated for model problems [18] and [19]. In recent years, polynomial chaos has been applied to measurement uncertainty [20] and [21], entropy multivariate analysis [22], control design [23], [24] and [25], design of a two-planar manipulator [26] and polynomial chaos based observers for use in control theory [27]. Ref. [28] presents an overview of applications of polynomial chaos theory to electrical engineering. Among other possible applications, circuit simulation by means of polynomial chaos expansion is introduced. In that paper the problem is solved for a specific topology without providing a generalized algorithm for the solution of uncertain systems based on a SPICE-like netlist and information about the uncertainty of some parameters. The major contribution of this present work is in effect the formalization of the automatic solution process using resistive companion method [29]. In particular, a systematic algorithm to automatically define the conductance matrix of an uncertain system is presented. This theory yields the definition of a new approach to nodal analysis where the topology of a network is mapped to a multilayer topology where each layer represents one level of polynomial expansion. This approach helps to pave the way for a new generation of uncertainty based CAD tools.

The examples in this paper demonstrate that polynomial chaos theory can be used to generate accurate simulation results for uncertain systems. The primary technical innovation within this paper is the derivation of a formal modeling theory based on polynomial chaos theory. This new modeling theory is an enabler for new CAD simulators focused on the evaluation and propagation of uncertainty. This paper demonstrates multiple advantages that this new modeling theory has over other uncertainty evaluation methods. The first advantage is that the polynomial chaos theory can calculate the entire PDF of each variable during only one execution of the simulation. This allows statistical information about the circuit to be available after each time step of the simulation. Monte Carlo analysis requires all of the simulations to be performed in parallel in order to have the full statistics after each time step. If the Monte Carlo iterations are not executed in parallel then it must wait for each entire simulation to be completed before the statistics are available. Secondly, the information required for the reconstruction of the PDF can be compactly stored by polynomial chaos in the form of the coefficients of the multi-variable polynomial basis. Monte Carlo analysis, on the other hand, requires the storage of all of the data resulting from each Monte Carlo iteration. A third advantage is that polynomial chaos automatically allows for irregular shapes of PDFs to be used for inputs, making it easier to represent complicated uncertainties. This paper introduced and validated a new modeling theory based on uncertainty evaluation using polynomial chaos theory. While this paper has focused mainly upon linear electric circuit analysis capabilities of the resistive companion equation, the resistive companion method applies to numerous different disciplinary fields of study. The polynomial chaos expanded resistive companion equation can be used to simulate multi-disciplinary systems just as easily as it simulates electric circuits. This method is currently being incorporated into the Virtual Test Bed simulation environment (VTB) [33] at the University of South Carolina.