Modelling and Simulation of Large-Scale Multiparameter Computer Aided Dynamical Control Systems

Abstract

Modelling and simulation of large-scale Multidimensional multiparameter dynamical systems require the use of large-scale computers to generate feed-back control laws, especially when model uncertainties exist. Vow robust control theory is concerned with the problem of analyzing and synthesizing control systems that provide and acceptable level of performance where many model parameters or uncertainties may exist, since mathematical models of physical systems are usually never exact due to the presence of such parameters.The need to be able to design robust feedback control laws is very important in such systems. Usually a physical model will have significant structural information about the interconnection of components and subsytems but leas information concerning their integrated system performance. Hence, many variations of parameters must be carried out on supercomputers in order to determine the more significant and sensitive parameters which must be adjusted very rapidly to accomplish a desired level of performance.Dynamical systems of the form: Eθxt, θ=Aθxt, θ+Bθut, θ;xt, θ=dxt, θdtyt, θ=Cθxt, θ+Dθut, θ;t ≥0 are considered.The elements of matrices E(θ), A(θ). B(θ), C(θ), D(θ) belong to the ring of polynomials R[θ] where θ = (θ1, θ2, θ3,...,θq) is a multiparameter and the coefficients of the polynomials belonging to C[θ] have elements belonging to the field C of complez numbers, or the elements of such multiparameter matrices may be the form f(θ) = a(θ)∣b(θ) where the polynomials a(θ). b(θ). & c(θ). for b(θ) ≠ 0. E(θ) may be a singular matrix for possible parameter values of θ. The parameter θ may also be holomorphic function of a single complex variable z for z belonging to simply-connected bounded regions in the z-plane. The basic question of stabilization, controllability, observability, etc., in the presence of changes in subsystems as regards the overall dynamical systems needs to be treated in response to changing parameters in subsystems.Fast numerical methods requiring parallel proceasing are necessary to compute adjustments as time t changes. Transfer function matrices, controllability matrices, observability matrices, feedback control laws need to be recomputed as parameters change. Such matrices may be multiparameter matrices and may allow for improvement of control laws such as in cases where E(θ) may become singular matrix and the dynamical systems require considerable fast changes in feedback control laws.Use is made of recent results o£ J. Jones. Jr. concerning generalized inverses of such multiparameter matrices to aid in computer aided changes to carry out modelling, simulation and analysis of such dynamical systems.