Model Order Reduction Methods Based On Orthogonal Polynomials And Balanced Truncation For Linear Systems

Many problems arising in nature and engineering can be modeled by dynamical systems.With the rapid development of science and technology,many systems in the engineering fields become much larger and more complex.It needs higher dimensional mathematical models to describe such systems.As a result,analyzing such large-scale systems is a difficult task in design,simulation and optimal control.It is necessary to propose a fast and effective algorithm to simplify the complexity and reduce the simulation time.Fortunately,model order reduction(MOR)is an effective way to solve this problem.It aims to change a large-scale system into a lower dimensional one that preserves the main properties of the original system as much as possible.It can accelerate the simulation and is easier to analyze by reducing the complexity and computational cost.Model order reduction methods are a class of efficient numerical methods to handle large-scale complex systems.In the literature,most traditional MOR methods often only consider the input-output effect but ignore the impact of the initial conditions or make the restrictive assumption that the initial date are zero.However,often these methods may destroy the initial conditions information.Motivated by the above discussion,we discuss the time-domain MOR methods based on general orthogonal polynomials for linear systems with inhomogeneous initial conditions.A time domain model order reduction method based on Legendre orthogonal polynomials is presented.The basic procedure is to use the expansion coefficient matrix under the orthogonal polynomial space satisfied a simple recurrence formula to generate a projection matrix.Then,the reduced model can match a desired number of the expansion coefficients of the original system.Two practical examples are simulated to illustrate the effectiveness of the proposed method.Present a time domain MOR method by general orthogonal polynomials.The basic procedure is to use the expansion coefficient matrix in the orthogonal polynomial space satisfying a simple recurrence formula to generate a projection matrix,which is produced by a modified Arnoldi algorithm.Then,the resulting reduced model matches a desired number of the expansion coefficients of the original system.The approximate error estimate of the reduced model is given.Since the initial conditions are well represented by the subspaces constructed by our algorithm,it can well deal with those systems with inhomogeneous initial conditions.Two benchmark examples in real applications are simulated to illustrate the effectiveness of the proposed method.