Model Order Reduction Methods For Bilinear Systems Based On Orthogonal Polynomials

Nowadays,more and more large-scale dynamic systems,such as power systems,control systems,mechanical systems,thermal conversion and fluid flow,are produced in many engineering applications.The equations involved in these large-scale systems are very large in scale or dimension,resulting in large storage capacity and low computational efficiency.Therefore,how to reduce data storage and the amount of computation in large-scale systems has been widely concerned.In consequence,model order reduction(MOR)technology has been gradually formed and developed.The main idea is to replace the large-scale system with a simpler low-order system and retain some important features of the original system,such as stability,passivity and structure.In this way,MOR can not only reduce data storage and improve computational efficiency,but also maintain some special structures and properties of the original system.In this paper,MOR methods for bilinear systems are proposed based on general orthogonal polynomials.On the basis of Legendre orthogonal polynomials,finite-time MOR for bilinear systems is studied.Based on Laguerre orthogonal polynomials,this paper presents a series of structure preserving MOR methods for K-power bilinear systems on infinite time interval.The main research includes the following contents:(1)For bilinear systems,a finite-time MOR method based on Legendre polynomials is presented.The main idea of this approach is to expand the impulse response in the space spanned by Legendre polynomials and calculate the approximate low-rank factors of the finite-time controllability and observability Gramian of bilinear systems by solving the block tridiagonal linear system,then generate reduced order models by using the balanced truncation method.The proposed approach does not need to solve Lyapunov equation directly,which makes it computationally efficient and adaptive.In addition,combined with the dominant subspace projection method,MOR process is modified to alleviate the shortcomings that above method may unexpectedly result in unstable systems although the original one is stable,and two modified algorithms are proposed.Finally,two numerical examples are given to demonstrate the effectiveness of the proposed algorithms.(2)For K-power bilinear systems,structure preserving MOR methods for K-power bilinear systems are proposed based on Laguerre polynomials and equivalent bilinear systems form.The preserving structure here refers to the structural preservation of each subsystem of the original K-power bilinear system.Firstly,the method aims to transform the K-power bilinear system into a general bilinear system and calculate the approximate low-rank factors of the cross Gramian of the bilinear system by Laguerre functions expansion of the matrix exponential function.After that,the matrix partitioning technique is used for low-rank factors,then the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem.Finally,structure preserving reduced order model is obtained by truncating the states corresponding to smaller singular values.At the same time,two modified algorithms to preserve stability are obtained according to research content(1),and the stability is analyzed and verified.Finally,numerical simulation results are given to verify the effectiveness of the proposed algorithms.