NSU CTRG 2019-2020
Model Reduction of Second-Order Descriptor Systems over Finite-Frequency Interval (CTRG-19/SEPS/05)
- Funded by: North South University
- Principle Investigator: Dr. Mohammad Monir Uddin (monir.uddin@northsouth.edu)
- Co-Principle Investigator: Dr. Xin Du (duxin@shu.edu.cn)
- Duaration: 1 years (Started from October 2019)
Project Overview
Before implementing new ideas or decisions in different disciplines of science, engineering, and technology, an experiment is required. The classical approach of this experiment would require a laboratory with a lot of new equipment, which is an expensive method to demonstrate a concept. The modern approach, rather less expensive and often easier to apply than experiments, to explore scientific ideas to convince others of their validity is through computer simulation. In simulation, one needs to convert a physical model into a mathematical model. Often also in reallife applications the mathematical models are represented by linear time-invariant (LTI) continuous-time systems. In many cases, these systems are in second-order form i.e., the system are represented by set of second order ordinary differentiaol equations. Sometimes they are subject to additional algebraic constraints, leading to differential-algebraic equations (DAEs) or descriptor systems. The mathematical models are generated in many different ways. In many applications, the systems are obtained by finite element (FEM) or finite difference (FDM) discretization. In order to model a system accurately, a sufficient number of grid points must be generated because many geometrical details must be resolved. Sometimes physical systems consist of several bodies and each body is composed of a large number of disparate devices. Therefore, the mathematical models become more detailed and different coupling effects must be included. In either case, the resulting systems are typically very large and sparse. Moreover, often they might be well-structured. A large-scale system leads to additional memory requirements and enormous computational efforts. They also prevent frequent simulations which is often required in many applications. Sometimes, the generated systems are too large to store due to the restriction of computer memory. To circumvent these complexities reducing the size of the systems is unavoidable. The method to reduce a higher dimensional to a lower one is called model order reduction (MOR).
The fundamental aim of MOR is to replace the high dimensional dynamical systems by substantially lower dimensional systems, while the dynamics of the original and reduced systems should be approximated to the largest possible extent. In some cases, some important features such as stability, passivity, definiteness, symmetry and so forth of the original system must be preserved in the reduced systems. There are many techniques for MOR, namely optimal Hankel norm approximation, singular perturbation approximation, dominant subspaces projection, balanced truncation and Iterative rational Krylov Algorithms etc.. Among all the aforementioned methods, currently balanced truncation (BT) is the most commonly used techniques for the model reduction of large-scale dynamical systems. In general the BT method preserves the stability of the original system i.e., if the original system is stable then the reduced model is also stable. Moreover, the method has a global error bound which can be defined prior by an user. That mens the error of the reduced order model can be bounded by providing a tolerance. This prominent method is updated by minimizing the error of the reduced system over a certain frequency interval. In the literature this is known as frequency-limited balanced truncation (FLBT). Currently frequency-limited balanced truncation has taken lot of attentions since in engineering and technology in some applications it is important to minimize the error of the system over a certain frequency interval. Moreover frequency limited balancing based reduced model has meaningful physical interpretation and provides more accurate approximation. The FLBT method is well established for generalized state space model. Recently it is also generalized for second-order standard and first-order descriptor systems. Until now there is no investigation of such model reduction method for second order descriptor systems. This research is mainly devoted to fill this gap.
This project focuses on developing mathematical algorithm for the model reduction of large-scale second-order descriptor systems based on frequency-limited balanced truncation. The efficiency and capability of the proposed method will be checked to apply this to a very large-scale (around 300 0000 dimensional) real world data of a structural FEM model of a micro-mechanical piezo-actuators based adaptive spindle support (ASS). The data of this model is already in our hand which is provided by the Fraunhofer Institute for machine tools and forming technology in Dresden (Germany).
Team
Project Outcome
In this project we proposed to develop algorithms for the frequency limited model reduction of large-scale sparse descriptor systems. The proposed research work has been successfully done theoretically and some numerical experiments also have been carried out. The results have been shown in the attached papers. Some numerical experiments are being carried out at a automation lab of Shanghai University, China. The project was supposed to be ended by September 2020. Due to pandemic problem it was delayed by three months. In future work we will generalize the ideas obtained here for limited time interval case.
Sumulation & Results
Publications
Time Restricted Balanced Truncation for Index-I Descriptor Systems with Non-homogeneous Initial Condition
Proceedings of International Joint Conference on Advances in Computational Intelligence
In this paper, we discuss on the time restricted balanced truncation of index-I descriptor system with non-homogeneous initial condition in order to find better low-rank approximation of large-scale systems on nominated time interval.BOOK CHAPTERS
Time Restricted Balanced Truncation for Index-I Descriptor Systems with Non-homogeneous Initial Condition
About The Publication
In this paper, we discuss on the time restricted balanced truncation of index-I descriptor system with non-homogeneous initial condition in order to find better low-rank approximation of large-scale systems on nominated time interval. At first, we create a basis matrix spanning the null space with a view to extend the input matrix what makes the whole procedure working efficiently with non-homogeneous initial condition. Next, the low-rank Gramian factors are computed by solving the time restricted Lyapunov equations of index-I system using rational Krylov subspace method (RKSM) what is followed by reducing the time restricted large sparse system through balanced truncation (BT) technique. We select various data models of index-I system for numerical experiments to demonstrate the efficiency of the proposed techniques on restricted time interval.
Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications
Numerical Algebra, Control & Optimization
This paper discusses model order reduction of large sparse second-order index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA).JOURNAL
Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications
About The Publication
This paper discusses model order reduction of large sparse second-order index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA). In general, such DAEs arise in constraint mechanics, multibody dynamics, mechatronics and many other branches of sciences and technologies. By deflecting the algebraic equations the second-order index-3 system can be altered into an equivalent standard second-order system. This can be done by projecting the system onto the null space of the constraint matrix. However, creating the projector is computationally expensive and it yields huge bottleneck during the implementation. This paper shows how to find a reduce order model without projecting the system onto the null space of the constraint matrix explicitly. To show the efficiency of the theoretical works we apply them to several data of second-order index-3 models and experimental resultants are discussed in the paper.
Frequency Limited H2 Optimal Model Reduction of Large-Scale Sparse Dynamical Systems
arXiv
We mainly consider the frequency limited H2 optimal model order reduction of large-scale sparse generalized systems. For this purpose we need to solve two Sylvester equations. This paper proposes efficient algorithm to solve them efficiently. The ideas are also generalized to index-1 descriptor systemsJOURNAL
Frequency Limited H2 Optimal Model Reduction of Large-Scale Sparse Dynamical Systems
About The Publication
We mainly consider the frequency limited H2 optimal model order reduction of large-scale sparse generalized systems. For this purpose we need to solve two Sylvester equations. This paper proposes efficient algorithm to solve them efficiently. The ideas are also generalized to index-1 descriptor systems. Numerical experiments are carried out using Python Programming Language and the results are presented to demonstrate the approximation accuracy and computational efficiency of the proposed techniques.
On the Frequency Limited H2 Optimal Model order Reduction of Large-Scale Dynamical Systems
arXiv
We consider the problem of frequency limited optimal model order reduction for large-scale sparse linear systems. A set of first-order optimality conditions are derived for the frequency limited model order reduction problem.SUBMITTED
On the Frequency Limited H2 Optimal Model order Reduction of Large-Scale Dynamical Systems
About The Publication
We consider the problem of frequency limited optimal model order reduction for large-scale sparse linear systems. A set of first-order optimality conditions are derived for the frequency limited model order reduction problem. These conditions involve the solution of two frequency limited Sylvester equations that are known to be computationally complex. We discuss a framework for solving these matrix equations efficiently. The idea is also extended to the frequency limited optimal model order reduction of index-1 descriptor systems. Numerical experiments are carried out using Python programming language and the results are presented to demonstrate the approximation accuracy and computational efficiency of the proposed technique.