Numerical methods for model reduction of periodic dynamical systems: Review and applications

In the second-half of the past century the expeditious development of systems and control theory together with the achievements of digital control and signal processing have set the stage for a renewed interest in the study of periodic dynamical systems, specially in aerospace realm, control of industrial processes, mechanical systems, modeling of periodic time-varying filters and networks, circuit simulation, and multirate sampleddata systems, etc. These complicated systems are composed of large numbers of separate devices and they are described by very large mathematical models consisting of more and more mathematical systems with very large dimensions. Simulations of such systems can be unacceptably expensive and time-consuming due to limited computer memory and CPU consumption. The idea of model reduction is that the large models should be replaced by smaller models which are amenable to fast and efficient simulation and which still capture the devices’ inputoutput behavior to an accepted accuracy. In this paper we review the different approaches for model reduction of time varying systems, and depict the numerical results showing the advantages and disadvantages of these approaches.

[1]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[2]  Peter Benner,et al.  Model Reduction of Periodic Descriptor Systems Using Balanced Truncation , 2011 .

[3]  J. Phillips,et al.  Model reduction of time-varying linear systems using approximate multipoint Krylov-subspace projectors , 1998, 1998 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers (IEEE Cat. No.98CB36287).

[4]  Wen-Wei Lin,et al.  Projected Generalized Discrete-Time Periodic Lyapunov Equations and Balanced Realization of Periodic Descriptor Systems , 2007, SIAM J. Matrix Anal. Appl..

[5]  Daniel Kressner.  An Efficient and Reliable Implementation of the Periodic QZ Algorithm , 2001 .

[6]  Valeria Simoncini,et al.  Krylov subspace methods for projected Lyapunov equations , 2012 .

[7]  Geir E. Dullerud,et al.  Model reduction of periodic systems: a lifting approach , 2005, Autom..

[8]  A. Varga.  Balanced truncation model reduction of periodic systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[9]  Jaijeet Roychowdhury,et al.  Reduced-order modeling of time-varying systems , 1999 .

[10]  J. Roychowdhury,et al.  Reduced-order modelling of linear time-varying systems , 1998, 1998 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers (IEEE Cat. No.98CB36287).

[11]  Mohammad-Sahadet Hossain,et al.  Numerical Methods for Model Reduction of Time-Varying Descriptor Systems , 2011 .

[12]  Mohammad-Sahadet Hossain,et al.  Projection-based model reduction for time-varying descriptor systems: New results , 2016 .

[13]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[14]  Youcef Saad,et al.  Overview of Krylov subspace methods with applications to control problems , 1989 .

[15]  A. Varga.  Computation of generalized inverses of periodic systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[16]  Michel S. Nakhla,et al.  Efficient model reduction of linear time-varying systems via compressed transient system function , 2002, Proceedings 2002 Design, Automation and Test in Europe Conference and Exhibition.

[17]  Mohammad-Sahadet Hossain,et al.  SVD-Krylov based model reduction for time-varying periodic descriptor systems , 2015, 2015 International Conference on Electrical Engineering and Information Communication Technology (ICEEICT).

[18]  Serkan Gugercin,et al.  An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems , 2008 .

[19]  Peter Benner,et al.  Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems , 2014, Numerical Algorithms.

[20]  Michel S. Nakhla,et al.  Efficient model reduction of linear periodically time-varying systems via compressed transient system function , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[21]  Mohammad-Sahadet Hossain,et al.  On Model Reduction of Periodic Descriptor Systems Exploiting the Generalized Inverses of Periodic Matrix Pairs , 2015 .