Model reduction of synchronized homogeneous Lur'e networks with incrementally sector-bounded nonlinearities

Abstract This paper proposes a model order reduction scheme that reduces the complexity of diffusively coupled homogeneous Lur’e systems. We aim to reduce the dimension of each subsystem and meanwhile preserve the synchronization property of the overall network. Using the Laplacian spectral radius, we characterize the robust synchronization of the Lur’e network by a linear matrix inequality (LMI), whose solutions then are treated as generalized Gramians for the balanced truncation of the linear component of each Lur’e subsystem. It is verified that, with the same communication topology, the resulting reduced-order network system is still robustly synchronized, and an a priori bound on the approximation error is guaranteed to compare the behaviors of the full-order and reduced-order Lur’e subsystems.

[1]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[2]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[3]  S. Bhattacharyya,et al.  Synthesis of absolutely stabilising controllers , 2008 .

[4]  Karl Henrik Johansson,et al.  Clustering-Based Model Reduction of Networked Passive Systems , 2016, IEEE Transactions on Automatic Control.

[5]  Jacquelien M. A. Scherpen,et al.  Reduction of Second-Order Network Systems With Structure Preservation , 2017, IEEE Transactions on Automatic Control.

[6]  M. Kanat Camlibel,et al.  Stability and synchronization preserving model reduction of multi-agent systems , 2013, Syst. Control. Lett..

[7]  Mario di Bernardo,et al.  Adaptive Pinning Control of Networks of Circuits and Systems in Lur'e Form , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..

[9]  Brian D. O. Anderson,et al.  Singular perturbation approximation of balanced systems , 1989 .

[10]  Petar Mlinarić.  Clustering-based model order reduction for multi-agent system , 2016 .

[11]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[12]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[13]  J. M. A. Scherpen,et al.  Balancing for nonlinear systems , 1993 .

[14]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[15]  J. Wen,et al.  Robustness analysis of LTI systems with structured incrementally sector bounded nonlinearities , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[16]  Takayuki Ishizaki,et al.  Model Reduction and Clusterization of Large-Scale Bidirectional Networks , 2014, IEEE Transactions on Automatic Control.

[17]  Nathan van de Wouw,et al.  An error bound for model reduction of Lur'e-type systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[18]  G. Dullerud,et al.  A Course in Robust Control Theory: A Convex Approach , 2005 .

[19]  E.M. Atkins,et al.  A survey of consensus problems in multi-agent coordination , 2005, Proceedings of the 2005, American Control Conference, 2005..

[20]  Jacquelien M. A. Scherpen,et al.  Model Reduction of Multiagent Systems Using Dissimilarity-Based Clustering , 2019, IEEE Transactions on Automatic Control.

[21]  Xiaodong Cheng,et al.  Balanced truncation of networked linear passive systems , 2017, Automatica.

[22]  Leon O. Chua,et al.  Synchronization of Chua's circuits with time-varying channels and parameters , 1996 .

[23]  Jacquelien M. A. Scherpen,et al.  Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis , 2010, SIAM J. Control. Optim..

[24]  Alessandro Astolfi,et al.  Families of moment matching based, low order approximations for linear systems , 2014, Syst. Control. Lett..

[25]  P. Moylan,et al.  Connections between finite-gain and asymptotic stability , 1980 .

[26]  Alessandro Astolfi,et al.  Model Reduction by Moment Matching for Linear and Nonlinear Systems , 2010, IEEE Transactions on Automatic Control.

[27]  Fan Zhang,et al.  Fully distributed robust synchronization of networked Lur'e systems with incremental nonlinearities , 2014, Autom..

[28]  R. M. Murray,et al.  Model reduction of interconnected linear systems , 2009 .

[29]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[30]  W. Shiu,et al.  The Laplacian spectral radius of some graphs , 2009 .

[31]  Nathan van de Wouw,et al.  Model Reduction for Nonlinear Systems by Incremental Balanced Truncation , 2014, IEEE Transactions on Automatic Control.

[32]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[33]  Lingsheng Shi.  Bounds on the (Laplacian) spectral radius of graphs , 2007 .

[34]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[35]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[36]  Jacquelien M. A. Scherpen,et al.  Robust Synchronization Preserving Model Reduction of Lur’e Networks , 2018, 2018 European Control Conference (ECC).