Model Reduction by Differential Balancing Based on Nonlinear Hankel Operators

In this paper, we construct balancing theory for nonlinear systems in the contraction framework. First, we define two novel controllability and observability functions via prolonged systems. We analyze their properties in relation to controllability and observability, and use them for so-called differential balancing, and its application to model order reduction. One of the main contribution of this paper is showing that differential balancing has close relationships with the Fréchet derivative of the nonlinear Hankel operator. Inspired by [3], we provide a generalization in order to have a computationally more feasible method. Moreover, error bounds for model reduction by generalized balancing are provided.

[1]  Arjan van der Schaft,et al.  On Differential Passivity , 2013, NOLCOS.

[2]  Jacquelien M. A. Scherpen,et al.  Minimality and local state decompositions of a nonlinear state space realization using energy functions , 2000, IEEE Trans. Autom. Control..

[3]  L. Silverman,et al.  Controllability and Observability in Time-Variable Linear Systems , 1967 .

[4]  Rodolphe Sepulchre,et al.  On Differentially Dissipative Dynamical Systems , 2013, NOLCOS.

[5]  Alessandro Astolfi,et al.  Dynamic generalized controllability and observability functions with applications to model reduction and sensor deployment , 2014, Autom..

[6]  W.S. Gray,et al.  Nonlinear balanced realizations , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

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

[8]  Rodolphe Sepulchre,et al.  A Differential Lyapunov Framework for Contraction Analysis , 2012, IEEE Transactions on Automatic Control.

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

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

[11]  D. Bao,et al.  An Introduction to Riemann-Finsler Geometry , 2000 .

[12]  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..

[13]  Jacquelien M. A. Scherpen,et al.  On differential balancing: Energy functions and balanced realization , 2015, 2015 European Control Conference (ECC).

[14]  Erik I. Verriest,et al.  FLOW BALANCING NONLINEAR SYSTEMS , 2000 .

[15]  Jacquelien M. A. Scherpen,et al.  Singular Value Analysis Of Nonlinear Symmetric Systems , 2011, IEEE Transactions on Automatic Control.

[16]  W.S. Gray,et al.  Hankel operators and Gramians for nonlinear systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[17]  Jacquelien M. A. Scherpen,et al.  Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators , 2005, IEEE Transactions on Automatic Control.

[18]  L. Weiss.  The concepts of differential controllability and differential observability , 1965 .

[19]  Alessandro Astolfi,et al.  Moment matching for nonlinear port Hamiltonian and gradient systems , 2013 .

[20]  Ian R. Manchester,et al.  Control Contraction Metrics: Differential L2 Gain and Observer Duality , 2014, ArXiv.

[21]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[22]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

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

[24]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[25]  J. Scherpen,et al.  Model reduction by generalized differential balancing , 2015 .

[26]  Ian R. Manchester,et al.  Control Contraction Metrics and Universal Stabilizability , 2013, ArXiv.