The structure of characteristic values for classes of linear time-invariant systems

ABSTRACT In this paper, a particular structure of the characteristic values found in some classes of systems is dealt with. In particular, for the linear time-invariant systems in minimal form (A, B, C, D) with transfer function matrix G(s) for which if μ i is a characteristic value, then also is. Consequently, when the system order is odd, a singular value is equal to one. The classes of systems for which this property holds are characterised by the existence of specific state-space representations. Moreover, the same structure of the characteristic values can be also retrieved for the class of even, (G(s) = G T (−s)), and odd (G(s) = −G T (−s)) systems. Several examples related to Multi Input Multi Output (MIMO) systems are reported, also referring to real physical systems.

[1]  R. Karl Rethemeyer,et al.  Network analysis , 2011 .

[2]  W. Gawronski.  Dynamics and control of structures : a modal approach , 1998 .

[3]  Edmond A. Jonckheere,et al.  LQG balancing and reduced LQG compensation of symmetric passive systems , 1985 .

[4]  B. De Moor,et al.  A note on closed-loop balanced truncation , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

[6]  Shodhan Rao,et al.  Higher-order linear lossless systems , 2008, Int. J. Control.

[7]  Giovanni Muscato,et al.  Closed-loop balancing and singular perturbation: a new result , 1991 .

[8]  Ian R. Petersen,et al.  A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems , 2010, IEEE Transactions on Automatic Control.

[9]  Duncan McFarlane,et al.  Balanced canonical forms for minimal systems: A normalized coprime factor approach , 1989 .

[10]  Li Li,et al.  A Gramian-based approach to model reduction for uncertain systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[11]  Louis Weinberg,et al.  Network Analysis and Synthesis , 1962 .

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

[13]  Wodek Gawronski,et al.  Balanced Control of Flexible Structures , 1995 .

[14]  Edmond A. Jonckheere,et al.  A new set of invariants for linear systems--Application to reduced order compensator design , 1983 .

[15]  Giovanni Muscato,et al.  Closed-loop balanced realizations of LTI discrete-time systems , 1993 .

[16]  Wodek Gawronski,et al.  Dynamics and Control of Structures , 1998 .

[17]  R. E. Kalman,et al.  Linear system theory-The state space approach , 1965 .

[18]  M Maarten Steinbuch,et al.  Iterative model and controller reduction using closed-loop balancing, with application to a Compact Disc mechanism , 1999 .

[19]  Luigi Fortuna,et al.  Invariance of characteristic values and L∞ norm under lossless positive real transformations , 2016, J. Frankl. Inst..

[20]  Jing Sun,et al.  Closed-loop model order reduction and MPC for diesel engine airpath control , 2015, 2015 American Control Conference (ACC).

[21]  Luigi Fortuna,et al.  Optimal and Robust Control : Advanced Topics with MATLAB , 2012 .

[22]  Franklin Fa-Kun Kuo,et al.  Network analysis and synthesis , 1962 .

[23]  Luigi Fortuna,et al.  Continuous time LTI systems under lossless positive real transformations: open-loop balanced representation and truncated reduced-order models , 2017, Int. J. Control.

[24]  Luigi Fortuna,et al.  New parameterization of MIMO systems , 1989 .