### “New Schemes for Positive Real Truncation”

**Authors:**Kari Unneland, Paul Van Dooren and Olav Egeland,

**Affiliation:**NTNU, Department of Engineering Cybernetics and Université Catholique de Louvain (Belgium)

**Reference:**2007, Vol 28, No 3, pp. 53-65.

**Keywords:**Model Reduction, Balanced Truncation, Positive Realness, Frequency Weighting

**Abstract:**Model reduction, based on balanced truncation, of stable and of positive real systems are considered. An overview over some of the already existing techniques are given: Lyapunov balancing and stochastic balancing, which includes Riccati balancing. A novel scheme for positive real balanced truncation is then proposed, which is a combination of the already existing Lyapunov balancing and Riccati balancing. Using Riccati balancing, the solution of two Riccati equations are needed to obtain positive real reduced order systems. For the suggested method, only one Lyapunov equation and one Riccati equation are solved in order to obtain positive real reduced order systems, which is less computationally demanding. Further it is shown, that in order to get positive real reduced order systems, only one Riccati equation needs to be solved. Finally, this is used to obtain positive real frequency weighted balanced truncation.

PDF (215 Kb) DOI: 10.4173/mic.2007.3.1

**DOI forward links to this article:**

[1] François Rongère, Jean-Michel Kobus, Aurélien Babarit and Gérard Delhommeau (2011), doi:10.1051/lhb/2011052 |

[2] Peter Benner and Tatjana Stykel (2017), doi:10.1007/978-3-319-46618-7_3 |

[3] A. Buscarino, L. Fortuna, M. Frasca and M.G. Xibilia (2017), doi:10.1016/j.jfranklin.2017.04.009 |

[4] Umair Zulfiqar, Waseem Tariq, Li Li and Muwahida Liaquat (2017), doi:10.1109/TCSII.2017.2685440 |

**References:**

[1] Antoulas, A. (2005). Approximation of Large-Scale Dynamical Systems, SIAM.

[2] Benner, P., Mehrmann, V., Sorensen, D. (2005). Dimension Reduction of Large-Scale Systems, Springer doi:10.1007/3-540-27909-1

[3] Chen, C.-T. (1999). Linear System Theory and Design, Oxford University Press.

[4] Desai, U. Pal, D. (1984). A transformation approach to stochastic model reduction, IEEE Transactions on Automatic Control, AC-29:1097-1100 doi:10.1109/TAC.1984.1103438

[5] Glover, K. (1984). All optimal Hankel-norm approximations of linear multivariable systems and their L-Infinity-error bounds, International Journal of Control, 39:1115-1193 doi:10.1080/00207178408933239

[6] Green, M. (1988). Balanced Stochastic Realizations, Linear Algebra and its Applications, 98:211-247 doi:10.1016/0024-3795(88)90166-8

[7] Gugercin, S. Antoulas, A. (2004). A survey of model re- duction and some new results, International Journal of Control, 77:748-766 doi:10.1080/00207170410001713448

[8] Harshavardhana, P., Jonckheere, E., Silverman, L. (1984). Stochastic balancing and approximation-stability and minimality, IEEE Transaction on Automatic Control, 29:744-746 doi:10.1109/TAC.1984.1103631

[9] Moore, B. (1981). Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Transactions on Automatic Control, 23:17-32 doi:10.1109/TAC.1981.1102568

[10] Mullis, C. Roberts, R. (1976). Synthesis of minimum roundoff noise fixed point digital filters, IEEE Transactions on Circuits and Systems, CAS-23:551-562 doi:10.1109/TCS.1976.1084254

[11] Obinata, G. Anderson, B. (2001). Model Reduction for Control System Design, Springer.

[12] Opdenacker, P. Jonckheere, E. (1986). A state space approach to approximation by phase matching, Modelling, Identification and Robust Control.

[13] Phillips, J., Daniel, L., Silveira, L. (2003). Guaranteed passive balancing transformations for model order reduction, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 22:1027-1041 doi:10.1109/TCAD.2003.814949

[14] Willems, J. (1971). Dissipative dynamical systems, Part I: General Theory, Arch. Rat. Mech. An., 45:321-351 doi:10.1007/BF00276493

[15] Yan, W.-Y. Lam, J. (1999). An approximate approach to H2 optimal model reduction, IEEE Transaction on Automatic Control. 44:1341-1358 doi:10.1109/9.774107

**BibTeX:**

@article{MIC-2007-3-1,

title={{New Schemes for Positive Real Truncation}},

author={Unneland, Kari and Van Dooren, Paul and Egeland, Olav},

journal={Modeling, Identification and Control},

volume={28},

number={3},

pages={53--65},

year={2007},

doi={10.4173/mic.2007.3.1},

publisher={Norwegian Society of Automatic Control}

};