Advances in Computational Mathematics

, Volume 44, Issue 6, pp 1845–1886 | Cite as

Balanced truncation for linear switched systems

  • Ion Victor GoseaEmail author
  • Mihaly Petreczky
  • Athanasios C. Antoulas
  • Christophe Fiter
Open Access


We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.


Model order reduction Switched systems Balanced truncation Infinite Gramians Controllability Observability 

Mathematics Subject Classification (2010)

93A15 93A30 93B11 93C05 93C10 



Open access funding provided by Max Planck Society.


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Data-Driven System Reduction and Identification (DRI) GroupMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.Center de Recherche en InformatiqueSignal et Automatique de Lille (CRIStAL), UMR CNRS 9189, CNRS, Ecole Centrale de LilleVilleneuve d’ AscqFrance
  3. 3.Department of Electrical and Computer EngineeringRice UniversityHoustonUSA
  4. 4.Baylor College of MedicineHoustonUSA
  5. 5.CNRS CRIStAL UMR 9189Université de Lille 1, Sciences et TechnologiesVilleneuve d’ AscqFrance

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