Ensemble Observability of Linear Systems

Shen Zeng, Steffen Waldherr, Christian Ebenbauer, Frank Allgöwer

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments, which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual viewpoints, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.

Original languageEnglish
Article number7174989
Pages (from-to)1452-1465
Number of pages14
JournalIEEE Transactions on Automatic Control
Issue number6
StatePublished - Jun 2016


  • Ensemble control
  • moment dynamics
  • observability
  • polynomial systems
  • tomography


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