Duality for Nonlinear Filtering I: Observability

Jin W. Kim, Prashant G. Mehta

Research output: Contribution to journalArticlepeer-review


This article is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulas, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table II).

Original languageEnglish (US)
Pages (from-to)699-711
Number of pages13
JournalIEEE Transactions on Automatic Control
Issue number2
StatePublished - Feb 1 2024
Externally publishedYes


  • Nonlinear filtering
  • observability
  • stochastic systems

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Control and Systems Engineering
  • Computer Science Applications


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