## Observability and Identifiability of Linear Hybrid Systems

Rene Vidal, Alessandro Chiuso^{1}, and Stefano Soatto^{2}

(Professor S. Shankar Sastry)

(ONR) N00014-00-1-0621

Modeling time series is a fundamental problem in many scientific and engineering disciplines. In particular, in computer vision the problem arises when one tries to characterize the dynamics of visual processes, for instance, the appearance of smoke, fire, foliage of trees, or walking gaits. All of these examples can be modeled as a linear hybrid system, that is, a system whose evolution is determined by a collection of linear models connected by switches among a number of discrete states.

In [1] and [2] we analyze the observability of the continuous and discrete states of a class of linear hybrid systems. We derive rank conditions that the structural parameters of the model must satisfy in order for filtering and smoothing algorithms to operate correctly.
Our conditions are simple rank tests that exploit the geometry of the observability subspaces generated by the output of a linear hybrid system. We also derive weaker rank conditions that guarantee the uniqueness of the reconstruction of the state trajectory from a specific output, even when the hybrid system is unobservable.
We also study the identifiability of the model parameters by characterizing the set of models that produce the same output measurements. Finally, when the data is generated by a model in the class, we give conditions under which the true model can be identified.

- [1]
- R. Vidal, A. Chiuso, and S. Soatto, "Observability and Identifiability of Jump Linear Systems,"
*Int. Conf. Decision and Control,* Las Vegas, NV, December 2002.
- [2]
- R. Vidal, A. Chiuso, and S. Soatto, "Observability of Linear Hybrid Systems,"
*Workshop on Hybrid Systems Computation and Control,* 2003 (submitted).

^{1}Professor, Universita di Padova

^{2}Professor, University of California at Los Angeless

More information (http://www.eecs.berkeley.edu/~rvidal/) or

Send mail to the author : (rvidal@eecs.berkeley.edu)

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