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Realization Theory of Jump-Markov Linear Systems

Realization Theory of Jump-Markov Linear Systems. René Vidal Center for Imaging Science Johns Hopkins University. Mihaly Petreczky Eindhoven University of Technology, Netherlands. Realization Theory of Jump-Markov Linear Systems. Mihaly Petreczky Eindhoven University of Technology.

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Realization Theory of Jump-Markov Linear Systems

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  1. Realization Theory of Jump-Markov Linear Systems René Vidal Center for Imaging Science Johns Hopkins University Mihaly Petreczky Eindhoven University of Technology, Netherlands

  2. Realization Theory of Jump-Markov Linear Systems Mihaly Petreczky Eindhoven University of Technology René Vidal Center for Imaging Science Johns Hopkins University

  3. Realization theory • Given an input-output map Y, find • Conditions for existence of a state-space representation of Y • Characterizations of minimalilty of such representations • Algorithms for computing such representations • Motivation • Systems identification, model reduction, control design • Comparison of dynamical systems • Historical overview: last 40 years • Deterministic linear/bilinear systems: Kalman, Isidori, Sontag, Fliess, etc. • Finite-state automata: Kleene, Gluskov, Nerode etc. • Polynomial systems: Sontag, Bartuszewicz, etc. • General nonlinear systems: Jakubczyk, Sussmann etc. • Stochastic linear systems: Anderson, Faurre, Akaike, Lindquist, Picci etc. • Stochastic bilinear systems: Desai, Frahzo

  4. Realization theory of hybrid systems • Realization theory of hybrid systems is not as well developed • First work but no formal theory: Grossman and Larson ‘95 • Series of works on the deterministic case by Petreczky et al. • Linear and bilinear hybrid and switched systems: without guards • Nonlinear hybrid systems: conditions for existence • Piecewise-affine discrete-time hybrid systems: conditions for existence • Stochastic Jump-Markov Systems: necessary conditions for existence of a realization • Missing from the previous works • Conditions for existence and minimality, and realization algorithm for Stochastic Jump-Markov Systems

  5. Paper contributions • Realization theory for Stochastic Jump-Markov Linear Systems • Necessary and sufficient conditions for existence of a realization • Characterization of minimality in terms of reachability and observability • Realization algorithm • Realization theory for Stochastic Generalized Bilinear Systems • The realization theory of stochastic JMLS depends on realization theory of stochastic generalized bilinear systems with non white-noise inputs

  6. Outline of the approach • Realization Theory of Jump-Markov Systems • A Jump-Markov System is a bilinear system with the discrete states as observed inputs (not white-noise) • Realization theory of stochastic bilinear systems • Covariance sequence of future outputs with products of past outputs and inputs is a rational formal power series • Project the future outputs and inputs onto the past outputs and inputs • The structure of the covariance sequence is used to obtain a recursive formula the projection. • The recursive formula yields a bilinear system whose state is the projection.

  7. Stochastic Jump-Markov Linear Systems • A JMLS is

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