293 research outputs found
Average reachability of continuous-time Markov jump linear systems and the linear minimum mean square estimator
. In this paper we study the average reachability gramian for continuous-time linear
systems with additive noise and jump parameters driven by a general Markov chain. We define a
rather natural reachability concept by requiring that the average reachability gramian be positive
definite. Aiming at a testable condition, we introduce a set of reachability matrices for this class
of systems and employ invariance properties of the null space of the noise coefficient matrices to
show that the system is reachable if and only if these matrices are of full rank. We also show for
reachable systems that the state second moment is positive definite. One consequence of this result
in the context of linear minimum mean square state estimation for reachable systems is that the
expectation of the error covariance matrix is positive definite. Moreover, the average boundedness of
the error covariance matrix is invariant to a type of perturbation in the noise model, meaning that
the estimates are not overly sensitive, which consists in a property that is desirable in applications
and sometimes referred to as stability of the estimator
Fidelity is a sub-martingale for discrete-time quantum filters
Fidelity is known to increase through any Kraus map: the fidelity between two
density matrices is less than the fidelity between their images via a Kraus
map. We prove here that, in average, fidelity is also increasing for any
discrete-time quantum filter: fidelity between the density matrix of the
underlying Markov chain and the density matrix of its associated quantum filter
is a sub-martingale. This result is not restricted to pure states. It also
holds true for mixed states
A new approach to detectability of discrete-time infinite Markov jump linear systems
This paper deals with detectability for the class of discrete-time Markov jump linear systems (MJLS) with the underlying Markov chain having countably infinite state space. The formulation here relates the convergence of the output with that of the state variables, and due to the rather general setting, a novel point of view toward detectability is required. Our approach introduces invariant subspaces for the autonomous system and exhibits the role that they play. This allows us to show that detectability can be written equivalently in term of two conditions: stability of the autonomous system in a certain invariant space and convergence of general state trajectories to this invariant space under convergence of input and output variables. This, in turn, provides the tools to show that detectability here generalizes uniform observability ideas as well as previous detectability notions for MJLS with finite state Markov chain, and allows us to solve the jump-linear-quadratic control problem. In addition, it is shown for the MJLS with finite Markov state that the second condition is redundant and that detectability retrieves previously well-known concepts in their respective scenarios. Illustrative examples are included.4362132215
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