The filtering equations revisited

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of--probability-measure method originally introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by a martingale-problem formulation. Two specific applications are treated

Memory functions and Correlations in Additive Binary Markov Chains

A theory of additive Markov chains with long-range memory, proposed earlier in Phys. Rev. E 68, 06117 (2003), is developed and used to describe statistical properties of long-range correlated systems. The convenient characteristics of such systems, a memory function, and its relation to the correlation properties of the systems are examined. Various methods for finding the memory function via the correlation function are proposed. The inverse problem (calculation of the correlation function by means of the prescribed memory function) is also solved. This is demonstrated for the analytically solvable model of the system with a step-wise memory function.Comment: 11 pages, 5 figure

Control for a Class of Hybrid Systems

A hybrid control system is a control theoretic model for a computer controlled engineering system. A definition of a hybrid control system is formulated that consists of a product of a finite state automaton and of a family of continuous control systems. An example of a transportation system consisting of a line of conveyor belts is used as a running example. The realization problem for this class of systems is discussed. Control synthesis of hybrid systems is in a first approach based on supervisory control of discrete event systems

Biochemical reaction systems – system theory and decomposition

Biochemical reaction networks may be modeled as biochemical reaction systems consisting of differential equations with rational functions. Biochemical reaction systems are defined as rational positive dynamic systems with inputs and outputs, and illustrated by examples. This formulation makes available the results from algebraic system theory for rational systems and a relation with computer algebra. It is shown how to decompose networks into subsystems and how to relate them to graphs. The realization problem for this class of systems is briefly discussed. Finally, control problems for biochemical reaction networks are formulated

Observability of hybrid systems and Turing machines

In this paper we discuss the observability of hybrid systems and Turing machines. We give an elementary example to show that observability is undecidable for Turing machines with output. Since many classes of hybrid systems simulate Turing machines, this immediately shows these classes are undecidable. We discuss the observability of piecewise-affine hybrid systems, and give a number of examples illustrating different observability propertie

Approximating the minimal-cost sensor-selection for discrete-event systems

This paper discusses the approximation of solutions to several NP-complete optimization problems related to the supervisory control of discrete-event systems. Approximation calculations for the minimal-cost sensor-selection problem in a partial observation, centralized control setting is first discussed. It is shown that approximate solutions to this problem cannot always be calculated with a given degree of accuracy in polynomial time. An efficient construction method is shown to convert this sensor selection problem into a novel type of graph cutting problem. Several heuristic algorithms are then shown to approximate solutions to this problem. Approximation methods for computationally difficult communicating decentralized controller problems and actuator selection problems are also discussed. It is shown how to convert these problems into graph cutting problems

The weak and strong Gaussian probabilistic realization problem

A classification is given of all [sigma]-algebras that make two given [sigma]-algebras conditionally independent in the case that the [sigma]-algebras are generated by finite dimensional Gaussian random variables. In addition a classification is given of all Gaussian measures that have the conditional independence property and such that restricted to a subspace, they coincide with a given measure.Conditional independence Gaussian random variables canonical variable representation sufficient statistics stochastic systems