A Contraction Analysis of the Convergence of Risk-Sensitive Filters

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the N-fold composition of the Riccati map is strictly contractive with respect to the Riemannian metric of positive definite matrices, when N is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.Comment: 22 pages, 6 figure

Observability and nonlinear filtering

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces

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

Sequential Detection with Mutual Information Stopping Cost

This paper formulates and solves a sequential detection problem that involves the mutual information (stochastic observability) of a Gaussian process observed in noise with missing measurements. The main result is that the optimal decision is characterized by a monotone policy on the partially ordered set of positive definite covariance matrices. This monotone structure implies that numerically efficient algorithms can be designed to estimate and implement monotone parametrized decision policies.The sequential detection problem is motivated by applications in radar scheduling where the aim is to maintain the mutual information of all targets within a specified bound. We illustrate the problem formulation and performance of monotone parametrized policies via numerical examples in fly-by and persistent-surveillance applications involving a GMTI (Ground Moving Target Indicator) radar

Review of selection criteria for sensor and actuator configurations suitable for internal combustion engines

This literature review considers the problem of finding a suitable configuration of sensors and actuators for the control of an internal combustion engine. It takes a look at the methods, algorithms, processes, metrics, applications, research groups and patents relevant for this topic. Several formal metric have been proposed, but practical use remains limited. Maximal information criteria are theoretically optimal for selecting sensors, but hard to apply to a system as complex and nonlinear as an engine. Thus, we reviewed methods applied to neighboring fields including nonlinear systems and non-minimal phase systems. Furthermore, the closed loop nature of control means that information is not the only consideration, and speed, stability and robustness have to be considered. The optimal use of sensor information also requires the use of models, observers, state estimators or virtual sensors, and practical acceptance of these remains limited. Simple control metrics such as conditioning number are popular, mostly because they need fewer assumptions than closed-loop metrics, which require a full plant, disturbance and goal model. Overall, no clear consensus can be found on the choice of metrics to define optimal control configurations, with physical measures, linear algebra metrics and modern control metrics all being used. Genetic algorithms and multi-criterial optimisation were identified as the most widely used methods for optimal sensor selection, although addressing the dimensionality and complexity of formulating the problem remains a challenge. This review does present a number of different successful approaches for specific applications domains, some of which may be applicable to diesel engines and other automotive applications. For a thorough treatment, non-linear dynamics and uncertainties need to be considered together, which requires sophisticated (non-Gaussian) stochastic models to establish the value of a control architecture

Phase Transitions in Nonlinear Filtering

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite-dimensional problems are outside its scope. Far from being a technical issue, the infinite-dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur. We also discuss closely related problems in the setting of conditional Markov random fields.Comment: 51 page

Stability of Non-linear Filter for Deterministic Dynamics

This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal.Comment: 24 pages, 2 figures. In V3, few subsections are added and several typos are correcte