Maciej Patan

D-optimal design of a monitoring network for parameter estimation of distributed systems

This paper addresses the design of a network of observation locations in a spatial domain that will be used to estimate unknown parameters of a distributed parameter system. We consider a setting where we are given a finite number of possible sites at which to locate a sensor, but cost constraints allow only some proper subset of them to be selected. We formulate this problem as the selection of the gauged sites so as to maximize the log-determinant of the Fisher information matrix associated with the estimated parameters. The search for the optimal solution is performed using the branch-and-bound method in which an extremely simple and efficient technique is employed to produce an upper bound to the maximum objective function. Its idea consists in solving a relaxed problem through the application of a simplicial decomposition algorithm in which the restricted master problem is solved using a multiplicative algorithm for optimal design. The use of the proposed approach is illustrated by a numerical example involving sensor selection for a two-dimensional convective diffusion process.

Sensor network design for the estimation of spatially distributed processes

Sensor network design for the estimation of spatially distributed processes In a typical moving contaminating source identification problem, after some type of biological or chemical contamination has occurred, there is a developing cloud of dangerous or toxic material. In order to detect and localize the contamination source, a sensor network can be used. Up to now, however, approaches aiming at guaranteeing a dense region coverage or satisfactory network connectivity have dominated this line of research and abstracted away from the mathematical description of the physical processes underlying the observed phenomena. The present work aims at bridging this gap and meeting the needs created in the context of the source identification problem. We assume that the paths of the moving sources are unknown, but they are sufficiently smooth to be approximated by combinations of given basis functions. This parametrization makes it possible to reduce the source detection and estimation problem to that of parameter identification. In order to estimate the source and medium parameters, the maximum-likelihood estimator is used. Based on a scalar measure of performance defined on the Fisher information matrix related to the unknown parameters, which is commonly used in optimum experimental design theory, the problem is formulated as an optimal control one. From a practical point of view, it is desirable to have the computations dynamic data driven, i.e., the current measurements from the mobile sensors must serve as a basis for the update of parameter estimates and these, in turn, can be used to correct the sensor movements. In the proposed research, an attempt will also be made at applying a nonlinear model-predictive-control-like approach to attack this issue.

D-optimal trajectory design of heterogeneous mobile sensors for parameter estimation of distributed systems

An approach is proposed to joint optimization of trajectories and measurement accuracies of mobile nodes in a sensor network collecting measurements for parameter estimation of a distributed parameter system. The problem is cast as maximization of the log-determinant of the information matrix associated with the estimated parameters over the set of all feasible information matrices, which yields a formulation in terms of convex optimization. This makes it possible to employ powerful tools of optimum experimental design theory to characterize the optimal solution and adapt the Wynn-Fedorov algorithm to construct its numerical approximation. As a crucial subtask in each iteration, a nontrivial optimal control problem must be solved, which is accomplished using the MATLAB PDE toolbox and the RIOTS_95 optimal control toolbox which handles various constraints imposed on the sensor motions. The effectiveness of the method is illustrated with a numerical example regarding a two-dimensional diffusion equation.

Configuring A Sensor Network for Fault Detection in Distributed Parameter Systems

Configuring A Sensor Network for Fault Detection in Distributed Parameter Systems The problem of fault detection in distributed parameter systems (DPSs) is formulated as that of maximizing the power of a parametric hypothesis test which checks whether or not system parameters have nominal values. A computational scheme is provided for the design of a network of observation locations in a spatial domain that are supposed to be used while detecting changes in the underlying parameters of a distributed parameter system. The setting considered relates to a situation where from among a finite set of potential sensor locations only a subset can be selected because of the cost constraints. As a suitable performance measure, the Ds-optimality criterion defined on the Fisher information matrix for the estimated parameters is applied. Then, the solution of a resulting combinatorial problem is determined based on the branch-and-bound method. As its essential part, a relaxed problem is discussed in which the sensor locations are given a priori and the aim is to determine the associated weights, which quantify the contributions of individual gauged sites. The concavity and differentiability properties of the criterion are established and a gradient projection algorithm is proposed to perform the search for the optimal solution. The delineated approach is illustrated by a numerical example on a sensor network design for a two-dimensional convective diffusion process.

Optimal observation strategies for model-based fault detection in distributed systems

In model oriented diagnostics of real-world systems, the problems of structural identification and parameter estimation are of crucial importance. They require a properly designed schedule of measurements in such a way as to obtain possibly the most informative observational data. The aim of this work is to develop a novel approach to fault detection in distributed systems based on the maximization of the power of parametric hypothesis test, which verifies the nominal state of the considered system. The optimal locations of sensors are determined using the performance index operating on the Fisher Information Matrix. A general scheme is then proposed and tested on a computer example regarding an advection-diffusion problem.

OPTIMAL ACTIVATION STRATEGY OF DISCRETE SCANNING SENSORS FOR FAULT DETECTION IN DISTRIBUTED-PARAMETER SYSTEMS

Abstract The problem under consideration is to determine an activation policy of discrete scanning sensors in such a way as to maximize the power of a simple parametric hypothesis test, which verifies the nominal state of the considered distributed system specified over a given multi-dimensional spatial domain. The optimal locations of sensors are determined based on the D s -optimality criterion defined on the respective Fisher Information Matrix. The proposed approach exploits the notion of directly constrained design measures recently introduced in modern optimum experimental design theory which leads to an extremely fast iterative procedure of exchange type. In this work, a general scheme of such an approach leading to maximization of the fault detection efficiency in distributed-parameter systems is delineated and tested via computer simulations regarding an advection-diffusion problem.

Resource-Constrained Sensor Routing for Parameter Estimation of Distributed Systems

Abstract We consider a setting where mobile nodes with sensing capacity form a network whose mission is to collect measurements for parameter estimation of a distributed parameter system (DPS). Two techniques to optimize node motions are presented which constitute a trade-off between the achievable accuracy of parameter estimates and limited motion resources of sensor network nodes. The framework is based on the use of the D-optimality criterion defined on the Fisher information matrix associated with the estimated parameters as a measure of the information content in the measurements. Restrictions on maximal distances traveled by sensor nodes are imposed so as to guarantee realizable solutions. The approach is to convert the problem to a canonical optimal control one in Mayer form, in which both the control forces of the sensors and the initial sensor positions are optimized. Numerical solutions are then obtained using the M atlab PDE toolbox and the RIOTS_95 optimal control toolbox which handles various constraints imposed on the node motions. Illustrative numerical experiments with the proposed techniques are presented.

Distributed scheduling of sensor networks for identification of spatio-temporal processes

Distributed scheduling of sensor networks for identification of spatio-temporal processes An approach to determine a scheduling policy for a sensor network monitoring some spatial domain in order to identify unknown parameters of a distributed system is discussed. Given a finite number of possible sites at which sensors are located, the activation schedule for scanning sensors is provided so as to maximize a criterion defined on the Fisher information matrix associated with the estimated parameters. The related combinatorial problem is relaxed through operating on the density of sensors in lieu of individual sensor positions. Then, based on the adaptation of pairwise communication algorithms and the idea of running consensus, a numerical scheme is developed which distributes the computational burden between the network nodes. As a result, a simple exchange algorithm is outlined to solve the design problem in a decentralized fashion.

Optimal location of discrete scanning sensors for parameter estimation of distributed systems

Abstract We investigate possibilities of choosing an activation policy of discrete scanning sensors in such a way as to maximize the accuracy of parameter estimation of a distributed system defined in a given multidimensional domain. A general functional defined on the Fisher information matrix is used as the design criterion. The setting examined here corresponds to situations where one has many sensors and activates only some of them during a given time interval, or alternatively, has several sensors which are mobile. The proposed approach, which has been suggested by Fedorovs idea of directly constrained design measures, consists in imposing constraints on the sensor density in a given spatial domain. As a result, an extremely fast iterative procedure is obtained whose each step reduces to replacing less informative sensor locations with points which furnish more informaton about the parameters. The performance of the proposed algorithm is evaluated by numerical experiments.

Optimal activation policies for continuous scanning observations in parameter estimation of distributed systems

The problem of determining an optimal measurement scheduling for identification of unknown parameters in distributed systems described by partial differential equations is discussed. The discrete-scanning observations are performed by an optimal selection of measurement data from spatially fixed sensors. In the adopted approach, the sensor scheduling problem is converted to a constrained optimal control problem. In this framework, the control value represents the selected sensor configuration. Thus the control variable is constrained to take values in a discrete set and switchings between sensors may occur in continuous time. By applying the control parameterization enhancing transform technique, a computational procedure for solving the optimal scanning measurement problem is obtained. The numerical scheme is then tested on a computer example regarding an advection-diffusion problem.