Showing papers on "State vector published in 2011"

Kalman-filter-based orientation determination using inertial/magnetic sensors: observability analysis and performance evaluation.

Abstract: In this paper we present a quaternion-based Extended Kalman Filter (EKF) for estimating the three-dimensional orientation of a rigid body. The EKF exploits the measurements from an Inertial Measurement Unit (IMU) that is integrated with a tri-axial magnetic sensor. Magnetic disturbances and gyro bias errors are modeled and compensated by including them in the filter state vector. We employ the observability rank criterion based on Lie derivatives to verify the conditions under which the nonlinear system that describes the process of motion tracking by the IMU is observable, namely it may provide sufficient information for performing the estimation task with bounded estimation errors. The observability conditions are that the magnetic field, perturbed by first-order Gauss-Markov magnetic variations, and the gravity vector are not collinear and that the IMU is subject to some angular motions. Computer simulations and experimental testing are presented to evaluate the algorithm performance, including when the observability conditions are critical.

Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition

Abstract: We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, mom...

Statistical Software for State Space Methods

Abstract: In this paper we review the state space approach to time series analysis and establish the notation that is adopted in this special volume of the Journal of Statistical Software. We first provide some background on the history of state space methods for the analysis of time series. This is followed by a concise overview of linear Gaussian state space analysis including the modelling framework and appropriate estimation methods. We discuss the important class of unobserved component models which incorporate a trend, a seasonal, a cycle, and fixed explanatory and intervention variables for the univariate and multivariate analysis of time series. We continue the discussion by presenting methods for the computation of different estimates for the unobserved state vector: filtering, prediction, and smoothing. Estimation approaches for the other parameters in the model are also considered. Next, we discuss how the estimation procedures can be used for constructing confidence intervals, detecting outlier observations and structural breaks, and testing model assumptions of residual independence, homoscedasticity, and normality. We then show how ARIMA and ARIMA components models fit in the state space framework to time series analysis. We also provide a basic introduction for non-Gaussian state space models. Finally, we present an overview of the software tools currently available for the analysis of time series with state space methods as they are discussed in the other contributions to this special volume.

Distributed Estimation via Iterative Projections with Application to Power Network Monitoring

Abstract: This work presents a distributed method for control centers to monitor the operating condition of a power network, i.e., to estimate the network state, and to ultimately determine the occurrence of threatening situations. State estimation has been recognized to be a fundamental task for network control centers to ensure correct and safe functionalities of power grids. We consider (static) state estimation problems, in which the state vector consists of the voltage magnitude and angle at all network buses. We consider the state to be linearly related to network measurements, which include power flows, current injections, and voltages phasors at some buses. We admit the presence of several cooperating control centers, and we design two distributed methods for them to compute the minimum variance estimate of the state given the network measurements. The two distributed methods rely on different modes of cooperation among control centers: in the first method an incremental mode of cooperation is used, whereas, in the second method, a diffusive interaction is implemented. Our procedures, which require each control center to know only the measurements and structure of a subpart of the whole network, are computationally efficient and scalable with respect to the network dimension, provided that the number of control centers also increases with the network cardinality. Additionally, a finite-memory approximation of our diffusive algorithm is proposed, and its accuracy is characterized. Finally, our estimation methods are exploited to develop a distributed algorithm to detect corrupted data among the network measurements.

Optimal Trajectory Generation With Probabilistic System Uncertainty Using Polynomial Chaos

Abstract: In this paper, we develop a framework for solving optimal trajectory generation problems with probabilistic uncertainty in system parameters. The framework is based on the generalized polynomial chaos theory. We consider both linear and nonlinear dynamics in this paper and demonstrate transformation of stochastic dynamics to equivalent deterministic dynamics in higher dimensional state space. Minimum expectation and variance cost function are shown to be equivalent to standard quadratic cost functions of the expanded state vector. Results are shown on a stochastic Van der Pol oscillator.

A distributed method for state estimation and false data detection in power networks

Abstract: This work presents a distributed method for control centers to monitor the operating condition of a power network. Specifically we consider (static) state estimation problems, in which the state vector consists of the voltage magnitude and angle at all network buses. We consider the state to be linearly related to network measurements, which include power flows, current injections, and voltages phasors at some buses. We admit the presence of several cooperating control centers, and we design two distributed methods for them to compute the minimum variance estimate of the state given the network measurements. The two distributed methods rely on different modes of cooperation among control centers: in the first method an incremental mode of cooperation is assumed, whereas, in the second method, a diffusive interaction is implemented. These estimation methods, which are proved to converge in finite time, are further exploited to develop a distributed algorithm to detect corrupted data among network measurements.

Offline Detection, Identification, and Correction of Branch Parameter Errors Based on Several Measurement Snapshots

Abstract: This paper proposes a three-stage offline approach to detect, identify, and correct series and shunt branch parameter errors. In Stage 1 the branches suspected of having parameter errors are identified through an Identification Index (II). The II of a branch is the ratio between the number of measurements adjacent to that branch, whose normalized residuals are higher than a specified threshold value, and the total number of measurements adjacent to that branch. Using several measurement snapshots, in Stage 2 the suspicious parameters are estimated, in a simultaneous multiple-state-and-parameter estimation, via an augmented state and parameter estimator which increases the V-θ state vector for the inclusion of suspicious parameters. Stage 3 enables the validation of the estimation obtained in Stage 2, and is performed via a conventional weighted least squares estimator. Several simulation results (with IEEE bus systems) have demonstrated the reliability of the proposed approach to deal with single and multiple parameter errors in adjacent and non-adjacent branches, as well as in parallel transmission lines with series compensation. Finally the proposed approach is confirmed on tests performed on the Hydro-Quebec TransEnergie network.

Localization of Mobile Nodes in Wireless Networks with Correlated in Time Measurement Noise

Abstract: Wireless sensor networks are an inherent part of decision making, object tracking, and location awareness systems. This work is focused on simultaneous localization of mobile nodes based on received signal strength indicators (RSSIs) with correlated in time measurement noises. Two approaches to deal with the correlated measurement noises are proposed in the framework of auxiliary particle filtering: with a noise augmented state vector and the second approach implements noise decorrelation. The performance of the two proposed multimodel auxiliary particle filters (MM AUX-PFs) is validated over simulated and real RSSIs and high localization accuracy is demonstrated.

Model Adaptation for Prognostics in a Particle Filtering Framework

Abstract: One of the key motivating factors for using particle filters for prognostics is the ability to include model parameters as part of the state vector to be estimated. This performs model adaptation in conjunction with state tracking, and thus, produces a tuned model that can used for long term predictions. This feature of particle filters works in most part due to the fact that they are not subject to the "curse of dimensionality", i.e. the exponential growth of computational complexity with state dimension. However, in practice, this property holds for "well-designed" particle filters only as dimensionality increases. This paper explores the notion of wellness of design in the context of predicting remaining useful life for individual discharge cycles of Li-ion batteries. Prognostic metrics are used to analyze the tradeoff between different model designs and prediction performance. Results demonstrate how sensitivity analysis may be used to arrive at a well-designed prognostic model that can take advantage of the model adaptation properties of a particle filter.

Solving ill-posed Image Processing problems using Data Assimilation

Abstract: Data Assimilation is a mathematical framework used in environmental sciences to improve forecasts performed by meteorological, oceanographic or air quality simulation models. It aims to solve an evolution equation, describing the temporal dynamics, and an observation equation, linking the state vector and observations. In this article we use this framework to study a class of ill-posed Image Processing problems, usually solved by spatial and temporal regularization techniques. An approach is proposed to convert an ill-posed Image Processing problem in terms of a Data Assimilation system, solved by a 4D-Var method. This is illustrated by the estimation of optical flow from a noisy image sequence, with the dynamic model ensuring the temporal regularity of the result. The innovation of the paper concerns first, the extensive description of the tasks to be achieved for going from an image processing problem to a data assimilation description; second, the theoretical analysis of the covariance matrices involved in the algorithm; and third a specific discretisation scheme ensuring the stability of computation for the application on optical flow estimation.

Decentralized Optimal Control of a Class of Interconnected Nonlinear Discrete-Time Systems by Using Online Hamilton-Jacobi-Bellman Formulation

Abstract: In this paper, the direct neural dynamic programming technique is utilized to solve the Hamilton-Jacobi-Bellman equation forward-in-time for the decentralized near optimal regulation of a class of nonlinear interconnected discrete-time systems with unknown internal subsystem and interconnection dynamics, while the input gain matrix is considered known. Even though the unknown interconnection terms are considered weak and functions of the entire state vector, the decentralized control is attempted under the assumption that only the local state vector is measurable. The decentralized nearly optimal controller design for each subsystem consists of two neural networks (NNs), an action NN that is aimed to provide a nearly optimal control signal, and a critic NN which evaluates the performance of the overall system. All NN parameters are tuned online for both the NNs. By using Lyapunov techniques it is shown that all subsystems signals are uniformly ultimately bounded and that the synthesized subsystems inputs approach their corresponding nearly optimal control inputs with bounded error. Simulation results are included to show the effectiveness of the approach.

A state space model for online polyphonic audio-score alignment

Abstract: We present a novel online audio-score alignment approach for multi-instrument polyphonic music. This approach uses a 2-dimensional state vector to model the underlying score position and tempo of each time frame of the audio performance. The process model is defined by dynamic equations to transition between states. Two representations of the observed audio frame are proposed, resulting in two observation models: a multi-pitch-based and a chroma-based. Particle filtering is used to infer the hidden states from observations. Experiments on 150 music pieces with polyphony from one to four show the proposed approach outperforms an existing offline global string alignment-based score alignment approach. Results also show that the multi-pitch-based observation model works better than the chroma-based one.

Inference on core surface flow from observations and 3‐D dynamo modelling

Abstract: SUMMARY We show how a 3-D, self-consistent numerical model of the geodynamo can be used as the subjective prior information for the determination of Earth’s core surface flows from the geomagnetic field and its secular variation. This is achieved by estimating those parts of the numericalmodelstatevectorhiddenfromtheobservations,throughastandardKalmanfiltering (or stochastic inverse) procedure, where the Kalman gain matrix is based on the multivariate statisticsofthegeodynamomodel.Toallowforadirectcomparisonwithobservations,thefield variablesenteringthesestatisticsarescaledfollowingtwoofthescalinglawsthathaverecently come to the fore in numerical dynamo modelling, which express the dependency of the secular variation timescale and the magnetic energy density on their respective control parameters. We perform test experiments with noisy synthetic data, showing good to excellent recovery of the hidden parts of the state vector. A geomagnetic field model parent to a candidate model to the 2010 release of IGRF is then used for a core surface flow estimation. The estimated flow confirms the presence of convective columns underneath America, whereas exhibiting a high level of equatorial symmetry. We suggest that the discrete state estimation problem considered here (in connection with the classical core flow problem) could be used generically as a means to assess the degree of geophysical realism of a given geodynamo model. More generally, this study opens the way to using scaling laws and multivariate statistics from numerical models in the broader context of geomagnetic data assimilation.

Minimizing Euclidian State Estimation Error for Linear Uncertain Dynamic Systems Based on Multisensor and Multi-Algorithm Fusion

Abstract: In this paper, a multisensor linear dynamic system with model uncertainty and bounded noises is considered. Based on previously developed set-valued estimation methods in terms of convex optimization, we propose several efficient algorithms of centralized sensor fusion, distributed sensor fusion, and multi-algorithm fusion to minimize the Euclidian estimation error of the state vector. Obviously, an ellipsoid/box with a larger “size” cannot be in general guaranteed to contain another ellipsoid/box with a smaller “size” since centers and shapes of the two ellipsoids/boxes may be different from each other. This fact and the complementary advantages of multiple sensors and multiple algorithms motivate us to construct multiple estimation ellipsoids/boxes squashed along each entry of the state vector as much as possible respectively by using the technique of multiple differently weighted objectives. Then intersection fusion of these estimation ellipsoids/boxes yields a final Euclidian-error-minimized state estimate. Numerical examples show that the new method with multi-algorithm at both the sensors and the fusion center can significantly reduce the Euclidian estimation error of the state.

Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors

Abstract: Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor. We apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise. The mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling. This article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck).

Target Tracking With Particle Filters Under Signal Propagation Delays

Abstract: Signal propagation delays are hardly a problem for target tracking with standard sensors such as radar and vision due to the fact that the speed of light is much higher than the speed of the target. This contribution studies the case where the ratio of the target and the propagation speed is not negligible, as in the case of sensor networks with microphones, geophones or sonars for instance, where the signal speed in air, ground and water causes a state dependent and stochastic delay of the observations. The proposed approach utilizes an augmentation of the state vector with the propagation delay in a particle filtering framework to compensate for the negative effects of the delays. The model of the physics rules governing the propagation delays is used in interaction with the target motion model to yield an iterative prediction update step in the particle filter which is called the propagation delayed measurement particle filter (PDM-PF). The performance of PDM-PF is illustrated in a challenging target tracking scenario by making comparisons to alternative particle filters that can be used in similar cases.

Information geometry of density matrices and state estimation

Abstract: Given a pure state vector |x and a density matrix , the function defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher–Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.

Generalised minimum variance control state-space design

Abstract: The main goal of this study is to propose a state-space design technique for the generalised minimum variance control. In this sense, the same results achieved in the transfer function design framework are granted by the state-space method. It simplifies the design procedure while avoiding the solution of the Diophantine equation. Instead, the minimum variance predictor is obtained by the direct feed-through of an estimated state vector using a Kalman filter designed directly from the state-space model and without the need to solve an algebraic Riccati difference equation. In this way, even when dealing with systems with long time delays, the design procedure requires only a small amount of work as compared to the classical Diophantine-dependent technique. The proof of equality between the transfer function and state-space methods is easily verified by simple linear algebra, showing that the Diophantine equation results are intrinsically embedded in the gains of the state-space predictor derived, which means that resultant polynomials of the Diophantine equation can also be obtained by construction with the new design method. Two simulation examples are given to demonstrate the proposed technique.

Functional observers for linear impulsive systems

Abstract: The problem of designing functional observers for linear impulsive systems is addressed. By applying suitable matrix transformations, sufficient conditions for the existence of the functional observers that are capable of exponentially estimating any given function of the state vector are derived. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Bang―bang control design for quantum state transfer based on hyperspherical coordinates and optimal time―energy control

Abstract: We present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang?bang or square-wavefunction controls. Optimal time?energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.

Dual optimal filters for parameter estimation of a multivariate autoregressive process from noisy observations

Abstract: This study deals with the estimation of a vector process disturbed by an additive white noise. When this process is modelled by a multivariate autoregressive (M-AR) process, optimal filters such as Kalman or H∞ filter can be used for prediction or estimation from noisy observations. However, the estimation of the M-AR parameters from noisy observations is a key issue to be addressed. Off-line or iterative approaches have been proposed recently, but their computational costs can be a drawback. Using on-line methods such as extended Kalman filter and sigma-point Kalman filter are of interest, but the size of the state vector to be estimated is quite high. In order to reduce this size and the resulting computational cost, the authors suggest using dual optimal filters. In this study, the authors propose to extend to the multi-channel case the so-called dual Kalman or H∞ filters-based scheme initially proposed for single-channel applications. The proposed methods are first tested with a synthetic M-AR process and then with an M-AR process corresponding to a mobile fading channel. The comparative simulation study the authors carried out with existing techniques confirms the effectiveness of the proposed methods.

Angular resolution limit for vector-sensor arrays: Detection and information theory approaches

Abstract: The Angular Resolution Limit (ARL) on resolving two closely spaced polarized sources using vector-sensor arrays is considered in this paper. The proposed method is based on the information theory. In particular, the Stein's lemma provides, asymptotically, a link between the probability of false alarm and the relative entropy between two hypothesis of a given statistical binary test. We show that the relative entropy can be approximated by a quadratic function in the ARL. This property allows us to derive and analyze a closed-form expression of the ARL. To illustrate the interest of our approach, the ARL, in the sense of the detection theory, is also derived. Finally, we show that the ARL is only sensitive to the norm of the polarization state vector and not to the particular values of the polarization parameters.

Method for the computer-assisted modeling of a technical system

Abstract: A method for computer-assisted modeling of a technical system is disclosed. At multiple different operating points, the technical system is described by a first state vector with first state variable(s) and by a second state vector with second state variable(s). A neural network comprising a special form of a feed-forward network is used for the computer-assisted modeling of said system. The feed-forward network includes at least one bridging connector that connects a neural layer with an output layer, thereby bridging at least one hidden layer, which allows the training of networks with multiple hidden layers in a simple manner with known learning methods, e.g., the gradient descent method. The method may be used for modeling a gas turbine system, in which a neural network trained using the method may be used to estimate or predict nitrogen oxide or carbon monoxide emissions or parameters relating to combustion chamber vibrations.

A Bootstrap Algebraic Multilevel Method for Markov Chains

Abstract: This work concerns the development of an algebraic multilevel method for computing state vectors of Markov chains. We present an efficient bootstrap algebraic multigrid (AMG) method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, and least squares fitting of test vectors. Our adaptive variational strategy for computation of the state vector of a given Markov chain is then a combination of this multilevel eigensolver and an associated additive multilevel preconditioned correction process. We show that the bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the steady state vector. An additional benefit of the bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes. This in turn allows for the use of the resulting multigrid hierarchy as a preconditioner to accelerate the generalized minimal residual (GMRES) iteration for computing an additive correction equation for the approximation to the steady state vector. Unlike other existing multilevel methods for Markov chains, our method does not employ any special processing of the coarse-level systems to ensure that stochastic properties of the fine-level system are maintained there. The proposed approach is applied to a range of test problems involving nonsymmetric M-matrices arising from stochastic matrices and showing promising results.