Initial State Vector Research Articles

Quantum brachistochrone problem for a spin-1 system in a magnetic field

We study the quantum brachistochrone problem for a spin-1 system in a magnetic field of constant absolute value. Such a system gives us the possibility to examine in detail the statement that the state vectors realizing evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006); Brody and Hook, J. Phys. A 39, L167 (2006)]. Considering an explicit example, we show the existence of a quantum brachistochrone with the minimal possible time; however, the state vector we study leaves the subspace spanned by the initial and final state vectors during evolution. This is the result of our choice of a more constrained Hamiltonian than the one assumed in the general quantum brachistochrone problem. It is worth noting that such an evolution, being more complicated, is time optimal but with larger time than in the general case. This might be important for experiments, where a general Hamiltonian with all the allowed parameters is difficult to implement, but a constrained one, depending on the magnetic field, can be realized. However, for the preconstrained Hamiltonian not all final states are accessible. The present result does not contradict the general statement of the quantum brachistochrone problem, but gives additional insight into possible realization of the time-optimal passage.

State Estimation With Initial State Uncertainty

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The problem of state estimation with initial state uncertainty is approached from a statistical decision theory point of view. The initial state is regarded as deterministic and unknown. It is only known that the initial state vector belongs to a specified parameter set. The (frequentist) risk is considered as the performance measure and the minimax approach is adopted. Minimax estimators are derived for some important cases of unbounded parameter sets. If the parameter set is bounded, a method of finding estimators whose maximum risk is arbitrarily close to that of a minimax estimator is provided. This method is illustrated with an example in which an estimator whose maximum risk is at most 3% larger than that of a minimax estimator is derived. </para>

A novel health assessment technique with minimum information

SUMMARY A novel structural health assessment method is proposed for detecting defects at the element level using only minimum measured response information considering imperfect mathematical model representing the structure, various sources of uncertainty in the mathematical model, and uncertainty in the measured response information. It is a linear time domain finite element-based system identification procedure capable of detecting defects at the local element level using only limited response measurements and without using any information on the exciting dynamic forces. The technique is a combination of the modified iterative least-squares technique with unknown input excitation (MILS-UI) proposed earlier by the research team at the University of Arizona and the extended Kalman filter with a weighted global iteration (EKF-WGI) procedures to address different sources of uncertainty in the problem. The authors denote the new method as GILS-EKF-UI. To implement the concept, a two-stage substructure approach is used. In the first stage, a substructure is selected that satisfies all the requirements for the MILS-UI procedure. This provides information on the initial state vector and the input excitation required to implement any EKF-based procedure. In the second stage, the EKF-WGI is applied to identify the whole structure. This way the whole structure, defect free or defective, can be identified with limited response measurements in the presence of uncertainty and without using excitation information. The theoretical concept behind this novel method is presented in this paper. Copyright # 2007 John Wiley & Sons, Ltd.

On the outputs of linear control systems

This paper studies autonomous, single-input, single-output linear control systems on finite time intervals. The object of interest is the output operator O , which associates to each input function and initial state vector the corresponding system output. Main result: If the system has relative degree r < ∞ , then for any “admissible” Banach space U of inputs, O is a bounded operator taking U × C n onto the “Sobolev space” of complex functions f ∈ C ( r − 1 ) ( [ 0 , T ] ) for which the ( r − 1 ) -order derivative f ( r − 1 ) is absolutely continuous, with f ( r ) ∈ U . This completes recent results of Jönsson and Martin [Ulf Jönsson, Clyde Martin, Approximation with the output of linear control systems, J. Math. Anal. Appl. 329 (2007) 798–821] who showed that if the system is minimal and U is either L 2 ( [ 0 , T ] ) or C ( [ 0 , T ] ) , then O : U × C n → U has dense range.

Low computation and low latency algorithms for distributed sensor network initialization

In this paper, we show how an underlying system’s state vector distribution can be determined in a distributed heterogeneous sensor network with reduced subspace observability at the individual nodes. The presented algorithm can generate the initial state vector distribution for networks with a variety of sensor types as long as the collective set of measurements from all the sensors provides full state observability. Hence the network, as a whole, can be capable of observing the target state vector even if the individual nodes are not capable of observing it locally. Initialization is accomplished through a novel distributed implementation of the particle filter that involves serial particle proposal and weighting strategies that can be accomplished without sharing raw data between individual nodes. If multiple events of interest occur, their individual states can be initialized simultaneously without requiring explicit data association across nodes. The resulting distributions can be used to initialize a variety of distributed joint tracking algorithms. We present two variants of our initialization algorithm: a low complexity implementation and a low latency implementation. To demonstrate the effectiveness of our algorithms we provide simulation results for initializing the states of multiple maneuvering targets in smart sensor networks consisting of acoustic and radar sensors.

Huygens’ entry and descent through Titan's atmosphere—Methodology and results of the trajectory reconstruction

The European Space Agency's Huygens probe separated from the NASA Cassini spacecraft on 25 December 2004, after having been attached for a 7-year interplanetary journey and three orbits around Saturn. The probe reached the predefined NASA/ESA interface point on 14 January 2005 at 09:05:52.523 (UTC) and performed a successful entry and descent sequence. The probe softly impacted on Titan's surface on the same day at 11:38:10.77 (UTC) with a speed of about 4.54 m/s. The probe entry and descent trajectory was reconstructed from the estimated initial state vector provided by the Cassini Navigation team, the probe housekeeping data, and measurements from the scientific payload. This paper presents the methodology and discuss the results of the reconstruction effort. Furthermore the probe roll rate was reconstructed prior to the main entry phase deceleration pulse and throughout the entire descent phase under the main and drogue parachute.

Ballistic missile trajectory prediction using a state transition matrix

A method for the determination of the trajectory of a ballistic missile over a rotating, spherical Earth given only the launch position and impact point has been developed. The iterative solution presented uses a state transition matrix to correct the initial conditions of the ballistic missile state vector based upon deviations from a desired set of final conditions. A six-degree-of-freedom simulation of a ballistic missile is developed to calculate the resulting trajectory. Given the initial state vector of the ballistic missile, the trajectory is simulated and the state transition matrix propagated along the trajectory to the impact point. The error in the final state vector is calculated and elements of the initial state vector are corrected using the state transition matrix. The process is repeated until the ballistic missile impacts the target location within a pre-defined miss distance tolerance. The result of this research is an analysis tool which accurately solves for the initial state vector of the ballistic missile.

A Gravity Field Model Derived From the Short Dynamical Arcs of Champ

AbstractThe dynamical method for recovering gravity field model based on the dynamical orbits and accelerometer data of CHAMP is discussed and the mathematical algorithm is educed, in which the accelerometer scale and bias, the satellite's initial state vector and the model coefficients are estimated simultaneously. The gravity field model TJCHAMP01S has been computed with this algorithm from the 120‐day CHAMP data including dynamical orbits and accelerometer data, and the model is validated using various criteria. The results show that the model TJCHAMP01S is more accurate than EGM96 and EIGEN‐1S model of the same degree and order.

Harnessing Machine Learning to Improve the Success Rate of Stimuli Generation

The initial state of a design under verification has a major impact on the ability of stimuli generators to successfully generate the requested stimuli. For complexity reasons, most stimuli generators use sequential solutions without planning ahead. Therefore, in many cases, they fail to produce a consistent stimuli due to an inadequate selection of the initial state. We propose a new method, based on machine learning techniques, to improve generation success by learning the relationship between the initial state vector and generation success. We applied the proposed method in two different settings, with the objective of improving generation success and coverage in processor and system level generation. In both settings, the proposed method significantly reduced generation failures and enabled faster coverage

Statistical reconstruction of qutrits

We discuss a procedure of measurement followed by the reproduction of the quantum state of a three-level optical system - a frequency- and spatially degenerate two-photon field. The method of statistical estimation of the quantum state based on solving the likelihood equation and analyzing the statistical properties of the obtained estimates is developed. Using the root approach of estimating quantum states, the initial two-photon state vector is reproduced from the measured fourth moments in the field . The developed approach applied to quantum states reconstruction is based on the amplitudes of mutually complementary processes. Classical algorithm of statistical estimation based on the Fisher information matrix is generalized to the case of quantum systems obeying Bohr's complementarity principle. It has been experimentally proved that biphoton-qutrit states can be reconstructed with the fidelity of 0.995-0.999 and higher.

The Inclusion of 3D Prognostic Cloud and Precipitation Variables in Adjoint Calculations

Abstract The potential of the inclusion of two prognostic variables for cloud condensate and precipitation content in adjoint calculations has been investigated in the framework of Meteo-France's operational spectral global model, ARPEGE. The tangent linear and adjoint versions of the recently developed explicit large-scale cloud and precipitation scheme have been coded and tested. In particular, the validity of the linear hypothesis has been checked when finite-size perturbations are applied to the initial model state vector. Then, the new explicit cloud and precipitation scheme and its adjoint version have been integrated in a set of sensitivity experiments. These allowed the assessment of the relative impacts of initial perturbations imposed on each model variable on the variations of various meteorological aspects of the forecast that relate either to the dynamics or to the cloud and precipitating activity of some extratropical cloud systems.

Satellite orbit determination using satellite gravity gradiometry observations in GOCE mission perspective

Abstract. Between the years 2004 and 2005 the launch of the first gradiometric satellite is planned. This satellite will be an important element of the Gravity Field and Steady – State Ocean Circulation Explorer Mission (GOCE). This mission is one of the reasons for performing the simulation research of the Satellite Gravity Gradiometry. Our work contains the theory description and simulation results of the satellite orbit determination using the gravity tensor observations. In the process of the satellite orbit determination the initial dynamic state vector corrections are obtained. These corrections are estimated by means of the gravity gradiometry measurements. The performed simulations confirm the possibility of satellite orbit determination by means of the gravity tensor observations.Key words. satellite geodesy, satellite gradiometry, satellite orbits

On the joint inversion of SGG and SST data from the GOCE mission

Abstract. The computation of spherical harmonic coefficients of the Earth’s gravity field from satellite-to-satellite tracking (SST) data and satellite gravity gradiometry (SGG) data is considered. As long as the functional model related to SST data contains nuisance parameters (e.g. unknown initial state vectors), assembling of the corresponding normal matrix must be supplied with the back-substitution operation, so that the nuisance parameters are excluded from consideration. The traditional back-substitution algorithm, however, may result in large round-off errors. Hence an alternative approach, back-substitution at the level of the design matrix, is implemented. Both a stand-alone inversion of either type of data and a joint inversion of both types are considered. The conclusion drawn is that the joint inversion results in a much better model of the Earth’s gravity field than a standalone inversion. Furthermore, two numerical techniques for solving the joint system of normal equations are compared: (i) the Cholesky method based on an explicit computation of the normal matrix, and (ii) the pre-conditioned conjugate gradient method (PCCG), for which an explicit computation of the entire normal matrix is not needed. The comparison shows that the PCCG method is much faster than the Cholesky method.Key words. Earth’s gravity field, GOCE, satellite-tosatellite tracking, satellite gravity gradiometry, backsubstitution

Filtering and smoothing of state vector for diffuse state‐space models

Abstract. This paper presents exact recursions for calculating the mean and mean square error matrix of the state vector given the observations for the multi‐variate linear Gaussian state‐space model in the case where the initial state vector is (partially) diffuse.

The state vector methods for space axisymmetric problems in multilayered piezoelectric media

The state vector equations for space axisymmetric problems of transversely isotropic piezoelectric media are established from the basic equations. Using the Hankel transform, the state vector equations are reduced to a system of ordinary differential equations. An analytical solution of the problems in the Hankel transform space is presented in the form of the product of initial state vector and transfer matrix. The transfer matrices are given for the three distinct eigenvalues. Applications of the solutions are discussed. An analytical solution for the transversely isotropic semi-infinite piezoelectric media subjected to concerted point loads on the surface z=0 is presented in the Hankel transform space. Using transfer matrix and the continuity conditions at the layer interfaces, the general solution formulation of N-layered transversely isotropic piezoelectric media is given. A selected set of numerical solutions is presented for a layered semi-infinite piezoelectric solid.

A parametric convex optimal control problem for a linear system

The dependence of the solutions of a terminal optimal control problem on a parameter in the initial state vector is investigated. Attention is devoted mainly to the behaviour of the solution in the neighbourhood of a non-regular point. On the basis of the results, a method is proposed for constructing solutions of the problem for all parameter values. In practical work it is often important to know not only the solution of an optimal control problem for fixed parameter values, but also the dependence of the solution on the parameters, which enables one to estimate how the solution may vary when the parameters fluctuate. In addition, a knowledge of the dependence of the solutions of optimal control problems on the parameters provides the basis for methods of constructing feedback controls [1, 2], as well as stabilization and estimation methods based on the moving horizon strategy [3–5]. Numerical solutions of such problems are generally achieved by continuation of the solution with respect to a parameter [6–9]. The greatest difficulties in applying such methods arise in the case when the “actual” value of the parameter is a non-regular point. Therefore, in most publications devoted to sensitivity analysis and to investigating the parameter-dependence of the solutions, it is assumed that all parameter values are regular, or of degree of non-regularity one. In this paper no such assumptions are made.

Discrete‐Time Risk‐Sensitive Filters with Non‐Gaussian Initial Conditions and Their Ergodic Properties

ABSTRACTIn this paper, we study asymptotic stability properties of risk‐sensitive filters with respect to their initial conditions. In particular, we consider a linear time‐invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk‐sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk‐sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non‐Gaussian initial conditions, we derive the expression for the risk‐sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon‐Nikodym derivative, we show that a suboptimal risk‐sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk‐sensitive filter for non‐Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims.

Geometric phases, symmetries of dynamical invariants and exact solution of the Schrödinger equation

We introduce the notion of the geometrically equivalent quantum systems (GEQSs) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQSs. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T -periodic, the corresponding class of GEQSs includes a system with a T -periodic Hamiltonian. We apply our general results to study the classes of GEQSs that include a system with a cranked Hamiltonian H( t) = e −iKt H0e iKt . We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator.

An algorithmic synthesis of discrete optimal singular adaptive observers for multiple-input single-output linear systems

An algorithmic synthesis of a new generation of discrete adaptive observers, including optimal estimators of the initial state vector, identifiers of parameters and discrete optimal singular (full and degenerate) adaptive observers of the current state vector for multiple-input single-output linear discrete systems is suggested. The estimation of the initial state vector elements is not typical of the methods for synthesis of discrete adaptive observers, based only on identifiers. The synthesized algorithm has features such as large-scale granularity and natural parallelism, which also allow its software and hardware implementation on a parallel computer architecture.

An algorithmic synthesis of discrete optimal singular adaptive observers for multiple-input single-output linear systems

An algorithmic synthesis of a new generation of discrete adaptive observers, including optimal estimators of the initial state vector, identifiers of parameters and discrete optimal singular (full and degenerate) adaptive observers of the current state vector for multiple-input single-output linear discrete systems is suggested. The estimation of the initial state vector elements is not typical of the methods for synthesis of discrete adaptive observers, based only on identifiers. The synthesized algorithm has features such as large-scale granularity and natural parallelism, which also allow its software and hardware implementation on a parallel computer architecture.