Showing papers on "State vector published in 1971"

Optimal state-vector estimation for non-Gaussian initial state-vector

Abstract: The optimal estimate, in the mean-square-error sense, of state-vector of a linear system excited by zero-mean white Gaussian noise with non-Gaussian initial state-vector is obtained. Both the optimal estimate and the corresponding error covariance matrix are given. It is shown that the optimal estimator consists of two parts: a linear estimator that is obtained from a Kalman filter and a nonlinear estimator. In addition, the a posteriori probability p(x_{k}/\lambda_{k}) is also given.

Optimal control of systems with state-dependent time delay†

Abstract: Necessary conditions for optimal control of systems containing a time delay that is a function of the state of the system and of time are derived by utilizing calculus of variations. The time delay may be in the state vector and in the control vector. The state vector and the control vector can be constrained by inequality constraints. A transformation to eliminate state variable inequality constraints by increasing the dimensions of state space, developed by Jacobson for an undelayed system, is extended to a system with time delays. Necessary conditions to obtain an optimal delay are shown, and an example of finding an optimal delay is included. A gradient algorithm for systems with state dependent time delays has been developed.

DC solution of nonlinear state-space equations in circuit analysis

Abstract: A reliable algorithm is described for calculation of the initial condition state vector as formulated in network transient analysis programs. The algorithm is based on a combination of Davidenko's parameter-stepping method and Broyden's method. It has been found to give fast reliable solutions even in cases for which most Newton-Raphson methods fail to converge, as well as for circuits that produce singular systems of equations at the starting point.

The estimation of the constituent densities of the upper atmosphere by means of a recursive filtering algorithm

Abstract: The structure of the upper atmosphere can be indirectly probed by light in order to determine the global density structure of ozone, aerosols, and neutral atmosphere. Scattered and directly transmitted light is measured by a satellite and is shown to be a nonlinear function of the state which is defined to be a point-wise decomposition of the density profiles. Dynamics are imposed on the state vector and a structured estimation problem is developed. The estimation of these densities is then performed using a linearized Kalman-Bucy filter and a linearized Kushner-Stratonovich filter.

Sub-optimal solutions of linear control and eigenvalue pattern problems†

Abstract: The present paper deals with sub-optimal solutions for problems of minimizing integral cost fimetioiuils which consider the state variables alone for stationary linear processes. The control vector and its time derivative are assumed to be bounded. The resulting control policies are linear with respect to the state vector. The performance of the proposed sub-optimal policies is compared with that of classical control systems and a procedure for sub-optimal eigenvalue patterns is presented for systems of any order.

Smooth Empirical Bayes Estimation of Observation Error Variances in Linear Systems

Abstract: A filter set is developed for estimating the state vector and observation error variances in a discrete-time linear system by use of empirical Bayes techniques. The error variances are assumed to be random and to vary over time. No initial conditions or distributional assumptions are required for the error variances, but all other assumptions for the Kalman filter are assumed to hold. The treatment is analytical, and a Monte Carlo simulation is used to verify the results. Graphs are presented which compare performance with the ideal case of known variances. The filter was found to converge fairly rapidly for the examples considered.

A Second- and Higher Order Perturbation Analysis of Two-Body Trajectories

Abstract: An approach to the development of a second- and higher-order perturbation theory for two- body trajectories is presented. It is shown that the higher order analysis can be developed in a systematic manner in terms of series solutions to the two-body problem as functions of the time variable. Simple algebraic recursive formulas for the determination of the series coeffi- cients are derived and the radii of convergence for these series solutions are determined. Their accuracy and the rate of convergence are investigated in a number of numerical cases. ingly more precise mission objectives and with long flight times. It is the purpose of this paper to present an approach to the higher order perturbation analysis of two-body trajec- tories. A second-order theory, involving second partial derivatives of the six-dimensional state vector of a spacecraft at a given time with respect to the state vector at an initial epoch, is con- sidered in detail. It is shown that a second as well as higher order theory can be developed in a simple and systematic manner using power series solutions of the two-body problem as functions of the time variable, a technique suggested by the theory of Lie series.1 In this paper, the series representa- tions associated with the second-order theory are developed; they are uniformly valid for all two-body conies. It is shown that the coefficients of these series expansions can be generated by simple algebraic recursive formulas, and their radii of con- vergence can be determined analytically. The rate of convergence and truncation error associated with these series representations are investigated in a number of numerical cases. Results indicate that the convergence characteristics and accuracies can in general be estimated in terms of the eccentricity and the position vector at the initial epoch. Generally speaking, the results exhibit good con- vergence characteristics for elliptical orbits covering less than one orbital revolution. Results for hyperbolic orbits are less favorable.

Quantum theory of state reduction and measurement.

Abstract: The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrodinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrodinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.

Applications of Generalized Inverses to Estimation in Dynamic Systems

Abstract: Generalized matrix inverses are used to obtain an estimation procedure for estimation of the state vector of a dynamic system. This sequential procedure is studied analytically with respect to the choice of an arbitrary vector. The covariance matrix of the estimator is determined and compared to the optimal Kalman type procedure. A numerical example illustrates the procedure and compares it to the optimal one.

An analysis of approach navigation accuracy and guidance requirements for the grand tour mission to the outer planets

Abstract: The navigation and guidance process for the Jupiter, Saturn and Uranus planetary encounter phases of the 1977 Grand Tour interior mission was simulated. Reference approach navigation accuracies were defined and the relative information content of the various observation types were evaluated. Reference encounter guidance requirements were defined, sensitivities to assumed simulation model parameters were determined and the adequacy of the linear estimation theory was assessed. A linear sequential estimator was used to provide an estimate of the augmented state vector, consisting of the six state variables of position and velocity plus the three components of a planet position bias. The guidance process was simulated using a nonspherical model of the execution errors. Computation algorithms which simulate the navigation and guidance process were derived from theory and implemented into two research-oriented computer programs, written in FORTRAN.

Design of desirable airplane handling qualities via optimal control

Abstract: The technique known as implicit model following allows desired closed-loop characteristics of a system to be included in an optimal control algorithm. Previous authors have used a model of the form x = Lx9 where x is the state vector. The optimal control algorithm then computes the feedback which makes the system response optimally close to the model. This paper extends this approach to include all desired handling qualities by using the model x = Lx + 7V8, where 8 is the vector of pilot commands. The algorithm which is derived computes both optimal feedback and optimal feedforward from 8 to the controls. Algorithms are given for both sampled-data and continuous control. General guidelines for choosing L and TV are presented. An example is given for the design of the landing approach control for a short takeoff and landing (STOL) airplane.

Calculation of trajectories using constant and slowly varying functions

Abstract: A method is presented for calculating trajectories for the restricted problem of three bodies which utilizes conic propagation of the state vector with frequency correction of position and velocity by means of a constant or slowly varying function. This method of calculating trajectories was applied to the planar circular restricted three body problem, the planar elliptic restricted problem, and the ephemeral restricted problem. Two methods (the refined method and the straight forward method) of determining the direction of the position correction are presented for the circular restricted problem and the elliptic restricted problem of three bodies. Only the straight forward method was used with the ephemeral restricted problem. The earth, the moon, and a space vehicle comprise the restricted three body model that is used.

A computer-aided optimization of a high pressure control system

Abstract: The development of a high pressure control system for isostatic compaction processes is described. The transient response of the system, which is designed to follow continuous pressure-time cycles, is optimized with respect to an index of performance. The input and the system transfer function are expressed in state vector form for use with a polar direct search technique, which is accelerated by the imposition of stability constraints. The system is generally applicable to processes where accurate, continuous pressure control.is essential.