Positive Definite Weighting Research Articles

An Algebraic Solution of the GPS Equations

The global positioning system (GPS) equations are usually solved with an application of Newton's method or a variant thereof: Xn+1 = xn + H-1(t - f(xn)). (1) Here x is a vector comprising the user position coordinates together with clock offset, t is a vector of tour pseudorange measurements, and H is a measurement matrix of partial derivatives H = fx· In fact the first fix of a Kalman filter provides a solution of this type. If more than four pseudoranges are available for extended batch processing, H-1 may be replaced by a generalized inverse (HTWH)-1HTW, where W is a positive definite weighting matrix (usually taken to be the inverse of the measurement covariance matrix). This paper introduces a new method of solution that is algebraic and noniterative in nature, computationally efficient and numerically stable, admits extended batch processing, improves accuracy in bad geometric dilution of precision (GDOP) situations, and allows a "cold start" in deep space applications.

Design of linear feedback laws for bilinear systems

Abstract A method for the design of linear state feedback laws for bilinear systems is presented. Using a Liapunov approach, a class of stabilizing feedback amplifier matrices is constructed, depending on three positive definite weighting matrices which can be selected freely. The computation of the amplifiers requires the solution, by successive approximations, of an algebraic Riccati equation for which a positive definite solution must be found. The relative magnitudes of the weighting matrices, expressed in terms of scaling factors, constitute the main design parameters. The influence of these parameters on some important dynamic characteristics of the system is analysed, thus allowing to adapt the parameter choice to the specifications of the design.

S-Orthogonal Projection Operators as Asymptotic Solutions of a Class of Matrix Differential Equations

A class of nonlinear autonomous matrix differential equations is considered. Two special cases of this class yield the differential equations of adaptive network models which were initially introduced in connection with idealized neuron networks. It is shown that these equations exhibit an asymptotic behavior that is intimately related with the matrix form of the Gram–Schmidt orthogonalization algorithm in a real vector space, with respect to an inner product with an arbitrary positive definite symmetric weight matrix S.

Recent advances in gradient algorithms for optimal control problems

This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice University. The following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is given. Next, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional restoration. Then, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration algorithm. Next, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this seems to be only a minor modification of the basic problem. In practice, the change is considerable in that it enlarges dramatically the number and variety of problems of optimal control which can be treated by gradient-restoration algorithms. Indeed, by suitable transformations, almost every known problem of optimal control theory can be brought into this scheme. This statement applies, for instance, to the following situations: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with equality constraints on the time rate of change of the state, (iv) problems with control inequality constraints, (v) problems with state inequality constraints, and (vi) problems with inequality constraints on the time rate of change of the state. Finally, the simultaneous presence of nondifferential constraints and multiple subarcs is considered. The possibility that the analytical form of the functions under consideration might change from one subarc to another is taken into account. The resulting formulation is particularly relevant to those problems of optimal control involving bounds on the control or the state or the time derivative of the state. For these problems, one might be unwilling to accept the simplistic view of a continuous extremal arc. Indeed, one might want to take the more realistic view of an extremal arc composed of several subarcs, some internal to the boundary being considered and some lying on the boundary. The paper ends with a section dealing with transformation techniques. This section illustrates several analytical devices by means of which a great number of problems of optimal control can be reduced to one of the formulations presented here. In particular, the following topics are treated: (i) time normalization, (ii) free initial state, (iii) bounded control, and (iv) bounded state.

Optimal design of geodetic nets, 2

Let the configuration or first-order design matrix A of a geodetic net be given and let the weight or second-order design matrix P of m observations be unknown. For some predetermined choice of the dispersion matrix Σ of the n unknown net coordinates, the unique minimum Euclidean norm solution for the weight matrix P is P = (A′)+Σ−1A+. A′ indicates the transpose and A+ the pseudoinverse of the (in general) rectangular matrix A. For a general first-order design matrix A and a given Σ, there does not exist a solution for a diagonal positive definite weight matrix P. Our solution for P belongs to the first category of an optimal design of C. R. Rao. The designs P = I and P = (A′)+Σ−1A+ yield the same best linear unbiased estimator for the unknowns, but the variance covariance matrix for these designs is different. Examples, especially the structure by G. I. Taylor and T. Karman for the homogeneous and isotropic geodetic net, illustrate theoretical results.

Two-Photon Counting Statistics for Laser and Chaotic Radiation

The relation between the two-quantum photocounting distribution p(n, T) and the intensity fluctuations of the radiation incident on the detector is obtained and compared with the one-quantum results. The actual statistics are evaluated for several cases of interest, including a chaotic source, an amplitude-stabilized wave, and the output of a single-mode laser (Van der Pol oscillator). The two-quantum count rate 〈n〉/T is found to depend on the mean-square intensity of the radiation, in contrast to the one-quantum count rate which is proportional to the mean intensity. The effects of photon correlations in the radiation beam become apparent since the two-quantum distributions manifest a distinctly more positive second derivative than the corresponding one-quantum distributions. Both the low-and the high-count probabilities are therefore increased at the expense of counts near the mean. The quantum-theoretical treatment is found to be equivalent to the semiclassical treatment for density operators possessing a positive-definite weight function in the P-representation. Some possible experiments to verify the theory are discussed and shown to be feasible.

Correlation Functions for Coherent Fields

A factorization condition which must be satisfied by the first $n$ correlation functions for the electromagnetic field operators has been used to define nth-order coherence. The first-order coherence condition has been shown to imply maximum fringe contrast in interference patterns. In the present paper we investigate the mathematical consequences of assuming the condition for maximum fringe contrast. By considering the correlation functions as scalar products and formulating rigorous inequalities for them we are able to show that the assumed condition in turn implies factorization of the first-order correlation function. By extending the same methods we are able to show that all of the higher order correlation functions factorize into forms similar to those required for full coherence, but differing from them through the inclusion of a sequence of constant numerical factors. These coefficients are shown to furnish a convenient description of the higher-order coherence properties of the field. Their values are presented for some typical examples. We derive a number of inequalities satisfied by the coefficients for the case of fields which possess positive-definite weight functions in the $P$ representation. Some inequalities obeyed by the correlation functions for such fields are derived as well.