Least-squares Solutions of Linear Differential Equations

This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second order differential equations. The proposed method has two steps. The first step consists of writing a constrained expression, introduced in Ref. \cite{Mortari}, that has embedded the differential equation constraints. These expressions are given in term of a new unknown function, $g (t)$, and they satisfy the constraints, no matter what $g (t)$ is. The second step consists of expressing $g (t)$ as a linear combination of $m$ independent known basis functions, $g (t) = \mathbf{\xi}^T \mathbf{h} (t)$. Specifically, Chebyshev orthogonal polynomials of the first kind are adopted for the basis functions. This choice requires rewriting the differential equation and the constraints in term of a new independent variable, $x\in[-1, +1]$. The procedure leads to a set of linear equations in terms of the unknown coefficients vector, $\mathbf{\xi},$ that is then computed by least-squares. Numerical examples are provided to quantify the solutions accuracy for initial and boundary values problems as well as for a control-type problem, where the state is defined in one point and the costate in another point.Comment: Study partially presented as: Mortari, D. "Least-squares Solutions of Linear Differential Equations,? AAS 17-256, 2017 AAS/AIAA Space Flight Mechanics Meeting Conference, San Antonio, TX, February 5-9, 201

Non-dimensional Star-Identification

This study introduces a new "Non-Dimensional" star identification algorithm to reliably identify the stars observed by a wide field-of-view star tracker when the focal length and optical axis offset values are known with poor accuracy. This algorithm is particularly suited to complement nominal lost-in-space algorithms, which may identify stars incorrectly when the focal length and/or optical axis offset deviate from their nominal operational ranges. These deviations may be caused, for example, by launch vibrations or thermal variations in orbit. The algorithm performance is compared in terms of accuracy, speed, and robustness to the Pyramid algorithm. These comparisons highlight the clear advantages that a combined approach of these methodologies provides.Comment: 17 pages, 10 figures, 4 table

Theory of functional connections applied to nonlinear programming under equality constraints

This paper introduces an efficient approach to solve quadratic programming problems subject to equality constraints via the Theory of Functional Connections. This is done without using the traditional Lagrange multipliers approach, and the solution is provided in closed-form. Two distinct constrained expressions (satisfying the equality constraints) are introduced. The unknown vector optimization variable is then the free vector \B{g}, introduced by the Theory of Functional Connections, to derive constrained expressions. The solution to the general nonlinear programming problem is obtained by the Newton's method in optimization, and each iteration involves the second-order Taylor approximation, starting from an initial vector \B{x}^{(0)} which is a solution of the equality constraint. To solve the quadratic programming problems, we not only introduce the new approach but also provide a numerical accuracy and speed comparisons with respect to MATLAB's \verb"quadprog". To handle the nonlinear programming problem using the Theory of Functional Connections, convergence analysis of the proposed approach is provided.Comment: 21 pages, 1 figure, 1 table, submitted to Journal of Computational and Applied Mathematic

ICNPAA-2000

Abstract A general mathematical formulation of the n × n proper Orthogonal matrix, that corresponds to a rigid rotation in n-dimensional real Euclidean space, is given here. It is shown that a rigid rotation depends on an angle (principal angle) and on a set of (n − 2) principal axes. The latter, however, can be more conveniently replaced by only 2 Orthogonal directions that identify the plane of rotation. The inverse problem, that is, how to compute these principal rotation parameters from the rotation matrix, is also treated. In this paper, the Euler Theorem is extended to rotations in n-dimensional spaces by a constructive proof that establishes the relationship between orientation of the displaced Orthogonal axes in n dimensions and a minimum sequence of rigid rotations. This fundamental relationship, which introduces a new decomposition for proper Orthogonal matrices (those identifying an orientation), can be expressed either by a product or a sum of the same rotation matrices. A similar decomposition in terms of the Skew-Symmetric matrices is also given. The extension of the rigid rotation formulation to n-dimensional complex Euclidean spaces, is also provided. Finally, we introduce the Ortho-Skew real matrices, which are simultaneously proper Orthogonal and Skew-Symmetric and which exist in even dimensional spaces only, and the Ortho-Skew-Hermitian complex matrices which are Orthogonal and Skew-Hermitian. The Ortho-Skew and the Ortho-SkewHermitian matrices represent the extension of the scalar imaginary to the matrix field

Ortho-Skew and Ortho-Sym Matrix Trigonometry

Abstract This paper introduces some properties of two sets of matrices, the OrthoSkew, which are simultaneously Orthogonal and Skew-Hermitian, and the real Ortho-Sym matrices, which are Orthogonal and Symmetric. The introduced relationships, all demonstrated, consist of closed-form compact expressions of trigonometric and hyperbolic functions that show these matrices working as angles in the matrix field. The analogies with trigonometric and hyperbolic functions, such as the periodicity of the trigonometric functions and some properties of the inverse functions are all shown. Additional expressions are derived for some other functions of matrices such as the logarithm, exponential, inverse, and power functions. All these relationships show that the Ortho-Skew and the Ortho-Sym matrices can be considered as the extension to the matrix field of the imaginary and the real units, respectively

Ortho-Skew and Ortho-Sym Matrix Trigonometry

Abstract This paper introduces some properties of two families of matrices: the Ortho-Skew, which are simultaneously Orthogonal and Skew-Hermitian, and the real Ortho-Sym matrices, which are Orthogonal and Symmetric. These relationships consist of closed-form compact expressions of trigonometric and hyperbolic functions that show that multiples of these matrices can be interpreted as angles. The analogies with trigonometric and hyperbolic functions, such as the periodicity of the trigonometric functions, are all shown. Additional expressions are derived for some other functions of matrices such as the logarithm, exponential, inverse, and power functions. All these relationships show that the Ortho-Skew and the Ortho-Sym matrices can be respectively considered as matrix extensions of the imaginary and the real units