New Results and Matrix Representation for Daehee and Bernoulli Numbers and Polynomials

In this paper, we derive new matrix representation for Daehee numbers and polynomials, the lambda-Daehee numbers and polynomials and the twisted Daehee numbers and polynomials. This helps us to obtain simple and short proofs of many previous results on Daehee numbers and polynomials. Moreover, we obtained some new results for Daehee and Bernoulli numbers and polynomials

New Results on Higher-Order Daehee and Bernoulli Numbers and Polynomials

We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and short proofs of many previous results on higher order Daehee numbers and polynomials. Moreover, we obtained recurrence relation, explicit formulas and some new results for these numbers and polynomials. Furthermore, we investigated the relation between these numbers and polynomials and Stirling numbers, Norlund and Bernoulli numbers of higher order. The results of this article gives a generalization of the results derived very recently by El-Desouky and Mustafa [6]

ALGORITHMS AND OPTIMAL CONTROL FOR SPACECRAFT MAGNETIC ATTITUDE MANEUVERS

This study focused on providing applicable control solutions for spacecraft magnetic attitude control system. Basically, two main lines are pursued; first, developing detumbling control laws and second, an improvement in the three-axis attitude control schemes by extending magnetic rods activation time. Spacecraft, after separation from the launching mechanism, experiences a tumbling phase due to an undesired angular momentum. In this study, we present a new efficient variant of the B-dot detumbling law by introducing a substitute of the spacecraft angular velocity, based on the ambient magnetic field data. This B-dot law preserves the orthogonality, among the applied torque, dipole moment and magnetic field vectors. Most of the existing variants of the B-dot law in the literature don\u27t preserve this orthogonality. Furthermore, the problem of minimum-time spacecraft magnetic detumbling is revisited within the context of optimal control theory. Two formulations are presented; the first one assumes the availability of the angular velocity measurements for feedback. The second formulation assumes the availability of only the ambient magnetic field measurements in the feedback; the latter is considered another optimal-based B-dot law. A reduction in detumbling time is fulfilled by the proposed laws along with less power consumption for the proposed B-dot laws. In magnetic attitude maneuvers, magnetic rods and magnetometers usually operate alternatively, to avoid the magnetic rods\u27 noise effect on magnetometers measurements. Because of that, there will be no control authority over the spacecraft during the magnetometer measurement period. Hence longer maneuver times are usually experienced. In this study, a control scheme that enables the extension of the magnetic rods’ activation time is developed, regardless of the attitude control law. The key concept is replacing the real magnetic field measurement by a pseudo measurement, which is computed based on other sensors measurements. By applying a known command to the spacecraft and measuring the spacecraft response, it is possible to compute the ambient magnetic field around the spacecraft. The system mathematical singularity is solved using the Tikhonov regularization approach. Another developed approach estimates the magnetic field, using a relatively simple and fast dynamic model inside a Multiplicative Extended Kalman Filter. A less maneuver time with less power consumption are fulfilled. These control approaches are further validated using real telemetry data from CASSIOPE mission. This dissertation develops a stability analysis for the spacecraft magnetic attitude control, taking into consideration the alternate operation between the magnetic rods and the magnetometers. It is shown that the system stability degrades because of this alternate operation, supporting the proposed approach of extending the operation time of the magnetic rods