Comparative stability analysis and performance of magnetic controllers for bias momentum satellites

Abstract

This paper reexamines magnetic controllers for roll/yaw control of Earth-pointing bias momentum satellites in circular orbits using pitch dipoles. A general magnetic controller employs three gains, first for controlling precession, second for damping nutations, and third for stiffening the roll/yaw motion directly, using attitude angles and rates for feedback. For an orbit-averaged magnetic field, dependence on these gains of the closed-loop roots associated with precession and nutation and their damping coefficients is investigated via eigenanalysis and root loci. Stability inequalities constraining the three gains are developed by applying Routh-Hurwitz criteria, and the stability boundaries so obtained are spot checked with Floquet theory. The stability and performance of a three-gain controller is compared with that of 1) the classical A/t x B controller and 2) controllers having zero stiffness gain with or without yaw feedback for precession. Faster transient performance and improved pointing accuracy attributed to yaw feedback and stiffness gain are clearly brought forth in this study. Simple formulas are developed to calculate the gains for desired precession rate and nutation damping coefficient.

Magnetic precession and product-of-inertia nutation damping of bias momentum satellites

“…Hence, a more practical and efficient control scheme without these shortcomings is desired. Since a linear magnetic controller with constant gains for a circular orbit 10 does not suit an elliptic orbit, a closed-loop bang-bang scheme along with an active nutation damper recommends itself for consideration.…”

A New Iterative Algorithm to Solve Periodic Riccati Differential Equations With Sign Indefinite Quadratic Terms

“…In many applications in which periodic models arise (see, e.g., [17]), the averaging method [20] has been used to deal with time-variability. In the averaging method, the actual time-periodic dynamic model is approximated with a time-invariant one by taking time averages over one period.…”

“…As the variability of the geomagnetic field along the orbit is almost periodic, the resulting linearized models turn out to be time-periodic. Classical methods for the design of magnetic attitude control laws rely on averaged models [17], [31], [32], where the use of averaging was specifically developed to deal with the stabilization problem for the coupled roll/yaw dynamics of a momentum biased spacecraft using a magnetic torquer aligned with the pitch axis. In such a situation, viable alternatives are several recently developed design methods able to handle fully periodic models, with the significant advantage of guaranteeing closed loop stability a priori.…”

Optimal Discrete-Time Design of Three-Axis Magnetic Attitude Control Laws

“…Similarly, the assumptions of a circular orbit and of a diagonal inertia matrix are made only for ease of presentation, but they are by no means necessary for the applicability of the proposed design approach. Indeed, unlike existing design methods based on averaging (see, e.g., [2]), inertial coupling between roll/yaw and pitch dynamics can be handled in the design problem. The angular kinematics and dynamics of the spacecraft are modelled using as state variables the quaternion describing the attitude of the body axes with respect to the orbital axes, and the inertial angular velocity vector , with respect to the body axes.…”

“…Digital Object Identifier 10.1109/TCST.2009.2024757 verify a posteriori that the designed controller actually stabilizes the original time-varying dynamics with a satisfactory performance level; the approach is applicable only to configurations for which the averaged model is completely controllable; averaging implies limitations in closed-loop performance. This approach, originally proposed in [1], was further developed in [2] to deal with the (relatively simple) stabilization problem for the coupled roll/yaw dynamics of a momentum biased spacecraft using a magnetic torquer aligned with the pitch axis. • Methods based on full periodic models.…”