This thesis explores the attitude estimation and control problem of a magnetically controlled
small satellite in initial acquisition phase. During this phase, large data uncertainties
pose estimation challenges, while highly nonlinear dynamics and inherent limitations
of the magnetic actuation are primary issues in control. We aim to design algorithms,
which can improve performance compared to the state of the art techniques and remain
tractable for practical applications.
Static attitude estimation, which is an essential part of a satellite control system,
uses vector information and solves a constrained weighted least-square problem. With
large data uncertainties, this technique results in large errors rendering divergence or
infeasibility in dynamic filtering and control. When static estimation is the primary
source of attitude, these errors become critical; for example in low budget small satellites.
To address this issue, we formulate a robust static estimation problem with norm-bounded
uncertainties, which is a difficult optimization problem due to its unfavorable convexity
properties and nonlinear constraints. By deriving an analytical upper bound for the
convex maximization, the robust min-max problem is approximated with a minimization
problem with quadratic cost and constraints (a QCQP), which is non-convex. Semidefinite
relaxation is used to upper bound the non-convex QCQP with a semi-definite
program, which can efficiently be solved in a polynomial time. Furthermore, it is shown
that the derived upper bound has no gap in solving the robust problem in practice.
Semi-definite relaxations are also applied to solve the robust formulations of a more
general class of problems known as the orthogonal Procrustes problem (OPP). It is shown
that the solution of the relaxed OPP is exact when no uncertainties are considered;
however, for the robust case, only a sub-optimal solution can be obtained.
Finally, a satellite rate damping in initial acquisition phase is addressed by using
nonlinear model predictive control (NMPC). Standard NMPC schemes with guaranteed
stability show superior performance than existing techniques; however, they are computationally
expensive. With large initial rates, the computational burden of NMPC
becomes prohibitively excessive. For these cases, an algorithm is presented with an additional
constraint on the cost reduction that allows an early termination of the optimizer
based on the available computational resources. The presented algorithm significantly
reduces the de-tumbling time due to the imposed cost reduction constraint.