Semidefinite Relaxations of Robust Binary Least Squares under Ellipsoidal Uncertainty Sets

The problem of finding the least squares solution s to a system of equations Hs = y is considered, when s is a vector of binary variables and the coefficient matrix H is unknown but of bounded uncertainty. Similar to previous approaches to robust binary least squares, we explore the potential of a min-max design with the aim to provide solutions that are less sensitive to the uncertainty in H. We concentrate on the important case of ellipsoidal uncertainty, i.e., the matrix H is assumed to be a deterministic unknown quantity which lies in a given uncertainty ellipsoid. The resulting problem is NP-hard, yet amenable to convex approximation techniques: Starting from a convenient reformulation of the original problem, we propose an approximation algorithm based on semidefinite relaxation that explicitly accounts for the ellipsoidal uncertainty in the coefficient matrix. Next, we show that it is possible to construct a tighter relaxation by suitably changing the description of the feasible region of the problem, and formulate an approximation algorithm that performs better in practice. Interestingly, both relaxations are derived as Lagrange bidual problems corresponding to the two equivalent problem reformulations. The strength of the proposed tightened relaxation is demonstrated by pertinent simulations

Robust estimation and sub-optimal predictive control for satellites

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