Convexification of Robust Trajectory Planning Problems With Nominal State and Control Dependent Uncertainties

We consider the problem of robust trajectory optimization of constrained discrete-time linear systems under bounded uncertainty. This paper extends our previous work on robust trajectory planning for linear systems with control-dependent uncertainties to a more general class of uncertainties, including nominal-state-dependent uncertainties. In particular, we show that if strong duality holds for the robust state constraints, and if the uncertainties are bounded by convex elementwise nonnegative functions of the nominal state and control, the robust constraints can be equivalently reformulated as deterministic convex constraints, enabling globally optimal solutions with no conservatism. We first use convex duality theory to reformulate robust linear inequality state constraints as deterministic biconvex constraints. We then exploit the elementwise nonnegativity of the uncertainty bounds to remove the biconvexity in closed form, resulting in convex deterministic constraints that can be handled by off-the-shelf solvers. These two lemmas are our main contribution, and lead to our final result, and equivalent convex reformulation of the original robust optimization problem which allows efficient trajectory optimization solutions under control and state dependent uncertainties. We then demonstrate the practical applicability of our method via numerical simulations.