Robust Output Feedback of Nonlinear Systems through the Efficient Solution of Min-Max Optimization Problems

TL;DR

This work addresses robust output feedback control for discrete-time nonlinear systems with bounded uncertainties by formulating a min-max optimization over a finite horizon to compute gains that minimize a performance cost while enforcing nonlinear constraints for all uncertainty realizations. It avoids explicit state estimation by constructing an implicit description of the feasible state space from system dynamics and past measurements, and solves the resulting semi-infinite program via a local reduction method that generates finite scenario sets. The approach is demonstrated on a two-dimensional quadrotor, where open-loop control fails to meet tracking and safety constraints, while 1-step and especially 2-step output-feedback policies achieve robust performance with no constraint violations within tested uncertainty samples. Key contributions include implicit feasible-set formulation, semidefinite-programming-agnostic min-max optimization via local reduction, and demonstrated robustness in a nonlinear, uncertain setting with practical computation times. This framework offers a viable alternative to state-estimation-based control in scenarios where guarantees under bounded uncertainties are critical and measurements are imperfect.

Abstract

We examine robust output feedback control of discrete-time nonlinear systems with bounded uncertainties affecting the dynamics and measurements. Specifically, we demonstrate how to construct semi-infinite programs that produce gains to minimize some desired performance cost over a finite prediction horizon for the worst-case realization of the system's uncertainties, while also ensuring that any specified nonlinear constraints are always satisfied. The solution process relies on an implicit description of the feasible state space through prior measurements and the system dynamics, and assumes that the system is always in the subset of the feasible space that is most detrimental to performance. In doing so, we can guarantee that the system's true state will meet all of the chosen performance criteria without resorting to any explicit state estimation. Under some smoothness assumptions, we also discuss solving these semi-infinite programs through local reduction techniques, which generate optimal scenario sets for the uncertainty realizations to approximate the continuous uncertainty space and speed up the computation of optima. When tested on a two-dimensional nonlinear quadrotor, the developed method achieves robust constraint satisfaction and tracking despite dealing with highly uncertain measurements and system dynamics.

Figures (3)

• Figure 1: Comparison of the real feasible state space for $x_1$ (random samples drawn in red) to a box approximation (shown in grey) under the dynamics of \ref{['eqn:ex_sys_feasible']} given the measurements $y_0 = [0,0]^\top$ and $y_1 = [1,2]^\top$, and the control inputs $u_0 = [1,1]^\top$.
• Figure 2: Sampled trajectories with end positions marked in red and the target marked in green.
• Figure 3: Illustration of the quadrotor states at time step $k$.