Smooth operator: Control using the smooth robustness of temporal logic

Modern control systems, like controllers for swarms of quadrotors, must satisfy complex control objectives while withstanding a wide range of disturbances, from bugs in their software to attacks on their sensors and changes in their environments. These requirements go beyond stability and tracking, and involve temporal and sequencing constraints on system response to various events. This work formalizes the requirements as formulas in Metric Temporal Logic (MTL), and designs a controller that maximizes the robustness of the MTL formula. Formally, if the system satisfies the formula with robustness r, then any disturbance of size less than r cannot cause it to violate the formula. Because robustness is not differentiable, this work provides arbitrarily precise, infinitely differentiable, approximations of it, thus enabling the use of powerful gradient descent optimizers. Experiments on a temperature control example and a two-quadrotor system demonstrate that this approach to controller design outperforms existing approaches to robustness maximization based on Mixed Integer Linear Programming and stochastic heuristics. Moreover, it is not constrained to linear systems.