Functional Gradient Descent Method for Metric Temporal Logic Specifications

Metric Temporal Logic (MTL) specifications can capture complex state and timing requirements. Given a nonlinear dynamical system and an MTL specification for that system, our goal is to find a trajectory that violates or satisfies the specification. This trajectory can be used as a concrete feedback to the system designer in the case of violation or as a trajectory to be tracked in the case of satisfaction. The search for such a trajectory is conducted over the space of initial conditions, system parameters and input signals. We convert the trajectory search problem into an optimization problem through MTL robust semantics. Robustness quantifies how close the trajectory is to violating or satisfying a specification. Starting from some arbitrary initial condition and parameter and given an input signal, we compute a descent direction in the search space, which leads to a trajectory that optimizes the MTL robustness. This process can be iterated to reach local optima (min or max). We demonstrate the method on examples from the literature.