An MILP approach for real-time optimal controller synthesis with Metric Temporal Logic specifications

Abstract

The fundamental idea of this work is to synthesize reactive controllers such that closed-loop execution trajectories of the system satisfy desired specifications that ensure correct system behaviors, while optimizing a desired performance criteria. In our approach, the correctness of a system's behavior can be defined according to the system's relation to the environment, for example, the output trajectories of the system terminate in a goal set without entering an unsafe set. Using Metric Temporal Logic (MTL) specifications we can further capture complex system behaviors and timing requirements, such as the output trajectories must pass through a number of way-points within a certain time frame before terminating in the goal set. Given a Mixed Logical Dynamical (MLD) system and system specifications in terms of MTL formula or simpler reach-avoid specifications, our goal is to find a closed-loop trajectory that satisfies the specifications, in non-deterministic environments. Using an MILP framework we search over the space of input signals to obtain such valid trajectories of the system, by adding constraints to satisfy the MTL formula only when necessary, to avoid the exponential complexity of solving MILP problems. We also present experimental results for planning a path for a mobile robot through a dynamically changing environment with a desired task specification.

Smooth operator: Control using the smooth robustness of temporal logic

“…As seen in Table I, SOP is consistently faster than BluSTL in Boolean mode, and displays smaller variances in runtimes. Note also that the problem solved here is very similar to the one used in [3], which uses another MILP-based method. While the underlying dynamics differ and their numbers are reported on a more power machine, SOP numbers compare favourably with those in [3].…”

“…The first method automatically creates a MIP from the semantics of the MTL formula and solves the MIP to yield a satisfying control sequence [2], [3], [4]. See Related Work and the Experiments.…”

“…Note also that [2] requires all constraints to be linear, so all atomic propositions must involve half-spaces (p : a x ≤ b), which is not a restriction in the current work. Another MILP based approach is presented in [3] where constraints are added when necessary, in order to reduce MILP complexity. The work closest to this paper is [13], [14].…”

“…Related work. Current approaches to optimizing the robustness fall into four categories: the use of heuristics like Simulated Annealing [10], cross-entropy [11] and RRTs [8]; non-smooth optimization [12]; Mixed Integer Linear Programming (MILP) [2], [3]; and iterative approximations [13], [14], [15]. Black-box heuristics are the most commonly used approach.…”

Coordinated Control of Wind Turbine Generator and Energy Storage System for Frequency Regulation under Temporal Logic Specifications

“…The performance measure J(u(·)) can be set as the control effort u(·) 2 or u(·) 1 . For linear systems, the above optimization problem can be converted to a linear program (LP) problem [20], [21] and it can be solved efficiently by LP solvers. As we only focus on the time horizon [0, T end ] (e.g.…”