Model Predictive Control for Signal Temporal Logic Specifications with Time Interval Decomposition (2211.08031v1)

Abstract: In this paper, we investigate the problem of Model Predictive Control (MPC) of dynamic systems for high-level specifications described by Signal Temporal Logic (STL) formulae. Recent works show that MPC has the great potential in handling logical tasks in reactive environments. However, existing approaches suffer from the heavy computational burden, especially for tasks with large horizons. In this work, we propose a computationally more efficient MPC framework for STL tasks based on time interval decomposition. Specifically, we still use the standard shrink horizon MPC framework with Mixed Integer Linear Programming (MILP) techniques for open-loop optimization problems. However, instead of applying MPC directly for the entire task horizon, we decompose the STL formula into several sub-formulae with disjoint time horizons, and shrinking horizon MPC is applied for each short-horizon sub-formula iteratively. To guarantee the satisfaction of the entire STL formula and to ensure the recursive feasibility of the iterative process, we introduce new terminal constraints to connect each sub-formula. We show how these terminal constraints can be computed by an effective inner-approximation approach. The computational efficiency of our approach is illustrated by a case study.

• The paper introduces a novel time interval decomposition that significantly reduces MPC's computational demand for STL specifications.
• It applies an iterative shrinking horizon approach with terminal constraints to ensure recursive feasibility across sub-tasks.
• The framework demonstrates improved efficiency in real-time robot motion planning compared to traditional MILP methods.

Model Predictive Control for Signal Temporal Logic Specifications with Time Interval Decomposition

The paper presents a novel approach to enhance the efficiency of Model Predictive Control (MPC) in fulfilling Signal Temporal Logic (STL) specifications. Signal Temporal Logic is known for its capability to express complex temporal properties in control systems, particularly in the synthesis of controllers for cyber-physical systems (CPS). These specifications ensure that systems behave according to high-level requirements in terms of signals evolving over time.

Problem and Proposed Approach

A significant challenge in using STL with MPC lies in the computational complexity, particularly for tasks with long prediction horizons. Traditional methods primarily rely on encoding STL satisfaction constraints directly into a mixed-integer linear programming (MILP) framework. While complete and globally optimal, these methods suffer from an exponential increase in computational burden with growing task horizons.

The authors propose an innovative decomposition framework that partitions STL tasks into smaller sub-tasks with disjoint time intervals. Their primary contribution is the restructuring of the MPC problem by applying the shrinking horizon principle iteratively to these decomposed sub-tasks. This method leverages time interval decomposition and uses terminal constraints between sub-tasks to ensure the overall task's satisfaction and recursive feasibility.

Key Contributions

1. Time Interval Decomposition: The paper divides a STL formula into several sub-formulae, each associated with disjoint time intervals. By applying the shrinking horizon MPC approach iteratively to these sub-formulae, computational demand is significantly reduced.
2. Terminal Constraints: To ensure transitory feasibility between the decomposed tasks, the authors introduce terminal constraints. These constraints connect each sub-formula and are computed via an inner-approximation method. This allows the framework to handle subsequent tasks feasibly even under uncertainties.
3. Robust Optimization Framework: By structuring the satisfactory problem as a robust optimization problem, the framework efficiently manages logical tasks in the presence of disturbances, while minimizing computational overhead during online control application.

Numerical Results and Implications

The authors demonstrate the computational efficiency and scalability of their decomposed framework through a case paper involving robot motion planning. They illustrate that compared to the baseline MILP approach, their method reduces online computation time considerably while maintaining a satisfactory level of performance balance.

The results are pivotal for applications in domains like autonomous vehicles, robotics, and intelligent infrastructure systems, where rapid and reliable decision-making is paramount. The reduction in computation enables real-time feasibility, broadening the use of STL in practical and complex environments.

Future Directions

This paper potentially opens up several avenues for future exploration. Extending this decomposition approach to involve learning techniques could provide adaptive performance improvements in unknown or rapidly changing environments. Additionally, further work could explore hybrid systems where continuous STL synthesis is combined with discrete decision-making models.

In conclusion, this paper makes a significant theoretical and practical contribution to the field of control synthesis. It successfully circumvents the complexity challenges inherent in traditional STL-MPC approaches, thus expanding the applicability and efficiency of STL in dynamic and reactive environments.