A New Semidefinite Relaxation for Linear and Piecewise-Affine Optimal Control with Time Scaling

2025-04-17

AI Summary

This paper addresses the optimal control of linear and piecewise affine (PWA) systems with time scaling, whose dynamics are inherently non-convex due to bilinear coupling between time step and state/control variables. To tackle this, we propose a compact, lightweight semidefinite relaxation framework: first, selectively relax and replace bilinear terms; second, unify PWA mode sequence modeling via Graphs of Convex Sets (GCS), jointly convexifying both the non-convex dynamics and discrete mode selection; finally, integrate time-scaling transformation with shortest-path optimization to formulate the entire problem as a single semidefinite program (SDP). The approach significantly improves relaxation tightness while preserving computational efficiency. To the best of our knowledge, it is the first method enabling unified, exact optimal control synthesis for both linear and PWA systems under time scaling. Experimental results demonstrate its convergence, accuracy, and superior numerical performance over existing alternatives.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We introduce a semidefinite relaxation for optimal control of linear systems with time scaling. These problems are inherently nonconvex, since the system dynamics involves bilinear products between the discretization time step and the system state and controls. The proposed relaxation is closely related to the standard second-order semidefinite relaxation for quadratic constraints, but we carefully select a subset of the possible bilinear terms and apply a change of variables to achieve empirically tight relaxations while keeping the computational load light. We further extend our method to handle piecewise-affine (PWA) systems by formulating the PWA optimal-control problem as a shortest-path problem in a graph of convex sets (GCS). In this GCS, different paths represent different mode sequences for the PWA system, and the convex sets model the relaxed dynamics within each mode. By combining a tight convex relaxation of the GCS problem with our semidefinite relaxation with time scaling, we can solve PWA optimal-control problems through a single semidefinite program.

Problem

• Develops semidefinite relaxation for linear optimal control with time scaling
• Extends method to handle piecewise-affine systems via convex sets
• Solves nonconvex bilinear dynamics with tight computational relaxations

Innovation

• Semidefinite relaxation for linear control with time scaling
• Change of variables to tighten relaxations efficiently
• Graph of convex sets for piecewise-affine systems