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

TL;DR

This work tackles nonconvex optimal control problems for linear and piecewise-affine systems with time scaling by developing a specialized semidefinite relaxation (SDR) that preserves key problem structure to achieve tight, computationally efficient relaxations. It introduces a time-flexible relaxation (TFR) and a sparse variant to handle bilinear terms involving the time step, while extending the approach to PWA systems via a Graph of Convex Sets (GCS) formulation that jointly optimizes mode sequences and trajectories. The framework yields spectrahedra-based convex sets for each mode and leverages a GCS relaxation to identify promising mode sequences, subsequently refined with nonlinear programming to recover feasible, time-scaled trajectories. Experimental results on an inverted pendulum with elastic walls and a 7-DoF robot arm demonstrate tighter relaxations, competitive numerical performance, and the ability to handle mode switches and obstacle constraints that challenge traditional NLP or MICP methods. Overall, the method provides a scalable, globally-minded alternative for time-scaled trajectory optimization in linear and PWA systems, with potential impact on robotics and time-critical control tasks.

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.

Figures (8)

• Figure 1: Minimum-time trajectories for a 7D double integrator moving among obstacles with robot arm geometry. The collision-free configuration space is approximately decomposed into convex polytopes. Our algorithm efficiently navigates these polytopes while respecting input limits. Videos of the trajectories can be found at https://lujieyang.github.io/ctfoc/comparison_LB_RB.html and https://lujieyang.github.io/ctfoc/comparison_RB_AS.html.
• Figure 2: Comparison of different SDRs for the minimum-time double integrator. (a) Ground truth state-space trajectories. (b) Trajectories from dense TFR. (c) Trajectories from sparse TFR. (d) Trajectories from standard SDR.
• Figure 3: Effect of the coupling variable on balancing the inverted pendulum with the wall in minimum time for different initial conditions. The vertical dashed line indicates the angular limit beyond which the pendulum collides with the wall. The legend indicates the time duration for each trajectory.
• Figure 4: Time-optimal trajectories for the inverted pendulum with wall. (a) GCS+TFR+NLP solutions. The vertical dashed line represents the angle at which the pendulum makes contact with the wall. (b) GCS+TFR solutions. (c) Convex relaxation of MICP formulation with fixed $h$. The different segment colors indicate mode transitions.
• Figure 5: Performance comparison for balancing the inverted pendulum with contact in minimum time. The pendulum starts from the initial conditions $(0.09, \dot \theta_0)$. (a) Final time. (b) Relaxation gap. The proposed GCS+TFR formulation is much tighter than the relaxation of the standard MICP.

Theorems & Definitions (2)

• Example IV.1: Minimum-time double integrator
• Example V.1: Minimum-time Inverted Pendulum with Wall