Geometric Conditions for Lossless Convexification in Fuel-Optimal Control of Linear Systems with Discrete-Valued Inputs

TL;DR

The paper addresses real-time trajectory optimization for linear time-invariant systems with discrete-valued inputs using a lossless convexification framework. It develops an epigraph-based reformulation from Lagrange-form to Mayer-form and proves normality under cross-polytope input geometry, enabling an extreme-point relaxation that yields an exact convex program equivalent to the original mixed-integer problem. Through a spacecraft rendezvous CW-model, it demonstrates that the optimizer produces discrete-valued controls with sub-second solve times, validating real-time applicability for safety-critical guidance. The approach generalizes to hands-off optimization and lays groundwork for extending to state-constrained and MPC settings, offering a practical route to real-time discrete-valued control in aerospace and related domains.

Abstract

Trajectory generation for autonomous systems with discrete-valued actuators is challenging due to the mixed-integer nature of the resulting optimization problems, which are generally intractable for real-time, safety-critical applications. Lossless convexification offers an alternative by reformulating mixed-integer programs as equivalent convex programs that can be solved efficiently with guaranteed convergence. This paper develops a lossless convexification framework for the fuel-optimal control of linear systems with discrete-valued inputs. We extend existing Mayer-form results by showing that, under simple geometric conditions, system normality is preserved when reformulating Lagrange-form problems into Mayer-form. Furthermore, we derive explicit algebraic conditions for normality in systems with cross-polytopic input sets. Leveraging these results and an extreme-point relaxation, we demonstrate that the fuel-optimal control problem admits a lossless convexification, enabling real-time, discrete-valued solutions without resorting to mixed-integer optimization. Numerical results from Monte Carlo simulations confirm that the proposed approach consistently yields discrete-valued control inputs with computation times compatible with real-time implementation.

Figures (8)

• Figure 1: Lossless convexification enables real-time, fuel-optimal trajectory generation by solving a convex program instead of a Mixed-Integer Convex Program.
• Figure 2: Discrete input set ${\mathcal{U}}$ and its convex relaxation $\tilde{{\mathcal{U}}}$ which forms a cross-polytope. The epigraph-based augmented set $\tilde{{\mathcal{U}}}_e$ yields a compact convex polytope.
• Figure 3: The discrete-input OCP (MICP) with running costs is transformed via the epigraph formulation and convex relaxation steps, leading to the final equivalence stated in \ref{['thm:CP_for_MICP']}.
• Figure 4: System states during rendezvous, showing convergence to the desired terminal conditions.
• Figure 5: Optimal control inputs with respect to the admissible discrete set ${\mathcal{U}}$. The dashed lines represent the admissible inputs.

Theorems & Definitions (17)

• Definition 1
• proof
• Definition 2
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• proof