Geometric Conditions for Lossless Convexification in Linear Optimal Control with Discrete-Valued Inputs
This work addresses the problem of real-time, safety-critical control for systems with discrete inputs, offering an incremental improvement by extending existing convexification results to linear systems with geometric conditions.
The paper tackles the challenge of solving optimal control problems with discrete-valued inputs, which are typically intractable for real-time use, by developing a lossless convexification method for linear systems; it shows that under specific geometric conditions, the convex relaxation yields the exact solution, enabling real-time computation as confirmed by Monte Carlo simulations with compatible computation times.
Optimal control problems with discrete-valued inputs are 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 convex programs that can be solved efficiently. This paper develops a lossless convexification for optimal control problems of linear systems. We extend existing results by showing that system normality is preserved when reformulating Lagrange-form problems into Mayer-form via an epigraph transformation, and under simple geometric conditions on the input set the solution to the relaxed convex problem is the solution to the original non-convex problem. These results enable real-time computation of optimal discrete-valued controls without resorting to mixed-integer optimization. Numerical results from Monte Carlo simulations confirm that the proposed algorithm consistently yields discrete-valued control inputs with computation times compatible with safety-critical real-time applications.