Synthesis of Discrete-time Control Barrier Functions for Polynomial Systems Based on Sum-of-Squares Programming
Erfan Shakhesi, W. P. M. H. Heemels, Alexander Katriniok
TL;DR
This work introduces an SOS-based alternating-descent framework to synthesize polynomial discrete-time Control Barrier Functions (DTCBFs) and polynomial control policies for discrete-time, control-affine polynomial systems with polyhedral input constraints and semi-algebraic safety sets. A key challenge is the bilinearity in terms like $h(f(x)+g(x)\pi(x))$, which the authors address by introducing auxiliary variables and a modified DTCBF $\tilde h_+$, accompanied by a suite of SOS constraints and Propositions to preserve linearity and guarantee safety on the current invariant set. The method is extended to higher-degree DTCBFs via two strategies: iterative bilinear-term handling and fixing a base quadratic DTCBF to construct higher-degree functions, both demonstrated in numerical case studies on linear and nonlinear systems. Results show near-maximal invariant sets under safety constraints and illustrate practical feasibility, with future work targeting robustness under disturbances. The approach advances systematic, computationally tractable DTCBF synthesis for safe operation of polynomial discrete-time systems.
Abstract
Discrete-time Control Barrier Functions (DTCBFs) are commonly utilized in the literature as a powerful tool for synthesizing control policies that guarantee safety of discrete-time dynamical systems. However, the systematic synthesis of DTCBFs in a computationally efficient way is at present an important open problem. This article first proposes a novel alternating-descent approach based on Sum-of-Squares programming to synthesize quadratic DTCBFs and corresponding polynomial control policies for discrete-time control-affine polynomial systems with input constraints and semi-algebraic safe sets. Subsequently, two distinct approaches are introduced to extend the proposed method to the synthesis of higher-degree polynomial DTCBFs. To demonstrate its efficacy, we apply the proposed method to numerical case studies.