Convex synthesis and verification of control-Lyapunov and barrier functions with input constraints

Frank Permenter; Hongkai Dai

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Convex synthesis and verification of control-Lyapunov and barrier functions with input constraints

Hongkai Dai, Frank Permenter

Robotics · Submitted 2 Oct 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This yields a CLF and CBF synthesis technique that does not rely on a nominal controller.

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arXiv ID2210.00629 (version 1)

Pith IDX4UUFJB6, Pith's permanent record of this paper and its review

CitationFrank Permenter, Hongkai Dai. "Convex synthesis and verification of control-Lyapunov and barrier functions with input constraints." arXiv:2210.00629 (2022). https://pith.science/paper/X4UUFJB6

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Control Lyapunov functions (CLFs) and control barrier functions (CBFs) are widely used tools for synthesizing controllers subject to stability and safety constraints. Paired with online optimization, they provide stabilizing control actions that satisfy input constraints and avoid unsafe regions of state-space. Designing CLFs and CBFs with rigorous performance guarantees is computationally challenging. To certify existence of control actions, current techniques not only design a CLF/CBF, but also a nominal controller. This can make the synthesis task more expensive, and performance estimation more conservative. In this work, we characterize polynomial CLFs/CBFs using sum-of-squares conditions, which can be directly certified using convex optimization. This yields a CLF and CBF synthesis technique that does not rely on a nominal controller. We then present algorithms for iteratively enlarging estimates of the stabilizable and safe regions. We demonstrate our algorithms on a 2D toy system, a pendulum and a quadrotor.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Control Lyapunov functions (CLFs) and control barrier functions (CBFs) are widely used tools for synthesizing controllers subject to stability and safety constraints.	background 1.00	0.04	0.03	asserted 1.00
s1	Paired with online optimization, they provide stabilizing control actions that satisfy input constraints and avoid unsafe regions of state-space.	background 1.00	0.08	0.04	asserted 1.00
s2	Designing CLFs and CBFs with rigorous performance guarantees is computationally challenging.	background 0.77	0.08	0.03	asserted 1.00
s3	To certify existence of control actions, current techniques not only design a CLF/CBF, but also a nominal controller.	background 0.86	0.05	0.03	asserted 1.00
s4	This can make the synthesis task more expensive, and performance estimation more conservative.	background 0.59	0.08	0.04	asserted 0.55
s5	In this work, we characterize polynomial CLFs/CBFs using sum-of-squares conditions, which can be directly certified using convex optimization.	method 0.65	0.82	0.30	asserted 1.00
s6	This yields a CLF and CBF synthesis technique that does not rely on a nominal controller.	contribution 0.76	0.91	0.47	asserted 1.00
s7	We then present algorithms for iteratively enlarging estimates of the stabilizable and safe regions.	method 0.54	0.94	0.12	asserted 1.00
s8	We demonstrate our algorithms on a 2D toy system, a pendulum and a quadrotor.	method 0.44	0.96	0.06	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.76.

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Original abstract

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) are widely used tools for synthesizing controllers subject to stability and safety constraints. Paired with online optimization, they provide stabilizing control actions that satisfy input constraints and avoid unsafe regions of state-space. Designing CLFs and CBFs with rigorous performance guarantees is computationally challenging. To certify existence of control actions, current techniques not only design a CLF/CBF, but also a nominal controller. This can make the synthesis task more expensive, and performance estimation more conservative. In this work, we characterize polynomial CLFs/CBFs using sum-of-squares conditions, which can be directly certified using convex optimization. This yields a CLF and CBF synthesis technique that does not rely on a nominal controller. We then present algorithms for iteratively enlarging estimates of the stabilizable and safe regions. We demonstrate our algorithms on a 2D toy system, a pendulum and a quadrotor.

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References

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7 arXiv papers cite this in 13 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 8 sentences without a confident label

Uses its method or tool 1 paper

Inspired by [ref] , we will derive an equivalent condition for <ref> by using contraposition.

Parameter-Dependent Control Lyapunov Functions for Stabilizing Nonlinear Parameter-Varying Systems · Verification of a PD-CLF Under Input Constraints via SOS Programming · 0.83

Cites it as background 4 papers

For systems with polynomial dynamics and simple polyhedral bounds on the control, current works convert the verification problem into a convex sum-of-squares (SOS) program [ref] and the synthesis problem into a nonconvex bilinear SOS program [ref] .

Verification and Synthesis of Robust Control Barrier Functions: Multilevel Polynomial Optimization and Semidefinite Relaxation · Introduction · 0.92
Existing CBF synthesis methods typically leverage certain relative degree properties [ref] , learning-based approaches [ref] , sum-of-squares programs for polynomial dynamics [ref] , or reachability-based constructions from Hamilton–Jacobi value functions [ref] .

Safety-Critical Control via Recurrent Tracking Functions · Introduction · 0.98
SOS optimization based CBF synthesis has been studied in [ref] without considering input constraints, and in [ref] with consideration of input constraints.

Parameter-Dependent Control Lyapunov Functions for Stabilizing Nonlinear Parameter-Varying Systems · Introduction · 0.95

What it cites · 3 sentences without a confident label

Compares with 1 paper

In this paper, we provide a new synthesis technique with formal guarantees that is conceptually simpler than previous formal methods [ref] .

Safety Verification and Controller Synthesis for Systems with Input Constraints · INTRODUCTION · 0.95

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