Backup Control Barrier Function Synthesis using Sum-of-Squares Reachability

Jungbae Chun; Shima Sadat Mousavi; David E. J. van Wijk; Ersin Daş; Aaron D. Ames; Felix Biertümpfel

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Backup Control Barrier Function Synthesis using Sum-of-Squares Reachability

Jungbae Chun, Shima Sadat Mousavi, David E. J. van Wijk, Ersin Daş, Aaron D. Ames, Felix Biertümpfel

arXiv:2608.27916v1Aug 28, 2026eess.SY

TL;DR

The paper targets conservatism in bCBF safety filtering when the backup set and controller are prescribed. It uses finite-horizon SOS backward reachability to synthesize a certified set and controller, then establishes conditions for their use as a backup pair. The resulting integration certifies larger safe sets and enables less conservative filtering.

Problem
bCBF performance is limited by the prescribed backup set–backup controller pair, especially when the pair is small or conservative.
Method
Finite-horizon SOS backward reachability synthesizes a set and controller satisfying safety and input constraints, with conditions making the set valid for a piecewise backup controller.
Results
The synthesized backup pair integrates into the bCBF framework and certifies larger safe sets with less conservative safety filtering.
Takeaways & Limitations
The method reduces bCBF conservatism without assuming the final backup set–controller pair is given a priori.

Abstract

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Backup control barrier functions (bCBFs) enforce safety for input-constrained nonlinear systems using a pre-certified backup set and controller, but their performance depends strongly on this prescribed pair. This letter develops a constructive method for synthesizing a less conservative backup pair via finite horizon sum-of-squares (SOS) backward reachability. Starting from an initial backup set, we compute an SOS-certified finite horizon backward reachable set and controller that satisfy safety and input constraints while steering trajectories to the original backup set. We then provide conditions under which this certified set becomes a valid backup set for a piecewise backup controller. The resulting backup pair is integrated into the bCBF framework to certify larger safe sets.

I. INTRODUCTION

The paper addresses the conservatism of backup control barrier functions caused by prescribing the backup set–controller pair. It proposes finite-horizon SOS backward reachability to synthesize an enlarged pair and integrate it into bCBF safety filtering.

Input-constrained systems require sufficiently large controlled invariant sets to maintain safety and feasibility.
Existing bCBF methods construct implicit safe sets by propagating a backup controller toward a forward invariant backup set.
The certified bCBF region depends fundamentally on the selected backup set–backup controller pair.
Finite-horizon SOS backward reachability constructs a polynomial backward reachable set and associated controller under safety and input constraints.
Under additional invariance and containment conditions, the certified set becomes a valid backup set for a piecewise backup controller.
Integrating the synthesized pair into bCBFs yields less conservative safety filtering than approaches using an a priori specified pair.

A. Control Barrier Functions

Control barrier functions provide forward-invariance conditions for safe sets, but constructing a CBF under bounded inputs is difficult because the safe set must be controlled invariant.

Safety is represented by a set CS = {x ∈ X : h(x) ≥ 0}, which must remain forward invariant under the closed-loop system.
A control barrier function provides conditions that enable synthesis of controllers rendering the safe set forward invariant.
The CBF theorem guarantees forward invariance for locally Lipschitz controllers satisfying the CBF condition.
A QP can minimally modify a primary controller while enforcing the CBF constraint.
Under input constraints, constructing a CBF is challenging because CS must be controlled invariant by some admissible controller.

B. Backup Control Barrier Functions

bCBFs expand a backup set through finite-horizon backup-controller prediction, producing a controlled-invariant region, but its size depends on the prescribed backup pair.

A backup controller is a continuously differentiable controller that renders the backup set forward invariant.
The bCBF flow evolves the system under the backup controller over a fixed horizon T.
The expanded set CI contains states that safely reach the backup set within time T while remaining in the safe set.
The expanded set is controlled invariant, and the backup controller renders it forward invariant.
The bCBF framework yields a QP safety filter that enforces safety on the expanded set CI.
The size of CI depends on the chosen backup set–backup controller pair.

C. Backward Reachability

Backward reachability characterizes states that can remain within a target tube under an admissible controller. Polynomial storage functions and SOS conditions provide tractable inner approximations under input constraints.

A target tube defined by r(t, x) induces an associated backward reachable set.
The backward reachable set contains initial conditions from which an admissible controller keeps trajectories within the target tube.
The associated state-feedback controller is time-invariant and memoryless, with inputs constrained to a polytope.
Sublevel sets of a reachability storage function V(t, x) provide inner approximations of the backward reachable set through a dissipation argument.
Under the stated storage-function and controller conditions, trajectories starting in the certified sublevel set remain in the target tube over the finite horizon.
Polynomial system and storage-function assumptions enable formulation of a polynomial optimization problem for synthesizing a backup set–controller pair.

III. MAIN RESULT

This section constructs an enlarged backup set using SOS-based backward reachability, then establishes conditions for using it as a valid bCBF backup pair.

The method first synthesizes an SOS-certified initial set and associated controller through backward reachability.
It then provides conditions under which the synthesized set defines a valid backup pair for the bCBF framework.

A. Offline Sum-of-Squares Controller Synthesis

The offline synthesis seeks a polynomial reachability storage function, controller, and level parameter whose certified sublevel set safely reaches the original invariant backup set. An iterative SOS procedure enlarges this set, but generally guarantees only a local optimum of the relaxation.

The synthesis searches over a storage function V, polynomial control law k, and scalar γ for a fixed horizon T1 and original invariant set CB.
The terminal target condition ensures the finite-horizon backward reachable set reaches CB, while safety conditions make the initial sublevel set itself safe.
The certified initial sublevel set ΩV 0,γ can serve as a backup set when the stated containment and invariance conditions hold.
The optimization maximizes the volume of the inner approximation ΩV 0,γ because a larger certified set is preferred.
Polynomial dynamics, storage functions, and controllers allow the formulation to become computationally tractable through the S-procedure and SOS constraints.
Bilinear terms make the SOS problem nonconvex, so the method solves it iteratively through alternating γ and V steps.
The iterative SOS relaxation generally converges only to a local optimum, although each individual step is convex and the γ-step can use bisection.
The examples initialize the time-independent storage function with −hb, which satisfies the initial constraints at γ = 0.

B. Extended Backup Controller

The synthesized backward reachable set is paired with an extended backup controller that follows the SOS controller outside the original backup set and the original controller inside it. Theorem 2 establishes controlled invariance, enabling integration with the bCBF safety filter.

The extended controller uses a smooth saturation function and can be made smooth by interpolation at the boundary of the original backup set.
The extended backup dynamics are defined by f_e(x)=f(x)+g(x)k_e(x), with flow ϕ_e(τ,x) generated by this controller.
The BRS-based pair can therefore be used in the backup-set method and its minimally invasive safe control calculation.
Theorem 2 states that a feasible Algorithm 1 solution makes ΩV 0,γ controlled invariant under the extended backup controller.
For states in the original backup set CB, the extended controller agrees with kb, while states outside CB reach CB within the finite horizon T1.
In the double-integrator simulation, the figure compares original and BRS-based bCBF-QPs, showing their trajectories and safe control inputs in separate plots.
Applying the same bCBF procedure requires the extended controller to be C1 so that the flow sensitivity matrix exists.

A. Double Integrator

The double-integrator example compares the original bCBF-QP with an approach using an inner-approximated backward reachable set. The BRS-based method enlarges the certified invariant set, delays safety-filter intervention, and respects input bounds.

Setup: The double integrator uses position and velocity states, bounded input u ∈ [−1, 1], and a left-half-plane safe set.The primary controller drives the system toward the unsafe right half-plane, while the backup controller renders the prescribed backup set forward invariant.
Setup: The backup controller is u = −1, and the original backup set CB is defined by a quadratic inequality and rendered forward invariant.The simulation uses horizon T = 2 s for both the bCBF-QP and the SOS algorithm, with polynomial degrees six, two, and two for V, k, and s.
Results: The inner-approximated BRS ΩV 0,γ contains CB by construction, producing a noticeably larger controlled invariant set CeI than CI.This enlarges the set of initial states from which the backup strategy can be certified over the finite horizon.
Results: The BRS-based bCBF-QP allows the primary controller to remain active longer before the safety constraint becomes active.The original bCBF-QP begins modifying the control shortly after simulation starts.
Results: Both approaches satisfy the control input constraint U = [−1, 1] throughout the simulation.The comparison includes safe control inputs generated by the original and BRS-based bCBF-QPs.

B. Quadrotor with Vertical Dynamics Neglected

The quadrotor example evaluates the synthesized BRS-based backup pair against the original pair using slice plots, trajectories, backup flows, and safe inputs. The BRS-based controller yields a larger certified invariant set while satisfying the actuator bounds.

Setup: The quadrotor model uses horizontal position, velocity, roll angle, and roll rate with bounded thrust and desired-roll inputs.The safe set bounds all four states, and the backup pair is constructed from a quadratic Lyapunov level set and an LQR controller.
Setup: The comparison uses one-second horizons for both the backup CBF-QP and the SOS algorithm.The primary controller is fixed at [g/K, π/12]⊤, while the initial backup controller is an LQR state-feedback controller.
Certified sets: The inner-approximated BRS ΩV 0,γ contains CB, resulting in an extended controlled invariant set CeI that is noticeably larger than CI.The figure shows two-dimensional slices in the (x1, x2) and (x3, x4) subspaces, with the remaining coordinates fixed at zero.
Certified sets: Backup flows confirm that trajectories starting in CI and CeI reach their respective backup sets within the finite horizon T.The controlled invariant sets are visualized through slices even though the underlying state space is four-dimensional.
Visualization caveat: Two-dimensional slices can make trajectories appear to exit the invariant sets, but individual state trajectories identify this as a visual artifact.The apparent exits do not reflect violations of the four-dimensional controlled invariant sets.
Control inputs: Both controllers satisfy the thrust and desired-roll input constraints throughout the simulation, while the BRS-based bCBF-QP keeps the primary controller active slightly longer.The bottom row displays the safe control inputs computed by the original and BRS-based bCBF-QPs.

V. CONCLUSION AND FUTURE WORK

The paper concludes that SOS backward reachability can synthesize enlarged backup pairs for input-constrained bCBFs. It identifies SDSOS optimization as future work for improving scalability to higher-dimensional polynomial systems.

Conclusion: The proposed SOS method synthesizes enlarged backup sets for input-constrained backup control barrier functions.The method certifies a set as a valid backup set under a piecewise backup controller.
Conclusion: The synthesized backup pair reduces conservatism in bCBF safety filtering.The pair is integrated into the bCBF framework rather than assumed to be given a priori.
Future work: Future work will use scaled diagonally dominant sum-of-squares optimization to compute control invariant sets for higher-dimensional polynomial systems.This stated direction targets the scalability of the approach.