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

• Input-constrained systems require sufficiently large controlled invariant sets to maintain safety and feasibility.Source: §I · p. 1
• Existing bCBF methods construct implicit safe sets by propagating a backup controller toward a forward invariant backup set.Source: §I · p. 1
• The certified bCBF region depends fundamentally on the selected backup set–backup controller pair.Source: §I · p. 1
• Finite-horizon SOS backward reachability constructs a polynomial backward reachable set and associated controller under safety and input constraints.Source: §I · p. 1
• Under additional invariance and containment conditions, the certified set becomes a valid backup set for a piecewise backup controller.Source: §I · p. 1
• Integrating the synthesized pair into bCBFs yields less conservative safety filtering than approaches using an a priori specified pair.Source: §I · pp. 1–2

• Safety is represented by a set CS = {x ∈ X : h(x) ≥ 0}, which must remain forward invariant under the closed-loop system.Source: A. Control Barrier Functions · p. 2
• A control barrier function provides conditions that enable synthesis of controllers rendering the safe set forward invariant.Source: A. Control Barrier Functions · p. 2
• The CBF theorem guarantees forward invariance for locally Lipschitz controllers satisfying the CBF condition.Source: A. Control Barrier Functions · p. 2
• A QP can minimally modify a primary controller while enforcing the CBF constraint.Source: A. Control Barrier Functions · p. 2
• Under input constraints, constructing a CBF is challenging because CS must be controlled invariant by some admissible controller.Source: A. Control Barrier Functions · p. 2

• A backup controller is a continuously differentiable controller that renders the backup set forward invariant.Source: B. Backup Control Barrier Functions · p. 2
• The bCBF flow evolves the system under the backup controller over a fixed horizon T.Source: B. Backup Control Barrier Functions · p. 2
• The expanded set CI contains states that safely reach the backup set within time T while remaining in the safe set.Source: B. Backup Control Barrier Functions · p. 2
• The expanded set is controlled invariant, and the backup controller renders it forward invariant.Source: B. Backup Control Barrier Functions · p. 2
• The bCBF framework yields a QP safety filter that enforces safety on the expanded set CI.Source: B. Backup Control Barrier Functions · p. 2
• The size of CI depends on the chosen backup set–backup controller pair.Source: B. Backup Control Barrier Functions · p. 3

• A target tube defined by r(t, x) induces an associated backward reachable set.Source: C. Backward Reachability · p. 3
• The backward reachable set contains initial conditions from which an admissible controller keeps trajectories within the target tube.Source: C. Backward Reachability · p. 3
• The associated state-feedback controller is time-invariant and memoryless, with inputs constrained to a polytope.Source: C. Backward Reachability · p. 3
• Sublevel sets of a reachability storage function V(t, x) provide inner approximations of the backward reachable set through a dissipation argument.Source: C. Backward Reachability · p. 3
• Under the stated storage-function and controller conditions, trajectories starting in the certified sublevel set remain in the target tube over the finite horizon.Source: C. Backward Reachability · p. 3
• Polynomial system and storage-function assumptions enable formulation of a polynomial optimization problem for synthesizing a backup set–controller pair.Source: C. Backward Reachability · p. 3

• The method first synthesizes an SOS-certified initial set and associated controller through backward reachability.Source: §III · p. 3
• It then provides conditions under which the synthesized set defines a valid backup pair for the bCBF framework.Source: §III · p. 3

• 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.Source: A. Offline Sum-of-Squares Controller… · p. 3
• The terminal target condition ensures the finite-horizon backward reachable set reaches CB, while safety conditions make the initial sublevel set itself safe.Source: A. Offline Sum-of-Squares Controller… · p. 3
• The certified initial sublevel set ΩV 0,γ can serve as a backup set when the stated containment and invariance conditions hold.Source: A. Offline Sum-of-Squares Controller… · p. 3
• The optimization maximizes the volume of the inner approximation ΩV 0,γ because a larger certified set is preferred.Source: A. Offline Sum-of-Squares Controller… · p. 3
• Polynomial dynamics, storage functions, and controllers allow the formulation to become computationally tractable through the S-procedure and SOS constraints.Source: A. Offline Sum-of-Squares Controller… · p. 4
• Bilinear terms make the SOS problem nonconvex, so the method solves it iteratively through alternating γ and V steps.Source: A. Offline Sum-of-Squares Controller… · p. 4
• The iterative SOS relaxation generally converges only to a local optimum, although each individual step is convex and the γ-step can use bisection.Source: A. Offline Sum-of-Squares Controller… · p. 4
• The examples initialize the time-independent storage function with −hb, which satisfies the initial constraints at γ = 0.Source: A. Offline Sum-of-Squares Controller… · p. 4

• The extended controller uses a smooth saturation function and can be made smooth by interpolation at the boundary of the original backup set.Source: B. Extended Backup Controller · p. 4
• 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.Source: B. Extended Backup Controller · p. 4
• The BRS-based pair can therefore be used in the backup-set method and its minimally invasive safe control calculation.Source: B. Extended Backup Controller · pp. 4–5
• Theorem 2 states that a feasible Algorithm 1 solution makes ΩV 0,γ controlled invariant under the extended backup controller.Source: B. Extended Backup Controller · p. 4
• 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.Source: B. Extended Backup Controller · p. 4
• In the double-integrator simulation, the figure compares original and BRS-based bCBF-QPs, showing their trajectories and safe control inputs in separate plots.Source: Figure 1 · p. 5
• Applying the same bCBF procedure requires the extended controller to be C1 so that the flow sensitivity matrix exists.Source: B. Extended Backup Controller · p. 5

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.Source: A. Double Integrator · p. 5
• 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.Source: A. Double Integrator · p. 5

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.Source: A. Double Integrator · p. 5
• 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.Source: A. Double Integrator · p. 5
• 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.Source: A. Double Integrator · p. 5

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.Source: B. Quadrotor with Vertical Dynamics… · p. 5
• 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.Source: B. Quadrotor with Vertical Dynamics… · p. 5

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.Source: B. Quadrotor with Vertical Dynamics… · p. 5, Figure 2 · p. 6
• 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.Source: B. Quadrotor with Vertical Dynamics… · p. 5

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.Source: B. Quadrotor with Vertical Dynamics… · p. 6

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.Source: B. Quadrotor with Vertical Dynamics… · p. 6, Figure 2 · p. 6

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.Source: §V · p. 6
• 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.Source: §V · p. 6

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.Source: §V · p. 6