Regions of Attraction of CBF-Based Safety Filters for Stable Planar Linear Systems

The gist In this paper we study the dynamical properties induced by safety filters based on control barrier functions (CBFs) for stable planar linear systems.

Introduction and Motivation

Safety filters are used to enforce hard constraints on controlled systems by minimally modifying a nominal controller only when necessary The resulting safety-filtered closed loop is piecewise defined, switching between the nominal controller and the corrective control induced by the filter Consequently, its global dynamics can differ substantially from those of the nominal closed loop. The paper addresses a central question regarding what is the region of attraction of the desired equilibrium when using CBF-based safety filters.

Methodology and Model Setup

The study focuses on a planar linear plant with an affine constraint, defined by the system dynamics x˙ = Ax + Bu and h(x) = c⊤x + d. A control barrier function (CBF) is characterized by the condition h˙(x, u) + αh(x) ≥ 0. The safety filter selects an admissible input u∗ using a quadratic program (QP) to find the input closest to the nominal command When the nominal input already satisfies the CBF condition, the safety filter leaves it unchanged. The resulting closed-loop vector field is continuous and piecewise affine, switching between two modes based on whether η(x) ≥ 0 or η(x) < 0.

Dynamical Characterization in Planar Systems

The analysis reduces the planar safety-filtered closed loop to a dimensionless normal form with three parameters: the damping ratio ζ of the stable nominal dynamics, a normalized invariant-zero parameter k, and the filter gain α. For k > 0, global exponential stability (GES) of the origin is proven for every α > 0. For k = 0, global asymptotic stability (GAS) is established but not GES. When k < 0, the filter creates a saddle equilibrium on the constraint boundary. The boundary of attraction depends on whether the CBF gain is below, at, or above a homoclinic transition value. For underdamped nominal dynamics, this boundary is an unstable periodic orbit at small gains, a homoclinic loop at a unique transition gain given by an explicit scalar equation, or the saddle’s stable manifold depending on whether the CBF gain is below, at, or above a homoclinic transition value.

Region of Attraction and Stability Conditions

The stability of the active mode is governed by the sign of z:= tr A − c⊤ABc⊤B. If z 0, an additional equilibrium p˜ on the safety boundary exists. The region of attraction B(0) depends on k:

-k = 0:

If k > 0, the origin is GES. If k = 0, it is GAS but not GES.

-k < 0:

The boundary is determined by the nominal dynamics and filter gain. For underdamped systems (0 < ζ < 1), there is a transition at a specific gain αhom where the boundary becomes a homoclinic loop, defining B(0) as its interior or stable manifold depending on the gain relative to αhom.

Extensions Beyond Planar Dynamics

The planar analysis relies on the Poincare–Bendixson theorem, which has no counterpart in higher dimensions In three dimensions, two Hurwitz modes no longer guarantee global stability even if both modes are Hurwitz. Furthermore, for an unstable plant in three dimensions with an unstable nominal loop, the filter can produce a safe chaotic attractor. This chaotic attractor is safe because it grazes the constraint but never crosses it.

Conclusion

A CBF safety filter can change the global behavior of an otherwise stable planar loop by creating an additional equilibrium, oscillations, and diverging trajectories. Recovery from arbitrarily large safe states is possible only if the nominal loop does not oscillate and the gain is at least the decay rate of its slow mode. The full three-dimensional classification remains open for future work.

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Regions of Attraction of CBF-Based Safety Filters for Stable Planar Linear Systems

Adam K. Kiss, Pol Mestres, Andras Bartfai, Shima Sadat Mousavi, and Aaron D. Ames

Abstract— In this paper we study the dynamical properties induced by safety filters based on control barrier functions (CBFs). These are obtained by minimally modifying a nominal controller with the goal of rendering it safe. We consider the simple setting of a planar linear plant with a stable linear nominal controller and an affine constraint. We show that even in this simple setting, the filtered system can oscillate or diverge while remaining safe. We fully characterize the parameters for which the origin is globally (exponentially or asymptotically) stable. For all other parameters, the filter creates a saddle on the constraint boundary, and we characterize the region of attraction of the origin. For underdamped nominal dynamics, its boundary is an unstable periodic orbit, a homoclinic loop, or the saddle’s stable manifold, depending on whether the CBF gain is below, at, or above a homoclinic transition value. A vehicle sideslip example and two three-dimensional examples illustrate the results and what changes beyond planar dynamics.

I. INTRODUCTION

Many controlled systems must respect hard constraints on their state: an aircraft must not exceed a sideslip or loadfactor limit, a vehicle must maintain a safe distance, and a robot arm must avoid forbidden regions. Safety filters provide a systematic way to enforce such constraints by modifying a nominal control input only when necessary. Control barrier functions (CBFs) [1], [2] characterize control inputs that render a prescribed safe set forward invariant. Since many inputs may satisfy the resulting CBF condition, a common approach is to use a quadratic program (QP) to select the admissible input closest to the nominal command [1]. When the nominal input already satisfies the CBF condition, the safety filter leaves it unchanged. The resulting safety-filtered closed loop is piecewise defined, switching between the nominal controller and the corrective control induced by the filter. Consequently, its global dynamics can differ substantially from those of the nominal closed loop. In [3], [4], it was shown that CBFbased safety filters can introduce undesired equilibria on the boundary of the safe set, while [5] characterized undesired equilibria together with other spurious behaviors, including unbounded trajectories and limit cycles, for general controlaffine systems. Adam K. Kiss, Pol Mestres, Shima Sadat Mousavi, and Aaron D. Ames are with the Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA. Adam K. Kiss and Andras Bartfai are with the Department of Applied Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Budapest, Hungary. Adam K. Kiss was supported by NKFIH (NKKP STARTING 153210) and the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences. Andras Bartfai is part of the MTA-BME Lendulet ¨ Machine Tool Vibration Research Group. Given the difficulty of characterizing such global phenomena for general nonlinear dynamics and constraints, several works have focused on more structured settings. For planar linear systems with an ellipsoidal safe set, [6] identified conditions under which the safety boundary has a single saddle equilibrium or multiple equilibria containing one which is asymptotically stable. For linear systems with affine constraints, [7] showed that the closed loop is piecewise affine with two modes, corresponding to the nominal and the filtered input. They characterized the undesired equilibria and gave sufficient conditions for the stability of the desired equilibrium. The four-state example of [8], in which both modes are stable and the closed loop still oscillates, shows that instability can arise when these conditions fail. Extensions to multiple affine constraints appear in [9], which characterizes the active-set spectra through invariant zeros, boundary equilibria and tangential unstable directions, and gives geometric and Lyapunov-based conditions for divergence, boundedness, and stability. For linear systems with multiple affine constraints, [10] characterizes QP feasibility. For constraints of full relative degree, [11] gave a frequency-domain positiverealness test that ensures global stability, and [12] provided minimum-phase conditions for safety-filter stability based on input-output linearization. Building on this framework, [13] constructed linear feedback laws without switching that satisfy the CBF condition everywhere. These results establish that a safety filter can create undesired equilibria, limit cycles, or unbounded trajectories even when the nominal closed loop is stable.

1. textbfProcessing of Unsafe States in Planar Systems (k < 0): The improved system can precisely characterize when a safety filter creates an undesired equilibrium or limit cycle by identifying the homoclinic transition value mentioned in the text, which is given by an explicit scalar equation.

2. textbfDefining Robust Regions of Attraction for Non-Minimum Phase Constraints (k < 0): The system can determine whether the region of attraction is bounded or unbounded for nonminimum phase constraints by checking if the nominal loop carries Qk to Qα via the condition in equation (18).

3. textbfReal-Time Safety Assessment and Predictive Control: The improved AI can use the derived conditions to predict if a trajectory will enter an unsafe region defined by the constraint boundary, allowing for proactive input modification based on whether the filter is active or inactive.

4. textbf Adaptive Gain Scheduling for Safety Filters: The system can implement gain scheduling based on nominal dynamics; specifically, it can determine that Increasing damping moves this transition to smaller gains, allowing the filter gain to be adjusted dynamically based on the current state's damping ratio.

5. textbf Enhanced Safety Margin Monitoring: The AI can continuously monitor the nominal CBF margin defined as c⊤(A0 + αI)x + αd, immediately flagging potential constraint violation when this margin drops below zero, signaling that the KKT conditions of (3) make the CBF constraint active.

In this paper we study the dynamical properties induced by safety filters based on control barrier functions (CBFs). These are obtained by minimally modifying a nominal controller with the goal of rendering it safe. We consider the simple setting of a planar linear plant with a stable linear nominal controller and an affine constraint. We show that even in this simple setting, the filtered system can oscillate or diverge while remaining safe. We fully characterize the parameters for which the origin is globally (exponentially or asymptotically) stable. For all other parameters, the filter creates a saddle on the constraint boundary, and we characterize the region of attraction of the origin. For underdamped nominal dynamics, its boundary is an unstable periodic orbit, a homoclinic loop, or the saddle's stable manifold, depending on whether the CBF gain is below, at, or above a homoclinic transition value. A vehicle sideslip example and two three-dimensional examples illustrate the results and what changes beyond planar dynamics.