⚡ electrical engineering

On the Stability of Undesirable Equilibria in the Quadratic Program Framework for Safety-Critical Control

This paper investigates the formation and stability of undesirable equilibrium points in CLF-CBF-QP safety controllers, deriving compatibility conditions and proposing a novel control strategy that dynamically adjusts CLF geometry to ensure safety while guaranteeing quasi-global convergence to the desired target.

Original authors: Matheus F. Reis, A. Pedro Aguiar

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matheus F. Reis, A. Pedro Aguiar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a self-driving car. Your goal is to get to a specific destination (let's call it the "Home Base") as quickly as possible. However, the road is full of obstacles: construction zones, potholes, and other cars. You need a computer system that does two things simultaneously:

Steer you toward Home Base (Stability).
Keep you from crashing (Safety).

In the world of robotics and engineering, mathematicians use two powerful tools to do this:

The "Gravity Hill" (CLF): Imagine the destination is the bottom of a hill. The car naturally wants to roll down to the bottom. This is the "Control Lyapunov Function." It pulls the car toward the goal.
The "Force Field" (CBF): Imagine invisible walls around the obstacles. If the car gets too close, the wall pushes back hard to keep it safe. This is the "Control Barrier Function."

The Problem: The "Deadlock" Trap

The paper addresses a sneaky problem that happens when you combine these two tools using a specific math framework (called a Quadratic Program).

Sometimes, the "Gravity Hill" and the "Force Field" fight each other.

The Gravity Hill says, "Roll down here!"
The Force Field says, "Stop! You're too close to the wall!"

In some cases, these two forces cancel each other out perfectly at a spot that is not your destination. The car stops moving, stuck in a "deadlock." It's safe (it won't crash), but it's also useless because it can never reach Home Base.

Even worse, this stuck spot might be stable. If you nudge the car slightly, it just rolls back into the trap. It's like a ball sitting in a small dip on a hill; it won't roll away on its own. The authors call these "Undesirable Equilibria."

The Discovery: Geometry is Key

The authors realized that whether these traps appear depends entirely on the shape of the Gravity Hill and the shape of the Force Fields.

If the hill is round and the obstacle is round, they might cancel out perfectly in a bad spot.
If you change the shape of the hill (make it flatter on one side, or steeper on another), the "canceling out" spot might disappear or become unstable (like a ball on a peak; if you nudge it, it rolls away).

They introduced a concept called "CLF Compatibility."

Incompatible: The hill and the wall create a trap.
Compatible: The hill and the wall are shaped in a way that no matter where the wall is, the only place the car can get stuck is at the destination (Home Base).

The Solution: The "Shape-Shifting" Car

The paper proposes a clever fix. Instead of using a static Gravity Hill that never changes, they suggest using a Shape-Shifting Hill.

Here is how the strategy works, using an analogy:

The Driver's Plan: The car starts with a standard, round hill pointing toward Home Base.
The Danger Zone: As the car drives, it approaches an obstacle. The computer calculates: "Uh oh, with this current hill shape, we are about to get stuck in a deadlock trap near this wall."
The Transformation: Before the car gets stuck, the computer instantly morphs the shape of the hill. It stretches or squishes the hill so that the "trap" disappears.
- Imagine the hill was a rubber sheet. The computer pulls the sheet so that the small dip where the car was about to get stuck turns into a slope that pushes the car away from the trap and toward the destination.
The Escape: The car rolls past the obstacle safely.
The Reset: Once the car is past the danger zone, the hill slowly morphs back to its original, perfect round shape to guide the car the rest of the way home.

Why This Matters

Previously, engineers had to be extremely conservative. They had to leave huge gaps between the car and obstacles to ensure no traps would form. This made the car drive very slowly and inefficiently.

This new method allows the car to drive much closer to obstacles (maximizing efficiency) because it can dynamically change its "gravity" to ensure it never gets stuck, even in tight spaces.

Summary in a Nutshell

The Issue: Combining safety and goal-seeking can accidentally create "deadlocks" where a robot gets stuck forever.
The Cause: The specific shapes of the safety zones and the goal zones clash.
The Fix: Don't just use a fixed shape. Use a smart, adaptive shape that changes in real-time to dissolve any potential traps, ensuring the robot always finds a path to its goal without crashing.

The authors proved mathematically when these traps happen and provided a recipe (an algorithm) for engineers to design these "shape-shifting" controllers so robots can be both safe and efficient.

1. Problem Statement

Safety-critical control systems often utilize Control Lyapunov Functions (CLFs) for stabilization and Control Barrier Functions (CBFs) for safety, unified through a Quadratic Program (QP) framework. While this framework guarantees that system trajectories remain within a safe set (forward invariance), it introduces a critical flaw: the closed-loop system may develop undesirable equilibrium points other than the global minimum of the CLF.

The Issue: These spurious equilibria can be asymptotically stable, causing the system to get "stuck" (deadlock) before reaching the target or near unsafe boundaries.
Limitations of Existing Solutions: Previous methods to address deadlocks often rely on modifying safety constraints (potentially violating safety) or computing complex non-linear "convexification" functions that are computationally expensive and difficult to generalize.
Goal: The authors aim to characterize the existence and stability of these equilibrium points and propose a method to ensure the CLF and CBFs are "compatible," thereby eliminating stable undesirable equilibria without compromising safety.

2. Methodology

A. Characterization of Equilibrium Points

The authors analyze nonlinear, control-affine systems ( $\dot{x} = f(x) + g(x)u$ ) with multiple CBFs.

Existence: They prove that equilibrium points occur either in the interior of the safe set or on the boundaries of the safe sets ( $\partial C_i$ ).
Stability Analysis: Using the KKT conditions of the QP, they derive the closed-loop Jacobian at boundary equilibrium points. They establish a necessary and sufficient condition for instability based on the geometry of the CLF and CBF gradients. Specifically, stability depends on the projection of the system's Jacobian onto the subspace orthogonal to the CBF gradient.

B. Concept of CLF-CBF Compatibility

The paper introduces the concept of CLF-CBF Compatibility:

A CLF is defined as compatible with a set of CBFs if the resulting closed-loop system has no stable equilibrium points other than the CLF's global minimum.
If a CLF is compatible, any boundary equilibrium points that exist are guaranteed to be unstable, ensuring trajectories will not converge to them.

C. Analytical Framework for Quadratic Systems

For specific classes of systems (Driftless full-rank systems and Linear Time-Invariant (LTI) systems) with quadratic CLFs and CBFs, the authors derive a rigorous algebraic framework:

Generalized Eigenvalue Problem (GEP): They formulate the equilibrium condition as a GEP involving a Linear Matrix Pencil (LMP).
Q-Function: They define a scalar rational function, $q(\lambda)$ , which encodes the existence of equilibrium points on the boundary. Roots of $q(\lambda) = 1$ correspond to equilibrium locations.
Stability Polynomial Matrix ( $S(\lambda)$ ): They derive a matrix polynomial $S(\lambda)$ $S (λ)$ whose negative semi-definiteness determines the stability of the equilibrium.
- If $S(\lambda)$ is negative semi-definite at a root of $q(\lambda)=1$ , the equilibrium is stable (undesirable).
- If $S(\lambda)$ is not negative semi-definite, the equilibrium is unstable (safe).

D. Proposed Control Strategy

To resolve incompatibility, the authors propose a two-step adaptive strategy:

Compatibilization Algorithm: An optimization problem is solved to find a new Hessian matrix ( $H_{V_i}$ ) for the CLF. This new Hessian is the "closest" to the original reference CLF but satisfies the compatibility conditions (ensuring $S(\lambda)$ is positive definite at equilibrium roots).
Adaptive CLF Controller: A secondary QP controller dynamically adjusts the geometry (Hessian) of the CLF level sets in real-time.
- When the system state enters a region where a stable boundary equilibrium might exist (the "deadlock region"), the controller morphs the CLF geometry toward the compatible Hessian.
- This induces a bifurcation in the system dynamics, destabilizing the unwanted equilibrium and allowing the trajectory to escape and converge to the target.
- Once the state leaves the danger zone, the CLF geometry reverts to the original reference.

3. Key Contributions

Theoretical Characterization: Provided a detailed analysis of the existence and stability conditions for equilibrium points in multi-CBF CLF-QP frameworks, generalizing previous single-CBF results.
CLF-CBF Compatibility: Introduced the formal definition of compatibility and derived necessary and sufficient conditions for quadratic CLFs/CBFs in LTI and driftless systems using matrix polynomial theory.
Q-Function and Stability Polynomials: Developed the $q(\lambda)$ and $S(\lambda)$ tools to analytically predict the number and stability of boundary equilibria without simulation.
Adaptive Control Strategy: Proposed a novel method to compute a compatible CLF and a dynamic controller that smoothly transitions the CLF geometry to eliminate deadlocks while maintaining safety.

4. Results

The authors validated their method through numerical simulations on two-dimensional LTI systems:

Scenario 1 (Full-rank system): A nominal controller with a fixed CLF resulted in trajectories converging to a stable boundary equilibrium (deadlock). The proposed adaptive strategy successfully modified the CLF level sets (ellipses) as the system approached the barrier, destabilizing the equilibrium and guiding the system to the origin.
Scenario 2 (Underactuated system): Similar results were observed. The adaptive controller dynamically altered the CLF curvature to avoid the stable equilibrium point, ensuring convergence to the target while respecting safety constraints.
Visual Evidence: The simulations (Figs. 3 and 4) clearly demonstrate the "bifurcation" effect where the stable point vanishes as the CLF geometry adapts, contrasting with the deadlock observed in the fixed-CLF approach.

5. Significance

This work addresses a fundamental limitation in safety-critical control: the trade-off between safety and liveness (reaching the goal).

Safety Preservation: Unlike previous deadlock-avoidance methods that might relax safety constraints, this approach guarantees safety (invariance) is never violated.
Liveness Guarantee: By ensuring the CLF is compatible, the method theoretically guarantees that the system will not get stuck in a stable deadlock, ensuring the task is completed.
Generalizability: While the rigorous proofs focus on quadratic functions and specific system classes, the geometric insights regarding the interaction between CLF and CBF gradients provide a foundation for extending these concepts to more complex, non-quadratic systems.
Practical Application: The proposed adaptive strategy offers a practical solution for robotics and autonomous systems operating in complex environments with multiple obstacles, where traditional safety filters often lead to system stagnation.