Control Lyapunov function

A decreasing quantity encodes convergence#

A Lyapunov function is often compared with an energy measure: it is positive away from a desired state and decreases as the system approaches that state. A control Lyapunov function adds a control choice. For each relevant state, the controller must be able to choose an input that makes the function decrease at the required rate.

The target may be a fixed posture, a trajectory, or a periodic walking gait. The function and decrease condition are tied to a particular model and objective. A CLF is therefore a certificate within stated assumptions, not a general label meaning that a robot is stable.

The condition can enter an optimisation#

One common implementation writes the CLF decrease requirement as a constraint in a quadratic program. Other objectives can then choose among inputs that meet it. Galloway and colleagues used this structure to incorporate torque limits into a walking controller and demonstrated it on the MABEL biped.

A full-body controller can combine the condition with robot dynamics and task costs. Reher and colleagues report an inverse-dynamics CLF formulation tested in simulation for walking and on hardware for dynamic crouching. These results support the specific controllers and experiments described in those papers.

Stability and safety are different claims#

A CLF expresses progress toward an objective. A control barrier function instead expresses invariance of a safe set. A controller can include both, often treating safety as a hard constraint and allowing the performance condition some relaxation when the two conflict.

Model error, state-estimation error, actuator delay, torque saturation, and infeasible constraints can weaken the expected behaviour. The chosen function must also represent the desired equilibrium or gait correctly. Checking one CLF inequality does not validate unrelated properties such as collision avoidance or hardware reliability.