Control Lyapunov Function

An energy-like function that, as long as some control choice can always make it decrease, proves the system can be steered to its goal.

A Lyapunov function V(x) behaves like a system's ‘energy’: zero at the goal, positive everywhere else. A plain Lyapunov function is used to analyze whether an already-designed closed loop is stable; a control Lyapunov function instead applies to a system that still has a free input u and no fixed control law yet: if, at every non-goal state, there exists some u making V̇ < 0, the system can be steered — stabilized — to the goal. This theory was developed by Artstein and Sontag in the 1980s, and Sontag also gave a general formula for constructing a control law directly from a CLF. Robotics commonly uses CLF-QP: each control cycle, solve for a u satisfying V̇ ≤ −λV (exponential convergence at rate λ) while keeping torque as small as possible, which makes it easy to add torque limits, friction cones, and similar constraints at the same time. Ames et al. applied it to bipedal walking; it is also often combined with a control barrier function in the same quadratic program, with the safety constraint kept hard and the convergence constraint relaxed as a soft constraint.

ExampleFor a first-order system ẋ = u, take V = x²/2, so V̇ = x·u. Requiring V̇ ≤ −V (i.e., λ = 1) at x = 2 becomes 2u ≤ −2, i.e., u ≤ −1. The CLF-QP picks the smallest-magnitude u satisfying that, u = −1, and the state converges to 0 at an exponential rate.

Also called

CLF, CLF-QP

Related

Lyapunov Stability · Control Barrier Function · Quadratic Programming · Hybrid Zero Dynamics · Feedback Linearization · Optimal Control

Sources

Wikipedia: Control-Lyapunov function
Ames, Galloway, Sreenath, Grizzle: Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics (IEEE TAC 2014)
Ames et al., Control Barrier Functions: Theory and Applications（CLF-CBF-QP 一节） (Chinese)