Lyapunov Stability

A theory for judging whether a disturbed system returns to equilibrium, typically proven using an ‘energy function’ that never increases.

Lyapunov stability comes from the Russian mathematician Lyapunov's 1892 doctoral thesis, and is the foundation for analyzing the stability of nonlinear systems. It comes in grades: a state starting near equilibrium that never wanders far is Lyapunov stable; if it not only stays close but eventually returns to equilibrium, that's asymptotic stability; if the return rate is at least exponential, that's exponential stability. The most commonly used tool is the second method (the direct method): rather than solving the differential equations, find a function V(x) that equals 0 at equilibrium and is positive everywhere else, and that never increases over time along the system's trajectories (dV/dt ≤ 0) — this proves stability; strictly decreasing proves asymptotic stability. V can be thought of as a generalized energy, but it doesn't have to be actual physical energy. The hard part is that there's no general recipe for constructing V. In robotics, stability proofs for PD plus gravity compensation, impedance control, and passivity-based control all rely on it, and both the control Lyapunov function and control barrier function are built on top of it.

ExampleA damped pendulum: taking V = kinetic energy + potential energy (zero at the lowest point), one can compute that along the motion dV/dt = −b·θ̇² (b the damping coefficient, θ̇ angular velocity) — energy only decreases, never increases, so the lowest point is stable without ever solving the pendulum's equations of motion; adding LaSalle's invariance principle further proves the pendulum eventually comes to rest there.

Also called

Lyapunov Function, Lyapunov's Second Method, Lyapunov's Direct Method

Related

Control Lyapunov Function · Control Barrier Function · Passivity-Based Control · Impedance Control · Dynamic Stability · Robust Control

Sources

Lyapunov stability - Wikipedia