Graduate control engineering · interactive classroom lab

Lyapunov stability & control functions

Watch states move through the phase plane and over a Lyapunov energy bowl. Compare stability, asymptotic convergence, a saddle, the LaSalle case, and feedback control. Change a gain or perturb the state while the animation runs.

Offline HTML · no dependencies · mouse / projector friendly

1 · Choose a dynamical system

Parameters update while running
0.50
Playback 2×

Initial state & disturbances

Click or drag in phase plane
Impulse disturbance (applied to the live state)

An impulse produces an instantaneous jump in V; the plotted V̇ describes the continuous dynamics between impulses.

2 · Phase portrait

t = 0.0 / 32 s
State trajectory V(x) contour✣ Equilibrium at origin

3 · Lyapunov energy landscape

Same V(x), different dynamics
State on energy surfaceHeight = V(x)Wireframe = candidate function

5 · Watch the proof evolve over time

Histories can change if you adjust gains or inject disturbances
V(x(t)) — candidate “energy”V ≥ 0
V̇(x(t)) — energy rateV̇ negative / zero / positive

Suggested 10-minute class exercise

1. Run the stable center, then the stable spiral. Both stay near the origin, but only the spiral converges.

2. Run the damped oscillator: V̇ is zero whenever x₂ = 0, even though trajectories ultimately converge. Discuss LaSalle's invariance principle.

3. Choose the CLF controller. Set K = 0.8, 1.0, and 1.6, resetting between trials. Ask students when V̇ becomes negative definite.

Reading the result correctly

V positive definite is not sufficient by itself: the same bowl appears for stable and unstable dynamics.

V̇ < 0 away from the equilibrium proves asymptotic stability under the relevant hypotheses. V̇ = 0 for a center demonstrates stability without convergence. For the damped oscillator, V̇ ≤ 0 needs an additional invariance argument to conclude convergence.