Learning objectives
After this laboratory, you should be able to distinguish stability from attraction, evaluate a candidate Lyapunov function and its derivative, explain a semidefinite derivative using invariance, and choose a stabilizing feedback gain.
1. Stability means more than bounded trajectories
Consider an autonomous system ẋ = f(x) with an equilibrium at the origin, f(0) = 0. The origin is Lyapunov stable if every sufficiently small initial perturbation remains arbitrarily small for all future time. It is attractive if nearby trajectories converge to the origin. It is asymptotically stable if it is both stable and attractive.
An undamped center illustrates stable but non-attractive motion. A stable spiral illustrates asymptotic—and in this model exponential—convergence.
2. Lyapunov's direct method
A differentiable scalar function V(x) can certify stability without solving the differential equation explicitly. A standard candidate satisfies V(0)=0 and V(x)>0 for nonzero x. Its derivative along trajectories is:
If V is positive definite and V̇ is nonpositive near the equilibrium, the origin is stable under the usual regularity assumptions. If V̇ is strictly negative for nonzero states in a neighborhood, asymptotic stability follows. Global statements additionally need appropriate global hypotheses such as radial unboundedness (properness) and global decrease.
Key distinction: A bowl-shaped V only tells us that V is positive. Whether the system stabilizes depends on the direction of motion relative to the contours, expressed through V̇.
3. Four visual cases
| System | Lyapunov observation | Conclusion |
|---|---|---|
| Center | V̇ = 0 | Stable; no attraction |
| Stable spiral | V̇ = −a‖x‖² for a > 0 | Exponential convergence |
| Damped oscillator | V̇ = −c x₂² ≤ 0 | Use LaSalle for c > 0 |
| Unstable saddle | V̇ is indefinite for the selected quadratic V | Unstable: x₁(t) = x₁(0)eat, a > 0 |
For the oscillator, V̇ = 0 on the entire set x₂ = 0, yet most points in that set are not invariant because ẋ₂ = −1.4x₁. The only invariant point on that set is the origin. For fixed c > 0, the weighted quadratic V = ½(1.4x₁² + x₂²) has compact positively invariant sublevel sets. LaSalle's invariance principle then establishes convergence to that largest invariant subset. An indefinite derivative of one candidate, by itself, would not prove instability of another system.
4. Worked example: a control Lyapunov function
Take the deliberately simple fully actuated unstable plant and proportional state feedback:
Choose V(x) = ½(x₁² + x₂²). Substituting the feedback into the derivative gives:
For K > 1, both coefficients are negative and the origin is globally exponentially stable for this linear example. At K = 1, the x₁ component stays constant: stability remains, but asymptotic convergence fails. When K < 1, at least one state direction is unstable. A control Lyapunov function is a function for which a suitable control can make the derivative negative; that does not mean every feedback law succeeds.
5. Guided practice · 20 minutes
- Prediction: Without using the simulation, classify a center, stable spiral, damped oscillator and saddle. Sketch the expected V(t).
- Run: Use the same initial state for the center and spiral. Compare the level sets, state norm and V̇.
- LaSalle: For the damped oscillator, pause when x₂ is near zero. Why is V̇ near zero even though the trajectory can keep moving?
- Controller design: Set K to 0.8, 1.0 and 1.6. Export results from each run and compare.
- Perturbation: Under stabilizing feedback, apply a state impulse. Describe the instantaneous jump in V and subsequent behavior.
Suggested study record
A small comparison table for K = 0.8, 1.0 and 1.6; a graph of V(t) from at least two runs; and 150–250 words explaining which conclusions follow mathematically and which are only observations from the simulation.
6. Lecture slides & deeper reading
- Kellett & Braun — Stability Notions (PDF lecture slides)
- Braun & Kellett — Control Lyapunov Functions (PDF lecture slides)
- Tedrake — Lyapunov Analysis (MIT notes)
The simulator uses idealized continuous-time two-state systems and numerical integration. Its energy surface illustrates a candidate V, not a physical 3D energy surface. Changing gains or injecting pulses during a run alters the autonomous-system assumptions.