Control Lyapunov Function (CLF)

A Control Lyapunov Function (CLF) is a tool used in nonlinear control to design a controller that ensures the system becomes stable. Unlike a regular Lyapunov function which simply checks if a system is stable, a CLF helps you construct a stabilizing control input.

What is a Control Lyapunov Function?

For a system with control input: 

      𝑥̇ = f(x) + g(x)u

A function V(x) is a Control Lyapunov Function if:

  • V(x) > 0 for all x ≠ 0, and V(0) = 0
  • There exists a control input u such that: 
              𝑑V/dt = ∇V(x)ᵗ (f(x) + g(x)u) < 0

This means we can always find some u that makes the "energy" of the system go down.

Simple Example

System:

      𝑥̇ = -x³ + u

Candidate Lyapunov Function:

      V(x) = ½ x²

Compute the Derivative:

      𝑑V/dt = x * 𝑥̇ = x(-x³ + u) = -x⁴ + x * u

Choose Control:

Let u = -k x where k > 0, then: 

      𝑑V/dt = -x⁴ - k x² < 0 ∀ x ≠ 0

✅ So, V(x) = ½ x² is a valid Control Lyapunov Function and u = -k x stabilizes the system.

Why is it Useful?

  • Guarantees stabilizing control exists at every non-zero state
  • Forms the basis of feedback design in nonlinear systems
  • Used in techniques like backstepping, adaptive control, and safety-critical control

Applications of CLF

  • Autonomous robots and drones
  • Adaptive cruise control and safety enforcement
  • Backstepping controller design
  • Combining with Control Barrier Functions (CBFs) for safe control