From Code to Motion: Simulating and Deploying Robots

📘 LaSalle's Invariance Principle in Control Systems

LaSalle’s Invariance Principle is a powerful tool in nonlinear control theory. It is used to analyze the long-term behavior (stability) of a dynamical system when the Lyapunov function’s derivative is non-strictly negative (i.e., V̇(x) ≤ 0 instead of V̇(x) < 0).

🔍 The Problem It Solves

In Lyapunov’s direct method, we typically show that a function V(x) (Lyapunov function) decreases over time (V̇(x) < 0) to conclude stability.
But what if V̇(x) = 0 for some trajectories?

That’s where LaSalle’s Principle comes in—it helps determine what happens even when V̇(x) is not strictly negative.

🧠 Statement of LaSalle’s Invariance Principle

Consider an autonomous system:

    𝑥̇ = f(x)
  

Let V(x) be a continuously differentiable function such that:

• V(x) ≥ 0 for all x
• V̇(x) ≤ 0 along system trajectories

Then, all system trajectories starting in a compact, positively invariant set Ω will approach the largest invariant set M contained in:

    E = { x ∈ Ω | V̇(x) = 0 }
  

💡 Key Concepts

• Invariant Set: A set where if a trajectory enters it, it stays inside forever.
• Limit Set: Where the system ends up as t → ∞.
• Lyapunov Function: Acts like a generalized "energy" function that decreases over time.

✅ Why LaSalle is Useful

• It relaxes the strict negativity requirement of V̇(x)
• Can conclude asymptotic stability where Lyapunov method only gives stability
• Helpful when V̇(x) is zero on a nontrivial set

🤖 Application in Robotics

In robotic systems (e.g., manipulators or drones), exact negativity of V̇(x) is hard due to nonlinearities, coupling, or underactuation.
LaSalle allows:

• Stability analysis of tracking controllers
• Proofs of convergence to desired trajectories
• Robustness analysis of adaptive and nonlinear control systems

🧮 Example Sketch

Suppose V(x) = x2 and V̇(x) = -x2 when x ≠ 0, but V̇(0) = 0. LaSalle's principle confirms that x(t) will approach 0, even though V̇ is not strictly negative everywhere.

📌 Summary

• LaSalle’s Principle helps analyze asymptotic behavior even if V̇(x) ≤ 0 only
• It identifies where trajectories "settle" — the largest invariant set inside {V̇(x) = 0}
• Widely used in nonlinear and adaptive control systems