: Its techniques, such as recursive backstepping and "Immersion & Invariance" (I&I), have been applied to spacecraft attitude stabilization missile autopilot design Mechanical & Electrical Systems
In the context of , this theory is inverted. Instead of analyzing a given system, the engineer constructs the control law $u$ specifically to make $\dotV$ negative. This is known as Lyapunov-based control design (often implemented via Control Lyapunov Functions, or CLFs). : Its techniques, such as recursive backstepping and
Choose sliding surface (s = x). Design (u = -g^-1(x)(f(x) + k, \textsgn(s))) with (k > D). Lyapunov function (V = \frac12 s^2) yields (\dotV = s(d - k,\textsgn(s)) \leq |s|D - k|s| \leq -\eta |s|), (\eta = k-D > 0). Hence finite‑time convergence to (s=0), i.e., robust stabilization. Choose sliding surface (s = x)
Imagine a ball in a bowl. If you can prove that the "energy" of the system is always decreasing toward a minimum point (the bottom of the bowl), you know the system is stable. In control design, we create a Lyapunov Function ( Hence finite‑time convergence to (s=0), i
Lyapunov’s direct method is the unsung hero. Instead of solving messy nonlinear ODEs, we ask: "Is there a scalar energy-like function that always decreases along system trajectories?"
The book serves as both a theoretical summary and a practical guide for engineers facing real-world nonlinearities. Amazon.com Aerospace & Robotics