Robust Nonlinear Control Design State Space And Lyapunov Techniques Systems Control Foundations Applications Instant

The main bottleneck of Lyapunov methods is that there is no universal recipe for (V(\mathbfx)). For linear systems, (V = \mathbfx^T \mathbfP \mathbfx) with (\mathbfP) solving the Lyapunov equation works. For nonlinear systems, researchers use:

Warning: This is not a "cookbook." You won’t find MATLAB scripts on every page. You will find theorems, proofs, and lemmas. But the applications chapters (robot manipulators, spacecraft, motors) ground the math in reality. The main bottleneck of Lyapunov methods is that

Consider a scalar system: (\dotx = f(x) + g(x)u + d(t)), with (|d(t)| \leq D). You will find theorems, proofs, and lemmas

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)

Trade‑off: Chattering due to signum → often smoothed (e.g., saturation or high‑order SMC).