Linear control traditionally relies on input-output transfer functions. For a nonlinear system, this approach collapses. Instead, the becomes the natural language. A nonlinear system is described as:
is a highlight. If you can find a Control Lyapunov Function ( V(x) ) (a positive definite function whose derivative can be made negative by choosing ( u )), Sontag’s formula gives you an explicit, universal feedback law: [ u(x) = -\fracL_f V + \sqrt(L_f V)^2 + (L_g V)^4L_g V ] (Yes, it looks intimidating. No, you don’t implement it by hand—but the theory is pure gold for nonlinear backstepping and adaptive control.) this approach collapses. Instead