A Phase-Space Energy-Coupled Sliding Surface for Quadrotor Attitude Control: Well-Posedness Conditions and a Dual-Regime Evaluation

Constant-slope sliding surfaces require a trade-off: steep slopes improve regulation under sustained disturbance but increase transient control effort, while shallow slopes reduce effort at the expense of accuracy. We investigate a surface whose slope is scheduled by a weighted phase-space measure of the tracking error. Because the schedule depends on error rate, differentiation introduces a state-dependent coefficient on the error acceleration; if it vanishes, the equivalent control becomes singular, and if it changes sign, the effective input gain reverses. We show that the coefficient’s deviation from unity is bounded by one quarter of the coupling gain times the square root of the velocity weight, independently of the energy scale. A single inequality therefore guarantees a well-posed control law and a uniquely solvable manifold. Finite-time reaching and exponential convergence follow, with practical stability under a boundary layer. In six-degree-of-freedom quadrotor simulations with exact kinematics, attitude-upset recovery, hover disturbance, and inertia uncertainty, evaluated over 20 held-out disturbance phases, the reference PSEC configuration reduces peak commanded torque by 51.7% at e0=0.3rad compared with c=24 at matched steady-state accuracy. However, the peak increases by 32.6% at e0=0.1rad. A procedure-selected PSEC configuration (origin slope 25.35) reduces the peak by 67–77% over e0=0.1–0.5rad. A tuned capped terminal surface is more accurate.