Define a sliding surface s(x) = 0. Choose a control law u = u_eq + u_sw, where u_eq satisfies the dynamics on the surface and u_sw = -K·sgn(s) enforces the reaching phase. Once s = 0 is reached, the system slides along the surface and its dynamics become insensitive to disturbances satisfying the matching condition.
How to design controllers robust to model uncertainties and disturbances without knowing their exact values — only an upper bound needs to be known.
Hyperplane s(x)=0 in state space defining the desired reduced-order dynamics.
Control component u_sw = -K·sgn(s) ensuring the surface is reached in finite time.
Control component u_eq that keeps the trajectory on the sliding surface once reached.
Rapid sgn(s) switching generates control oscillations that damage mechanical drive elements and excite high-frequency dynamics.
Excessively large K damages drives; too small fails to ensure robustness.
V. I. Utkin and colleagues formalise VSS and SMC theory at the Moscow Institute of Cybernetics.
Utkin's IEEE TAC paper introduces SMC to the Western literature.
A. Levant proposes HOSMC eliminating chattering by extending to higher orders; super-twisting algorithm (STA).
SMC becomes a standard robust control method for manipulators and drones in the robotics literature.
SMC runs on RT CPU; high sampling frequency (>1 kHz) is key for chattering reduction.
FPGAs for ultra-high frequencies (>10 kHz) in electric drives.
