Iteration-free implicit Euler realization of integral sliding mode control for integrator-chain systems.

A direct implicit Euler discretization of integral sliding mode control (ISMC) yields a coupled algebraic inclusion that depends on the unknown matched disturbance. This paper develops an iteration-free digital realization of implicit ISMC for matched nth-order integrator-chain systems. A nominal-state decomposition removes the disturbance from the online solve, reducing the inclusion exactly to the scalar projection of standard implicit first-order SMC. The resulting O(n) update has deterministic execution time and, for the implicit-Euler closed-loop model, an O(T) ultimate bound. Simulations and DC-motor experiments show tracking comparable to the iterative implicit implementation and 9--64× lower control-rate variation than explicit Euler in the sinusoidal cases, while revealing a trade-off between tracking accuracy and actuator activity at coarse sampling. Embedded benchmarks show a 9--19× controller-core speedup.