A note on discrete-time observer-based integral sliding mode predictive control

A fundamental issue in control applications is ensuring performance and stability in the presence of modeling uncertainties and external disturbances. In this regard, in recent years the integration of Sliding Mode Control (SMC) with Model Predictive Control (MPC) has proven particularly effective, combining the intrinsic robustness of SMC with MPC's ability to handle constraints and provide optimal performance. In this context, Integral Sliding Mode Predictive Control (ISMPC) has emerged as a promising approach. Despite this practical relevance, relatively few studies in the literature address discrete-time, observer-based implementations of ISMPC when the full state is not available and noisy output measurements must be used. In particular, the application of ISMPC to collocated electromechanical systems presents the issue of dealing with invariant zeros. In this work, three discrete-time observer implementations, namely, a standard Discrete-time Sliding Mode Observer (DSMO), a DSMO variant named the Direct Output Injection Estimator (DOIE), and an Unknown Input Observer (UIO), are analyzed and compared, when used in combination with ISMPC to control a simple two-mass system with friction. Under noisy measurement conditions, both the UIO and DOIE significantly outperform the DSMO; while the UIO provides the highest theoretical robustness, the DOIE delivers comparable performance with minimal tuning effort.