Sliding Mode Control for Delta Operator Sampled Systems

This paper addresses a class of sliding mode control (SMC) strategies for high-frequency sampled systems with uncertainties, utilizing a generalized Delta operator reaching law. First, Delta operator sampled systems are regarded as a bridge in the unified research framework of discrete-time systems and continuous-time systems. With the increase of sampling frequency, Delta operator sampled systems can maintain consistent stability with their corresponding continuous models. Then, a generalized reaching law by Delta operator is presented, which can degenerate into multiple existed reaching laws under certain conditions. And then, the reaching condition, quasi-sliding mode (QSM) band, and finite time reachability of the proposed generalized Delta operator reaching law are analyzed. Next, sliding mode controllers are designed separately for both matched uncertainty and unmatched uncertainty cases. With the effect of these controllers, Delta operator SMC systems exhibit desirable dynamic characteristics. Finally, two examples on truck-trailer system and ball-beam system demonstrate the effectiveness of Delta operator sliding mode controllers, and show efficient performance in stabilizing high-frequency sampled systems with uncertainties. These scenarios illustrate the potential of the proposed strategies for practical applications. Note to Practitioners—This work aims to develop sliding mode control strategy for Delta operator sampled systems, which is of fundamental significance in the fields of high-frequency sampled control. We propose a generalized Delta operator reaching law and provide a unified framework for high-frequency sampled discrete-time systems and their corresponding continuous-time systems, addressing practical challenges in areas, such as high-speed motors, supercomputing, and network transmission. The reaching condition, quasi-sliding mode bandwidth, and finite time reachability of the proposed generalized Delta operator reaching law are fully considered. Given that the effects of unknown matched or unmatched uncertainties, sliding mode controllers based on Delta operator are designed, which are applied to truck-trailer system and ball-beam system.