In this brief a new switching type reaching law for sliding mode control of discrete time systems is proposed. The proposed reaching law is a refined version of an earlier approach (introduced in the seminal work of Gao et al.) which enforces constant plus proportional decrease rate of change of the sliding variable. In our method, the proportional term is modified, so that the rate is always bounded and decreases slower for smaller values of the sliding variable than in the original approach. The refined reaching law proposed in this brief, on the one hand, ensures faster convergence and better robustness of the controlled plant than the earlier approach, and on the other hand, it helps satisfy constraints of important signals in the system. Furthermore, in the latter part of this brief a new nonswitching type reaching law is introduced, and it is demonstrated that it results in further improvement of the system robustness without increasing the magnitude of the critical signals in the system. Index Terms-Constraints, discrete time sliding mode control, nonswitching sliding mode controller, reaching law approach, sliding mode control, switching sliding mode controller.

In this paper, a discrete-time sliding mode inventory management strategy based on a novel non-switching type reaching law is introduced. The proposed reaching law eliminates undesirable chattering, and ensures that the sliding variable rate of change is upper bounded by a design parameter which does not depend on the system initial conditions. This approach guarantees fast convergence with non-negative, upper limited supply orders, and ensures that the maximum stock level may be specified a priori by the system designer. Furthermore, a sufficient condition for 100% customers' demand satisfaction is derived. The inventory replenishment system considered in this paper involves multiple suppliers with different lead times and different transportation losses in the delivery channels. Note to Practitioners-This paper presents a new periodic reviewinventory management strategy which prevents from exceeding the available storage capacity, ensures smooth order evolution and helps attenuate the bullwhip effect. The strategy is scalable, computationally efficient, and easy to implement in any typical inventory replenishment system. The strategy explicitly accounts for transportation losses and different lead times of commodity suppliers.

In this study, a new reaching law for sliding mode control of discrete time systems is proposed and applied to solve the problem of congestion avoidance in multi-source, connection oriented data transmission networks. Since the proposed reaching law does not require switching of the sliding variable between positive and negative values in each successive control step, it leads to chattering free operation, does not cause overshoots and helps achieve 100% exploitation of the bottleneck link available bandwidth. Furthermore, the proposed controller always generates bounded data transmission rates. The rates are limited by design parameters and they do not depend on the network initial conditions. The properties of the proposed controller are stated as three theorems, formally proved and verified in a simulation example.