In this paper, discrete-time quasi-sliding-mode control systems are considered. A new definition describing the quasi-sliding mode as a motion of the system, such that its state always remains in a certain band around the sliding hyperplane, is introduced. Then, two novel reaching laws satisfying conditions of the definition are proposed and applied to the design of appropriate linear control strategies which drive the state of the controlled system to a band around the sliding hyperplane. Consequently, the undesirable chattering and high-frequency switching between different values of the control signal are avoided. The strategies, when compared with previously published results, guarantee better robustness, faster error convergence, and improved steady-state accuracy of the system. Furthermore, better performance of the system is achieved using essentially reduced control effort.

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.

Three types of optimal, continuously time-varying sliding mode for robust control of second-order uncertain dynamic systems subject to input constraint are presented. Two of the modes incorporate straight sliding lines, and the third uses the so-called terminal slider, that is a curve that guarantees system error convergence to zero in finite time. At first, all three lines adapt themselves to the initial conditions of the system, and afterwards they move in such a way that, for each of them, the integral of the absolute value of the systems error is minimised over the whole period of the control action. By this means, insensitivity of the system to external disturbances and parameter uncertainties is guaranteed from the very beginning of the proposed control action, and the system error convergence rate can be increased. Performance of the three control algorithms is compared, and the Lyapunov theory is used to prove the existence of a sliding mode on the lines.

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.