Continuity and order of continuity in discrete-time higher order sliding mode

Discrete-time sliding mode is inherently discontinuous in nature. But a relaxed version of continuity can be provided with the help of continuity in discrete sets. In this paper, a definition of higher order sliding mode in discrete-time systems using the concept of relaxed version of continuity in discrete-sets, (p,q)-continuity, is proposed. Moreover, a generalized relationship between sliding order and (p,q)-continuity is defined as the order of the continuity. These definitions facilitates to define ideal sliding and real sliding in arbitrary order discrete-time sliding mode based on (p,q)-continuity concept. Finally, for validation of definitions introduced in this paper, some existing first-order and second-order discrete-time sliding mode control algorithms are used.