In this chapter, we propose a difference equation with a minima-based approach for developing discrete-time twisting-like and super-twisting-like algorithms. In contrast to most existing methods that rely on Euler discretization, the proposed algorithms are characterized by distinct roots. These algorithms are designed to handle both unperturbed and perturbed scenarios. Furthermore, we investigate the use of the discrete-time super-twisting-like algorithm in the design of a discrete-time super-twisting observer (STO), which facilitates finite-time state estimation. To validate the effectiveness of this approach, we present a case study involving a perturbed pendulum system with both known and unknown inputs, focusing on the state observation problem. Simulation results demonstrate the effectiveness of the proposed algorithm.