The thesis discusses stability analysis for different hold functions applied in digital control, with and without considering time delays within the control system. The stability analysis was carried out for the classical control problem of balancing an inverted pendulum with a discrete time PD controller. The stability of the control system was examined in case of the zero-order, first-order, second-order and for the system-matched hold functions. In many cases the controlled plant is continuous, therefore the conversion from the digital control signal into a continuous signal is needed for proper functioning. Hold functions are responsible for converting the discrete-time signal into a continuous-time signal. The most basic form of the hold function is the zero-order hold, which holds the current value of the input signal until the next signal arrives. The nth order hold uses the past n+1 discrete data to generate the hold function. Higher order hold functions are proven to have the ability to increase the accuracy and in some cases the stability of the control system. The system-matched hold (SMH) is a special form of the generalized sampled data hold function, where the hold function is determined from system dynamics. In the first three chapters the theoretical framework of the thesis is presented including the discussion of the necessity for hold functions in digital control systems, the method of plant discretization in state space and the stability criteria for control systems. In the fourth chapter the dynamic model of the inverted pendulum was constructed. Firstly the equation of motion was derived, then the time discretization of the control system was carried out using the matrix exponential method. As a result of the discretization the whole control system can be analyzed in discrete time.