Optimal Impulsive Disturbance Rejection in Oscillating Systems

The paper presents a new methodology for optimal disturbance rejection in the case where impulsive input signals are applied to a second-order linear system with oscillating behavior. Impulsive systems can be characterized as systems in which the control and disturbance inputs are specifically constrained to purely impulsive forms, while their activity over time is rather sporadic. Therefore, conventional approaches, which assume either continuous or piecewise constant inputs, cannot be used in such a scenario. To this end, a novel technique based on a quadratic integral criterion is developed to find the optimal magnitude and relative time shift of the compensating impulse with respect to the disturbance. The aim of the design is to ensure that the resulting response of the system to the disturbance is optimally rejected, hence its area is minimized. The formulation of the problem leads to an analytical solution of the cost function, while providing its partial derivatives results in the corresponding optimality conditions and their explicit solutions to obtain the parameters of the compensating impulse. The proposed technique is then validated within a simulation case study.