Error Bounds for PD-Controlled Mechanical Systems under Bounded Disturbances Using Interval Arithmetic

We present a numerical algorithm based on invariant set theory to evaluate the worst-case performance of PD-controlled mechanical systems affected by bounded disturbances. By performing a specific coordinate transformation, the search and computation of positive invariant sets is simplified. It is shown that, thanks to the preservation of problem structure, the proposed method allows to obtain tight, component-wise bounds on the states, which are especially useful for performance evaluation and tuning of a PD controller. The bounds are formally guaranteed and can be used for safety certification. The method is compared to ultimate boundedness, and the superior results are shown via numerical simulations.