Some remarks about the tradeoffs between exactness and robustness in control

The performance under uncertain dynamics (perturbation) and measurement noise of linear and discontinuous controllers is discussed. On one hand, discontinuous controllers are exact: they completely remove the effect of the perturbation in absence of noise; however, they are usually assumed to be more sensitive to noise than linear controllers. On the other hand, in absence of perturbation, linear controllers are usually considered optimal under Gaussian noise; however, they are unable to completely remove the effect of the perturbation. The question we pose is which of the two classes of controllers performs better when both noise and perturbation are present, and whether if there exists a better class of controllers than this two. By an elementary example, it is shown that, under the bounded disturbance hypothesis, linear and discontinuous algorithms are equally sensitive to noise by measuring their performance using the steady state error they produce. Therefore, since discontinuous controllers are additionally exact to perturbations, they are the optimal selection under both noise and perturbation. A discussion of this apparent counter-intuitive result is presented.