  Bode integrals of sensitivity and sensitivity-like functions along with
complementary sensitivity and complementary sensitivity-like functions are
conventionally used for describing performance limitations of a feedback
control system. In this paper, we investigate the Bode integral and evaluate
what happens when a fractional order Proportional-Integral-Derivative (PID)
controller is used in a feedback control system. We extend our analysis to when
fractal PID controllers are applied to irrational systems. We split this into
two cases: when the sequence of infinitely many right half plane open-loop
poles doesn't have any limit points and when it does have a limit point. In
both cases, we prove that the structure of the Bode Integral is similar to the
classical version under certain conditions of convergence. We also provide a
sufficient condition for the controller to lower the Bode sensitivity integral.
