Adaptive control of a time-varying system

where P (t, λ) is a Poisson process with rate λ and Di for i ≥ 0 are i.i.d. random variables whose logarithm is uniformly distributed: logDi = U(log 0.25, log 32.0) between the logarithms of 0.25 and 32.0. This results in a distribution where the density is inversely proportional to the value. This is very important for the results below, because different distributions would emphasize either small or large values of Kr too much and allow compromises to be made easier in a controller (another alternative would be to penalize large deviations from the setpoint by a more severe factor than square). In a simulation, this means that a new value is selected from the uniform distribution at each timestep with the probability 0.05∆t, and the exponential of this value is used as the new process gain. Similarly, the setpoint xr is defined by