Statistical Control for Performance Shaping Using Cost Cumulants

The cost cumulant statistical control method optimizes the system performance by shaping the probability density function of the random cost function. For a stochastic system, a typical optimal control method minimizes the mean (first cumulant) of the cost function. However, there are other statistical properties of the cost function such as variance (second cumulant), skewness (third cumulant) and kurtosis (fourth cumulant), which affect system performance. In this technical note, we extend the theory of traditional stochastic control by deriving the Hamilton-Jacobi-Bellman (HJB) equation as the necessary conditions for optimality. Furthermore, we derived the verification theorem, which is the sufficient condition, for higher order statistical control, and construct the optimal controller. In addition, we utilize neural networks to numerically solve HJB partial differential equations. Finally, we provide simulation results for an oscillator system to demonstrate our method.