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Abstract In the stochastic context, expected value of the cost function is optimized either by changing the mean values of the manipulated variables or by reducing their variance. An extension is to look for an optimal shape for the entire probability density function (PDF). Though the use of asymmetric PDFs is proposed in the literature, no formal proof that justifies their use has been provided. In this paper, it is shown that an asymmetric PDF is required if and only if the cost function is asymmetric and the manipulated variable is penalised. The proof uses an analytical solution of the Fokker-Planck-Kolmogorov equation derived to calculate the shape the output PDF for scalar systems. In particular, this analytical solution is adapted to a switching proportional controller. The theoretical concepts are illustrated on a simulation example, where the advantage of choosing an asymmetric PDF is shown. |
