Regulatory control design for stochastic processes: shaping the probability density function

This paper presents a regulatory controller synthesis technique that makes a pre-selected probability density function (PDF) for the closed-loop (CL) process the target of the design. The proposed design technique, referred to as PDF-shaping, approximately solves the integral equation giving the PDF of the CL process dynamics. The parameterization of the closed-loop process dynamics results in the parameterization of a controller that yields the required closed-loop PDF. Controller synthesis is then a matter of selecting, based on engineering or economic concerns, a target distribution, and then applying the proposed technique to find the approximate closed-loop dynamics. The special case of PDF-shaping using Gram-Charlier PDFs is presented, as well as the general case. Results of this novel approach to controller synthesis are demonstrated with numerical simulations for an example process.