Weighted least-squares, cost density-shaping, stochastic optimal control: A step towards total probabilistic control design

To date, both moment-based and cumulant-based stochastic control schemes have been proposed to constrain statistics of the random cost functional in the stochastic optimal control formalism. However, existing methodologies do not enable the designer to deliberately shape the probability density function of the random cost according to a pre-specified target density characterized by a finite number of cost statistics. Since the mean, variance, skew, and kurtosis of a variate are strongly associated with the appearance of its density function, a cumulant-based control paradigm that can steer four or more cost cumulants towards nominal target values would be ideal for cost density-shaping objectives. Bearing this idea in mind, we propose a novel weighted least-squares optimization problem of minimizing a weighted sum of squared differences between initial cost cumulants and target initial cost cumulants, for arbitrarily-many terms. The problem is solved using dynamic programming techniques adapted for the cost cumulant-generating equations of the Linear Quadratic Gaussian (LQG) framework. The Minimum Weighted Least-Squares, Cost Density-Shaping (MWLS-CDS) optimal controller results and is applied in a building protection problem. It is shown that MWLS-CDS controls can achieve target cost cumulants resultant from a family of 3CC controls to within a 0.5% margin of normalized error.