Complete statistical characterization of discrete-time LQG and cumulant control

As the performance index of the linear-quadratic-Gaussian (LQG) problem is governed by the noncentral generalized Chi-square distribution, solely controlling the expected value of the performance index, as the traditional LQG theory aims at, is insufficient to deliver a satisfactory solution in some situations. While the risk sensitive control does control a specific weighting sum of various moments of the performance index, the single degree of freedom in adjusting the weighting coefficients in this specific weighting sum of various moments of the performance index often prevents the risk sensitive control from generating a desired pattern of high order moment-distribution. We achieve in this note the complete statistical characterization of the performance index for the discrete-time LQG formulation. More specifically, we derive a recursive relationship to obtain cumulants of various orders of the performance index successively. Parameterized in feedback gain, the optimal feedback control law can be computed off-line by solving a static polynomial optimization problem, thus serving two design goals: i) To shape the probability density function (pdf) of the performance index to attain, at least approximately, a given desired pattern by regulating cumulants of various orders, and ii) to improve the performance measure of an incumbent solution (generated by the risk sensitive control, for example) by adjusting the levels of cumulants of various orders.