Cumulants and Risk-Sensitive Control: A Cost Mean and Variance Theory with Application to Seismic Protection of Structures
The risk-sensitive optimal stochastic control problem is interpreted in terms of managing the value of linear combinations of the cumulants of a traditional performance index. The coefficients in these linear combinations are fixed, explicit functions of the risk parameter. This paper demonstrates the possibility of controlling linear combinations of index cumulants with broader opportunities to choose the coefficients. In view of the considerable interest given to cumulants in the theories of signal processing, detection, and estimation over the last decade, such an interpretation offers the possibility of new insights into the broad modern convergence of the concepts of robust control in general. Considered in detail are the foundations for a full-state-feedback solution to the problem of controlling the second cumulant of a cost function, given modest constraints on the first cumulant. The formulation is carried out for a class of nonlinear stochastic differential equations, associated with an appropriate class of nonquadratic performance indices. A Hamilton-Jacobi framework is adopted; and the defining equations for solving the linear, quadratic case are determined. The method is then applied to a situation in which a building is to be protected from earthquakes. Densities of the cost function are computed, so as to give insight into the question of how the first and second cumulants affect a cost considered as a random variable.
KeywordsCost Function Linear Quadratic Gaussian Stochastic Differential Game Boundary Condition Versus Verification Theorem
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