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ON MINIMAL-VARIANCE CONTROL OF LINEAR SYSTEMS WITH QUADRATIC LOSS,
ILLINOIS UNIV URBANA COORDINATED SCIENCE LAB
The problem of minimizing the variance of a quadratic performance index, in the presence of control noise whose properties are known a priori, is studied for linear, constant systems with open-loop control. The expected value of the performance index is constrained to be a positive number, which cannot be less than the optimal mean index when variance is free. Solution is by means of the calculus of variations, which is applied to an equivalent noise-free problem. The necessary Euler equations are integro-differential and have a kernel matrix derived from output correlation functions. In general, these equations contain a forcing vector which depends upon the third-moment properties of the noise process. For systems in which the state vector can be chosen as the output, the existence of an inverse for the kernel matrix can be related to the total state controllability of an equivalent linear, noise-free plant which incorporates statistical data from the disturbance process. A maximum estimate for the number of eigenvalues is given, and is refined for the special case of single-input, state-output control. Necessary and sufficient conditions for a unique solution can be found in specific examples. Author