Linear control with incomplete state feedback and known initial-state statistics?

A method is proposed for the design of a linear, time-invariant state feedback control for a linear, time-invariant, finite-dimensional system with infinite duration of control (regulator problem), which makes use of only those state variables which are available for measurement, providing that these are sufficient to render the system stable. The expected value vector and the covariance matrix for the initial state are presumed to be known.

The cost function is quadratic and is expressed in terms of the initial state statistics and the cost-weighting matrix. The necessary conditions are derived for the minimization of the expected value of the cost. The minimization results in a set of m simultaneous polynomial equations in m unknowns where m is the product of the number of the available state variables and the number of control signals formed out of these. The theory is illustrated by a simple example of a third-order system.