Constrained optimal control

The explicit form of the optimal control law of a given linear, discrete-time, time-invariant process subject to a quadratic cost criterion is well known. In some applications it is desirable that the state of a controlled dynamic process be nonnegative, given a certain class of initial disturbances. Using the controllable block companion transformation, sufficient conditions on the weighting matrices of the cost criterion are derived to ensure that the closed-loop response of the original process with the standard, unconstrained optimal feedback law will be nonnegative. It is shown that the nondiagonal elements of the transformed weighting matrices can be chosen to ensure nonnegativity