LQ state-constrained control

Can a discrete-time, linear system be controlled so that if the initial state x(0) is nonnegative the state x(k) will remain nonnegative for all future k, while at the same time a quadratic cost criterion is minimized? It is known that the nondiagonal elements of the transformed weighting matrices of a finite-horizon cost criterion can be chosen to give a closed-loop matrix which contains no negative elements. The controllable block companion transformation which has to be used must be positive. Here this restrictive condition is removed, allowing the method to be applied to more general systems. The infinite-horizon problem is also solved. Examples are given to demonstrate the method