LQR problem of linear discrete time systems with nonnegative state constraints

In the paper the infinite-horizon Linear Quadratic Regulator (LQR) problem of linear discrete time systems with non-negative state constraints is presented. Such kind of constraints on the system determine the class of positive systems. They have big application in many fields like economics, biology, ecology, ICT and others. The standard infinite LQR-optimal state feedback law is used for solving the problem. In order to guarantee the nonnegativity of the system states, we define the admissible set of initial states. It is proven that, for each initial state from this set the nonnegative orthant is invariant set. Two cases are considered, first, when the initial state belongs to the admissible set, and the second, when the initial state does not belong to the admissible set. The procedures for solving the problem are given for two cases. In second case we use a dual-mode approach for solving the problem. The first mode is until the state trajectory enters the admissible set and after that the procedure for the first case is used. The illustrative examples are given for both cases.