On the Discrete-Time Algebraic Riccati Equation and Its Solution in Closed-Form

In the present paper we obtain a closed-form solution for the class of discrete-time algebraic Riccati equations (DTARE), whenever the eigenvalues of the A matrix are distinct. The obtained closed-form solution gives insight on issues such as loss of controllability and it might also prove comparable in terms of numerical precision over current solving algorithms. We also consider further extensions of the main result including, for example, a closed-form expression for the optimal regulator gain matrix K. Whenever possible we discuss the connection with earlier and established results.