Modified Algebraic Riccati Equation Closed-Form Stabilizing Solution

A modified discrete-time algebraic Riccati equation (MARE) is a discrete-time algebraic Riccati equation (DARE) for which the quadratic term is weighted by a modifying parameter
$\alpha $
. The MARE is known to arise for example when conducting estimation or stabilization of single-input single-output (SISO) systems subject to packet losses in networked control systems (NCSs). In the present paper we characterize the solution to the MARE in closed form that, to the best of the author’s knowledge, is a completely novel result. We then verify the already known critical value of the modifying parameter
$\alpha _{c}$
for which the MARE is solvable and propose closed-form expressions for the optimal state feedback gain matrix. We finally present examples to illustrate the obtained contributions.