{'fraction_non_alphanumeric': 0.024193548387096774, 'fraction_numerical': 0.0013440860215053765, 'mean_word_length': 4.866141732283465, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the structure of the solutions of discrete-time algebraic Riccati equation with singular closed-loop matrix

The classical discrete-time algebraic Riccati equation (DARE) is considered in the case when the corresponding closed-loop matrix is singular. It is shown that in this case all the symmetric solutions of the DARE coincide along some directions. A parametrization of the set of solutions in terms of a reduced-order DARE is then obtained. This parametrization provides an algorithm (that appears to be computationally very attractive when the multiplicity of the eigenvalue /spl lambda/=0 of the closed-loop matrix is large) for the computation of the solutions of the DARE. The same issue for the generalized DARE is also addressed.