{'fraction_non_alphanumeric': 0.023379383634431455, 'fraction_numerical': 0.003188097768331562, 'mean_word_length': 5.0, '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}}
Application of Matrix Generalized Inverses to the Computation of Zeros and Zero Directions of Singular Linear Control Systems

The purpose of this paper is to apply the notion of the matrix generalized inverse to the problem of finding inputs which generate zero outputs in linear time invariant systems whose state equation is of the form Edx/dt=Ax+Bu where E is a singular square matrix. The method developed makes use of the simplest matrix generalized inverse. Two approaches are presented: (1) General eigenvalue problems are obtained for the finding of states and inputs of the respective forms x=exp(rt)w and u=exp(rt)1(t)v leading to zero output. (2) Laplace transform is used to find inputs generating zero outputs whithout a priori assumptions on the form of the states and the inputs. Three simple numerical examples are included. A brief discussion of the simplest matrix generalized inverse is added to the paper as an Appendix.