{'fraction_non_alphanumeric': 0.014806378132118452, 'fraction_numerical': 0.0011389521640091116, 'mean_word_length': 5.104166666666667, '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}}
Representation and LQR of exact fractional order systems

This paper proposes an efficient mathematical operator called spatial product whereby fractional order systems (FOSs) can be represented as a standard state space form of partial differential equations. This infinite dimensional model represents the true state of the FOS and is adopted in this study, whereas the widely used pseudo state space equation with Caputo's definition is only an approximate model. We derive the exact state space solution of the FOS in terms of the spatial product and also reveal a separation property with respect to the initialization of FOSs. Furthermore, the linear quadratic regulator (LQR) and the H2 control of FOSs are studied in such rigorous frame. The results show that these optimal control problems can be formatted as Riccati-type equations in the sense of the spatial product.