id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
american_scientific_journal-3454	I. Ali, Hazem; Z. M. Ali, Mohammed	LQR and H_2 Controllers Design Using State Derivative Feedback for Multivariable Systems	2017	11	.pdf	application/pdf	3013	159	65	Volume 37, No 1, pp 118-128 120 𝐽𝐽 =∫ (�̇�𝑥𝑄𝑄𝑥𝑥 + (𝐾𝐾�̇�𝑥)𝑇𝑇𝑅𝑅(𝐾𝐾�̇�𝑥))𝑑𝑑𝑡𝑡 = ∫ �̇�𝑥𝑇𝑇(𝑄𝑄 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾)�̇�𝑥𝑑𝑑𝑡𝑡∞ 0 ∞ 0 (6) Suppose that a constant positive semidefinite symmetric matrix 𝑃𝑃 that satisfy equation (6) can be obtained, thus �̇�𝑥𝑇𝑇(𝑄𝑄 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾)�̇�𝑥=− 𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑥𝑥𝑇𝑇𝑃𝑃𝑥𝑥) = −�̇�𝑥𝑇𝑇𝑃𝑃𝑥𝑥 − 𝑥𝑥𝑇𝑇𝑃𝑃�̇�𝑥 (7) then, equation (7) can be rewritten as: �̇�𝑥𝑇𝑇(𝑄𝑄 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾)�̇�𝑥 = −�̇�𝑥𝑇𝑇(𝑃𝑃𝐴𝐴𝑐𝑐−1 + 𝐴𝐴𝑐𝑐−𝑇𝑇𝑃𝑃)�̇�𝑥 (8) where 𝐴𝐴𝑐𝑐−1 = 𝐴𝐴−1(𝐼𝐼 + 𝐵𝐵𝐾𝐾) � = ∫ (∞ 0 �̇�𝑥𝑇𝑇(𝑡𝑡)(𝑄𝑄𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅𝐾𝐾𝑚𝑚)𝑥𝑥(t))𝑑𝑑𝑡𝑡 (37) Suppose that, it can be found a constant positive simidefinite symmetric 𝑃𝑃 that satisfy equation (37), �̇�𝑥𝑇𝑇(𝑡𝑡)(𝑄𝑄𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅K𝑚𝑚)�̇�𝑥(t) = − 𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑥𝑥𝑇𝑇(𝑡𝑡)𝑃𝑃𝑥𝑥(𝑡𝑡)� = −�̇�𝑥𝑇𝑇(𝑡𝑡)𝑃𝑃𝑥𝑥(𝑡𝑡)	cache/american_scientific_journal-3454.pdf	txt/american_scientific_journal-3454.txt