Tham khảo tài liệu 'sabatier agrawal machado advances in fractional calculus episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | LMI CHARACTERIZATION OF FRACTIONAL SYSTEMS STABILITY 427 Theorem 5 13 sufficient condition Fractional system 7 is t r stable if matrix P 0 P M x M exists such that 1Y 1Ì Av P P Av 0. I J I J Proof See steps above. Also in 13 . Validity of the stability condition Figure 3 presents stability domain DS of a fractional system characterized using theorem 5 according to fractional order V and to arg spec A . A comparison between Fig. 3 and Fig. 1 reveals that the entire stability domain is not identified using theorem 5. It therefore leads to a sufficient but nonnecessary condition. arg spec A in n 2 rad Fig. 3. Stability domain DS u determined by criterion 2 according to the values of V and ớ . A simple explanation can be given. Systems 7 and 19 have strictly the same behavior. However transformations given by relations 16 to 18 produce a matrix Af whose eigenvalues are outside the left-half complex plane. The unstable modes thus created are compensated by zeros produced by matrix Bf A t thus leading to a stable response to nonzero initial conditions. Due to such a situation a method based on eigenvalue analysis of matrix Af can only produce pessimistic stability conditions. 428 Moze Sabatier and Oustaloup In order to analyze such a conservatism let Of be an arguments of an eigenvalue of matrix Aỉ v and o be the one of system 6 state transition matrix A of system 6 . Line D in Fig. 4 represents the function Fv that associates ớ to ớ f according to v Fv 0 tf 0 2tf 1 x x V 20 Fig. 4. Fv as a function of ớ and 0f I and deduced stable domain . As V decays towards 0 the slope of D increases significantly such that high values of ớ lead to detection of some instability within the fractional stability domain Ds which is thus reduced to Ds 0- Ụ 4 - 3 vy 4 - 1 vy . 21 i 1 2 . A method leading to a necessary and sufficient condition for stability of fractional systems is therefore necessary. LMI CHARACTERIZATION OF FRACTIONAL SYSTEMS STABILITY 429 6 Stability Theorem Based on
