Systems, Structure and Control 2012 Part 6

Tham khảo tài liệu 'systems, structure and control 2012 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Integral Sliding Modes with Block Control of Multimachine Electric Power Systems 93 Thus the trajectories of the last variables vector zr are asymptotically stable. Step r-1. Proceeding in similar way as in previous step the Lyapunov function Vr sr_1 sr is proposed then V -1 s _-Pr-1 xr-1 sigm sr-1 gr-1 f x . 41 In the region sr-JI r-1 the equation 41 becomes s - Pr-1 xr-1 sign Sr-1 gr-1 t x . . 42 1 Isr-11 l-Pr-1 xr-1 gr-1 t x Moreover under the condition Pr-1 xr-1 gr-1 f z sr-1 will be decreasing until it reaches the set JIsr-JI r-1 in a finite time and it remains inside. The upper bound of this reaching time can be calculated by using the comparison lemma Khalil 1996 as follows r-1 sr-1 0 - r-1. Furthermore the equivalent control xr-11eq fulfills S--1 Xr-ỉ ỉeq gr-1 f z r-1Y r-1 43 where r-1Yr-1 is the error introduced by using the control law 29 . To analyze the stability of the r-1 block of the system 38 the Lyapunov function Vr-1 12 zr T z r -1 is considered and its time derivative is given by Vr-1 z -1T . . z- X . s V - 7 . -kr-1zr-1 Er-1zr - Pr-1 xr-1 sism I r I gr-1 t z V r-1 J gr-1 f z . In the region sr-J r-1 the derivative Vr-1 becomes Vr-1 -MM 2 IMI kll - Pr -1sign l - 1J gr-1 S- z -kr-11 zr-1 2 1 zr-11 zJ s -1 and considering 43 it can be rewritten as Vr-1 -kr-JM2 IMI Hzll r-1Yr-1 . Suppose that r-1Yr-1 satisfies the following bound r-1Yr-1 ar-JIz -II fir-1 ar-1 fir-1 e R . Then it is possible to present the equation 44 of the form 44 94 Systems Structure and Control Vr-1 -kr-1 K J2 I z r-J z JI a-1 z -JI pr-1 - z-JI k-1 -ar-1 z r-J- z JI - P-1 which is negative in the region z JI p-J zJ -1 45 where õr-1 k ------ and k P - Moreover Sr-1 and Ằr-1 are positive for kr-1 ar-1 . Thus the trajectories of the vector state enter ultimately in the region defined by Hz r-JI Sr-JIz JI Pr-1 . Step i. The step r-1 can be generalized for the block i with i r-1 r-2 . 1. In the region sJI i the derivative of the Lyapunov function Vj s s is calculated as Vi

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