tailieunhanh - RECENT ADVANCES IN ROBUST CONTROL – NOVEL APPROACHES AND DESIGN METHODSE Part 10

Tham khảo tài liệu 'recent advances in robust control – novel approaches and design methodse part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | New Robust Tracking and Stabilization Methods for Significant Classes of Uncertain Linear and Nonlinear Systems 259 Proof. Note that if S a1 a- b 0 Va1 e a- and Vb e b b J then S a1 a- b 0 Va1 e a- a J Va- e a- a J and Vb e b b J . Moreover if S a- a- b 0 and S a a- b 0 Vb e b b then by using Theorem 10 the polynomials 3 . .- d s s a1 bk1 s a- bk- s bk3 d s s a bk1 s a- bk- s bk3 45 Vb e b b have a dominant real root. By taking into account the root loci with respect h of the polynomial 1 _ 1 1 7 z _ iĩ iĩ 1 7 d s s a- bk1 s a- bk- s bk3 hs 46 in the two cases of polynomial d s with all the roots real negative and of polynomial d s with a real negative root on the right of the remaining complex roots see Figs. 9 10 the proof easily follows. 2 - - 1 - - 0 - - -1 - - -2 - -4 -3 -2 -1 0 Fig. 9. Root locus of the polynomial 46 in the hypothesis that all the roots of d s are real. Fig. 10. Root locus of the polynomial 46 under the hypothesis that d s has a real negative root on the right of the remaining complex roots. Finally from Theorems 7 9 11 and by using the Routh criterion the next theorem easily follows. Theorem 12. Give the process 3 with limited uncertainties for n n 1 3 and assigned some design values of Kv and r ._d . Let be choose P p1 p- p- -1 - a ia - a-ia 3a1 Op with a 1 and a such that the roots of 260 Recent Advances in Robust Control - Novel Approaches and Design Methods n s s3 - d s d1s2 d2s d3 d s s - p1 s - p2 s - p3 47 are real . a and O a -2 and co . . Then said 5 a number not minus than b dn Pk Kvj such that S a3 a2 b 2 a1 bk1 - 9 a1 bk1 a2 bk2 27bk3 0 Vb e b b J and a1 a- a 2 a1 b k1 0 k1k2b b k1 k2 - k3 a1a2 0 Vb e b b where k Pd1 - a- k iPd2 - a2 1 b- 2 b- 48 49 50 51 i p3 b- under the hypothesis that the initial state of the control system of Fig. 2 with nc n 1 3 and k k is null and that r 0 - d 0 0 the error e t of the control system of Fig. 2 considering all the .