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RECENT ADVANCES IN ROBUST CONTROL – NOVEL APPROACHES AND DESIGN METHODSE Part 14

Tham khảo tài liệu 'recent advances in robust control – novel approaches and design methodse part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Partially Decentralized Design Principle in Large-Scale System Control 379 where qh t e Rnh uh t e Rrh yh t e Rmh Ahl e Rnhxnl Bh e Rrhxnh and Chl e Rmhxnh V lV and 17 p p Tỉ m p 777 respect v xy ax L x I n I 1 n l r I 1 r I I t m I 1 r ml Problem of the interest is to design closed-loop system using a linear memoryless state feedback controller of the form u t -Kq t 80 in such way that the large-scale system be stable and K K11 K12 K1p K21 K22 K2p Kp1 Kp2 Kpp p Khh Klh l 1 l h 81 p uh t -Khh qh t - Khl qil t h 1 2 p 82 l 1 l h Lemma 1. Unforced autonomous system 75 - 77 is stable if there exists a set of symmetric matrices P I Ph Ph 83 Phk IPkh pỉ such that p_ 1 p r Pk n 1 m r Pk n 1 E I ihk t P h Ph qhk t Vhk t P h Ph qhk t 0 84 h 1 k h 1 Pkh Pk Pkh Pk where ihk t AhAh qhk t Ể Mq t 85 Akh Akk l 1 l h k Akl qlk t Vh t ik t 86 Proof. Defining Lyapunov function as follows v q t qT t Pq t 0 87 where P PT 0 P e Rnxn then the time rate of change of v q t along a solution of the system 75 77 is r q t qT t Pq t qT t Pq t 0 88 Considering the same form of P with respect to K Le P P11 P12 P1M P21 P22 P2M Pm1 PM2 PMM. p Phh Plh l 1 l h 89 380 Recent Advances in Robust Control - Novel Approaches and Design Methods then the next separation is possible P 1 P12 P1 12 0 0 . 0 . 0 . . P1p 0 0 0 . 0 P1p . 0 0 . . 1 0 0 0 . 0. P p1 0 . . 0 Pp 0 . 0 0 0 P p-1 p 90 0 . 0 P 0 . 0 Pp p 1 P p-1 J Writing 78 as a t Ahh Ahk a. t V Ahl a i Bh .01 rMh i 91 Vhk t - _Akh Akk Vhk t L a Vl t 0 bJ uk t J 91 and considering that for unforced system there are Ui t 0 l 1 . p then 91 implies 85 . Subsequently with 90 91 the inequality 88 implies 84 . 5.2 Pairwise system description Supposing that there exists the partitioned structure of K as is defined in 81 82 then it yields uh t p r - Kh Khl l 1 l h Vh t q l t jsk i I Vh t D l r 1 Vh t 1tk ix I vf 1tl i Kh Khki Ax _ Kh Khli Ax uh t ._ uh t Lqk t J l 1 l h k _ql t J l 1 l h k 92 where for l 1 2 . p i h k uh t - lKh Khl Vh t Vl t 93 Defining with h