Discrete Time Systems Part 11

Tham khảo tài liệu 'discrete time systems part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Stability Criterion and Stabilization of Linear Discrete-time System with Multiple Time Varying Delay 289 TX1TQiX1 di - di XỈRiXi - X1 i 1 i 1 X2 - AX1 -2 P1 X2 X3 X3 XT 3 -Q1 0 . 0 - -1 0 AT - Ad1 0 0 0 - 0 -Q2 0 0 -Al. Ad1 0 . Ad1 Ad2 AdN J 0 0 . 0 0 -Qn _ .02 _AT Ad1 J Let Si Qi1 and T Rị1 we have -X1 AX1 - BF X2 X3 XT X2 X3 -X1 0 -S1ÁT1 0 -S1 0 -S2Ad2 - 0 -S2 0 0 X 0 . SNAdN S SN X1 0 0 -S1 0 X X1 0 - -SN - 0 - T d1 - d1 X1 0 0 0 0 0 0 Tv d. - d. UN dv 0 13 -X1 -- AX1 - BF X2 X3 XT X2 X3 -X1 0 -S1ÁT1 0 -S1 0 -S2 Ad2 0 -S2 0 0 0 -SnATn X -Sn X1 0 0 -S1 - 0 X X1 0 - -SN T . 0 A d1 - d1 X1 0 0 0 0 0 0 Tn d - d. dN dN 0 14 290 Discrete Time Systems Theorem 2 For a given set of upper and lower bounds d dị for corresponding time-varying delays dki if there exist symmetric and positive-definite matrices X1 e X n Si e X n and Tị e nxn i 1 . N and general matrices X2 and X3such that LMIs below hold the memoryless state-feedback gain is given by K FX-1. Proof Now we consider substituting system matrices of 12 into LMIs conditions 13 the LMIs-based conditions of the memoryless state-feedback problem can be obtained directly as 14 . Remark When these time delays are constant that is d di di i 1 . N theorem 2 is reduced to the following condition - X1 . . AXl X 2 X3 X3 . X 2 X3 X1 . 0 -S1 AT 0 -S1 0 -S2 Ad2 0 0 -S2 0 0 -SNAdN 0 0 -SN X1 0 0 0 0 -S1 0 X1 0 0 0 0 0 . 0 -Sn 0 15 The condition above is delay-independent which is more restrictive than delay-dependent conditions 14 . Remark When the time-varying delay of discrete systems is assumed to be unique in state variables that is N 1 these results in theorem 2 could be reduced to those obtained in Boukas E. K. 2006 . Remark As to robust control problem of discrete time systems with poytopic-type uncertainties robust state feedback synthesis can be considered by these new formulations. When system state matrices in 11 are assumed as A A k Adi U k B Ằ k X dj k Aj Adj Bj j 1 d j k 0 X d j k 1 j 1 Robust state feedback .