Discrete Time Systems Part 7

Tham khảo tài liệu 'discrete time systems part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Discrete Time Mixed LQR H Control Problems 169 -Ỵ-1 A X B U B X A AtU3B2U-1b2u3a T A2 X B U U-1ĨỈ1U-1ỈỈ1TU-1 U IXL u-BTu- U- B X A AtxJ B2X b IẠBXA We have AtXAk - Xx Z-2aTW BXAk cC Q ktrk AXA - X - AtxJ BtxJ IẠBX A C-C1 Q K U-1Bj U3 A t u2 k u-1b22 U3 A Also noting that the discrete time Riccati equation 13 and substituting the above equality for 15 we get X TO 1 I -- rp -- II II o II II ĩ Ỉ- I l -1 T I J 2xT k nx k Y -XV x k - z Y2 w - Y2 U 2 w - y 2U1-1bKx Akx k 0 k 0 I u22 K U-1bTu3A x 2 16 Based on the above it is clear that if K -U21b22U3 a then we get mll J xTXxx0 K -1lzll2 Y21wil2 - Y2 w lw - Y-U ÉKXAkx i 17 1 . y 1 ---9 1 -r 1 -rA r -r T A y 1 1 y 1 . L . 1 By letting w k Y 2U1 1BKXxAKx k for all k 0 we get that x k AKx0 with AK which belongs to L2 0 x since AK A - B Bt XxB R 1B2 XXA is stable. Also we have w k 2 Y-4xTXwx0 Iz k 2 xTXzx0 Then it follows from 17 that sup inf J xT X Y-2Xw - Xz x0 weÍ2 K Thus we conclude that there exists an admissible state feedback controller such that sup inf J xTX r-2Xw - Xz 0 subject to hwllX Y . weL1 K 4. Non-fragile controller In this section we will consider the non-fragile discrete-time state feedback mixed LQR HX control problem with controller uncertainty. This problem is defined as follows Consider the system 2 4 satisfying Assumption 1-3 with w e L2 0 x and x 0 x0 for a given number Y 0 and any admissible controller uncertainty determine an admissible non-fragile controller FX such that 170 New T rends in T echnologies sup  subject to lỊĩ z Y . weL2 where the controller uncertainty AF k considered here is assumed to be of the following structure AF k HKF k EK where HK and EK are known matrices of appropriate dimensions. F k is an uncertain matrix satisfying FT k F k I with the elements of F k being Lebesgue measurable. If this controller exists it is said to be a non-fragile discrete time state feedback mixed LQR H controller. In order to solve the problem defined in the above we first connect the its .