Discrete Time Systems Part 8

Tham khảo tài liệu 'discrete time systems part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Quadratic D Stabilizable Satisfactory Fault-tolerant Control with Constraints of Consistent Indices for Satellite Attitude Control Systems 199 J xTPX0 Y2p2 Zmax UTPU Y2p2 Theorem 2 Consider the system 1 and the cost function 7 for the given index q r and H norm-bound index Y if there exists symmetric positive matrix X matrix Y and scalars si 0 i 4 9 such that the following linear matrix inequality -X 0 AX BY T S21 -Y2I dt 0 0 . 1. X a cz w Sd w. 0 S22 _ 13 holds where S21 ex EY T X X YT YT YT YT YT YTJ s22 diag -I 7EJET -Q 1 -S4I -e5I e8J - 6J-1 - 7J-1 - 8J-1 9I - R-1 - 9I . Then for all admissible uncertainties and possible faults M the faulty closed-loop system 6 with satisfactory fault-tolerant controller u k Kx k M-1YX-1x k is asymptotically stable with an H norm-bound Y and the corresponding closed-loop cost function 7 is with J Zmax UTX-1U Y-p- . According to Theorem 1 and 2 the consistency of the quadratic D stabilizability constraint H performance and cost function indices for fault-tolerant control is deduced as the following optimization problem. Theorem 3 Given quadratic D stabilizability index o q r suppose the system 1 is robust fault-tolerant state feedback assignable for actuator faults case then LMIs 10 13 have a feasible solution. Thus the following minimization problem is meaningful. min y X Y Y i . LMIs 10 13 14 Proof Based on Theorem 1 if the system 1 is robust fault-tolerant state feedback assignable for actuator faults case then inequality aTPAc - P 0 has a feasible solution P K . And existing z 0 5 0 the following inequality holds z atpAc - p CctCc Q KtMRMK dI 0 15 Then existing a scalar Y0 when Y Y0 it can be obtained that at PjD y2i - dt p1d -1 dt p1 Ac ỐI where P1 ZP . Furthermore it follows that atP1Ac - P1 CctCc Q KTmrmk atp1d y2i -dtp1d -1 dtp1Ac 0 Using Schur complement and Theorem 2 it is easy to show that the above inequality is equivalent to linear matrix inequality 13 namely P1 K Y is a feasible solution of LMIs 200 Discrete .