Tham khảo tài liệu 'adaptive control 2011 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 168 Adaptive Control Johnson 1991 A I Dfni 1 4 1 _ A 1 C-1 _ _ nA 1RV 1 DA -1 A dCU A A d C DA B DA the proposed parameter estimation algorithm in Section 10 - 12 can be equivalently rewritten as õk p. 1 r PkPk zk TP . 1 P 1 P 1 r lym Cf . Pk Pk 1 rv ỉkVkYk 16 17 Suppose that Pk is initialized by p0I where p0 is a positive real value large enough and define rk tr Pk . The relation between rk and IP-1 can be established in the following lemma. Lemma . Thefollowing relation holds lnEIP 1 I O lnErk . Proof Using the formulae n n tr X A X and X nA X i 1 1 1 where n is the dimension of X we have EIP 1 Erk n. This completes the proof. The next lemma shows the convergence of two infinite series that will be useful later. Lemma . The following inequalities hold t M1E PỈP-P- lnE Pk 1 1 n0 ln p i 1 18 19 -1 lnE P-1 X . 20 where c 1. Proof The proof can be done along the similar way as Lemma 2 in Ding Chen 2004b and is omitted here. The following is the well-known martingale convergence theorem that lays the foundation for the convergence analysis of the proposed algorithms. Adaptive Control for Systems with Randomly Missing Measurements in a Network Environment 169 Theorem . Goodwin Sin 1984 Let Xk be a sequence of nonnegative random variables adapted to an increasing Ơ -algebras F k . If E Xk 1 Fk 1 4 Xk-ak Pk . where ak 0 Pk 0 EXo 1 Xl ữ1 1 1 and X 1_0 P- 1 almost surely . then Xk converges . to a finite random variable and N lim ai 1 . N-á Convergence analysis To carry out the convergence analysis of the proposed algorithms it is essential to appropriately construct a martingale process satisfying the conditions of Theorem . Main results on the convergence properties of the proposed algorithm are summarized in the following Theorem. Theorem . For the system considered in 3 assume that A1 vk F k is a martingale difference sequence satisfying E vk Fk-1 0 . 21 E vk2 Fk-1 r 1 . 22 11. A2 is strictly positive real A3 Bz is stable
