Chaotic Systems Part 8

Tham khảo tài liệu 'chaotic 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 | 164 Chaotic Systems Now define Xj k d pT . G Uj k d-1 F - Uj k G1 Gd n j k xj k d Oj k F T . G Uj k d-1 . GTjd - Uj k T 10 11 12 j F T G T T T 13 Using the above definitions Eq. 11 can be written as n j k . k J 14 j fc denotes the estimate of j and it is defined as j k F k T Gi k T . Gd k T T 15 The error vector can be written as s j k n j k -O T k j k -1 16 To obtain the estimated parameters j fc the least squares technique is used. Consider the following objective functions Jk Ek i n7 n - T n j k 2 17 T- 1 cc . . T- -1 r 1 . . zA 1 . . . 1 By differentiating Eq. 17 with respect to and set it to zero we have r oT 1 r nJ 1 J k Oj 1 - - Oj k o 1 - - Oj k - 18 k oT k nj k Let r rT nA 1 oT 1 . Ạ lOjt1 k Oj k . oT k 7 19 Adaptive Control of Chaos 165 Substituting Eq. 19 into Eq. 18 yields J k Pj k Ek i J n n J n Pj k fck-1o J n n J n o J k n J k Pj k P k -1 J k -1 Pj k O J k n J k Using Eq. 19 one can write P-1 k -1 P-I k -o J k oT k Substituting Eq. 21 into Eq. 20 results in J k J k -1 Pj k o J k J k After some matrix manipulations Eqs. 21 and 22 can be rewritten as p k-1 k T k p k-1 p k p k -1 - J p_r J J 1 o k Pj k -1 Oj k A Ar . p k-1 o k Si k 9. k . k -1 p prj p J J 1 OT k P k -1 o k 20 21 22 23 24 The above recursive equations can be solved using a positive definite initial matrix for Pj 0 and an arbitrary initial vector for j 0 . The identification method given by Eqs. 23 and 24 implies that OT k J k -1 oT k J ask 25 To show the property 25 note Eq. 19 implies that P fc is a positive definite matrix. Define Sj k J k - J 26 Consider the following Lyapunov function Vj k Sj k Pj-1 k d7. k 27 Using Eq. 24 we have Sj k Pj k Pj-1 k - 1 Sj k -1 28 AVj fc can be obtained as follows 166 Chaotic Systems Vj k - Vj k -1 sTj k Pj-1 k dj k -sT k - 1 Pj-1 k - 1 Ổj k -1 sT k Pj-1 k - i k -1 -sTj k - 1 p k -1 d j k -1 T k - T k - 1 Pj1 k -1 j k -1 - j 29 - n j k - T k j k -1 2 _ej k 1 T k Pj k - 1 J k 1 T k Pg k -1 g k Equation 29 shows that Vj fc is a decreasing .