tailieunhanh - Time Delay Systems Part 10

Tham khảo tài liệu 'time delay systems part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sliding Mode Control for a Class of Multiple Time-Delay Systems 169 When substituting the update laws 17 18 into the above Vn t further satisfies Vn t -kfS2 t - S t Ẽ x -d S t -kfS2 t - d- ẽ x S t -kfS t Since Vn t 0 and Vn t 0 we obtain the fact that Vn t Vn 0 which implies all S t 0 t and S t are bounded. In turn S t e L due to all bounded terms in the right-hand side of 16 . Moreover integrating both sides of the above inequality the error signal S t is L2 -gain stable as kf 0S2 T dT Vn 0 - Vn t Vn 0 where Vn 0 is bounded and Vn t is non-increasing and bounded. As a result combining the facts that S t S t e L and S t e L2 the error signal S t asymptotically converges to zero as t by Barbalat s lemma. Therefore according to Theorem 1 the state x t will is asymptotically sliding to the origin. The results will be similar when we replace another FNN 26 or NN 27 with the TS-FNN but the slight different to transient. 5. Simulation results In this section the proposed TS-FNN sliding mode controller is applied to two uncertain time-delay system. Example 1 Consider an uncertain time-delay system described by the dynamical equation 1 with x t x1 t x2 t x3 t -10 sin t 1 1 sin t A AA t 1 -8 - cos t 1 - cos t L 5 cos t 4 sin t 2 cos t 1 sin t 0 1 sin t Ad AAd t .0 1 cos t 1 cos t 3 sin t 4 cos t 2 sin t B 0 0 1 T g x 1and h x x x t-d sin t . It is easily checked that Assumptions 1 3 are satisfied for the above system. Moreover for Assumption 3 the uncertain matrices AA11 t AA12 t AAd11 t and AAd12 t are decomposed with 1 0 d D2 E11 E21 12 22 1 1 T C1 t sin t 0 0 - cos t C2 t sin t 0 0 cos t 170 Time-Delay Systems First let us design the asymptotic sliding surface according to Theorem 1. By choosing Ẽ and solving the LMI problem 8 we obtain a feasible solution as follows A P The error signal S is thus created from 6 . Next the TS-FNN 11 is constructed with ni 1 nR 8 and nv 4 . Since the T-S fuzzy