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Time Delay Systems Part 9

Tham khảo tài liệu 'time delay systems part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Resilient Adaptive Control of Uncertain Time-Delay Systems 149 Now let r5 be a positive scalar then using Fact 1 we have 0 0 2x t PAd J ụ t s Bo 1 x t s ds 2x 1 t PAd J Bo Iz t s ds T r 1 xT t PAdAdPx t r5 zT t s ITBoTBoIz t s ds. 35 Also if r- is a positive scalar then using Fact 1 we have r _ 2xT t PAd J E x t s ds T r- 1 xT t PAdAjPx t r- Ị E2 x t s E x t s ds. 36 It is known that 2ụ t xT t PBoIx t 2 PBoI ụ t x t 2. 37 Also using Assumption 2.1 it can be shown that 2xT t PE x t 2 P e x t 2. 38 Using equations 31 - 38 and equations 17 - 24 with the fact that 0 T T in 30 we have Va x xT t Sx t T r4 xT t AKT t Bor Bo AK t x t T r5zT t ITBg BoIz t T r-ET x t E x t 2p PBo x t 2 2 PBoI ụ t x t 2 2e P x t 2 2 ụ t ụ t . 39 where s PAod A dP PBoK Kt BoT P T r1 a Ao T r2 a Ad T r3 BoKKT bJ 1 1 1 1 1 1 T T r1 r2 r3 r4 r5 r6 PAdAd1 . 40 To guarantee that xT t sx t 0 it sufficient to show that s 0. Let us introduce the linearizing terms X P 1 y KX and Z XBoK. Also let 1 r 1 2 r 1 3 r 1 4 r 1 5 r 1 and 6 r 1. Now by pre-multiplying and post-multiplying s by X and invoking the Schur complement we arrive at the LMI 25 which guarantees that s 0 and consequently xT t sx t 0. Now we need to show that the remaining terms of 39 are negative definite. Using the definition of z t ụ t x t we know that T r5zT t ITBoTBoIz t T r5 ITbJBoI ụ2 t x t 2. 41 Also using Assumptions 2.1 and 2.2 we have T r-ET x t E x t T r- e 2 x t 2 42 and T r4xT t AKT t B2BoAK t x t T r4 p 2 bJBo x t 2. 43 150 Time-Delay Systems Now using 41 - 43 the adaptive law 26 and the fact that I ụ t I 1 equation 39 becomes IZa x XJ t sx t r r4 p 2 pj Bo11 x t 2 T r5 PtbJ BoIW ụ2 t x t 2 r r6 9 2 x t 2 2p PBo x t 2 2 PBoI wy x t 2 2Ỡ P x t 2 2X1 ụự x t 2 2X2 ụ2 t x t 2. 44 It can be easily shown that by selecting X1 and 2 as in 27 and 28 we guarantee that Va x xT t sx t 45 where s 0. Hence Va x 0 which guarantees asymptotic stabilization of the closed-loop system. I 3.2 Adaptive control when 9 is known and p is unknown .