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Laser Welding Part 13

Tham khảo tài liệu 'laser welding part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 234 Laser Welding PAm AmT -Q Q QT 0 28 Define the Lyapunov candidate as V eTPe ỔT-1 fl - flfl where fl is a positive constant and the desired value of fl V -fA e b fe x z ẾV-x n g e x Ki Pe e P. Ae bf e xd z fh-xd I m I d d d m g e xd V 29 -eT --- --- 2fl0 2 X- -2fld 2 2 i 2 I - 2- - 2flb eTPbia 2r C- I 2fle 2fle 2 -Ỉ- -2fld 2 -fl--Oaf ecfl-odfl-fl 2fle where V 2ỔT1 fl - flflịfl eTPb 2 f e xd z 1 e xj 2 1z2 fl-1 2r 2 Ke 1 - rafl c4 sup yr n c l g e xd I a-1 D tyf Accordingly V -eTQe -0cr fl- fl 2 M 30 M c 2 c2 c2 c.2 Ớ2 1 CTẾfl2 2fl0ỵ1 2 3 4 As a result the Lyapunov function V will decrease monotonically which means that e fl are bounded. The system is accordingly bounded asymptotically stable 12. Related to the Diode Laser Processing System Without u As shown in the identification our laser welding system can be represented by y t Ớ1 y ớ2 y t e3 eu eiU2 t 06u3 t flu4 t 31 Define the state function as x1 t y t x2 t y t z t u t z t V 32 Then the system can be represented by Laser welding techniques of real time sensing and control development 235 x1 t x2 t 33 x2 t 01x1 t 02x2 t 03 04z t 05z2 t 06z t 3 07z t 4 08V For simplicity we let the nonlinear function be f 01x1 t 02 x2 t 04 z t 05 z2 t 06 z t 07 z tỴ 34 Accordingly the amplitude limit function can be written as f 0 . .v. t 02 x2 t 04 z t 05 z 2 t 06z t 3 07 z t 4 35 f 0 4x4 4x4 zz2 z2 -77 where 0can be unknown Thus f 0 4 x12 4 x22 4z z 2 442 36 Equation 36 gives the boundary of the nonlinearity function. Let the tracking signal yr sin t the standard sinusoidal signal with amplitude 1 Then the error signal can be written by è1 t x1 t - yr t 37 è2 t x2 t - yr t or è1 t è2 t è2 t 01 08v 04z 05z2 06z3 07z4 -yr t 38 z t V or 0 0 1 0 1 è t f è yr z -yr t 05v 03 39 è 0 let A 0 1 and b 0 0 0_ _1_ 40 Then equation 9.3 can be rewritten by è t Aè b f è yr z - yr t 05V 03 41 With simpler substitution the error matrix is