Tham khảo tài liệu 'recent advances in robust control theory and applications in robotics and electromechanics part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Passivity Based Control for Permanent-Magnet Synchronous Motors 379 Deqe W2 qm qm R q U - L q 31 TA tT T- T rs r Dmqm - W qm q Rmqm q - Lvqm 32 ịT 1T where q qe qm I is the observer state qe qm represents the estimated current and estimated velocity respectively qe qe - qe qm qm - qm are the estimated current error and estimated velocity error where Le LT 0 Lv 0 33 The model 31 32 can be written under the following form D q C qm q Rq MU ệ-Lq 34 Where q qT qm J and L diag Le L v From the equation 12 and 34 we deduce the observer error dynamic __ _ _ . - __ D q C qm q R L q 03xi 35 In order to prove the asymptotic stability of the observer estimated error we choose the following desired energy error function 1 H q 1 qTD q 36 Taking the time derivative of Ho along the trajectory 35 we get H q -qT R L q 37 Since L IĨ 0 q 0 is asymptotically stable. Following the same procedure used in section we conclude that q f mo q 0 e 1 V t. 38 where mo Âmax D Anin D 0 - dT 0 We conclude that the observer 34 reconstructs asymptotically the current and velocity signals. Remark We can notice that the gain matrix L has the same effect than that of matrix K1 in 25 i. e L is the damping that is injected in the observer system to ensure the asymptotic stability of the observation error. Combined Controller-Observer Design The desired dynamics when only rotor position is measurable are Dq W2 qm qm Req U 39 380 Recent Advances in Robust Control - Theory and Applications in Robotics and Electromechanics Dmfim - W2 qm qe Rmi - - km em 40 Where km 0. We have the following result The controller law becomes U De q W2 qm qm Reqe-K2ee 41 In order to establish the stability of the closed loop system with presence of the observer we consider equation of state error 35 . We get from 25 16 40 and 41 De G qm e N qm q 0 42 Where G . _ Re K2 02x1 3 qm -WT qm Rm km2 N qm Le2 01x2 -W2 qm 3 Rm lm2 Proposition 2 Consider the PMSM model 1 - 3 in closed