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Advances in PID Control Part 5

Tham khảo tài liệu 'advances in pid control part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 70 Advances in PID Control Kp2 KpịKp ãKD - Kp K 3 1 Kp KiãKD 2 K Kp Kj 1 The last two conditions imply that Kp 1 ã Kị and Kp Kị Kị which are satisfied by conditions 6 . And the first condition implies to solve the equation Kp2 Kp Ki ã ỳ KKKiệ t 1 for Kp. Similar to the way in which conditions for positive value of the entries of matrix M Kp Kd Kị it follows that KKaKK J K KD 4 K 1 1 Kp ---------------------- ------------- p 2 That is conservatively satisfied by the condition Kp KD Kị given at Theorem 1 equation 6 . On the other hand for Kp to be real it is necessary that KD 3Kị 2yj2K2 sKa 1 and for Kd to be real it is required that Kị 1 all these conditions are clearly satisfied by those stated at Theorem 1 equations 6 . Therefore if the conditions given by 6 are satisfied the Lyapunov function results on a sum of quadratic terms V e m1 2 e1 e2 2 e2 e3 2 Ẳyỷ k2e22 k3e32 9 for positive parameters k2 k2 k3 thus concluding that V e 0 for e DO and V e o for e 0. Since the definition of the matrix entries 8 allows cancellation of all cross error terms on the time derivative of the Lyapunov function 7 then along the position error solutions it follows that V e eTMe am 2 Kpe22 K2e32 KpKD Kị Kd ầ2ỳ KD2 2 e2 K 10 To ensure that v e 0 it is required that KpKD Kị Kị ă 2ặ Kịỷ 0 which implies that 2K2aKDKf Kd2 F KD which is satisfied by the condition Kp KD Kị given at Theorem 1 equations 6 . Nonetheless to guaranteed that Kp is real it follows that 2Kị ă KpKf Kpf 0 that implies when considering equal to zero that the solutions are tựíự ãS Kn-----z ---- D 2 Thus for Kd to be real it is required that Kị 8 and finally the condition on KD results on A PI2D Feedback Control Type for Second Order Systems 71 Ki JkXk 8 D 2 Such that the above conditions are satisfied by considering those of Theorem 1 equation 6 . Therefore by satisfying conditions 6 it can be guaranteed that all coefficients of the derivative of the Lyapunov function v e are positive such that V e 0 for e DO and V e 0