Tham khảo tài liệu 'pid control implementation and tuning part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Multivariable PID control of an Activated Sludge Wastewater Treatment Process 13 dynamics at zero frequency this method is expected to provide good decoupling characteristics at low frequencies. Penttinen - Koivo method The Penttinen- Koivo is slightly more advanced than the Davison method. A proportional term has been added to the control law giving u s Kce s Kị1 e s 19 s where Kc p CBp 1 and K sG 1 0 . The Davison and Penttinen are similar in the sense that the integral gains of both controllers are linearly related to the inverse of the plant dynamics at zero frequency and both controllers are therefore expected to provide good control-loop decoupling characteristics at low frequencies. Unlike the Davison the Penttinen controller also includes proportional control action where the feedback gain is linearly related to the inverse of the plant dynamics at high frequencies. Therefore by following the same line of reasoning as above the latter controller is expected to exhibit good decoupling characteristics at high frequencies. The term CBp represents the initial slope of the step output response . CBp - - ym 1 - yl m V ym m 20 where m is the system order and ỳị j is the initial slope of output i in response to a step at input j. It can be shown that CGp is the inverse of the plant dynamics at high frequencies by writing the Laurent series expansion of the transfer function G s as follows G s -- . CG cFGGỉL - 21 s s2 s3 A good approximation of G s at high frequencies is G s CBpỊs is given by 21 . As Kị s terms are also negligible at high frequencies compared to Kc so it can be concluded that G s Kc I s thus giving the following closed-loop transfer function Hi s --- 0 H s I GK - GK for large s 22 0 The tuning parameters p and s can be used to tune the proportional and integral gains. Maciejowski method M3 extends M2 to non-zero frequencies and hence the controller gains are linearly related to the inverse of the plant dynamics at a particular .
