tailieunhanh - PID Control Implementation and Tuning Part 11

Tham khảo tài liệu 'pid control implementation and tuning part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Adaptive PID Control for Asymptotic Tracking Problem of MIMO Systems 193 where Fp1 Fd1 Fp2 Fd2 Rmm are arbitrary positive definite matrices YI is positive scalar. Tr - denotes trace of a square matrix. Here put V t V Ị t y t tyI t Yp1 t Yp2 t YD1 t Yd2 t for simplicity. The derivative of 37 along the trajectories of the error system 30 32d can be calculated as V t 2 t Pĩ 2 t 2ứi t T Y 1 CAB 1ibi t V t 2 y t _0 py_ _y t 2tyI t YI CAB tyI t Tn 1 1 Tr 1 1 2Tr _ p 1 t r p1 CAB P1 t 2Tr _ D1 t rD1 CAB D1 t _ T 1 1 T 1 1 2Tr rp2 t Ip2 CAB Tp2 t 2Tr TD2 t i D2 cab rD2 t t T p Aị BK Aị BK T Pị Pị Q1 QT Py Ĩ t .n t . _ PịQ1 QT Pn T Pn Ay AT Py .n t . 2tyI t T bT PịĨ t Y CAB -1 ty I t 2Tr YP1 t T bTPịỉ ì Ĩ1 t T r CAB -11XP1 t 2Tr D1 t T bTpẽỉ ì Ĩ2 t T FD1 CAB -1ixD1 t 2Tr yK t T bTpg t yM t r CAB 1Ỳp2 t 2Tr D2 t r bTpg t ýM t rD1 CAB 1ỶD2 t a s t . T p Aị BK Aị BK T Pị Pị Q1 QT Py Ĩ t .n t . _ PịQ1 QTPy T Py Ay at Py .y t . 2tyI t T bT Pịỉ t Y 1 KI0 Ĩ1 t Kp0 Ĩ2 t 2Tr YP1 t T bTPịỉ t ĩ1 t T r Kp1 t 2Tr D1 t T bTPịĨ ì Ĩ2 t T Fd1Kd1 t 2Tr Yp2 t T bTp ĩ t yM t T FpJKp2 t 2Tr YD2 tV BTpg t ýM tỹ rD1 D2 t . 38 Therefore from ĩ t T Ỉ2 T ey iy T and B p H1 H2 giving the constant gain matrices Ki0 Kp0 as 18 20 and the adaptive tuning laws of Kpi t K-Di t i 1 2 as 19a 19d 20 we can get 38 be V t a s t . T p Aị BK Aị BK T Pị Pị Q1 QT Py Ĩ t .y t . _ PịQ1 QTPy T Py Ay ATPy .y t . 39 Here the symmetric matrix of 39 can be expressed as 2e I2m p Q1 QỈ py Pị Q1 Q2 Py T In 2m 40 194 PID Control Implementation and Tuning from 36 34 . Using Schur complement we have the following necessary and sufficient conditions such that 40 is negative definite 2eỈ2m 0 41 - In 2m PQ1 QTPn T1PQ1 QTPn 0 42 2e where _ P ppj Q23 H2 Q23 43 from 27 35 . Obviously the first inequality 41 is hold. The second inequality 42 is also T achieved under large e 0 because Q2Pn and p Q1 are independent of e . At this time 40 becomes negative definite matrix and 39 is V t Ĩ t T r _ _ T -1 2eI2m P Q1 Q2 Pn n t _ PỊQ1 Q2 Pn T