tailieunhanh - Essentials of Process Control phần 10

Tham khảo tài liệu 'essentials of process control phần 10', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | CHAPTER 15 Stability Analysis of Sampled-Data Systems 525 On the Cr contour z Rea where R co and Ớ goes from 7T through 0 to -7T. Substituting into Eq. gives MmiA --------------Re _ b As R 0. Therefore the infinite circle in the z plane maps into the origin in the plane. The contour is just the reverse of the contour going from the origin out along the negative real axis. The contour is just the reflection of the contour over the real axis. So just as in a continuous system we only have to plot the C contour. If it goes around the - point this sampled-data system is closedloop unstable since P 0. The farther the curve is from the - point the more stable the system. We can use exactly the same frequency-domain specifications we used for continuous systems phase margin gain margin and maximum closedloop log modulus. The last is obtained by plotting the function 1 HG For this process with To Kp 1 with a proportional sampled-data controller and a sampling period of minutes the controller gain that gives a phase margin of 45 is Kc . The controller gain that gives a 2-dB maximum closedloop log modulus is Kc . The ultimate gain is Ku . EX A M PL E . If a deadtime of one sampling period is added to the process considered in the previous example Kc 2 b We substitute eia Ts for z in Eq. and let d go from 0 to Íiự2. At t 0 where 1 the Nyquist plot starts at on the positive real axis. At where z the curve ends at - KcK 1 - G 0 5 38 tiGM ill U ia - _ b -------fỳ b------- This is on the positive real axis. Figure shows the complete curve in the plane. At some frequency the curve crosses the negative real axis. This occurs when the real part of HGiẠ ji ư D ía is equal to for the numerical case considered in the previous example. Thus the ultimate gain is Note that this is smaller than the ultimate gain for the process with no deadtime. The controller gain that gives a maximum closedloop log modulus is . The final .