K c < K u K c = K u K c > K u step 4 Calculate and implement PID parameters using the the ZieglerNichols tuning tables: 30


 Elwin Bryant
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1 1.5 QUANTITIVE PID TUNING METHODS Tuning PID parameters is not a trivial task in general. Various tuning methods have been proposed for dierent model descriptions and performance criteria CONTINUOUS CYCLING METHOD Frequently called ZieglerNichols method since it was rst proposed by Ziegler and Nichols (1942). Also referred to as loop tuning or the ultimate sensitivity method. Procedure: step1 Under Pcontrol, set K c at a low value. Be sure to choose the right (direct/reverse) mode. step 2 Increase K c slowly and monitor y(t) whether it shows oscillating response. If y(t) does not respond to K c change, apply a short period of small pulse input on r(t). step 3 Increase K c until y(t) shows continuous cycling. Let K u be K c at this condition. Also let T u be the period of oscillation under this condition. 29
2 K c < K u K c = K u K c > K u step 4 Calculate and implement PID parameters using the the ZieglerNichols tuning tables: 30
3 ZieglerNichols Controller Settings Controller K c T I T D P 0:5K u,,,, PI 0:45K u T u =1:2,, PID 0:6K u T u =2 T u =8 Remarks : We call { K u ultimate gain { T u ultimate period (! co = 2=T u critical frequency) ZieglerNichols tuning is based on the process characteristic at a single point where the closedloop under Pcontrol shows continuous cycling. At! co, jg c K c j!co = 1 y(t) is 180 o (phase lag) behind u(t): 31
4 K u is the largest K c for closedloop stability under Pcontrol. When K c = 0:5K u under Pcontrol, the closedloop approximately shows 1/4 (Quarter) decay ratio response. This roughly gives 50% overshoot. 32
5 An alternative way toemperically nd K u and T u is to use relay feedack control (sometimes called bangbang control). Under relay feedback, the following repsonse is obtained: K u = 4(u max, u min ) ; T u : from the repsonse A 33
6 Typical Responses of ZN Tuning Process: G(s) = 1 (s +1) 4 34
7 Since 50% overshoot is considered too oscillatory in chemical process control, the following modied ZieglerNichols settings have been proposed for PID contollers: Modied ZieglerNichols Settings K c T I T D Original(1/4 decay ratio) 0:6K u T u =2 T u =8 Some Overshoot 0:33K u T u =2 T u =3 No Overshoot 0:2T u T u T u =3 Process: G(s) = 1 (s +1) 4 35
8 Eects of Tuning Parameters: G(s) =1=(s +1) 4 36
9 37
10 1.5.2 REACTIONCURVEBASED METHOD Not all but many SISO(singleinput singleoutput) chemical processes show step responses which can be well approximated by that of the FirstOrder Plus Dead Time(FOPDT) process. The FOPDT process is represented by three parameters K p d steady state gain dened by y ss =u ss dead time (min), no response during this period 38
11 time constant, represents the speed of the process dynamics. G(s) = K pe,ds s+1 Procedure step 1 Wait until the process is settled at the desired set point. step 2 Switch the A/M toggle to the manual position and increase the CO (u(t)) by u ss stepwise. step 3 Record the output reponse and nd an approximate FOPDT model using one of the following methods: 39
12 Drawing a tangent is apt to include signicant error, especially when the measurement is noisy. To avoid this trouble, the following method is recommended: 1. Obtain K p = y ss =u ss. 40
13 2. Estimate the area A Let + d = A 0 =K p and estimate the area A Then = 2:782A 1 =K p and d = A 0 =K p, step 4 Once a FOPDT model is obtained, PID setting can be done based on a tuning rule in the next subsection. Popular tuning rules are Quaterdecay ratio setting and Integral error criterionbased setting FOPDTBASED TUNING RULES The following PID tuning rules are applicable for FOPDT processes with 0:1 < d= < 1. 1/4 Decay Ratio Settings ZN tuning for the FOPDT model. Controller K c T I T D P (=K p d),,,, PI 0:9(=K p d) 3:33d,, PID 1:2(=K p d) 2:0d 0:5d 41
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