1、1Process ControlCollege of AutomatonChongqing University重慶大學自動化學院2Design &Tuning of Single Loop SystemsOutlinenTuning controller Based on PRC基于脈沖比控制的調優(yōu)控制器pCohen-Coon Relation科恩 - 庫恩的關系Ziegler-Nichols Relation,齊格勒 - 尼科爾斯關系、pITAE relation, pCiancone-Marlin relationnDirect Synthesis直接合成nInternal Mo

2、del Control內?？刂?Design &Tuning of Single Loop SystemsPID Tuning協(xié)調 trial實驗 P(t)4Design &Tuning of Single Loop SystemsTuning Controllers with Empirical RelationsnA practice of designing the controller settings設置pMaking an open (or closed) loop experimentpFitting the system response to a model擬

3、合模型的系統(tǒng)響應nCalculating計算 the controller gains pusing the fitting curve擬合曲線 (empirical model經驗模型)pFollowing some empirical relationsnEmpirical tuning gives a quick starting pointpThough it doesnt dictate system dynamic specifications盡管它不能決定系統(tǒng)動態(tài)參數pIt is seldom optimal最佳的, involving包括 field tuning現(xiàn)場調試1.

4、Cohen-Coon Relation2. Ziegler-Nichols Relation, 3. ITAE relation, 4. Ciancone-Marlin relation5Design &Tuning of Single Loop SystemsTwo Remarks評論 on Empirical Tuning經驗調整nTuning relations調整關系 are based onpFitting open loop data to the 1st order with dead time model擬合一階開環(huán)數據的延遲時間模型 Systems is suppos

5、ed to被期望 be self-regulating自動調節(jié)的 Dead time can approximate multi-stage process, or measuring lag延遲時間可以近似的多階段過程，或測量滯后pSome tuning relations, e.g., Cohn and Coon being developed（ in term of在方面） decay ratio衰退率 in handling disturbance being not suitable for適合 servo problems (why?)6Design& Tuning of

6、Single Loop SystemsController settings Based on PRCnObtaining獲取 process reaction curve過程響應曲線 (PRC)pDisabling the controllerp禁用控制器pIntroducing a step change to the actuator執(zhí)行機構引入階躍變化pMeasuring the open loop step response, resulting in導致 PRC7Design& Tuning of Single Loop SystemsController settings

7、 Based on PRCnIn this open loop step test開環(huán)階躍測試pThe lumped總的 response can be represented表現(xiàn) bynUsing GPRC forpDesign of the required controller if the process models are not known in advance預先pA reduced order model降階模型 for the product乘積 of Ga, Gp, and Gm8Design& Tuning of Single Loop SystemsContr

8、oller settings Based on PRCnExample: A self-regulating自調整 capacity processpStep test is shown in the figure below階躍測試結果如下表Note that注意 we assume that the process is self-regulating9Design& Tuning of Single Loop SystemsController settings Based on PRCnExample: A self-regulating multi-capacity proc

9、ess Step test is shown in the figure below 一個自我調節(jié)的多容量的過程測試步驟如下圖所示Open loop test data1st order with dead timeTime constant decided by the initial slope and a visual estimation of dead time td 時間常數由最初的斜坡和目測的延遲時間TD決定sigmoidal formS形10Design& Tuning of Single Loop SystemsController settings Based on

10、 PRCnExample (cont.): Why does FOPDT純滯后過程work?pThe real time data usually takes the sigmoidal form S形pProcess unit operations are generally open-loop stable過程單元操作通常都是開環(huán)穩(wěn)定, thus, the approximation works in realitypReminder提示: there is no underdamped response due to the taking out of拿出 the controller

11、in the step testpThe empirical relations such as Z-N, C-C, C-M, and IAE are usually be used in this case這種情況下11Design& Tuning of Single Loop SystemsController settings Based on PRCnComments on評論 the minimum error最小誤差 integral criteria 積分標準pCalculating計算 the error by E = R-C, where C is based GPR

12、CpMinimizing極小化的 the error over time超時 (e.g., by Lagrange Multiplier)pThe forms of integral積分的 errorWhy so many forms?12Design& Tuning of Single Loop SystemsController settings Based on PRCnComments on the minimum error integral 積分過程(cont.)pIntegral of the absolute error絕對誤差pIntegral time-weight

13、ed加權的 absolute errorIAE puts the equal weight重要性 to large and small error1. It puts a heavy penalty on errors persisting for long periods of time 很大的誤差將持續(xù)很長一段時間1. Resulting controller allows for得到 low settling times建立時間13Design& Tuning of Single Loop SystemsController settings Based on PRCnComme

14、nts on the minimum error integral (cont.)pIntegral of the square error平方誤差Squaring e makes it even smaller when e 2 35Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnFurther consideration to（進一步考慮） the desired responseWhen the dead time is inevitably不可避免的 in the process, the

15、 desired response can be defined as:Synthesis:前面的方程36Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnExample 3: Derive the controller function for a system with a first order process（推導一個一階系統(tǒng)的控制器函數） with dead time , andthe desired system response Solution: Assuming =tdThis i

16、s a PI controllerGuideline to choosing c:1. c 1.7 for PI Controller, 2. c 0.25 for PID Controller37Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnSupposing the desired system response iswith the system natural period , and the damping ratio阻尼比 Synthesis:When intentionally s

17、implifying the process當故意簡化了過程 Gp, we can obtain獲得 Gc as a PID controller38Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnExample 4: Derive the controller function for a system with a second order overdamped process推導一個二階過阻尼過程的傳遞函數 , and an underdamped system response Solut

18、ion:39Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnExample 4 (cont.):Simplifying簡化:Suppose that 2 is related to the slower pole, i.e., 21, Set f=2 in order to the effect of pole-zero cancellation 為了實現(xiàn)零極點對消f=2the PI controllerThe final form:The only tunable parameter可調的參數

19、is the damping ratio 40Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnPole-zero (p-z) cancellation (cancellation compensation)1. Real life problems can not have a perfect p-z cancellation2. The analysis below gives a theoretical理論的 understanding of p-zWhat is pole-zero canc

20、ellation? Canceling the (undesirable open-loop) poles of our process Replacing them with a desirable closed-loop pole用我們想要的閉環(huán)極點取代開環(huán)極點41Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnPole-zero (p-z) cancellation (cancellation compensation)How pole-zero cancellation occurs?Gc

21、 is sort of是一種 reciprocal倒數 of Gp, i.e., , thusthe zeros of Gp is the poles of Gc. Therefore, we haveThe closed-loop poles only depend on c due to 1+GcGp=0 42Design& Tuning of Single Loop SystemsDirect Synthesis of Controllers控制器的融合nPole-zero (p-z) cancellation (cancellation compensation)Behavio

22、rs作用 of the closed-loop system with p-z cancellationSupposing that a system (Ga=Gm=1) uses PI controller and Gp is the first order function,If choosing I=p, we have System characteristic equation:43Design& Tuning of Single Loop SystemsDirect Synthesis of ControllersnPole-zero (p-z) cancellation

23、(cancellation compensation)Behaviors of the closed-loop system with p-z cancellationRecall example 1, we have Kc=p/(Kpc) and I=pChecking GcGp with s=-1/ c and Kc=p/(Kpc)c1sGiving also us the consistent一致的 results 44Design& Tuning of Single Loop SystemsInternal Model Control內?？刂?(IMC)nIMC conside

24、rs the following以下的 factors:pThe model is only an approximation of the actual真實的 processpMeasurements測量值 of the process parameters is not accurateGc consists of由組成 p*cG and GIMC （Scheme概形） 45Design& Tuning of Single Loop SystemsInternal Model Control (IMC)nIMC with PID controllers:The error is T

25、he closed-loop functions based on IMC scheme基于內?？刂频拈]環(huán)傳遞函數The model controller outputRearranging46Design& Tuning of Single Loop SystemsInternal Model Control (IMC)nIMC with PID controllers:The relationship between Gc and IMC functionsThis will be used to retrieve檢索 the PID gainsThe system output:

26、The closed-loop equation47Design& Tuning of Single Loop SystemsInternal Model Control (IMC)nHow to choose *cGSet the approximate function to avoid the above situation: Recall the basic fact from the direct synthesis直接合成: The poles of Gc are “inherited繼承的” from the zeros of Gp, i.e., If Gp has po

27、sitive zeros, the Gc function with positive poles. where contains包含 all positive zeros, and The controller is designed on the basis of 基于 only pG-pG48Design& Tuning of Single Loop SystemsInternal Model Control (IMC)nDefining the controllerObservations:1. The only tunable可調的 parameter is closed-l

28、oop time constant c 2. An integer power整數的大小 of r is used to ensure that the controller is physically Realizable物理可實現(xiàn)3. The example below violate this intention下面的例子違反這個意圖just for obtain獲得 results that resemble類似 an ideal理想的 PID controller.49Design& Tuning of Single Loop SystemsInternal Model Co

29、ntrol (IMC)nExample 5: Derive a controller function for a system with a first order process with dead time using IMC獲得一個一階系統(tǒng)控制器函數與延遲時間使用內?？刂芐olution:The experiments has experimental errors實驗誤差 or uncertainties不確定性 The measured model function is written as50Design& Tuning of Single Loop SystemsIn

30、ternal Model Control (IMC)nExample 5 (cont.):Using Pad approximation for the dead time functionPadwhere The controller by IMCr = 151Design& Tuning of Single Loop SystemsInternal Model Control (IMC)nExample 5 (cont.):Rearranging the controller function, we have These are the tuning parameters of

31、the ideal PID controller52Design& Tuning of Single Loop SystemsInternal Model Control (IMC)nExample 6 What is the PID controller settings for the dye mixing problem if using IMC-based tuning relations?如果利用內?？刂普{整關系，pid控制器設置什么關于燃料混合問題Solution:Using the same dead time in the dye mixing problem, Observations:The integral time constant is relatively相當的 high, thus less integrating acti