OptimalPIandPIDParametersfora Batch of Benchmark Process Models Representative for the Process Industry

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1 OptimalPIandPIDParametersfora Batch of Benchmark Process Models Representative for the Process Industry Olof Garpinger August 27, 205 Abstract ThisreportpresentsalargenumberofoptimalPIandPIDparameters for a batch of 34 benchmark process models representative for the process industry. This information can e.g. be used as a reference for other controller design methods. The optimal parametershavebeenderivedusingthematlab -basedsoftwarecalled SWORD, which determines the PI or PID controller that gives the best load disturbance performance given some closed-loop robustness. The performance measure used here is the integrated absolute error during a step disturbance on the process input, and the robustness constraints are given by upper limits on maximum sensitivity and complementary sensitivity. It has previously been shown that SWORD can be used to find near-optimal controllers also with respect to an additional noise sensitivity constraint. A controller design methodbasedonthisfindinggivesasetofpiandpidcontrollers all with the same robustness, but instead with a trade-off between performance and noise sensitivity. Five such sets of controllers are presented here for process models with varying dynamics. There are alsoalargenumberofoptimal,unfiltered,swordpiandpidparameters presented, and related to four different robustness values. Alldatahasbeencollectedintableswhicharegivenattheendofthis report.thesametableshavebeencollectedinmatlab datafiles, Excel spreadsheets, as well as in a tex-file.. Introduction ThepurposeofthisdocumentistoprovidealargenumberofIAEoptimal PI and PID controllers for a batch of benchmark process models, which can be used for comparison with other design methods. Both for the unfiltered case(with regards to four different robustness levels) and for the filtered, noise sensitivity constrained, case. The first few sections are used to define the optimization problem used to determine the controllers, and to present the batch of benchmark models. 77 tables filled with optimal PID parametersandrelateddataaregiveninsections8to0.thetableshavealso

2 r Σ d n e u z y C(s) Σ P(s) Σ F(s) Figure Aloaddisturbance,d,measurementnoise,n,andset-point,r,acton the closed-loop system with process P(s), controller C(s) and measurement filter F(s). beencollectedinmatlab mat-files,excelspreadsheets(differentsheets fordifferentprocesstypes),aswellasinatex-file.thesecanbedownloaded from: 2. The closed-loop system The single-input single-output feedback loop in Fig. will be used here to setupaconstrainedoptimizationproblemforthedesignofpiandpid controllers. The process, P(s), is manipulated by a controller, C(s), such thatthecontrolledvariable,z,iskeptascloseaspossibletoaset-point,r, inordertominimizethecontrolerror,e.theprocessisaffectedbyaload disturbance, d, at the process input. The measurements of the controlled variable, y, typically contain noise, n, and are fed through a low-pass filter, F(s),tokeepthenoiselevelofthecontrolsignal,u,low. Assuming regulatory control around a constant set-point, r = 0, the closed-loop system can be described by three equations Z(s)= Y(s)= P(s) P(s)C(s)F(s) D(s) N(s), () +P(s)C(s)F(s) +P(s)C(s)F(s) P(s) +P(s)C(s)F(s) D(s)+ N(s), (2) +P(s)C(s)F(s) U(s)= P(s)C(s)F(s) +P(s)C(s)F(s) D(s) C(s)F(s) N(s), (3) +P(s)C(s)F(s) with frequency-domain signals in capital letters. 2

3 It is well-known that the four transfer functions S(s)= +P(s)C(s)F(s), (4) T(s)= P(s)C(s)F(s) +P(s)C(s)F(s), (5) S p (s)= S c (s)= P(s) +P(s)C(s)F(s), (6) C(s)F(s) +P(s)C(s)F(s), (7) are sufficient to describe the closed-loop system in Fig.. S(s) is called the sensitivity function and T(s) the complementary sensitivity function. 3. Controllers and noise filters In this document, optimal PI and PID parameters will be presented, sometimes together with a corresponding noise filter time constant. There are several PID and filter forms commonly used, and it is therefore important todefinetheonesusedhere.thepicontrollerwillbegivenbythetransfer function C PI (s)=k ( + ), (8) st i where K istheproportionalgainand T i istheintegraltime.forpid controltheparallelformhasbeenchosensinceitismoregeneralthanthe series form. It has the transfer function C PID (s)=k ( + ) +st d. (9) st i Thederivativepartcomeswiththederivativetimeparameter,T d.lowpass filters are used here to decrease closed-loop noise throughput. This is especially important for PID control, but also useful for PI control. A first-order low-pass filter has been used for PI control, F PI (s)= st f +, (0) which gives the controller high-frequency roll-off. A second order filter F PID (s)= (st f ) 2 /2+sT f +, () is used for PID control, also for the sake of high-frequency roll-off. The filtertimeconstantt f istheonlyparameterthatneedstobesetinboth F PI andf PID. 3

4 P (s)= e s +st, T=0.02,0.05,0.,0.2,0.3,0.5,0.7,, P 2 (s)=.3,.5,2,4,6,8,0,20,50,00,200,500,000 e s (+st) 2, T=0.0,0.02,0.05,0.,0.2,0.3,0.5,0.7,, P 3 (s)=.3,.5,2,4,6,8,0,20,50,00,200,500 (s+)(+st) 2, T=0.005,0.0,0.02,0.05,0.,0.2,0.5,2,5,0 P 4 (s)= P 5 (s)= (s+) n, n=3,4,5,6,7,8 (+s)(+ αs)(+ α 2 s)(+ α 3 s), α=0.,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 P 6 (s)= s(+st ) e sl, L =0.0,0.02,0.05,0.,0.2,0.3,0.5,0.7,0.9,.0, T +L = P 7 (s)= (+st)(+st ) e sl, T +L =, T=,2,5,0 L =0.0,0.02,0.05,0.,0.3,0.5,0.7,0.9,.0 P 8 (s)= αs (s+) 3, α=0.,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,.0,. P 9 (s)= (s+)((st) 2 +.4sT+), T=0.,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,.0 Figure 2 Test batch of 34 processes representative for the process industry. 4. Abatchofbenchmarkmodels A batch of 34 benchmark process models, representative for the process industry,wasintroducedin[]andcanbefoundinfig.2,onthenextpage. Sections8to0holdsoptimalPIandPIDparametersforthisbatch. In addition to these models, the second order time-delayed model P 2,mod (s)= e 0.s (20s+) 2, (2) willalsobeusedtoderiveasetofoptimalcontrollerparameters(with 4

5 corresponding filters). The benchmark models have been approximated with first-order timedelayed(fotd) models P m (s)= K p st+ e sl, (3) in order to classify them based on the normalized time delay, τ= L L+T. (4) 5. Criteria for PID controller design Arationalwaytostartthedesignofacontrolleristofirstderiveaprocess model and a collection of requirements. Constrained optimization can then be applied to make a trade between often conflicting requirements. Tuning of PID controllers for the process industry is seldom done this way since theeffortthatcanbedevotedtoasingleloopisseverelylimited. Requirements typically include specifications on load disturbance attenuation, robustness to process uncertainty, noise sensitivity and set-point tracking. Load disturbance attenuation is a primary concern in process control where steady-state regulation is a key issue, while set-point tracking is a major concern in motion control. Set-point tracking can, however, be treated separately by using a control architecture having two degrees of freedom. Therefore, the controllers listed here will be optimal with respect to load disturbance attenuation. 5. Performance Closed-loop performance, i.e. load disturbance attenuation will here be characterized by the integrated absolute error(iae) IAE= 0 e(t) dt, (5) foraunitsteploaddisturbanceontheprocessinput,seefig Robustness Itisveryimportanttoensurethatallplantprocessesaresafeandstable around their set-points. For this reason, the primary goal of many PID design methods is to secure the robustness of the closed-loop system, i.e. to keep good margins to the point of instability. Robustness can be captured by the sensitivity function, S(s), and the complementary sensitivity function, T(s), see Eqs.(4 5). The maximum values of these functions S(iω) M s, T(iω) M t, ω R +, (6) i.e. the H -norms, will be used here as robustness constraints on the closed-loopsystem.largevaluesof M s and M t willleadtopoorrobustness, while values close to give very high robustness. Reasonable robustnessistypicallygivenwithm s andm t between.2and2.0.forsimplicity,m s andm t willhavethesamevaluesineachnewoptimizationhere (M s =M t =.2,.4,.6,and.8).Notethat ωisthefrequencyin[rad/s] and R + denotesthesetofallnon-negativerealnumbers. 5

6 5.3 Noise sensitivity Large control signal activity generated by measurement noise could cause undesirable actuator wear and tear. The impact of the transfer function frommeasurementnoisetocontrolaction, S c (s),dependsonmanyfactors, with the controller parameters and low-pass filter being particularly important.inthisdocument,theh 2 -normofs c (s)(seeeq.7) S c (s) 2 κ u, (7) will be used to constrain the noise sensitivity. Assuming continuous-time whitegaussianmeasurementnoisewithunitspectraldensity, S c (s) 2 can be derived using the integral formula S c (s) 2 = 2π S c (iω) 2 dω. (8) 6. Optimal PID design using SWORD TheMATLAB softwaretoolcalledsword(software-basedoptimalrobustdesign),wasfirstpresentedin[2].thistoolisusedtoderiveoptimal PI and PID controllers with respect to the optimization problem minimize K,T i,t d R + 0 e(t) dt, subjectto S(iω) M s, T(iω) M t, ω R +. (9) The low-pass filtertime constant, T f,isfixed in each optimization. To simplify the problem solving, the software finds the controller that makes at least one of the robustness constraints active. This will give the optimal solutionalsoto(9),unlessm s andm t arequitelarge(i.e.whensword startsgivinghighlyoscillatingresponses).thistypicallyhappensform s andm t greaterthan2.0. Sections8and9holdtableswithunfilteredSWORDPIandPIDparametersforthebenchmarkmodelsinFig.2,withrespecttoM s =M t =.2,.4,.6,and.8. The SWORD software tool is freely available at: AnolderversionoftheSWORDsoftwareisalsoavailableonthiswebpage.Ithasbeentestedmorethoroughlyandcane.g.beusedifthemost recent SWORD version would not work. 6

7 7. Using SWORD to determine the noise filter In[3],itisshownthatSWORDcanprovidecontrollersthatarecloseto optimal also with respect to the extended optimization problem minimize K,T i,t d,t f R + 0 e(t) dt, subjectto S(iω) M s, T(iω) M t, ω R +, S c (s) 2 κ u. (20) Thisisaccomplishedbyvaryingthelow-passfiltertimeconstant,T f,as wellasthecontrollertype(pid,piori).theresultisasetofcontrollers andlow-passfilters.theusercanthenusethissettochoosethebest possible controller giving an acceptable level of the control signal activity. This level can be determined by visual feedback of the control signal. Section 0 presents tables with controller sets for five different models (withm s =M t =.4),containing:lag-dominant,balanced,delay-dominant, integrating, and dominant second-order processes. 7

8 8. Optimal PI parameters for the test batch The following tables hold process data, optimal unfiltered SWORD PI parameters, and corresponding IAE-values, for the benchmark models. Section8.presentstablesforaclosed-looprobustnessgivenbyM s =M t =.2, Section8.2forM s =M t =.4,Section8.3forM s =M t =.6,andSection 8.4forM s =M t =.8.ThetableshavealsobeencollectedinMATLAB mat-files, Excel spreadsheets(different sheets for different process types), aswellasinatex-file.thesecanbedownloadedfrom: 8. M s =M t =.2 Table P (s)= e s (st+),unfilteredpi,m s=m t =.2 T τ K T i k p k i IAE

9 e s Table2 P 2 (s)= (st+) 2,UnfilteredPI,M s=m t =.2 T τ K T i k p k i IAE

10 Table3 P 3 (s)= (s+)(st+) 2,UnfilteredPI,M s=m t =.2 T τ K T i k p k i IAE Table4 P 4 (s)= (s+) n,unfilteredpi,m s=m t =.2 n τ K T i k p k i IAE Table5 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPI,M s=m t =.2 α τ K T i k p k i IAE

11 e sl Table6 P 6 (s)= s(st +),UnfilteredPI,M s=m t =.2 T L τ K T i k p k i IAE

12 e sl Table7 P 7 (s)= (st+)(+st ),UnfilteredPI,M s=m t =.2 T T L τ K T i k p k i IAE

13 Table8 P 8 (s)= sα (s+) 3,UnfilteredPI,M s=m t =.2 α τ K T i k p k i IAE Table9 P 9 (s)= (s+)((st) 2 +.4sT+),UnfilteredPI,M s=m t =.2 T τ K T i k p k i IAE

14 8.2 M s =M t =.4 Table0 P (s)= e s (st+),unfilteredpi,m s=m t =.4 T τ K T i k p k i IAE

15 e s Table P 2 (s)= (st+) 2,UnfilteredPI,M s=m t =.4 T τ K T i k p k i IAE

16 Table2 P 3 (s)= (s+)(st+) 2,UnfilteredPI,M s=m t =.4 T τ K T i k p k i IAE Table3 P 4 (s)= (s+) n,unfilteredpi,m s=m t =.4 n τ K T i k p k i IAE Table4 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPI,M s=m t =.4 α τ K T i k p k i IAE

17 e sl Table5 P 6 (s)= s(st +),UnfilteredPI,M s=m t =.4 T L τ K T i k p k i IAE

18 e sl Table6 P 7 (s)= (st+)(+st ),UnfilteredPI,M s=m t =.4 T T L τ K T i k p k i IAE

19 Table7 P 8 (s)= sα (s+) 3,UnfilteredPI,M s=m t =.4 α τ K T i k p k i IAE Table8 P 9 (s)= (s+)((st) 2 +.4sT+),UnfilteredPI,M s=m t =.4 T τ K T i k p k i IAE

20 8.3 M s =M t =.6 Table9 P (s)= e s (st+),unfilteredpi,m s=m t =.6 T τ K T i k p k i IAE

21 e s Table20 P 2 (s)= (st+) 2,UnfilteredPI,M s=m t =.6 T τ K T i k p k i IAE

22 Table2 P 3 (s)= (s+)(st+) 2,UnfilteredPI,M s=m t =.6 T τ K T i k p k i IAE Table22 P 4 (s)= (s+) n,unfilteredpi,m s=m t =.6 n τ K T i k p k i IAE Table23 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPI,M s=m t =.6 α τ K T i k p k i IAE

23 e sl Table24 P 6 (s)= s(st +),UnfilteredPI,M s=m t =.6 T L τ K T i k p k i IAE

24 e sl Table25 P 7 (s)= (st+)(+st ),UnfilteredPI,M s=m t =.6 T T L τ K T i k p k i IAE

25 Table26 P 8 (s)= sα (s+) 3,UnfilteredPI,M s=m t =.6 α τ K T i k p k i IAE Table27 P 9 (s)= (s+)((st) 2 +.4sT+),UnfilteredPI,M s=m t =.6 T τ K T i k p k i IAE

26 8.4 M s =M t =.8 Table28 P (s)= e s (st+),unfilteredpi,m s=m t =.8 T τ K T i k p k i IAE

27 e s Table29 P 2 (s)= (st+) 2,UnfilteredPI,M s=m t =.8 T τ K T i k p k i IAE

28 Table30 P 3 (s)= (s+)(st+) 2,UnfilteredPI,M s=m t =.8 T τ K T i k p k i IAE Table3 P 4 (s)= (s+) n,unfilteredpi,m s=m t =.8 n τ K T i k p k i IAE Table32 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPI,M s=m t =.8 α τ K T i k p k i IAE

29 e sl Table33 P 6 (s)= s(st +),UnfilteredPI,M s=m t =.8 T L τ K T i k p k i IAE

30 e sl Table34 P 7 (s)= (st+)(+st ),UnfilteredPI,M s=m t =.8 T T L τ K T i k p k i IAE

31 Table35 P 8 (s)= sα (s+) 3,UnfilteredPI,M s=m t =.8 α τ K T i k p k i IAE Table36 P 9 (s)= (s+)((st) 2 +.4sT+),UnfilteredPI,M s=m t =.8 T τ K T i k p k i IAE

32 9. Optimal PID parameters for the test batch The following tables hold process data, optimal unfiltered SWORD PID parameters, and corresponding IAE-values, for the benchmark models. Section9.presentstablesforaclosed-looprobustnessgivenbyM s =M t =.2, Section9.2forM s =M t =.4,Section9.3forM s =M t =.6,andSection 9.4forM s =M t =.8.ThetableshavealsobeencollectedinMATLAB mat-files, Excel spreadsheets(different sheets for different process types), aswellasinatex-file.thesecanbedownloadedfrom: 9. M s =M t =.2 Table37 P (s)= e s (st+),unfilteredpid,m s=m t =.2 T τ K T i T d k p k i k d IAE

33 e s Table38 P 2 (s)= (st+) 2,UnfilteredPID,M s=m t =.2 T τ K T i T d k p k i k d IAE

34 Table39 P 3 (s)= (s+)(st+) 2,UnfilteredPID,M s=m t =.2 T τ K T i T d k p k i k d IAE Table40 P 4 (s)= (s+) n,unfilteredpid,m s=m t =.2 n τ K T i T d k p k i k d IAE Table4 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPID,M s,m t =.2 α τ K T i T d k p k i k d IAE

35 e sl Table42 P 6 (s)= s(st +),UnfilteredPID,M s=m t =.2 T L τ K T i T d k p k i k d IAE

36 e sl Table43 P 7 (s)= (st+)(+st ),UnfilteredPID,M s=m t =.2 T T L τ K T i T d k p k i k d IAE

37 Table44 P 8 (s)= sα (s+) 3,UnfilteredPID,M s=m t =.2 α τ K T i T d k p k i k d IAE Table45 P 9 (s)= (s+)((st) 2 +.4sT+),UnfilteredPID,M s=m t =.2 T τ K T i T d k p k i k d IAE

38 9.2 M s =M t =.4 Table46 P (s)= e s (st+),unfilteredpid,m s=m t =.4 T τ K T i T d k p k i k d IAE

39 e s Table47 P 2 (s)= (st+) 2,UnfilteredPID,M s=m t =.4 T τ K T i T d k p k i k d IAE

40 Table48 P 3 (s)= (s+)(st+) 2,UnfilteredPID,M s=m t =.4 T τ K T i T d k p k i k d IAE Table49 P 4 (s)= (s+) n,unfilteredpid,m s=m t =.4 n τ K T i T d k p k i k d IAE Table50 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPID,M s,m t =.4 α τ K T i T d k p k i k d IAE

41 e sl Table5 P 6 (s)= s(st +),UnfilteredPID,M s=m t =.4 T L τ K T i T d k p k i k d IAE

42 e sl Table52 P 7 (s)= (st+)(+st ),UnfilteredPID,M s=m t =.4 T T L τ K T i T d k p k i k d IAE

43 Table53 P 8 (s)= sα (s+) 3,UnfilteredPID,M s=m t =.4 α τ K T i T d k p k i k d IAE Table54 P 9 (s)= (s+)((st) 2 +.4sT+),UnfilteredPID,M s=m t =.4 T τ K T i T d k p k i k d IAE

44 9.3 M s =M t =.6 Table55 P (s)= e s (st+),unfilteredpid,m s=m t =.6 T τ K T i T d k p k i k d IAE

45 e s Table56 P 2 (s)= (st+) 2,UnfilteredPID,M s=m t =.6 T τ K T i T d k p k i k d IAE

46 Table57 P 3 (s)= (s+)(st+) 2,UnfilteredPID,M s=m t =.6 T τ K T i T d k p k i k d IAE Table58 P 4 (s)= (s+) n,unfilteredpid,m s=m t =.6 n τ K T i T d k p k i k d IAE Table59 P 5 (s)= (s+)(sα+)(sα 2 +)(sα 3 +),UnfilteredPID,M s,m t =.6 α τ K T i T d k p k i k d IAE

47 e sl Table60 P 6 (s)= s(st +),UnfilteredPID,M s=m t =.6 T L τ K T i T d k p k i k d IAE

48 e sl Table6 P 7 (s)= (st+)(+st ),UnfilteredPID,M s=m t =.6 T T L τ K T i T d k p k i k d IAE