The Basic Feedback Loop. Design and Diagnosis. of the Basic Feedback Loop. Parallel form. The PID Controller. Tore Hägglund. The textbook version:

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1 The Basic Feedback Loop Design and Diagnosis l n of the Basic Feedback Loop sp Σ e x Controller Σ Process Σ Tore Hägglnd Department of Atomatic Control Lnd Institte of Technolog Lnd, Sweden The PID Controller Parallel form The textbook version: e P e I Σ D D I P = K (e+ Ti ) de e(t)dt + T d dt t = K (e + Ti ) de e(t)dt + T d dt

2 Series form Relations between parallel and series form Series form parallel form: e D Σ e I P Σ K = K T i + T d T i T i = T i + T d T d = T i T d T i + T d e = e + T d de dt = K (e + T i ) e (t)dt Parallel form series form (Reqirement: T i > 4T d ): ( ) K = K T i = T i T d = T i ( ( + + 4T d T i 4T d T i 4T d T i ) ) Setpoint handling Noise handling = K (br + Ti ) d e(t)dt T d dt or better U = K ( + ) st d + Y st i + st d /N One additional parameter: b Also: Filters, ramping modles, feed-forward,... U = K ( + ) + st d st i ( + st d /N) Y One additional parameter: N

3 Specifications Ziegler-Nichols step response method l n sp Σ e x Controller Σ Process Σ K A Load distrbance rejection Setpoint following Measrement noise amplification Robstness with respect to process variations a L Design criterion: Deca ratio.5 Two parameters: a and L Ziegler-Nichols step response method Example: ZN step response method Controller K T i T d P /a PI.9/a 3L PID./a L.5L Process: Controller: G(s) = (s+) 3 K = 5.5 T i =.6 T d =.43 sp

4 Optimization methods Load distrbances l n sp G ff Σ Σ G Σ G c Specifications on: Load distrbance rejection Setpoint following Measrement noise amplification Robstness with respect to process variations Tning parameter Minimize IE = at load distrbances e(τ)dτ = k i = T i K Robstness ImL(iω ) k i Robstness R Re L(iω ) ω r/r r/r k R = /M s M s is a sefl tning parameter. Range [.,]. r(ω )= G p (iω) 4

5 Robstness MIGO Design.3 M-constrained Integral Gain Optimization.5. Maximize k i (i.e., minimize IE) sch that L(iω ) is otside the M s -circle. ki k M s tning parameter. M s =.4 good defalt vale. Reqires accrate process model. AMIGO Design Approximate M-constrained Integral Gain Optimization Approximate the MIGO design b Using simple process models Based on step responses or freqenc responses Three parameters needed Fitting controller parameters to a large test batch Using simple tning rles like Ziegler-Nichols Process model: AMIGO Design G p (s) = Processes with integration: Relative time dela: τ = G p (s) = K v s e sl L L + T Dnamic gain: a = K p L T = K vl K p +st e sl K v = K p T 5

6 AMIGO PI Test batch AMIGO PI Design e s P (s) =, +st T =.,.5,.,.3,.5,,, 3, 5,,, e s P (s) =, (+st) T =.,.5,.,.,.3,.5,.7,,.3,.5,, 4, 6, 8,,, P 3 (s) =, (s+)( + st) T =.,.,.5,.,.,.5 P 4 (s) = (s+) n, n=, 3, 4, 5, 6, 7, 8 P 5 (s)= (+s)( + α s)( + α s)( + α 3, s) α =.,.,.5,.7 P 6 (s) = αs (s+) 3, α =.,.,.5,, P 7 (s)= s, +ζs+ ζ =.5,.7, KK p vs τ T i /T vs τ ak vs τ T i /L vs τ AMIGO PI Design MIGO PID Design ( K = T ( ) ) T K p L L + T T i =.35L + 3 parameters needed! 6.7LT T + LT + L k i k k d

7 .8.6 MIGO PID Design k d = k d = k d = MIGO PID Design Problems with PID control. Additional constraints reqired k d = 3 k d = 3. k d = x MIGO PID Design Problems with PID control. Additional constraints reqired. MIGO PID Design Use M circle instead of M s circle Step in setpoint Step in load distrbance M s = max ω S(iω ) = max ω +PC(iω) (Dashed) PC(iω) M p = max ω T(iω ) = max ω (Dotted) +PC(iω) 7

8 MIGO PID Design Sggested additional constraints: T i = α T d L(iω ) has negative crvatre and monotone phase k i / k = (Used in the following) AMIGO PID Test batch P (s) = e s, +st T =.,.5,.,.,.3,.5,.7,,.3,.5,, 4, 6, 8,,, 5,,, 5, e s P (s) =, (+st) T =.,.,.5,.,.,.3,.5,.7,,.3,.5,, 4, 6, 8,,, 5,,, 5 P 3 (s) =, T =.5,.,.,.5,.,.,.5,, 5, (s+)( + st) P 4 (s) = (s+) n, n=3, 4, 5, 6, 7, 8 P 5 (s)= (+s)( + α s)( + α s)( + α 3, α =.,.,.3,, 4,.5,.6,.7,.8,.9 s) P 6 (s) = s(+st ) e sl, L =.,.,.5,.,.3,.5,.7,.9,., T + L = T P 7 (s) = (+st)( + st ) e sl, T + L =, T =,,5, L =.,.,.5,.,.3,.5,.7,.9,. P 8 (s) = αs (s+) 3, α =.,.,.3,.4,.5,.6,.7,.8,.9,.,. P 9 (s)= (s+)((st), T =.,.,.3,.4,.5,.6,.7,.8,.9,. +.4sT + ) MIGO PID Design AMIGO PID Design 4 KK p vs τ ak vs τ Wh large spread of controller parameters for small τ? Processes with transfer fnctions T i /T vs τ T d /T vs τ T i /L vs τ T d /L vs τ P(s) = K v s(+st ) and P(s) = K p (+st )( + st ) can be controlled with arbitraril high gains in the PID controller. These processes have τ <.3, with eqalit when T = T

9 Conseqence PI or PID? Modeling with the strctre P(s) = K p +st e sl imposes fndamental limitations that ma not be present in the tre process! k i [PID]/k i [PI] vs τ Average Residence Times Ratio T i /T d 3 PI 3 PID T i /T d vs τ T ar/t cl ar ol τ T ar/t cl ar ol τ

10 AMIGO PID Design KK p vs τ 5 ak vs τ AMIGO PID Design K = ( T ) K p L T i /T vs τ T d /T vs τ T i /L vs τ T d /L vs τ Efficient for τ >.. T i = T d = Conservative for τ <...4L +.8T L +.T.5LT.3L + T L AMIGO PID Design Example Lag-dominant process P(s) = (+s)( +.s)( +.s)( +.s) Solid = AMIGO-PID, Dashed = MIGO-PID, Dashed-dotted = MIGO-PI t

11 Example Balanced lag and dela P(s) = (s+) 4 Example Dela dominant P(s) = (+.5s) e s Solid = AMIGO-PID, Dashed = MIGO-PID, Dashed-dotted = MIGO-PI t Solid = AMIGO-PID, Dashed = MIGO-PID, Dashed-dotted = MIGO-PI t AMIGO Smmar Ziegler-Nichols freqenc response method PI ( K = T ( ) ) T K p L L + T Ultimate point Im G(iω) PID T i =.35L + 6.7LT T + LT + L K = ( T ) K p L ϕ a ω Re G(iω) T i = T d =.4L +.8T L +.T.5LT.3L + T L Design criterion: Deca ratio.5 Two parameters: K 8 and T 8

12 AMIGO Freqenc response Relation between κ and τ Process model: K p = G p () K 8 = G p (iω 8 ) T 8 = π ω 8 For FOTD process: κ Integrating processes: K 8 and T 8 Gain ratio: κ = K 8 K p τ = τ π arctan /κ π arctan /κ + /κ AMIGO Freqenc response Is the gain too high? PI PID KK 8 =.5 T i.8 = T κ K =(.3.κ 4 )/K 8 T i =.6 + κ T KK p vs τ T i /T vs τ ak vs τ T i /L vs τ T d =.5( κ ).95κ T 8 Efficient for processes with κ >

13 AMIGO Detning PI AMIGO Detning PI.4.3 T = T = T = k i = k i α + KK p α +K K p β (α +KK p) K p (L+T) for KK p k i K p(l+t) β(α +K K p ) α for KK p < k i K p(l+t) β(α +K K p ) α ki. ki ki k. 3 4 k 4 k α = M s M s ) M s (M s + Ms / for design based on M s β = M(M ) for design based on M AMIGO Detning PI, Example AMIGO Detning PI, Example AMIGO Design: P(s) = +s e s.5.5 K = 349 T i = Gain redction factors:,.5,.,.5,., and.5 3

14 AMIGO Detning PI, Example AMIGO Detning PI, Test Batch 5 k i vs k T i vs k.5 K /K =.5 K /K = K /K = K /K = AMIGO Detning PID AMIGO Detning PID.8 K = K PI + k d kd PID (K PID K PI ) k i.6.4 k i = k PI i + k d kd PID (k PID i k PI i )..5 k k d

15 AMIGO Detning PID kd/ kd PI D = kd/ kd PI D = investigate the process! Process = Everthing otside the PID algorithm! Are there an scaling factors? Are there an filters?.5.5 kd/ kd PI D =..5.5 Before o start tning kd/ kd PI D =.5 Avoid dead-times Sensors and actators OK? Friction or hsteresis? Other nonlinearities? Controller series or parallel? Is dnamics reall the limiting factor? Control with friction (Stick-slip motion) Diagnosis of friction

16 Control with hsteresis Diagnosis of hsteresis Atomatic performance monitoring We have improved process control. We have lost hman performance monitoring. We need atomatic performance monitoring. Reasons for poor control loop performance Eqipment problems Stiction in valves Sensor falts Poor controller tning Never tned? Nonlinear plant Time-varing plant Oscillating load distrbances Two monitoring tools: Detection of oscillating control loops Detection of slggish control loops 6

17 Oscillation detection Oscillation detection Determine ti IAE = e(t) dt t i between zero crossings of the control error. Good control: IAE small Load distrbances: IAE large Oscillation detection Example Plp concentration control The loop is oscillating if the rate of load distrbances becomes high. The loop is oscillating if more than n lim load distrbances are detected dring a spervision time T sp. 3 parameters: IAE lim, n lim, T sp. Sggestion: IAE lim = T i /π n lim = T sp = 5T i Water Plp QIC FIC QT FT 7

18 Example Diagnosis: Pt the controller in manal mode Process otpt and estimated set point sp IAE and IAE lim No Still oscillating? Yes Check the valve! Make small changes in and see if follows The distrbances are generated otside the loop. Search for the sorce! Control signal 4 6 Rate of load detections x and rate limit n lim Yes Friction? Possible to eliminate? Yes 8 Make a valve maintenance No No Eliminate the distrbances Check if the controller is properl tned Possible to feed forward? Yes No Use feed forward Tne the controller so that the effects of the distrbances are minimized Frövi Paper Mill Oscillation detection procedre sed in Honewell TDC3. 9% of the loops in the carton board mill are spervised. Each loop has an Oscillation index that is increased ever time a detection is made. The Oscillation index is reset to zero at maintenance or tning. Top-ten list presents the worst loops. Pressre control loop PV % 5 5 Frövi Paper Mill OP %

19 Flow control loop Frövi Paper Mill l/s % 4 Maintenance 3 PV FLOW [l/s] Ato mode OP CONTROL SIGNAL [%] OSC INDEX Man mode Detection of slggish control loops Good and bad control of load distrbances:.4 Set point sp and measred variable Control signal I i = t pos t neg t pos + t neg I i large Slggish control Idle Index I i [,] Previos example: I i =.8 and I i =.68 Recrsive version Filtering important Atomatic temperatre valve position Control of a heat exchanger { tpos + h if > t pos = t pos if { tneg + h if < t neg = t neg if K =. T i = 3s I i =.8 9

20 temperatre valve position Control of a heat exchanger K =.5 T i = 8s I i =.3 Implemented where? The DCS sstem or an external compter? What is needed? Measrement signal Measrement signal range Setpoint Control signal Control signal range Controller parameters Control mode: Man/Ato/Tracking Filters etc. Sampling interval This information is normall available in the DCS sstem onl. Design and Diagnosis of the Basic Feedback Loop Design 4 considerations 3 parameters needed No niversal tning rle Diagnosis Before o start The loop will change Control the control References [] Karl Johan Åström and Tore Hägglnd. PID Controllers: Theor, Design, and Tning. Instrment Societ of America, Research Triangle Park, North Carolina, 995. [] Tore Hägglnd and Karl Johan Åström. Revisiting the Ziegler-Nichols tning rles for PI control. Asian Jornal of Control, 4(4):364 38, December. [3] T. Hägglnd and K. J. Åström. Revisiting the Ziegler-Nichols tning rles for PI control Part II the freqenc response method. Asian Jornal of Control, sbmitted, 4. [4] Tore Hägglnd and Karl Johan Åström. Revisiting the Ziegler-Nichols step response method for PID control. Jornal of Process Control, 4(6):635 65, 4. [5] T. Hägglnd and K. J. Åström. Revisiting the Ziegler-Nichols freqenc response method for PID control. xxx, to be pblished, 4. [6] Tore Hägglnd. A control-loop performance monitor. Control Engineering Practice, 3:543 55, 995. [7] Tore Hägglnd. Atomatic detection of slggish control loops. Control Engineering Practice, 7:55 5, 999.