# Controller Design Based on Transient Response Criteria. Chapter 12 1

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1 Controller Design Based on Transient Response Criteria Chapter 12 1

2 Desirable Controller Features 0. Stable 1. Quik responding 2. Adequate disturbane rejetion 3. Insensitive to model, measurement errors 4. Avoids exessive ontroller ation 5. Suitable over a wide range of operating onditions Impossible to satisfy all 5 unless self-tuning. Use optimum sloppiness" Chapter 12 2

3 Chapter 12 3

4 Alternatives for Controller Design 1.Tuning orrelations most limited to 1st order plus dead time 2.Closed-loop transfer funtion - analysis of stability or response harateristis. 3.Repetitive simulation (requires omputer software like MATLAB) 4.Frequeny response - stability and performane (requires omputer simulation and graphis) 5.On-line ontroller yling (field tuning) Chapter 12 4

5 Controller Synthesis - Time Domain Time-domain tehniques an be lassified into two groups: (a) Criteria based on a few points in the response (b) Criteria based on the entire response, or integral riteria Approah (a): settling time, % overshoot, rise time, deay ratio (Fig an be viewed as losed-loop response) θ s Ke Proess model Gs ( ) = (1st order) τ s + 1 Several methods based on 1/4 deay ratio have been proposed: Cohen-Coon, Ziegler-Nihols Chapter 12 5

6 Chapter 12 6

7 Comparison of Ziegler-Nihols and Cohen-Coon Equations for Controller Tuning (1940 s, 50 s) Controller Ziegler-Nihols Cohen-Coon Proportional KK = ( τ ) C KK = ( τ ) + 1 θ C θ 3 Proportional ( ) + KK = 0.9 τ KK ( τ C θ C = 0.9 ) θ Integral τ [ ( )] ( θ ) θ I τ θ I = 3.33 = τ τ ( θ ) τ τ τ Proportional ( ) + KK = 1.2 τ KK ( τ C θ C = 1.35 ) θ Integral ( ) + τ I = 2.0 θ [ ( θ τ θ )] I Derivative τ τ = ( ) τ τ θ τ τ ( θ ) D = 0.5 τ τ ( θ τ 0.37 ) D = τ τ ( θ ) τ Chapter 12 7

8 Approah (b) 1. Integral of square error (ISE) ISE = [ e(t) ] 2. Integral of absolute value of error (IAE) IAE = 3. Time-weighted IAE 0 0 ITAE = 0 e(t) dt 2 dt t e(t) dt Pik ontroller parameters to minimize integral. IAE allows larger deviation than ISE (smaller overshoots) ISE longer settling time ITAE weights errors ourring later more heavily Approximate optimum tuning parameters are orrelated with K, θ, τ (Table 12.3). Chapter 12 8

9 Chapter 12 9

10 Chapter 12 10

11 Chapter 12 11

12 Summary of Tuning Relationships 1. K C is inversely proportional to K P K V K M. 2. K C dereases as θ/τ inreases. 3. τ I and τ D inrease as θ/τ inreases (typially τ D = 0.25 τ I ). 4. Redue K, when adding more integral ation; inrease K, when adding derivative ation 5. To redue osillation, derease K C and inrease τ I. Chapter 12 12

13 Disadvantages of Tuning Correlations 1. Stability margin is not quantified. 2. Control laws an be vendor - speifi. 3. First order + time delay model an be inaurate. 4. K p, τ, and θ an vary. 5. Resolution, measurement errors derease stability margins. 6. ¼ deay ratio not onservative standard (too osillatory). Chapter 12 13

14 Diret Synthesis Y Y GCG = 1 G G sp + C ( G inludes G m, G v ) 1. Speify losed-loop response (transfer funtion) Y Y sp 2. Need proess model, G (= G P G M G V ) d 3. Solve for G, G C Y 1 Y sp d = G Y 1 Y sp d (12-3b) 14

15 Speify Closed Loop Transfer Funtion s Y θ e = (12 6) Y sp τ s+ 1 d (first order response, no offset) ( τ = speed of response, θ = proess time delay in G) But other variations of (12-6) an be used (e.g., replae time delay with polynomial approximation) 1 1 If θ = 0, then (12-3b) yields G = ) (12-5) G τ s K τs+1 τ 1 For G =, G = = + (PI) τs+1 Kτ s Kτ Kτ s 15

16 Derivation of PI Controller for FOPTD Proess Consider the standard first-order-plus-time-delay model, ( ) G s = θs Ke τs + 1 (12-10) Speify losed-loop response as FOPTD (12-6), but approximate - θ e s - 1-θs. Substituting and rearranging gives a PI ontroller, ( ) G = K 1+ 1/τ s, K I 1 τ =, τi = τ (12-11) K θ+ τ with the following ontroller settings: Chapter 12 16

17 Derivation of PID Controller for FOPTD Proess: let Y 1 θ s = 2 Y sp τ s+ 1 d θ s Ke K(1 θ s) Gs () = 2 τ s + 1 ( τ s + 1)(1 + θ s) 2 Y 1 Y sp d G = G Y 1- Y sp d 1-θ s ( )( ) 2 τs θ s 1 τs s 2 τs+ G 2 = = K( 1-θ s) 1-θ s K +τ s - 2 τ s+ 1 K τ θ θ τ = τ = + τ τ = K τ ( τ ) + 1 θ θ + I D ( )( θ ) ( θ ) (12-3b) (12-2a) (12-30) 17

18 Seond-Order-plus-Time-Delay (SOPTD) Model Consider a seond-order-plus-time-delay model, Use of FOPTD losed-loop response (12-6) and time delay approximation gives a PID ontroller in parallel form, where K 1 ( ) G s = Ke θs ( τ s+ 1)( τ s+ 1) G = K 1+ + τ Ds (12-13) τi s τ + τ τ τ, τ τ τ, τ (12-14) = I = 1+ 2 D = K τ + θ τ1+ τ2 (12-12) Chapter 12 18

19 Example 12.1 Use the DS design method to alulate PID ontroller settings for the proess: s 2e G = 10s+ 1 5s+ 1 ( )( ) Consider three values of the desired losed-loop time onstant: τ = 1, 3, and 10. Evaluate the ontrollers for unit step hanges in both the set point and the disturbane, assuming that G d = G. Perform the evaluation for two ases: a. The proess model is perfet ( G = G). b. The model gain is K = 0.9, instead of the atual value, K = 2. This model error ould ause a robustness problem in the ontroller for K = 2. s 0.9e G = 19 10s+ 1 5s+ 1 ( )( )

20 The ontroller settings for this example are: K K = ( 2) ( 0.9) K K = τ I τ D τ = 1 τ = 3 τ = Note only K is affeted by the hange in proess gain. Chapter 12 20

21 The values of K derease as τ inreases, but the values of and do not hange, as indiated by Eq τ D τ I Figure 12.3 Simulation results for Example 12.1 (a): orret model gain. 21

22 Figure 12.4 Simulation results for Example 12.1 (b): inorret model gain. 22

23 Chapter 12 23

24 Internal Model Control (IMC) It an be shown that G * * G = 1 GG GG GG Y = Y + D 1 G ( G G) 1 G ( G G ) * * 1 * sp * G = G Y = G GY + (1 G G ) D * * sp Chapter 12 24

25 IMC Design 1) G = G G + 2) * 1 1 G = f f G = ( τ s + 1) r Y = G fy + (1 fg ) D + sp + Chapter 12 25

26 Chapter 12 26

27 Controller Tuning Relations Model-based design methods suh as DS and IMC produe PI or PID ontrollers for ertain lasses of proess models, with one tuning parameter τ (see Table 12.1) How to Selet τ? Several IMC guidelines for τ have been published for the model in Eq : 1. τ / θ > 0.8 and τ > 0.1τ (Rivera et al., 1986) 2. τ> τ > θ (Chien and Fruehauf, 1990) 3. τ θ (Skogestad, 2003) = Chapter 12 27

28 Tuning for Lag-Dominant Models First- or seond-order models with relatively small time delays θ / τ<<1 are referred to as lag-dominant models. ( ) The IMC and DS methods provide satisfatory set-point responses, but very slow disturbane responses, beause the value of is very large. ( τ = τ ) τ I Fortunately, this problem an be solved in three different ways. Method 1: Integrator Approximation θs Ke K * e Approximate Gs ( ) = by Gs ( ) = τ s+ 1 s where K* = K/ τ. I θs Then an use the IMC tuning rules (Rule M or N) to speify the ontroller settings. 28

29 Method 2. Limit the Value of τ I Skogestad (2003) has proposed limiting the value of : ( ) { } τ = min τ,4 τ + θ (12-34) I 1 where τ 1 is the largest time onstant (if there are two). τ I Method 3. Design the Controller for Disturbanes, Rather Set-point Changes The desired CLTF is expressed in terms of (Y/D) d, rather than (Y/Y sp ) d Referene: Chen & Seborg (2002) Chapter 12 29

30 Example 12.4 Consider a lag-dominant model with θ / τ = 0.01: 100 G ( s) = e 100s + 1 Design three PI ontrollers: a) IMC ( τ = 1) ( ) b) IMC τ 2 based on the integrator approximation = ) IMC with Skogestad s modifiation (Eq ) τ 1 ( ) = s Chapter 12 30

31 Evaluate the three ontrollers by omparing their performane for unit step hanges in both set point and disturbane. Assume that the model is perfet and that G d (s) = G(s). Solution The PI ontroller settings are: Controller K τ I (a) IMC (b) Integrator approximation () Skogestad Chapter 12 31

32 Figure Comparison of set-point responses (top) and disturbane responses (bottom) for Example The responses for the integrator approximation and Chen and Seborg (disussed in textbook) methods are essentially idential. Chapter 12 32

33 On-Line Controller Tuning 1. Controller tuning inevitably involves a tradeoff between performane and robustness. 2. Controller settings do not have to be preisely determined. In general, a small hange in a ontroller setting from its best value (for example, ±10%) has little effet on losed-loop responses. 3. For most plants, it is not feasible to manually tune eah ontroller. Tuning is usually done by a ontrol speialist (engineer or tehniian) or by a plant operator. Beause eah person is typially responsible for 300 to 1000 ontrol loops, it is not feasible to tune every ontroller. 4. Diagnosti tehniques for monitoring ontrol system performane are available. Chapter 12 33

34 Controller Tuning and Troubleshooting Control Loops Chapter 12 34

35 Ziegler-Nihols Rules: These well-known tuning rules were published by Z-N in 1942: ontroller K τ I τ D P PI PID 0.5 K CU 0.45 K CU 0.6 K CU - P U /1.2 P U /2 - - P U /8 Z-N ontroller settings are widely onsidered to be an "industry standard". Z-N settings were developed to provide 1/4 deay ratio -- too osillatory? Chapter 12 35

36 Modified Z-N settings for PID ontrol ontroller K τ I τ D original Some overshoot No overshoot 0.6 K CU 0.33 K CU 0.2 K CU P U /2 P U /2 P U /3 P U /8 P U /3 P U /2 Chapter 12 36

37 Chapter 12 37

38 Chapter 12 38

39 Homework #9 (Due to 08/05/08) Exerises: 12.1, 12.2, 12.3, 12.5, Chapter 12 39

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