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 selftuning. 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.Closedloop 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.Online ontroller yling (field tuning) Chapter 12 4
5 Controller Synthesis  Time Domain Timedomain 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 losedloop response) θ s Ke Proess model Gs ( ) = (1st order) τ s + 1 Several methods based on 1/4 deay ratio have been proposed: CohenCoon, ZieglerNihols Chapter 12 5
6 Chapter 12 6
7 Comparison of ZieglerNihols and CohenCoon Equations for Controller Tuning (1940 s, 50 s) Controller ZieglerNihols CohenCoon 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. Timeweighted 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 losedloop 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 (123b) 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 (126) an be used (e.g., replae time delay with polynomial approximation) 1 1 If θ = 0, then (123b) yields G = ) (125) 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 firstorderplustimedelay model, ( ) G s = θs Ke τs + 1 (1210) Speify losedloop response as FOPTD (126), but approximate  θ e s  1θs. Substituting and rearranging gives a PI ontroller, ( ) G = K 1+ 1/τ s, K I 1 τ =, τi = τ (1211) 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 1θ s ( )( ) 2 τs θ s 1 τs s 2 τs+ G 2 = = K( 1θ s) 1θ 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 ) (123b) (122a) (1230) 17
18 SeondOrderplusTimeDelay (SOPTD) Model Consider a seondorderplustimedelay model, Use of FOPTD losedloop response (126) 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 (1213) τi s τ + τ τ τ, τ τ τ, τ (1214) = I = 1+ 2 D = K τ + θ τ1+ τ2 (1212) 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 losedloop 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 Modelbased 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 LagDominant Models First or seondorder models with relatively small time delays θ / τ<<1 are referred to as lagdominant models. ( ) The IMC and DS methods provide satisfatory setpoint 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 (1234) I 1 where τ 1 is the largest time onstant (if there are two). τ I Method 3. Design the Controller for Disturbanes, Rather Setpoint 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 lagdominant 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 setpoint 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 OnLine 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 losedloop 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 ZieglerNihols Rules: These wellknown tuning rules were published by ZN 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 ZN ontroller settings are widely onsidered to be an "industry standard". ZN settings were developed to provide 1/4 deay ratio  too osillatory? Chapter 12 35
36 Modified ZN 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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