3.1 Experimental Design

Transcription

1 3 Relay Feedback Åström and Hägglnd [1] sggest the relay feedback test to generate sstained oscillation as an alternative to the conventional continos cycling techniqe. It is very effective in determining the ltimate gain and ltimate freqency. Lyben [2] olarizes the relay feedback method and calls this method ATV (atotne variation). The acronym also stands for all-terrain vehicle, since ATV rovides a sefl tool for the rogh and rocky road of system identification. As ointed ot by Lyben, the motivation for sing the relay feedback (ATV) has grown ot of a stdy of an indstrial distillation colmn. The distillation colmn is an imortant nit in chemical rocess indstries. It is rather difficlt to obtain a linear transfer fnction model for highly nonlinear colmns. Attemts have been made sing ste or lse tests. Unfortnately, the system reslts in an extremely long time constant, e.g. τ 870 h [2]. Moreover, very large deviations occr in the linear model as the size or direction of the int is changed. Simlation stdies also reveal that, sometimes, very small changes of magnitde (less than 0.01%) have to be made to get an accrate linear model. This immediately rles ot the se of this kind of int design in real lants becase lant data are never known to anywhere near this order of accracy. Lyben shows that the simle relay feedback tests rovide an effective way to determine linear models for sch rocesses. It has become a standard ractice in chemical rocess control, as can be seen in recent textbooks in rocess control [3,4]. Wang et al. [5] discss varios asects of the relay feedback. The distinct advantages of the relay feedback are: 1. It identifies rocess information arond the imortant freqency, the ltimate freqency (the freqency where the hase angle is π ). 2. It is a closed-loo test; therefore, the rocess will not drift away from the nominal oerating oint. 3. For rocesses with a long time constant, it is a more time-efficient method than conventional ste or lse testing. The exerimental time is roghly eqal to two to for times the ltimate eriod. 23

2 24 Atotning of PID Controllers 3.1 Exerimental Design Consider a relay feedback system where G(s) is the rocess transfer fnction, y is set the controlled ott, y is the SP, e is the error and is the manilated int (Figre 3.1A). An on off (ideal) relay is laced in the feedback loo. The Åström Hägglnd relay feedback system is based on the observation: when the ott lags behind the int by π radians, the closed-loo system may oscillate with a eriod P. Figre 3.1(B) illstrates how the relay feedback system works. A relay of magnitde h is inserted in the feedback loo. Initially, the int is increased by h. As the ott y starts to increase (after a dead time D ), the relay switches to the oosite osition, h. Since the hase lag is π, a limit cycle with a eriod P reslts (Figre 3.1). The eriod of the limit cycle is the ltimate eriod. Therefore, the ltimate freqency from this relay feedback exeriment is ω 2π (3.1) P From the Forier series exansion, the amlitde a can be considered to be the reslt of the rimary harmonic of the relay ott. Therefore, the ltimate gain can be aroximated as [1,6] 4h K (3.2) π a where h is the height of the relay and a is the amlitde of oscillation. These two vales can be sed directly to find controller settings. Notice that Eqations 3.1 Figre 3.1. (A) Block diagram for a relay feedback system and (B) relay feedback test for a system with ositive steady state gain

3 Relay Feedback 25 and 3.2 give aroximate vales of ω and K. A more accrate exression will be derived shortly. The relay feedback test can be carried ot manally (withot any atotner). The rocedre reqires the following stes. 1. Bring the system to steady state. 2. Make a small (e.g. 5%) increase in the manilated int. The magnitde of change deends on the rocess sensitivities and allowable deviations in the controlled ott. Tyical vales are between 3 and 10%. 3. As soon as the ott crosses the SP, the manilated int is switched to the oosite osition (e.g. 5% change from the original vale). 4. Reeat ste 2 ntil sstained oscillation is observed (Figre 3.1). 5. Read off ltimate eriod P from the cycling and comte 3.2. K from Eqation This rocedre is relatively simle and efficient. Physically, it imlies moving the manilated int against the rocess. Consider a system with a ositive steady state gain (Figre 3.1). When yo increase the int (as in ste 1), the ott y tends to increase also. As a change in the ott is observed, yo switch the int to the oosite direction. This is meant to bring the ott back down to the SP. However, as soon as the ott comes down to the SP, yo switch the int to the er osition. Conseqently, a continos cycling reslts, bt the amlitde of oscillation is nder yor control (by adjsting h ). More imortantly, in most cases, yo obtain the information yo need for tning of the controller. Several characteristics can be seen from the relay feedback test. Consider the most common FOPDT systems. G () s Ds Ke τ s+1 (3.3) where K is the steady state gain, D is the dead time and τ is the time constant. Figre 3.2 indicates that, if the normalized dead time D / τ is less than 0.28, the ltimate eriod is smaller than the rocess time constant. In terms of lant test, that imlies the relay feedback test is more time efficient than the ste test. The reason is that it takes almost 3τ to reach 95% of the steady state vale in a ste test and the time reqired for the relay feedback is also roghly eqal to 3P (to establish a stable oscillation). Therefore, the relay feedback system is more time efficient than the ste test for systems with D / τ < 0.28 (3.4) Since the dead time cannot be too large (it often comes from the measrement delay), the temeratre and comosition loos in rocess indstries seem to fall into this category. In other words, Eqation 3.4 is fairly tyical for many slow chemical rocesses, esecially for nits involved with comosition changes.

4 26 Atotning of PID Controllers 3 P / Time Constant 2 1 relay more efficient Dead Time / Time Constant Figre 3.2. P /τ as fnction of the normalized dead time D/τ 3.2 Aroximate Transfer Fnctions: Freqency-domain Modeling After the relay feedback exeriment, the estimated ltimate gain Kˆ and ltimate freqency ωˆ can be sed directly to calclate controller arameters. Alternatively, it is ossible to back-calclate the aroximated rocess transfer fnctions. The other data sefl in finding the transfer fnction are the dead time D and/or the steady state gain K. In theory, the steady state gain can be obtained from lant data. One simle way to find K is to comare the int and ott vales at two different steady states. That is: K y/ (3.5) where y denotes the change in the controlled variable and stands for the deviation in the manilated int. However, recations mst be taken to make sre that the sizes of the changes in are made small enogh sch that the gain in Eqation 3.5 trly reresents the linearized gain. For highly nonlinear rocesses, these changes are tyically as small as 10 3 to 10 6 % of the fll range [2]. Sch small changes wold only be feasible sing a mathematical model. Trying to obtain reliable steady state gains from lant data is sally imractical. The dead time D in the transfer fnction can be easily read off from the initial art of the relay feedback test. It is simly the time it takes for y to start resonding to the change in (Figre 3.1). For the FOPDT system, it is simly the time to reach the eak amlitde in a half eriod, as will be shown in Chater 4. Therefore, it is more likely that we will have information on the dead time rather than the steady state gain.

5 Relay Feedback 27 Now we are ready to find an aroximate model. Tyical transfer fnctions in rocess control are assmed and arameters can be calclated. The transfer fnctions have the following forms: Model I (integrator ls dead time) Model P (re dead time) Ds K e G( s) (3.6) s Ds ( ) Ke G s (3.7) Model 1 (FOPDT) Ds K e G( s) τs+ 1 Model 2a (second-order ls dead time) ( ) G s Ds K e ( τs+ 1) 2 Model 2b (second-order ls dead time with two neqal lags) ( ) G s Ds K e ( τ1s+ 1)( τ2s+ 1) (3.8) (3.9) (3.10) In these five models, model I and model P have two nknown arameters, models 1 and 2a have three nknown arameters and model 2b has for nknown arameters. Therefore, additional information, sch as D or K, is needed if the last three models are emloyed. As ointed ot by Tyres and Lyben [7], the simlest integrator-ls-time-delay model (model I) rovides good aroximation for slow chemical rocesses, e.g. systems showing a small D / τ vale. It is the model we recommend for slow rocesses. The relay feedback exeriment has the following stes: 1. If necessary, the dead time D can be read off from the initial resonse, or the time to the eak amlitde, and the steady state gain can be obtained from steady state simlation. 2. The ltimate gain Kˆ and ltimate freqencyωˆ are comted (Eqations 3.1 and 3.2) after the relay feedback exeriment. 3. Different model strctres (Eqations ) are fitted to the data Simle Aroach Once the model is selected, we can back-calclate the model arameters from two eqations describing the ltimate gain and the ltimate freqency.

6 28 Atotning of PID Controllers Model I (Friman and Waller [8]) K ω 2π K KP π P D 2 ω 4 (3.11) (3.12) Notice that no a riori rocess knowledge is needed for this model. Moreover, comtation of K and D is qite straightforward. Model P K P 1 K (3.13) P D (3.14) 2 Similar to model I, no a riori rocess knowledge is necessary. Model 1 ( π Dω ) τ tan ω (3.15) ( K ) 2 K 1 τ (3.16) ω For model 1, either D or K is needed to solve for the time constant. For examle, if the dead time is read off from the relay test, then we can comte τ from Eqation Then, K can be fond by solving Eqation Model 2a tan( π Dω )/ 2 τ (3.17) ω τ ( ) K K ω 1 (3.18) The eqations describing model 2a are qite similar to those for model 1. Again, we need to know D or K before finding model arameters.

7 Relay Feedback 29 Model 2b ( 1) tan ( 2) D tan 1 1 π ω ωτ ωτ (3.19) 1 Kˆ K ( ωτ 1) ( ωτ 2) (3.20) Since we have for arameters in model 2b, both K and D have to be known in order to solve for the two time constants τ 1 and τ 2. This is the most comlex model strctre in or models, and it is often sfficient for rocess control alications. Let s se an FOPDT system to illstrate the arameter estimation rocedre. Examle 3.1 WB colmn [9] s 12.8e G( s) 16.8s + 1 This is the transfer fnction between the to comosition x D and the reflx flow R. From a relay feedback test, we obtain the following ltimate gain and ltimate freqency: K ˆ 1.71 and ω ˆ Note that these two vales are only an aroximation to the tre vales: K 2.1 and ω Parameters can be calclated for different model strctres: Model I (no rior knowledge on Model P (no rior knowledge on K and D ) 0.94 e G( s) s K and D ) 0.97s 1.94 ( ) 0.58e G s Model 1 (assme D is known, i.e. D 1) Model 2a (assme D is known) Model 2b (assme ( ) G s ( ) G s K and D are known) ( ) G s s 13.2e s ( 14.0s+ 1) 1.12 e s ( 0.59s+ 1) e s ( 13.5s+ 1)( s+ 1) Desite varying in model arameters, all these for models have the same ltimate gain and ltimate freqency. That is, the models are correct arond the ltimate

8 30 Atotning of PID Controllers freqency, which is imortant for the controller design. However, if we extraolate the model to different freqencies, e.g. ω 0, then the reslts can be comletely misleading. For examle, the steady state gain of model 2a is only 1.12, which is less than 10% of the tre vale. We have to be very catios when sing these models Imroved Algorithm In theory, if the model strctre is correct and the ltimate gain and ltimate freqency are correctly identified, then we cold have a very good aroximation of the transfer fnction. For examle, if the K and ω in the revios examle are close to the tre vales, then we will not have errors in the steady state gains and time constant for model 1. Unfortnately, since Eqations 3.1 and 3.2 only give aroximations to the ltimate gain and ltimate freqency, the arameters derived from Eqations 3.15 and 3.16 can deviate significantly from the tre system arameters. This imlies the observed ltimate eriod P ˆ and the comted ltimate gain are not the tre vales. In order to have a better aroximation of the transfer fnction, fndamental analysis of the relay feedback system is necessary. First, one wold like to know what the eriod of oscillation from the relay feedback exeriment really reresents. In other words, given a transfer fnction with known arameters, what is the exression for the eriod of oscillation observed from the relay feedback exeriment, P ˆ? The following theorem [1] rovides the answer. Theorem 3.1 Consider the relay feedback system with a transfer fnction G(s) and an ideal relay (Figre 3.1). Let HG( Ts, z ) be the lse transfer fnction of G( s ) with a samling time of T s. If there is a eriodic oscillation, then the eriod of oscillation P ˆ is given by ( ) HG Pˆ /2, 1 0 Åström and Hägglnd [1] rove the theorem starting form the discrete-time state-sace eqations. The reslt, HG( Pˆ /2, 1 ) 0, is obtained by finding the z- domain eqivalent. The continos-time resonse of an ideal relay (Figre 3.1) can be discretized at the oint when the relay switches. The z-transforms of the int and ott are h/ ( z+ 1) and 0 resectively. Since this is a self-oscillation system, the roagation of the int is described by the gain HG( Pˆ /2, 1 ) 0. This eqation can be sed to find the eriod of oscillation for a known system. In identification, P ˆ is observed from the resonse and one is able to se this to backcalclate system arameters. Unlike the continos-time analysis based on the rimary harmonic, the discrete-time exression gives a sond basis for finding the system arameters, since no assmtion is made in the derivation. Based on the theorem, a better relationshi between ω ˆ (or P ˆ ) and the system arameters can be derived. For the transfer fnctions of interest (models 1, 2a and 2b), the following reslts can be derived from the modified z-transform [10]:

9 Relay Feedback 31 Model 1 Model 2a π τ ωˆ ln 2ex 1 ( ( D τ) ) (3.21) π 2π m+ ( m 1ex ) ˆ τω τ π mπ π ωˆ 1+ ex ex 1 ex 2 τωˆ + τωˆ τωˆ (3.22) ˆ where Dω m 1 π Model 2b mπ mπ 2ex 2ex τωˆ τωˆ τ τ τ τ π π 1+ ex 1 ex τω 1ˆ + τω 2 ˆ (3.23) Eqations rovide alternative exressions between the observed ltimate eriod, e.g. ω ˆ, and system arameters. For examle, Eqation 3.21 relates ω ˆ to D and τ in a way that differs sbstantially from the standard hase angle eqation (i.e. Eqation 3.15). 1 π ωˆ D tan ( ωτ ˆ ) Again, we can derive a better exression for the amlitde ratio art at the ltimate freqency, since the exression in Eqation 3.2 is based on the first harmonic of the Forier series exansion. The sqare-wave resonse of (Figre 3.1) consists of many freqency comonents: 4h () t π n 0 (( n+ ) ωt) sin 2 1 2n+ 1 (3.24) Therefore, it becomes obvios that the amlitde observed in the relay feedback resonse is contribted from mltile freqencies, ω ωˆ, 3ωˆ, 5ωˆ, etc. In theory, one can have a better estimate of the amlitde ratio by emloying more terms. An iterative rocedre is necessary if more than one term is emloyed (e.g. finding G(s) from the single-term soltion and inclding the higher freqency information, ω 3ωˆ, to find a new G(s) and the rocedre is reeated ntil G(s) converges). However, exerimental reslts show that the estimation of system arameters can be imroved sbstantially by imroving the exression for eriod of oscillation alone, as shown in the next section. Frthermore, for higher order systems, there is little incentive to imrove the exression for the amlitde by inclding more terms, since higher order harmonics (e.g. ω 3ωˆ or ω 5ωˆ ) are attenated by

10 32 Atotning of PID Controllers the rocess. If only one term is emloyed, then the eqations describing the amlitde ratio are exactly the same as Eqations 3.16, 3.18 and Parameter Estimation From the ongoing analysis, the rocedre for the evalation of the transfer fnction has the following stes: 1. Select model strctre. 2. Comte model arameters according to Table 3.1. Table 3.1 smmarizes the information reqired and the corresonding eqations to find the aroximate transfer fnction. Most of these eqation sets can be solved seqentially. Notice that if the imroved algorithm is sed, then better estimates of the ltimate gain and ltimate freqency can be calclated from the model. For model 2b, if some information is not known, then a different rocedre shold be emloyed. For examle, if K is not available, we can erform a second relay feedback test [11] or se a biased relay (Chaters 7 and 12) to find additional information. Nonetheless, the eqations noted in Table 3.1 are generally alicable regardless of the rocedre Examles Several examles are sed to illstrate the advantages of the imroved algorithm. Consider a first-order ls dead time system. Examle 3.2 FOPDT rocess 10s 16.5e G( s) 20s + 1 From a relay feedback exeriment with h 0.04 we have P ˆ and a If D and/or K are available, we can back-calclate τ. The τ vales calclated from Eqations 3.15 and 3.16 are τ 16.3 and resectively. The imroved algorithm (Eqation 3.21) gives a better estimate in τ, τ 19.97, by imroving the exression in the eriod of oscillation alone. The reslt from Eqation 3.21 is almost exact (the difference may have reslted from reading off a and P from the resonse crve). Figre 3.3 shows the mltilicative modeling errors, e ( ( ) ˆ( )) ˆ m Giω Giω Gi ( ω), for the transfer fnction Ĝ estimated from Eqations 3.15, 3.16 and The reslts show that the error e m is significantly less when τ is calclated from Eqation 3.21 alone. In the following examles, we assme K and D are known and the time constant τ for models 1 and 2a is obtained by taking the average of the vales calclated from the corresonding eqations for the case of the simle algorithm. Next, the effects of dead time on the estimation of the ltimate gain and ltimate freqency are also investigated. In the original ATV method, K ˆ is calclated from

11 Relay Feedback 33 Table 3.1. Eqations for different model strctres Model Simle algorithm Imroved algorithm Prior information Model I Model P Model 1 Model 2a Model 2b Eqations 3.11 and 3.12 Eqations 3.13 and 3.14 Eqations 3.15 and 3.16 Eqations 3.17 and 3.18 Eqations 3.19 and 3.20 None None Eqations 3.21 and 3.16 Eqations 3.22 and 3.18 Eqations 3.23 and 3.20 D or K D or K D and K Eqation 3.21 Eqation 3.16 Eqation 3.15 Figre 3.3. Mltilicative errors of an FOPDT system obtained from Eqations 3.15, 3.16 and 3.21 Eqation 3.2 and ω ˆ is derived from Eqation 3.1. In the roosed method, K and ω are back-calclated from the estimated transfer fnction Ĝ( s ). Again, this is shown in the following transfer fnction: Examle 3.3 Variable dead time Ds 16.5e G( s) 20s + 1

12 34 Atotning of PID Controllers The ercentage errors in K and ω are comared for these two methods over a range of dead time (D ). The reslts (Figre 3.4) show that the errors in K for the simle method are qite significant (5 20%). Frthermore, the error in ω is almost nil for the imroved method. Similar behavior can also be observed for a second-order lag with time-delay system. Examle 3.4 Second-order system with two neqal lags ( ) G s 37.3e Ds ( 7200s+ 1)( 2s+ 1) Ĝ s can be achieved over a range of D ( D < 60). Again, imrovements can be made in finding the correct K and ω by sing a more accrate exression in the eriod of oscillation. Since the estimated transfer fnction is tyically emloyed in the analysis and design of a feedback control system, the imact of the modeling errors in closedloo erformance is evalated. A model-based controller, IMC, is emloyed to analyze the erformance. One of the advantages of the IMC is that we can secify the desired trajectory in the design. Figre 3.6 comares the SP resonses of IMC when different models G ˆ ' are emloyed in the design of the controllers. Consider Figre 3.5 shows that a better estimation of ( ) Figre 3.4. Percentage errors in K and ω for the FOPDT system over a range of dead time D

13 Relay Feedback 35 the FOPDT system 10s 16.5e G( s) 20s+ 1 Ĝ s from the simle algorithm, tends to be more slggish than the desired trajectory (Figre 3.6). The roosed method imroves this sitation, as shown in Figre 3.6. Desite the fact that a tighter resonse can be achieved by shortening the closed-loo time constant nder modeling errors, one has to realize that the vale of a model-based controller is that one can foresee the closed-loo resonse. In other words, a good model always hels. Generally, the roosed method imroves the estimation in G(s) at the nominal The SP resonse of the control system, designed according to ( ) condition (with erfect knowledge of K and D). The robstness with resect to errors in the dead time is investigated. Since the imroved method calclates K and ω by finding the transfer fnction Ĝ( s ) first, followed by solving the corresonding eqations for them, it is more sensitive to the errors in the dead time than the original method. Let s take another FOPDT system as an examle. Examle 3.5 Error in the observed dead time s 16.5e G( s) 20s+ 1 Figre 3.5. Percentage errors in K and ω for a second-order ls dead time system over a range of delay time D

14 36 Atotning of PID Controllers Figre 3.6. SP resonses of IMC designed according to the estimated transfer fnctions Ĝ(s) (the closed-loo constant is 20 for the desired trajectory) Figre 3.7 shows the estimate of K and ω for both methods when the ercentage errors in dead time range from 50% to 50%. Desite the fact that the errors in K and ω are less for the imroved method over a reasonable range of errors in dead time, it is more sensitive to the error in D. Therefore, care shold be taken in reading off the dead time from the initial resonses or the time to the eak amlitde. 3.3 Aroximate Transfer Fnctions: Time-domain Modeling U to this oint, the model identification is based on the freqency domain aroach, which is based on the describing fnctions. A method to derive FOPDTtye systems was roosed by Wang et al. [12] sing a single relay test. In a searate attemt, Majhi and Atherton [3] roosed a techniqe to identify lant arameters, bt the method needs a correct initial gess and convergence is not garanteed. Kaya and Atherton [14] describe another method (A-locs) to identify loworder rocess arameters from relay atotning. Panda and Y [15] develo analytical models to reresent relay resonses rodced by different systems. The relay ott consists of a series of ste changes in manilated variables (with oosite sign). Hence, the stabilized ott is a sm of infinite terms of ste resonses de to those ste changes. For systems with dead time D, the actal relay ott

15 Relay Feedback 37 Figre 3.7. Percentage errors in K and ω for a first-order system over a range of variation in the dead time lags behind the int by a time nit D. The ints and otts can be synchronized by shifting the ott forward in time by an amont D, as shown in Figre 3.8B, and, in doing this, the dead time D can be eliminated from the exression for relay resonses, as will be shown later. The shifted version of a tyical relay feedback resonse rovides the basis for the derivation. It is assmed that the relay resonse is formed by n-nmber of ste changes, of oosite directions ( ± ), in int. The switching eriod for each ste change is P 2, excet for the initial ste change. In Figre 3.9, in the first interval, as time changes from t 0 to t D, the resonse y 1 is rodced de to the first ste change 1. Again, in the second interval, time rogressing from D to D+ P /2, resonse y 2 reslts de to the combined effects of ste changes 1 and 2. Similarly, the effect of 1, 2 and 3 rodces y 3 dring the third time interval ( D+ P /2 to D+ P ). Two half eriods ( P /2) are of secial interest in Figre 3.9. The even vales of n reslt in descending half eriod y 2n, and the odd vales of n formlate the ascending half eriods y 2n + 1. It is interesting to note that the

16 38 Atotning of PID Controllers Figre 3.8. Schematic reresentation of the shifted version of relay feedback resonse for the develoment of their analytical exressions: (A) original relay feedback resonses and (B) ott y shifted by D Figre 3.9. Shifted version of relay int and ott y resonse of a tyical SOPDT system

17 Relay Feedback 39 generalized resonse term y n slowly forms a convergent series. Let s se a second-order system to illstrate the derivation as they are rich in system dynamics Derivation for a Second-order Overdamed System The transfer fnction of an SOPDT system with a daming coefficient greater than one can be exressed as G( s ) Ds Ke ( τ1s+ 1)( τ2s+ 1), where K is the steady state gain, τ 1 and τ 2 are rocess time constants with τ1 > τ2, and D is the dead time. The original ste resonse of an overdamed SOPDT can be given by where a 1 and b 1 are given by y K ae + be a [1 ( t D )/ τ1 ( t D )/ τ2] τ and b1 τ1 τ2 τ1 τ 2 Under the shifted version (Figre 3.8B), the first segment of the relay resonse y 1 is simly the ste resonse withot dead time in the time index: y K ae + be 1 1 t t 1 1 τ / τ1 / τ 2 (3.25) At the second instant, the time is reset to zero at the initial oint. The ste resonse (relay ott) is given by (i.e. introdcing a time shift by D amont in Eqation 3.22) t+ D t+ D t t y K 1 ae 1 τ + be 1 τ 2K 1 ae 1 τ + be 1 τ Here, the first term reresents the effect of the first ste change (occrred at D time earlier) and the second term shows the effect of the second ste int, switching to the oosite direction. The above eqation can be simlified to t D t D y [ ] K 1 2 ae τ 1 e τ 2 be τ 1 e τ + 2 (3.26) The relay resonse at the third interval is the reslt of three ste changes, lags by an amont D+ P /2 from int. After introdcing a time shift of D+ P /2 in Eqation 3.22, the net effect becomes y K ae be t+ D+ P /2 t+ D+ P / τ + 1 τ t+ P /2 t+ P /2 t t 2 1 ae τ + b1e τ ae 1 τ + be 1 τ which can be simlified frther as 1 One may ski the derivation in Section and refer directly to Tables 3.2 and 3.3 for the reslts.

18 40 Atotning of PID Controllers y3 K + ae 1 e e + + be 1 [ 1 2 2] τ 1 τ 1 2 2τ 1 2 τ 2 D+ P /2 P e τ 2 2e 2τ t D+ P /2 P t (3.27) It can be seen that the terms in the right-hand side (RHS) of the above eqation are slowly forming a series. With the rogress of time, the resonse becomes stabilized and the general exression for the nth term can be described as t D+ ( n 2) P /2 y [ ] 1 1 n K + ae 1 τ e τ ( n 2) P ( n 1) P P 2e 2τ1 + 2e 2τ1 + 2e 2τ1 2 + be e e + e t D+ ( n 2) P /2 ( n 2) P ( n 1) P τ τ 2 τ 2 τ P + 2e 2τ 2 2 (3.28) The RHS of Eqation 3.28 has three arts, and each art consists of an infinite series, F 1, F 2 and F 3. If n is odd, the first series F 1 is simly The second series becomes: e ν t t y 1 2 n K F1 - ae 1 τ F2 be 1 τ + F3 F 1 [ ] 1 F e r r r r r D τ n 2 n + 2 n 2 n where r and ν1 P 2τ1. This above series is convergent and can be t into the following form (note that terms are rearranged from the back side of the above exression): 2 D/ τ n ( e ) 2( 1 ) F lim r + r + r r + n r r r r 1+ e P /2τ1

19 Relay Feedback 41 In a similar way, the F 3 of the RHS of Eqation 3.28 can also be simlified. Ultimately, the resonse can be given by t t 2 2 yn K 1 ae 1 τ1 be 1 τ + 2 P /2 P /2 1+ e τ1 1+ e τ2 (3.29) This reresents the ascending resonse (n is odd). Since this resonse is dissymmetric, the general form can be emloyed as t t 2 2 n yn K 1 ae 1 τ1 be 1 τ + 2 /2 /2 ( 1 P ) P (3.30) 1+ e τ1 1+ e τ2 One can refer to Panda and Y [15] for the derivations for critically damed and nderdamed SOPDT systems, as well as for high-order systems Reslts Different tyes of transfer fnction are considered, and the analytical exressions for their relay feedback ott resonse are develoed following the above rocedre. Table 3.2 gives a list of first-, second-, and third-order ls dead time rocesses and their corresonding mathematical exressions for the stabilized relay feedback ott resonses. These eqations y n denote the ward or ascending trend (or sometimes, crves in the lower art of midline for higher order systems) of relay feedback ott (while time t changes from 0 to P /2). The downward or descending trend can be obtained by reversing the sign of the ott ( yn ). In Table 3.2, the individal exressions, for relay feedback resonses of first-, second- and third-order systems contain terms similar to those of the corresonding eqations for the ste resonses, excet that they differ only in weighting factor ( 2/1 ( + e P /2τ ) ). If we comare the terms of the exressions of the relay feedback resonse with those of ste resonse of a rocess, we see that they differ by a weighting factor of 2/1 ( + e P /2τ ). For an FOPDT system, the resonse starts ( t 0 ) from the minimal oint, at y a, and ends ( t P /2) at the maximal oint, at y a. Also note that, for an nstable FOPDT system, stable limit cycles can occr only if D / τ < ln(2). For the lead/lag second-order system (No. 6 in Table 3.2), the exression is alicable to systems with left-half lane ( τ 3 > 0 ) or right-half lane ( τ 3 < 0 ) zero. Analytical exressions of relay feedback ott resonses for higher order systems are resented in Table 3.3. They are of mch interest becase, when we see, for examle, the exression for fifth-order rocess, the eqation contains mainly five terms (excet 1 ) and each of these terms reresents corresonding lower order rocesses. The first term inside the third bracket of the first line/row aears to be for an FOPDT. The second term (having two terms inside the first bracket) is for an SOPDT (critically damed). The third term (having three terms inside the first bracket) is for a third-order rocess. The terms in the second row/line (having

20 42 Atotning of PID Controllers Table 3.2. Time resonse yn of relay feedback for FOPDT, SOPDT and third-order rocesses

21 Table 3.3. Time resonse yn of relay feedback for forth and high-order rocesses Relay Feedback 43

22 44 Atotning of PID Controllers for terms inside) are for a forth-order rocess. In the third or last row/line there are five terms for a fifth-order rocess. Hence, the nmber of terms (size of the series) for a articlar order of rocess is rhythmic. These tables are similar to the tables of inverse Lalace transform and will hel in finding an eqation for relay feedback resonses Validation Two kinds of resonse can be observed in the analytical exressions in Tables 3.2 and 3.3. These resonses are tablated in Figre Systems with serial nmbers 1 and 2 in Table 3.2 always rodce a monotonic resonse, where, at t 0, the resonse from the model starts at the lowermost (or ermost) oint (A or B) and, at t P /2, it ends at the other extreme oint (B or C). Processes with serial nmbers 3, 4, 5 and 6 in Table 3.2 may give a non-monotonic resonse, as shown in Figre The third tye is higher order systems withot dead time (i.e. n 3 ). For this tye of system, this vale occrs at the mid-oint of the half eriod, as also shown in Figre Figre 3.10 shows the correctness of the derived mathematical models. If the relay height is other than nity, then the model for the relay ott resonse will be jst mltilied by actal vale of relay height h. 3.4 Conclsion In this chater the relay feedback test is introdced and the stes reqired to erform the exeriment are also given. It can be carried ot with or withot a commercial atotner. Once yo have obtained the information on the ltimate freqency, the controller settings can be decided sing the original or modified Ziegler Nichols methods. Yo can also go a ste frther to find an aroriate transfer fnction for the rocess. This can be sefl for imlementing MPC or dead time comensator (Smith redictor). Better aroximation can be achieved sing the imroved algorithm. Finding transfer fnctions sing the biased relay ls hysteresis was discssed by Wang et al. [12]. Finally, analytical exressions for relay feedback resonses are tablated for different tyes of rocess. This can be sefl if the model strctre is known.

23 Relay Feedback 45 Transfer fnction Resonses Transfer fnction Resonses 5s 1.0e 2s + 1 1s 1.0( 1 s) e ( 3s+ 1)( 2s+ 1) 1s 1.0e 1 2s ( 2s + 1) 3 1s 1.0e ( 4 + 1)( 3 + 1)( 2 + 1) ( 8s 1)( s 1) 1.0e 5s ( 2s + 1) ( 2s + 1) 4 10S 1.0e s + 0.4s+ 1 ( 2s + 1) 5 Figre Validation of analytical exressions for relay ott of different systems: solid line is relay ott and dashed line is model ott. (A denotes starting of one cycle that ends at B. Again from B next cycle starts and ends at C).

24 46 Atotning of PID Controllers 3.5 References 1. Åström KJ, Hägglnd T. Atomatic tning of simle reglators with secifications on hase and amlitde margins. Atomatica 1984;20: Lyben WL. Derivation of transfer fnctions for highly nonlinear distillation colmns. Ind. Eng. Chem. Res. 1987;26: Seborg DE, Edgar TF, Mellicham DA. Process dynamics and control. 2nd ed. New York: Wiley; Lyben WL, Lyben ML. Essentials of rocess control. New York: McGraw-Hill; Wang QG, Lee TH, Lin C. Relay feedback. London: Sringer-Verlag; Ogata K. Modern control engineering. Prentice-Hall: Englewood Cliffs; Tyres BD, Lyben WL. Tning PI controllers for integrator/dead time rocesses. Ind. Eng. Chem. Res. 1992;31: Friman M, Waller KV. Atotning of mltiloo control systems. Ind. Eng. Chem. Res. 1994;33: Wood RK, Berry MW. Terminal comosition control of a binary distillation colmn. Chem. Eng. Sci. 1973;28: Chang RC, Shen SH, Y CC. Derivation of transfer fnction from relay feedback systems. Ind. Eng. Chem. Res. 1992;31: Li W, Eskinat E, Lyben WL. An imroved atotne identification method. Ind. Eng. Chem. Res. 1991;30: Wang QG, Hang CC, Zo B. Low-order modeling from relay feedback. Ind. Eng. Chem. Res. 1997;36: Majhi S, Atherton DP. Ato-tning and controller design for rocesses with small time delays. IEE Proc. Control Theory Al. 1999;146(3): Kaya I, Atherton DP. Parameter estimation from relay ato-tning with asymmetric limit cycle data. Process Control 2001;11: Panda RC, Y CC. Analytical exressions for relay feedback resonses. J. Process Control 2003;13:48.

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