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Alternative Identification Method using Biased Relay Feedback
Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Available online at www.sciencedirect.com Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Proceedings,16th IFAC Symposium on Information Control in Manufacturing Bergamo, Italy, JuneProblems 11-13, 2018 Information Control in Manufacturing Bergamo, Italy, JuneProblems 11-13, 2018 Bergamo, Italy, June 11-13, 2018
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IFAC PapersOnLine 51-11 (2018) 891–896 Alternative Identification Method using Biased Relay Feedback Alternative Identification Method using Biased Relay Feedback Alternative Identification Method using Biased Relay Feedback M. Hofreiter Alternative using Alternative Identification Identification Method Method using Biased Biased Relay Relay Feedback Feedback M. Hofreiter
M. Hofreiter M. Hofreiter M. Hofreiter Czech Technical University in Prague, Department of Instrumentation and Control Engineering, Department of Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty ofofMechanical Engineering, Czech Republic, (e-mail: [email protected]) Department Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty of Mechanical Engineering, Czech Republic, (e-mail: [email protected]) Department of Mechanical Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty of Engineering, Czech Republic, (e-mail: [email protected]) Department of Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty of Mechanical Engineering, Czech Republic, (e-mail: [email protected]) Faculty of Mechanical Czech Abstract: This paper presents aEngineering, method which can Republic, be used to(e-mail: [email protected]) dynamical systems using a biased Abstract: This paper presents a method which can be used to identify dynamical systems using biased relay feedback for control purposes. The method is applicable for systems describable by a aatransfer Abstract: This paper presents a method which can be used to identify dynamical systems using biased relay feedback for control purposes. The method is applicable for systems describable by a transfer function if This therepaper are control sustained inmethod the biased-relay test. It describable enables to estimate up to Abstract: presents aoscillations methodThe which can be used to feedback identify dynamical systems using atransfer biased relay feedback for purposes. is applicable for systems by a Abstract: presents aoscillations method which can berelay usedtest. to feedback identify dynamical systems using a biased function if This therepaper are sustained in the biased-relay test. can It enables tofor estimate up to three frequency response points from a single biased These points be used fitting up relay feedback for control purposes. The method is applicable for systems describable by a transfer function if therefor arecontrol sustained oscillations in the biased-relay feedback test. can Itdescribable enables tofor estimate up to to relay feedback purposes. The method is applicable for systems by a transfer three frequency response points from a single biased relay test. These points be used fitting up to five parameters of process transfer function with various structures. HowIt toenables improve thefitting parameter function if there response area sustained oscillations in the biased-relay feedback test. tofor estimate up to three frequency points from a single biased relay test. These points can be used up to function if of there area sustained oscillations in the biased-relay feedback test. Ittoenables to estimate up to to five parameters of process using transfer function with various structures. How improve the enables parameter estimation process a transport delay shown as well. points The proposed technique three frequency response pointstransfer from a function single biased relay test.structures. These can used forthe fitting up five parameters of a models process withis various How to be improve parameter three frequency response points from a single biased relay test. These points can be used for fitting up estimation of process models using a transport delay is shown as well. The proposed technique enables to improve theofestimate of a static gain for a plantwith under a staticstructures. load disturbance. applicability of the five parameters of a process transfer function various How to The improve the parameter estimation process models using is shown as well. The proposed technique enables to five parameters of a process transfer function with various structures. How to The improve theLevitation”. parameter improve the estimate of a static gaina transport for a plantdelay under aa static load disturbance. applicability of the method is demonstrated on simulated examples and on laboratory apparatus called “Water estimation of process models using a transport delay is shown as well. The proposed technique enables to improveisthe estimate of on a static gain for a plant under a laboratory staticasload disturbance. Thetechnique applicability of the estimation of process models using a transport delay is shown well. The proposed enables to method demonstrated simulated examples and on a apparatus called “Water Levitation”. Matlab/Simulink programming environment wasand usedonfor all experiments. improve estimate of aonstatic gain for a plant under aa static load disturbance. The “Water applicability of the method isthe demonstrated simulated examples laboratory apparatus called Levitation”. improve the estimate of a static gain for a plant a static load disturbance. The applicability of the Matlab/Simulink programming environment was under used for all experiments. method is demonstrated on simulated examples and onfor a laboratory apparatus called “Water Levitation”. Matlab/Simulink programming environment was used all experiments. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. response, All rights Levitation”. reserved. method is demonstrated on simulated examples and on a laboratory apparatus called “Water Keywords: System identification, relay control, parameter estimation, frequency oscillation, Matlab/Simulink programming environment was used for all estimation, experiments. Keywords: System identification, relay control, parameter Matlab/Simulink programming environment was used for all experiments.frequency response, oscillation, transport delay. Keywords: System identification, relay control, parameter estimation, frequency response, oscillation, transport delay. Keywords: System identification, relay control, parameter estimation, frequency response, oscillation, transport delay. Keywords: System identification, relay control, parameter estimation, frequency response, oscillation, transport delay. transport delay. second test runs with an additional dead time. A parasitic 1. INTRODUCTION second an additional deadmultiple time. A parasitic relay andtest theruns FFT with algorithm can estimate points on a 1. INTRODUCTION second test runs with an additional dead time. A parasitic and the FFT algorithm can estimate multiple points on A relay feedback test is a simple technique, which can be relay 1. INTRODUCTION process frequency response from a single feedback test; seeaa second test runs with an additional dead time. A parasitic A relay feedback test is a simple technique, which can be relay and theruns FFTwith algorithm can estimate multiple points on second test an additional dead time. A parasitic 1. INTRODUCTION process frequency response from a single feedback test; see used for a process modelling and for a design and tuning of et and al. frequency (1997). possibility is using an attenuation A relay test is a simple which can of be Bi relay the FFTAnother algorithm can estimate multiple points onsee a 1. INTRODUCTION used for afeedback process modelling and afortechnique, a design and tuning process response from a single feedback test; relay and the FFT algorithm can estimate multiple points on a Bi et al. (1997). Another possibility is using an attenuation controllers. A closed loop where process is under a relay A relay test is a simple technique, which can be offrequency the exponential expression theusing Laplace transform, used for feedback a process modelling and afor a design and tuning of factor process response from a in single feedback test; see A relay feedback test is a simple technique, which can be controllers. A closed loop where process is under a relay Bi et al. (1997). Another possibility is an attenuation process frequency response from a single feedback test; see factor of the exponential expression in the Laplace transform, control is illustrated by the block diagram in Fig. 1, where w used for a process modelling and for a design and tuning of see Chidambaram and Sathe (2014). in Since the relay feedback controllers. A closed where a process under a relay et al. Another possibility isthe using an attenuation used foristhe aillustrated process modelling for a design and of control byloop the block in is Fig. 1,tuning where w Bi factor of (1997). the exponential expression Laplace transform, Bi et al. (1997). Another possibility is using an attenuation see Chidambaram and Sathe (2014). Since the relay feedback denotes desired yand thediagram variable, the controllers. A closedvariable, loop where a controlled processinisFig. under a urelay test belongs among the most popular methods used in control is illustrated by the block diagram 1, where w factor of the exponential expression in the Laplace transform, controllers. closedvariable, loop where process isvariable, under a eurelay denotes the A desired y thea controlled the factor see Chidambaram and the Sathe (2014). Since the relaytransform, feedback of the exponential expression in the Laplace test belongs among most popular methods used in manipulated variable, d the disturbance variable and the control is illustrated by the block diagram in Fig. 1, where w engineering applications for a closed-loop identification, denotes the desired variable, y the controlled variable, u the see Chidambaram and Sathe (2014). Since the relay feedback control error. is illustrated by block diagram inproposed Fig. 1,and where wa see manipulated variable, dthe the disturbance variable eusethe testChidambaram belongs applications among the for most popular methods used in and Sathe (2014). Since the relay feedback engineering a closed-loop identification, control Åström and Hägglund (1984) to denotes the desired variable, y the controlled variable, u the there are also applications several about relay autotuning manipulated variable, d the disturbance variable and e the the test belongs among monographs the for mosta popular methods used for in denotes the desired variable, y the controlled variable, u control error. Åström and Hägglund (1984) proposed to use a engineering closed-loop identification, test belongs among the most popular methods used in are also several monographs about relay autotuning for symmetric relay for tuning PID controllers instead thea there manipulated variable, d the disturbance variable andtoofeuse identification and control, e.g. Yu (1999), Liu and Gao control error. Åström and Hägglund (1984) proposed engineering applications for a closed-loop identification, manipulated variable, d the disturbance variable and e symmetric relay for tuning PID controllers instead of the there are also several monographs about relay autotuning for engineering applications for a closed-loop identification, identification and control, e.g. Yu (1999), Liu and Gao Ziegler-Nichols method, see Ziegler and Nichols (1943). The control error.relay Åström Hägglund (1984) proposed toofuse a there (2012), al. (2013), Chidambaram and autotuning Sathe (2014). symmetric for and tuning PID controllers instead the areLiu alsoetseveral monographs about relay for control error. Åström and Hägglund (1984) proposed to use Ziegler-Nichols method, Ziegler and Nichols (1943). The identification and control, e.g. Yu (1999), Liu and Gao areLiu alsoetseveral monographs about autotuning for (2012), al. (2013), Chidambaram and Sathe (2014). relay method enables tosee find ultimate gain and thea there symmetric relay for tuning PIDthe controllers instead of The The mentioned identification methods arerelay proposed for lowZiegler-Nichols method, see Ziegler and Nichols (1943). identification and control, e.g. Yu (1999), Liu and Gao symmetric relay for tuning PID controllers instead of relay method enables to find the ultimate gain and the (2012), Liu et al. (2013), Chidambaram and Sathe (2014). identification and control, e.g. Yu (1999), Liu and Gao The mentioned identification methods are proposed for lowultimate frequency like the Ziegler-Nichols method but in a Ziegler-Nichols method, to seefind Ziegler Nicholsgain (1943). The models. more complex models methods mostly relay method enables the and ultimate and in the (2012), Liu etFor al. (2013), Chidambaram and Sathe for (2014). Ziegler-Nichols method, see Ziegler-Nichols Ziegler and Nichols (1943). The ultimate frequency likewithout the method but a order The mentioned identification methods arethe proposed lowLiu etFor al. (2013), Chidambaram and Sathe (2014). order models. more complex models the methods mostly controllable manner prior information relay method enables to find the ultimate gainabout and thea (2012), require additional a prior knowledge of some parameters or ultimate frequency like the Ziegler-Nichols method but in The mentioned identification methods are proposed for lowrelay method enables to find the ultimate gain and the controllable manner without prior information about order models. For more complex models the methods mostly mentioned identification methodsof aresome proposed for lowrequire additional a prior knowledge parameters or process and inmanner a short experimental time. The ultimate ultimate frequency likewithout the Ziegler-Nichols method but gain inthe a The more relay feedback experiments. controllable prior information about order models. For more complex models the methods mostly ultimate frequency likeexperimental the can Ziegler-Nichols method but gain in a order process and in a frequency short time. The ultimate require additional a experiments. prior knowledge of the some parameters or models. For more complex models methods mostly more relay feedback and the ultimate be utilized for the estimation controllable manner without prior information about the process and in afrequency shortwithout experimental time. Thethe ultimate gain require additional a prior knowledge of some parameters or controllable manner prior information about the and the ultimate can be utilized for estimation more relay feedback experiments. knowledge of some parameters or of frequency response and fitting process and inona the shortprocess experimental time. The ultimate gain require y wadditional e a prior u andone thepoint ultimate frequency can be utilized for the estimation relay feedback experiments. process and inona the short experimental time. The ultimate gain more of one point process frequency response and fitting Relay Process y w feedback e u more relay experiments. model parameters. However, the knowledge of this frequency and the ultimate frequency can be utilized for the estimation of one point on frequency the process frequency response and fitting Relay Process y and theparameters. ultimate can be utilized for estimation w e model However, the knowledge of the thisparameters. frequency u point the estimation of tworesponse model of oneonly pointallows on the process frequency and fitting Relay Process y w e model parameters. However, the knowledge of this frequency u of one point on the process frequency response and fitting point only allows the estimation of two model parameters. Relay w e Process y u Therefore models with low number of parameters are model parameters. However, the knowledge of this frequency Relay Process point only allows However, the estimation of twoofmodel parameters. model parameters. knowledge this frequency Therefore models with lowthenumber parameters are predominantly usedthe forestimation modelling. itofis theparameters. first order point only allows ofMostly two of model Block diagram of a process under relay feedback. Therefore models with low number parameters are Fig. 1. point only allows the estimation of two model parameters. predominantly used for modelling. Mostly it is the first order 1. Block diagram of a process under relay feedback. time delayed model called the number FOTD model orthethe second Therefore models with low ofit parameters are Fig. predominantly used for modelling. Mostly is first order Fig. 1. Block diagram of a process under relay feedback. Therefore models with low of model, parameters are time delayed modelmodel calledcalled the number FOTD model or the second order time delayed theMostly SOTD which are predominantly used for modelling. it is or thethe first order Fig. 1. Block diagram of a process under relay feedback. time delayed model called the FOTD model second predominantly used for modelling. Mostly it is the first order order time delayed model called the SOTD model, which are Fig. 1. Block diagram2.ofBACKGROUND a process under relay feedback. sufficient modelling many processes. In time model calledof the FOTD model or the second 2. BACKGROUND orderdelayed time for delayed model called the industrial SOTD model, which are time delayed model called the FOTD model or the second sufficient for modelling of many industrial processes. In Let us consider a time invariant process described by the to obtain model thesemodel, simple models 2. BACKGROUND order time delayed modelparameters called the of SOTD which are sufficient for modelling of many industrial processes. In Let us consider a time invariant process described order time delayed model called the SOTD model, which are to obtain model parameters of these simple models 2. BACKGROUND where ωby is the the process frequency response function GP(jω), from the relay feedback test, Luyben (1987) estimated the sufficient for modelling of many industrial processes. In process Let us consider a time invariant process described by the 2. BACKGROUND order to obtain model parameters of these simple models (jω), where ω is the frequency response function G P sufficient for modelling of many industrial processes. In from the relay feedback test, Luyben (1987) estimated the angular frequency. Three points on the Nyquist frequency Let us consider a time invariant process described by the transport delay from the initial response and assumed that the order to obtain model parameters of these simple models (jω), where ω is the process frequency response function G P from the relay from feedback test, response Luyben (1987) estimated the Let us consider a time invariant process described the angular frequency. Three points on the Nyquist order to delay obtain model parameters of al. these simple models transport thepriori. initial and assumed that the characteristic of the process can be estimated fromfrequency a by single where ω is the process frequency response function G static gain is known a Shen et (1996) proposed to P(jω), from the relay feedback test, Luyben (1987) estimated the angular frequency. Three points on the Nyquist frequency transport delay from initial response and assumed that the the where is the frequency function GP(jω), characteristic ofexperiment theresponse processby can estimated from ω a single from relay feedback test, Luyben (1987) estimated static gain is known athe priori. Shen et al. (1996) proposed to process relay feedback so be called shifting method, angular frequency. points on estimated thethe Nyquist frequency use anthe asymmetric for estimation the static gain. In this transport delay fromrelay the initial response and assumed that the characteristic ofexperiment theThree process can be from a single static gain is known a priori. Shen et al. (1996) proposed to angular frequency. Three points on the Nyquist frequency relay feedback by so called the shifting method, transport delay from the initial response and assumed that the use an asymmetric relay for estimation the static gain. In this see (2016), (2017). The shifting for the process estimated from uses amethod, single manner twoispoints estimated on the Nyquist static knownare a priori. Shen et al. proposed toa characteristic relayHofreiter feedbackof experiment bycan so be called the method shifting use angain asymmetric relay for estimation the(1996) staticcurve gain. from In this of the process can be estimated from uses a single see Hofreiter (2016), (2017). The shifting method for static gain knownare a priori. Shen et al. (1996) proposed toa characteristic manner twoispoints estimated onestimation the Nyquist curve from this purpose a biased relay with hysteresis. The process under relay feedback experiment by so called the shifting method, single relay experiment. It allows of three model use an asymmetric relay for estimation the static gain. In this see Hofreiter (2016), (2017). The shifting method uses for manner two points are estimated on the Nyquist curve from a relay feedback experiment by so called the shifting method, this purpose a biased relay with hysteresis. The process under use an relay asymmetric relay for estimation the static gain. In this the single experiment. It allows estimation of three model relay feedback is depicted in Fig.shifting 1. TheThe frequency points Hofreiter (2016), (2017). The method uses for parameters a proportional system. in the case manner two of points are estimated onestimation theHowever, Nyquistofcurve from a see this purpose a biased relay with hysteresis. process under single relay experiment. It allows three model see Hofreiter (2016), (2017). The shifting method uses for the relay feedback is depicted in Fig. 1. The frequency points manner points are estimated on theHowever, Nyquist curve from parameters a proportional system. in the case a proportional system for angular this purpose a biased withinare hysteresis. The process under where ittwo isof necessary toIt use higher order of model and toa of single relay experiment. allows estimation three model the relay feedback is relay depicted Fig.estimated 1. The frequency points parameters of a proportional system. However, in the case this purpose a biased relay with hysteresis. The process under of a proportional system are estimated fortheangular single relay experiment. It allows estimation of three model where it is necessary to use higher order model and to =0, ω =2π/T , ω =4π/T , where T period frequencies ω the relay feedback is depicted in Fig. 1. The frequency points 0 1 system p 2 are estimated p p is estimate more parameters, finding other applicable parameters of a proportional system. However, in the case of a proportional for angular where it is necessary to use higher order model and to the relay feedback is depicted in Fig. 1. The frequency points =0, ω =2π/T , ω =4π/T , where T is the period frequencies ω 0 1 p 2 p p parameters of a proportional system. However, in the case estimate more parameters, findingis required. other applicable a astable oscillation in the ,biased-relay feedback These proportional system relationships for parameter estimation Liand et al. where ismore necessary to use higher order model to of ω2are =4π/Testimated Tpfor istest. theangular period frequencies ω0=0, ω1=2π/T p, where estimateit parameters, findingis other applicable proportional system are estimated for angular a astable oscillation in thep biased-relay feedback test. These where itsuggested is necessary to use higher order model to of relationships for parameter estimation required. Liand et the al. =0, ω =2π/T , ω =4π/T , where T is the period frequencies ω (1991) using two relay feedback tests where estimate more parameters, findingis required. other applicable of a stable oscillation in thepp, biased-relay feedback These relationships for parameter Li etthe al. frequencies ω22=4π/Tpp, where Tpp is test. the period ω00=0, ω11=2π/T estimate more parameters, finding other applicable (1991) suggested using two estimation relay feedback tests where a stable oscillation in the biased-relay feedback test. These relationships for parameter estimation is required. Li et the al. of (1991) suggested using two relay feedback tests where relationships for parameter estimation is required. Li et al. of a stable oscillation in the biased-relay feedback test. These (1991) suggested using two relay feedback tests where the 910 Copyright © 2018 IFAC (1991) suggested using two relayFederation feedbackoftests whereControl) the Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Automatic Copyright © 2018 IFAC 910 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 910Control. 10.1016/j.ifacol.2018.08.491 Copyright © 2018 IFAC 910 Copyright © 2018 IFAC 910
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GP(jω2), which were obtained by the shifting method, are shifted to lower frequencies., see Fig. 4. Due to the additional transport delay D the phase shift of the point GP(jω1) varies by ω1⋅D and the phase shift of the point GP(jω2) by ω2⋅D.
three frequency points can be estimated using the following formulas computed by a numerical integration. t +T p
GP ( jω0 ) = GP (0) = K P =
ò
y (τ ) dτ
t t +T p
ò
, t ³ tL
(1)
e
w
u (τ ) dτ
u
Relay
Delay
Process
y
t
t +T p
GP ( jω1 ) =
ò
y (τ ) e- jω1 dτ
ò
u (τ ) e- jω1 dτ
t t +T p
, t ³ tL
Fig. 3. Block diagram of a process under relay feedback with a transport delay.
(2)
Im
GP(0)
t
t +T p
GP ( jω2 ) =
ò
æ ö çç y (τ ) + y çæçt - T p ÷÷ö÷÷ e- jω2 dτ ÷ ÷ ç 2 ø÷÷ø èçç èç
ò
æ æ T öö çççu (τ ) + u çççt - p ÷÷÷÷÷÷ e- jω2 dτ çè çè 2 ø÷÷ø
t t +T p t
, t ³ tL
(3)
GP(jω1)
GP(jω1)
Im
GP(jω)
Fig. 4. The Nyquist curve of a process and the points obtained by the shifting method by means of a transport delay.
where tL is the time of the achievement of a stable oscillation and KP is the static gain. GP(jω2)
Re
GP(jω2)
GP(0)
The real and imaginary parts of the values GP(0), GP(jω1) and GP(jω2) can be used for estimation up to five model parameters. In some cases, it is possible to derive explicit formulas for fitting model parameters. In other cases estimated parameters including the transport delay can be obtained e.g. by minimizing of the criterion
Re
GP(jω) Fig. 2. The Nyquist curve of a process and the points obtained by the shifting method.
2
2
Kr = å (G ( jωi ) - M ( jωi )) ,
(4)
i=1
where M(jω) is the frequency transfer function of a model. One can use a numerical method for this purpose. The static gain K of a proportional system can be estimated directly by the relationship
This approach is applicable for various models described by a transfer function. Model parameters can be estimated by fitting the selected transfer function to the points given by points GP(0), GP(jω1) and GP(jω2), see Fig. 2. The knowledge of these three points can be used for estimation up to five model parameters from a single relay test. The method is less sensitive to noise if the integrals (1), (2) and (3) are calculated over multiple oscillation periods. The noise is also suppressed by means of a hysteresis in the relay. For modelling and control, the frequency range corresponding to the phase lag 0°-180° is very important. The disadvantage of the shifting method consists in a position of the frequency point GP(jω2), because the phase lag for the frequency ω2 is greater than 180°, see Fig. 2. Therefore in the next section it is shown how to improve the position of the frequency points by means of a delay or an integrator when using the shifting method.
K = G ( 0 ) = M ( 0) .
(5)
If we know the transport delay from the initial response of the system then the other parameters can be estimated using the least square method. Via inserting the additional transport delay D into the closed loop circuit (see Fig. 3), two new points on the Nyquist curve from each relay test can be obtained for the chosen value of D. This way it is possible to find multiple points on the process frequency response. Li et al. (1991) used a similar procedure but without the shifting method and therefore they received only one new point on the Nyquist curve from each relay test. Since the frequency response points are calculated by formulas (2) and (3), the presence of a static load disturbance with a magnitude of dA does not have any influence on the calculation as it holds
2. SHIFTING METHOD WITH ADDITIONAL TRANSPORT DELAY To receive a better position of the points GP(jω1) and GP(jω2) we can insert an additional chosen transport delay into the closed loop circuit, see Fig. 3 Then the points GP(jω1) and
t +T p
ò t
911
- jωi
dA ⋅e
t +T p
dτ = d A
ò t
e- jωi dτ , i = 1, 2 .
(6)
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Hence the points GP(jω1) and GP(jω2) may be also used for the static gain estimation. This approach can be used for various LTI (Linear Time Invariant) isochronic and anisochronic models because the positions of the estimated points are calculated by the formulas (1), (2) and (3) which do not depend on a model structure.
æ ö æ ö÷ 1 ÷÷- arg (G ( jω ))÷÷÷ ççç arg ççç l ÷ ç çç a jω 2 + a ⋅ jω + 1÷÷÷ ÷÷ è 2( l) ø 1 ççç 2 l 1 ÷÷ Td = çå ÷÷ (15) ωl 2 çç l =1 ÷÷ çç ÷÷ çç ÷÷ çè ø
Example #1 A process with the transfer function (7) is controlled by the asymmetrical relay, see Fig. 1. The time courses of the manipulated variable u and the controlled variable y are shown in Fig. 6 provided, that the process was initially in a steady state. 4 u,y
3. SIMULATED EXAMPLES To demonstrate the introduced method we utilize the following three process transfer functions, which are taken from Berner et. al. (2016). 1 , (7) P1 ( s ) = ( s + 1)(0.1s + 1)(0.01s + 1)(0.001s + 1) P2 ( s ) = P3 ( s ) =
1 4
( s + 1) 1
2
(0.05s + 1)
,
(8)
e-s ,
(9)
3 1 -1 -2
uB
e
2 ⋅ ω12
a1 =
1
ω1
K = G ( 0 ) = M ( 0) ,
(12)
æ ö÷ 4 K P2 1 çç K P2 ÷ çç + 3÷÷ , 2 2 ÷÷ 3 çç G ( jω ) G ( jω1 ) ÷ø è 2
(13)
K P2 G ( jω1 )
2
(
- 1- a2 ⋅ ω12
2
3
4 t [s]
5
G ( jω0 ) = G (0) = 1 ,
(17)
G ( jω1 ) = -0.055 - 0.063 j ,
(18)
G ( jω2 ) = -0.026 - 0.099 j ,
(19)
where
ω1 =
2
)
,
2π = 8.84 rad⋅s-1, Tp
ω2 = 2 ⋅ ω1 = 17.68 rad⋅s-1.
The parameters K, a2, a1, Td of the SODT model can be calculated from the estimated points GP(0), GP(jω1) and GP(jω2) by the following explicit formulas
a2 =
1
From these time courses we can determine the period of stable oscillation Tp (see Fig. 6) and by the shifting method utilizing formulas (1), (2), (3), we estimate the points G(0), G(jω1) and G(jω2) on the Nyquist curve from one relay feedback experiment. (16) T p = 0.711 s
Fig. 5 The static characteristic of an asymmetrical relay with hysteresis.
1
0
Fig. 6. The time courses of the relay output u and the process output y
u uA
�A
y
0
where K, a2, a1, Td are estimated parameters. In all the simulating examples the asymmetrical relay with a hysteresis has following parameters, see Fig. 5: (11) uA=2, uB=-1, εA=0.1, εB=-0.1
0
Tp
u
2
where s is the complex variable in L-transform. The first process is lag dominated, the second is balanced and the third is delay dominated. Assume that all these processes can be described by a SOTD model in the form K (10) M (s) = e-Td ⋅s , a2 s + a1s + 1
�B
893
(20) (21)
In Fig. 7 the estimated points G(0), G(jω1) and G(jω2) are depicted in the Nyquist diagram for the transfer function P1(s). From these points we can estimate up to five model parameters. If we want to obtain frequency responses for lower frequencies we can use a relay feedback with a transport delay or with an integrator (see Fig. 3). In this example the additional transport delay is chosen as D = 3.5 s. (22) The time courses of the relay output u and the output y of the process P1(s) are depicted in Fig. 8. From it follows that the period of the stable oscillation (23) T p = 8.87 s.
(14)
912
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0.2 0 -0.2
Milan Hofreiter / IFAC PapersOnLine 51-11 (2018) 891–896
Im
G(jω2)
0.2
G(0) Re
G(jω1)
-0.4
P1(jω)
-0.6 0
0.5
4 u,y 3
1
0 -1 1
20
30
40 t [s]
50
Fig. 8. The time courses of the relay output u and the process output y for the relay feedback with the additional transport delay.
4 u,y 3
Utilizing the formulas (1), (2), (3) and the observed time courses of u and y we can calculate G ( jω0 ) = G (0) = 1 ,
(24)
G ( jω1 ) = 0.625 - 0.522 j ,
(25)
G ( jω2 ) = 0.252 - 0.513 j ,
(26)
2
2π = 0.71 rad⋅s-1, Tp
ω2 = 2 ⋅ ω1 = 1.42 rad⋅s-1.
Tp
u
y
1 0 -1 -2 0
100 t [s] Fig. 10. The time courses of the relay output u and the process output y for the relay feedback with the additional transport delay.
where
ω1 =
M1(jω)
G(jω1)
Example #2 A process with the transfer function (8) is controlled by the asymmetrical relay with the additional transport delay, see Fig. 3. The time courses of the relay output u and the output y of the process P2(s) are shown in Fig. 10, provided that the process was initially in a steady state and the additional transport delay D = 3.6 s. (30) From the time courses u, y it follows that the period of stable oscillation (31) T p = 14.9 s.
Tp
u
y
0
G(jω2)
-0.8 -0.5 0 0.5 1 1.5 Fig. 9. The Nyquist diagram for the transfer functions P1(s), M1(s) and the estimated points G(0), G(jω1), G(jω2).
1.5
1
-2
P1(jω)
-0.6
Fig. 7. The Nyquist diagram for the transfer function P1(s) and the estimated points G(0), G(jω1), G(jω2).
2
Re
-0.2
-0.4
-0.8-0.5
G(0)
Im
0
(27) (28)
20
40
60
80
Utilizing the formulas (1), (2), (3) and the observed time courses of u and y we can calculate
The model transfer function estimated from the points G(0), G(jω1) and G(jω2) is 1 M1 ( s ) = e-0.033s . (29) 0.073s 2 + 1.083s + 1 The position of these points together with the Nyquist diagram of the transfer functions P1(s) and M1(s) are shown in Fig. 9. This figure shows a very good conformity of frequency responses of the identified process P1(s) and the estimated model M1(s). The example illustrates that from a single asymmetrical relay feedback test one can obtain three points of the frequency response by the shifting method. If we use an additional transport delay then we can obtain these points for lower frequencies. Therefore, from two relay tests we can receive five frequency points (the static gain is obtained in each relay feedback test).
G ( jω0 ) = G (0) = 1 ,
(32)
G ( jω1 ) = -0.018 - 0.721 j ,
(33)
G ( jω2 ) = -0.321- 0.114 j ,
(34)
where
ω1 =
2π = 0.42 rad⋅s-1, Tp
ω2 = 2 ⋅ ω1 = 0.84 rad⋅s-1.
(35) (36)
The model transfer function estimated from the points G(0), G(jω1) and G(jω2) is 1 M 2 (s) = e-0.808s . (37) 2 3.203s + 3.128s + 1
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Milan Hofreiter / IFAC PapersOnLine 51-11 (2018) 891–896
ω2 = 2 ⋅ ω1 = 2.87 rad⋅s-1.
The position of these points together with the Nyquist diagram of the transfer functions P2(s) and M2(s) are shown in Fig. 11. This figure shows a very good conformity of frequency responses of the identified process P2(s) and the estimated model M2(s). 0.2
Re
G(jω2)
-0.2 -0.4 -0.6
-1
P2(jω)
G(jω1)
-0.8 -0.4
-0.2
M2(jω) 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 11. The Nyquist diagram for the transfer functions P2(s), M2(s) and the estimated points G(0), G(jω1), G(jω2). Example #3 A process with the transfer function (9) is controlled by the asymmetrical relay with the additional transport delay, see Fig. 3. The time courses of the relay output u and the output y of the process P3(s) are shown in Fig. 12, provided that the process was initially in a steady state and the additional transport delay D = 1.1 s. (38) From the time courses u, y it follows that the period of the stable oscillation T p = 4.38 s. (39) 4 u,y 3
y u
1 0 -1 5
10
15
20
25 t [s] 30
Fig. 12. The time courses of the relay output u and the process output y for the relay feedback with the additional transport delay.
(40)
G ( jω1 ) = -0.007 - 0.995 j ,
(41)
G ( jω2 ) = -0.980 - 0.012 j ,
(42)
stand water-flow pump
2π = 1.43 rad⋅s-1, Tp
ball jet reservoir
where
ω1 =
4. LABORATORY EXPERIMENT The laboratory apparatus “Water Levitation” consists of a water reservoir, a pump, a jet and an ultrasonic sensor, see Fig. 14. The pump sucks water from the reservoir and pumps water into the jet. The water streams out from the jet and lifts up the ball. The flow from the jet depends on the actual power of the pump which is controlled by the manipulated variable u. The controlled variable y (the position of a ball) is measured by the sensor. The task is to estimate by the shifting method the parameters of model (10) describing the relationship between the manipulated variable u and the controlled variable y around the working point. The time courses of y and u observed during the relay feedback test on the laboratory apparatus are depicted in Fig. 15. This test was realised with the additional transport delay (D=1.2s) in the relay feedback, see Fig. 3. sensor
Utilizing the formulas (1), (2), (3) and the observed time courses of u and y we can calculate G ( jω0 ) = G (0) = 1 ,
Im P3(jω), M3(jω) 1 0.8 0.6 0.4 G(0) 0.2 0 -0.2 Re -0.4 ) G(jω -0.6 2 -0.8 G(jω1) -1 -1 -0.8 -0.6-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fig. 13. The Nyquist diagram for the transfer functions P3(s), M3(s) and the estimated points G(0), G(jω1), G(jω2).
2
-20
(44)
The model transfer function estimated from the points G(0), G(jω1) and G(jω2) is 1 M 3 (s) = e-s . (45) 2 0.00252s + 0.1s + 1 The position of these points together with the Nyquist diagram of the transfer functions P3(s) and M3(s) are shown in Fig. 13. The process transfer function P3(s) is roughly the same as the estimated model transfer function M3(s). Therefore, the Nyquist diagrams of P3(s) and M3(s) look like a single line.
G(0)
Im
0
895
Fig. 14. The laboratory model “Water Levitation”
(43)
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Milan Hofreiter / IFAC PapersOnLine 51-11 (2018) 891–896
response points from each relay test than the usual approach. The method is also applicable for integrating systems where only frequency response points G(jω1) and G(jω2) are used for fitting model parameters.
u[V],y[cm]
25 20
y
15
Acknowledgment
u
10 5 0 -5
0 5 10 15 20 25 30 35 40 Fig. 15 The observed input u and the output y.
45
This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS16/210/OHK2/3T/12 and by the Institutional Resources of CTU for research (RVO12000).
50
REFERENCES Åström, K. J. and Hägglund, T. (1984). Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica, 20 (5), pp. 645-651. Berner, J., Hägglund, T. and Åström, K. J. (2016). Improved relay autotuning using normalized time delay. American Control Conference (ACC), IEEE, pp. 1869-1875. Bi Q., Wang Q. G. and Hang C. C. (1997). Relay-based estimation of multiple points on process frequency response. Automatica, 33 (9), pp. 1753-1757. Chidambaram M. and Sathe V. (2014). Relay Autotuning for Identification and Control. Cambridge University Press, Cambridge. Hofreiter, M. (2016). Shifting method for relay feedback identification. IFAC-PapersOnLine, 49 (12), pp. 19331938. Hofreiter, M. (2017). Biased-relay feedback identification for time delay systems, IFAC-PapersOnLine, 50 (1), pp. 1462-14625. Li W., Eskinat E. and Luyben W. L. (1991). An improved auto-tune identification method, Ind. Eng. Chem. Res. 30 (7), pp. 1530-1541. Liu T. and Gao F. (2012). Industrial process identification and control Design: Step-test and relay-experiment-based Method. Advances in Industrial Control. Springer-Verlag, London. Liu T., Wang Q. G. and Huang H. P. (2013). A tutorial review on process identification from step or relay feedback test. Journal of Process Control, 23 (10), pp. 1597-1623. Luyben, W. L. (1987). Derivation of transfer functions for highly nonlinear distillation columns. Ind. Eng. Chem. Res. 26 (12), pp. 2490-2495. Shen, S. H., Wu, J. S. and Yu C. C. (1996). Use of biasedrelay feedback for system identification. AIChE. J. 42 (4), pp. 1174-1180. Yu C. C. (1999). Autotuning of PID Controllers, ch. 2 and 3. Springer-Verlag, London. Ziegler J. G. and Nichols N. B. (1943). Optimum settings for automatic controllers. Trans. ASME, vol. 65, pp. 433-444.
From the courses y and u and formulas (1), (2) and (3) it was determined T p = 4.7 s, (46)
ω1 =
2π = 1.34 rad⋅s-1, Tp
(47)
ω2 = 2 ⋅ ω1 = 2.68 rad⋅s-1,
(48)
-1
G (0) = 2 cm⋅V ,
(49)
G ( jω1 ) = 0.552 -1.723 j ,
(50)
G ( jω2 ) = -0.618 -1.161 j .
(51)
Using relationships (49), (50) and (51) the SOTD model is 2 M (s) = ⋅ e-0.315 s . (52) 2 0.105s + 0.561s + 1 Fig. 16 shows a good conformity between the step responses of the identified process and the model (52). u[V], yP[cm], yM[cm]
25 20
yM yP u
15 10 5 0 0
5
10
15
20
25
30
35
40
45
50
Fig. 16. The process step response yP and the model step response yM to the step u. 5. CONCLUSIONS The combination of the shifting method and the additional delay in the relay feedback loop enables to obtain three frequency response points for the phase lag in the range from 0 to 180° using a single relay test. These points are more suitable for fitting the model parameters and they can be used for fitting up to five parameters of a model transfer function with various structures. If we want to obtain more frequency response points then we can select different values of the additional transport delay for each relay feedback test. This approach makes it possible to estimate twice more frequency 915
ScienceDirect
IFAC PapersOnLine 51-11 (2018) 891–896 Alternative Identification Method using Biased Relay Feedback Alternative Identification Method using Biased Relay Feedback Alternative Identification Method using Biased Relay Feedback M. Hofreiter Alternative using Alternative Identification Identification Method Method using Biased Biased Relay Relay Feedback Feedback M. Hofreiter
M. Hofreiter M. Hofreiter M. Hofreiter Czech Technical University in Prague, Department of Instrumentation and Control Engineering, Department of Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty ofofMechanical Engineering, Czech Republic, (e-mail: [email protected]) Department Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty of Mechanical Engineering, Czech Republic, (e-mail: [email protected]) Department of Mechanical Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty of Engineering, Czech Republic, (e-mail: [email protected]) Department of Instrumentation and Control Engineering, Czech Technical University in Prague, Faculty of Mechanical Engineering, Czech Republic, (e-mail: [email protected]) Faculty of Mechanical Czech Abstract: This paper presents aEngineering, method which can Republic, be used to(e-mail: [email protected]) dynamical systems using a biased Abstract: This paper presents a method which can be used to identify dynamical systems using biased relay feedback for control purposes. The method is applicable for systems describable by a aatransfer Abstract: This paper presents a method which can be used to identify dynamical systems using biased relay feedback for control purposes. The method is applicable for systems describable by a transfer function if This therepaper are control sustained inmethod the biased-relay test. It describable enables to estimate up to Abstract: presents aoscillations methodThe which can be used to feedback identify dynamical systems using atransfer biased relay feedback for purposes. is applicable for systems by a Abstract: presents aoscillations method which can berelay usedtest. to feedback identify dynamical systems using a biased function if This therepaper are sustained in the biased-relay test. can It enables tofor estimate up to three frequency response points from a single biased These points be used fitting up relay feedback for control purposes. The method is applicable for systems describable by a transfer function if therefor arecontrol sustained oscillations in the biased-relay feedback test. can Itdescribable enables tofor estimate up to to relay feedback purposes. The method is applicable for systems by a transfer three frequency response points from a single biased relay test. These points be used fitting up to five parameters of process transfer function with various structures. HowIt toenables improve thefitting parameter function if there response area sustained oscillations in the biased-relay feedback test. tofor estimate up to three frequency points from a single biased relay test. These points can be used up to function if of there area sustained oscillations in the biased-relay feedback test. Ittoenables to estimate up to to five parameters of process using transfer function with various structures. How improve the enables parameter estimation process a transport delay shown as well. points The proposed technique three frequency response pointstransfer from a function single biased relay test.structures. These can used forthe fitting up five parameters of a models process withis various How to be improve parameter three frequency response points from a single biased relay test. These points can be used for fitting up estimation of process models using a transport delay is shown as well. The proposed technique enables to improve theofestimate of a static gain for a plantwith under a staticstructures. load disturbance. applicability of the five parameters of a process transfer function various How to The improve the parameter estimation process models using is shown as well. The proposed technique enables to five parameters of a process transfer function with various structures. How to The improve theLevitation”. parameter improve the estimate of a static gaina transport for a plantdelay under aa static load disturbance. applicability of the method is demonstrated on simulated examples and on laboratory apparatus called “Water estimation of process models using a transport delay is shown as well. The proposed technique enables to improveisthe estimate of on a static gain for a plant under a laboratory staticasload disturbance. Thetechnique applicability of the estimation of process models using a transport delay is shown well. The proposed enables to method demonstrated simulated examples and on a apparatus called “Water Levitation”. Matlab/Simulink programming environment wasand usedonfor all experiments. improve estimate of aonstatic gain for a plant under aa static load disturbance. The “Water applicability of the method isthe demonstrated simulated examples laboratory apparatus called Levitation”. improve the estimate of a static gain for a plant a static load disturbance. The applicability of the Matlab/Simulink programming environment was under used for all experiments. method is demonstrated on simulated examples and onfor a laboratory apparatus called “Water Levitation”. Matlab/Simulink programming environment was used all experiments. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. response, All rights Levitation”. reserved. method is demonstrated on simulated examples and on a laboratory apparatus called “Water Keywords: System identification, relay control, parameter estimation, frequency oscillation, Matlab/Simulink programming environment was used for all estimation, experiments. Keywords: System identification, relay control, parameter Matlab/Simulink programming environment was used for all experiments.frequency response, oscillation, transport delay. Keywords: System identification, relay control, parameter estimation, frequency response, oscillation, transport delay. Keywords: System identification, relay control, parameter estimation, frequency response, oscillation, transport delay. Keywords: System identification, relay control, parameter estimation, frequency response, oscillation, transport delay. transport delay. second test runs with an additional dead time. A parasitic 1. INTRODUCTION second an additional deadmultiple time. A parasitic relay andtest theruns FFT with algorithm can estimate points on a 1. INTRODUCTION second test runs with an additional dead time. A parasitic and the FFT algorithm can estimate multiple points on A relay feedback test is a simple technique, which can be relay 1. INTRODUCTION process frequency response from a single feedback test; seeaa second test runs with an additional dead time. A parasitic A relay feedback test is a simple technique, which can be relay and theruns FFTwith algorithm can estimate multiple points on second test an additional dead time. A parasitic 1. INTRODUCTION process frequency response from a single feedback test; see used for a process modelling and for a design and tuning of et and al. frequency (1997). possibility is using an attenuation A relay test is a simple which can of be Bi relay the FFTAnother algorithm can estimate multiple points onsee a 1. INTRODUCTION used for afeedback process modelling and afortechnique, a design and tuning process response from a single feedback test; relay and the FFT algorithm can estimate multiple points on a Bi et al. (1997). Another possibility is using an attenuation controllers. A closed loop where process is under a relay A relay test is a simple technique, which can be offrequency the exponential expression theusing Laplace transform, used for feedback a process modelling and afor a design and tuning of factor process response from a in single feedback test; see A relay feedback test is a simple technique, which can be controllers. A closed loop where process is under a relay Bi et al. (1997). Another possibility is an attenuation process frequency response from a single feedback test; see factor of the exponential expression in the Laplace transform, control is illustrated by the block diagram in Fig. 1, where w used for a process modelling and for a design and tuning of see Chidambaram and Sathe (2014). in Since the relay feedback controllers. A closed where a process under a relay et al. Another possibility isthe using an attenuation used foristhe aillustrated process modelling for a design and of control byloop the block in is Fig. 1,tuning where w Bi factor of (1997). the exponential expression Laplace transform, Bi et al. (1997). Another possibility is using an attenuation see Chidambaram and Sathe (2014). Since the relay feedback denotes desired yand thediagram variable, the controllers. A closedvariable, loop where a controlled processinisFig. under a urelay test belongs among the most popular methods used in control is illustrated by the block diagram 1, where w factor of the exponential expression in the Laplace transform, controllers. closedvariable, loop where process isvariable, under a eurelay denotes the A desired y thea controlled the factor see Chidambaram and the Sathe (2014). Since the relaytransform, feedback of the exponential expression in the Laplace test belongs among most popular methods used in manipulated variable, d the disturbance variable and the control is illustrated by the block diagram in Fig. 1, where w engineering applications for a closed-loop identification, denotes the desired variable, y the controlled variable, u the see Chidambaram and Sathe (2014). Since the relay feedback control error. is illustrated by block diagram inproposed Fig. 1,and where wa see manipulated variable, dthe the disturbance variable eusethe testChidambaram belongs applications among the for most popular methods used in and Sathe (2014). Since the relay feedback engineering a closed-loop identification, control Åström and Hägglund (1984) to denotes the desired variable, y the controlled variable, u the there are also applications several about relay autotuning manipulated variable, d the disturbance variable and e the the test belongs among monographs the for mosta popular methods used for in denotes the desired variable, y the controlled variable, u control error. Åström and Hägglund (1984) proposed to use a engineering closed-loop identification, test belongs among the most popular methods used in are also several monographs about relay autotuning for symmetric relay for tuning PID controllers instead thea there manipulated variable, d the disturbance variable andtoofeuse identification and control, e.g. Yu (1999), Liu and Gao control error. Åström and Hägglund (1984) proposed engineering applications for a closed-loop identification, manipulated variable, d the disturbance variable and e symmetric relay for tuning PID controllers instead of the there are also several monographs about relay autotuning for engineering applications for a closed-loop identification, identification and control, e.g. Yu (1999), Liu and Gao Ziegler-Nichols method, see Ziegler and Nichols (1943). The control error.relay Åström Hägglund (1984) proposed toofuse a there (2012), al. (2013), Chidambaram and autotuning Sathe (2014). symmetric for and tuning PID controllers instead the areLiu alsoetseveral monographs about relay for control error. Åström and Hägglund (1984) proposed to use Ziegler-Nichols method, Ziegler and Nichols (1943). The identification and control, e.g. Yu (1999), Liu and Gao areLiu alsoetseveral monographs about autotuning for (2012), al. (2013), Chidambaram and Sathe (2014). relay method enables tosee find ultimate gain and thea there symmetric relay for tuning PIDthe controllers instead of The The mentioned identification methods arerelay proposed for lowZiegler-Nichols method, see Ziegler and Nichols (1943). identification and control, e.g. Yu (1999), Liu and Gao symmetric relay for tuning PID controllers instead of relay method enables to find the ultimate gain and the (2012), Liu et al. (2013), Chidambaram and Sathe (2014). identification and control, e.g. Yu (1999), Liu and Gao The mentioned identification methods are proposed for lowultimate frequency like the Ziegler-Nichols method but in a Ziegler-Nichols method, to seefind Ziegler Nicholsgain (1943). The models. more complex models methods mostly relay method enables the and ultimate and in the (2012), Liu etFor al. (2013), Chidambaram and Sathe for (2014). Ziegler-Nichols method, see Ziegler-Nichols Ziegler and Nichols (1943). The ultimate frequency likewithout the method but a order The mentioned identification methods arethe proposed lowLiu etFor al. (2013), Chidambaram and Sathe (2014). order models. more complex models the methods mostly controllable manner prior information relay method enables to find the ultimate gainabout and thea (2012), require additional a prior knowledge of some parameters or ultimate frequency like the Ziegler-Nichols method but in The mentioned identification methods are proposed for lowrelay method enables to find the ultimate gain and the controllable manner without prior information about order models. For more complex models the methods mostly mentioned identification methodsof aresome proposed for lowrequire additional a prior knowledge parameters or process and inmanner a short experimental time. The ultimate ultimate frequency likewithout the Ziegler-Nichols method but gain inthe a The more relay feedback experiments. controllable prior information about order models. For more complex models the methods mostly ultimate frequency likeexperimental the can Ziegler-Nichols method but gain in a order process and in a frequency short time. The ultimate require additional a experiments. prior knowledge of the some parameters or models. For more complex models methods mostly more relay feedback and the ultimate be utilized for the estimation controllable manner without prior information about the process and in afrequency shortwithout experimental time. Thethe ultimate gain require additional a prior knowledge of some parameters or controllable manner prior information about the and the ultimate can be utilized for estimation more relay feedback experiments. knowledge of some parameters or of frequency response and fitting process and inona the shortprocess experimental time. The ultimate gain require y wadditional e a prior u andone thepoint ultimate frequency can be utilized for the estimation relay feedback experiments. process and inona the short experimental time. The ultimate gain more of one point process frequency response and fitting Relay Process y w feedback e u more relay experiments. model parameters. However, the knowledge of this frequency and the ultimate frequency can be utilized for the estimation of one point on frequency the process frequency response and fitting Relay Process y and theparameters. ultimate can be utilized for estimation w e model However, the knowledge of the thisparameters. frequency u point the estimation of tworesponse model of oneonly pointallows on the process frequency and fitting Relay Process y w e model parameters. However, the knowledge of this frequency u of one point on the process frequency response and fitting point only allows the estimation of two model parameters. Relay w e Process y u Therefore models with low number of parameters are model parameters. However, the knowledge of this frequency Relay Process point only allows However, the estimation of twoofmodel parameters. model parameters. knowledge this frequency Therefore models with lowthenumber parameters are predominantly usedthe forestimation modelling. itofis theparameters. first order point only allows ofMostly two of model Block diagram of a process under relay feedback. Therefore models with low number parameters are Fig. 1. point only allows the estimation of two model parameters. predominantly used for modelling. Mostly it is the first order 1. Block diagram of a process under relay feedback. time delayed model called the number FOTD model orthethe second Therefore models with low ofit parameters are Fig. predominantly used for modelling. Mostly is first order Fig. 1. Block diagram of a process under relay feedback. Therefore models with low of model, parameters are time delayed modelmodel calledcalled the number FOTD model or the second order time delayed theMostly SOTD which are predominantly used for modelling. it is or thethe first order Fig. 1. Block diagram of a process under relay feedback. time delayed model called the FOTD model second predominantly used for modelling. Mostly it is the first order order time delayed model called the SOTD model, which are Fig. 1. Block diagram2.ofBACKGROUND a process under relay feedback. sufficient modelling many processes. In time model calledof the FOTD model or the second 2. BACKGROUND orderdelayed time for delayed model called the industrial SOTD model, which are time delayed model called the FOTD model or the second sufficient for modelling of many industrial processes. In Let us consider a time invariant process described by the to obtain model thesemodel, simple models 2. BACKGROUND order time delayed modelparameters called the of SOTD which are sufficient for modelling of many industrial processes. In Let us consider a time invariant process described order time delayed model called the SOTD model, which are to obtain model parameters of these simple models 2. BACKGROUND where ωby is the the process frequency response function GP(jω), from the relay feedback test, Luyben (1987) estimated the sufficient for modelling of many industrial processes. In process Let us consider a time invariant process described by the 2. BACKGROUND order to obtain model parameters of these simple models (jω), where ω is the frequency response function G P sufficient for modelling of many industrial processes. In from the relay feedback test, Luyben (1987) estimated the angular frequency. Three points on the Nyquist frequency Let us consider a time invariant process described by the transport delay from the initial response and assumed that the order to obtain model parameters of these simple models (jω), where ω is the process frequency response function G P from the relay from feedback test, response Luyben (1987) estimated the Let us consider a time invariant process described the angular frequency. Three points on the Nyquist order to delay obtain model parameters of al. these simple models transport thepriori. initial and assumed that the characteristic of the process can be estimated fromfrequency a by single where ω is the process frequency response function G static gain is known a Shen et (1996) proposed to P(jω), from the relay feedback test, Luyben (1987) estimated the angular frequency. Three points on the Nyquist frequency transport delay from initial response and assumed that the the where is the frequency function GP(jω), characteristic ofexperiment theresponse processby can estimated from ω a single from relay feedback test, Luyben (1987) estimated static gain is known athe priori. Shen et al. (1996) proposed to process relay feedback so be called shifting method, angular frequency. points on estimated thethe Nyquist frequency use anthe asymmetric for estimation the static gain. In this transport delay fromrelay the initial response and assumed that the characteristic ofexperiment theThree process can be from a single static gain is known a priori. Shen et al. (1996) proposed to angular frequency. Three points on the Nyquist frequency relay feedback by so called the shifting method, transport delay from the initial response and assumed that the use an asymmetric relay for estimation the static gain. In this see (2016), (2017). The shifting for the process estimated from uses amethod, single manner twoispoints estimated on the Nyquist static knownare a priori. Shen et al. proposed toa characteristic relayHofreiter feedbackof experiment bycan so be called the method shifting use angain asymmetric relay for estimation the(1996) staticcurve gain. from In this of the process can be estimated from uses a single see Hofreiter (2016), (2017). The shifting method for static gain knownare a priori. Shen et al. (1996) proposed toa characteristic manner twoispoints estimated onestimation the Nyquist curve from this purpose a biased relay with hysteresis. The process under relay feedback experiment by so called the shifting method, single relay experiment. It allows of three model use an asymmetric relay for estimation the static gain. In this see Hofreiter (2016), (2017). The shifting method uses for manner two points are estimated on the Nyquist curve from a relay feedback experiment by so called the shifting method, this purpose a biased relay with hysteresis. The process under use an relay asymmetric relay for estimation the static gain. In this the single experiment. It allows estimation of three model relay feedback is depicted in Fig.shifting 1. TheThe frequency points Hofreiter (2016), (2017). The method uses for parameters a proportional system. in the case manner two of points are estimated onestimation theHowever, Nyquistofcurve from a see this purpose a biased relay with hysteresis. process under single relay experiment. It allows three model see Hofreiter (2016), (2017). The shifting method uses for the relay feedback is depicted in Fig. 1. The frequency points manner points are estimated on theHowever, Nyquist curve from parameters a proportional system. in the case a proportional system for angular this purpose a biased withinare hysteresis. The process under where ittwo isof necessary toIt use higher order of model and toa of single relay experiment. allows estimation three model the relay feedback is relay depicted Fig.estimated 1. The frequency points parameters of a proportional system. However, in the case this purpose a biased relay with hysteresis. The process under of a proportional system are estimated fortheangular single relay experiment. It allows estimation of three model where it is necessary to use higher order model and to =0, ω =2π/T , ω =4π/T , where T period frequencies ω the relay feedback is depicted in Fig. 1. The frequency points 0 1 system p 2 are estimated p p is estimate more parameters, finding other applicable parameters of a proportional system. However, in the case of a proportional for angular where it is necessary to use higher order model and to the relay feedback is depicted in Fig. 1. The frequency points =0, ω =2π/T , ω =4π/T , where T is the period frequencies ω 0 1 p 2 p p parameters of a proportional system. However, in the case estimate more parameters, findingis required. other applicable a astable oscillation in the ,biased-relay feedback These proportional system relationships for parameter estimation Liand et al. where ismore necessary to use higher order model to of ω2are =4π/Testimated Tpfor istest. theangular period frequencies ω0=0, ω1=2π/T p, where estimateit parameters, findingis other applicable proportional system are estimated for angular a astable oscillation in thep biased-relay feedback test. These where itsuggested is necessary to use higher order model to of relationships for parameter estimation required. Liand et the al. =0, ω =2π/T , ω =4π/T , where T is the period frequencies ω (1991) using two relay feedback tests where estimate more parameters, findingis required. other applicable of a stable oscillation in thepp, biased-relay feedback These relationships for parameter Li etthe al. frequencies ω22=4π/Tpp, where Tpp is test. the period ω00=0, ω11=2π/T estimate more parameters, finding other applicable (1991) suggested using two estimation relay feedback tests where a stable oscillation in the biased-relay feedback test. These relationships for parameter estimation is required. Li et the al. of (1991) suggested using two relay feedback tests where relationships for parameter estimation is required. Li et al. of a stable oscillation in the biased-relay feedback test. These (1991) suggested using two relay feedback tests where the 910 Copyright © 2018 IFAC (1991) suggested using two relayFederation feedbackoftests whereControl) the Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Automatic Copyright © 2018 IFAC 910 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 910Control. 10.1016/j.ifacol.2018.08.491 Copyright © 2018 IFAC 910 Copyright © 2018 IFAC 910
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GP(jω2), which were obtained by the shifting method, are shifted to lower frequencies., see Fig. 4. Due to the additional transport delay D the phase shift of the point GP(jω1) varies by ω1⋅D and the phase shift of the point GP(jω2) by ω2⋅D.
three frequency points can be estimated using the following formulas computed by a numerical integration. t +T p
GP ( jω0 ) = GP (0) = K P =
ò
y (τ ) dτ
t t +T p
ò
, t ³ tL
(1)
e
w
u (τ ) dτ
u
Relay
Delay
Process
y
t
t +T p
GP ( jω1 ) =
ò
y (τ ) e- jω1 dτ
ò
u (τ ) e- jω1 dτ
t t +T p
, t ³ tL
Fig. 3. Block diagram of a process under relay feedback with a transport delay.
(2)
Im
GP(0)
t
t +T p
GP ( jω2 ) =
ò
æ ö çç y (τ ) + y çæçt - T p ÷÷ö÷÷ e- jω2 dτ ÷ ÷ ç 2 ø÷÷ø èçç èç
ò
æ æ T öö çççu (τ ) + u çççt - p ÷÷÷÷÷÷ e- jω2 dτ çè çè 2 ø÷÷ø
t t +T p t
, t ³ tL
(3)
GP(jω1)
GP(jω1)
Im
GP(jω)
Fig. 4. The Nyquist curve of a process and the points obtained by the shifting method by means of a transport delay.
where tL is the time of the achievement of a stable oscillation and KP is the static gain. GP(jω2)
Re
GP(jω2)
GP(0)
The real and imaginary parts of the values GP(0), GP(jω1) and GP(jω2) can be used for estimation up to five model parameters. In some cases, it is possible to derive explicit formulas for fitting model parameters. In other cases estimated parameters including the transport delay can be obtained e.g. by minimizing of the criterion
Re
GP(jω) Fig. 2. The Nyquist curve of a process and the points obtained by the shifting method.
2
2
Kr = å (G ( jωi ) - M ( jωi )) ,
(4)
i=1
where M(jω) is the frequency transfer function of a model. One can use a numerical method for this purpose. The static gain K of a proportional system can be estimated directly by the relationship
This approach is applicable for various models described by a transfer function. Model parameters can be estimated by fitting the selected transfer function to the points given by points GP(0), GP(jω1) and GP(jω2), see Fig. 2. The knowledge of these three points can be used for estimation up to five model parameters from a single relay test. The method is less sensitive to noise if the integrals (1), (2) and (3) are calculated over multiple oscillation periods. The noise is also suppressed by means of a hysteresis in the relay. For modelling and control, the frequency range corresponding to the phase lag 0°-180° is very important. The disadvantage of the shifting method consists in a position of the frequency point GP(jω2), because the phase lag for the frequency ω2 is greater than 180°, see Fig. 2. Therefore in the next section it is shown how to improve the position of the frequency points by means of a delay or an integrator when using the shifting method.
K = G ( 0 ) = M ( 0) .
(5)
If we know the transport delay from the initial response of the system then the other parameters can be estimated using the least square method. Via inserting the additional transport delay D into the closed loop circuit (see Fig. 3), two new points on the Nyquist curve from each relay test can be obtained for the chosen value of D. This way it is possible to find multiple points on the process frequency response. Li et al. (1991) used a similar procedure but without the shifting method and therefore they received only one new point on the Nyquist curve from each relay test. Since the frequency response points are calculated by formulas (2) and (3), the presence of a static load disturbance with a magnitude of dA does not have any influence on the calculation as it holds
2. SHIFTING METHOD WITH ADDITIONAL TRANSPORT DELAY To receive a better position of the points GP(jω1) and GP(jω2) we can insert an additional chosen transport delay into the closed loop circuit, see Fig. 3 Then the points GP(jω1) and
t +T p
ò t
911
- jωi
dA ⋅e
t +T p
dτ = d A
ò t
e- jωi dτ , i = 1, 2 .
(6)
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Hence the points GP(jω1) and GP(jω2) may be also used for the static gain estimation. This approach can be used for various LTI (Linear Time Invariant) isochronic and anisochronic models because the positions of the estimated points are calculated by the formulas (1), (2) and (3) which do not depend on a model structure.
æ ö æ ö÷ 1 ÷÷- arg (G ( jω ))÷÷÷ ççç arg ççç l ÷ ç çç a jω 2 + a ⋅ jω + 1÷÷÷ ÷÷ è 2( l) ø 1 ççç 2 l 1 ÷÷ Td = çå ÷÷ (15) ωl 2 çç l =1 ÷÷ çç ÷÷ çç ÷÷ çè ø
Example #1 A process with the transfer function (7) is controlled by the asymmetrical relay, see Fig. 1. The time courses of the manipulated variable u and the controlled variable y are shown in Fig. 6 provided, that the process was initially in a steady state. 4 u,y
3. SIMULATED EXAMPLES To demonstrate the introduced method we utilize the following three process transfer functions, which are taken from Berner et. al. (2016). 1 , (7) P1 ( s ) = ( s + 1)(0.1s + 1)(0.01s + 1)(0.001s + 1) P2 ( s ) = P3 ( s ) =
1 4
( s + 1) 1
2
(0.05s + 1)
,
(8)
e-s ,
(9)
3 1 -1 -2
uB
e
2 ⋅ ω12
a1 =
1
ω1
K = G ( 0 ) = M ( 0) ,
(12)
æ ö÷ 4 K P2 1 çç K P2 ÷ çç + 3÷÷ , 2 2 ÷÷ 3 çç G ( jω ) G ( jω1 ) ÷ø è 2
(13)
K P2 G ( jω1 )
2
(
- 1- a2 ⋅ ω12
2
3
4 t [s]
5
G ( jω0 ) = G (0) = 1 ,
(17)
G ( jω1 ) = -0.055 - 0.063 j ,
(18)
G ( jω2 ) = -0.026 - 0.099 j ,
(19)
where
ω1 =
2
)
,
2π = 8.84 rad⋅s-1, Tp
ω2 = 2 ⋅ ω1 = 17.68 rad⋅s-1.
The parameters K, a2, a1, Td of the SODT model can be calculated from the estimated points GP(0), GP(jω1) and GP(jω2) by the following explicit formulas
a2 =
1
From these time courses we can determine the period of stable oscillation Tp (see Fig. 6) and by the shifting method utilizing formulas (1), (2), (3), we estimate the points G(0), G(jω1) and G(jω2) on the Nyquist curve from one relay feedback experiment. (16) T p = 0.711 s
Fig. 5 The static characteristic of an asymmetrical relay with hysteresis.
1
0
Fig. 6. The time courses of the relay output u and the process output y
u uA
�A
y
0
where K, a2, a1, Td are estimated parameters. In all the simulating examples the asymmetrical relay with a hysteresis has following parameters, see Fig. 5: (11) uA=2, uB=-1, εA=0.1, εB=-0.1
0
Tp
u
2
where s is the complex variable in L-transform. The first process is lag dominated, the second is balanced and the third is delay dominated. Assume that all these processes can be described by a SOTD model in the form K (10) M (s) = e-Td ⋅s , a2 s + a1s + 1
�B
893
(20) (21)
In Fig. 7 the estimated points G(0), G(jω1) and G(jω2) are depicted in the Nyquist diagram for the transfer function P1(s). From these points we can estimate up to five model parameters. If we want to obtain frequency responses for lower frequencies we can use a relay feedback with a transport delay or with an integrator (see Fig. 3). In this example the additional transport delay is chosen as D = 3.5 s. (22) The time courses of the relay output u and the output y of the process P1(s) are depicted in Fig. 8. From it follows that the period of the stable oscillation (23) T p = 8.87 s.
(14)
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0.2 0 -0.2
Milan Hofreiter / IFAC PapersOnLine 51-11 (2018) 891–896
Im
G(jω2)
0.2
G(0) Re
G(jω1)
-0.4
P1(jω)
-0.6 0
0.5
4 u,y 3
1
0 -1 1
20
30
40 t [s]
50
Fig. 8. The time courses of the relay output u and the process output y for the relay feedback with the additional transport delay.
4 u,y 3
Utilizing the formulas (1), (2), (3) and the observed time courses of u and y we can calculate G ( jω0 ) = G (0) = 1 ,
(24)
G ( jω1 ) = 0.625 - 0.522 j ,
(25)
G ( jω2 ) = 0.252 - 0.513 j ,
(26)
2
2π = 0.71 rad⋅s-1, Tp
ω2 = 2 ⋅ ω1 = 1.42 rad⋅s-1.
Tp
u
y
1 0 -1 -2 0
100 t [s] Fig. 10. The time courses of the relay output u and the process output y for the relay feedback with the additional transport delay.
where
ω1 =
M1(jω)
G(jω1)
Example #2 A process with the transfer function (8) is controlled by the asymmetrical relay with the additional transport delay, see Fig. 3. The time courses of the relay output u and the output y of the process P2(s) are shown in Fig. 10, provided that the process was initially in a steady state and the additional transport delay D = 3.6 s. (30) From the time courses u, y it follows that the period of stable oscillation (31) T p = 14.9 s.
Tp
u
y
0
G(jω2)
-0.8 -0.5 0 0.5 1 1.5 Fig. 9. The Nyquist diagram for the transfer functions P1(s), M1(s) and the estimated points G(0), G(jω1), G(jω2).
1.5
1
-2
P1(jω)
-0.6
Fig. 7. The Nyquist diagram for the transfer function P1(s) and the estimated points G(0), G(jω1), G(jω2).
2
Re
-0.2
-0.4
-0.8-0.5
G(0)
Im
0
(27) (28)
20
40
60
80
Utilizing the formulas (1), (2), (3) and the observed time courses of u and y we can calculate
The model transfer function estimated from the points G(0), G(jω1) and G(jω2) is 1 M1 ( s ) = e-0.033s . (29) 0.073s 2 + 1.083s + 1 The position of these points together with the Nyquist diagram of the transfer functions P1(s) and M1(s) are shown in Fig. 9. This figure shows a very good conformity of frequency responses of the identified process P1(s) and the estimated model M1(s). The example illustrates that from a single asymmetrical relay feedback test one can obtain three points of the frequency response by the shifting method. If we use an additional transport delay then we can obtain these points for lower frequencies. Therefore, from two relay tests we can receive five frequency points (the static gain is obtained in each relay feedback test).
G ( jω0 ) = G (0) = 1 ,
(32)
G ( jω1 ) = -0.018 - 0.721 j ,
(33)
G ( jω2 ) = -0.321- 0.114 j ,
(34)
where
ω1 =
2π = 0.42 rad⋅s-1, Tp
ω2 = 2 ⋅ ω1 = 0.84 rad⋅s-1.
(35) (36)
The model transfer function estimated from the points G(0), G(jω1) and G(jω2) is 1 M 2 (s) = e-0.808s . (37) 2 3.203s + 3.128s + 1
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ω2 = 2 ⋅ ω1 = 2.87 rad⋅s-1.
The position of these points together with the Nyquist diagram of the transfer functions P2(s) and M2(s) are shown in Fig. 11. This figure shows a very good conformity of frequency responses of the identified process P2(s) and the estimated model M2(s). 0.2
Re
G(jω2)
-0.2 -0.4 -0.6
-1
P2(jω)
G(jω1)
-0.8 -0.4
-0.2
M2(jω) 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 11. The Nyquist diagram for the transfer functions P2(s), M2(s) and the estimated points G(0), G(jω1), G(jω2). Example #3 A process with the transfer function (9) is controlled by the asymmetrical relay with the additional transport delay, see Fig. 3. The time courses of the relay output u and the output y of the process P3(s) are shown in Fig. 12, provided that the process was initially in a steady state and the additional transport delay D = 1.1 s. (38) From the time courses u, y it follows that the period of the stable oscillation T p = 4.38 s. (39) 4 u,y 3
y u
1 0 -1 5
10
15
20
25 t [s] 30
Fig. 12. The time courses of the relay output u and the process output y for the relay feedback with the additional transport delay.
(40)
G ( jω1 ) = -0.007 - 0.995 j ,
(41)
G ( jω2 ) = -0.980 - 0.012 j ,
(42)
stand water-flow pump
2π = 1.43 rad⋅s-1, Tp
ball jet reservoir
where
ω1 =
4. LABORATORY EXPERIMENT The laboratory apparatus “Water Levitation” consists of a water reservoir, a pump, a jet and an ultrasonic sensor, see Fig. 14. The pump sucks water from the reservoir and pumps water into the jet. The water streams out from the jet and lifts up the ball. The flow from the jet depends on the actual power of the pump which is controlled by the manipulated variable u. The controlled variable y (the position of a ball) is measured by the sensor. The task is to estimate by the shifting method the parameters of model (10) describing the relationship between the manipulated variable u and the controlled variable y around the working point. The time courses of y and u observed during the relay feedback test on the laboratory apparatus are depicted in Fig. 15. This test was realised with the additional transport delay (D=1.2s) in the relay feedback, see Fig. 3. sensor
Utilizing the formulas (1), (2), (3) and the observed time courses of u and y we can calculate G ( jω0 ) = G (0) = 1 ,
Im P3(jω), M3(jω) 1 0.8 0.6 0.4 G(0) 0.2 0 -0.2 Re -0.4 ) G(jω -0.6 2 -0.8 G(jω1) -1 -1 -0.8 -0.6-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fig. 13. The Nyquist diagram for the transfer functions P3(s), M3(s) and the estimated points G(0), G(jω1), G(jω2).
2
-20
(44)
The model transfer function estimated from the points G(0), G(jω1) and G(jω2) is 1 M 3 (s) = e-s . (45) 2 0.00252s + 0.1s + 1 The position of these points together with the Nyquist diagram of the transfer functions P3(s) and M3(s) are shown in Fig. 13. The process transfer function P3(s) is roughly the same as the estimated model transfer function M3(s). Therefore, the Nyquist diagrams of P3(s) and M3(s) look like a single line.
G(0)
Im
0
895
Fig. 14. The laboratory model “Water Levitation”
(43)
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Milan Hofreiter / IFAC PapersOnLine 51-11 (2018) 891–896
response points from each relay test than the usual approach. The method is also applicable for integrating systems where only frequency response points G(jω1) and G(jω2) are used for fitting model parameters.
u[V],y[cm]
25 20
y
15
Acknowledgment
u
10 5 0 -5
0 5 10 15 20 25 30 35 40 Fig. 15 The observed input u and the output y.
45
This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS16/210/OHK2/3T/12 and by the Institutional Resources of CTU for research (RVO12000).
50
REFERENCES Åström, K. J. and Hägglund, T. (1984). Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica, 20 (5), pp. 645-651. Berner, J., Hägglund, T. and Åström, K. J. (2016). Improved relay autotuning using normalized time delay. American Control Conference (ACC), IEEE, pp. 1869-1875. Bi Q., Wang Q. G. and Hang C. C. (1997). Relay-based estimation of multiple points on process frequency response. Automatica, 33 (9), pp. 1753-1757. Chidambaram M. and Sathe V. (2014). Relay Autotuning for Identification and Control. Cambridge University Press, Cambridge. Hofreiter, M. (2016). Shifting method for relay feedback identification. IFAC-PapersOnLine, 49 (12), pp. 19331938. Hofreiter, M. (2017). Biased-relay feedback identification for time delay systems, IFAC-PapersOnLine, 50 (1), pp. 1462-14625. Li W., Eskinat E. and Luyben W. L. (1991). An improved auto-tune identification method, Ind. Eng. Chem. Res. 30 (7), pp. 1530-1541. Liu T. and Gao F. (2012). Industrial process identification and control Design: Step-test and relay-experiment-based Method. Advances in Industrial Control. Springer-Verlag, London. Liu T., Wang Q. G. and Huang H. P. (2013). A tutorial review on process identification from step or relay feedback test. Journal of Process Control, 23 (10), pp. 1597-1623. Luyben, W. L. (1987). Derivation of transfer functions for highly nonlinear distillation columns. Ind. Eng. Chem. Res. 26 (12), pp. 2490-2495. Shen, S. H., Wu, J. S. and Yu C. C. (1996). Use of biasedrelay feedback for system identification. AIChE. J. 42 (4), pp. 1174-1180. Yu C. C. (1999). Autotuning of PID Controllers, ch. 2 and 3. Springer-Verlag, London. Ziegler J. G. and Nichols N. B. (1943). Optimum settings for automatic controllers. Trans. ASME, vol. 65, pp. 433-444.
From the courses y and u and formulas (1), (2) and (3) it was determined T p = 4.7 s, (46)
ω1 =
2π = 1.34 rad⋅s-1, Tp
(47)
ω2 = 2 ⋅ ω1 = 2.68 rad⋅s-1,
(48)
-1
G (0) = 2 cm⋅V ,
(49)
G ( jω1 ) = 0.552 -1.723 j ,
(50)
G ( jω2 ) = -0.618 -1.161 j .
(51)
Using relationships (49), (50) and (51) the SOTD model is 2 M (s) = ⋅ e-0.315 s . (52) 2 0.105s + 0.561s + 1 Fig. 16 shows a good conformity between the step responses of the identified process and the model (52). u[V], yP[cm], yM[cm]
25 20
yM yP u
15 10 5 0 0
5
10
15
20
25
30
35
40
45
50
Fig. 16. The process step response yP and the model step response yM to the step u. 5. CONCLUSIONS The combination of the shifting method and the additional delay in the relay feedback loop enables to obtain three frequency response points for the phase lag in the range from 0 to 180° using a single relay test. These points are more suitable for fitting the model parameters and they can be used for fitting up to five parameters of a model transfer function with various structures. If we want to obtain more frequency response points then we can select different values of the additional transport delay for each relay feedback test. This approach makes it possible to estimate twice more frequency 915