Modified relay feedback method for improved system identification

K. Srinivasan, M. Chidambaram *

Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India

Received 19 November 2001; received in revised form 14 October 2002; accepted 11 November 2002

Abstract

The asymmetrical relay feedback method is modified to get improved parameters estimates of a first order plus time delay transfer

function model. The method uses a single relay test. Analytical solution is provided for the model parameters. The method

overcomes the problem of getting a negative time constant by the conventional symmetrical relay testing for modeling higher order

system. Simulation results are given for stable and unstable transfer function models. The proposed method gives model parameters

close to the actual values and gives closed loop ISE values similar to that of the actual system. The method is also compared with

that of Huang et al. (AIChE J., 42 (1996) 2687) and Shen et al. (AIChE J., 42 (1996) 1174).

# 2002 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Relay tuning; Asymmetric; Stable; Unstable system; FOPTD

1. Introduction

Relay feedback testing has received considerable

attention in the area of system identification. Astrom

and Hagglund (1984) have suggested the use of an ideal

(on�/off) relay to generate sustained closed loop oscilla-

tions. The ultimate gain (ku) and ultimate frequency

(vu) can be found out easily [using ku�/4h /(pa ), where

h is the relay height and ‘a ’ is the amplitude of the

closed loop oscillation]. From these ultimate values, the

settings of the conventional PID controller can be

obtained. Luyben (1987) has employed the relay feed-

back method to identify a transfer function model. Since

a first order plus delay (FOPTD) transfer function

model is adequate for designing controllers, the method

aims to estimate the model parameters of FOPTD (kp,

td and t ) from ku and vu. Since only ku and vu are

available (giving two equations relating the three model

parameters), an additional process information such as

process gain or time delay is required to calculate the

model parameters. Li, Eskinat and Luyben (1991) have

indicated that there is an error involved in calculating ku

from [4h /(pa )]. Li et al. (1991) and Leva (1993) haveproposed the use of two relay feedback tests (one

conventional test and a second test with addition of a

known delay). An excellent review of work done in the

relay feedback method is given by Yu (1999). Shen, Wu

and Yu (1996) have used a biased relay (Fig. 1) for

getting the model parameters using a single relay test. In

this method, the process gain (kp) is calculated from u(t)

and y (t) data as:

kp�g2p

0

e(t)d(vt)

g2p

0

u(t)d(vt)

(1)

Here e(t)�/yr�/y(t), where yr is the set point and for

relay test yr�/0. By substituting the values of kp, vu, and

calculated ku [from the relation ku�/4h /(pa )] in the

amplitude criterion and the phase angle criterion, the

model parameters t and td are calculated analyticallyfrom:

ku��

1

kp

�[1�t2v2

u]0:5 (2)* Corresponding author.

E-mail address: chidam@iitm.ac.in (M. Chidambaram).

Computers and Chemical Engineering 27 (2003) 727�/732

www.elsevier.com/locate/compchemeng

0098-1354/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved.

PII: S 0 0 9 8 - 1 3 5 4 ( 0 2 ) 0 0 2 5 7 - 0

p�tan�1 (tvu)�tdvu�0 (3)

Shen et al. (1996) have shown that the method gives

improved model parameters with a single test. In the

method of Shen et al. (1996), though a good estimation

of kp is obtained, there is an error in calculating ku

because of using the relation ku�/4h /(pa ). They have

also used the observed frequency of oscillation for the

ultimate frequency of oscillation. Shen et al. (1996) havegiven simulation results only for stable systems. Huang,

Chen, Lai and Wang (1996) have used a asymmetrical

relay test to get output response for three or four cycles

followed by turning off the relay and recording the

output response till the output is constant (the later part

of the test is an open loop test). There is no need to

calculate ku in this method. Equations are given for

calculating the FOPTD model parameters. It should benoted that the relay used by Shen et al. (1996) is shifting

the (ideal) relay horizontally whereas that of Huang et

al. (1996) is shifting the ideal relay vertically. Huang et

al. (1996) method is not applicable for identifying

unstable FOPTD model since the method requires an

open loop response also. In the present work, using only

a single asymmetrical relay test (without turning off the

relay), a method is proposed for formulating thenecessary equations so as to get better estimates of the

FOPTD model parameters than that of Shen et al.

(1996) and Huang et al. (1996).

2. Proposed method

In the present method, we consider an asymmetrical

relay as shown in Fig. 1 where for e(t)]/0, u�/gh and

when e (t)B/0, u�/�/h , where g is the displacement in

the relay height (h). g�/1 gives a symmetrical relay and

g�/1 (or B/1) gives an asymmetrical relay. With this

relay, the responses in y and u are recorded. Let us

denote G (s) as the transfer function model to beidentified. Since y(t) and u(t) are piece-wise continuous

and periodic functions, we can write the expression for

the Laplace transform of y (t ) as (Kreysizig, 1996):

L[y(t)]�y(s)��

1

(1 � expf�Psg)

�gP

0

y(t)exp(�st)dt (4)

Similar expression can be written for u (s ) also. Hence,

we can write

G(s)�y(s)

u(s)�gP

0

y(t)exp(�st)dt

gP

0

u(t)exp(�st)dt

(5)

G (jv ) can be written by substituting s�/jv in the above

equation, where P�/2p /v and v is the frequency of

oscillation observed in the output response. By using the

Euler’s equation for exp(�/jvt ), Eq. (5) can be written

as:

G(jv)��

c1 � jd1

c2 � jd2

�(6)

where

Nomenclature

G process transfer functionh relay heightkp process gainkc controller gaina real part of G (jv )

b imaginary part of G (jv )

c1, d1 defined by Eq. (7a)c2, d2 defined by Eq. (7b)P period of output oscillationt process time constanttd process time delaytI integral timetD derivative timev frequency of oscillationvu ultimate frequency of oscillationg relay displacement

Fig. 1. Block diagram for the asymmetric relay feedback system.

K. Srinivasan, M. Chidambaram / Computers and Chemical Engineering 27 (2003) 727�/732728

c1�gP

0

u(t)cos(vt)dt; d1�gP

0

u(t)sin(vt)dt (7a)

c2�gP

0

y(t)cos(vt)dt; d2�gP

0

y(t)sin(vt)dt (7b)

Eqs. (7a) and (7b) can be evaluated numerically using

Simpson’s rule. Eq. (6) can be written as

G(jv)�a� jb (8)

where

a��

c1c2 � d1d2

c22 � d2

2

�(9a)

b��

d2c1 � d1c2

c22 � d2

2

�(9b)

Let us identify the parameters of a FOPTD model:

G(s)�kpexp(�tdS)(tS�1) (10)

For this model, we can write G (jv ) as:

a� jb�kp

�cos(tdv) � jsin(tdv)

1 � jtv

�(11)

On cross multiplying and equating the real part and

imaginary part separately to zero we get:

a�btv�kpcos(tdv)�0 (12)

atv�b�kpsin(tdv)�0 (13)

Substituting the value of kp obtained from Eq. (1), and

substituting vu for v , an analytical solution of above

two equations (refer to Appendix A) gives the values for

t and td.The modifications proposed in the present work are:

use of Eq. (1) for kp and the use of Eq. (6) to formulate

the two equations [Eqs. (12) and (13)] for the model

parameters. Whereas the equations [Eqs. (2) and (3)]

used by Shen et al. (1996) assume that all the higher

harmonics die out and the frequency of oscillation

corresponds to that of the ultimate frequency. As stated

earlier, Shen et al. (1996) have calculated ku value from4h /(pa ). Hence the use of amplitude and phase angle

criteria is made by Shen et al. (1996). Li et al. (1991),

Shen et al. (1996) and Huang et al. (1996) considered

only stable transfer function. In the present method, we

do not use the equation ku�/4h /(pa ) and also do not

need vu. We will show the improvement of the proposed

method in the following case studies. The general

guidelines reported in literature for selecting the valueof g is 2�/4 for stable systems and 1.05�/1.1 for unstable

systems. In the present work g�/2 is used for stable

systems and g�/1.1 is used for unstable systems.

3. Simulation results

3.1. Case study 1

Consider the transfer function given by Huang et al.

(1996): G (s )�/2 exp(�/0.1s)/[(2s�/1)(s�/1)]. The asym-

metrical relay (with h�/1 and g�/2) is used and the

oscillation in the output y (t ) and u (t ) are noted. The use

of Eqs. (1), (12) and (13) gives the model parameters as

kp�/2.19, t�/7.92 and td�/0.443, whereas the method

of Huang et al. (1996) gives kp�/2.0, t�/2.67 and td�/

0.42, respectively. Application of Li et al. (1991) method

gives negative value for the time constant (kp�/13.5, t�/

�/0.1, td�/0.1). From the phase angle criterion for the

actual system, we get by numerical solution vu�/3.841

and ku�/15.28. Using these values and that of kp in the

phase angle and amplitude criteria equations for a

FOPTD we get the model parameters: kp�/2, t�/8

and td�/0.42. The present method gives model para-meters closer to these values. Using the identified model

parameters, PID controllers are designed by Ziegler�/

Nichols ultimate cycling method. The closed loop

responses are evaluated on the actual system. The

responses are shown in Fig. 2. Table 1 shows the

controller settings and the ISE values. Table 1 and

Fig. 2 show that the present method gives closed

response very close to the performance based on thecontroller designed for the actual system.

3.2. Case study 2

Consider the transfer function G (s)�/exp(�/2s)/

(10s�/1). The present method gives the model para-meters as kp�/1.03, t�/10.3 and td�/2.3, whereas the

ideal relay method of Li et al. (1991) gives kp�/0.988,

t�/8.02 and td�/2, respectively. In the Li et al. method,

Fig. 2. Closed loop response comparisons for case study 1 solid,

actual; dash, proposed method; chain, Huang et al.; dot, Shen et al.

K. Srinivasan, M. Chidambaram / Computers and Chemical Engineering 27 (2003) 727�/732 729

the value of the delay is noted from the initial response

as 2. Li et al. (1991) have suggested to carry out an open

loop step test to get the parameters of the delay time. A

significant improvement is obtained by the present

method. The use of Shen et al. (1996) method gives

the model parameters as kp�/0.999, t�/8.118 and td�/

2.005. Fig. 3 and the Table 1 show that the present

method gives ISE values of the closed loop system

similar to that of the actual system. The effect of

measurement noise is studied by adding a random noise

(with 0.5% S.D.) and the corrupted signal is used for

feedback. The present method gives the model para-

meters as kp�/0.9336, td�/2.03 and t�/9.3638. These

values are close to the values obtained without any

measurement noise. This indicates that the present

method is robust for measurement noise.

3.3. Case study 3

Consider the transfer function given by Li et al.

(1991): exp(�/2s)/(10s2�/11s�/1). Application of ideal

relay method by Li et al. (1991) gives the model

parameters of FOPTD as kp�/�/0.501, td�/2 and t�/

�/5.030. We get negative values for time constant and

gain. According to Li et al. (1991), under such a case, the

FOPTD model is not valid and recommend to fit a

SOPTD model. Instead, the proposed method is able to

identify a FOPTD model (with kp�/1.017, td�/2.872

and t�/11.71). The application of Shen et al. (1996)

method gives the model parameters as kp�/0.998, t�/

10.82 and td�/2.0. PID controllers are designed for the

identified models and also on the actual system. Fig. 4

Table 1

Closed loop performance comparisons of different methods of identification with that of the actual system

Case study Method (kckp)max vu kc tI tD ISEa

1 Actual 30.7 3.841 18.44 0.8174 0.204 0.4928

Proposed 28.11 3.547 16.86 0.886 0.222 0.4694

Huang et al. 10.03 3.738 6.01 0.84 0.21 0.9508

Shen et al. 31.88 3.673 18.75 0.836 0.218 0.4684

Li et al. (gives negative time constant, not applicable)

2 Actual 7.91 0.785 4.748 4.0 1.000 3.3826

Proposed 7.10 0.683 4.261 4.6 1.150 3.2380

Li et al. 6.38 0.785 3.725 4.0 1.000 3.5146

Shen et al. 6.43 0.783 3.861 4.01 1.000 3.5037

3 Actual 7.07 0.6 4.242 5.23 1.308 4.5172

Proposed 6.48 0.5466 3.887 5.744 1.436 4.2917

Shen et al. 8.55 0.785 5.13 4.0 1.000 6.7261

Li et al. (gives negative time constant, not applicable)

a ISE evaluated on the actual system using the above PID settings.

Fig. 3. Closed loop response comparisons for case study 2 solid,

actual; dash, proposed method; chain, Li et al.; dot, Shen et al.

Fig. 4. Closed loop response comparisons for case study 3 solid,

actual; dash, proposed method; chain, Huang et al.

K. Srinivasan, M. Chidambaram / Computers and Chemical Engineering 27 (2003) 727�/732730

and Table 1 show that the present method gives the

closed loop response close to that of the actual system.

3.4. Case study 4

Let us consider an unstable FOPTD system as kp

exp(�/tds)/(ts�/1) with kp�/1 and td�/0.5 and t�/1.

The asymmetrical relay test is conducted (with h�/1.0,

for e ]/0, u�/h , for e B/0, u�/�/gh , with g�/1.1). By the

proposed method we get kp�/1.005, t�/1.0094 and td�/

0.61. The model parameters obtained by the method ofLi et al. (1991) are kp�/0.94 and t�/0.77 and td�/0.5. It

should be noted that the value of td is assumed in the Li

et al. method. Shen et al. method gives kp�/1.005, t�/

0.7953 and td�/0.5082. These values are similar to that

obtained for Li et al. (1991). It should be noted that

appropriate equation [tan�1(tvu)�/tdvu�/0] for un-

stable system is used instead of Eq. (3). As stated

earlier, Huang et al. (1996) method is not applicable forunstable systems. Based on the FOPTD model, PID

settings are calculated based on ISE minimization

method (Jhunjhunwala & Chidambaram, 2001). PID

controller is also designed on the actual system para-

meters. The PID controller settings and the closed loop

ISE values are given in Table 2. Fig. 5 shows the closed

loop responses. Table 2 and Fig. 5 show that the

identified model by the proposed method gives aresponse closer to that of the actual system.

4. Conclusions

The proposed modification in the asymmetrical relaytest gives improved values of the parameters of the

FOPTD model. Analytical solutions are given for the

evaluation of the model parameters. The method needs

only one relay test to identify the parameters of a

FOPTD model. The method is also applicable for

identifying unstable FOPTD model. Simulation results

are given for stable and unstable transfer function

models. The proposed method gives better closed loopresponse than that of Huang et al. (1996) and Shen et al.

(1996).

Appendix A

Analytical solution for Eqs. (12) and (13) is derived

here.Eqs. (12) and (13) are written as:

a�bx1�kpcos(x2)�0 (A:1)

ax1�b�kpsin(x2)�0 (A:2)

where

x1�tv (A:3)

and

x2�tdv (A:4)

Multiplying Eq. (A.1) by a and Eq. (A.2) by b and

then adding the resulting equations we get:

bkpsin(x2)�dkpcos(x2)�q (A:5)

where d�/�/a , q�/�/(a2�/b2).

The above equation can be written as

Msin(x2�c)�q (A:6)

where

M�k2p(b2�d2) (A:7)

Table 2

PID controller settings and ISE for case study 5 (unstable system)

Serial number Settings Actual system Present Li et al. Shen et al.

1 kc 2.381 2.033 2.533 1.971

2 tI 2.379 2.951 1.832 1.815

3 tD 0.290 0.348 0.223 0.293

4 ISEa 2.357 2.617 2.933 3.182

PID settings by the method of Jhunjhunwala and Chidambaram (2001).a ISE values calculated on the actual system with the above PID settings.

Fig. 5. Closed loop servo response of the unstable FOPTD system

solid, actual system; dash, present work; chain, Shen et al.; dot, Li et

al.

K. Srinivasan, M. Chidambaram / Computers and Chemical Engineering 27 (2003) 727�/732 731

c� tan�1

�b

d

�(A:8)

Eq. (A.6) gives

x2�sin�1

�q

M

��c (A:9)

Using this value of x2 in Eq. (A.1) gives

x1�[a � kpcos(x2)]

b(A:10)

Hence t and td are obtained as

t�x1

v(A:11)

td�x2

v(A:12)

Here v is the known from the closed loop oscillation.

References

Astrom, K. J., & Hagglund, T. (1984). Automatic tuning of simple

regulators with specification on phase and amplitude margin.

Automatica 20 , 645.

Huang, H. P., Chen, C.-L., Lai, C.-W., & Wang, G.-B. (1996). Auto

tuning for model based PID controllers. American Institute of

Chemical Engineering Journal 42 , 2687.

Kreysizig, E. (1996). Advanced engineering mathematics (5th ed, p.

235). New York: Wiley.

Jhunjhunwala, M., & Chidambaram, M. (2001). PID controller tuning

for unstable system by optimization method. Chemical Engineering

Communication 185 , 91.

Leva, A. (1993). PID auto tuning algorithm based on relay feedback.

IEE Proceedings-CTA 140 , 328.

Luyben, W. L. (1987). Derivation of transfer function model for highly

nonlinear distillation column. Industrial Engineering and Chemical

Research 26 , 2490.

Li, W., Eskinat, E., & Luyben, W. L. (1991). An improved auto tune

identification method. Industrial Engineering Research and Design

30 , 1530.

Shen, S.-H., Wu, J.-S., & Yu, C.-C. (1996). Use of biased relay

feedback for system identification. American Institute of Chemical

Engineering Journal 42 , 1174.

Yu, C.-C. (1999). Auto tuning of PID controllers: relay feedback

approach . Berlin: Springer.

K. Srinivasan, M. Chidambaram / Computers and Chemical Engineering 27 (2003) 727�/732732