PID Controller Design for Controlling Integrating Processes with Dead Time using Generalized Stability Boundary Locus

Proceedings of the 3rd IFAC Conference on Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Available online at www.sciencedirect.com Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Ghent, Belgium, May 9-11, 2018 Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018

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IFAC PapersOnLine 51-4 (2018) 924–929

PID Controller Design for Controlling Integrating Processes with Dead Time PID Controller Design for Controlling Integrating Processes with Dead Time PID Controller Design for Controlling Integrating Processes Dead Time using Generalized Stability Boundary Locus with PID Controller Design for Controlling Integrating Processes using Generalized Stability Boundary Locus with Dead Time using Generalized Stability Boundary Locus using Generalized Stability Boundary Locus

Serdal Atic*, Erdal Cokmez**, Fuat Peker**, Ibrahim Kaya** Serdal Atic*, Erdal Cokmez**,  Fuat Peker**, Ibrahim Kaya** Serdal Atic*, Erdal Cokmez**,  Fuat Peker**, Ibrahim Kaya** Serdal Atic*, Erdal Cokmez**, Ibrahim Kaya**  Fuat Peker**, * Electricity and Energy Department, Vocational High School, Batman University,  * Electricity and Energy Department, Vocational High School, Batman University, Batman, Turkey, (e-mail: [email protected]). * Electricity and EnergyTurkey, Department, Vocational High School, Batman University, Batman, (e-mail: [email protected]). * Electricity and Energy Department, Vocational High School, Batman University, ** Electrical and Electronics Engineering Department, Dicle University, Batman, Turkey, (e-mail: [email protected]). ** Electrical and Electronics Engineering Department, Dicle University, Batman, Turkey, (e-mail: [email protected]). Diyarbakır, Turkey, (e-mail: [email protected], [email protected], [email protected]) ** Electrical Electronics Engineering [email protected], Department, Dicle University, Diyarbakır, Turkey, (e-mail:and [email protected], [email protected]) ** Electrical Electronics Engineering [email protected], Department, Dicle University, Diyarbakır, Turkey, (e-mail:and [email protected], [email protected]) Diyarbakır, Turkey, (e-mail: [email protected], [email protected], [email protected]) Abstract: This paper proposes a method so that all PID controller tuning parameters, which are Abstract: This paper proposes a method so that all PIDcan controller tuning by parameters, which are satisfying stability of any integrating time delay processes, be calculated forming the stability Abstract: stability This paper proposes a method so that all PIDcan controller tuningby parameters, which are satisfying of any integrating time delay processes, be calculated forming the stability Abstract:loci. ThisProcesses paper proposes method so transfer that allfunction PIDcan controller parameters, which are boundary having aahigher order must firsttuning be modeled by anthe integrating satisfying stability of any integrating time delay processes, be calculated by forming stability boundary loci. Processes having a higher order transfer function must first bethe modeled byLater, anthe integrating satisfying stability of any integrating time delay processes, can be calculated by forming stability plus first order plus dead time (IFOPDT) transfer function in order to apply method. IFOPDT boundary loci. Processes having a higher order transfer function must first bethe modeled an integrating plus firsttransfer order plus dead and timethe (IFOPDT) transfer function inare order to apply method.by Later, IFOPDT boundary loci. Processes having acontroller higher order transfer function must first bethe modeled byforms an integrating process function transfer function converted to normalized to obtain plus first order plus dead time (IFOPDT) transfer function in order to apply method. Later, IFOPDT process transfer function and the controller transfer function are converted to normalized forms to obtain 2 2 plus first order plus dead time (IFOPDT) transfer function in order to apply the method. Later, IFOPDT ,  KK c T are and to the stability boundary locus , KKconverted T (T 2 / Ti ), forms KK c Td to T , KK c (transfer T 2 / Ti ) function process transfer function andinthe controller normalized obtain  KK  planes  KK c d  c c ,  KK c T are and to planes the stability boundary locus , KKconverted T KK (T 2 / Ti ), forms KK c Td to KK T , KK c (transfer T 2 / Ti ) function process transfer function andinthe controller normalized obtain    c d  c c and  KKof , KK c Td  stability the stability boundary locusPID in controller (T 2 / control Ti ), KK csystem Td  planes KK c T , KKparameter (T 2 / Ti )  ,values  KK c Tachieving for PID controller design. can c the c , and planes KK T , KK T the stability boundary locus in KK ( T / T ), KK T KK T , KK ( T / T )  Thec achieving for PID controller design. PID stability controller parameter of can  method c d  stability c the control i csystem d  c c i  values be determined by the obtained boundary loci. advantage of the given in this study for PID controller design. PID stability controllerboundary parameter values achieving stability of the control system can be determined by the obtained loci. The advantage of the method given in this study for PID controller design. PID controller parameter values achieving stability of the control system can compared with by previous studiesstability in this subject is toloci. remove need ofof re-plotting thegiven stability boundary be determined the obtained boundary The the advantage the method in this study compared with previous studies in this subject is to remove the need of re-plotting the stability boundary be determined by the transfer obtainedfunction stability changes. boundaryThat loci. is, Thetheadvantage ofresults the method given ingeneralized this study locus as the process approach in somehow compared with previous studiesfunction in this subject is to remove theapproach need of re-plotting the stability boundary locus as boundary the process changes. That is, the in somehow generalized compared with previous studies in this plus subject is delay to remove theapproach need of re-plotting the stability boundary stability locitransfer for integrating time processes under aresults PID controller. Application of locus as the process transfer function changes. That is, the results in somehow generalized stability loci for integrating plus time delay processes under aresults PID controller. Application of locus as boundary thehas process transfer function changes. That is, the approach in somehow generalized the method been clarified with examples. stability boundary loci for integrating plus time delay processes under a PID controller. Application of the method has been clarified with examples. stability boundary loci for integrating plus time delay processes under a PID controller. Application of the method has(International been clarified with PID examples. © 2018, IFAC Federation ofcontroller, Automatictransfer Control) Hosting dead by Elsevier Ltd. All rights reserved. Keywords: Stability, PI controller, function, time, modeling. the method has been clarified with examples. Keywords: Stability, PI controller, PID controller, transfer function, dead time, modeling. Keywords: Stability, PI controller, PID controller, transfer function, dead time, modeling. Keywords: Stability, PI controller, PID controller, transfer function, dead time, modeling. controller tuning parameters. Shafiei and Shenton (1997) and  1. INTRODUCTION controller parameters. Shafiei and Shenton (1997) and  Huang andtuning Wang (2000) provided graphical solutions for 1. INTRODUCTION controller tuning parameters. Shafiei and Shenton (1997) and Huang and Wang (2000) provided graphical solutions for 1. INTRODUCTION controller tuning parameters. Shafiei and Shenton (1997) and determination of all stabilizing PID controller parameter Researchers have always been interested in PID controllers Huang and Wang (2000) provided graphical solutions for 1. INTRODUCTION determination of all stabilizing PID controller parameter Researchers have always been interested insystems PID controllers Huang and (2000) provided graphical solutions for values. Tan Wang et of al. all (2003) and TanPID (2005) suggested a new which are generally used industrial control owing to determination stabilizing controller parameter Researchers have always been interested insystems PID controllers values. Tan et of al. all (2003) and TanPID (2005) suggested a PI new which are generally used industrial control owing to determination stabilizing controller parameter approach providing a faster calculation of all stabilizing or Researchers have always been interested in PID controllers their simple structureused andindustrial performing robustly. Compared Tan et al. (2003) and Tan (2005) suggested a PI new which are generally control systems owing to to values. approach providing a faster calculation of all stabilizing or their simple structure and performing robustly. Compared to values. Tan et al. (2003) and Tan (2005) suggested a new PID controller tuning parameters, based on stability boundary which are generally used industrial control systems owing to PD controllers, PI controllers have a robustly. larger usage. For this providing a faster calculation of all stabilizing PI or their simple structure and performing Compared to approach PID controller tuning parameters, based on stability boundary PD controllers, PI controllers have a larger usage. For this providing a faster calculation of stabilizing PI or locus calculation. This approach has been used in boundary different their simple structure and performing Compared to approach reason, determination of tuning parameters a PI For or PID PID controller tuning parameters, based onall stability PD controllers, PI controllers have a robustly. larger of usage. this locus calculation. This approach has been used in different reason, determination of tuning parameters of a PI2001). or PID PID controller tuning parameters, based on stability boundary studies up to date. Zàvackà et al. (2013) suggested a robust PI PD controllers, PI controllers have a larger usage. For this controller is quite important (Aström and Hagglund, locus calculation. This approach has been used in different reason, determination of tuning parameters of a PI2001). or PID studies up to date. for Zàvackà et al. (2013) suggested a robust PI controller is quite important (Aström and Hagglund, locus calculation. Thisa approach has been used reactor in different controller design continuous stirred tank with reason, determination of tuning parameters of a PI or PID studies up to date. Zàvackà et al. (2013) suggested a robust PI Most commonly used methods forand determination of PID controller design for a continuous stirred tank reactor with controller is quite important (Aström Hagglund, 2001). studies upsteady-states. to date. Zàvackà et al. (2013) suggested a robust PI multiple Sandeep and Yogesh (2014) gave Most commonly used methods for determination of PID controller is quite important (Aström and Hagglund, 2001). controller design for a continuous stirred tank reactor with controllers are Ziegler Nicholsfor(1942), Cohen and multiple steady-states. Sandeep and Yogesh (2014) gave Most commonly used and methods determination of Coon PID design controller design for a continuous stirred tank reactor with of a PID controller for an inverted pendulum. Yogesh controllers are Ziegler and Nichols (1942), Cohen and Coon Most commonly used methods determination of Coon PID design multipleof steady-states. Sandeep and Yogesh (2014) gave (1953) and are Aström andand Haggland (1984) methods. Methods a PID controller for an design inverted controllers Ziegler Nicholsfor (1942), Cohen and multipleprovided steady-states. Sandeep and Yogesh gave (2016) a PI controller forpendulum. one (2014) joint Yogesh robotic (1953) and Aström andand Haggland (1984) methods. Methods controllers are Ziegler Nichols (1942), Cohen and Coon design of a PID controller for an inverted pendulum. Yogesh based on integral performance criteria (Zhuang and Atherton, (2016) provided aal.PI(2016) controller design forpendulum. one joint Yogesh robotic (1953)on and Aström and Haggland (1984) methods. Methods arm. design of a PID controller for an inverted based integral performance criteria (Zhuang and Atherton, Deniz et recommended an integer order (2016) Deniz provided aal.PI(2016) controller design for one joint robotic (1953)on andamong Aström and standard Haggland (1984) methods. Methods 1993) are very approaches asand well. Other arm. etmethod an integer order based performance criteria (Zhuang Atherton, (2016) provided a PI controller one jointlocus robotic 1993) areintegral among very standard approaches asand well. Other approximation basedrecommended on design stabilityforboundary for based on integral performance criteria (Zhuang Atherton, arm. Deniz et al. (2016) recommended an integer order methods that used for calculating PID controller tuning approximation method based on stability boundary locus for 1993) are among very standard approaches as well. Other arm. Deniz et al. (2016) recommended an integer order fractional order derivative/integrator operators. All of the methods that used for calculating PID controller tuning 1993) arethat among very standard approaches as (Morari well. tuning Other method based on stability boundaryAlllocus for parameters are used Internal Model Control (IMC) and approximation fractional order derivative/integrator operators. of the methods for calculating PID controller approximation method based on stability boundary locus for mentioned above consider the case of a specific plant parametersand arecontroller Internal Model Control (IMC) (Morari and studies fractional order derivative/integrator operators. All of the methods that used for calculating PID controller tuning Zafiriou) synthesis (Smith and Corripio, 1997) studies mentioned above consider the case of a specific plant parameters are Internal Model Control (IMC) (Morari and fractional order derivative/integrator operators. All of the function. above consider the case of a specific plant Zafiriou) controller (Smith (IMC) and Corripio, studies mentioned parametersand Internal synthesis Model Control (Morari1997) and transfer methods. transfer function. above consider the case of a specific plant Zafiriou) andarecontroller synthesis (Smith and Corripio, 1997) studies mentioned methods. transfer function. Zafiriou) and controller synthesis (Smith and Corripio, 1997) methods. transfer function. methods. In this the approach suggested by Kaya and Atic Special interest has been paid to determination of all In this paper, paper, the approach suggested by Kaya toand Atic (2016) for obtaining all stabilizing PI controllers control Special interest has been paid to determination of all this for paper, the approach suggested by Kaya toand Atic stabilizing PI and PID controller parameters after the study of In (2016) obtaining all stabilizing PI controllers control Special interest has been paid to determination of all this paper, thetime approach suggested bybeen Kayaextended and Atic stabilizing PI and PID controller parameters after the to study of In open loop stable delay processes has to (2016) for obtaining all stabilizing PI controllers to control Special interest has been paid to determination of all Ho et al. (1996, 1997a, 1997b, 1997c). Thanks these open loop stable time delay processes has been extended to stabilizing PI and PID controller parameters after the to study of all (2016) for obtaining all stabilizing PI controllers to control stabilizing PID controllers to control integrating and time Ho et al. (1996, 1997a, 1997b, 1997c). Thanks these stabilizing and PID controller parameters after theofto study of all open loop stable time delay processes has been extended to studies, allPI integral and derivative gain values a these PID Ho et al. (1996, 1997a, 1997b, 1997c). Thanks stabilizing PID controllers to control integrating and time open loop stable time processes has been extended to processes. In thisdelay approach, modelling of higher order studies, allcan integral and derivative gain plane values ofto a fixed PID delay all stabilizing PID controllers to control integrating and time Ho et al. (1996, 1997a, 1997b, 1997c). Thanks these controller be shown in the same for a processes. Incontrollers this approach, modelling of higher order studies, allcan integral and derivative gain plane valuesforof a a fixed PID delay all stabilizing PID to control integrating and time processes by a first order plus integrating plus dead controller be shown in the same studies, all integral and derivative gain valuesforofprovides PID processes delay processes. In this approach, modelling plus of higher proportional gain value. Although the plane method bymodel a first order plus integrating dead order time controller be shown in the same a a fixed delay processes. In is this approach, of higher order (IFOPDT) required. Itmodelling is assumed that relay proportionalcan gain value. Although the plane method provides processes by a first order plus integrating plus dead time controller can be shown in the same for a fixed calculation of all PI and PID controller tuning parameters, (IFOPDT) model is required. It is assumed that relay proportionalofgain value. Although the tuning methodparameters, provides feedback processes identification by a first order plusofintegrating plus dead(2001) time calculation all PI and PID controller method Kaya and Atherton proportional gain value. Although the method provides (IFOPDT) model is required. It is assumed that relay application of allthePI method time. tuning For that reason, feedback identification method of Kaya and Atherton (2001) calculation of and PIDtakes controller parameters, (IFOPDT) model is required. It is assumed that relay application of the method takes time. For that reason, can be used for this purpose. The relay feedback method calculation of PI method and PID controller tuning parameters, feedback identification method of Kaya andfeedback Athertonmethod (2001) researchers have gravitated totakes develop different approaches. can beexact used for this ifpurpose. The relay application of allthe time. For that reason, gives feedback identification method Kaya andfeedback Atherton (2001) solutions there areof no measurement errors and researchers have gravitated toand develop different approaches. application of the method takes time. For that reason, can be used for this purpose. The relay method Munro and Söylemez (2000) Söylemez et al. (2003) find researchers have gravitated toand develop different approaches. gives exact solutions ifpurpose. there areThe nosystem. measurement errors and can be used for this relay feedback method disturbances entering the control Process transfer Munro and Söylemez (2000) Söylemez et al. (2003) find gives exact solutions ifthe there are nosystem. measurement errors and researchers have toand different approaches. out a method thatgravitated provided a develop faster calculation of all PID disturbances entering control Process transfer Munro and Söylemez (2000) Söylemez et al. (2003) find gives exact solutions if there are notransfer measurement errors and function model and the controller function are first out a method that provided a faster calculation of all PID Munro and Söylemez (2000) and Söylemez et al. (2003) find disturbances entering the control system. Process transfer function model and thethe controller transfer function are first out a method that provided a faster calculation of all PID disturbances entering control system. Process transfer out a method that provided a faster calculation of all PID function model and the controller transfer function are first function model and the controller transfer function are first 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 924Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 924Control. 10.1016/j.ifacol.2018.06.104 Copyright © 2018 IFAC 924 Copyright © 2018 IFAC 924

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converted into normalized forms and then used to form stability boundary loci for obtaining all stabilizing PID controller tuning parameters for varying normalized dead time    / T . The advantage of the method is to eliminate the need of re-plotting the stability boundary locus whenever the transfer function changes so that calculation of all stabilizing PID controllers becomes easier. The rest of paper is organized as follows. Next section gives the procedure to obtain stability boundary locus in  KK c (T 2 / Ti ), KK c Td  plane for a fixed value of K K c T to

G ( j ) 

925

2

2

2

2

N e (   )  j N o (   ) D e (   )  j D o (   )

(6)

.

Dropping the dash over  for simplicity, the characteristic equation can be written as:   j    j  K K c T Ti cos    2



  K K c T Ti sin      K K c T cos    2

3



 jK K c T sin       K K c T Ti T d cos    3

2

 j  K K c T Ti T d si n    2

obtain all stabilizing PID controllers. In Section 3, the application of method is illustrated with several examples. Conclusions are given in Section 4.

3





(7)

2

 j  T Ti   T Ti  R   jI   0 .

By equating the real and imaginary parts of the characteristic equation to zero, the following equations are obtained:

2. PID CONTROLLER DESIGN FOR THE INTEGRATING PROCESSES

K K cT

K K c T   sin      

Consider single-input single-output control system depicted in Fig. 1.

2 2  cos       K K c T d    cos     

Ti

(8)

2



K K cT

K K c T   cos      

Ti

2 2   sin       K K c T d   sin     

(9)

3

 .

Defining the following equations,

Fig. 1. SISO control system

Q ( )   sin    ,

and G ( s ) are the controller and the process transfer functions, respectively. Transfer function for ideal PID controller is: C (s)

  1 C (s)  K c 1   Td s  T s i  

R     cos    ,

F       cos     , 2

X        K K c Td cos     , 2

(1)

H      2

and the IFOPDT model of process transfer function is assumed to be given by: G (s) 

Ke

(2)



G ( s) 

K Te 2

 T

s s



K Te

B      sin    , 2

Y        KK c Td sin    , 3

(3)

N     

2

 s

s s

.

(4)

KK c T 2 Ti se  s  KK c T 3 e  s  KK c TTi Td s 2 e  s  TTi s 3  T Ti s 2

2

K K cT

2

sin     .

Ti

(11)

Equations (8) and (9) are rewritten as follows:

Here, the aim is to calculate all controller parameter values in (1) to satisfy the stability of the control system shown in Fig. 1. Closed-loop characteristic equation of the system is 1  C ( s) G ( s) . Hence, substituting C ( s) and G ( s) , correspondingly, from (3) and (4), the closed loop characteristic equation can be found to be given by:  (s ) 

(10)

U      sin     ,

3

s

cos     .

Ti

S      cos     ,

By substituting T s  s in (1) and (2), the normalized controller and process transfer functions were obtained:  sT d  T C (s )  K c 1    Ti s T  

2

K K cT

and

 s

s  T s  1

2

K K c T Q ( )  K K c

T

K K c T S ( )  K K c

T

2

R ( )  X ( ),

Ti 2

U ( )  Y ( ).

Ti

(12)

and

(5) K K c T Q ( )  K K c Td F ( )  H ( ),

The numerator and the denominator of (2) have been decomposed into their even and odd parts and s  j is replaced in order to achieve

K K c T S ( )  K K c Td B ( )  N ( ).

925

(13)

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Equations (12) and (13) can be solved to obtain the following expressions: K K cT 

KK c

T

X ( )U ( )  Y ( ) R ( ) Q ( )U ( )  R ( ) S ( )

2



Ti

(14)

,

Y ( ) Q ( )  X (  ) S (  ) Q ( )U ( )  R ( ) S (  )

(15)

,

and K K c Td 

N ( ) Q ( )  H (  ) S (  )

(16)

Q ( ) B ( )  F ( ) S (  )

Equations (10) and (11) are substituted into (14), (15) and (16) to gain the following equations: 2 (17) K K c T   sin      cos(  ), KK c

T

Fig. 3. Stability boundary locus in  KK c T , KK c Td  plane for

2

Ti

   sin       cos(  )   K K c T d , 3

2

2

(18)

fixed values of K K c (T 2 / Ti ) .

and K K c T d   sin      cos(  )  

2

K K cT



i

  0.142 ,

l3 : KK c Td  1.603 KK c  T / T 

  0.143 ,

l1 : KK c Td  1.602 KK c  T / T 

2

(19)

.

l 2 : KK c  T / T 2

Ti 2

Stability boundary loci in ( K K c T , K K c (T / Ti )) plane for the normalized dead time value of   1 and fixed K K c Td values of 1 and 0.5 are drawn, by using (17) and (18), in Fig. 2. In Fig. 3 illustrates the stability boundary loci in ( K K c T , K K c Td )) plane for the normalized dead time value of



i

2

 0,



2



i



l 4 : KK c Td  0.308 KK c  T / Ti   1.987 . 2

(20)

Using the above obtained linear equations, stability boundary locus in  KK c (T 2 / Ti ), KK c Td  plane has been formed for the

  1 and fixed K K c (T 2 / Ti ) values of 1 and 0.5, by the use

normalized dead time   1 and fixed value of K K c T  1 . The result is depicted in Fig. 4.

of (17) and (19). Also, it is worth mentioning that plotting stability boundary locus for   [0,  c ] will be enough since the controller operates in this frequency range (Tan, 2005). Here,  c is the critical frequency value where the Nyquist plot of a plant transfer function intersects the negative real axis, or open loop transfer function phase is equal to  1 8 0 o . Therefore, with the help of these graphs, the following four linear equations are obtained by using K K c (T 2 / Ti ) and K K c Td values corresponding to a constant K K c T value.

Fig 4. Stability region for

 KK  T c

Fig. 2. Stability boundary locus in

 KK T , KK  T c

c

2

/ Ti 

2

/ Ti  , KK c Td



K K cT  1

and   1

in

plane.

Similar computations are carried out for normalized dead time values of   0 .7 5 ,   0 .5 and   0 .2 5 so that generalized stability boundary locus are formed in 2 plane. Stability boundary loci ( K K c (T / Ti ), K K c Td )) corresponding to those cases are presented in Fig. 5. The stability boundary loci given in Fig. 5 can be considered as generalized, since, once the IFOPDT model is known, all stabilizing PID controller tuning parameters can be found from Fig. 5 for the fixed value of K K c T  1 and varying values of normalized dead time. If it is required, stability boundary loci can be plotted for different normalized dead



plane for fixed values of K K c Td . 926

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927

time and K K c T values. By this way, the approach can be made more generalized.

Fig. 5. Stability region in

 KK  T

2

c

/ Ti  , KK c Td



Fig. 6. Step input responses for determined PID controllers for example 1.

plane for

different normalized dead time ratios and K K c T  1 .

3.2 Example 2: In this example, let's take a higher order process transfer function given by  0.2 s G (s)  e / s (0.1s  1) ( s  1 . 2) . This process transfer function is modelled as IFOPDT model of  0.299 s G m ( s )  0 . 8 43e / s (1.072 s  1) by using relay feedback identification method of Kaya and Atherton (2001). Obtained IFOPDT model transfer function has the normalized dead time of   0.2789 . Before determining all stabilizing PID controller parameters for this example, it would be appropriate to show the matching between the stability boundary locus of the actual process transfer function and the stability boundary locus of IFOPDT model transfer function. This matching is shown in Fig. 7. As it is seen, a very close matching has been achieved and the stability boundary locus obtained by the actual process transfer function includes the stability boundary locus obtained by the IFOPDT model transfer function. This is a general case observed from many different experiences. This means that the PID controller tuning parameters which are determined by each point taken from the corresponding stability boundary locus will make the system stable.

3. EXAMPLES 3.1 Example 1: Let’s consider a process transfer function of G (s)  e

s

/ s ( s  1) . The normalized dead time value for this

transfer function is   1 . Since the actual system transfer function exactly matches the IFOPDT model transfer function, the relay feedback identification method (Kaya and Atherton, 2001) will give exact solutions for the IFOPDT model. In Fig. 5, the region remaining inside of   1 can be used to determine all stabilizing PID controller tuning parameters. Some points taken from the stability region corresponding to   1 and the resultant PID tuning parameters are summarized in Table 1. Note that the controller gain K c  1 in all cases, as K  1 , T  1 for this example. Fig. 6, shows the unit step responses of the closed loop system for the determined PID controllers. The figure proves the validity of the obtained stability region. Table 1. Some calculated tuning parameters for example 1 Calculated tuning parameters

Selected points case KK c  T / Ti  2

a b c d e f

0.2 0.4 0.6 0.8 1 1.2

K K c Td

1.2 1.4 1.6 1.8 2 2.2

Ti

Td

5 2.5 1.66 1.25 1 0.83

1.2 1.4 1.6 1.8 2 2.2

Fig. 7. Stability regions for actual system and IFOPDT model transfer function of example 2.

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So, the stability region obtained in Fig. 5 for the value of   0 .2 5 , which is the closest value to the normalized dead time value of the IFOPDT model transfer function, is used to determine the PID controller tuning parameters. Table 2 summarizes the results for this example. In this example, the controller gain K c  1.106 in all cases, as K  0 .8 4 3 , T  1.072 . In Fig. 8, unit step responses are given for the determined PID controller parameter values. Again, the validity of the design approach has been verified.

Since K  1 and T  1.756 for this example, hence the controller gain K c  0.569 in all cases. Fig. 9 illustrates unit step responses for designed PID controllers. The validity of the approach has been confirmed once again. Table 3. Some calculated tuning parameters for example 3

case

Table 2. Some calculated tuning parameters for example 2 Selected points case KK c  T / Ti  2

a b c d e f

0.5 1.5 2 2.5 1 3.5

K K c Td

1 3 4.2 4.5 5 5.2

KK c  T / Ti  2

Calculated tuning parameters Ti

2.144 0.714 0.536 0.428 1.072 0.306

Calculated tuning parameters

Selected points

a b c d e f

Td

1.072 3.216 4.502 4.824 5.36 5.574

0.25 0.6 0.4 0.9 1.2 1.5

K K c Td

0.5 1.1 2.2 2 1.7 2.5

Ti

7.024 2.926 4.39 1.951 1.463 1.17

Td

0.878 1.931 3.863 3.512 2.985 4.39

Fig. 9. Step input responses for determined PID controller parameter values for example 3. 4. CONCLUSIONS In this study, a generalized method has been given for determining all stabilizing PID controllers for stability of integrating plus time delay processes. In order to implement the method, the IFOPDT model of the actual process transfer function has to be obtained. If the actual process and the IFOPDT model transfer functions matches exactly, then obtained stability regions will give exact solutions. If the actual process transfer function is a high-order transfer one, there will be a small mismatch between the stability regions obtained from the actual model transfer functions, but this will not cause any serious problem because it has been shown that the stability boundary locus of the IFOPDT model process transfer function always lies inside the stability boundary locus of the actual transfer function. Thus, the proposed approach removes the necessity of redrawing the stability boundary locus each time as the process transfer function changes.

Fig. 8. Step input responses for determined PID controller parameter values for example 2. 3.3 Example 3: In this example, another high order transfer function of G ( s )  e  0.5 s / s ( s  1)(0.5 s  1)(0.2 s  1(0.1s  1) is studied. This IFOPDT model was obtained as  1.123 s by using relay feedback Gm (s )  e / s (1.756 s  1) identification method of Kaya and Atherton (2001). IFOPDT model has the normalized dead time of   0.6395 . In Fig. 5, the stability region for the normalized dead time of   0 .7 5 can be used to determine all stabilizing PID controllers. The points and corresponding PID controller parameters taken from the inside of stability region corresponding to the normalized dead time of   0.6395 are given in Table 3. 928

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ACKNOWLEDGEMENT

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Tan, N. (2005). Computation of stabilizing PI and PID controllers for processes with time delay. ISA Transactions, 44, 213-223. Kaya, I. and Atherton, D.P. (2001). Parameter estimation from relay autotuning with asymmetric limit cycle data. Journal of Process Control, 11, 429–439. Kaya, I. and Atiç, S. (2016). PI Controller Design based on Generalized Stability Boundary Locus. International Conference on System Theory, Control and Computing (ICSTCC), 20, 24–28. Zàvackà, J., Bakošovà, M. and Matejičkovà, K. (2013). Robust PI controller design for a continuous stirred tank reactor with multiple steady-states. International Conference on Process Control (PC), 468-473. Sandeep, D.H. and Yogesh, V.H. (2014). Design of PID controller for inverted pendulum using stability boundary locus. Annual IEEE India Conference (INDICON). Yogesh, V.H. (2016). PI Controller Design for One Joint Robotic Arm. International Conference on Control, Automation, Robotics & Vision (ICARCV), 14. Deniz F.N., Alagoz B.B., Tan, N. and Atherton D.P. (2016). An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators. ISA Transactions, 62, 154-163.

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