Chemical Engineering Science 76 (2012) 66–72

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Chemical Engineering Science

0009-25

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Enhanced VRFT design of adaptive PID controller

Xin Yang a, Yan Li a, Yasuki Kansha b, Min-Sen Chiu a,n

a Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576, Singaporeb Collaborative Research Center for Energy Engineering, Institute of Industrial Science, The University of Tokyo, Tokyo 1538505, Japan

a r t i c l e i n f o

Article history:

Received 15 July 2011

Received in revised form

18 February 2012

Accepted 3 April 2012Available online 21 April 2012

Keywords:

Control

System engineering

Process control

Simulation

Virtual reference feedback tuning

Adaptive PID controller

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2012.04.007

esponding author. Tel.: þ65 65162223; fax:

ail address: checms@nus.edu.sg (M.-S. Chiu).

a b s t r a c t

In this paper, an enhanced virtual reference feedback tuning (EVRFT) design method with specific

application to adaptive PID controller design is developed. The proposed EVRFT design is an extension

of conventional VRFT design and adaptive VRFT (AVRFT) design previously developed in the literature.

The EVRFT design makes use of a second-order reference model instead of the first-order reference

model used by both VRFT and AVRFT designs. In addition, unlike the AVRFT design, the parameter of

reference model in EVRFT design is updated at each sampling instant to further enhance the

performance of resulting adaptive PID controller. Simulation results demonstrate that the adaptive

PID controller using the proposed EVRFT design gives better response than those obtained by the VRFT

and AVRFT designs.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Model-based proportional–integral–derivative (PID) control-lers have been an active research topic in the last severaldecades. Among them, a number of PID tuning formulas such asZiegler–Nichols continuous cycling method (Ziegler and Nichols,1942), direct synthesis method (Chen and Seborg, 2002; Seborget al., 2004; Panda, 2008), internal model control method (Garciaand Morari, 1982; Rivera et al., 1986; Morari and Zafiriou, 1989;Chien and Fruehauf, 1990; Lee et al., 1998; Skogestad, 2003;Vilanova, 2008), relay feedback approach (Yu, 1999; Wang et al.,2003; Huang et al., 2005; Astrom and Hagglund, 2006), frequencyresponse approach (Wang et al., 1997; Tan et al., 1999; Sung andLee, 2000; Matausek and Sekara, 2011), and optimizationapproach (Toscano, 2005) were developed. These model-based PIDcontroller design methods generally follow a two-step procedure inwhich the first step is to identify a reasonably accurate processmodel and the second step is to design the controller based on themodel thus obtained.

Different from the model-based PID controller design methodsdiscussed above, designing controllers directly based on a set ofmeasured process input and output data in one step is an attractivealternative. Toward this end, the virtual reference feedback tuning(VRFT) design method (Campi et al., 2000, 2002) determines theparameters of a controller in discrete time by using process data

ll rights reserved.

þ65 67791936.

collected from open-loop tests without resorting to the identifica-tion of a process model. As a result, the resulting controller isexpected to perform well in the vicinity of operating space close tothe operating condition where such process data is generated.Therefore, the VRFT design framework was developed for linearsystems and the resulting controller has limited performance fornonlinear processes. This design framework is recently extended tocontinuous time systems with application to PID controller design(Yang et al., 2011).

To alleviate the aforementioned limitation of VRFT design, anadaptive version of the VRFT (AVRFT) design (Kansha et al., 2008)was proposed to extend the VRFT design to nonlinear processes.In the AVRFT design, the off-line database employed in theconventional VRFT design is continuously updated by addingthe current process data into the database, by which PID para-meters are then updated by the VRFT method at each samplinginstant using the relevant data set selected from the currentdatabase according to the k-nearest neighborhood criterion.However, the parameter of reference model employed in theAVRFT design is not updated corresponding to the changingdynamics due to process nonlinearity.

To improve the AVRFT design to better cope with processnonlinearity, an enhanced VRFT (EVRFT) design with specificapplication to adaptive PID controller design is developed in thispaper. In the proposed EVRFT design, a second-order referencemodel is adopted instead of the first-order reference model usedby both VRFT and AVRFT designs. In addition, unlike the AVRFTdesign, the parameter of reference model in EVRFT design isupdated at each sampling instant to further improve the resulting

X. Yang et al. / Chemical Engineering Science 76 (2012) 66–72 67

control performance. Simulation results are presented to illus-trate the performance of the proposed EVRFT design and acomparison with the VRFT and AVRFT designs is made.

2. EVRFT and its application to adaptive PID controller design

In this section, a new formulation for VRFT design of PIDcontrollers using a second-order reference model is derived,which paves the way for the ensuing discussion of the proposedenhanced VRFT (EVRFT) design and its specific application toadaptive PID controller design.

2.1. New VRFT-based PID design using a second-order

reference model

The VRFT design approximately solves the model-referenceproblem in discrete time domain as depicted in Fig. 1, where thereference model T(z�1) describes the desired behavior of the closed-loop system consisting of a linear time-invariant process G(z�1) anda parameterized controller C(z�1;y), where y is a vector consisting ofthe controller parameters, as shown in Fig. 2 (Campi et al., 2000,2002). Assume that G(z�1) is unknown and only a set of processinput and output data, {u(k)}k¼1�n and {y(k)}k¼1�n where n denotesthe number of process data, are available, the design goal is to solvey such that the feedback control system in Fig. 2 behaves as closelyas possible to the pre-specified reference model T(z�1).

Given the measured output signal {y(k)}k¼1�n, the corre-sponding set-point signal f~rðkÞgk ¼ 1�n in Fig. 2 is obtained by

~rðz�1Þ ¼ T�1ðz�1Þyðz�1Þ ð1Þ

where ~rðz�1Þ and y(z�1) are the Z-transform of f~rðkÞgk ¼ 1�n and{y(k)}k¼1�n, respectively.

In the proposed EVRFT design, the following second-ordertransfer function, instead of the commonly used first-ordertransfer function, is adopted as the reference model:

1

ðlsþ1Þ2Ue�NDts ð2Þ

where NDt denotes the process time delay, Dt is the samplingperiod, and l determines the speed of closed-loop response. Asthe VRFT design developed by Campi et al. (2000, 2002) isformulated for the discrete time systems, the discrete timereference model T(z�1) corresponding to Eq. (2) is derived asfollows:

Tðz�1Þ ¼yðz�1Þ

rðz�1Þ¼ Z

1�e�Dts

sU

1

ðlsþ1Þ2Ue�NDts

( )

¼ z�Nð1�z�1Þ1

1�z�1�

1

1�e�ðDt=lÞz�1�

Dte�ðDt=lÞz�1

lð1�e�ðDt=lÞz�1Þ2

" #

¼ðaþbz�1Þz�N�1

1�2Az�1þA2z�2ð3Þ

)( 1−zTr~ y

Fig. 1. Reference model.

)( 1−zr + ;�)( 1−zC )( 1−zu )( 1−zG )( 1−zy)( 1−ze

-

Fig. 2. Feedback control system.

where A¼e�(Dt/l) is the tuning parameter, a¼ 1�AþAlnA, andb¼ A2

�A�AlnA.Consider a discrete time PID controller given by:

uðkÞ ¼ uðk�1ÞþKP ½eðkÞ�eðk�1Þ�þKIeðkÞþKD½eðkÞ�2eðk�1Þþeðk�2Þ�

ð4Þ

where u(k) is the manipulated variable at the kth samplinginstant, e(k) is the feedback error at the kth sampling instant,and KP, KI and KD are controller parameters. The correspondingcontroller transfer function C(z�1) is obtained as:

Cðz�1Þ ¼ KpþKI

1�z�1þKDð1�z�1Þ ð5Þ

In the VRFT design framework, the controller’s output isformulated as:

~uðz�1Þ ¼ Cðz�1Þf~rðz�1Þ�yðz�1Þg ¼ Cðz�1ÞfT�1ðz�1Þ�1gyðz�1Þ ð6Þ

Substituting Eqs. (3) and (5) into Eq. (6) obtains

~fðz�1Þ ¼ KPþKI

1�z�1þKDð1�z�1Þ

� �

�1�2Az�1þA2z�2�ðaþbz�1Þz�N�1

z�N�1yðz�1Þ ð7Þ

where

~fðz�1Þ ¼ ðaþbz�1Þ ~uðz�1Þ ð8Þ

The following equation can be obtained from Eq. (7):

~fðkÞ ¼KTcðkÞ ð9Þ

where

cðkÞ ¼ cPðkÞ cIðkÞ cDðkÞh iT

ð10Þ

K¼ KP KI KD� �T

ð11Þ

cPðkÞ ¼ yðkþNþ1Þ�2AyðkþNÞþA2yðkþN�1Þ�ayðkÞ�byðk�1Þ

ð12Þ

cIðkÞ ¼

yðkþ1Þ�Að1þ lnAÞyðkÞ for N¼ 0

yðkþ2Þ�ð2A�1Þyðkþ1ÞþbyðkÞ for N¼ 1

yðkþNþ1Þ�ð2A�1ÞyðkþNÞþð1�AÞ2XN�1

i ¼ 1

yðkþ iÞþbyðkÞ for NZ2

8>>>><>>>>:

ð13Þ

cDðkÞ ¼ yðkþNþ1Þ�ð2Aþ1ÞyðkþNÞþðA2þ2AÞyðkþN�1Þ

�A2yðkþN�2Þ�ayðkÞþða�bÞyðk�1Þþbyðk�2Þ ð14Þ

Based on Eq. (8), PID controller design is equivalent to solvingthe following minimization problem:

J1ðKÞ ¼minK

Xn

k ¼ 1

ffðkÞ�KTcðkÞg2¼min

K:U�KTW:2

ð15Þ

where

fðkÞ ¼ auðkÞþbuðk�1Þ ð16Þ

U¼ fð1Þ � � � fðnÞh i

ð17Þ

W¼

cPð1Þ cPð2Þ � � � cPðnÞ

cIð1Þ cIð2Þ � � � cIðnÞ

cDð1Þ cDð2Þ � � � cDðnÞ

264

375 ð18Þ

Consequently, PID parameters are obtained by solving the leastsquare problem stated above. It is evident that this solution notonly depends on the database used for the VRFT design but alsothe design parameter A of the reference model.

Table 1Model parameters and nominal operating condition for CSTR.

CA¼0.1 mol/L T¼438.54 K

CAf¼1 mol/L E/R¼104 K

Tf¼350 K V¼100 L

Tcf¼350 K r¼103 g/L

f¼100 L/min rc¼103 g/L

hA¼7�105 cal/(min K) Cp¼1 cal/(g K)

k0¼7.2�1010 min�1 Cpc¼1 cal/(g K)

DH¼�2�105 cal/mol fc¼103.41 L/min

X. Yang et al. / Chemical Engineering Science 76 (2012) 66–7268

2.2. Enhanced VRFT (EVRFT) design

Similar to the AVRFT design (Kansha et al., 2008), the proposedEVRFT design updates the original off-line database by adding thecurrent process data at each sampling instant so that theexpanded database can cover new operating space where itsdynamics is not available in the construction of original database.This expanded database is then used to obtain PID parameters ateach sampling instant by the new VRFT design equation discussedin the previous section. To do so, the relevant data set in theexpanded database that resembles the current process conditionis determined using the k-nearest neighborhood criterion basedon the following distance metric:

di ¼ :xonline�xi: ð19Þ

where : � : denotes the Euclidean norm, xi ¼ yðiÞ uðiÞh iT

is the ithprocess data in the present database, and xonline is the on-lineprocess data that has similar definition of xi.

Based on Eq. (19), those xi corresponding to the k smallest di

are selected as the relevant data set, by which the constrainedleast squares problem in Eq. (15) is solved to obtain PID para-meters and calculate the control action for the current samplinginstant. Subsequently, the database for EVRFT design is updatedby the current process data and the same design procedurerepeats at the next sampling instant.

While retaining some features of AVRFT design as discussedabove, the proposed EVRFT design enhances the capability to copewith process nonlinearity by updating the second-order referencemodel, instead of maintaining a fixed first-order reference modeladopted in the AVRFT design. This is achieved by updating thetuning parameter A of second-order reference model at eachsampling instant as it has a direct impact on the PID parametersobtained using the VRFT method.

To adjust the parameter A at each sampling instant, thefollowing quadratic function is considered in the ensuing discus-sion of the updating equation for A,

J2ðkÞ ¼12 ð1�wÞ½rðkþ1Þ�yðkþ1Þ�2þw½uðkÞ�uðk�1Þ�2� �

ð20Þ

where wA[0, 1] is a weight parameter and yðkþ1Þ is the desiredoutput calculated by the reference model at the (kþ1) th sampl-ing instant:

yðkþ1Þ ¼ 2AyðkÞ�A2yðk�1Þþarðk�NÞþbrðk�N�1Þ ð21Þ

Because the parameter A is constrained between 0 and 1, thefollowing sigmoid function is employed to map the set [0 1] to theset of real number B(k)AR so that B(k) can be readily updated byEq. (23) given below,

AðkÞ ¼1

1þe�BðkÞð22Þ

By the steepest descent method, the updating algorithm for Bis obtained as:

Bðkþ1Þ ¼ BðkÞ�Z @J2ðkÞ

@BðkÞ ð23Þ

where Z is the learning rate.The following outlines the computational algorithm for the

proposed EVRFT design of an adaptive PID controller.

Step 1: Given process input and output data that characterizethe dynamics of nonlinear system, the off-line database forEVRFT design is constructed as ðxiÞi ¼ 1 � n;Step 2: At the kth sampling instant, the relevant data set isselected from the current database for EVRFT design accordingto Eq. (19) and PID parameters are then obtained by solving

the minimization problem in Eq. (15). Subsequently, themanipulated variable u(k) is calculated by Eq. (4).Step 3: The database for EVRFT design is updated by append-ing the current process data y(k) and u(k), while the tuningparameter A is updated using Eqs. (22) and (23).Step 4: Set k¼kþ1 and go to step 2.

3. Examples and results

Example 1. Consider a continuous stirred tank reactor (CSTR)with a coolant stream where an irreversible, first-order, andexothermic reaction, A-B, takes place. The following first-prin-ciples equations describe the CSTR dynamics (Lightbody andIrwin, 1995; Tan and Chiu, 2001).

dCA

dt¼

f

VðCAf�CAÞ�k0CAe�E=RT

dT

dt¼

f

VðTf�TÞ�

DHk0

rCpCAe�E=RTþ

rcCpc

rCpVf cð1�e�ðhA=ðrcCpcf c ÞÞÞðTcf�TÞ

ð24Þ

where CA and CAf are the concentrations of component A in thereactor and feed stream, respectively, T and Tf are the tempera-tures of the reactor and feed stream, respectively, Tcf is thetemperature of coolant stream, f and fc are the respective flowrates of the feed and coolant streams, r and rc are the density ofthe reactant and coolant streams, Cp and Cpc are the specific heatsof the reactant and coolant, k0 and hA are the reaction rate andheat transfer coefficient, DH and E/R are the heat of reaction andactivation energy term, and V is the reactor volume. The modelparameters and nominal operation condition are given in Table 1.

The control objective is to regulate reactor concentration CA bymanipulating coolant flow rate fc, i.e., y¼CA and u¼ fc. To obtainthe database for EVRFT design, random steps around the nominaloperating condition of process input fc are generated and thecorresponding output response is given in Fig. 3. The tuningparameters used for EVRFT design are specified as follows: theinitial value of parameter A¼0.5, learning rate Z¼0.004, andweight parameter in Eq. (20) w¼0.6.

To evaluate the performance of adaptive PID controllerdesigned using the EVRFT method, successive set-point changesin a wide range of operating conditions are conducted. Forcomparison purpose, two PID controllers based on the VRFTmethods and one adaptive PID controller using the AVRFT designare designed. It is clear from Fig. 4 that the VRFT design isinadequate to provide satisfactory performance as CSTR dynamicsvary considerably in the concerned operating space due to theset-point changes. Specifically, the PID controller designed by theVRFT method using the first-order reference model (Campi et al.,2000, 2002) is not able to stabilize this process regardless of thevalue of tuning parameter A chosen, as evidenced by the oscilla-tory and unstable response observed for the last set-point change.By using the second-order reference model given in Eq. (3), thecorresponding VRFT design of a PID controller is obtained by

X. Yang et al. / Chemical Engineering Science 76 (2012) 66–72 69

Eq. (15). Fig. 4 shows that stable response is achieved when thetuning parameter A is designed conservatively (A tuned to begreater than 0.97), but at the expense of extremely sluggishresponse for the first three set-point changes. For the AVRFTdesign, Fig. 5 shows that the resulting adaptive PID controllergives stable and improved performance as compared with the twoVRFT designs discussed above. However, its performance isinferior to that obtained by the proposed EVRFT design. Fig. 6shows the corresponding updating of tuning parameter A andPID parameters of the EVFRT design for the set-point changesaforementioned.

Fig. 4. Servo response of the

Fig. 3. Input–output data used for constructing the database.

Example 2. Consider an isothermal free-radical polymerizationof methyl methacrylate (MMA) that takes place in a jacket CSTRusing azo-bis-isobutyronitrile (AIBN) as initiator and toluene assolvent. The control objective is to regulate the product numberaverage molecular weight (NAMW) by manipulating the flow rateof the initiator (FI), i.e., y¼D1/D0 and u¼FI. The dynamics of thereactor can be described by the following equations (Doyle et al.,1995):

dCm

dt¼�ðkpþkf mÞCmP0þ

FðCmin�CmÞ

V

dCI

dt¼�kICIþ

FICIin�FCI

V

dD0

dt¼ ð0:5kTc

þkTdÞP2

0þkf mCmP0�

FD0

V

dD1

dt¼Mmðkpþkf m

ÞCmP0�FD1

Vð25Þ

EVRFT and VRFT designs.

Fig. 5. Servo response of the EVRFT and AVRFT designs.

Fig. 6. Updating of tuning parameters in the EVRFT design.

Table 2Model parameters for polymerization reactor.

kTc¼ 1.3281�1010 m3/(kmol h) F¼1.00 m3/h

kTd¼ 1.0930�1011 m3/(kmol h) V¼0.1 m3

kI¼1.0225�10�1 L/h CIin¼ 8.0 kmol=m3

kp¼2.4952�106 m3/(kmol h) Mm¼100.12 kg/kmol

kf m¼ 2.4522�103 m3/(kmol h) Cmin

¼ 6.0 kmol=m3

fn¼0.58

Table 3Nominal operating condition of polymerization reactor.

Cm¼5.506774 kmol/m3 D1¼49.38182 kmol/m3

CI¼0.132906 kmol/m3 u¼FI¼0.016783 m3/h

D0¼0.0019752 kmol/m3 y¼D1/D0¼25,000.5 kg/kmol

Fig. 7. Input–output data used for constructing the database.

Fig. 8. Servo response of the EVRFT and AVRFT designs.

Fig. 9. Updating of tuning parameters in the EVRFT design for set-point changes

from 25,000.5 to 40,000 (left) and to 12,500 (right).

X. Yang et al. / Chemical Engineering Science 76 (2012) 66–7270

where

P0 ¼2f nkICI

kTdþkTc

� �0:5

:

The model parameters and nominal operation condition aregiven in Tables 2 and 3, respectively.

To obtain the database for EVRFT design, an open-looptest given in Fig. 7 is conducted by introducing random stepsaround the nominal operating condition of process input FI. Toproceed the EVRFT design, the tuning parameters are specified asfollows: the initial value of parameter A¼0.6, learning rateZ¼0.06, and weight parameter w¼0.7. To evaluate the perfor-mance of adaptive PID controller using the EVRFT design, two set-point changes from 25,000.5 to 40,000 and 12,500 are conducted,respectively. For comparison purpose, an adaptive PID controllerusing the AVRFT design is considered. As evident from Fig. 8, the

X. Yang et al. / Chemical Engineering Science 76 (2012) 66–72 71

adaptive PID controller designed by the EVRFT method outperformsits counterpart designed by the AVRFT method, resulting in 57%reduction in the mean absolute error (MAE) index. Fig. 9 shows theupdating of tuning parameter A and PID parameters of the EVFRTdesign for the abovementioned servo response.

Next, to evaluate the sensitivity of the proposed methodwith respect to noise, both process input and output data arecorrupted with 5% Gaussian white noise as shown in Fig. 10.It can be seen in Fig. 11 that the proposed design yields reason-ably good performance in the presence of process noise.Finally, suppose that there exists time delay in the outputmeasurement of two sampling times. In this case, the tuningparameters for EVRFT design are given as follows: A¼0.6, Z¼0.07,and w¼0.7. Fig. 12 illustrates that the proposed EVRFT designalso gives good performance for controlling processes with timedelay.

Fig. 10. Input–output data used for constructing the database (with noise).

Fig. 11. Servo response of EVRFT design in the presence of process noise.

Fig. 12. Servo response of the EVRFT design in the presence of time delay.

4. Conclusion

An enhanced virtual reference feedback tuning (EVRFT) designmethod with specific application to adaptive PID controller designis developed in this paper. Unlike the previous work, the proposedmethod enables the updating of controller parameters throughadjusting the parameter A of reference model at each samplinginstant. Simulation results demonstrate that the proposed EVRFTdesign gives better control performance than the VRFT design aswell as the AVRFT design.

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