# Shams Closed-loop Tuning Method

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Closed-Loop PI/PID Controller Tuning for Stable and IntegratingProcess with Time DelayMohammad Shamsuzzoha

Department of Chemical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Kingdom of Saudi Arabia

ABSTRACT: The objective of this study is to develop a new online controller tuning method in closed-loop mode. Theproposed closed-loop tuning method overcomes the shortcoming of the well-known Ziegler-Nichols (1942) continuous cyclingmethod and it can be an alternative for the same. This is a simple method to obtain the PI/PID setting which gives the acceptableperformance and robustness for a broad range of the processes. The method requires a closed-loop step set-point experimentusing a proportional only controller with gain Kc0. On the basis of simulations for a range of first-order with time delay processes,simple correlations have been derived to give PI/PID controller settings. The controller gain (Kc/Kc0) is only a function of theovershoot observed in the set-point experiment. The controller integral and derivative time (τI and τD) is mainly a function of thetime to reach the first peak (tp). The simulation has been conducted for a broad class of stable and integrating processes, and theresults are compared with a recently published paper of Shamsuzzoha and Skogestad (2010).1 The proposed tuning method givesconsistently better performance and robustness for a broad class of processes.

1. INTRODUCTIONThe proportional, integral, and derivative (PID) controller iswidely used in the process industries due to its simplicity,robustness, and wide ranges of applicability in the regulatorycontrol layer. The stable and integrating processes are verycommon in process industries in flow, level, and temperatureloop. On the basis of a survey of more than 11 000 controllersin the process industries, Desborough and Miller2 reported thatmore than 97% of the regulatory controllers utilize the PI/PIDalgorithm. A recent survey of Kano and Ogawa3 shows thatthe ratio of applications of a different type of controller, forexample, PI/PID control, conventional advanced control, andmodel predictive control is about 100:10:1. Although thePI/PID controller has only few adjustable parameters, they aredifficult to be tuned properly in real processes. One reason isthat tedious plant tests are required to obtain improved con-troller settings. Because of this reason, finding a simple PI/PIDtuning approach with a significant performance improvementhas been an important research issue for process engineers.Therefore, the objective of this paper is to develop a methodthat should be simpler with enhanced performance in closed-loop mode.There are variety of controller tuning approaches reported in

the literature,4−21 and among those two are widely used forcontroller tuning; one may use open-loop or closed-loop planttests. Most tuning approaches are based on open-loop plantinformation, typically the plant’s gain (k), time constant (τ),and time delay (θ). One popular approach is direct synthesis(Seborg et al.4) and the direct synthesis for the disturbance(DS-d) method proposed by Chen and Seborg,5 in which theyobtained the PI/PID controller parameters by computing theideal feedback controller which gives a predefined desiredclosed-loop response. The IMC based PI/PID tuning methodwas proposed by Rivera et al.,6 Skogestad,7 and Shamsuzzohaand Lee8,9 for different types of processes. Although the idealcontroller for both the approach are often more complicatedthan the PI/PID controller for time delayed processes, the

controller form can be reduced to either a PI/PID controller ora PID controller cascaded with a low order filter by performingappropriate approximations of the dead time in the processmodel. The PI/PID tuning method based on both the approachesis simpler in use with significantly improved performance. It iswell-known that the PID tuning based on both the methodsgive very good performance for set-point changes but sluggishresponses to input (load) disturbances for lag-dominant(including integrating) processes with θ/τ < 0.125. To improveload disturbance rejection, Skogestad7 proposed the modifiedSIMC method where the integral time is reduced for processeswith a large value of the time constant τ. The SIMC rule hasone tuning parameter similar to IMC, the closed-loop timeconstant τc, and for “fast and robust” control it is recommendedto choose τc = θ, where θ is the (effective) time delay. Shamsuzzohaand Lee9 developed the PID controllers in series with lead/lagcompensators for stable, integrating, and unstable processes.This method gives significantly better performance for differenttypes of second order process.However, these approaches require that one first obtains

an open-loop model of the process and then tuning of thecontrol-loop. There are two problems here. First, an open-loopexperiment, for example a step test, is normally needed to getthe required process data. This may be time-consuming andmay upset the process and even lead to process runaway.Second, approximations are involved in obtaining the processparameters (e.g., k, τ, and θ) from the data.The main alternative is to use closed-loop experiments. One

approach is the classical method of Ziegler−Nichols,10 whichrequires very little information about the process; namely, theultimate controller gain (Ku) and the period of oscillations (Pu)

Received: December 23, 2012Revised: August 7, 2013Accepted: August 12, 2013Published: August 12, 2013

Article

pubs.acs.org/IECR

© 2013 American Chemical Society 12973 dx.doi.org/10.1021/ie401808m | Ind. Eng. Chem. Res. 2013, 52, 12973−12992

which are obtained from a single experiment. For a PI-controllerthe recommended settings are Kc = 0.45Ku and τI = 0.83Pu.However, there are several disadvantages. First, the systemneeds to be brought to its limit of instability and a number oftrials may be needed to bring the system to this point. To avoidthis problem one may induce sustained oscillation with an on−off controller using the relay method of Åstrom and Hagglund,.11

However, this requires that the feature of switching toon/off-control has been installed in the system. Anotherdisadvantage is that the Ziegler−Nichols10 tunings do not workwell on all processes. It is well-known that the recommendedsettings are quite aggressive for lag-dominant (integrating)processes (Tyreus and Luyben,12) and quite slow for delay-dominant process (Skogestad7). To get better robustness forthe lag-dominant (integrating) processes, Tyreus and Luyben12

proposed to use less aggressive settings (Kc = 0.313Ku andτI = 2.2Pu), but this makes the response even slower for delay-dominant processes (Skogestad7). This is a fundamental prob-lem of the Ziegler−Nichols10 method because it uses only twopieces of information about the process (Ku, Pu), which cor-respond to the critical point on the Nyquist curve. This doesallow one to distinguish, for example, between a lag-dominantand a delay-dominant process. A fix is to use additional closed-loop experiments, for example, an experiment with anintegrating controller (Schei13), and this does allow one todistinguish between a lag-dominant and a delay-dominantprocess. A third disadvantage of the Ziegler−Nichols10 methodis that it can only be used on processes for which the phase lagexceeds −180 deg at high frequencies. For example, it does notwork on a simple second-order process.Luyben14 proposed modified Relay−Feedback method for

the identification of the process by using information of theshapes of the response curve. The method providesapproximate model for the processes that can be describedby a first-order lag with dead time. His method works on somehigher-order systems, but it is not applicable for inverse-response and unstable processes.Recently, Shamsuzzoha and Skogestad1 developed a new

procedure for PI/PID tuning method in the closed-loop mode.Their method is based on the SIMC tuning rule and providessatisfactory results for both performance and robustness. Forthe PID tuning parameter they need to repeat the experimentwith the PD controller based on the prior information obtainedfrom the P controller test. They recommended adding thederivative action only for a dominant second-order process.Haugen15 developed the “Good Gain” method in which one

must find the suitable controller gain in closed-loop mode. Likein the “Set Point Overshoot” method1 the system is notbrought into marginal stability during the tuning, and that is theadvantage of this method. The Good Gain method has asignificant drawback, as the method may not be quick to usebecause of the number of trials needed to find a good value ofthe controller gain and eventually suitable tuning parameters.Dale’s closed-loop16 PI tuning technique is mainly for an

industrial practitioner, and it is based on the trial and errorapproach in which one should have controller gain (Kcd) in aclosed-loop for the critically damped output response. In arepetitive process the suitable controller gain (Kcd) for criticallydamped output response is obtained, and then the finalcontroller gain is given based on the desired response. Thesuggested final controller gains are Kc ≈ 1.2Kcd and Kc ≈ 0.8Kcdfor desired underdamped and overdamped responses, respec-tively. A large integral time (τI) is recommended for the offset

removal, and if required derivative action can be added in thefinal setting.Hu and Xiao17 have tried to develop an analytical PI tuning

method, which resembles “Set Point Overshoot” method.1

They derived an analytical PI-tuning rule for integral plus timedelay (ITD) and first-order plus time delay (FOTD) processesusing the Set Point Overshoot method.1 The rule expresses thePI parameters in terms of the steady-state offset, peak time, andovershoot or rise time, as recorded in a closed-loopexperiment.1 The rule turns out to be applicable to a broadrange of processes typical for process control, and it givescomparable performance to the PI tuning rule proposed in therecent work of Shamsuzzoha and Skogestad.1

Yuwana and Seborg18 originally proposed a two-step tuningprocedure based on a closed-loop set-point experiment with aP-controller. They identified a first-order with delay model bymatching the closed-loop set-point response with a standardoscillating second-order step response. They used first-orderPade approximation for the time delay term in the process.They identified first overshoot and undershoot and secondovershoot from the set-point response, but the method may bemodified to not using the second overshoot, as in the presentstudy. In next step for the controller setting they used theZiegler−Nichols10 tuning rules, which as mentioned earlier maygive a rather aggressive setting.Veronesi and Visioli19 recently published another two-step

approach, where the idea is to assess and possibly retune anexisting PI controller. From a closed-loop set point ordisturbance response using the existing PI controller, theyidentify a first-order with delay model and time constant anduse this to assess the closed-loop performance. If theperformance is worse than what could be expected, then thecontroller is retuned, for example, using the SIMC method. Inanother paper, Seki and Shigemasa20 proposed to retune thecontroller based on a comparison of closed-loop responsesobtained with two different controller settings.It is important to note that often it is difficult to carryout

open-loop tests. There is always the possibility that a controlvariable may drift, and an operator may need to intervene toprevent product qualities off-specification. In the case of closed-loop tests, one can easily maintain control on the processduring the experiment and reduce the effect of disturbances tothe process operation.The PI/PID controller design method was discussed

extensively in the literature, and it shows that most of thetuning method is based on the two-steps procedure. The firststep is to find the process parameters (e.g., k, τ, and θ) by usingan open-loop or closed-loop test. The second step is to use asuitable tuning method to obtain the PI/PID controller setting.The design method, which gives the PI/PID controller

setting in a simple and effective way has always been animportant research issue for process engineers.Therefore, the present study is focused on the design of the

PI/PID controller to fulfill the various objectives: (i) Proposedcontroller tuning method should be in closed-loop mode. (ii)The PI/PID tuning rule should be simple, analytically derived,and applicable to different types of processes with a wide rangeof process parameters in a unified framework. (iii) Theproposed closed-loop tuning method should overcome theshortcoming of the Ziegler−Nichols continuous cyclingmethod. (iv) The method should be applicable to the widerange of the overshoot (approximately 10−60%) with the initialcontroller gain Kc0.

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2. IMC-PID CONTROLLER TUNING RULE

The motivation of this section is to give a brief description ofthe concept of the IMC-PID (Seborg al.4) controller tuning fora first order process with time delay.In Figure 1, the block diagram of a conventional feedback

control system is shown, where g denotes the process transfer

function and c the feedback controller. The other variables arethe manipulated variable u, the measured and controlled outputvariable y, the set point ys, and the disturbance d, which isassumed to be a “load disturbance” at the plant input. Theclosed-loop transfer functions from the set point and loaddisturbance to the output are

=+

++

ycg

cgy

gcg

d1 1s (1)

In process control, a first-order process with time delay is acommon representation of the process dynamics:

τ=

+

θ−g s

ks

( )e

1

s

(2)

where k is the process gain, τ is the lag time constant, and θ isthe time delay. Most processes in the chemical industries can besatisfactorily controlled using a PID controller:

ττ= + +

⎛⎝⎜

⎞⎠⎟c s K

ss( ) 1

1c

ID

(3)

The other structure of the PID controller-like series formof the PID can easily be transformed from eq 3 (Seborg et al.4).The following relation can express the conventional feedbackcontroller, which is equivalent to the IMC controller.

=−

c sq

gq( )

1 (4)

where g denotes the process transfer function and c and q arethe conventional controller and IMC controller, respectively.The IMC controller is designed in two steps (details areavailable in Seborg et al.4):Step 1: The process model g is decomposed into two parts:

=g p pM A (5)

where pM and pA are the portions of the model inverted andnot inverted, respectively, by the controller (pA is usually anonminimum phase and contains dead times and/or right halfplane zeros); pA(0) = 1.Step 2: The IMC controller is designed by

= −q p fM1

(6)

The IMC filter f is usually given as f = 1/(τcs+1)r where τc is an

adjustable parameter that controls the trade-off between theperformance and robustness; r is selected to be large enough tomake the IMC controller semiproper. The first order Padeapproximation has been utilized for the approximation of thedead time term in eq 2.

τ=

−

+ +

θ

θ

( )( )

g sk s

s s( )

1

( 1) 1

2

2 (7)

The resulting IMC-PID tuning formula (Seborg et al.4) aftersimplification is obtain in eq (8) for the first order process withtime delay in eq 2.

τ θτ θ

= ++

Kk

2(2 )c

c (8a)

τ τ θ= +2I (8b)

τ τθτ θ

=+2D (8c)

The IMC-PID controller designed on the basis of the IMCprinciple provides excellent set-point tracking, but has asluggish disturbance response, especially for processes with asmall θ/τ ratio.1,4−8 To improve the load disturbance response,Skogestad7 recommended modification of the integral time as

τ τ θ= +4( )I c (9)

In the proposed method, the objective is to obtain the improveddisturbance rejection response. Therefore, the integral timein eq 8b is modified similar to SIMC7 for the improveddisturbance and given as

τ τ θ τ θ= + +⎜ ⎟⎧⎨⎩

⎛⎝

⎞⎠

⎫⎬⎭min2

, 4( )I c(10)

τc = θ is the recommend setting for this tuning rule which givesmaximum sensitivity (Ms) = 1.70, approximately. The resultingsimplified tuning rule for the PID controller setting after τc = θis given as

τ θθ

= +K

k2

3c (11a)

τ τ θ θ= +⎜ ⎟⎧⎨⎩

⎛⎝

⎞⎠

⎫⎬⎭min2

, 8I(11b)

τ = τθτ + θ2D (11c)

3. CLOSED-LOOP EXPERIMENTThis section is devoted for the development of the PI/PIDcontroller based on the closed-loop data which resembles thePID tuning method in eq (11). The simplest closed-loopexperiment is probably a set-point step test (Figure 2) whereone maintains full control of the process, including the changein the output variable. The simplest to observe is the time tp toreach the (first) overshoot and its magnitude, and thisinformation is therefore the basis for the proposed method.The proposed procedure is as follows:1

(1.) Switch the controller to P-only mode (for example,increase the integral time τI to its maximum value or set theintegral gain KI to zero). In an industrial system, with bumplesstransfer, the switch should not upset the process.(2) Make a set-point change that gives an overshoot between

0.10 (10%) and 0.60 (60%); about 0.30 (30%) is a good value.Record the controller gain Kc0 used in the experiment. Mostlikely, unless the original controller was quite tightly tuned, one

Figure 1. Block diagram of feedback control system.

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will need to increase the controller gain to get a sufficientlylarge overshoot.Note that small overshoots (less than 0.10) are not

considered because it is difficult in practice to obtain fromexperimental data accurate values of the overshoot and peaktime if the overshoot is too small. Also, large overshoots (largerthan about 0.6) give a long settling time and require moreexcessive input changes. For these reasons we recommendusing an “intermediate” overshoot of about 0.3 (30%) for theclosed-loop set-point experiment.(3) From the closed-loop set-point response experiment,

one can obtain the following values (see Figure 2): Controllergain, Kc0; overshoot = (Δyp − Δy∞)/Δy∞; time from set-pointchange to reach peak output (overshoot), tp; relative steadystate output change, b = Δy∞/Δys.Here the output variable changes are given as: set-point change,

Δys = ys − y0; peak output change (at time tp), Δyp = yp − y0;steady-state output change after set-point step test Δy∞ = y∞ − y0.To find Δy∞ one needs to wait for the response to settle,

which may take some time if the overshoot is relatively large(typically, 0.3 or larger). In such cases, one may stop the experi-ment when the set-point response reaches its first minimumand record the corresponding output, Δyu, (Shamsuzzoha andSkogestad1).

Δ = Δ + Δ∞y y y0.45( )p u (12)

To make the proposed set-point experiment more under-standable, simulation has been conducted for six differentcontroller gains Kc0. The resulting closed-loop response isshown in Figure 3, which gives the overshoots of 0.10, 0.20,0.30, 0.40, 0.50, and 0.60. A typical process g(s) = e−s/(10s + 1)is considered for this analysis which has a unit time delay(θ = 1) and has a 10 times larger time constant (τ = 10).As expected, the closed-loop response gets faster and

more oscillatory as the overshoot increases. As mentionedearlier the recommended intermediate overshoot of about0.3 (30%) is the best choice for the closed-loop set-pointexperiment.Figure 4 shows set-point responses when the P-controller

gain Kc0 has been adjusted to give an overshoot of 0.3 for awide range of first-order plus delay processes with a unit timedelay (θ = 1), g(s) = e−s/(τs + 1). The process time constant τvaries from 0 (pure delay process) to 100 (almost integratingprocess). The time to reach the first peak (tp) increasessomewhat as we increase τ, but the most striking difference isthat the steady-state output change (b-value) approaches 1 as τincrease. Thus, the b-value provides an indirect measure of thevalue of τ/θ, which will be utilized in the next section.

4. CORRELATION BETWEEN CLOSED-LOOP SETPOINT RESPONSE AND THE PID SETTINGS

The objective of this paper is to provide a procedure in closed-loop for controller tuning similar to the Shamsuzzoha andSkogestad1 and Ziegler-Nichols10 method. Thus, the goal is toderive a correlation, preferably as simple as possible, betweenthe set-point response data (Figure 2) and the PID settings ineq (11), initially with the choice τc = θ. For this purpose,consider 15 first-order with delay models g(s) = ke−θs/(τs + 1)that cover a wide range of processes; from delay-dominant tolag-dominant (integrating):

τ/θ = 0.1, 0.2, 0.4, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, 7.5, 10.0,20.0, 50.0, 100.0

It is always possible to scale time with respect to the timedelay (θ) and since the closed-loop response depends on theproduct of the process and controller gains (kKc), so with-out loss of generality k = 1 and θ = 1 were used in allsimulations.

Figure 2. Closed-loop step set-point response with P-only control.

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For each of the 15 process models (different values of τ/θ),the PID settings were obtained using eq (11) with the choiceτc = θ. Furthermore, for each of the 15 processes, sixclosed-loop step set-point responses were generated usingP-controllers that give different fractional overshoots.

overshoots = 0.10, 0.20, 0.30, 0.40, 0.50, and 0.60

In total, it has then 90 set-point responses, and for each of thesefour data were recorded: the P-controller gain Kc0 used in theexperiment, the fractional overshoot, the time to reach the over-shoot (tp), and the relative steady-state change, b = Δy∞/Δys.Controller Gain (Kc). Initially the aim is to obtain a

relationship between the above four data and the correspondingproposed controller gain Kc. Indeed, as illustrated in Figure 5,where kKc was plotted as a function of kKc0 for 90 set-point

experiments, the ratio Kc/Kc0 is approximately constant for afixed value of the overshoot, independent of the value of τ/θ.Thus, it is possible to write

=KK

Ac

c0 (13)

where the ratio A is a function of the overshoot only. In Figure 6,the plot of the value of A as a function of the overshoot is given,which is obtained as the best fit of the slopes of the lines inFigure 5. The following equation (solid line in Figure 6) fits thedata in Figure 5 well and is given as

= − +A [1.55 (overshoot) 2.159 (overshoot) 1.35]2

(14)

Figure 3. Step set-point responses with various overshoots for first-order plus time delay process, g = e−s/(10s + 1).

Figure 4. Step set-point responses with overshoot of 0.3 (30%) for eight first-order plus time delay processes with τ/θ ranging from 0 to100 (g = e−θs/(τs + 1), θ = 1).

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Conclusion. The controller gain (Kc) from the closed-loopstep test is obtained from the following final eq 15. It is only afunction of initial controller gain (Kc0) and overshoot.

= − +K K [1.55 (overshoot) 2.159 (overshoot) 1.35]c c02

(15)

Integral Time (τI). It is interesting to find a simple cor-relation for the integral time. The PID method in eq 11b uses aminimum of two values for the integral time. Therefore, it isreasonable to search a similar relationship, that is, to find one

value (τI1 = τ + θ/2) for processes with a relatively large delay,and another value (τI2 = 8θ) for processes with a relativelysmall delay including integrating processes.

Case I: (Process with Relatively Large Delay). This casearise when processes have a relatively large delay i.e., τ/θ < 8,the integral action in the proposed tuning rule is to useτI = (τ + θ/2). Rearrangement of eq 11a is given as

τθ θ

=−kK3

2c

(16)

Figure 5. Relationship between P-controller gain kKc0 used in set-point experiment and corresponding IMC-PID controller gain kKc in eq 11a.

Figure 6. Variation of A with overshoot using data (slopes) from Figure 5.

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Adding both the side θ/2 in eq 16 and substitute (τ + θ/2) = τI,the resulting equation is

τ θ= kK1.5I c (17)

In eq 17, it is also required to balance the value of the processgain k, and to this effect write

= ·kK kK K K/c c0 c c0 (18)

Here, the value of the loop gain kKc0 for the P-control set-pointexperiment is given from the value of b:

=−

kK bb(1 )c0

(19)

Substituting kKc from eq 18 and Kc/Kc0 = A into eq 17, it isgiven as

τ θ=−

Ab

b1.5

(1 )I(20)

To prove this, the closed-loop set-point response is Δy/Δys =gc/(1 + gc) and a P-controller with gain Kc0, the steady-statevalue is Δy∞/Δys = kKc0/(1 + kKc0) = b. The absolute value isincluded to avoid the problems if b > 1, as may occur for anunstable process or because of inaccurate data.It is possible to obtain the value of time delay θ directly from

the closed-loop set-point response, but usually this is not alwaysan easy task. The reasonable correlation has been developed byShamsuzzoha and Skogestad1 for θ and the set-point peak timetp, which is easier to observe.For processes with a relatively large time delay (τ/θ < 8), the

ratio θ/tp varies between 0.27 (for τ/θ = 8 with overshoot = 0.1)and 0.5 (for τ/θ = 0.1 with all overshoots). For the intermediateovershoot of 0.3, the ratio θ/tp varies between 0.32 and 0.50. Aconservative choice would be to use θ = 0.5tp because a largevalue increases the integral time. However, to improve per-formance for processes with smaller time delays, it is reasonableto use θ = 0.43tp which is only 14% lower than 0.50 (the worstcase).In summary, the integral time (τI) for a process with a

relatively large time delay is

τ =−

Ab

bt0.645

(1 )I p(21)

Case II: (Process with Relatively Small Delay). Theproposed tuning rule and the Shamsuzzoha and Skogestad1

method have the same integral action for the lag-dominantprocess. For the integral time for a lag-dominant (includingintegrating) process with τ/θ > 8, the recommended tuningrule has the integral time

τ θ= 8I2 (22)

For τ/θ > 8, Figure 7 shows that the ratio θ/tp varies between0.25 (for τ/θ = 100 with overshoot = 0.1) and 0.36 (for τ/θ = 8with overshoot 0.6). It is reasonable to select the average valueθ = 0.305tp which is only 15% lower than 0.36 (the worst case).Also note that for the intermediate overshoot of 0.3, the ratioθ/tp varies between 0.30 and 0.32. In summary, the integraltime for a lag-dominant process is

τ = t2.44I2 p (23)

Conclusion. Therefore, the integral time τI is obtainedas the minimum of the above two values as recommended ineq 11b:

τ =‐

⎛⎝⎜

⎞⎠⎟A

bb

t tmin 0.645(1 )

, 2.44I p p(24)

Derivative Time (τD). A significant number of the PIDcontrollers switched off their derivative part, but proper useof derivative action can increase stability and improve theclosed-loop performance. The derivative action is veryimportant for slow moving loops where overshoot isundesirable, for example, temperature loop. The motivationof this section is to develop the approach for inclusion of thederivative action from closed-loop data. In this study the deriva-tive action is recommended for the process having τ/θ ≥ 1.The addition of the derivation action in that kind of slowprocess could be useful for the performance and stabilityimprovement.Substitute the value of τ = τI − 0.5θ into τ/θ ≥ 1, and after

rearrangement the resulting equation is given as

τ θθ

−≥

( 0.5 )1I

(25)

After simplification it is τI/θ ≥ 1.5 and the resulting constraintis kKc ≥ 1.0. The corresponding closed-loop condition for thederivative action is given as

−≥A

bb(1 )

1(26)

Case I. For an approximately integrating process (τ ≫ θ),where integral time is τI = 8θ, in the closed-loop the time delayand tp relation is θ= 0.305tp, the derivative time τD1 in eq 11ccan be approximated as

τ τθτ θ

τθτ

θ=+

≈ = = =t

t2 2 2

0.305

20.15D1

pp (27)

Case II. The process with a relatively large delay, for thiscase integral time τI = (τ + 0.5θ), and time delay in a closed-loop is θ = 0.43tp. For such cases, the derivative action is re-commended only if τ/θ ≥ 1. Assuming the case when τ = θ, theτD2 is given from eq 11c as

Figure 7. Ratio of process time delay (θ) and set-point overshoot time(tp) as a function of overshoot for four first-order with delay processes(solid lines). Dotted lines are values of θ/tp used in final correlations,(Shamsuzzoha and Skogestad1).

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Table

1.PI/PID

Con

trollerSettingforPropo

sedMetho

d(F

=1)

andCom

parisonwiththeSetPoint

Oversho

otmetho

d(hereafter,SO

Mmetho

d1)

resulting

PI/PID

-controllerwith

performance

androbustness

index

P-controlset-pointexperim

ent

setpoint

load

disturbance

case

processmodel

methods

Kc0

overshoot

t pb

Kc

τ Iτ D

Ms

IAE(y)

TV(u)

IAE(y)

TV(u)

E1+

+s

s1

(1)

(0.2

1)SO

M15.0

0.322

0.393

0.937

9.03

0.958

1.74

0.30

23.72

0.11

1.81

proposed

15.0

0.322

0.393

0.937

12.29

0.958

0.055

1.20

0.26

27.19

0.78

1.35

E2−

++

++

++

+

ss

ss

ss

s

((0.

31)

(0.0

81)

)/((

21)

(1)

(0.4

1)(0

.21)

(0.0

51)

)3

SOM

0.85

0.131

5.31

0.46

0.688

3.14

1.41

4.56

1.20

4.57

1.01

proposed

0.85

0.131

5.31

0.46

1.263

3.62

0.623

1.87

2.89

2.13

2.87

1.043

E3+

++

+s

ss

s2(

151)

(20

1)(

1)(0

.11)

2SO

M5.0

0.314

0.527

0.909

3.043

1.287

1.70

0.43

7.16

0.43

1.48

proposed

5.0

0.314

0.527

0.909

4.14

1.29

0.074

1.36

0.37

8.64

0.312

1.21

E4+

s1

(1)

4SO

M1.25

0.304

5.25

0.556

0.77

3.49

1.56

4.50

1.49

4.50

1.09

proposed

1.25

0.304

5.250

0.556

1.05

3.55

0.735

1.37

3.43

1.69

3.38

1.0

E5+

++

+s

ss

s1/

((1)

(0.2

1)(0

.04

1)(0

.008

1))

SOM

6.50

0.292

0.615

0.867

4.093

1.50

1.59

0.46

9.13

0.37

1.42

proposed

6.50

0.292

0.615

0.867

5.57

1.50

0.086

1.26

0.347

11.11

0.27

1.15

E6+

++

ss

ss

(0.1

71)

(1)

(0.0

281)

2

2

SOM

0.80

0.301

4.987

1.0

0.496

12.17

1.77

4.74

1.29

24.51

1.81

proposed

0.80

0.301

4.987

1.0

0.675

12.17

12.17

1.28

4.39

1.47

18.03

1.41

E7−

++s

s21

(1)

3SO

M0.40

0.309

5.98

0.286

0.245

1.263

2.13

7.04

1.57

8.62

1.83

proposed

0.40

0.309

5.98

0.286

0.334

1.286

3.15

8.80

2.77

10.56

3.10

E8+

ss

1(

1)2

SOM

0.58

0.307

6.19

1.0

0.357

15.10

1.75

6.21

0.90

42.33

1.72

proposed

0.58

0.307

6.19

1.0

0.485

15.10

0.87

1.34

5.73

1.05

31.13

1.38

E9+−

se

(1)s

2SO

M1.0

0.321

3.85

0.50

0.603

1.995

1.58

3.31

1.27

3.31

1.04

proposed

1.0

0.321

3.85

0.50

0.82

2.033

1.92

3.04

2.09

2.53

1.43

E10

++

−

ss

e(2

01)

(21)

sSO

M8.0

0.301

8.425

0.889

4.966

20.56

1.62

5.92

10.99

4.14

1.34

proposed

8.0

0.301

8.425

0.889

6.754

20.56

1.18

1.35

5.69

13.74

3.042

1.11

E11

−+

++−

ss

s(

1)e

(61)

(21)s

2SO

M1.40

0.344

13.67

0.583

0.817

9.602

1.59

11.72

1.60

11.78

1.09

proposed

1.40

0.344

13.67

0.583

1.112

9.786

1.91

1.44

9.27

1.91

8.831

1.06

E12

++

++

+−s

ss

ss

(61)

(31)

e(1

01)

(81)

(1)s

0.3

SOM

15.0

0.308

0.836

0.938

9.22

2.04

1.75

0.92

21.54

0.23

1.26

proposed

15.0

0.308

0.836

0.938

12.54

2.04

0.12

1.92

0.82

33.60

0.167

1.53

E13

++

+

−s

ss

(21)

e(1

01)

(0.5

1)

sSO

M4.75

0.302

2.20

0.826

2.9

5.367

1.76

2.88

6.60

1.85

1.20

proposed

4.75

0.302

2.20

0.826

4.0

5.367

0.308

2.56

2.51

14.98

1.35

2.58

E14

−+

s s1

SOM

nooscillatio

nwith

P-controller,methoddoes

notapply

proposed

nooscillatio

nwith

P-controller,methoddoes

notapply

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Table

1.continued

resulting

PI/PID

-controllerwith

performance

androbustness

index

P-controlset-pointexperim

ent

setpoint

load

disturbance

case

processmodel

methods

Kc0

overshoot

t pb

Kc

τ Iτ D

Ms

IAE(y)

TV(u)

IAE(y)

TV(u)

E14(a)

−+

−s

s(

1)e

s0.

1SO

M0.70

0.285

1.655

1.0

0.445

4.04

2.01

3.58

1.74

11.63

3.40

proposed

0.70

0.285

1.655

1.0

0.60

4.04

2.91

2.54

2.87

8.26

4.14

E15

−+ +s s

1(

1)SO

Mno

oscillatio

nwith

P-controller,methoddoes

notapply

proposed

nooscillatio

nwith

P-controller,methoddoes

notapply

E15(a)

−+ +

−s s

(1)

e(

1)

s0.

2SO

M0.51

0.31

1.55

0.338

0.314

0.418

3.90

4.12

3.88

5.26

4.41

proposed

0.51

0.31

1.55

0.338

0.433

0.432

12.29

11.78

14.52

13.2

15.22

E16

+s

1(

1)SO

Mno

oscillatio

nwith

P-controller,methoddoes

notapply

proposed

nooscillatio

nwith

P-controller,methoddoes

notapply

E16(a)

+− se (1)s

0.05

SOM

16.0

0.309

0.174

0.941

9.819

0.425

1.63

0.16

22.08

0.043

1.32

proposed

16.0

0.309

0.174

0.941

13.36

0.425

0.0244

1.73

0.15

30.64

0.032

1.26

E17

+−

se(5

1)

sSO

M4.0

0.298

3.05

0.80

2.494

6.538

1.56

2.62

4.96

2.62

1.04

proposed

4.0

0.298

3.05

0.80

3.391

6.658

0.427

1.66

1.97

7.60

1.96

1.22

E18

+−

se(

1)sSO

M0.90

0.326

2.40

0.474

0.538

1.111

1.58

2.09

1.23

2.06

1.03

proposed

0.90

0.326

2.40

0.474

0.733

1.132

1.93

2.02

1.98

1.64

1.38

E19

+− se(0

.21)

sSO

M0.30

0.292

2.0

0.231

0.189

0.325

1.67

1.88

1.12

1.87

1.10

proposed

0.30

0.292

2.0

0.231

0.257

0.331

2.08

1.93

1.58

1.90

1.55

E20

+− se(0

.05

1)

s

2SO

M0.30

0.30

2.0

0.231

0.187

0.321

1.61

1.74

1.02

1.74

1.01

proposed

0.30

0.30

2.0

0.231

0.254

0.327

1.98

1.69

1.39

1.69

1.39

E21

− es

SOM

0.30

0.30

2.0a

0.231

0.187

0.321

1.53

1.72

1.07

1.72

1.02

proposed

0.30

0.30

2.0a

0.231

0.254

0.327

1.84

1.46

1.35

1.46

1.35

E22

+−

s10

0e10

01s

SOM

0.80

0.301

3.293

0.99

0.496

8.034

1.68

3.79

1.18

16.19

1.50

proposed

0.80

0.301

3.293

0.99

0.675

8.034

0.461

1.72

3.36

1.70

11.9

1.51

E23

+ +

−s

ss

(10

1)e

(21)

sSO

M0.26

0.303

2.563

1.0

0.161

6.255

1.96

3.85

0.43

42.74

1.51

proposed

0.26

0.303

2.563

1.0

0.217

6.255

0.359

2.35

3.29

0.75

30.91

2.22

E24

− ses

SOM

0.80

0.302

3.282

1.0

0.496

8.008

1.70

3.94

1.21

16.15

1.55

proposed

0.80

0.302

3.282

1.0

0.675

8.008

0.46

1.72

3.47

1.73

11.87

1.53

E25

++

+s

ss

s(

6)(

1)(

36)

2

2

SOM

0.80

0.304

4.989

1.0

0.495

12.173

1.77

4.76

1.29

24.61

1.81

proposed

0.80

0.304

4.989

1.0

0.673

12.173

1.718

1.28

4.41

1.48

18.1

1.41

E26

−−

++

ss

s1.

6(0.

51)

(31)

SOM

−0.25

0.296

9.685

1.0

−0.156

23.632

1.77

9.46

0.41

151.3

1.82

proposed

−0.25

0.296

9.685

1.0

−0.213

23.632

1.356

1.27

8.68

0.502

111.2

1.57

Industrial & Engineering Chemistry Research Article

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Table

1.continued

resulting

PI/PID

-controllerwith

performance

androbustness

index

P-controlset-pointexperim

ent

setpoint

load

disturbance

case

processmodel

methods

Kc0

overshoot

t pb

Kc

τ Iτ D

Ms

IAE(y)

TV(u)

IAE(y)

TV(u)

E27

− ses 2

SOM

Not

possibleto

stabilize

with

PIcontroller

proposed

Not

possibleto

stabilize

with

PIcontroller

E28

−+

+ss

(2

1)(

1)3

SOM

0.40

0.309

5.98

0.286

0.246

1.263

2.14

7.04

1.56

8.62

1.83

proposed

0.40

0.309

5.98

0.286

0.334

1.286

3.15

8.80

2.77

10.56

3.10

E29

−+ +

−s s

(1)

e(

1)

s2

5

SOM

0.40

0.304

11.99

0.286

0.247

2.547

1.70

11.66

1.17

11.63

1.18

proposed

0.40

0.304

11.99

0.286

0.336

2.594

1.15

12.28

1.74

11.87

1.69

E30

++

+s

ss

9(

1)(

29)

2SO

M1.25

0.322

1.40

0.556

0.752

0.905

1.72

1.26

1.57

1.23

1.21

proposed

1.25

0.322

1.40

0.556

1.023

0.922

0.196

1.62

1.03

1.97

0.92

1.23

E31

++

+s

ss

9(

1)(

9)2

SOM

0.75

0.31

1.40

0.429

0.460

0.554

2.18

1.53

1.53

1.60

1.77

proposed

0.75

0.31

1.40

0.429

0.626

0.564

3.72

1.89

3.40

2.0

3.74

E32

++

−+

+

++

+

−s

ss

s

ss

s

((2

9)(

21)

(1)

)e)/

((0.

51)

(51)

)

s2

2

22

SOM

0.12

0.30

15.04

0.519

0.074

8.667

1.61

12.74

0.16

119.4

1.17

proposed

0.12

0.30

15.04

0.519

0.101

8.826

1.97

12.12

0.23

91.13

1.59

E33

+−

se(5

1)

sSO

M4.0

0.30

3.67

1.333

2.487

7.852

2.33

7.96

10.15

3.81

3.12

proposed

4.0

0.30

3.67

1.333

3.383

7.996

0.514

1.80

5.75

10.88

2.44

2.24

aFo

rpure

timedelayprocess(E21),obtain

t pas

endtim

eof

thepeak

(oraddasm

alltim

econstant

intheprocessforthesimulation).N

ote:

Detuningisrequiredforthecase

E15

(a).

Industrial & Engineering Chemistry Research Article

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τ τθτ θ

θθ θ

θθ

θ=+

≈+

= = = =t

t2 2 3 3

0.43

30.1433D2

2 2p

p

(28)

Note: The derivative action is only recommended for theprocesses which have τ/θ ≥ 1. The resulting criteria in theclosed-loop to add derivative action is A|b/(1 − b)| ≥ 1.Summary. The derivative action for both cases, that is, τD1

and τD2 are approximately the same, and the conservative choicefor the selection of τD is recommended as

τ =−

≥t A bb

0.14 if(1 )

1D p(29)

5. SELECTION OF PROPORTIONAL CONTROLLERGAIN (KC0)

It is mentioned earlier that the proposed method is valid for theovershoot between 0.1 to 0.6; however, an overshoot of around0.3 is recommended for a better response. Sometimes achievingthe P-controller gain (Kc0) via trial and error that gives theovershoot around 0.3 can be time-consuming.Therefore, an effective approach to get the value of Kc0 that

gives the overshoot around 0.3 is very significant for the proposedmethod. It is important to note that this procedure requires initialinformation of the first closed-loop experiment. Let us assumethat the first closed-loop test has a P-controller gain of Kc01, and a

Figure 8. Responses of the simple second-order process (1/((s + 1)(0.2s + 1))) E1. Set-point change at t = 0; load disturbance of magnitude 1 at t = 5.

Figure 9. Responses of high-order process (1/((s + 1)(0.2s + 1)(0.04s + 1)(0.008s + 1))) E5, Set-point change at t = 0; load disturbance ofmagnitude 1 at t = 10.

Industrial & Engineering Chemistry Research Article

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resulting overshoot OS1 is achieved that is between 0.1 to 0.60;this is not close to the recommended value of overshoot 0.30.Let the target overshoot be OS and the target P-controller

gain be Kc0. In the proposed closed-loop tuning method thegoal is to match the performance with the PID tuning rule. Thiscan be achieved only by maintaining a constant proportionalgain Kc, regardless of the overshoot that resulted from theclosed-loop set-point test. Ideally, Kc should be the same as thatdetermined with different overshoots from various closed-loopset-point tests and the resulting correlation is given as

− +

= − +

K

K

[1.55(OS ) 2.159(OS ) 1.35]

[1.55(OS) 2.159(OS) 1.35]1

21 c01

2c0 (30)

The above eq 30 gives a general guideline for choosing theP-controller gain for the next closed-loop set-point test. Asmentioned earlier, the proposed method is in good agreementwith the PID setting for the overshoot around 0.3. Therefore,the overshoot in eq 30 is set as 0.30, and after simplification thegain for the next closed-loop test is recommended as

= − +K K1.19(1.45(OS ) 2.02(OS ) 1.27)c0 12

1 c01 (31)

Note: It is not so important to achieve the precise fractionalovershoot of 0.3; therefore a few trials are sufficient to get thedesire overshoot around 0.3 from eq 31.A high order process given in example E2 is considered to show

the effectiveness of the proposed eq 31 for the calculation of the

Figure 10. Responses of third-order integrating process (1/(s(s + 1)2)) E8. Set-point change at t = 0; load disturbance of magnitude 1 at t = 100.

Figure 11. Responses of a third-order with positive zero and time delay process (((−s + 1)e−s)/((6s + 1)(2s + 1)2)) E11. Set-point change at t = 0;load disturbance of magnitude 1 at t = 100.

Industrial & Engineering Chemistry Research Article

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desired overshoot in the step test experiment. First trial: Let ussuppose that the P-controller is applied with an initial controllergain Kc01 = 0.85, and after the step test the resulting overshootcomes out to be OS1 = 0.13. From eq 31, the resulting controllergain for the next trial would be 1.042. Second trial: similar to firsttrial, use a controller gain of 1.042 in the second test and theresulting overshoot would be 0.18. On the basis of these two newpieces of information the controller gain would be 1.182, andcorresponding to this controller gain the overshoot will be 0.22.

6. FINAL CHOICE OF THE CONTROLLER SETTINGS(DETUNING)

The proposed method has been derived to match the per-formance with the PID tuning rule in eq (11). It is based

on the closed-loop time constant equal to the time delay(τc = θ). In real practice one may want to use less aggres-sive (detuned) settings (τc > θ), or one may even want tospeed up the response (τc < θ). To this end, we want tointroduce a detuning factor F, where F > 1 correspondsto less aggressive settings and F < 1 to more aggressivesettings.1,21

The detuning factor F has been included in the controllergain and integral time, and in conclusion the final tuningformulas for the proposed method are

=K K A F/c c0 (32)

τ =−

⎛⎝⎜

⎞⎠⎟A

bb

t F t Fmin 0.645(1 )

, 2.44I p p(33)

τ =−

≥t Ab

b0.14 if

(1 )1D p

(34)

where A = [1.55 (overshoot)2 −2.159 (overshoot) + 1.35] andF is a detuning parameter. F = 1 gives the “fast and robust”PI/PID settings corresponding to τc = θ. To detune theresponse and get more robustness one can select F > 1, but inspecial cases one may select F < 1 to speed up the closed-loopresponse.

7. SIMULATION STUDYThis section deals with the simulation study conducted on thedifferent types of representative model to cover several classesof process. The closed-loop simulations have been conductedfor 33 different processes and the proposed tuning methodprovides in all cases acceptable controller settings with respectto both performance and robustness. Several performance androbustness measures have been calculated for all 33 processesand are listed in Table 1. The brief overview of the performanceand robustness measures is mentioned here.

Figure 12. Responses of first-order with time delay process(e−s/(5s + 1)) E17. Set-point change at t = 0; load disturbance ofmagnitude 1 at t = 40.

Figure 13. Responses of pure time delay process e−s E21. Set-point change at t = 0; load disturbance of magnitude 1 at t = 15.

Industrial & Engineering Chemistry Research Article

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Output performance (y) is quantified by computing theintegrated absolute error, IAE = ∫ 0

∞|y − ys|dt. Manipulatedvariable usage is quantified by calculating the total variation(TV) of the input (u), which is the sum of all its moves up anddown. If we discretize the input signal as a sequence [u1, u2,u3 , ..., ui] then TV = ∑i = 1

∞ |ui+1 − ui|. Note that TV isthe integral of the absolute value of the derivative of theinput, TV = ∫ 0

∞|du/dt|dt, so TV is a good measure of thesmoothness.1,7−9 To evaluate the robustness, we computethe maximum closed-loop sensitivity, defined as Ms = maxω|1/[1 + g c(jω)]|. Since Ms is the inverse of the shortest distancefrom the Nyquist curve of the loop transfer function to thecritical point (−1,0), a small Ms-value indicates that the control

system has a large stability margin. It is recommended to haveIAE, TV, and Ms all to be small, but for a well tuned controllerthere is a trade-off, which means that a reduction in IAE impliesan increase in TV and Ms (and vice versa).For each process, PI/PID settings were obtained on the basis

of step response experiments with three different overshoots(about 0.1, 0.3, and 0.6) and compared with the recentlypublished method the Set Point Overshoot method.1 Theresults in Table 1 are only listed for the case of an overshootaround 0.3, but one can easily obtain the result for otherovershoots. The closed-loop performance is evaluated byintroducing a unit step change in both the set-point and loaddisturbance (ys = 1 and d = 1).

Figure 14. Responses of integrating process with time delay (e−s/s) E24. Set-point change at t = 0; load disturbance of magnitude 1 at t = 50.

Figure 15. Responses of first-order unstable process g=(e−s/(5s − 1)) E33. Set-point change at t = 0; load disturbance of magnitude 1 at t = 40.

Industrial & Engineering Chemistry Research Article

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The results for 33 example processes, without detuning(F = 1), which were studied by Shamsuzzoha and Skogestad1

are listed in Table 1. For first-order processes (E14, E15, E16),a small delay must be added (E14a, E15a, E16a) to be able toget the closed-loop overshoot needed to apply the proposedmethod.The comparison of the performance and robustness matrix

for an overshoot around 30% shows that the proposed con-troller setting response gives both smaller overshoot and fasterdisturbance rejection than the set point overshoot method. Italso gives significant advantage in overshoot and settling time,particularly in disturbance rejection. The closed−loop responsefor both the set-point tracking and disturbance rejectionconfirms the superior response of the proposed method.It provides the better controller setting for all cases withrespect to both the performance and robustness. To show the

effectiveness of the proposed method eight cases of thesimulation are shown below, which covers a wide range of theprocesses. The simulations illustrated in the figures for twodifferent overshoots (around 0.3 and 0.6) are compared withthe Set Point Overshoot method1 for the following examples.

+ +s s1

( 1)(0.2 1) (E1)

+ + + +s s s s1

( 1)(0.2 1)(0.04 1)(0.008 1) (E5)

+s s1

( 1)2(E8)

Figure 16. MV plots of E5. Set-point change at t = 0; load disturbance of magnitude 1 at t = 10.

Figure 17. MV plots of E8. Set-point change at t = 0; load disturbance of magnitude 1 at t = 100.

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− ++ +

−ss s( 1)e

(6 1)(2 1)

s

2 (E11)

+

−

se

(5 1)

s

(E17)

−e s (E21)

−

se s

(E24)

−

−

se

(5 1)

s

(E33)

Figures 8−15 present a comparison of the proposed methodby introducing a unit step change in the set point and an unitstep change of load disturbance at plant input. It is clearfrom Figures 8−15 that the proposed method constantly givesbetter closed-loop response for several types of processes.There are significant performance improvements in all the casesfor the disturbance rejection while the set-point performance ismaintained.Figures 16−18 show the manipulated variable (MV) response

of E5, E8, and E17 as the representative cases. In the beginningof Figure 16, the sharp spikes in the manipulated variable is dueto the derivative action. As mentioned earlier, TV is a goodmeasure of the smoothness of an output signal. The values of TVare also provided in Table 1 for all 33 processes.

Figure 18. MV plots of E17. Set-point change at t = 0; load disturbance of magnitude 1 at t = 40.

Figure 19. Responses of first-order with time delay process (e−0.1s/(s + 1)). Set-point change at t = 0; load disturbance of magnitude 1 at t = 2.

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The proposed method has been also compared to theLubyen14 Relay-Feedback test method for a first-order lagprocess with k = τ = 1 and deadtimes of θ = 0.1, 1, and 10. Theparameters of the PI controller settings for the Ziegler−Nichols(ZN), IMC, and Tyreus−Luyben (TL) were taken fromLubyen.14 Although the results for the proposed method havebeen compared for three different overshoots (around 0.1, 0.3,and 0.6), only overshoot around 0.3 is shown in Figure 19−21.For the result of the lag dominant process, that is, θ/τ = 0.1, theZN method shows aggressive response while IMC and TLexhibit similar responses. For the large θ/τ ratio, the closed-loop response of the ZN and TL methods are very sluggish asshown in Figure 21. From Figures 19−21, it is clear that theproposed method consistently gives better performance for awide range of θ/τ ratio.

Even though the response is not shown, simulation has beenconducted for the process g(s) = (1/8)e−θs/(s + 1)3 fordeadtime θ = 0.1, 1, and 10. It clearly shows that the proposedmethod has a significant advantage over the Lubyen14 methodfor the high-order process as well.It is important to mention that the overshoot around 0.1

typically gives slower and more robust PI/PID-settings,whereas a large overshoot around 0.6 gives more aggressivesettings. It is good because a more careful step response resultsin more careful tunings settings.The effect of using the detuning factor F is illustrated in

Figure 22 using a first order process with time delay (E18). Asexpected, using F > 1 results in more robust controller settings.A standard practice (Shamsuzzoha and Lee;8,9 Chen and

Seborg5) of using a lead-lag set-point filter is recommended to

Figure 20. Responses of first-order with time delay process (e−s/(s + 1)). Set-point change at t = 0; load disturbance of magnitude 1 at t = 10.

Figure 21. Responses of first-order with time delay process (e−10s/(s + 1)). Set-point change at t = 0; load disturbance of magnitude 1 at t = 60.

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remove the excessive overshoot from the set-point response inthe proposed method if it is required.

8. ANALYSISThe proposed closed-loop method is based on the IMC-PIDtuning rule given in eq 11. Therefore, it is interesting tocompare the results of both the methods and ensure theeffectiveness of the proposed closed-loop method.To compare the results of both methods, three typical

process models have been considered:

− ++ +

−ss s( 1)e

(6 1)(2 1)

s

2 (E11)

+

−

se

5 1

s

(E17)

+

−

s100e

100 1

s

(E22)

E17 and E22 are first-order plus delay processes, similar tothose used to develop the method. E22 is almost an integrating-with-delay process. The output responses of the proposedmethod are similar to the IMC-PID responses which are shownin Figures 23 and 24. It seems that the response is almostindependent of the value of the overshoot in all three cases.The comparison of the proposed and IMC-PID method has

been conducted for the high-order process E11, and the resultis shown in Figure 25. The model reduction technique (halfrule, Skogestad7) has been utilized to obtain the first-order plusdelay process, and the resulting process parameters areobtained as k = 1, τ = 7, and θ = 5. As expected, the outputresult of the proposed method and approximated IMC-PID isclose enough, its agreement with the IMC-PID method is bestfor the intermediate overshoot (around 0.3).The proposed tuning method is based on the IMC-PID

tuning rule given in eq (11) whereas the Set Point Overshootmethod1 is based on the SIMC rule.7 It is important to notethat the performance of both the proposed method and the Set

Point Overshoot method mainly depends upon their originaltuning rule.The performance of the SIMC and IMC-PID has been

compared and also shown in Figure 25 for the high orderprocess plus time delay E11. The figure clearly shows that theIMC-PID tuning rule gives better performance than the SIMCrule. The same observations have been found for the severalother processes, though it is not shown. It is assumed that thebest controller tuning method results in the best closed-loopoutput response. However, since both the methods utilize somekind of model reduction techniques to convert the PI/PIDcontroller to the closed-loop method, an approximation errornecessarily occurs. On the basis of the above observation, it isclear that the proposed method has better performance becauseof superior performance in its original IMC-PID tuning rule.The proposed method has advantage over other PI/PID

tuning method because of its simplicity and consistently betterperformance and robustness for a broad class of the processes.

Figure 22. Effect of detuning factor: Responses of first order process with time delay (e−s/(s + 1)) (E18). Set-point change at t = 0; load disturbanceof magnitude 1 at t = 20.

Figure 23. Responses of first-order with time delay process (e−s/(5s + 1))E17. Set-point change at t = 0; load disturbance of magnitude 1 at t = 40.

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It also has limitation because of the step test experiment in theset-point change, which might perturb the process even for ashort period of time.Sometimes in the chemical process industries, the set-point

step test experiment is not desirable due to several reasons. Forexample, changing the set point of a column temperature loopis not recommended because of off-specification of the prod-ucts. Because of this reason, occasionally we may have limita-tions in the set-point step test in chemical process industries.The proposed method is based on the step test in a closed-loopwith proportional controller (Kc0). Suitable selection of initialcontroller gain (Kc0) and subsequently number of trials cansignificantly reduce the time of the step test experiment andeventually off-specification in the product. One can stop theclosed-loop experiment just after obtaining the information offirst peak and valley. The required information (overshoot, tp)can be obtained after the first peak and valley, and then eq 12can be utilized to obtain parameter b. Along with these linesone can reduce the off-specification of the product during

controller tuning. It is not recommended to use the large testsignal amplitudes because that will cause off-specification ofproduct and/or will excite nonlinearity.

9. CONCLUSIONA simple approach has been developed for PI/PID controllertuning by the closed-loop set-point step experiment using aP-controller with gain Kc0. The PI/PID-controller settings areobtained directly from three values from the set-pointexperiment:

• overshoot, (Δyp − Δy∞) /Δy∞• time to reach overshoot (first peak), tp• relative steady state output change, b = Δy∞/Δys.

If one does not want to wait for the system to reach steady stateand speed up the closed-loop experiment, it is recommended touse the estimate Δy∞ = 0.45(Δyp + Δyu).In conclusion, the final tuning formula for the proposed

“Shams closed-loop tuning method” is summarized as

=K K A F/c c0

τ =−

⎛⎝⎜

⎞⎠⎟A

bb

t F t Fmin 0.645(1 )

, 2.44I p p

τ =‐

≥t if Ab

b0.14

(1 )1D p

where, A = [1.55 (overshoot)2 −2.159 (overshoot) + 1.35]F is a detuning parameter. F = 1 gives the “fast and robust”

PI/PID settings corresponding to τc = θ. To detune theresponse and get more robustness one can select F > 1, but inspecial cases one may select F < 1 to speed up the closed-loopresponse.An overshoot of around 0.3 is recommended for the better

response in the proposed method. The initial controller gain(Kc01) which gives an overshoot around 0.3 in the closed-looptest can be obtained from

= − +K K1.19(1.45(OS ) 2.02(OS ) 1.27)c c0 12

1 01

The proposed method works well for a wide variety of theprocesses typical for process control applications, including thestandard first-order plus delay processes as well as integrating,high-order, inverse response, unstable, and oscillating process.

■ AUTHOR INFORMATIONNotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe author would like to acknowledge the support (ProjectNumber: IN101012) provided by the Deanship of ScientificResearch at King Fahd University of Petroleum and Minerals(KFUPM). The research facilities by KFUPM are gratefullyacknowledged. I would also like to thank Prof. William L.Luyben for his helpful comments.

■ REFERENCES(1) Shamsuzzoha, M.; Skogestad, S. The setpoint overshoot method:A simple and fast closed-loop approach for PID tuning. J. ProcessControl 2010, 20, 1220−1234.(2) Desborough, L. D.; Miller, R. M. Increasing customer value ofindustrial control performance monitoringHoneywell’s experience;

Figure 24. Responses of the first-order with time delay process (approx-imately integrating process with time delay) (100e−s/(100s + 1)) E22. Set-point change at t = 0; load disturbance of magnitude 1 at t = 50.

Figure 25. Responses of third-order with positive zero and time delayprocess (((−s + 1)e−s)/((6s + 1)(2s + 1)2)) E11. Set-point change att = 0; load disturbance of magnitude 1 at t = 100.

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