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Journal of Process Control 15 (2005) 371–382

A simple method for detecting valve stiction in oscillatingcontrol loops

Ashish Singhal *, Timothy I. Salsbury

Controls Research Group, Johnson Controls, Inc., 507 E. Michigan Street, Milwaukee, WI 53202, USA

Received 30 June 2004; received in revised form 7 October 2004; accepted 7 October 2004

Abstract

This paper presents a simple and new method for detecting valve stiction in an oscillating control loop. The method is based onthe calculation of areas before and after the peak of an oscillating signal. The proposed method is intuitive, requires very little com-putational effort, and is easy to implement online. Analytical results are derived to show the theoretical basis of the new method andfield results are presented to show its effectiveness on real world control loops.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Control; Stiction; Oscillation diagnosis; Valves (mechanical); Actuators

1. Introduction

Surveys in the process industry have revealed that al-most 30% of control loops are oscillating [1,2]. Oscillat-ing loops are undesirable because they increasevariability in product quality, accelerate equipmentwear, and may cause oscillations in other interactingloops. Thus, detection, diagnosis, and correction ofoscillations are important activities in control loopsupervision and maintenance.

Some common causes of oscillations are (i) externaloscillating disturbances, (ii) poor controller tuning, (iii)nonlinearities in the actuator/plant such as static and/or dynamic nonlinearities, and stiction, or (iv) a combi-nation of these. In this paper, we focus on distinguishingoscillations caused by valve stiction from those causedby poor loop tuning, and assume that oscillations have

0959-1524/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.jprocont.2004.10.001

* Corresponding author. Tel.: +1 414 524 4688/4660; fax: +1 414524 5810.

E-mail addresses: ashish.singhal@jci.com (A. Singhal), timothy.i.salsbury@jci.com (T.I. Salsbury).

been detected by other methods such as those describedin [3,4].

A number of researchers have studied the valve stic-tion problem and suggested methods for detecting it.Horch and Isaksson [5] presented a fairly complexmethod for detecting stiction by calculating log-likeli-hood ratios for multiple models. Their method requiresknowledge of the nonlinear plant and stiction modelsand extended Kalman filtering. Stenman et al. [6]also proposed a complicated method based on ‘‘multi-model mode estimation’’ and change detection. Theirmethod requires identifying time-series models andperforming optimization to obtain the log-likelihoodratio.

Horch presented two more methods for detectingstiction in oscillating loops [7,8]. The first method de-tected valve stiction by analyzing the cross-correlationfunction (CCF) between the controller output (u) andthe plant output (y). He proposed that a sticking valveresults in a phase lag of 90� (odd CCF) between u andy, while an aggressive controller or an oscillating distur-bance results in a phase lag of 180� (even CCF). Thephase lag is 180� for an aggressive controller when the

0

time

cont

rol e

rror

Valve stiction

0

time

cont

rol e

rror

Aggressive control

A1

A2

A1

A2

A1

A2

____ > 1 ____A

1

A2

~ 1

Fig. 1. Control error signal shapes for valve stiction and aggressive control.

1 The control error is the difference between the setpoint and theprocess variable being controlled.

2 Plant with all left-half plane poles.

372 A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382

loop cycles due to controller output saturation.However, when stiction is present and the controlleroutput is not saturated, the phase lag can lie between90� and 180� for a PI controller.

Horch�s second method detected the differencesbetween the shapes of the signal oscillating because ofstiction and aggressive control using probability distri-butions. The method involved calculating filtered deriv-atives of the plant output and then analyzing the shapeof the probability distribution for the derivative signalby either performing a nonlinear fit to two probabilitydistributions (one for stiction, and one for aggressivecontrol/sinusoidal disturbance), or manually observingthe shapes of the two distributions. Although filteringthe plant output signal was recommended before calcu-lating derivatives, we found that even after filtering, thecalculation of derivatives amplified moderate amountsof noise and blurred the distinction between the shapesof the two probability distributions.

Choudhury et al. [9] used bicoherence to detect valvestiction by identifying non-Gaussian and nonlinear com-ponents in the signal. They presented simulation resultsfor detecting nonlinearities in the signals using a stictionmodel [10] and also detected signal nonlinearities inindustrial data. A manual inspection of the controlledvariable–controller output (pv–op) plot was then re-quired to determine the cause of the nonlinearity.According to Choudhury et al., nonlinearity in the sig-nal could be present because of stiction, dead-zones,hysteresis in the control valve, or the nonlinear natureof the process itself. Thus, the bicoherence test detectedthe presence of signal nonlinearities, and not specificallystiction.

Gerry and Ruel have published several papers ondetecting and measuring valve stiction by manuallyinspecting the shapes of the control error and the con-troller output signals during sustained oscillations [11–13]. They suggested that the controller output wouldbe a saw-tooth or triangular wave for a sticking valveand a sinusoid for an aggressive controller. Addition-ally, a sticking valve was assumed to produce a

‘‘square-shaped’’ control error signal, while an aggres-sive controller produced a sinusoidal signal. Ruel alsoproposed a test to quantify the amount of stiction byputting the controller in manual mode, and then execut-ing a series of small step-changes in the controller out-put until the controlled variable showed a change inits steady-state value.

Our new automated method is also based on distin-guishing between the shapes of the signals caused byan aggressive controller and a sticking valve using theratio of the areas before and after the peak of the con-trol error signal 1 as shown in Fig. 1. The idea is simple,easy to implement online, and requires little computa-tional effort.

This paper presents the proposed method in Section2. The stiction model of Choudhury et al. [10] is dis-cussed in Section 3 and a theoretical analysis of themethod is presented in Section 4. We focus on present-ing a theoretical framework for analyzing oscillationscaused by stiction in closed loops and illustrate the ideausing a simplified stiction model and popular plant mod-els. The same framework could be used to analyze oscil-lations with more complicated models.

A comparison between the results obtained using asimplified stiction model and the complete Choudhurymodel is presented in Section 5. Practical considerationsare discussed in Section 6 and a field result is presentedin Section 7.

2. Proposed method

For self-regulating plants 2 with a monotone step-re-sponse, aggressive control usually results in a sinusoidalcontrol error signal, while for a sticking valve, the signaltypically follows exponential decay and rise as shown in

A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382 373

Fig. 1. The reason for this behavior is that while theplant input is continuous for aggressive control (exceptwhen the controller output is saturated), valve stictionresults in a discontinuous plant input that closely resem-bles a rectangular pulse signal.

The new stiction detection methodology distinguishesbetween the shapes of the two signals in Fig. 1 by calcu-lating the ratio of the areas before and after the peaks.This quantity is called R and is defined as

R ,A1

A2

:

The decision rule is summarized as

R > 1 ) Sticking valve;

R � 1 ) Aggressive control

The proposed idea is very easy to implement online sothat stiction detection can be performed by a field con-troller at faster sampling rates compared to download-ing and analyzing data at an operator workstation.Also, online implementation in a field controller wouldresult in reduced traffic on a control network.

The assumptions for using the proposed method todetect stiction are: (1) the controller output is not cy-cling from one saturation limit to the other, and (2)the oscillations in the control loop are not caused byan external periodic disturbance. Violation of theseassumptions can result in R > 1 even when stiction isabsent.

The first assumption can be verified by observing ifthe controller output hits saturation limits. Satisfyingthe second assumption requires overriding and holdingthe controller output at its current value and againdetecting the presence of sustained oscillations. If thecontrol error signal still exhibits oscillations, an externalperiodic disturbance is likely to be causing them.

3. Plant and stiction models

3.1. Plant model

The proposed stiction detection method is designedfor self-regulating plants with monotone step-response.These plants can be represented by the nth-order trans-fer function,

Gp ¼Kpe

�Ls

ðT 1sþ 1ÞðT 2sþ 1Þ . . . ðT nsþ 1Þ ; ð1Þ

where n is the plant order, Kp is the plant gain, L is thedelay in the plant, and Ti are time-constants of differentdynamic components. The delay, L, is the amount ofdead-time in the plant, i.e., the time during which thereis no plant response to an input change. It is common

practice to approximate the nth-order transfer functionin Eq. (1) to first-order for controller tuning. Thisapproximation results in the popular first-order plustime-delay (FOPTD) transfer function,

Gp ¼Kpe

�~hs

ssþ 1; ð2Þ

where s is an approximation of the plant dynamics,and ~h represents a combination of the pure delay (L)and an apparent delay (h) caused by the higher orderdynamics.

It is important to distinguish between the true andapparent time-delay in order to understand the stictionmodel. First we present expressions for the apparenttime-delay and time-constant for an nth-order plantand subsequently show the effect of apparent time-delayon the stiction model.

In Eq. (1), the system gain can be set to unity withoutloss of generality. Because the focus of this section is toanalyze the effect of the plant�s order and not its differentdynamic components, we set Ti = T (i = 1, . . . ,n). Thissimplification also reduces the number of free parame-ters in the nth-order model and allows us to derive ana-lytical expressions for s and ~h. The nth-order plant nowbecomes:

Gp ¼e�Ls

ðTsþ 1Þn : ð3Þ

We use Astrom and Hagglund�s method [14] to calculates and ~h. Their method requires first calculating the aver-age residence time of the higher-order plant. The appar-ent time-delay is then calculated by finding theintersection of the tangent drawn through the inflectionpoint of the unit-step response with the time-axis. Theapparent time-constant is finally calculated by subtract-ing the apparent time-delay from the average residencetime. By following this procedure, the apparent time-constant and time-delay for the nth-order plant de-scribed by Eq. (3) are calculated as

s ¼ T 1þ1� e�ðn�1Þ Pn�1

i¼0

ðn�1Þii!

� �ðn�1Þn�1e�ðn�1Þ

ðn�1Þ!

� �2664

3775 ð4Þ

and

h ¼ T ðn� 1Þ �1� e�ðn�1Þ Pn�1

i¼0

ðn�1Þii!

� �ðn�1Þn�1e�ðn�1Þ

ðn�1Þ!

� �2664

3775: ð5Þ

Adding the pure delay, L, to the apparent time-delay re-sults in the effective time-delay ~h ¼ Lþ h. We define aquantity k as the ratio of the effective time-delay to theapparent time-constant:

Table 1k values for nth-order plants (L = 0)

n 1 2 3 4 5 6 7k 0 0.16 0.37 0.55 0.72 0.88 1.03

374 A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382

k ,

~hs: ð6Þ

The ratio k usually indicates the difficulty of control—alarge k means that the plant is more difficult to controlwith a PI controller. The variation of k with n for plantswith zero pure-delay is presented in Table 1. Theeffect of k on the input–output (I/O) characteristics ofa sticking valve will be discussed in the followingsection.

3.2. Stiction model

Understanding the type of oscillations caused by asticking valve in a control loop requires a good graspof the stick-slip phenomenon. Several researchers havemodeled stiction in mechanical systems and then ana-lyzed the behavior of oscillations caused by stiction ina closed loop. Armstrong-Helouvry et al. [15] presenteda comprehensive review of models, methods and controlof mechanical systems with friction. They list contribu-tions from tribology, lubrication, physics and control.Many researchers model friction as a combination ofstatic, coulomb and viscous friction and have analyzedoscillations in position control systems with friction[15–18].

The disadvantage of physics-based models is thatthey require several physical parameters such as theamount of static, coulomb and viscous friction, springtension, system mass, etc. to model the process accu-rately. In most cases, calibrating these parameters isnot an easy task. Thus, Choudhury et al. [10] introducedan empirical valve stiction model that has behavior sim-ilar to the physics-based models. Their empirical modelhas only two parameters that are intuitive and easy todefine. In this paper, we use the Choudhury model toanalyze valve-stiction in an oscillating control loop.We also use a simplified form of the model to deriveanalytical expressions and validate our approach, anduse numerical simulation to test the method with the fullmodel.

Choudhury et al. [10] developed an empirical modelfor valve stiction that produces input–output (I/O)behavior similar to that of more complicated physics-based models. The I/O characteristics of a sticking con-trol valve are presented in Fig. 2(a). For most plantswith k > 0, the I/O characteristics contain a dead-band + stickband, a slip jump, and a sliding part wherethe valve moves with the controller output. However, ifk = 0 (i.e., for a pure first-order plant), only the dead-

band and the slip jump part can be seen. This processis shown by the solid line in Fig. 2 and has the samecharacteristics as a relay with hysteresis. The stictionmodels used by Stenman et al. [6] and Horch [7] alsohave the characteristics of a relay.

The sliding part shown in Fig. 2(b) appears when atime-delay is added to the plant. For small values of k,the relay is a good approximation of the Choudhurymodel. Note that the Choudhury model becomes a relaywith hysteresis when k = 0.

For a second-order plus time-delay (SOPTD) plant,the relay approximation becomes less accurate becausethe k values are larger as shown in Fig. 2(c). Becausek > 0 for a SOPTD plant, the relay model is always anapproximation for this system. Fig. 2(d) presents theI/O characteristics of a sticking valve when the puretime-delay is zero. This figure shows that for higher-or-der plants, the relay model becomes less accurate com-pared to the Choudhury model because k increaseswith the plant order. When k is small, the relay is a goodapproximation of the stiction behavior.

The results in this section also demonstrate thatalthough L and h are different in nature, both result inthe emergence of the sliding part of stick-slip behavior.Fig. 2 shows that the I/O characteristics of a stickingvalve in closed-loop depend not only on the deadbandand slip-jump parameters, but also on the plant dynam-ics. We will show later in the paper that a pure time-delay contributes more to the differences in the twomodels than higher-order dynamics.

4. Analysis for first and second-order plus time-delay

plants

In this section, we analyze the behavior of first andsecond-order plus time-delay plants with valve stiction.The relay approximation is used to model stiction in or-der to derive analytical expressions for the ratio R.

4.1. First-order plus time-delay (FOPTD) plant

The FOPTD plant considered in this analysis is,

GpðsÞ ¼e�s

Tsþ 1; 1 6 T 6 10 ð7Þ

so that 0.1 6 k = 1/T 6 1.Analytical expressions for R are obtained by the fol-

lowing steps: (1) calculation of the oscillation frequency,(2) calculation of the steady-state periodic plant output,and (3) calculation of R from the steady-state periodicplant output.

Let Gc(s) be the transfer function of the controller.An estimate of the oscillation frequency is obtained bysolving for the frequency at which the Nyquist curveof Gc(jx) · Gp(jx) intersects the negative inverse of the

slip jump

deadband+stickband

valve input/controller output (u)

valv

e ou

tput

(x)

plant with time-delay (Choudhurymodel)

plant withno time-delay(relay)

valve input (u)

valv

e ou

tput

(x) λ = 0

λ = 0.1

λ = 0.5

λ = 0.83

valve input (u)

valv

e ou

tput

(x)

λ = 0.16

λ = 0.22

λ = 0.45

λ = 0.63

relay(λ = 0)

valve input (u)

valv

e ou

tput

(x)

firstorder

secondorder

thirdorder

pure timedelay = 0

(a) Sticking valve I/O (b) FOPTD plant

(d) Higher order plants(c) SOPTD plant

Fig. 2. Closed-loop I/O characteristics of a sticking valve.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2-1.5

-1

-0.5

0

0.5

1

1.5

imag

inar

y ax

is

increasing a

Gp = e-s

Ts+1

Gc =Kc(τI s+1)

τI s

osc

ωosc

Gc(jω) Gp(jω)

-1N(a)

-πα4

increasing ω

real axis

Fig. 3. Estimation of the oscillation frequency using a Nyquist plot.

A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382 375

describing function 3 of a relay [14] as shown in Fig. 3.Let Kc and sI be the proportional gain and integral timefor the PI controller, and a be the ratio of the deadband(e) to the slip-jump (d) parameters, then the oscillationfrequency is calculated by solving the following nonlin-ear equation for x:

3 Describing function: NðaÞ ¼ p4d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � e2

pþ j pe

4d

h i�1.

4Kcðcosxþ xsI sinxÞ þ 4KcxT ð� sinxþ xsI cosxÞ¼ paxsIð1þ x2T 2Þ: ð8Þ

The solution of Eq. (8) is xosc and the oscillation half-period is b = p/xosc.Remarks:

(1) The describing function approach provides anapproximate analysis and the actual oscillation fre-quency may be different from the calculated one.The magnitude of this difference depends on thenature of the nonlinearity and the spectral charac-teristics of GcGp. The reader may refer to Section7.2 of [19] for details regarding the describing func-tion method.

(2) The oscillation frequency xosc depends on the ratioa = e/d, and not on the individual magnitudes of thedeadband and slip-jump parameters. A larger a(large deadband, small slip-jump) results in a loweroscillation frequency and a smaller a (small dead-band, large slip-jump) results in a higher oscillationfrequency. Additionally, if there is no stiction, thenthere are no oscillations due to stiction and a = e/d = 0/0 is undefined.

We use LePage�s method [20] to determine the steadystate response of a first-order plant given by Eq. (7) to a

0 2 4 6 8 10

1

5

10

b /T

R

Fig. 5. Variation of the area ratio, R, with b/T for a FOPTD plant.

376 A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382

rectangular wave input. The Laplace transform of arectangular wave of unit amplitude and period 2b istanh(bs/2)/s. After solving for the system output, drop-ping the transient response and e�2bs terms, and takinginverse Laplace transform we obtain the steady-stateperiodic system output as

pðtÞ ¼ p0ðt � k � 2bÞ; k ¼ bt=2bc; ð9Þwhere k is the integer part of t/2b, and p0 is the repeatingpart of p(t) given by

p0ðtÞ ¼ 1� 2

1þ e�b=T

� �e�t=T � 2ð1� e�t=T Þhðt � bÞ;

0 6 t < 2b; ð10Þ

where h(t � b) is the Heaviside step function with a lagof b. The function p0(t) for b = 1 and T = 1 is plottedin Fig. 4.

The areas before and after the peak are denoted by A1

and A2. Using Fig. 4 and symmetry, the two areas arecalculated as

A1 ¼Z b

tz

p0ðtÞdt and A2 ¼ �Z tz

0

p0ðtÞdt; ð11Þ

where tz ¼ T logeð1þ K1Þ and K1 ¼ ðeb=T�1Þ2ðe2b=T�1Þ. After substi-

tuting Eq. (10) in Eq. (11), the expression for the ratioR = A1/A2 is found to be:

R ¼bT �

tzT

� �þ ð1þ K1Þðe�b=T � e�tz=T Þ

ð1þ K1Þð1� e�tz=T Þ � tzT

: ð12Þ

Because tz/T and K1 depend only on b/T, the expressionfor R in Eq. (12) also depends only on the ratio b/T, thatis, the ratio of the areas, R, is a function of the ratio ofthe oscillation period and the plant time constant. Alsonote that R depends on the ratio, a, of the deadband and

0 0.5 1 1.5 2–0.5

–0.25

0

0.25

0.5

t

p0(t)

b = 1T = 1

A1

A2

– A2

t z b

Fig. 4. Steady-state response of a first-order plant to a rectangular-wave.

slip-jump parameters and not their individualmagnitudes.

A plot of R for different values of b/T is presented inFig. 5 that shows R P 1. The reader may verify thatwhen b/T ! 0, R! 1. For aggressive controllers, b issmall. Thus, as the controller becomes more aggressive(smaller b/T), R becomes smaller.

To observe the effect of controller tuning on R, we se-lected four different PI controller tuning methods to cal-culate Kc and sI using the parameters ~h ¼ 1 and s = T.The different PI controller tuning methods can be foundin [21] and were:

(1) ApproximateMS-constrained Integral Gain Optimi-zation (AMIGO) tuning rule [22].

(2) Chien–Hrones–Reswick (CHR) rule for setpointchange with 0% overshoot.

(3) Ziegler–Nichols (ZN) tuning rule.(4) Cohen–Coon (CC) tuning rule.

When the controller and plant transfer functions areknown, the control performance can be measured usingthe maximum values of the absolute sensitivity and com-plimentary sensitivity functions. These maximum valuesare denoted byMS andMT. For satisfactory control,MS

should be in the range of 1.2–2.0, and MT should be inthe range of 1.0–1.5; higher values mean aggressive con-trol, while lower values mean sluggish control [23]. TheCHR and AMIGO methods result in well-tuned con-trollers, while Ziegler–Nichols and Cohen–Coon meth-ods result in aggressive control. Fig. 6 shows thevariation of R for different controller tuning methods.

Fig. 6 shows that for aggressively tuned controllers,such as the ones tuned using the Ziegler–Nichols andCohen–Coon methods, the value of R is closer to unity

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

5

10 AMIGO

λ

R

CHR(0% overshoot)

ZieglerNichols

CohenCoon

Fig. 6. Variation of the area ratio, R, with 1/T for a FOPTD plant anddifferent PI controller tuning rules.

0 0.5 1 1.5 2

–0.1

0

0.1

t

p0(t)

–A2–A1,1

A1,2

b = 1T = 1

A2

A1

tp

tz b

b + tp

Fig. 7. Steady-state response of a second-order plant to a rectangular-wave.

A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382 377

compared to more conservative tuning using the AMI-GO and CHR methods.

Out of the four curves of Fig. 6, only the CHR curveshows non-monotonic behavior. The reason for thisbehavior is that the CHR tuning rule uses only thetime-constant to calculate the integral time while theother three methods use the time-delay information aswell. Thus, for small k (or large T), the CHR rule resultsin larger sI compared to the other three tuning methodsand consequently results in larger values of R.

4.2. Second-order plus time-delay (SOPTD) plant

The SOPTD plant considered in this analysis is [14]:

GpðsÞ ¼e�s

ðTsþ 1Þ2; 1 6 T 6 10: ð13Þ

Following the procedure described in Section 4.1, theperiodic part of the steady-state output for the SOPTDplant is found to be:

p0ðtÞ ¼ 1� ½ð1þ K2Þ þ ð1þ K3Þt=T �e�t=T

� 2½1� ð1þ t=T Þe�t=T �hðt � bÞ; ð14Þ

where

K2 ¼ � ð2b=T Þð1� e2b=T Þðeb=T � e2b=T Þð1� e2b=T Þ2

" #

� ð1� eb=T Þ2ð1� e2b=T � ð2b=T Þe2b=T Þð1� e2b=T Þ2

" #; ð15Þ

K3 ¼ðeb=T � 1Þ2

ðe2b=T � 1Þ

" #: ð16Þ

Eq. (14) is plotted in Fig. 7 for the values b = 1 andT = 1.

Using Fig. 7 and symmetry, the areas A1 and A2 arecalculated as

A1 ¼ �Z tp

0

p0ðtÞdt|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}A1;1

þZ b

tz

p0ðtÞdt|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}A1;2

; ð17Þ

A2 ¼ �Z tz

tp

p0ðtÞdt; ð18Þ

where tp is the location of the first trough of the response(0 6 tp 6 b), and tz is the time when p0(t) = 0(0 6 tz 6 b). Calculation of tz requires solving the non-linear equation (14), while tp is calculated as

tp ¼ TK3 � K2

1þ K3

� �: ð19Þ

Substituting Eq. (14) in Eqs. (17) and (18) we have,

A1 ¼ ðb� tz � tpÞ þ T ð1þ K2Þ½1þ e�b=T � e�tz=T

� e�tp=T � þ T ð1þ K3Þ½1þ ð1þ b=T Þe�b=T

� ð1þ tz=T Þe�tz=T � ð1þ tp=T Þe�tp=T � ð20Þ

and

A2 ¼ ðtp � tzÞ þ T ð1þ K2Þ½e�tp=T � e�tz=T � þ T ð1þ K3Þ� ½ð1þ tp=T Þe�tp=T � ð1þ tz=T Þe�tz=T �: ð21Þ

The ratio of the areas, R, is simply:

R ¼ A1

A2

: ð22Þ

R is plotted for different values of b/T in Fig. 8. Aninteresting observation from Fig. 8 is that R < 1 forb/T < 2.46. Recall that a smaller value of b means a fas-ter/more aggressive controller. Thus, Fig. 8 suggests thatfor a second-order plant, R can be less than one foraggressive controllers when valve stiction is present.

0 2 4 6 8 10

1

5

10

b / T

R

2nd order system

1st order system

Fig. 8. Variation of the area ratio, R, with b/T for a SOPTD plant.

378 A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382

This phenomenon appears to be caused by the change inthe curvature of the step response of a second-order sys-tem. Assuming a positive plant gain, the second-deriva-tive of the step-response of a second-order system ispositive before the inflection point, and negative after-wards. For small values of b/T, a major proportion ofthe periodic response is before the inflection point (posi-tive second-derivative). By symmetry, the response afterthe peak of the signal also has a positive second-deriva-tive. These signal shapes cause the ratio R to become lessthan unity. However, for large b/T the major fraction ofthe periodic response is after the inflection point and wehave R > 1.

To observe the effect of controller tuning on R for aSOPTD plant, we apply tuning rules to determine con-troller settings. Because tuning rules commonly rely onFOPTD parameters, the SOPTD model is reduced toa FOPTD form to calculate the controller settings.By substituting n = 2 in Eqs. (4) and (5), the calcu-lated effective time-delay and the apparent time-constant for the SOPTD plant are ~h ¼ 1þ ð3� eÞTand s = (e � 1)T, respectively. The controller is tunedusing these apparent FOPTD plant parameters and thetuning methods listed in Section 4.1.

The variation of R with k for different tuning meth-ods is presented in Fig. 9. The figure shows that aggres-sive tuning methods such as ZN and CC result in smallerR values, while AMIGO and CHR methods that pro-duce satisfactory control performance result in largerR values.

In addition to calculating R for different tuning meth-ods, the sensitivity function,MS, and the complimentarysensitivity function MT are also presented in Fig. 9. Thefigure shows that for small values of k, MS and MT arelarge for Ziegler–Nichols and the Cohen–Coon tuningmethods. Thus, Ziegler–Nichols and Cohen–Coon

methods result in R < 1 because they produce control-lers that are too aggressive.

Fig. 9 shows that although the R-curves for ZN andCC methods are close to each other, their MS and MT

curves are far apart. The reason for this discrepancy ap-pears to be the difference in the controller setttings cal-ulated by the two methods. While both ZN and CCmethods result in very similar controller gains, the CCmethod calculates shorter integral times. Thus, the MS

andMT values are larger for the CC method. Also, it ap-pears that shorter integral time affects R less than thesensitivity functions.

5. Comparison of relay and Choudhury models for

valve stiction

In this section, we compare results in Section 4 usingthe Choudhury model. Because the describing functionfor the Choudhury stiction model is much more compli-cated than the relay model [24], obtaining analyticalexpressions for R is difficult. Thus, we present resultsby simulating a closed loop with stiction described bythe Choudhury model and calculating R numerically.To improve the accuracy of the numerical calculations,the sampling period for control and measuring processvariables is set to a 200th of the plant time-constant.

Fig. 10 presents the variation of R with k for the Cho-udhury and the relay stiction models when the controlleris tuned using the AMIGO method. Although, the curvefor the Choudhury model is lower than the curve for therelay model, the value of R is greater than unity for bothmodels. This result shows that the proposed method candetect valve stiction described by both models when thecontroller is well tuned.

The reason for the lower R values for the Choudhurymodel can be explained as follows: when the originalChoudhury model is used to simulate stiction in a closedloop, the valve output (or the plant input) is not a rect-angular wave but a combination of a step and a ramp asshown in Fig. 11. The step part of the signal correspondsto the slip-jump, while the ramp part is the sliding partof the I/O characteristic as shown in Fig. 2. As the delayincreases, the sliding part becomes larger compared tothe slip-jump and the valve output moves with the con-troller output for a longer period of time and results in asmaller R value.

The value of R increases with increasing k because thecontrollers become less aggressive (decreasing MS andMT in Fig. 9). The reason for the increasing differencebetween the two curves with k is that the difference be-tween the I/O characteristics of a relay and the Choudh-ury model increases with k as shown in Fig. 2. For thesame reason, the difference in the two curves is also lar-ger for the SOPTD plant compared to the FOPTDplant.

0.2 0.4 0.6 0.8 1

1

5

10

λ

R

(a) FOPTD plant

0.3 0.4 0.5 0.6 0.7

1

5

10

λ

R

(b) SOPTD plant

relay model

Choudhurymodel

relay model

Choudhurymodel

Fig. 10. Comparison of the area ratio, R using the relay and the Choudhury models with the AMIGO PI controller tuning rule.

0.2 0.3 0.4 0.5 0.6 0.71

3

5

MS

0.2 0.3 0.4 0.5 0.6 0.71

3

5

λ

MT

0.2 0.3 0.4 0.5 0.6 0.7

1

5

10

R

AMIGO

ZN

CHR

CC

AMIGO CHR CCZN

AMIGO CHR ZN

CC

Fig. 9. Variation of the area ratio, R, with k for a SOPTD plant and different PI controller tuning rules.

A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382 379

Fig. 12 shows the variation of R for different plant or-ders (cf. Eq. (3)) with no pure time-delay. In this situa-

tion, the difference between the relay and the originalstiction models is smaller, and the two curves do not

0 (1) 0.16 (2) 0.37 (3) 0.55 (4) 0.72 (5) 0.88 (6) 1.03 (7)

1

5

10

λ (n)

Rrelay model

Choudhurymodel

L = 0

Fig. 12. Comparison of R for the relay and the Choudhury models.The controller for every plant-order is tuned using the AMIGOmethod and the plant-orders are shown in parentheses with their kvalues.

270 280 290 3000

0.04

0.08

0.12

relay

valv

e ou

tput

Choudhurystiction model

time

Fig. 11. Valve output using the Choudhury model for a FOPTD plant(k = 0.5, deadband = slip-jump = 0.1).

380 A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382

diverge as much as in Fig. 10. Thus, it appears that apure time-delay contributes more to the differences inthe two models than higher-order dynamics.

6. Practical considerations

The stiction detection method proposed in this paperis designed for single-input single-output (SISO) controlloops and self-regulating plants with monotone step-response. The methodology is not designed for integrat-ing plants because stiction results in a triangular-wavewith R = 1. In this situation, other methods such asthe one proposed by [8] or [9] may be used, however,the user may still have to contend with noise and com-putational issues.

Nonlinear plants having high gain ratios, and changein dynamics with change in plant-input direction canalso result in R > 1 even though no valve-stiction maybe present. For such plants, more complex methodssuch as [9], may be used to diagnose the problem. A de-tailed analysis for distinguishing between nonlinearplant behavior and stiction is proposed as futureresearch.

Because the stiction detection method presented inthis paper is based on calculating areas under the con-trol error signal, factors that affect the area calculation,also affect the stiction detection method. Two major fac-tors: sampling period and noise, influence the effective-ness of the proposed method. Large sampling periodshide key features of the signal such as its curvatureand the location of the peak and adversely affect thearea calculation, while fast sampling reveals these fea-tures. Thus, to reliably calculate the estimate of theareas, the error signal must be sampled many timesper oscillation period.

In most practical applications, the control error sig-nal contains noise that can corrupt its key featuressuch as the location of the peak and the points ofzero-crossings. The zero-crossings are points at whicha signal crosses zero or its expected value. Presenceof noise can result in multiple zero-crossings whenthe control error signal is close to zero. Because theproposed method calculates areas between the zero-crossings and the peaks, the areas calculated betweenthe zero-crossings of noise can result in misleading Rvalues.

We hypothesize that the effect of noise on the locationof the peak is less severe because of the following reasonstated without proof. Consider stationary and zero-mean autoregressive noise on the oscillating signal thatcorrupts the actual location of the peak. Let the uncer-tainty in the location of the peak for every half-oscilla-tion be Dtp. Because the noise is zero-mean andstationary, the expected value or a statistical averageof Dtp over several oscillations is zero. Because of thisreason, averaging R over a few oscillation periods willreveal its expected or mean value for the oscillatingsignal.

Still, the effect of noise must be reduced for the pro-posed method to be effective for practical applications.Horch [8] suggested using a low-pass digital filter to re-duce noise on the signal. He suggested a filter cutoff fre-quency of three times the oscillation frequency. If xosc isthe oscillation frequency, and Ts is the sampling period,the filter transfer function is,

Hf ðq�1Þ ¼ 1� c1� cq�1

; c ¼ e�3xoscT s ; ð23Þ

where q�1 is a backward shift operation. In addition tofiltering, the value of R may be averaged over a few

500 600 700 800 900 1000

–1

0

1

Before detuning (average R ≈ 1.1)

cont

rol e

rror

time [min]3200 3300 3400 3500 3600

–1

0

1

After detuning (average R ≈ 2.5)

time [min]

cont

rol e

rror

noisy signal

filtered signal(solid line)

Fig. 13. Detection of the presence of stiction in a control loop using the proposed method.

A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382 381

oscillation periods to reduce its variability and improvethe stiction detection process.

In Section 2, we stated that if R > 1 then the oscilla-tions are caused by stiction, while R � 1 means aggres-sive control. For practical implementation, theboundary value of R = 1 will result in too many falsealarms. Thus, to improve the robustness of stictiondetection, we recommend the decision rule,

oscillation diagnosis ¼stiction if R > 1þ d;

aggressive control otherwise;

where d is a threshold that determines the sensitivity ofthe stiction detection method. A small value of d will re-sult in high sensitivity and high probability of falsealarms, while a larger d will result in reduced sensitivityand a lower probability of false alarms. A value of d be-tween half and one was found to provide a satisfactorytrade-off.

7. Field result

The proposed stiction detection method was used todiagnose the cause of oscillations in a room temperaturecontrol loop in a commercial building. The loop wasoscillating with a period of approximately 13 min.Detuning the controller increased the period to 62 minbut did not eliminate the oscillations. Fig. 13 presentsthe oscillating behavior of the control error signal beforeand after the detuning. The average R value for the per-iod after detuning is about 2.5, which is sufficiently largeto conclude that the oscillations are caused by stiction.Further examination of the actuator�s movement con-firmed that the oscillations were caused by stick-slipbehavior.

8. Conclusions and future research

A new, simple and effective method for detecting stic-tion in an oscillating control loop has been presented.The method is based on calculating the ratio of areas,R, before and after the peak of an oscillating control er-ror signal, and does not require measuring the controlleroutput. The method is simple and easy to implement on-line (e.g., in a field controller). By approximating thestiction behavior as a relay with hysteresis, analyticalexpressions were derived that demonstrate R > 1 whenoscillations are caused by valve stiction. Numerical sim-ulations with the full Choudhury stiction model confirmthat the proposed method also results in R > 1 whenstiction is present.

A natural extension of this research would involvecombining the stiction detection method presented inthis paper with automated methods for measuring andcompensating stiction using methods such as Hgglund�s‘‘stiction knocker’’ [25]. Future research would also fo-cus on performing a detailed analysis to investigate theeffect of variable gain, dynamic nonlinearity and stictionon loop oscillations. The analysis would help in differen-tiating between poor loop performance caused by thedifferent types of nonlinearities.

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