Journal of Process Control 15 (2005) 371–382 www.elsevier.com/locate/jprocont
A simple method for detecting valve stiction in oscillating control loops Ashish Ashish Singhal 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 on the calculation of areas before and after the peak of an oscillating signal. The proposed method is intuitive, requires very little computational effort, and is easy to implement online. Analytical results are derived to show the theoretical basis of the new method and field results results are presented to show its effectiven effectiveness ess 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 almost 30% of control loops are oscillating [1,2] [1,2].. Oscillating ing loop loopss are are unde undesi sira rabl blee beca becaus usee they they incr increa ease se variabi variability lity in produc productt quality quality,, acceler accelerate ate equipm equipment ent wear, wear, and may cause cause oscilla oscillatio tions ns in other other interac interactin tingg loops. loops. Thus, Thus, detecti detection, on, diagno diagnosis, sis, and correc correctio tion n of oscilla oscillation tionss are importa important nt activit activities ies in contro controll loop loop supervision and maintenance. Some common causes of oscillations are (i) external oscillating disturbances, (ii) poor controller tuning, (iii) nonlinearities in the actuator/plant such as static and/ or dynamic nonlinearities, and stiction, or (iv) a combination of these. In this paper, we focus on distinguishing oscillations caused by valve stiction from those caused by poor loop tuning, and assume that oscillations have
*
Corresponding author. Tel.: +1 414 524 4688/4660; fax: +1 414 524 5810.
[email protected] (A. Singhal), timothy.i. E-mail addresses: addresses:
[email protected] [email protected] (T.I.
[email protected] (T.I. Salsbury). 0959-1524/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2004.10.001
been detected by other methods such as those described in [3,4] in [3,4].. A number of researchers have studied the valve stiction tion proble problem m and sugges suggested ted method methodss for detect detecting ing it. Horch Horch and Isaksso Isaksson n [5] presen presented ted a fairly fairly comple complexx method for detecting stiction by calculating log-likelihood ratios for multiple models. Their method requires knowledge of the nonlinear plant and stiction models and and exten extende ded d Ka Kalma lman n filte filterin ring. g. Sten Stenma man n et al. al. [6] also proposed a complicated method based on ‘‘multimodel mode estimation’’ estimation’’ and change change detection. detection. Their method method require requiress identi identifyin fyingg time-s time-serie eriess models models and perform performing ing optimiz optimizatio ation n to obtain obtain the log-lik log-likeli elihoo hood d ratio. Horch Horch presen presented ted two more more method methodss for detecti detecting ng stiction in oscillating loops [7,8]. [7,8]. The first method detected valve stiction by analyzing the cross-correlation function (CCF) between the controller output (u) and the plant output ( y). He proposed that a sticking valve results in a phase lag of 90 (odd CCF) between u and aggressive controller or an oscillating oscillating distur y, while an aggressive bance results in a phase lag of 180 (even CCF). The phase lag is 180 for an aggressive controller when the
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A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382 Valve stiction
Aggressive control
A
1
A
A
A
2
1
2
r o r r e l o 0 r t n o c
r o r r e l o0 r t n o c A 1 ____ ~1 A
A 1 ____ >1 A
2
2
time
time
Fig. 1. Control error signal shapes for valve stiction and aggressive control.
loop cycles due to controller output saturation. However, when stiction is present and the controller output is not saturated, the phase lag can lie between 90 and 180 for a PI controller. Horch s second method detected the differences between the shapes of the signal oscillating because of stiction and aggressive control using probability distributions. The method involved calculating filtered derivatives of the plant output and then analyzing the shape of the probability distribution for the derivative signal by either performing a nonlinear fit to two probability distributions (one for stiction, and one for aggressive control/sinusoidal disturbance), or manually observing the shapes of the two distributions. Although filtering the plant output signal was recommended before calculating derivatives, we found that even after filtering, the calculation of derivatives amplified moderate amounts of noise and blurred the distinction between the shapes of the two probability distributions. Choudhury et al. [9] used bicoherence to detect valve stiction by identifying non-Gaussian and nonlinear components in the signal. They presented simulation results for detecting nonlinearities in the signals using a stiction model [10] and also detected signal nonlinearities in industrial data. A manual inspection of the controlled variable–controller output (pv–op) plot was then required to determine the cause of the nonlinearity. According to Choudhury et al., nonlinearity in the signal could be present because of stiction, dead-zones, hysteresis in the control valve, or the nonlinear nature of the process itself. Thus, the bicoherence test detected the presence of signal nonlinearities, and not specifically stiction. Gerry and Ruel have published several papers on detecting and measuring valve stiction by manually inspecting the shapes of the control error and the controller output signals during sustained oscillations [11– 13]. They suggested that the controller output would be a saw-tooth or triangular wave for a sticking valve and a sinusoid for an aggressive controller. Additionally, a sticking valve was assumed to produce a
‘‘square-shaped’’ control error signal, while an aggressive controller produced a sinusoidal signal. Ruel also proposed a test to quantify the amount of stiction by putting the controller in manual mode, and then executing a series of small step-changes in the controller output until the controlled variable showed a change in its steady-state value. Our new automated method is also based on distinguishing between the shapes of the signals caused by an aggressive controller and a sticking valve using the ratio of the areas before and after the peak of the control error signal 1 as shown in Fig. 1. The idea is simple, easy to implement online, and requires little computational effort. This paper presents the proposed method in Section 2. The stiction model of Choudhury et al. [10] is discussed in Section 3 and a theoretical analysis of the method is presented in Section 4. We focus on presenting a theoretical framework for analyzing oscillations caused by stiction in closed loops and illustrate the idea using a simplified stiction model and popular plant models. The same framework could be used to analyze oscillations with more complicated models. A comparison between the results obtained using a simplified stiction model and the complete Choudhury model is presented in Section 5. Practical considerations are discussed in Section 6 and a field result is presented in Section 7.
2. Proposed method
For self-regulating plants 2 with a monotone step-response, aggressive control usually results in a sinusoidal control error signal, while for a sticking valve, the signal typically follows exponential decay and rise as shown in
1
The control error is the difference between the setpoint and the process variable being controlled. 2 Plant with all left-half plane poles.
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Fig. 1. The reason for this behavior is that while the plant input is continuous for aggressive control (except when the controller output is saturated), valve stiction results in a discontinuous plant input that closely resembles a rectangular pulse signal. The new stiction detection methodology distinguishes between the shapes of the two signals in Fig. 1 by calculating 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
) Sticking valve; R 1 ) Aggressive control R > 1
The proposed idea is very easy to implement online so that stiction detection can be performed by a field controller at faster sampling rates compared to downloading and analyzing data at an operator workstation. Also, online implementation in a field controller would result in reduced traffic on a control network. The assumptions for using the proposed method to detect stiction are: (1) the controller output is not cycling from one saturation limit to the other, and (2) the oscillations in the control loop are not caused by an external periodic disturbance. Violation of these assumptions can result in R > 1 even when stiction is absent. The first assumption can be verified by observing if the controller output hits saturation limits. Satisfying the second assumption requires overriding and holding the controller output at its current value and again detecting the presence of sustained oscillations. If the control error signal still exhibits oscillations, an external periodic disturbance is likely to be causing them.
3. Plant and stiction models
practice to approximate the nth-order transfer function in Eq. (1) to first-order for controller tuning. This approximation results in the popular first-order plus time-delay (FOPTD) transfer function, ~
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 order dynamics. It is important to distinguish between the true and apparent time-delay in order to understand the stiction model. First we present expressions for the apparent time-delay and time-constant for an nth-order plant and subsequently show the effect of apparent time-delay on the stiction model. In Eq. (1), the system gain can be set to unity without loss of generality. Because the focus of this section is to analyze the effect of the plant s order and not its different dynamic components, we set T i = T (i = 1, . . . , n). This simplification also reduces the number of free parameters in the n th-order model and allows us to derive analytical expressions for s and ~ h. The n th-order plant now becomes:
1
ð3Þ
n
˚ stro¨m and Ha¨gglund s method [14] to calculate We use A s and ~h. Their method requires first calculating the average residence time of the higher-order plant. The apparent time-delay is then calculated by finding the intersection of the tangent drawn through the inflection point of the unit-step response with the time-axis. The apparent time-constant is finally calculated by subtracting the apparent time-delay from the average residence time. By following this procedure, the apparent timeconstant and time-delay for the nth-order plant described by Eq. (3) are calculated as
2 P 3 75 ¼ 64 þ 2 P ¼ 64ð Þ 1
The proposed stiction detection method is designed for self-regulating plants with monotone step-response. These plants can be represented by the n th-order transfer function,
¼ ðT s þ Þð þ Þ ð þ 1Þ ;
Ls
¼ ðTseþ 1Þ :
G p
s
G p
ð2Þ
¼ þ
G p
3.1. Plant model
K p e Ls 1 T 2 s 1 . . . T n s
K p eh s ; s s 1
T 1
eðn1Þ
where n is the plant order, K p is the plant gain, L is the delay in the plant, and T i are time-constants of different dynamic components. The delay, L, is the amount of dead-time in the plant, i.e., the time during which there is no plant response to an input change. It is common
ðn1Þi i! i ¼0
ð4Þ
ðn1Þn1 eðn1Þ ðn1Þ!
and
1
ð1Þ
n 1
h
T
n
1
n 1
ðn1Þi eðn1Þ i! i ¼0 ðn1Þn1 eðn1Þ ðn1Þ!
3 75
:
ð5Þ
Adding the pure delay, L , to the apparent time-delay results in the effective time-delay ~ h L h. We define a quantity k as the ratio of the effective time-delay to the apparent time-constant:
¼ þ
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Table 1 k values for n th-order plants (L = 0) n k
1 0
2 0.16
~ h k , : s
3 0.37
4 0.55
5 0.72
6 0.88
7 1.03
ð6Þ
The ratio k usually indicates the difficulty of control—a large k means that the plant is more difficult to control with a PI controller. The variation of k with n for plants with zero pure-delay is presented in Table 1. The effect of k on the input–output (I/O) characteristics of a sticking valve will be discussed in the following section. 3.2. Stiction model
Understanding the type of oscillations caused by a sticking valve in a control loop requires a good grasp of the stick-slip phenomenon. Several researchers have modeled stiction in mechanical systems and then analyzed the behavior of oscillations caused by stiction in a closed loop. Armstrong-He´louvry et al. [15] presented a comprehensive review of models, methods and control of mechanical systems with friction. They list contributions from tribology, lubrication, physics and control. Many researchers model friction as a combination of static, coulomb and viscous friction and have analyzed oscillations in position control systems with friction [15–18]. The disadvantage of physics-based models is that they require several physical parameters such as the amount of static, coulomb and viscous friction, spring tension, system mass, etc. to model the process accurately. In most cases, calibrating these parameters is not an easy task. Thus, Choudhury et al. [10] introduced an empirical valve stiction model that has behavior similar to the physics-based models. Their empirical model has only two parameters that are intuitive and easy to define. In this paper, we use the Choudhury model to analyze valve-stiction in an oscillating control loop. We also use a simplified form of the model to derive analytical expressions and validate our approach, and use numerical simulation to test the method with the full model. Choudhury et al. [10] developed an empirical model for valve stiction that produces input–output (I/O) behavior similar to that of more complicated physicsbased models. The I/O characteristics of a sticking control valve are presented in Fig. 2(a). For most plants with k > 0, the I/O characteristics contain a deadband + stickband, a slip jump, and a sliding part where the valve moves with the controller output. However, if k = 0 (i.e., for a pure first-order plant), only the dead-
band and the slip jump part can be seen. This process is shown by the solid line in Fig. 2 and has the same characteristics as a relay with hysteresis. The stiction models used by Stenman et al. [6] and Horch [7] also have the characteristics of a relay. The sliding part shown in Fig. 2(b) appears when a time-delay is added to the plant. For small values of k, the relay is a good approximation of the Choudhury model. Note that the Choudhury model becomes a relay with hysteresis when k = 0. For a second-order plus time-delay (SOPTD) plant, the relay approximation becomes less accurate because the k values are larger as shown in Fig. 2(c). Because k > 0 for a SOPTD plant, the relay model is always an approximation for this system. Fig. 2(d) presents the I/O characteristics of a sticking valve when the pure time-delay is zero. This figure shows that for higher-order plants, the relay model becomes less accurate compared to the Choudhury model because k increases with the plant order. When k is small, the relay is a good approximation of the stiction behavior. The results in this section also demonstrate that although L and h are different in nature, both result in the emergence of the sliding part of stick-slip behavior. Fig. 2 shows that the I/O characteristics of a sticking valve in closed-loop depend not only on the deadband and slip-jump parameters, but also on the plant dynamics. We will show later in the paper that a pure timedelay contributes more to the differences in the two models than higher-order dynamics.
4. Analysis for first and second-order plus time-delay plants
In this section, we analyze the behavior of first and second-order plus time-delay plants with valve stiction. The relay approximation is used to model stiction in order 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, e s
ð Þ ¼ Ts þ 1 ;
G p s
1 6 T 6 10
ð7Þ
so that 0.1 6 k = 1/T 6 1. Analytical expressions for R are obtained by the following 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 periodic plant output. Let G c(s) be the transfer function of the controller. An estimate of the oscillation frequency is obtained by solving for the frequency at which the Nyquist curve of G c( j x) · G p( j x) intersects the negative inverse of the
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λ = 0.5 λ = 0.1
) x ( t u p t u o e v l a v
plant with time-
) delay (Choudhury x ( t model) u p t u o plant with e v no time-delay l a (relay) v
λ = 0.83
λ = 0
slip jump
deadband+stickband valve input/controller output ( u) valve input ( u)
(b) FOPTD plant
(a) Sticking valve I/O second order
λ = 0.45 λ = 0.22
) x ( t u p t u o e v l a v
third order
λ = 0.63
λ = 0.16
relay (λ = 0)
) x ( t u p first t u order o e v l a v
pure time delay = 0
valve input (u)
valve input ( u)
(c) SOPTD plant
(d) Higher order plants
Fig. 2. Closed-loop I/O characteristics of a sticking valve.
4 K c cos x
ð þ xs sin xÞ þ 4 K xT ð sin x þ xs cos xÞ ¼ paxs ð1 þ x T Þ: ð8Þ
1.5 G p = 1 Gc = s i x a y r a n i g a m i
e-s Ts+1
I
K c(τ I s+1) τ I s
0.5
c
2
I
2
The solution of Eq. (8) is xosc and the oscillation halfperiod is b = p/xosc. Remarks:
increasing ω osc
0
-0.5
-1 N (a)
Gc( jω ) G p( jω)
-πα 4
increasing a -1
-1.5 -1
ω osc
-0.8
-0.6
-0.4 real axis
-0.2
0
0.2
Fig. 3. Estimation of the oscillation frequency using a Nyquist plot.
describing function 3 of a relay [14] as shown in Fig. 3. Let K c and s I be the proportional gain and integral time for the PI controller, and a be the ratio of the deadband (e) to the slip-jump (d ) parameters, then the oscillation frequency is calculated by solving the following nonlinear equation for x : 3
I
h p ffiffi ffi ffiffi ffi ffiffi ffi þ i ð Þ¼
Describing function: N a
p 4d
a2
e2
j pe 4d
1
(1) The describing function approach provides an approximate analysis and the actual oscillation frequency may be different from the calculated one. The magnitude of this difference depends on the nature of the nonlinearity and the spectral characteristics of G cG p. The reader may refer to Section 7.2 of [19] for details regarding the describing function method. (2) The oscillation frequency x osc depends on the ratio a = e/d , and not on the individual magnitudes of the deadband and slip-jump parameters. A larger a (large deadband, small slip-jump) results in a lower oscillation frequency and a smaller a (small deadband, large slip-jump) results in a higher oscillation frequency. Additionally, if there is no stiction, then there are no oscillations due to stiction and a = e/ d = 0/0 is undefined. We use LePage s method [20] to determine the steady state response of a first-order plant given by Eq. (7) to a
.
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A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382
rectangular wave input. The Laplace transform of a rectangular wave of unit amplitude and period 2 b is tanh(bs/2)/s. After solving for the system output, dropping the transient response and e 2bs terms, and taking inverse Laplace transform we obtain the steady-state periodic system output as
ð Þ ¼ p ðt k 2bÞ; k ¼ bt =2bc;
p t
ð9Þ
0
10
5
R
where k is the integer part of t /2b, and p0 is the repeating part of p(t) given by
ðÞ¼ 1
p 0 t
1 0 6 t < 2 b;
þ
2 et =T eb=T
t =T
2ð1 e Þhðt bÞ; ð10Þ
where h(t b) is the Heaviside step function with a lag of b. The function p0(t) for b = 1 and T = 1 is plotted in Fig. 4. The areas before and after the peak are denoted by A1 and A2. Using Fig. 4 and symmetry, the two areas are calculated as
Z ¼
Z ¼
b
A1
t z
p 0 t dt and
ðÞ
t z
A2
0
p 0 t dt ;
ðÞ
b=T
ð11Þ
2
where t z T loge 1 K 1 and K 1 ððee2b=T 11ÞÞ. After substituting Eq. (10) in Eq. (11), the expression for the ratio R = A 1/A2 is found to be:
¼
R
ð þ Þ
¼ þ ð þ t z T
¼
K 1 eb=T
t z =T
Þð e Þ : ð1 þ K Þð1 e Þ
b T
1
ð12Þ
t z T
t z =T
1
Because t z/T and K 1 depend only on b /T , the expression for R in Eq. (12) also depends only on the ratio b /T , that is, the ratio of the areas, R , is a function of the ratio of the oscillation period and the plant time constant. Also note that R depends on the ratio, a, of the deadband and
0.5
b = 1 T = 1
0.25
A
p (t )
A
1
0
2
0
– A
2
–0.25
t z
b
–0.5 0
0.5
1
1.5
2
t
Fig. 4. Steady-state response of a first-order plant to a rectangularwave.
1
0
2
4
6
8
10
b / T
Fig. 5. Variation of the area ratio, R , with b /T for a FOPTD plant.
slip-jump parameters and not their individual magnitudes. A plot of R for different values of b /T is presented in Fig. 5 that shows R P 1. The reader may verify that when b/T 0, R 1. For aggressive controllers, b is small. Thus, as the controller becomes more aggressive (smaller b/T ), R becomes smaller. To observe the effect of controller tuning on R , we selected four different PI controller tuning methods to calculate K c and sI using the parameters ~ h 1 and s = T . The different PI controller tuning methods can be found in [21] and were:
!
!
¼
(1) Approximate M S-constrained Integral Gain Optimization (AMIGO) tuning rule [22]. (2) Chien–Hrones–Reswick (CHR) rule for setpoint change with 0% overshoot. (3) Ziegler–Nichols (ZN) tuning rule. (4) Cohen–Coon (CC) tuning rule. When the controller and plant transfer functions are known, the control performance can be measured using the maximum values of the absolute sensitivity and complimentary sensitivity functions. These maximum values are denoted by M S and M T. For satisfactory control, M S should be in the range of 1.2–2.0, and M T should be in the range of 1.0–1.5; higher values mean aggressive control, while lower values mean sluggish control [23]. The CHR and AMIGO methods result in well-tuned controllers, while Ziegler–Nichols and Cohen–Coon methods result in aggressive control. Fig. 6 shows the variation of R for different controller tuning methods. Fig. 6 shows that for aggressively tuned controllers, such as the ones tuned using the Ziegler–Nichols and Cohen–Coon methods, the value of R is closer to unity
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A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382
10
b = 1 T = 1
ZieglerNichols
CHR (0% overshoot)
0.1
A
1
AMIGO
R 5
A
A
2
1,2
p (t ) 0
0 CohenCoon
–A 1,1
–A2
1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
–0.1
1
t
λ
t
compared to more conservative tuning using the AMIGO and CHR methods. Out of the four curves of Fig. 6, only the CHR curve shows non-monotonic behavior. The reason for this behavior is that the CHR tuning rule uses only the time-constant to calculate the integral time while the other three methods use the time-delay information as well. Thus, for small k (or large T ), the CHR rule results in larger s I compared to the other three tuning methods and 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]:
s
ð Þ ¼ ðTseþ 1Þ ;
1 6 T 6 10:
2
ð13Þ
Following the procedure described in Section 4.1, the periodic part of the steady-state output for the SOPTD plant is found to be: 2
3
0.5
1
2
Fig. 7. Steady-state response of a second-order plant to a rectangularwave.
Z Z ¼ ðÞ þ ðÞ |fflffl ffl ffl ffl {zfflffl ffl ffl ffl } |fflffl ffl ffl{zfflffl ffl ffl} Z ¼ ðÞ t p
b
p 0 t dt
A1
0
p 0 t dt ;
ð17Þ
t z
A1;1
A1;2
t z
p 0 t dt ;
A2
ð18Þ
t p
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 nonlinear equation (14), while tp is calculated as
¼
t p
K 1 þ K
K 3
T
2
3
ð19Þ
:
Substituting Eq. (14) in Eqs. (17) and (18) we have, b=T
¼ ðb t t Þ þ T ð1 þ K Þ½1 þ e e e þ T ð1 þ K Þ½1 þ ð1 þ b=T Þe ð1 þ t =T Þe ð1 þ t =T Þe
A1
ð14Þ
where
1.5
t
p
z
2
t p =T
t =T
b=T
3
z
t z =T
t z =T
p
t p =T
ð20Þ
and
"ð Þð Þð Þ# ¼ "ð Þð ð Þ ð Þ Þ# ð Þ "ð Þ # e2b=T eb=T
2b=T 1
K 2
eb=T
2
e2b=T
1
1
eb=T 1 2 : e2b=T 1
¼ ð
Þ
e2b=T
e2b=T 2
2b=T e2b=T
t p =T
t z =T
¼ ðt t Þ þ T ð1 þ K Þ½e e þ T ð1 þ K Þ ½ð1 þ t =T Þe ð1 þ t =T Þe : ð21Þ
A2
e2b=T 2
1
1
K 3
0
t =T
ð Þ ¼ 1 ½ð1 þ K Þ þ ð1 þ K Þt =T e 2½1 ð1 þ t =T Þe hðt bÞ;
p 0 t
p
p
Fig. 6. Variation of the area ratio, R, with 1/T for a FOPTD plant and different PI controller tuning rules.
G p s
b + t
b
z
p
2
z
p
;
ð15Þ
Eq. (14) is plotted in Fig. 7 for the values b = 1 and T = 1. Using Fig. 7 and symmetry, the areas A1 and A 2 are calculated as
z
3
t z =T
The ratio of the areas, R, is simply: R
ð16Þ
t p =T
¼ A A
1 2
:
ð22Þ
R is plotted for different values of b/T in Fig. 8. An interesting observation from Fig. 8 is that R < 1 for b/T < 2.46. Recall that a smaller value of b means a fas-
ter/more aggressive controller. Thus, Fig. 8 suggests that for a second-order plant, R can be less than one for aggressive controllers when valve stiction is present.
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A. Singhal, T.I. Salsbury / Journal of Process Control 15 (2005) 371–382
methods result in R < 1 because they produce controllers that are too aggressive. Fig. 9 shows that although the R-curves for ZN and CC methods are close to each other, their M S and M T curves are far apart. The reason for this discrepancy appears to be the difference in the controller setttings calulated by the two methods. While both ZN and CC methods result in very similar controller gains, the CC method calculates shorter integral times. Thus, the M S and M T values are larger for the CC method. Also, it appears that shorter integral time affects R less than the sensitivity functions.
10 st
1 order system 5
R
nd
2
order system
1
0
2
4
6
8
10
5. Comparison of relay and Choudhury models for valve stiction
b / T
Fig. 8. Variation of the area ratio, R , with b /T for a SOPTD plant.
This phenomenon appears to be caused by the change in the curvature of the step response of a second-order system. Assuming a positive plant gain, the second-derivative of the step-response of a second-order system is positive before the inflection point, and negative afterwards. For small values of b/T , a major proportion of the periodic response is before the inflection point (positive second-derivative). By symmetry, the response after the peak of the signal also has a positive second-derivative. These signal shapes cause the ratio R to become less than unity. However, for large b /T the major fraction of the periodic response is after the inflection point and we have R > 1. To observe the effect of controller tuning on R for a SOPTD plant, we apply tuning rules to determine controller settings. Because tuning rules commonly rely on FOPTD parameters, the SOPTD model is reduced to a FOPTD form to calculate the controller settings. By substituting n = 2 in Eqs. (4) and (5), the calculated effective time-delay and the apparent timeconstant for the SOPTD plant are ~ h 1 3 e T and s = (e 1)T , respectively. The controller is tuned using these apparent FOPTD plant parameters and the tuning methods listed in Section 4.1. The variation of R with k for different tuning methods is presented in Fig. 9. The figure shows that aggressive tuning methods such as ZN and CC result in smaller R values, while AMIGO and CHR methods that produce satisfactory control performance result in larger R values. In addition to calculating R for different tuning methods, the sensitivity function, M S, and the complimentary sensitivity function M T are also presented in Fig. 9. The figure shows that for small values of k , M S and M T are large for Ziegler–Nichols and the Cohen–Coon tuning methods. Thus, Ziegler–Nichols and Cohen–Coon
¼ þð Þ
In this section, we compare results in Section 4 using the Choudhury model. Because the describing function for the Choudhury stiction model is much more complicated than the relay model [24], obtaining analytical expressions for R is difficult. Thus, we present results by simulating a closed loop with stiction described by the Choudhury model and calculating R numerically. To improve the accuracy of the numerical calculations, the sampling period for control and measuring process variables is set to a 200th of the plant time-constant. Fig. 10 presents the variation of R with k for the Choudhury and the relay stiction models when the controller is tuned using the AMIGO method. Although, the curve for the Choudhury model is lower than the curve for the relay model, the value of R is greater than unity for both models. This result shows that the proposed method can detect valve stiction described by both models when the controller is well tuned. The reason for the lower R values for the Choudhury model can be explained as follows: when the original Choudhury model is used to simulate stiction in a closed loop, the valve output (or the plant input) is not a rectangular wave but a combination of a step and a ramp as shown in Fig. 11. The step part of the signal corresponds to the slip-jump, while the ramp part is the sliding part of the I/O characteristic as shown in Fig. 2. As the delay increases, the sliding part becomes larger compared to the slip-jump and the valve output moves with the controller output for a longer period of time and results in a smaller R value. The value of R increases with increasing k because the controllers become less aggressive (decreasing M S and M T in Fig. 9). The reason for the increasing difference between the two curves with k is that the difference between the I/O characteristics of a relay and the Choudhury model increases with k as shown in Fig. 2. For the same reason, the difference in the two curves is also larger for the SOPTD plant compared to the FOPTD plant.
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R
CC ZN 1 0.2
0.3
0.4
0.5
0.6
0.7
5 AMIGO
M
ZN
CHR
S
CC
3
1 0.2
0.3
0.4
0.5
0.6
0.7
5 AMIGO
M
T
ZN
CHR
3
CC
1 0.2
0.3
0.4
0.5
0.6
0.7
λ
Fig. 9. Variation of the area ratio, R , with k for a SOPTD plant and different PI controller tuning rules.
(a) FOPTD plant
10
(b) SOPTD plant
relay model
10
relay model 5
5
R
R
Choudhury model Choudhury model 1
1 0.2
0.4
0.6
0.8
λ
1
0.3
0.4
0.5
0.6
0.7
λ
Fig. 10. Comparison of the area ratio, R using the relay and the Choudhury models with the AMIGO PI controller tuning rule.
Fig. 12 shows the variation of R for different plant orders (cf. Eq. (3)) with no pure time-delay. In this situa-
tion, the difference between the relay and the original stiction models is smaller, and the two curves do not
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0.12
Choudhury stiction model
t u p t u 0.08 o e v l a v
relay
0.04
0 270
280
time
290
300
Fig. 11. Valve output using the Choudhury model for a FOPTD plant (k = 0.5, deadband = slip-jump = 0.1).
10
L = 0 5 relay model
R
Choudhury model 1 0 (1)
0.16 (2)
0.37 (3)
0.55 (4) 0.72 (5) 0.88 (6) 1.03 (7)
λ (n)
Fig. 12. Comparison of R for the relay and the Choudhury models. The controller for every plant-order is tuned using the AMIGO method and the plant-orders are shown in parentheses with their k values.
diverge as much as in Fig. 10. Thus, it appears that a pure time-delay contributes more to the differences in the two models than higher-order dynamics.
6. Practical considerations
The stiction detection method proposed in this paper is designed for single-input single-output (SISO) control loops and self-regulating plants with monotone stepresponse. The methodology is not designed for integrating plants because stiction results in a triangular-wave with R = 1. In this situation, other methods such as the one proposed by [8] or [9] may be used, however, the user may still have to contend with noise and computational issues.
Nonlinear plants having high gain ratios, and change in dynamics with change in plant-input direction can also result in R > 1 even though no valve-stiction may be present. For such plants, more complex methods such as [9], may be used to diagnose the problem. A detailed analysis for distinguishing between nonlinear plant behavior and stiction is proposed as future research. Because the stiction detection method presented in this paper is based on calculating areas under the control error signal, factors that affect the area calculation, also affect the stiction detection method. Two major factors: sampling period and noise, influence the effectiveness of the proposed method. Large sampling periods hide key features of the signal such as its curvature and the location of the peak and adversely affect the area calculation, while fast sampling reveals these features. Thus, to reliably calculate the estimate of the areas, the error signal must be sampled many times per oscillation period. In most practical applications, the control error signal contains noise that can corrupt its key features such as the location of the peak and the points of zero-crossings. The zero-crossings are points at which a signal crosses zero or its expected value. Presence of noise can result in multiple zero-crossings when the control error signal is close to zero. Because the proposed method calculates areas between the zerocrossings and the peaks, the areas calculated between the zero-crossings of noise can result in misleading R values. We hypothesize that the effect of noise on the location of the peak is less severe because of the following reason stated without proof. Consider stationary and zeromean autoregressive noise on the oscillating signal that corrupts the actual location of the peak. Let the uncertainty in the location of the peak for every half-oscillation be Dtp. Because the noise is zero-mean and stationary, the expected value or a statistical average of Dtp over several oscillations is zero. Because of this reason, averaging R over a few oscillation periods will reveal its expected or mean value for the oscillating signal. Still, the effect of noise must be reduced for the proposed method to be effective for practical applications. Horch [8] suggested using a low-pass digital filter to reduce noise on the signal. He suggested a filter cutoff frequency of three times the oscillation frequency. If x osc is the oscillation frequency, and T s is the sampling period, the filter transfer function is,
ð Þ ¼ 1 1cqc ;
H f q1
1
c
¼ e
3xosc T s
;
ð23Þ
where q 1 is a backward shift operation. In addition to filtering, the value of R may be averaged over a few
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After detuning (average R ≈ 2.5) noisy signal
1
1
r o r r e l 0 o r t n o c
r o r r e l 0 o r t n o c
–1
–1
500
600
700
800
900
1000
filtered signal (solid line)
3200
time [min]
3300
3400
3500
3600
time [min]
Fig. 13. Detection of the presence of stiction in a control loop using the proposed method.
oscillation periods to reduce its variability and improve the stiction detection process. In Section 2, we stated that if R > 1 then the oscillations are caused by stiction, while R 1 means aggressive control. For practical implementation, the boundary value of R = 1 will result in too many false alarms. Thus, to improve the robustness of stiction detection, we recommend the decision rule,
oscillation diagnosis
¼
stiction if R > 1 d; aggressive control otherwise;
þ
where d is a threshold that determines the sensitivity of the stiction detection method. A small value of d will result in high sensitivity and high probability of false alarms, while a larger d will result in reduced sensitivity and a lower probability of false alarms. A value of d between half and one was found to provide a satisfactory trade-off.
7. Field result
The proposed stiction detection method was used to diagnose the cause of oscillations in a room temperature control loop in a commercial building. The loop was oscillating with a period of approximately 13 min. Detuning the controller increased the period to 62 min but did not eliminate the oscillations. Fig. 13 presents the oscillating behavior of the control error signal before and after the detuning. The average R value for the period after detuning is about 2.5, which is sufficiently large to conclude that the oscillations are caused by stiction. Further examination of the actuator s movement confirmed that the oscillations were caused by stick-slip behavior.
8. Conclusions and future research
A new, simple and effective method for detecting stiction 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 error signal, and does not require measuring the controller output. The method is simple and easy to implement online (e.g., in a field controller). By approximating the stiction behavior as a relay with hysteresis, analytical expressions were derived that demonstrate R > 1 when oscillations are caused by valve stiction. Numerical simulations with the full Choudhury stiction model confirm that the proposed method also results in R > 1 when stiction is present. A natural extension of this research would involve combining the stiction detection method presented in this paper with automated methods for measuring and compensating stiction using methods such as Hgglund s ‘‘stiction knocker’’ [25]. Future research would also focus on performing a detailed analysis to investigate the effect of variable gain, dynamic nonlinearity and stiction on loop oscillations. The analysis would help in differentiating between poor loop performance caused by the different types of nonlinearities.
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