Control valves and process variability

ISA Transactions 39 (2000) 265±271

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Control valves and process variability Stephen R. Wilton * Syncrude Canada Ltd., MD X102, PO Bag 4009, Fort McMurray, Alberta, Canada T9H 3L1

Abstract A theoretical analysis is developed to explain some experimental results for a ¯ow loop from the literature. Process variability is shown to be dependent on the disturbance or noise parameters, the closed loop time constant, the loop dead time and the control valve dead band. The results can be extended to other process variables. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Control valve; Process variability; Noise; Dead time; Dead band

1. Introduction Consider the ¯ow circuit of Fig. 1 consisting of a pump, two valves, a ¯ow transmitter and piping connected in series in a closed loop. One valve and the transmitter are connected to a ¯ow controller. The other valve is connected to a recorded noise or disturbance signal. The ¯ow controller is to hold the ¯ow as steady as possible in the presence of disturbances caused by the valve connected to the noise source. Refs. [1±3] report some experimental measurements on a system of this type. They use twice the standard deviation of the ¯ow divided by its mean as a measure of the variability. They plot variability against the closed loop time constant lambda. Their curves for di€erent valves lie between the open loop or manual mode variability and a calculated minimum variability for an ideal control valve. The basis for their calculation is not given. Fig. 2 is typical of their results. The variability for good valves is only slightly higher than for the * Tel.: +1-780-790-7948; fax: +1-780-790-7919. E-mail address: [email protected]

ideal valve. The variability for poor valves is higher and may approach that for the open loop or manual mode. This leads to the following questions. . . . . .

How restricted or general are these results? What about transmitters other than ¯ow? What about other noise records? How do process parameters a€ect the results? What small signal valve parameters most a€ect the results?  Time constants  Dead time  Dead band, backlash, stiction, hysteresis . What theory explains the ideal valve curve? . Can it be extended to explain the real valve curves? This paper presents a theoretical basis for the curves of Fig. 2 and an answer to the questions. 2. Ideal control valve The system of Fig. 1 can be modeled with the small signal block diagram of Fig. 3. The ¯ow variance or

0019-0578/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0019-0578(00)00002-1

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S.R. Wilton / ISA Transactions 39 (2000) 265±271

Solving for the error e…s† ˆ SP ÿ f…s† ˆ

SP ÿ Gn …s† 1 ‡ Gp …s†Gc …s†

…3†

When t ! 1; if the controller includes reset e…s† ! Fig. 1. Experimental ¯ow loop.

ÿGn …s† ˆ ÿGn …s†Gcl …s† 1 ‡ Gp …s†Gc …s†

…4†

A PID controller is assumed.   1 ‡ Td s Gc …s† ˆ Kc 1 ‡ Ti s

…5†

If the control valve is ideal, it is linear and has constant gain with no dead time. Up to two valve or process lag time constants can be cancelled by the controller reset and rate time constants. Then, if there are no other valve or process time constants Gp …s†Gc …s† ˆ Kp Kc

Fig. 2. Flow variability for several valves.

Gcl …s† ˆ

1 ls ˆ 1 ‡ Gp …s†Gc …s† 1 ‡ ls

Gcl …!†Gcl …!† ˆ

Fig. 3. System block diagram.

average of the error squared is the frequency integral of the square of the error magnitude [4] … 1 1 e…!†e …!†d! …1† 2 ˆ 2 ÿ1 where e…!† ˆ e…s†jsˆj! and e …!† ˆ e…s†jsˆÿj!

f…s† ˆ Gn …s† ‡ Gp …s†Gc …s†e…s†

…2†

l2 ! 2 1 ‡ l2 ! 2

…6† …7†

…8†

Note that l is the closed loop time constant. The shape of the ideal valve curve in Fig. 2 requires a noise ®lter block transfer function with one zero and two poles. This is explained later (with Fig. 5). The ®lter block includes valve, process and recorded noise parameters. It can be determined from the spectrum of the ¯ow when the controller is in manual. G n …s† ˆ

N…1 ‡ s† …1 ‡ T1 s†…1 ‡ T2 s†

Gn …!†Gn …!†

From the block diagram, the ¯ow

1 1 Ti ˆ where l ˆ Kp Kc Ti s ls

ÿ
N2 1 ‡  2 !2
ÿ
ˆÿ 1 ‡ T21 !2 1 ‡ T22 !2

…9† …10†

Substituting (7) and (9) in (4) and this in (1) and separating denominator factors

S.R. Wilton / ISA Transactions 39 (2000) 265±271

2 ˆ

1 2

267

…1  ÿ1

 A B C ‡ ‡ d! 1 ‡ l2 !2 1 ‡ T21 !2 1 ‡ T22 !2

…11† 1  ˆ 2  1 A B C arctan l! ‡ arctan T1 ! ‡ arctan T2 ! l T1 T2 ÿ1 2

…12† 2 ˆ

  1 A B C ‡ ‡ 2 l T1 T2

After determining A, B and C " ÿ 2
2  ÿ l2 lN 2  ˆ ÿ 2
ÿ
2 l ÿ T21 l2 ÿ T22 ÿ
l  2 ÿ T21 1 ÿ ÿ 2
ÿ
T1 l ÿ T21 T21 ÿ T22 # ÿ
l  2 ÿ T22 1 ‡ ÿ 2
ÿ
T2 l ÿ T22 T21 ÿ T22 

2 ˆ

2

Fig. 5. Normalized process variability.

…15†

For the manual or open loop mode, Kc=0 so l=1 and  2   2 N ‡1 N2  2 T1 T2 2 ˆ ˆ lˆ1 2T1 T2 T 2…T1 ‡ T2 † …16† T1 ‡ T2 Tˆ T1 T2 1‡ 2  The normalized variance 2 2 lˆ1

T 1‡ l  ˆ T1 T2 1‡ 1‡ l l

Fig. 4. Normalized ®lter gain (t=).

…14†



 …T ‡ T 2 † ‡ 1 l ‡ 2 2 1 T1 T2 … ‡ T1 T2 † …l ‡ T1 †…l ‡ T2 † 2 … T1 ‡ T 2 †

lN2

…13†

…17†

The ideal control valve and manual mode (l ˆ 1) curves in the references can be closely ®t by the following noise parameters

N/=0.606 where  is the mean or average ¯ow which is its set point =21.1 T1=3.01 T2 =490.2 T=114.3 s The normalized noise ®lter block and process variability log±log curves and their asymptotes are shown in Figs. 4 and 5. The respective asymptote break points for the two curves are related. The variability increases continuously towards its manual mode value with an increasing l closed loop time constant. Fig. 5 requires at least three break points in the asymptote for a close approximation to the curve. Since the break points are related to those of Fig. 4, this explains the previous statement about one zero and two poles being required for the noise ®lter block. The ideal valve variability is a function of only lambda and the noise ®lter block parameters. It is otherwise independent of the process. An extra time constant, dead time and dead band will be analyzed to ®nd an explanation for the real valve curves.

268

S.R. Wilton / ISA Transactions 39 (2000) 265±271

3. Extra time constant If the valve or process has an extra time constant T that the controller does not cancel, then (6) and (7) change to Gp …s†Gc …s† ˆ Kp Kc Gcl …s† ˆ

1 1 1 1 ˆ Ti s 1 ‡ Ts ls 1 ‡ Ts

1 ls…1 ‡ Ts† ˆ 1 ‡ Gp …s†Gc …s† 1 ‡ ls ‡ lTs2

…18† …19†

As shown in Fig. 6, a large time constant is required to signi®cantly a€ect the low l variability. This is unlikely as an uncancelled time constant will be smaller than Ti and, from (6), l will be of the same order as Ti or greater [5]. The e€ect of a third valve or process time constant is not signi®cant. 4. Dead time If the valve or process has dead time D, Eqs (6)± (8) change to Gp …s†Gc …s† ˆ Kp Kc Gcl …s† ˆ

eÿDs eÿDs ˆ Ti s ls

1 ls ˆ 1 ‡ Gp …s†Gc …s† eÿDs ‡ ls

Gcl …!†Gcl …!† ˆ

l2 ! 2 1 ÿ 2l! sin D! ‡ l2 !2

The variance integral no longer has a closed form solution. If l=D ˆ 2=, then for some frequency, D! ˆ =2 and l! ˆ 1, the denominator is zero, and the variance is in®nite as shown in the Appendix. This may be approached at low l for the poor control valve in Fig. 2. The good control valve had a smaller dead time. The references did not go to lower values of l where in®nite variance would be approached for the good valve. As shown in Fig. 7, the dead time causes a minimum in the variability versus lambda curve resulting in a tuning optimum at a value somewhat higher than l ˆ 2D=. Dead time does not explain the departure of the real valve curves from ideal at larger lambdas. 5. Control valve with dead band Fig. 8 shows that the output of a deadband nonlinearity follows a di€erent path depending on whether the input is increasing or decreasing.

…20† …21†

…22†

Fig. 6. Variability with an uncompensated process pole.

Fig. 7. E€ect of dead time on process variability.

Fig. 8. Deadband nonlinearity.

S.R. Wilton / ISA Transactions 39 (2000) 265±271

Beginning with a normalized input of ÿ1, as the input increases the output remains the same until the input change exceeds the deadband. Then the output follows the input change until the input reaches its maximum. Similarly the output stays constant as the input decreases until the input change exceeds the deadband. Then the output follows the input change. Fig. 9 shows one cycle of the response of a deadband to a continuous sine wave input. The output is decreased in amplitude, shifted in phase, and distorted relative to the input. The output can be reproduced as the sum of an in®nite number of sine and cosine waves at whole number multiples of the input frequency. Their amplitude is given by Fourier analysis of the output response f(t) of a valve with dead band to an input v=V sin(!t). This produces the following results. an ˆ 2V bn ˆ 2V

… ÿX ÿX

… ÿX ÿX

sin…nx†f…x†dx

269

b1 1 ˆ ÿ cos2 …X† V  gain ˆ

1 1 2 a1 ‡ b21 2 V

…25† phase ˆ arctan

  b1 a1

Odd harmonics  an 1 1 1 ˆ sin…n ÿ 1†X ÿ sin…n ‡ 1†X V  nÿ1 n‡1  2 ‡ cos…nX† sin…X† n …27† " nÿ1 bn 1 2…ÿ1† 2 1 cos…n ‡ 1†X ˆ ÿ V  n…n ÿ 1†…n ‡ 1† n ÿ 1 # 1 2 cos…n ‡ 1†X ‡ sin…nX†sin…X† ‡ n‡1 n …28†

…23† cos…nx†f…x†dx

Fundamental   a1 1  1 ‡ X ‡ sin…2X† ˆ 2 V  2   DB where X ˆ arcsin 1 ÿ V

1 2P 1ÿ
32 2 2 a ‡ bn 6n>1 n 7 7 % harmonics ˆ 6 4 a2 ‡ b2 5 1 1

n odd

…24†

Fig. 9. Deadband response to a sine wave for DB/V=1.5.

…26†

…29†

The even harmonics are zero. Fig. 10 is a plot of these results. As frequency increases, the input to the valve will decrease, increasing the DB/V ratio. The gain will decrease and the phase and % harmonics increase until DB/V is greater than 2 and the output is zero.

Fig. 10. Response of a value with dead band.

270

S.R. Wilton / ISA Transactions 39 (2000) 265±271

From Fig. 3 and Eqs. (4), (7) and (9), the PI controller output to the valve is   1 ls v ˆ Gc e ˆ ÿGc Gcl Gn ˆ ÿKc 1 ‡ Ti s 1 ‡ ls …30† 1 ‡ s N …1 ‡ T1 s†…1 ‡ T2 s† The valve gain, phase and harmonics depend on DB/V as shown by Fourier analysis. They are 1, 0 and 0 when DB/V <<1 and are 0, ÿ/2, and 0 when DB/V52. When the valve and loop gain is 0 the tuning parameter l is 1 and Gcl=1. As a rough approximation, Gcl can be considered in two frequency bands, the ®rst where DB/V<1 and Gclls/(1+ls) and the second DB/V51 and Gcl1

This equation has a closed form solution. When only the ®rst harmonic is considered, typical graphs of DB/V, and process variability for valves with dead band are shown in Figs. 11±13. The increase in variability at small l is predicted. The increase at larger l in Fig. 2 is due to harmonic generation adding to the noise. Fig. 13 supports the control valve speci®cation [6] that combined backlash/stiction not exceed 1% of input signal span. The departure of the real valve curves of Fig. 2 from the ideal control valve curve is due mainly to dead band. The dead band causes lost motion between the controller output and the valve position. This makes the problem nonlinear and much more dicult to analyze exactly. An approximate analysis has been given based on the ®rst harmonic. The approximation is valid only for

with the division between them at ! ˆ where DB/V = 1. This splits the variance integral (1) into two parts. … … 1 1 1 1 … †d! ˆ … †d! 2 ˆ 2 ÿ1  0 1   2 3 …1 …

6 7 … †j d!5  4 … †j ls d! ‡ 0

Gcl ˆ

1 ‡ ls



Gcl ˆ1

…31†

Fig. 11. DB/V variation with frequency for a valve with dead band, l=0.3, Kc=1 and Ti=1. Frequency ! ˆ where DB/ V=1.

Fig. 12. versus DB for l =Ti and Kc=1.

Fig. 13. Process variability for a valve with dead band due to the ®rst harmonic only.

S.R. Wilton / ISA Transactions 39 (2000) 265±271

small dead bands. The gain is reduced and the phase shift and harmonics increased as dead band increases. 6. Summary A theoretical basis has been developed for some experimental results for control valves in a ¯ow loop. Process variability is de®ned as twice the standard deviation divided by the mean. It has been shown to be a function of the noise parameters, the closed loop time constant, the loop dead time and the valve dead band. Minor process time constants have only a small e€ect. The results are directly applicable to loops for other variables beside ¯ow if the block diagram has the same form. Appendix. Dead time in®nite variance  In …22† let
ˆ ÿ D! 2

…A1†

ˆ  2  l 

2 ÿ
D 2  2   2 l  l  ÿ
cos
‡ ÿ
1ÿ2 D 2 D 2

…A2†

Gcl …!†Gcl …!† 

Gcl …!†Gcl …!† 

1  if
is small 4 2 ‡ 1
2

Gcl …!†Gcl …!†  

…A7†

If
is small, Gn is constant and can be taken outside the variance integral. Near the peak of the integrand

2 D

… ‡ 2

 2ÿ

Gcl …!†Gcl …!†dD! ˆ

… 0‡ 0



1



4 ‡ 1
2 2

d
ˆ 1

…A8†

l 2 ˆ : D  …A3†

l 2 ˆ D 

…A6†

The peak has in®nite area so the variance is in®nite when

 2  2 l  ÿ
D 2    2    2 l 
2 l  ÿ
1ÿ ÿ
1ÿ2 ‡ 2 D 2 D 2

When

Gcl …!†Gcl …!†   2  2 2  ÿ
 2     4 4 4 1 ÿ 2 ÿ
ÿ
2 ‡ 1 ÿ
‡ 2
2   

1 D

Then Gcl …!†Gcl …!†

271

…A4†

 2  2 2  ÿ
 2  2  2    2   2 2  2 ÿ
ÿ
‡ ÿ 
‡
1ÿ2  2 4  4 …A5†

References [1] D. Beckman, F. Jury, Reducing process variability with control valves, Chemical Processing November (1997). [2] P. Studebaker, Valves make the di€erence in advanced control, Control November (1997). [3] Anon., Think of This as a Pro®t Opportunity, Fisher Controls International, Inc, Fisher-Rosemount, form 8596. [4] J.G. Truxall, Control system synthesis, McGraw±Hill, 1955. [5] S.R. Wilton, Controller tuning, ISA Transactions 38 (1999). [6] Anon. Control Valve Dynamic Speci®cation, Version 2.1, EnTech Control Inc., Toronto, ON, Canada, March 1994.