Controller Output Based, Single Number Statistics for Loop Status Monitoring

IFAC

Copyright 0 IFAC On-Line Fault Detection and Supervision in the Chemical Process Industries, Jejudo Island, Korea, 2001

~

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CONTROLLER OUTPUT BASED, SINGLE NUMBER STATISTICS FOR WOP STATUS MONITORING

Chunming Xia and John RoweD

Department ofMechanical Engineering University ofGlasgow, Glasgow G12 8QQ, Scotland, UK Email: [email protected]

Abstract: Two single number statistics are proposed that, when used with others, enable the basic characteristics of a closed loop trend to be categorised to indicate that the loop is in one of a number of different statuses: well-behaved & in steady state, undergoing a short-term transient, cycling at a fundamental frequency similar to the natural frequency of the loop, cycling at a relatively low fundamental frequency, or undergoing a trend that is disturbed in some non-stationary manner. These classifications are slightly different for PI and Pill controllers, only the former is discussed here. As is normal with CLP approaches, only the outer loop of each cascade control system is considered because of the continual set-point changes that arise in its inner loop. Copyright © 2001 IFAC Keywords: closed loop assessment, process control, distributed detection

resource intensive whilst a group of single number statistics would be useful to at least provide a first pass indication that a problem exists. Kozub sees the potential for both detecting and determining the extent of underdamped, or cyclical, response trend characteristics and views the automation of controller performance monitoring and diagnosis as an important new challenge. By providing a means of categorising loop status, the single number statistics proposed here can be viewed as a step towards this vision.

1. INTRODUCTION It has been reported that as many as 60% of all industrial controllers have some kind of a performance problem. such as an oscillation (Harris, et al., 1999; Hligglund, 1995). Possible causes include sensor and lor actuator faults, such as a bias or a sticking valve, poor tuning, errors in the models used to design the controller, abnormal disturbances, such as feedstoek variability and product changes and so on. This has encouraged the development of closed loop performance (CLP) assessment techniques (Harris, 1989; Thornhill, 1999; Huang, 1999; Thornhill et al., 2(00), which are now so well established that various vendors are offering commercial analysis products based on them. Various industry users (Kozub, 1997; OweD, et al., 1996) have published their experiences. In particular Kozub has compared experiences with time series based approaches with those with single statistic based approaches. Although the use of a single number statistic can hardly be as effective as analysis tools that offer far more detail, the latter are very

Current controlled variable based indices can be viewed as providing a 'grey' indication because a good index value does not always imply that the status of the loop hasn't changed. In particular such instances arise when the controller is able to reject the fault or disturbance sufficiently that the index remains within range. In these circumstances an index based on controDer output would be more appropriate. A second index, which compares a controller output index with the corresponding controDed variable index, is another possibility. This

III

1992), which is based on a d-step ahead prediction method for estimating Then

paper shows that, by viewing all these indices, it is possible to categorise, loosely, loop status providing some form of partial diagnosis. These classifications are slightly different for PI and Pill controllers; only the former is discussed here, the latter is available in a companion paper (Xia & Howell, 2(01).

y.

(7)

3. NEW INDICES BASED ON CONTROLLER OUTPUT 2. BACKGROUND The Harris index can be viewed as being a 'grey' indicator because a good index value does not always imply that the status of the loop hasn't changed. For instance the controller can reject certain kinds of disturbance or fault, such as a small process drift, sufficiently well to cause the index value to remain within the range that indicates good performance. Other indicators are therefore required.

Harris (1989) has described an approach to controlloop performance assessment that is based on routine operating data and does not require any detailed information of the process model except knowledge of its time delay (d). Like others (Kozub 1993) he compares the controlled variable error variance with the potentially best achievable variance for the measured conditions under minimum variance control (MVC). For a closed loop control system with time delay d, the controlled variable deviation Yt can be modelled as a MA process such as: I"

I"

-I

I"

Yt = (. )0 + Jlq +...+Jd-Iq

-d+l)

It is well known that the controller output might contain complementary information. Certain small faults and disturbances, like a process drift, have a more marked affect on controller output than on the controlled variable. So for the purpose of the FDI of control loops, indications extracted from a controller output might be more important than those from the controlled variable. An appropriate indicator, which is based on the controller output, is then

at, + (1)

I" -d ( )dq

I" -d-I + )d+lq +...) at,

y

CT

where

CT

/l 2 is the variance of controller output (u) and

2

is the variance of the residual, which is estimated by subtracting the predictable component u from the controller output. CTe.

Furthermore it has been found that the ratio R:

For any other control system the actual variance will be: CT/

2

=(1+ f0 + !J2+···+fd_1 + f l + fd+1

2

2

+ •• )CTa (3)

RE

." -1 Y

"'/l-1 The control loop performance index can be defmed as (4)

[::: -1] [::.: -1]

(T.2 y

CT 2 e, CT .2

/l

(S I N)of Y (9) (SI N)ofu

CT 2 e.

where SIN means signal-to-noise ratio, provides a means of categorising various types of trend behaviour, in particular, oscillatory type behaviour. If the process has an oscillatory trend of frequency co, then it is shown in the Appendix that

or alternatively as ."y =CTe.. 2/ CT y 2

(8)

CT e•

(2)

2

2

"'/l=~

where at is a white noise sequence, e"", is called the minimum variance output independent of feedback control (Harris, 1989), and yare predictable components. When the loop is under MVC, the output only contains e""" and the theoretical variance is as follows:

(5)

or as

(6)

R Deftnitions (5) and (6) are normalised indices, Le. their values are within [0,1]. This paper uses defInition (4), whose value is within [1,00]. When the loop is performing well, ", is small and very near to unity, and becomes much larger for a poorly performing loop. This deftnition is more sensitive to changes than others. A fast. simple, on-line method for estimating index ", has been proposed (Harris,

Q

1+_1_

(10)

(T;(JJ)2

where Tj is the integral time constant and Q is a constant with respect to the ratio of the data sampling time, T. to T;. A normalised value of R can be then be deftned as:

112

Table I Loop Status (IS) Criteria

R,,=.!i

1 1+_1_ (T;(tJ)2

Q

(11)

Steady

This relationship can also be extended to more complicated periodic trends, where (J) is now the fundamental frequency of the trend.

It can be seen from Equation 11 that when an oscillation is of high frequency, R is virtually constant and the normalised Rn tends to 1. When the frequency is low, the value of Rn will be smaller than one. An example of R and Rn' estimated by simulation, is shown in Fig. I.

Un-normalised R

Compensated

TJy=r

Shortterm Transient

TJy>=r, TJIl>r, R,,<= 1-;1

Decaying Transient

T-TDS
Nonstationary Longrandom term _(:.:..A::!..) Transient Lowfrequency cyclic (B)

R...

.r-

Criteria

Loop Status

I

Ultimate cyclic

1'-

(

I

~~

.: TI/1i_l/J

I

"':TJ/Ti-JnO

Critical

TJy>=r, TJu>r R,,<=1-;1

5TD(R,,) is large

>=;3

T-TDS>Tst

TJy>=r, TJIl>r 1-;1=r,TJIl>r. R,;>=1+~

or: TJy>=r, TJIl
10

15

20

25

30

~

1.0

-

paragraphs below and the various tests are given in Table 1. Note that the tests involve a number of parameters, the values of which have been chosen by analysing various data sets. Loops performing in a transient manner can be sub-categorised on the basis of further tests, one involving three time parameters T, Tos and T Sl and another involving uR" (i.e. the

Normalised R,.

u 1.2 1

.,.

r

... o.a

tbt£on tical

T

..0

standard deviation of Rn or S1D(R,,)). This is explained below. Examples of the various statuses are given in Table 2.

~rve

... o o

10

15

_(TNI

20

.

Table 2 Some causes

30

LS

Fig. 1. Simulation test results for R and Rn against frequency

Compensated

4. WOP STATUS ASSESSMENT AND PARTIAL DIAGNOSIS This leads on to an extension of loop performance assessment, which is called Loop Stiltlls (LS) Assessment. Again the focus is on the analysis of an individual SISO loop or on the outer loop of a The slightly cascade loop and on PI control. different interpretation that is required for PlO control is discussed in a companion paper (Xia & Howell, 2001).

It has been found that it should be possible to recognise that a loop is performing in one of a number of different ways (statuses) by applying various tests to the following indicators: 1]y. TJIl and Rn. The various statuses are outlined in the

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Cause Small disturbance, small fault such as sensor null shift, slow drift

Short-term transient

Step load disturbance ( loop dynamic change), sensor bias, sensor large drift, short-term large disturbance,

Long-term transient

Non-stationary disturbance, feedstock variability, raw Material variability Low-frequency cyclic disturbance

Ultimate cyclic

Valve problem. u1ti.mate-cyclic disturbance, bad tuning

Critical

Critical change in process

S. SIMULAnONS Sla1Us returns

Fig. 3 shows results obtained for three separate tests: (A) a high frequency cyclic disturbance at time lOoos, (B) a slow sensor drift from lOoos, (C) a nonstationary disturbance from lOOOs. It can be seen that LS responds quickly and possible candidates can be partially diagnosed according to Table 2.

Data-set blocks

Critical

Fig. 2. Settling time threshold explanation

I

U. Cyclic L. Transient S. Transient

Normal steady operation. Both y(t) and u(t) should have little predictable component, so 1}y
I

A

Compensated Steady • Critical

Cony>ensated The disturbance or fault is rejected effectively with the net effect that there is little predictable component in y(t), whilst there is a lot in u(t), so 11y=Y.

--

U. Cyclic L. Transient S. Transient

Compensated

All other statuses arise when both y(t) and u(t) have deterministic predictable components i.e. f1y>'( and 11">'(·

Steady

•

I

II

B

Critical

Short-term transients. In these situations, after a short period of time the loop status will return to a lower level status (steady or compensated). A transient is then deemed to be short-term if its time duration, T (i.e. the time the loop status is deemed to be 'transient'), is small relative to the loop settling time Tat. A correction has to be made to accommodate the fact that the analysis occurs over data blocks of duration T DS so that the test becomes T-T DS
STD(Rn) =0.1158

U. Cyclic L. Transient S. Transient

n I

c

Compensated Steady

•

Fig. 3. Test Results of LS

Long-term transients. If the transient status lasts longer than the settling time limit Tat, i.e. T-Tos>Tat , the loop could either be exhibiting a non-stationary random response or a more deterministic like cyclic trend. With the former Rn will be much smaller than 1, and the standard deviation STD(RJ will be much larger than that for an oscillation. Although a low frequency oscillation might have a similar small Rn value, it would also have a smaller STD(RJ.

6. CONCLUSIONS Two new indices have been proposed that are based on controller oUlput. When analysed together with a controlled variable index, it is possible to categorise loop performance into a number of broad statuses. If implemented in 'near real-time' on a plant, this could provide the operator with a relatively up-to-date understanding of the current operation of all the control loops. One possibility is that control loops on the plant schematic could change colour in response to changes in current statuses.

Ultimate cyclic. Some form of long term oscillation with the frequency similar to natural frequency. Critical. The controller lacks controllability e.g. the loop might diverge or the final control element might stick. If the controller is near sticking and the loop is uncontrolled, 1}y>=r. 11,,=r. 11..>Y and Rn> =1+~.

The default values of thresholds in Table 1 are: r=1.25, ~J=O.2, ~rO·2, ~.05, Tst=24d.

114

where B=lI2T, (T. is sampling time interval) and B o is relatively close to O. Define Kg: Bo = BlKg ,( typically Kg =1(0).

REFERENCES Hagglund, T. (1995). A control-loop performance monitor. Control Eng. Practice, 3,1543-1551. Harris, TJ. (1989). Assessment of closed loop performance. Can. J. Chem. Eng., 67, 856-861. Harris, TJ. and L. Desborough (1992). Performance assessment measures for uoivariate feedback control. Can. J. Chem. Eng., 70,1186-1197. C.T. Seppala and L.D. Harris, TJ., Desborough(1999). A review of performance monitoring and assessment techniques for uoivariate and multivariate control systems. Journal ofProcess Control, 9, 1-17. Huang, B. and S.L. Shah (1999). Performance Assessment of Control Loops. Springer Verlag, Lodon. Kosub, D.J. and C. Garcia (1993, Nov. 9). Monitoring and diagnosis of automated controllers in the chemical process industries. St. Louis, MO. MCHE annual Meeting. Kosub, DJ. (1997). Controller performance monitoring and diagnosis. Proceedings of Fifth International Conference on Chemical Process Control, CACHE Corp, pp. 83-96. Owen, J.G., D. Read, H. Blekkenhorst. and A. Roche (1996). A mill prototype for automatic monitoring of control loop performance. In: Proc. Control Systems'96, pp. 171-178 Thornhill, N.F., M. Oettinger, and P. Fedenczuk(1999). Refinery-wide control loop performance assessment. J. Proc. Control, 9, 109-124 Thornhill, N.F., S.L. Shah and B. Huang (2000). Controller performance assessment in set point tracking and regulatory control. IFAC Symposium ADCHEM 2000, Pisa, Italy. Xia, C. and J. Howell (2001). Loop status statistics. 4 th IFAC Workshop on On-line Fault Detection and Supervision in the Chemical Process Industries (CHEMFAS-4), Korea.

The controller output also contains two parts, the wave trend 0, which is generated from and the noise part ell' from er The auto-eorrelation function of ey is:

y,

r 2 11Bo T] =J::~(f)COS(21tfT)df =[Sin 2J1BT-sin 2HT G B

R~(T)

£1\

2HT

'" 0)

21dJ T (13)

The variance of ey is: U

~

2=R (O)-p 2=limR (T)=GB ~

~

1"--+0

~

(14)

According to the input/output auto-spectrum relationship, the auto-spectrum of ell can be given by:

Gell(f)=IH(211/j~2G~(f)=K/(I+

2 1 2)Gry(f) T; (21tf)

(15) where H(s) is the controller TF, Kp and Tj are PI controller parameters, and the variance of ell is:

r~ell(f)df = GK/ JrBa (1 + T; 2 (21tf) 1 2 \r JBa r B

u e/ = Reil (0) =

=GK/[(B-Ho)+ 4JT;2

(~ - ~ )]::::GBK/Q (16)

where K ,.z 1 =1+--::T 4 1; BHo ,.

Q=I+

B

2

~

2

(17)

[

1; ]

Assume y(t)=Asin(ar), the s-domain expression of y(t) is: A( ) Ys =

2

Aw

A

A

+w

1;s

APPENDIX

(18)

2

[I 1 S

2 Aw 2

U(s)=H(s)y(s)=Kpl+-

S

)

+w

(19)

Oft) can be given by Laplace inverse transformation,

The relationship between R and the frequency of sine wave periodic trend will be described here.

O(t) = Kp[ASinar

When a closed loop has a sine wave style oscillation, the controller error can be divided into two parts, deterministic sine wave yand wide-band noise ey , which is independent with E(ey)=O and

+~(I-COSar)] 1;w

(20)

The deviation of Oft) is: u(t)

y,

Var(e y ) = 0;,. The auto-spectrunl, Gey(f), of e y is

where M

assumed to be uniform over a wide bandwidth B, that is, Gry (f)=G =0

(0: Ho

::::G sin2J1BT = GJ sin 21dJT)

= K~JM sin(ar + jJ)

(21)

=1+_1_2 ,and P=_tan- 1(_I_), (1;w)

1;w

The variance of u(t) and y(t) can be given by:

(12)

u. 2 =.!..K 2A 2M Il

115

2

p

(22)

CT. 2 y

Substituting

the

2 =!A 2

above

CT y2 ,CT~ 2 , CTU2and CTeu 2·mto

(23)

four equations · 9 )'le ·lds equatIOn

for

!A 2

_2_

R=

=R= __Q~_

GB

.!..K

2A 2 M

M

(24)

1+_1_

2 P GBK/Q

(T;OJ)2

Q is a constant and just a little bigger than 1. It can be seen that R has a relation with the sine wave trend frequency co. When CO is bigger than the cut-off

frequency and near to the ultimate frequency, R will be near saturation. Normalised ratio Rn can be

defmed by

R,.

=-R E (0,1) Q

(25)

116