Diagnosis of Closed Control Loop Status Using Performance Analysis

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Ind. Eng. Chem. Res. 2005, 44, 5660-5671

Diagnosis of Closed Control Loop Status Using Performance Analysis Based Approach Yuezhi Yea and Junghui Chen* R&D Center for Membrane Technology, Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan, 3203, Republic of China

A novel approach for systematically assessing the current feedback controller performance, diagnosing, and removing the root causes resulting in performance deterioration is proposed. This fault diagnosis methodology does not require the traditional complex physical model and/ or a prior external input to perturb the operating system in order to achieve accurate fault identification. This is achieved by a series of the statistical hypothesis procedure, testing an impulse response of the control output which includes the feedback invariant (FBI) and the feedback dependent (FBD) terms. Both term variances via the statistic hypothesis testing are performed by comparing the current control and benchmark operating conditions. If the current performance has significant discrepancy from the benchmarked one, then the just-identified fault element (the process dead time, the controller tuning, the process, or the unmeasured disturbance) is isolated first and then the same diagnostic procedure based on FBI and FBD for exploring the new possible fault is successively applied. According to the characteristics of the closed loop output response, a diagnosis tree structure that has the characters of hierarchy and integrated knowledge is established. It is organized based on the sequence from general to the special. The capability of the proposed method is illustrated through a simulation case and a level tank system, including the single and the multiple fault problems. 1. Introduction In the past, chemical industries have made a concerted effort to streamline operation. The goal was simply to produce products as many as possible. Nowadays, the consistency of the product quality is essential to success as the market is highly competitive worldwide. The work of process engineers is not only to design reliable control strategy but also to ensure the control objectives accomplishing within the specified performance. Even though many processes have been around for years and engineers have acquired lots of experience, many operational problems still go undiagnosed for prolong period of time. If the deterioration of controller performance cannot be diagnosed and corrected timely, then the malfunction would make the damage worse in the company’s operating margin. Although the operating data in the most modern chemical processes can be accessible at any time, it is difficult for operators to fully understand how well the process is operated and to screen out the fault problem by examining timesequenced data. Thus, developing a firm grasp of the data mining technique to detect, identify, and correct the abnormal condition is strongly required to produce the product and maintain the good performance of the operating process. Past control research mainly focused on designing new strategies gave little attention to assessment and maintenance of installed systems.1,2 The aims of the controller performance assessment include determining the capability of control systems, developing the suitable statistics for measuring the performance, and establishing the techniques in diagnosing root causes * To whom correspondence should be addressed. E-mail: [email protected].

for the degradation in performance.3 The assessment of the control performance, such as mean squares error,4 settling time5 or time delay,6 can be used to monitor and evaluate the improving potential from the routine measured data. In the past decade, the use of minimum variance of the controlled output regarded as the performance benchmark for feedback loop performance assessment has been extensively applied.4 By comparing the difference between the minimum variance of the controlled output and the current variance of the controlled output, the performance of the current controlled system is evaluated. Several researchers extended this theory to defining the various performance indices in assessing current control relative to minimum variance control.7-9 Assessing whether current output variance is significantly beyond benchmark is solely to indicate the poor performance in the control loop, not to find out and remove the fault causes associated with the performance degradation. In the past decade, the fault detection and isolation problem is in general considered from an open loop point of view10,11 despite many cases placed in the closed loop feedback. Gustafsson and Graebe (1998) considered a deterministic disturbance within the closed-loop bandwidth frequency and utilized statistical hypothesis testing to assess whether the root cause was from the disturbance or the system change.12 Huang (1999) proposed local approach to detect small changes in system dynamics.13 Cinar et al. (1992) and Stanfelj et al. (1993) built a decision tree structure with small perturbations in the setpoint to monitor the feedback system in the IMC structure,14,15 but the change of the unmeasured disturbance was not considered. In the closed loop point of view, the acceptable control performance should be on the basis of all necessary control elements (process, disturbance and controller) in order to maintain the

10.1021/ie0492808 CCC: $30.25 © 2005 American Chemical Society Published on Web 06/17/2005

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5661

where GP(q-1), GL(q-1), and GC(q-1) represent the process, the unmeasured disturbance and the feedback controller. y and w are the process output and the unmeasured white noise; f is the integer dead time of the process. With the long division, the output variance of eq 1 can be depicted as

Figure 1. Feedback control system.

health of the control loop and to make the operation of the process profitable. Fault diagnosis based on the closed-loop performance is equivalent to the model validation and the controller validation.16 In the model validation, the model used for the controller design is validated; whereas, the controller validation is a validation of the performance of the closed-loop system. Therefore, the fault diagnosis methods based on the closed-loop performance cannot be used directly without the valid model. According to Harris’ work, the closed-loop behavior of the output in the feedback system can be expressed as an impulse response model which consists of the feedback invariant term and the feedback dependent term. Any fault in the feedback system might change either or both of the impulse response coefficients of the feedback invariant term and those of the feedback dependent term. Using these terms as performance fingerprints, a new diagnostic methodology is proposed in this paper. These terms can be computed from only the routine operating data of the closed loop system, and then a diagnostic reasoning tree which can locate and remove the root causes can be systematically derived. This paper is organized as follows. The second section defines the diagnosis problem of the control closed loop system. Because of the proposed fault diagnosis method based on the output variation, the background of performance assessment of the control loop is introduced in Section 2. The diagnostic procedure using impulse response analysis to detect the fault symptoms is derived in Sections 3 and 4. Then, in Section 5, the diagnostic reasoning tree based on the fault symptoms is derived to determine the location of the faults. A simulation case and a real level process with different types of faults are demonstrated in Section 6. Finally, the conclusions are made. 2. Problem Formulation A block diagram of the closed-loop system which describes the typical controlled system is shown in Figure 1. According to the description of the system given by Figure 1, the measured output can be derived as the following equation

y(k) )

GL(q-1)

w(k)

1 + q-(f+1)GP(q-1)GC(q-1)

(1)

where φi is the impulse response coefficient of y and w; σw2 is the variance of unmeasured white noise. On the basis of the routine operating data, Harris (1989) has indicated that the first (f + 1) terms (sFBI) of eq 2 are feedback invariant (FBI) where the others (sFBD) are feedback dependent (FBD).4 To assess feedback control performance, most work4,8-10 used minimum variance control as the performance benchmark which is the summation of the first (f + 1) terms in eq 2.

σmv2 ) (φ02 + φ12 + ‚‚‚ + φf2)σw2

(3)

However, due to the aggressive control inputs,17-18 Ko and Edgar (1998, 2004) took the achievable minimum variance under PI control as the performance benchmark.18,19 A similar idea is adopted for P or PID controllers except for the PI controller. Under achievable minimum variance, the actual variance of the feedback control system will be

σ/mv2 ) minσy2 ) (s/FBI + s/FBD)σw2

(4)

Gc

where

s/FBI ) σmv2 and s/FBD ) minsFBD Gc

In eq 4, any optimization procedure can be used to search for the optimal controller parameters resulting in the achievable minimum variance. Note that the benchmark of the control performance is not restricted to the minimum variance control. Other benchmarks like, LQG or GMV, can also be applied. Once the benchmark is established, it can just be viewed as being gray indicators because these index values only imply how poor the performance of the current control loop is, not indicating where the fault comes from. Due to different impulses (of eq 2) effect on the variance of the controlled output being exhibited in different faults, the FBI and the FBD terms obtained from the available historical data information can be used to analyze and find out different types of faults which can affect the achievable performance working condition. 3. Fault Diagnosis and Isolation Using the Output Variance To determine the location and the types of faults in the feedback controlled system, the relationship of process output and the control elements (GP, GL, and GC) in the closed-loop system should be derived. First, the disturbance transfer function (GL) can be

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Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005

decomposed as

Table 1. Fault Symptoms of the Feedback Control System

GL(q-1) ) F(q-1) + R(q-1)q-(f+1)

(5)

where F(q-1) is the polynomial in q-1 of order f, and R(q-1) is the remaining proper transfer function that satisfies the Diophantine identities. Substituting this identity (eq 5) into eq 1 and regrouping the terms into the FBI and the FBD terms. Equation 1 can be depicted as

y(k) ) yFBI(k) + yFBD(k) ) F(q-1)w(k) + q-(f+1)L(q-1)w(k) (6) where

L(q-1) )

R(q-1) - F(q-1)GP(q-1)GC(q-1) 1 + q-(f+1)GP(q-1)GC(q-1)

After matching the relationship of eq 1 and eq 6, the variances of the FBI and the FBD terms which are the functions of the corresponding control elements are obviously expressed as

σy2(GP, GL, GC, f) ) [sFBI(f, GL) + sFBD(f, GP, GL, GC)] (7) Because F(q-1) of eq 6 is the consequence of the feedback-invariant, it depends on the process time delay (f) and the disturbance transfer function (GL). The proper transfer function, L(q-1) of eq 6, is the feedbackdependent. The corresponding second term of the process output variance ( eq 6) depends on all control elements (f, GP, GL, and GC). The equivalent relations of eq 7 also indicate that FBI can be affected by the changes of the process dead time (f) or the disturbances (GL) only, but FBD would be changed when any fault (f, GP, GL, and GC) occurs.

∞

(φ/j )2 + ∑ (φ/j )2] ∑ j)0 j)f+1

σy*2(GP, GL, GC, f) ) [

/ / + s0,FBD ] σw2 ) [s0,FBI f

∞

c c (φcj )2 + ∑ (φcj )2]σw2 ) [s0,FBI + s0,FBD ] ∑ j)0 j)f+1

FBIr0 & FBDr0

at the benchmark condition are required before testing the current operation performance. Once there are faults in the feedback system, FBI and/or FBD would be changed. Because the disturbance is driven by a white noise sequence, the controlled outputs (yFBI and yFBD) from FBI and FBD ( eq 6) should be normally c / c /s0,FBI and s0,FBD / distributed. Thus, the ratios of s0,FBI / s0,FBD follow a distribution called the F distribution. On the basis of the quantified variation of both terms, the appropriate threshold for the F statistics based on the level of significance R, can be determined to identify what observations of the current operation are most relevant to the fault condition. Note that when two different disturbances (and/or processes) of the current and the benchmark systems have different impulse response coefficients, they may produce same value of the sum of squares of FBI and FBD. This means that the current variance is the same to the benchmark and the current performance is accepted. It is not necessary to further find out the faults, because our method is ignited when the current variance significantly deviates from the benchmark. Given m current observations for achievable benchmark and n observations for the current condition, the variances of FBI and FBD can be tested with the hypotheses where

(9)

or

To detect the root cause, the summations of square of impulse coefficients are compared between the achievable minimum variance control output (σy*2) and the current variance control output (σyc2).

σyc2 ) [

possible faults no faults the poor controller tuning (GC) the change of process dead time (f) the change of the process model (GP) the change of process dead time (f) the change of the disturbance model (GL) the poor controller tuning (GC) the change of the process model (GP)

c / FBI0: s0,FBI ) s0,FBI

4. Fault Symptoms Extraction from Impulse Response

f

symptoms FBIa0 & FBDa0 FBIa0 & FBDr0

(8)

where the superscript * and c represent each notation at the achievable and the current operating conditions, / f c f respectively. s0,FBI ) ∑j)0 (φ/j )2σw2, s0,FBI ) ∑j)0 (φcj )2σw2, / ∞ / 2 c ∞ c 2 2 s0,FBD ) ∑j)f+1 (φj ) σw and s0,FBD ) ∑j)f+1 (φj ) σw2 are the variances of FBI and FBD for the current and the achievable minimum variance operating conditions. Note the achievable minimum variance control output (σy*2) is directly computed by taking a long division of the closed-loop transfer function with benchmark condition. Thus, the process model and the disturbance model

c / ) s0,FBD FBD0: s0,FBD

indicates the possible faults to be occurred. The use of the two-sided hypothesis test for FBI0 and FBD0 statistics is calculated by means of F-distribution and FR(m - 1, n - 1) is the 1 - R percentile limit.20 With the above two null hypothesis testing, three possible fault symptoms would happen, (i) FBI0 is accepted (FBIa0) and FBD0 is accepted (FBDa0), (ii) FBI0 is accepted (FBIa0) and FBD0 is rejected (FBDr0), and (iii) FBI0 is rejected (FBIr0) and FBD0 is rejected (FBDr0). Table 1 lists these three symptoms and the corresponding possible occurred faults. It is noted that the symptom with the rejected FBI0 and the accepted FBD0 does not exist. Whenever any FBI0 fault occurs, FBD0 term must be unacceptable because the FBI0 fault coming from the change of process dead time or the change of the disturbance model will also affect the impulse coefficients of the FBD0 term.


Figure 2. Flowchart of the stepwise diagnosis procedure.

5. Generating the Diagnosis Tree to Isolate the Faults The previous section discusses the fault detection that identifies the output variance associated with the faulty subsystem. Now, a reasoning diagnosis tree is necessary to isolate the root cause associated with a particular control element assumed to occur. While this will narrow down the cause of the poor control performance operations, it will not equivocally diagnose the cause. Without any external input to change the current operating process, the diagnosis tree approach based on analysis of variances and analysis of causes is developed. A stepwise diagnosis procedure plotted in Figure 2 is proposed. First the fault detection and diagnosis approaches are ignited if and only if the current output variance (σyc2) significantly deviates from the initial benchmark (σy*2). In the upper part of Figure 2, using impulse response estimation coefficients obtained from c c and si,FBD ; the current closed-loop data calculates si,FBI / / in the lower part, si,FBI and si,FBD can be directly computed by taking a long division of the closed-loop transfer function with the benchmark condition (eq 1). Then, the hypothesis test is applied to check if the fault cause is accepted. As long as the value of the test statistics exceeds the threshold value, the effect of the just-identified fault is fixed. The method starts with the comparison between the current operation system and the benchmark system and then the benchmark models are successively replaced whenever any fault is identified. The new benchmark model obtained from the estimated sum of square of impulse response coefficients is successively replaced. The stepwise diagnosis procedure is repeated till all possible faults are detected. It starts only with early evidence of a given pathology, in the presence of the fault, being the conclusions eventually reached of an adductive nature. As more features of the fault become available, more deductive reasoning is enabled and new inferences become possible. It is actually practical and easy to do the operating process. On the basis of the stepwise diagnosis sequence, a diagnostic tree can be easily derived as follows. Assume the operating controller with the achievable performance is given and the fault from the poor tuning controller has been checked before further diagnosis; the possible fault sources would come from the process (GP) or the disturbance (GL). However, whenever the fault is from either GP or GL, the fault of the poor controller design (GC) would be indirectly affected. Thus, the

controller should certainly be re-designed. In the terms of FBI and FBD, because the process dead time is used to divide the process variance into FBI and FBD, the dead time should be first estimated for the current closed loop operating data. Here, the correlation analysis method is adopted.21 This time delay estimation technique is based on identifying the maximum value of the cross correlation between the process input and output of the closed loop data as the reference signal and the delay signal.22 By estimating the cross-correlation sequence, the highest value is considered as the process dead time.23 Other techniques can also help determining process dead time from the closed-loop data.24-25 After the dead time has been changed, the new FBI and FDI terms for the current data and the achievable model are computed by simply enclosing the properly number of the impulse response coefficients of the current and the achievable conditions based on the estimated dead time (f c)

σy*2(GP, GL, f c) ) fc

[

∑ j)0

(φ/j )2 +

∞

∑

/ / (φ/j )2]σw2 ) [s1,FBI + s1,FBD ]σw2

j)f c+1 fc

∞

(φcj )2 + ∑ ∑ j)0

σyc2 ) [

c c (φcj )2]σw2 ) [s1,FBI + s1,FBD ]σw2

j)f c+1

(10) where f c is the estimated process dead time from the c current operating data. The statistics of FBI1 (s1,FBI / / c / s1,FBI) and FBD1 (s1,FBD/s1,FBD) with the updating dead time are calculated. Thus, the use of the two-sided / ) hypothesis test, with the null hypothesis being s1,FBI c s1,FBI and the two-sided alterative hypothesis being / c * s1,FBI are applied here. On the basis of the fault s1,FBI symptom of Table 1 and the new estimated dead time, the diagnosis paradigm would put forward in the two classes of the fault symptoms. FBI0 is Accepted and FBD0 is Rejected (FBI0a & FBD0r). (a1) If the process dead time is unchanged, the only possible fault is the change of the process model (GP). This is an obvious outcome because FBIa0 tells us the disturbance model is no fault and the only fault is the process that causes FBDr0.

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Figure 3. Fault detection and diagnosis tree for the feedback control system.

(a2) If the process dead time is changed, the possible faults may have the change of the process model (GP) except for the change of the process dead (f). Here, the current dead time must be increased because it is under the FBIa0 condition. The fault from the process occurs based on the accepted or the rejected null hypothesis of FBD1. c / (a2-1) FBD1: s1,FBD ) s1,FBD cannot be rejected. One cannot accept the poor performance that results from the change of the process model. The only possible fault would be the process dead time. c / (a2-2) FBD1: s1,FBD ) s1,FBD is rejected. The change of the operating process that causes the poor control performance cannot be rejected. (b) FBI0 is Rejected and FBD0 is Rejected (FBIr0 & FBDr0). (b1) If the process dead time is unchanged, the possible faults may come from the change of the model (GP), but the GL fault must exist. One of the faults from the disturbance model exists because the dead time is unchanged and FBIr0 must come from the disturbance model (eq 7). To re-construct the unmeasured disturbance model, the approximate stochastic disturbance model realization18 is adopted here to identify the disturbance model (GcL). First, the first f + 1 terms of impulse response coefficients can be evaluated from the current closed-loop output data. Then finding a disturbance ARIMA model with a lower AR order such that the FIR sequence of the first f + 1 terms can meet the estimated impulse response coefficients within some constraints. The estimated model based on the approximate stochastic disturbance model

realization can be good enough when the pure process dead time is three or more.18 Thus, the accurate values of FBI and FBD depend on the process dead time; the effectiveness of the proposed method will be much better when the process dead time is three or more. After GcL substituting into the achievable model, the new FBI and FDI terms are computed as f

∑ j)0

σy*2(GP, GcL, f) ) [

(φ/j )2 +

∞

(φ/j )2]σw2 ) ∑ j)f+1 / / + s2,FBD ]σw2 (11) [s2,FBI

∞

f

∑ j)0

σyc2 ) [

(φcj )2 +

c c + s2,FBD ]σw2 ∑ (φcj )2]σw2 ) [s2,FBI

j)f+1

c / The new statistic FBD2 (s2,FBD /s2,FBD ) is computed based on the estimated disturbance model. c / (b1-1) FBD2: s2,FBD ) s2,FBD cannot be rejected. The only possible fault would be the disturbance model. c / (b1-2) FBD2: s2,FBD ) s2,FBD is rejected. The faults would be the process and the disturbance models. (b2) If the process dead time is changed, the faults may come from the change of the process model (GP) or the change of the disturbance model (GL) except for the change of the process dead (f). c / c / (b2-1) FBI1: s1,FBI ) s1,FBI and FBD1: s1,FBD ) s1,FBD cannot be rejected. The only possible fault would be the dead time. c / (b2-2) FBI1: s1,FBI ) s1,FBI is accepted and FBD1: c / s1,FBD ) s1,FBD is rejected. The faults come from the process and the dead time.

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5665 c / c / (b2-3) FBI1: s1,FBI ) s1,FBI and FBD1: s1,FBD ) s1,FBD are rejected. One of the possible faults is from the disturbance model because the dead time problem is fixed. Like (b1), the unmeasured disturbance model (GcL) should be re-estimated. With the estimated GcL and f c, the new FBI and FDI terms of the achievable model are computed as

σy*2(GP, GcL, f c) ) fc

∑ j)0

[

(φ/j )2 +

∞

/ / + s3,FBD ]σw2 ∑ (φ/j )2]σw2 ) [s3,FBI

(12)

j)fc+1

∞

fc

∑ j)0

σyc2 ) [

(φcj )2 +

c c + s3,FBD ]σw2 ∑ (φcj )2]σw2 ) [s3,FBI

j)fc+1

c / the new statistic FBD3 (s3,FBD /s3,FBD ) is also defined. c / (b2-3-1) FBD3: s3,FBD ) s3,FBD is accepted. The faults are from the disturbance model and the dead time. c / (b2-3-2) FBD3: s3,FBD ) s3,FBD is rejected. The faults come from the process model, the disturbance model and the dead time. In summary, the presented fault diagnosis system performs the impulse coefficient estimation and fault isolation at FBI and FBD terms. The values of the impulse coefficients are assumed to stay constant during the fault identification and the faults are assumed to be fixed during the impulse coefficient estimation. Figure 3 shows the summary of the feedback control diagnostic hierarchy to determine the location of the faults. Without prior external input to perturb the operating system, the proposed method can systematically and sequentially identify the possible faults. When the fault is from the process model, to tune the new controller of the good control performance, the closedloop identification should be conducted to re-estimate the new process model.

Figure 4. Closed-loop simulation in Example 1 of the controlled output responses of achievable benchmark (*), process model change (O), unmeasured disturbance change (×) and multiple faults (0).

6. Illustration Examples 6.1 Example 1: Simulation Case. In this example, the diagnosis procedure developed in the previous section is illustrated via a simulation example. A revised example based on Ko and Edgar (1998) problem18 is modeled as

Gp(q-1)q-(f+1) ) GL(q-1) )

1 q-6 -1 1 - 0.8q

1 + 0.6q-1 (13) (1 - 0.5q-1)(1 - 0.6q-1)(1 + 0.7q-1)

The unmeasured noise (w) with zero mean and unit variance enters the disturbance model. A PID controller is used to control this system

k1 + k2q-1 + k3q-2 (-y(k)) u(k) ) 1 - q-1

(14)

To obtain the achievable minimum variance control, the PID achievable performance bound can be estimated by minimizing the objective function ( eq 4). The calculated value is σ/mv2 ) 3.49 and the corresponding controller parameters are tuned as k1 ) - 0.1347, k2 ) 0.2505

Figure 5. Dead time estimation with process model error case in Example 1 (i).

and k3 ) - 0.1159. The achievable minimum variance benchmark can be separated as the feedback invariant / / ) 3.39 and the feedback dependent s0,FBD ) 0.10. s0,FBI Three different fault conditions are tested here, including the faults from (i) the process, (ii) the disturbance, and (iii) the process, the disturbance and the dead time together. The last case is a multiple fault problem. The simulated closed loop outputs for the achievable normal operation condition and the fault ones are plotted in Figure 4. From the figure, the systems under unmeasured disturbance change and multiple faults have large deviations from the system with benchmark control; however, it is hard to diagnose the root faults solely from the closed-loop output response. The proposed diagnosis procedure is used to find out the root causes. Change of the Process Model (GP). In this case, the process model is changed into GP(q-1) ) 2/(1 + 0.1q-1). The resulting output variance is 4.59 and the controller parameters are still based on the achievable benchmark parameters. Since the variance of the resulting output is 35% higher than that of the benchmark, the diagnostic procedure is adopted. First, the

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Figure 6. Fault detection and diagnosis tree whose bold line indicates the fault conditions for each testing case in Example 1.

hypothesis tests (FBI0 & FBD0) are applied onto the current data. At the 0.05 level of significance, the null hypothesis of FBI0 cannot be rejected (FBIa0) and the null hypothesis of FBD0 cannot be accepted (FBDr0)

F0.975 ) 0.91 < FBI0 )

3.54 ) 1.04 < F0.025 ) 1.10 3.39 (15)

and

FBD0 )

0.87 ) 8.70 > F0.025 ) 1.10 0.10

The symptom of FBIa0 & FBDr0 indicates that the faults may come from the process or the dead time. Following the diagnosis tree in Figure 3, the process dead time should be checked in the next stage. By applying the cross-correlation analysis onto input and output data, the process dead time is unchanged (f ) 5) (Figure 5). The diagnosis tree shown in Figure 6 provides the direction for finding out the fault location. Therefore, the root cause in this fault condition is the process model error. Change of the Process Model (GL). Here the disturbance model is changed into GL(q-1) ) (1 + 0.9q-1)/((1-0.8q-1)(1-0.6q-1)(1 + 0.7q-1)). Under the original achievable PID controller, the resulting output variance of the feedback system with disturbance model error is 16.39, which is 4.82 times bigger than that of the achievable benchmark (3.49). To diagnose the root

cause, the hypothesis tests (FBI0 & FBD0) are first used and both FBI0 and FBD0 cannot be accepted

FBI0 )

12.33 ) 3.64 > F0.025 ) 1.10 3.39

FBD0 )

4.35 ) 43.50 > F0.025 ) 1.10 0.10

(16)

and

The symptom can be classified as FBIr0 & FBDr0. The diagnosis tree of this case is shown in Figure 6. The next stage is to check if the process delay is changed using cross-correlation analysis on input-output data. Figure 7 indicates that the dead time is the same as its original value (f ) 5). Hence, one of the root causes resulting in FBDr0 must be the unmeasured disturbance model error. Re-identified disturbance model is then used to isolate the fault of the unmeasured disturbance model. A total of 1000 sets of the data are collected. The method identifies the disturbance model based on the first six (f ) 5) closed-loop impulse response coefficients obtained from time-series modeling of the output data

GL(q-1) )

1 + 0.8q-1 (1 - 0.8q )(1 - 0.6q-1)(1 + 0.6q-1) -1

(17)

The FBD2 statistic is then used to compare to the current operating conditon and the achievable


Figure 7. Dead time estimation with unmeasured disturbance model error case in Example 1 (ii).

Figure 9. Dead time estimation with multiple fault case in Example 1(iii).

original benchmark. The diagnosis tree shown in Figure 6 guides the fault location which is transparent. In the hypothesis test of FBI0 & FBD0, FBI0 and FBD0 are rejected

FBI0 )

9.50 ) 2.80 > F0.025 ) 1.10 3.39

FBD0 )

8.25 ) 82.5 > F0.025 ) 1.10 0.10

(19)

and

Figure 8. Impulse response coefficients of the current operation (0) and the achievable benchmark (*) after replacing the estimated disturbance model in Example 1 (ii).

benchmark FBD value that is obtained from the reestimated disturbance model.

3.11 F0.975 ) 0.91 < FBD2 ) ) 0.92 < F0.025 ) 1.10 3.39 (18) It indicates that the null hypothesis of the process change should not be rejected and the only fault is the unmeasured disturbance model error. Figure 8 also shows that the impulse response coefficients of the current operation and the achievable benchmark after replacing the re-estimated disturbance model are pretty similar. Change of the Process, the Disturbance and the Dead Time (GP, GL, f). The multiple faults that we consider in this case are the change of the process (GP(q-1) ) 2/(1 + 0.1q-1)), the change of the disturbance (GL(q-1) ) (1 + 0.9q-1/(1-0.8q-1)(1-0.6q-1)(1 + 0.7q-1))), and the change of the dead time (f ) 20). The control parameters of the closed loop are still based on the original achievable design values. The output variance of this fault condition (18.98) is far from that of the

Then in the next stage, the process dead time is checked. From Figure 9, it is obvious that the process dead time increases to 20; thus, a hypothesis test of FBI1 & FBD1 is needed. With the new dead time, the new FBI and FBD values of the current and the achievable benchmark conditions are re-computed. The hypothesis test of FBI1 & FBD1, FBI1, and FBD1 are still rejected

FBI1 )

14.65 ) 4.54 > F0.025 ) 1.10 3.23

FBD1 )

3.10 ) 34.44 > F0.025 ) 1.10 0.09

(20)

and

This implies the fault from the unmeasured disturbance model. Collecting the current operating data, the disturbance model is re-identified by the approximate stochastic disturbance model realization

GL(q-1) ) 1 + 0.6q-1 (1 - 0.9q-1)(1 - 0.4q-1)(1 + 0.4q-1)(1 + 0.1q-1) (21) The FBD3 statistic is then used to compare to the current operating condition and the achievable benchmark FBD value that is obtained from the reestimated disturbance model. Figure 10 also shows the impulse response coefficients of the current condition and the achievable benchmark after replacing the reestimated disturbance model and the new dead time.

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Figure 10. Impulse response coefficients of the current operation (0) and the achievable benchmark (*) after replacing the estimated disturbance model in Example 1 (iii).

Figure 11. Schematic diagram of the level control loop.

After the dead time is isolated, the significant differences of both impulse coefficients imply that the fault of the process exists. The hypothesis test of FBD3 statistics

FBD3 )

0.05 ) 0.016 < F0.975 ) 0.91 3.10

(22)

also indicates the fault is the process change. The above procedures are only used to find out the root cause to deteriorate the control performance, not to fix or improve the current performance. The last stage, but not the least one, is to recover the control performance after correcting the fault sources. In this case, one of the root causes is from plant/model mismatch. Without any difficulty, the conventional closedloop model identification26,27 is applied here directly to obtain the approximated process (1.832q-21/ (1 + 0.08781q-1)). When all faults are removed, the final stage is to calculate the new achievable benchmark (14.33) and the corresponding controller parameters (k1 ) -0.1347, k2 ) 0.2505 and k3 ) -0.1159). The performance bound will be used as the new benchmark to keep the monitoring and diagnosis of the next coming operation feedback system. 6.2 Example 2: Experimental Case. In this experiment, a pilot level in a gravity-drained tank shown in Figure 11 is studied. The objective is to assess the current control performance and diagnose the fault causes in order to maintain the level system at the desired performance in the presence of feed flow rate disturbances. The tank is equipped with differentialpressure-to-current (DP/I) transducers to provide a continuous measurement of the level. The computer is connected to a PCI-1710 analog/digital I/O expansion card from Advantech. The expansion board uses a 12bit converter, therefore, the digital signals are 12-bit. The analogue signals from the measured level are amplified and conditioned EDM35 (4-20 mA/0-5 V) modules. The data acquisition software and the PID controller algorithm are MATLAB of MathWorks, Inc. Assume that the original designed process is run under the achievable minimum variance (0.90) (Figure 12) and the corresponding controller parameters are k1

Figure 12. Experimental study in Example 2 of the controlled level responses of achievable benchmark (*) and the process with faults (0).

) -11.20, k2 ) 20.17, k1 ) -9.10. On the basis of a total of 500 sets of the collected data from the closed loop operation, the process and the disturbance models are also identified

GP ) GL )

0.02273 1 - 0.5161q-1 - 0.4628q-2

(23)

1 (1 - 0.9q )(1 - 0.3q-1)(1 + 0.1q-1) -1

respectively, and the estimated process delay is f ) 10. Now, the operating performance of the controlled system is diagnosed when the blockage or clog of the outlet valve of the tank occurs. This will constrict the outlet flow and degrade control performance if it goes undetected. Figure 12 shows that the liquid levels under achievable minimum variance control and under the fault condition, where the output variances are 0.90 and 4.68, respectively. Since the performance of the fault condition is unsatisfied, the proposed diagnosis is ap-


Figure 13. Fault detection and diagnosis tree whose bold line indicates the fault condition in Example 2.

Figure 15. Impulse response coefficients of the current operation (0) and the achievable benchmark (*) after replacing the estimated disturbance model in Example 2.

Figure 14. Dead time estimation in Example 2.

plied. When the hypothesis testing of FBI0 & FBD0 is applied

5.04 ) 3.73 > F0.025 ) 1.13 FBI0 ) 1.35

(24)

and

FBD0 )

15.94 ) 6.38 > F0.025 ) 1.13 2.50

both FBI0 and FBD0 cannot be accepted (FBIr0 and FBDr0). Following the diagnosis tree (Figure 13), the

next step is to check the process delay. Figure 14 shows that the process delay is unchanged. This indicates that the change of the disturbance model must exist. Then the disturbance model should be re-identified. The estimated model is

GL(q-1) )

1 (1 - 0.1q-1)(1 - 0.9q-1)

(25)

After isolating the fault from the disturbance model,

5670


Figure 16. Experimental study in Example 2 of the controlled level responses after the fault condition is identified.

the FBD2 statistic is then used

FBD2 )

15.94 ) 0.60 < F0.975 ) 0.88 26.69

(26)

It indicates that another fault from the process change also exists. Figure 15 also shows that the impulse response coefficients of the current operation and the achievable benchmark after replacing the re-estimated disturbance model are still significantly different. Like the previous discussion, after finding out and fixing the root cause to deteriorate the control performance, the process model should be re-estimated. Figure 16 shows the control response with the achievable re-design controller (k1 ) -10.88, k2 ) 19.99 and k3 ) -9.19). The new achievable benchmark (1.02) would be used to keep monitoring and diagnosing the next coming operation feedback system. 7. Conclusion Research shows that about 66% to 80% of the controllers of the industrial process are not performing as well as they are expected. This detrimental effect on the process profitability causes the product variance. Timely diagnosis of faults is instrumental to enhance safety and reliability of operation systems. A systematic monitoring and detection method is strongly needed to diagnose the operating fault from a cornucopia of data obtained from the computer control system. Furthermore, it can provide a bound on achievable control, detect and remove the root faults. It can manage the controller quality and the faults in performance degradation to get a more reliable control system. In this work, the diagnosis tree based analysis approach is employed to examine the possible faults in a feedback control system. Unlike the traditional physical models as the basis for diagnostic algorithm, this proposed data driven method without a major effort only on the basis of the closed-loop data can analyze the possible faults. The hypothesis test of the diagnosis is used to identify and isolate the system that contains the possible fault information. This is good for the complexity of a common industrial operating process. On the basis of the control output variances consisted

of the feedback invariant variance and the feedback dependent variance, the symptom recognition and diagnostic reasoning tree are developed. This is a stepwise diagnosis procedure. As long as the value of the test statistic exceeds the threshold value, the effect of the just-identified fault is eliminated. The diagnostic procedure will not be ignited even if there are minor faults because the output variance cannot be rejected under the defined critical region of the benchmark one. This is so-called the error risk in the testing statistical hypotheses. To improve the error risk, the level of significance can be increased by further reducing the critical regions. The proposed method is good for the practical operating process. It is easy to find out the changes in the operation system when the control performance is not acceptable. In this paper, the faults, such as process change, disturbance model error, process dead time change and poor tuning controller, can be examined. The proposed method can also be applied to the system containing multiple faults. The effective features of the analysis procedures are illustrated by a series of computer simulation and a actual pilot scale experiment, emphasizing how the hypothesis statistics based on the current and the achievable control output variances are re-computed to help detect and isolate the particular fault. Acknowledgment This work is sponsored by the National Science Council, Republic of China and by the Ministry of Economics, Republic of China. Literature Cited (1) Patwardhan, R. S.; Shah, S., L. Issues in Performance Diagnostic of Model-based Controllers, J. Proc. Cont. 2002, 12, 413. (2) Kozub, D. J. Controller Performance Monitoring and Diagnosis: Experiences and Challenges, CPC V Meeting; 1996. (3) Harris, T. J.; Seppala, C. T.; Desborough, L. D. A Review of Performance Monitoring and Assessment Techniques for Univariate and Multivariate Control Systems. J. Proc. Cont. 1999, 9, 1. (4) Harris, T. J. Assessment of Control Loop Performance. Can. J. Chem. Eng. 1989, 67, 856. (5) Swanda, A.; Seborg, D. Evaluating the Performance of PIDtype Feedback Control Loops using Normalized Settling Time. In ADCHEM 97; Banff, Canada, 1997. (6) Shinskey, F. G. Feedback Controllers for the Process Industries; McGraw-Hill: New York, 1994 (7) Thornhill, N. F.; Oettinger, M.; Fedenczuk, P. Refinery-wide Control Loop Performance Assessment. J. Proc. Cont. 1999, 9, 109. (8) Desborough, L.; Harris, T. J. Performance Assessment Measures for Univariate Feedback control. Can. J. Chem. Eng. 1993, 70, 1186. (9) Kozub, D. J.; Garcia, C. E. Monitoring and Diagnosis of Automated Controllers in the Chemical Process Industries. Proc. AIChE Ann. Meeting; St. Louis, 1993. (10) Basseville, M. Detecting Changes in Signals and Systems - A Survey. Automatica 1988, 24, 309. (11) Frank, P. M. Fault Diagnosis in Dynamic System Using Analytical and Knowledge Based Redundancy - A Survey. Automatica 1990, 26, 459. (12) Gustafsson, F.; Graebe, S. F. Closed-Loop Performance Monitoring in the Presence of System Changes and Disturbances. Automatica 1998, 34, 1311. (13) Huang, B. Process and Control Loop Performance Monitoring through Detection of Abrupt Parameter Changes, Electrical and Computer Engineering, 1999 IEEE Canadian Conference; 1999. (14) Cinar, A.; Marlin, T.; MacGregor, J. Automated Monitoring and Assessment of Controller Performance. IFAC Meeting; 1992.

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5671 (15) Stanfelj, N.; Marlin, T. E.; Macgregor, J. F. Monitoring and Diagnosing Process Control Performance: the Single-Loop Case. Ind. Eng. Chem. Res. 1993, 32, 301. (16) Brozenec, T. F.; Tsao, T. C.; Safonov, M. G. Controller Validation. Int. J. Adapt. Control Signal Proc. 2001, 15, 431. (17) Eriksson, P. G.; Isaksson, A. J. Some Aspects of Control Loop Performance Monitoring. IEEE Conference of Control Applications; Glasgow, 1994. (18) Ko, B. S.; Edgar, T. F. Assessment of Achievable PI Control Performance for Linear Process with Dead time. Proceedings of ACC; Philadelphia, 1998. (19) Ko, B.-S.; Edgar, T. F. PID Control Performance Assessment: The Single-Loop Case, AIChE J. 2004, 50, 1211. (20) Montgomery, D. C.; Runger, G. C. Applied Statistics and Probability for Engineers, 3rd ed.; John Wiley & Sons: New York, 2003. (21) Zheng, W. X.; Feng, C. B. Identification of Stochastic Time Lag Systems in the Presence of Colored Noise. Automatica 1990, 26, 769. (22) Carter, G.; Knapp, C. Time delay estimation. Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP 1976, 1, 357.

(23) Jacovitti, G.; Scarano, G. Discrete time techniques for time delay estimation. IEEE Transactions on Acoustics, Speech and Signal Processing 1993, 41, 525. (24) Gabay, E.; Merhav, S. J. Identification of Linear Systems with Time-delay Operating in a Closed-loop in the Presence of Noise. IEEE Transactions of Automatic Control 1976, AC-21, 711. (25) Etter, D. M.; Stearns, S. D. Adaptive Estimation of Time Delays in Samples Data Systems. IEEE Transactions on ASSP 1981, 29, 582. (26) So¨derstro¨m, T.; Stoica, P. System Identification; PrenticeHall: Englewood Cliffs, NJ, 1989. (27) Van Den Hof, P. M. J.; Schrama, R. J. P. An Indirect Method for Transfer Function Estimation from Closed Loop Data. Automatica 1993, 29, 1523.

Received for review August 10, 2004 Revised manuscript received March 31, 2005 Accepted April 18, 2005 IE0492808

Diagnosis of Closed Control Loop Status Using Performance Analysis

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