Diagnosis of Cascade Control Loop Status Using Performance

On the basis of the achievable performance of the control outputs, fault diagnosis of two cascade control systems, including series cascade control (S...
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Ind. Eng. Chem. Res. 2006, 45, 7540-7551

Diagnosis of Cascade Control Loop Status Using Performance Analysis Based Approach Junghui Chen,* Yuezhi Yea, and Cho-Kai Kong R&D Center for Membrane Technology and Department of Chemical Engineering, Chung-Yuan Christian UniVersity, Chung-Li, Taiwan 320, Republic of China

On the basis of the achievable performance of the control outputs, fault diagnosis of two cascade control systems, including series cascade control (SCC) and parallel cascade control (PCC), is developed. Without any prior knowledge of complicated operating processes and/or external inputs perturbing the operating system, the accurate fault identification can be achieved by a series of the statistical hypothesis procedures applied to the currently measured data. To isolate possible faults, the output variances of the primary and secondary loops are separated into cascade-invariant (CI) and cascade-dependent (CD) terms, respectively, by the Diophantine decompositions. After a sequence of the hypotheses tests is performed on the CI and the CD terms of current control and the achievable performance conditions, the hierarchical diagnosis trees for the primary and secondary outputs are established, respectively, to explore possible faults. The final faults can be inferred by merging both diagnosis trees. The statistical inference systems for SCC and PCC structures are the same. The performance of the proposed method is demonstrated via two examples, including a simulation case with a single fault and a pilot-level tank system with multiple faults. 1. Introduction Because of the external disturbance effect on the manipulated variables, another control loop is often added to the simple feedback control to reject the disturbance effect.1,2 This kind of approach is called cascade control, and it is widely used in process industries. Two kinds of control structures, series cascade control (SCC) and parallel cascade control (PCC), have been available for a long time. In SCC, an inner (or the secondary) loop is embedded within an outer (or the primary) loop. The secondary controller of the inner loop allows rapid rejection or reduction of the disturbances before the disturbance effects spill over to the primary loop, so there is little effect on the primary output. On the other hand, the primary and secondary loops are sometimes connected in a parallel cascade when the manipulated variables and the disturbances affect the primary and secondary outputs through the parallel action. Examples of SCC and PCC processes have been used, and the design methods of cascade control have been detailed.3,4 However, these research papers mainly focused on designing new strategies and gave little attention to monitoring and diagnosis of current operation systems. In the industrial plants, process engineers are responsible for not only designing reliable control strategies but also accomplishing their control objectives within the specified performance. As the performance of the control system often changes after long-time operation, the initial well-designed control system may become undesirably sluggish or aggressive due to many fault problems, such as changes in the process, external unknown disturbances, etc. If deterioration of the controller performance cannot be identified in time, the malfunction would cause monetary loss or even significant impacts on personnel, environmental, and equipment safety. Maintenance of the control system is generally the responsibility of either process control engineers or instrument technicians. It is tedious to consistently monitor a large number of loops, so some control problems may be overlooked. Particularly, poor performance often exists in an operating plant unnoticed for quite a long time. In the past, detecting poor perform* To whom correspondence should be addressed. E-mail: jason@ wavenet.cycu.edu.tw.

ance required expertise and experience. Today, this task is wellsuited for automation, because modern chemical processes are well instrumented with controllers and measurement sensors. The operating data already resides in distributed control systems (DCS) or a plant historical database. It can be used to explore the possible fault condition without plant tests. Monitoring techniques related to the control loop can be divided into two categories: performance assessment of the control loop and fault diagnosis of the control system. The former technique was originally developed by Harris5 based on the minimum variance control (MVC). Following Harris’ MVC approach, other monitoring control systems, such as feedforward/feedback systems,6 internal model control (IMC) control,7 multi-input/ multioutput (MIMO) feedback systems,8 cascade control systems9,10 and multiloop proportional-integral-derivative (PID) control systems,11 have also been proven useful in prioritizing the activities of process engineers. This can potentially benefit from assessing the controller performance of the current operating system. A comprehensive review of this area was found in the literature.7,12 However, assessing whether current output variance is significantly beyond the benchmark variance is only the initial phase in performance monitoring. The next important step is to find out and remove fault causes associated with performance degradation in hopes of bringing the system back to good performance. Some research work has been done to identify the root cause. Gustafsson and Graebe utilized the statistical hypothesis testing to assess the faults in the disturbance or the system change.13 Huang proposed a local approach to detect small changes in system dynamics.14 With prior process knowledge, the data-driven method was used to explore the root cause.15,16 Choudhury et al. generated graphical plots by higherorder statistical techniques to quantify non-Gaussanity and nonlinearity and diagnosed the cause of poor performance of the control loop.17 Under the IMC structure, Cinar et al. and Stanfelj et al. monitored the feedback and feedforward systems with a small perturbation at the setpoint via a decision tree.18,19 Using the routine operating data to construct a closed-loop impulse response model, Yea and Chen developed a diagnostic reasoning tree that could locate and remove root causes of poor performance in a feedback control system.20

10.1021/ie051181m CCC: $33.50 © 2006 American Chemical Society Published on Web 09/22/2006

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7541

outputs of the primary and secondary loops at the sampling time k. GL1 and GL2 are unmeasured disturbances to the primary output and the secondary output, respectively. w1(k) and w2(k) are the disturbances with white noise sequences. The goal of cascade control is to make y1(k) reach the set point as long as the constraints on y2(k) are respected. The inner-loop controller, Gc2, is used to regulate the constrained output y2(k). Gc2 is tuned to avoid overshooting the constrained variable y2(k). The outerloop controller, Gc1, is tuned to regulate the output y1(k) as close as possible to its set point. u1(k) and u2(k) are the controller outputs of Gc1 and Gc2. In the following two subsections, achievable performance bounds are derived for the primary outputs of SCC and PCC, respectively. 2.1. Performance Bound of SCC. When the primary setpoint is constant, the disturbances, w1(k) and w2(k), affect the output variables, y1(k) and y2(k), which are given by

y1 )

(1 + q-(f2+1)G/P2GC2)GL1

w1 + 1 + q-(f2+1)G/P2GC2 + q-(f1+f2+2)G/P1G/P2GC1GC2 q-(f2+1)G/P1GL2

w2 1 + q-(f2+1)G/P2GC2 + q -(f1+f2+2)G/P1G/P2GC1GC2 y2 ) -

1+q

-(f2+1)

Note that Gp1 and Gp2 are represented by Gp1 ) G/p1q-f1 and Gp2 ) G/p2q-f2, where q is the time shift operator; fi, i ) 1, 2, are the time delays; and G/pi, i ) 1, 2, are the process models without any time delay. To decompose y1 and y2, respectively, into the cascade-invariant and cascade-dependent terms, GL1 and GL2, G/P1 and G/P2 are replaced by the following polynomial division identities,

GL1 ) Q1 + R1q-(f1+f2+2) GL2 ) Q2 + R2q-(f2+1) GP1 ) q-(f1+1)G/P1

(2)

G/P1Q2 ) E + Fq-(f2+1) G/P2Q1 ) G + Hq-(f1+1) where Q1 and Q2 are polynomials in q-1 of degree f1 + f2 + 1 and f2, respectively; E and G are polynomials in q-1 of degree f1 and f2; and Ri, i ) 1, 2, F, and H are proper transfer functions. Substituting these into eq 1 gives

2. Performance Bound of Cascade Control Systems The block diagrams of SCC and PCC systems consist of two feedback loops, as shown in Figure 1. The process has two components, Gp1 and Gp2. In the figure, the control valve, the controlled process, and the sensor-transmitter are combined because the figure highlights the signals of the controller outputs and the transmitter signals in the system. Both signals can usually be observed and recorded. y1(k) and y2(k) are the process

G/P2GC2GC1GL1

w1 + G/P2GC2 + q-(f1+f2+2)G/P1G/P2GC1GC2 GL2 w2 -(f2+1) / 1+q GP2GC2 + q-(f1+f2+2)G/P1G/P2GC1GC2

Figure 1. (a) A series cascade control system with possible faults; (b) a parallel cascade control system with possible faults.

This article is an extension of our recent work,20 and it presents a diagnosis methodology for cascade control loops based on control performance. Like the single loop of feedback control, the primary and secondary output variances can be separated into cascade-invariant (CI) and cascade-dependent (CD) terms, respectively. Those terms are performance indices. Any fault in cascade control loops may affect either or both of the CI and CD terms of the primary and secondary output variances. With only the routine operating data of the closed loop system, the diagnostic reasoning tree can be systematically developed by combining the primary and secondary control loops, both of which can locate and remove the root cause of poor performance. This paper is structured as follows. The second section defines the diagnosis problem of the two cascade systems, SCC and PCC. Because the proposed fault diagnosis method is based on the output variation, the output variation for the performance assessments of SCC and PCC is derived first. In Section 3, the diagnostic reasoning tree of SCC is derived using impulse response analysis to detect the fault sources. In Section 4, without any significant modification, the same fault diagnosis procedure of SCC can be directly applied to PCC to determine the location of the faults. An index is derived from the operating data to separate SCC from PCC in Section 5. A simulation case with a single fault and a pilot-level tank system with multiple faults are demonstrated in Section 6. Finally, the conclusions are made.

q

(1)

-(f2+1)

where

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Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006

SSCC )

1 1+q

-(f2+1)

G/P2GC2

+q

-(f1+f2+2)

G/P1G/P2GC1GC2

In these equations, the effects of w1 and w2 on y1 and y2 are decomposed. It is obvious to see from the block diagram (Figure 1a) that Q1 is the cascade invariant of y1 because the disturbance w1 enters directly into the output y1. Because of the time delay (f2 + f1 + 1) of processes Gp1 and Gp2, R1 is a cascade-dependent effect of y1, which makes it possible for both controllers to make a change after f2 + f1 + 2 time delays. E is the cascade invariant of y1, because the disturbance w2 enters directly into the output y1 after the time delay (f1) of process Gp1. For y2, similar explanations can be used for the relationships between Q2 and R2. If minimum variance controllers are implemented in order to minimize the variances of the primary and secondary outputs, only the cascade-dependent (CDij, i, j ) 1, 2) terms of the outputs should be minimized, because CIij, i, j ) 1, 2, depends only on the disturbance characteristics and the cascade-dependent (CDij) term is a function of both controllers. The minimum variance control (MVC) law of SCC has been derived.9 The theoretical “best achievable performance” using minimum variance is very attractive because it can be estimated from the process operating data under the given process time delay. However, even though MVC is often used as a performance benchmark, it is unrealistic for most applications. Eriksson and Isaksson mentioned that MVC usually led to a large input action and it lacked robustness in control.21 The MVC benchmark bound can only indicate if the current performance is close to the MVC benchmark. There are no solutions to the potential improvement, because it is unknown how to retune the controller. Thus, a more practical performance benchmark for the specific controllers can be obtained by solving the following optimization problem:

(σSCC2)achievable-MV ) min σSCC,y12

where Q1 and Q2 are polynomials of degree f1; Q3 and Q4 are polynomials of degree f2; and Ri, i ) 1, ..., 4, are proper transfer functions. The outputs (y1(k) and y2(k)) can be expressed as CI terms plus CD terms

where SPCC ) 1/[1 + q-(f2+1)G/P2GC2 + q-(f1+1)G/P1GC1GC2]. Q1 is the cascade invariant of y1 because the disturbance (w1(k)) enters directly into the output y1. Because of the time delay (f1) of process Gp1, R1 is a cascade dependent effect of y1, which makes it possible for both controllers to make a change after f1 time delays. The identity for Q3 and R3 is used to describe the effects of w1 on y2, but both terms are cascade dependent because both controllers can change the terms. Similar explanations can be used for the relationships between Q2 and R2 and between Q4 and R4, respectively. The minimum variance control (MVC) law has been derived10 and the achievable performance benchmark for the specific controllers is also solved by the following optimization problem:

(4)

Gc1,Gc2

(σPCC2)achievable-MV ) min σPCC,y12

(8)

Gc1,Gc2

It is apparent that the variance of the output y1 is the function of the controller parameters Gc1 and Gc2. 2.2. Performance Bound of PCC. Like SCC, a PCC system also consists of two feedback loops. The only difference is that the manipulated variable and the disturbance affect the primary and secondary outputs through parallel transfer functions. When the primary setpoint is constant, the closed-loop response of the primary and secondary outputs (y1(k) and y2(k)) disturbed by the disturbances (w1(k) and w2(k)) can be expressed as

y1 )

(1 + q-(f2+1)GP2/GC2)GL1 1 + q-(f2+1)GP2/GC2 + q-(f1+1)G/P1GC1GC2 q-(f1+1)G/P1GC2GL2

1+q y2 ) -

-(f2+1)

GP2/GC2

+q

-(f1+1)

w1 -

w2 G/P1GC1GC2

q-(f2+1)G/P2GC2GC1GL1 1 + q-(f2+1)G/P2GC2 + q-(f1+1)G/P1GC1GC2

In eqs 4 and 8, the achievable minimum variance control that serves as an appropriate benchmark provides a measure of the current control-loop performance. Note that the control performance is not restricted to any specified control structure. Other benchmarks, like the linear quadratic Gaussian or the generalized minimum variance, can also be applied instead of the achievable minimum variance of the controlled output (eqs 4 and 8). Once the current performance is far from (σSCC2)achievable-MV (or (σPCC2)achievable-MV), the possible faults exist in the current control loop. 3. Fault Detection and Diagnosis of SCC

(5)

w1 +

Applying long division to eq 3, the output variances (σy12 and σy22) related to the impulse response coefficients can be expressed as

σy12 ) sy1,CI11(GL1)σw12 + sy1,CD11(G/P1, G/P2, GL1)σw12 +

(1 + q-(f1+1)G/P1GC1GC2)GL2

w2 1 + q-(f2+1)G/P2GC2 + q-(f1+1)G/P1GC1GC2

sy1,CD12(G/P1, G/P2, GL2)σw22 σ y2 ) 2

After substituting the following identities,

G/P2,

GL1)σw1 + sy2,CI22(GL2)σw2 + 2

2

sy2,CD22(G/P1, G/P2, GL2)σw22

GL1 ) Q1 + R1q-(f1+1) ) Q3 + R3q-(f2+1) GL2 ) Q2 + R2q-(f1+1) ) Q4 + R4q-(f2+1)

sy2,CD21(G/P1,

(6) where

(9)

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7543 ∞

f1

sy1,CI11 )

∑ j)0

(Rj)2 sy1,CI11 )

∑

j)f1+1

(Rj)2 ∞

sy1,CD12 )

(βj)2 ∑ j)f +1 1

∞

f2

sy2,CI22 )

(φj)2 ∑ j)0

sy2,CD21 )

(10)

∑ (φj)2σw 2 1

j)f2+1

∞

sy2,CD22 )

∑ (φj)2 j)f +1

(CI)10: sy/1,(CI)0 ) syc1,(CI)0 (CD)10: sy/1,(CD)0 ) syc1,(CD)0

2

where {Ri}, {βi}, {φi}, and {φi} are the impulse response coefficients from y1 and y2. Here, the unmeasured disturbances w1 and w2 are independently and identically distributed random variables with the same mean value of zeros; however, their variance values are different. To simplify the above expression and easily compute the impulse response coefficients from y1 and y2, σw2 ) a2σw12 ) b2σw22 are defined, where a and b are constant ratios for extracting a common unmeasured disturbance (w) for w1 and w2. The above equations can be written as

σy12 ) sy1,CI(GL1)σw2 + sy1,CD(G/P1, G/P2, GL1, GL2)σw2 σy22 ) sy2,CI(GL2)σw2 + sy2,CD(G/P1, G/P2, GL1,GL2)σw22

(11)

where ∞

f1

sy1,CI )

∑ j)0

(aRj)2 sy1,CD )

sy2,CI )

(bφj) ∑ j)0

[(aRj)2 + (bβj)2] ∑ j)f +1 1

(12)

∞

f2

2

sy2,CD )

∑ [(aφj)

j)f2+1

2

analysis on the variations of CI and CD. To get the accurate dead time, sufficient information should be provided. Other techniques can also help determining the process dead time from the closed-loop data.24,25 If prior knowledge of dead time is available, the user-defined benchmarks of CI and CD can be directly computed from eqs 11 and 12. Now, the rest of the possible fault causes (G/P1, G/P2, GL1, and GL2) can be found by examining corresponding subterms (sy1,CI, sy1,CD, sy2,CI, and sy2,CD) of σy12 and σy22 for the current and achievable conditions.

+ (bφj) ] 2

In eq 11, the output variance y1 can be classified into the variances from CI (sy1,CI) and from CD (sy1,CD). sy1,CI is the sum of the first (f1 + 1) square impulse response coefficients of the effect w1 on y1. sy1,CD is the sum of the square impulse response coefficients of the rest terms from f1 + 1, including part of CI11, CI12, CD11, and CD12. Because sy1,CI is computed from Q1, it is the function of GL1 only. A similar explanation is used for sy2,CI. sy2,CI is the function of GL2 only. However, sy2,CD is the function of all components (G/P1, G/P2, GL1, GL2). Assume the operating controllers (GC1 and GC2) with the achievable performance are given. The faults from these operating controllers can be checked first before starting diagnosis. The rest of the fault sources would come from the other elements (f1, f2, G/P1, G/P2, GL1, and GL2). However, whenever the fault comes from one of these elements, the controller design (GC1 and GC2) would be indirectly affected. In the terms of the output variances of y1 and y2, the dead times (f1 and f2) are used to divide the variance into CI and CD. The dead times for the current operating data should be estimated and checked first if the dead times are changed. With the available operating data, the correlation analysis method is adopted.22 The computational cost is often low, but the resolution may suffer from limited sampling time. Subsample resolution is possible using some form of interpolation between points. The dead time can be identified from the maximum value of the cross-correlation between the process input and output of the closed loop data as the reference signals and the delayed signals.23 From eq 9, the exact dead time is key to successful

(CI)20: sy/2,(CI)0 ) syc2,(CI)0 (CD)20: sy/2,(CD)0 ) syc2,(CD)0

(13)

where the superscripts * and c represent each notation at the achievable and current operating conditions, respectively. Note the achievable minimum variance terms (sy/1,CI, sy/1,CD, sy/2,CI, and sy/2,CD) are directly computed from the operating data under the accepted condition. These terms can also be computed by taking long division of the closed-loop transfer function when the process models and the disturbance models are available. Without modeling the current operation, the changes of CI and/ or CD can indicate the decay of the current performance once there are faults in the feedback system. As the disturbances are driven by a sequence of the normal random signals, CI and CD of y1 and y2, both of which are the linear combination of the disturbances, are also the normal random signals. Thus, the ratios syc1,CI0/sy/1,CI0, syc1,(CD)0/sy/1,(CD)0, syc2,CI0/sy/2,CI0, and syc2,(CD)0/sy/2,(CD)0 follow the F distribution.26 On the basis of the quantified variations of both terms, the appropriate threshold for the F statistics can be determined to identify the fault conditions in the current operation. 3.1. Separating Two Control Loops. Two diagnosis trees for the primary and secondary control loops are shown in parts a and b of Figure 2 , respectively. Each tree branches out into i,a i,a i,r three child nodes ((CI)i,a 0 & (CD)0 , (CI)0 & (CD)0 , and i,r i,r (CI)0 & (CD)0 ), where the superscripts a and r represent the accepted and rejected outcomes, respectively, and the superscript i represents the primary (i ) 1) or secondary output (i ) 2), respectively. i,a (a) (CI)i,a 0 & (CD)0 : It represents that no fault is found in the feedback loop (cs.p.0 in Figure 2a and cs.s.0 in Figure 2b). i,r (b) (CI)i,a 0 & (CD)0 : syi,CI is the function of the unmeasured disturbance model (GLi). (CI)i,a 0 shows that GLi has no indicates that the faults are possibly from fault. Thus, (CD)i,r 0 G/P1, G/P2, or G/Lj, j * i, error (cs.p.1 in Figure 2a and cs.s.1 in Figure 2b). i,r i,r i,r (c) (CI)i,r 0 & (CD)0 : (CI)0 & (CD)0 indicate GLi error must exist. To reconstruct the unmeasured disturbance model, the disturbance model (GLi) is identified based on the first fi + 1 closed loop impulse response coefficients obtained from timeseries modeling of the current output operating data. The approximate stochastic disturbance model realization (ASDR) is adopted here because the disturbance model can be identified using only routine output data.27 The method is good for the accurate disturbance model when there are three or more dead times. Other identifications of the disturbance model can also be applied.28,29 After GLi substituting into (CD)i, other faults can be further examined using the hypothesis test ((CD)i1), c / ) syi,(CD) (CD)i1: syi,(CD) 1 1

(14)

Thus, the above node can be further branched into two child nodes:

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Figure 2. Fault detection and diagnosis trees for (a) the primary control output and (b) the secondary control output. Table 1. Diagnosis Rules of a Cascade Control System by Merging the Primary and the Secondary Diagnosis Trees

(i) If (CD)i1 is accepted, the only fault is GLi (cs.p.2 in Figure 2a and cs.s.2 in Figure 2b). (ii) If (CD)i1 is rejected, it represents that faults come not only from GLi but also from G/P1, G/P2, or G/Lj, j * i (cs.p.3 in Figure 2a and cs.s.3 in Figure 2b). i,a Note that the condition ((CI)i,r 0 & (CD)0 ) does not exist. i Whenever any (CI)0 fault occurs, the (CD)i0 term must be unacceptable because the (CI)i0 fault from the change of the disturbance model will also affect the impulse coefficients of the (CD)i0 term. 3.2. Combining Two Individual Control Loops. The combined diagnosis rule can be created simply by taking the merge of the rules of the two individual diagnosis trees produced at the above step into a new set of rules. The combined set of rules is filled in the box of Table 1, in which the labeled columns and rows (cs.p.i and cs.s.j, i, j ) 1, 2, 3) represent three fault conditions of each individual control loop. Nine possible rules are found in the combined rule set. Note that the normal condition rule of each control output is not included. Because of interaction in the cascade control system, the control output must be accepted when the other control output satisfies the specified condition. In Table 1, interaction of any two individual

rules may be put into a specific fault condition. For example, when the symptom of the primary loop is cs.p.1 ((CI)1,a 0 & ) and the symptom of the secondary loop is cs.s.1 (CD)1,r 0 2,r & (CD) ), the former indicates that the fault G ((CI)2,a does L1 0 0 not exist and the latter shows that the fault GL2 does not happen, either. Thus, the faults in the current system would be from the process models (G/P1 or G/P2). However, the combination may also generate a conflicting condition. For example, when the symptom of the primary loop is cs.p.2 ((CD)1,a 1 ) and the symptom of the secondary loop is cs.s.3 ((CD)2,r ), the former 1 indicates that the only fault is GL1, but the latter shows that the other possible faults (GL1 or G/P1 or G/P2) must occur and the fault GL2 must exist. Thus, the conflicting condition should be removed. Similar explanations can be used for the other symptom pairs in the box of Table 1 to clearly indicate the corresponding faults, except for the condition when the symptoms 2,r of the control loop are cs.p.3 ((CD)1,r 1 ) and cs.s.3 ((CD)1 ). In this condition, both GL1 and GL2 have errors, but it is unknown whether G/P1 or G/P2 has any model error. Thus, the disturbance models (GL1 and GL2) should be updated to isolate these faults, and then the diagnostic tree of each loop is applied again. If the process models are mismatched, the final faults must be (G/P1 or G/P2) as well as (GL1 and GL2); otherwise, the only faults are GL1 and GL2. 4. Fault Detection and Diagnosis of PCC Like the above procedures of SCC, by applying long division to eq 7, the output variances (σy12 and σy22) of PCC can be also expressed as

σy12 ) sy1,CI(GL1)σw2 + sy1,CD(G/P1, G/P2, GL1, GL2)σw2 σy22 ) sy2,CI(GL2)σw2 + sy2,CD(G/P1, G/P2, GL1, GL2)σw2 where

(15)

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7545 ∞

f1

sy1,CI )

∑ j)0

(aRj)2 sy1,CD )

1

(aφj) ∑ j)0

(16)

∞

f2

sy2,CI )

[(aRj)2 + (bβj)2] ∑ j)f +1

2

sy2,CD )

∑ [(aφj)

j)f2+1

2

+ (bφj) ] 2

The structure of the control output variances of PCC (eqs 5 and 16) is the same as that of SCC (eqs 1 and 12). Therefore, the diagnostic procedures of SCC reasoning the faults can also be directly applied to the PCC system. Because of the limited length of this paper, the diagnosis procedures of PCC are not repeated. 5. Distinguishable Features between SCC and PCC Since the design of SCC and PCC depends on the characteristics of the process,4 improperly treated design of the control system would lead to unsatisfactory results.10 Therefore, it is necessary to determine the type of cascade control structure before assessing and diagnosing the control system. Reconsidering eqs 3 and 7, the primary outputs (y1) of SCC and PCC related to the impulse response coefficients are rearranged as

It is obvious to see that the second part (Rf1+1 + ‚‚‚ + Rf1+f2+1) of the parenthesis of w1 and the first part (βf1+1 + ‚‚‚ + βf1+f2+1) of the parenthesis of w2 in SCC and PCC have different features. In SCC, the above two terms are cascade invariant; in PCC, they are cascade dependent. Combining the above two terms, a new variance term (sy1,CS) is defined as f1+f2+1

σy1,CS2 )

∑ j)f +1

f1+f2+1

(Rj)2σw12 +

1

(βj)2σw 2 ) sy ,CSσw2 ∑ j)f +1 2

1

(18)

1

f1+f2+1 where sy1,CS ) ∑j)f [(aRj)2 + (bβj)2]. In view of this 1+1 equation, sy1,CS can be used to determine whether the current closed-loop system is SCC or PCC. If the two sets of controller parameters are implemented separately, the corresponding computed sy1,CS for the two sets are sy11,CS and sy21,CS, respectively. By checking the variances of sy11,CS and sy21,CS with the hypothesis,

CS: sy11,CS ) sy21,CS

(19)

If the ratio of sy11,CS to sy21,CS is close to unity, the control structure is SCC; otherwise, it is PCC. 6. Illustration Examples In this session, two examples, including a simulation process with a given system model and a pilot-scaled level-to-flow

Figure 3. Process delay estimations in Example 1: (a) the primary delay and (b) the secondary delay.

cascade experimental system without any prior knowledge of process models, are studied to demonstrate the applicability of the proposed diagnostic techniques. 6.1. Example 1: Simulation Case with Single Fault. Two process output transfer functions related to the controller outputs (u1(k) and u2(k)) and the unmeasurable disturbances (w1(k) and w2(k)) are

y1(k) )

1 1 y (k - 31) + w1(k) -1 2 1 - 0.2q 1 - 0.7q-1

y2(k) )

1 1 u (k - 11) + w2(k) -1 2 1 - 0.5q 1 - 0.1q-1

(20)

where the unmeasured disturbances w1(k) and w2(k) have zero means and the unit variances. Initially, a PID controller for the outer loop and a PI controller for the inner loop are

GC1 )

0.019 - 0.0164q-1 - 0.0085q-2 1 - q-1

GC2 )

0.002962 - 0.003274q 1 - q-1

(21)

-1

6.1.1. Determination of the Control Structure. Although the cascade system with SCC is known, the proposed scheme is applied here so as to determine which type of control structure is used when only the measured outputs are available. First, a new set of the controller parameters is implemented, so GC1 ) (0.042 - 0.12q-1 + 0.08q-2)/1 - q-1 and GC2 ) (0.04 0.039q-1)/1 - q-1. The responses of the control outputs (y1 and y2) are collected. To determine the control structure, the process delays (f1 and f2) should be computed first. By applying the cross-correlation analysis to y1-y2 and y2-u2 (shown in Figure 3), the process delays of f1 and f2 are 30 and 10, respectively. The hypothesis test (CS) is applied to the two sets of the closed-loop data obtained from the initial and new conditions,

F0.975 ) 0.8834 < CS )

0.0387 ) 0.0360 1.0750 < F0.025 ) 1.1319 (22)

The null hypothesis of CS cannot be rejected. It indicates that this control system is SCC.

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Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 1,r 2,a test of y1 is (CI)1,r 0 & (CD)0 and that of y2 is (CI)0 & 2,r (CD)0 . 1,r According to the symptom of (CI)1,r 0 & (CD)0 , the fault comes from the unmeasured disturbance (GL1) of the outer loop. To isolate the fault, the disturbance model is reidentified by ASDR from the current operating data,

GL1 )

1 + 0.6q-1 (1 - 0.7q-1)(1 - 0.1q-1)(1 + 0.2q-1)(1 + 0.1q-1) (25)

After the estimated disturbance model is updated, the (CD)11 statistic is used to detect if the other possible faults (G/P1, G/P2, and GL0,2) exist. At the 0.05 level of significance, the null hypothesis of (CD)11 cannot be rejected ((CD)1,a 1 ), Figure 4. Output responses under the normal condition (before the 200th sampling point) and the faulty condition (after the 200th sampling point) in Example 1.

6.1.2. Change of the Disturbance Model. In this system, the minimum achievable performance bound is 1.7454. This can be computed by minimizing the objective function (eq 4). The achievable minimum variance of this system can be further separated into the cascade-invariant and cascade-dependent terms,

sy/1,(CI)0 ) 1.9608, sy/2,(CI)0 ) 1.0101 sy/1,(CI)0 ) 4.9108 × 10-10, sy/2,(CI)0 ) 3.0978 × 10-12

(23)

In this case study, the disturbance model of the outer loop is changed into GL1(q-1) ) 1/(1 - 0.9q-1) after the 200th sampling point, but the controller parameters are still based on the achievable benchmark parameters. The resulting output variance is 4.6783, which is significantly larger than the benchmark. It is difficult to isolate the root fault solely from the output response plot (shown in Figure 4). The proposed diagnostic procedures are adopted to examine the capability. Since the process delays (f1 and f2) define the number of affected terms in cascade-invariant and cascade-dependent, the process delays should be rechecked to see if there are any changes. After the correlation analysis is performed, the outcome is the same as that in Figure 3. It indicates that the process delays of the inner and the outer loops remain unchanged. On the basis of the guides of the diagnosis trees (Figure 2), the hypothesis tests of (CI)10 & (CD)10 and (CI)20 & (CD)20 for the inner and outer loop outputs are conducted,

(CI)10 ) (CD)10 )

5.2555 ) 2.6803 > F0.025,1000,1000 ) 1.1319 1.9608

0.0095 ) 4.9108 × 10-10 1.9259 × 107 , F0.025,1000,1000 ) 1.1319

F0.975,1000,1000 ) 0.8834 < (CI)20 ) 1.0101 ) 1 < F0.025,1000,1000 ) 1.1319 1.0101 (CD)20 )

2.1001 × 10-11 ) 3.0978 × 10-12 6.7794 > F0.025,1000,1000 ) 1.1319 (24)

At the 0.05 level of significance, eq 24 depicts the hypothesis

F0.975 ) 0.8834 < (CD)11 )

0.4108 ) 0.4551 0.9027 < F0.025 ) 1.1319 (26)

Therefore, the output response of y1 indicates that the only fault is GL1 from the primary loop. 2,a According to the symptom of (CI)2,a & (CD)2,r 0 0 , (CI)0 2,r indicates there is no fault in GL2 and (CD)0 indicates the faults are from GP1, GP2, or GL1. The diagnosis trees of both loops are plotted in Figure 5. To sum up the outcomes from the two loops, Table 2 shows that the only fault in this SCC system is the unmeasured disturbance model error in GL1. 6.2. Example 2: Level Tank Controlled Process with Multifaults. A pilot level in three gravity-drained tanks shown in Figure 6 is studied. The primary loop is used to maintain the level in Tank 1, and the secondary loop rejects the effect of the disturbance (w2) in the manipulated variable (q0). q0 regulates the levels of Tank 1 and Tank 2 in a parallel way. Also, there is another unmeasured disturbance (w1) in the primary process. The objectives are to maintain the two levels at the desired set points and to assess the current control performance in the presence of feed flow rate disturbances. The tanks are equipped with differential-pressure-to-current (DP/I) transducers to provide continuous measurements of the levels. The computer is connected to a PCI-1710 analog/digital I/O expansion card from Advantech. The expansion board uses a 12-bit converter; therefore, the digital signals are 12-bit. The analog signals from the measured levels 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. In this example, two problems are addressed. First, the type of the cascade control structure should be identified. Even though it can be easily determined from the operation configuration, this simple experimental study demonstrates the effectiveness of the proposed method under the given configuration. In some processes, the cascade configuration is determined by the experts who have strong process knowledge.30 Second, multiple fault problems are demonstrated: two tanks are leaking and the rates of the upstream flows (w1 and w2) are increased. 6.2.1. Determination of the Control Structure. The output responses (y1 and y2) to two controller conditions are collected. The initial controllers (LC1 and LC2) are

GC1 )

1.4617 - 1.4610q-1 1 - q-1 and GC2 )

2.9672 - 2.8618q-1 (27) 1 - q-1

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Figure 5. Fault detection diagnosis trees of (a) the primary loop and (b) the secondary loop whose bold line indicates the fault condition in Example 1. Table 2. Overall Diagnostic Results from Both Loops in Example 1

and the other condition is

GC1 )

1.4617 - 1.4710q-1 1 - q-1 and GC2 )

2.9672 - 2.8418q-1 (28) 1 - q-1

To distinguish if the control structure is SCC or PCC, the dead times (f1 and f2) in these processes are examined. Figure 7 shows that both dead times are the same and equal to 3. Like the previous discussion in Example 1 (Section 6.1.1.), the hypothesis test (CS) is applied to the sets of the closed-loop data,

CS )

5.8938 ) 0.7028 < 0.8834 ) F0.975 8.3859

(29)

The null hypothesis of CS cannot be accepted. It obviously indicates that the operating level system is the PCC structure. 6.2.2. Change of the Process and the Disturbance Models. To assess and diagnose this PCC system, the benchmark of the control system should be set up first. The benchmark bound is determined based on the process and disturbance models. The identification procedures for the models can be found.10 The

Figure 6. Level tank control system.

identified models are gotten as follows:

GP1 )

0.0021q-1 - 0.0044q-2 u(k - 4) 1 - 0.5531q-1 - 0.4231q-2

GP2 ) GL1 )

0.0072q-1 + 0.0042q-1 u(k - 4) 1 - 0.5986q-1 - 0.376q-2

1 - 0.4468q

GL2 )

-1

(30)

1 - 0.2785q-2 - 0.1845q-3

1 1 - 0.4477q-1 - 0.182q-2 - 0.1352q-3

Thus, the estimated minimum variances of the primary and secondary outputs can be calculated from eq 8. The benchmark

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changed after the correlation analysis is conducted. The hypothesis tests of the primary and secondary outputs are performed,

2.3334 ) 1.2262 > F0.025 ) 1.1240 1.9030 54.1848 (CD)10 ) ) 29.6094 > F0.025 ) 1.1240 1.8300 (32) 2.0722 (CI)20 ) ) 1.3040 > F0.025 ) 1.1240 1.5894 25.5210 (CD)20 ) ) 98.9557 > F0.025 ) 1.1240 0.2579 (CI)10 )

1,r 2,r 2,r Both (CI)1,r 0 & (CD)0 and (CI)0 & (CD)0 cannot be accepted. On the basis of the diagnosis tree (Figure 9), there are unmeasured disturbance model errors in GL1 and GL2. Then the disturbance model should be reidentified. The estimated models are

Figure 7. Process delay estimations in Example 2: (a) the primary delay and (b) the secondary delay.

GL0,1 ) GL0,2 )

1 + 0.1q-1 (1 - 0.8q-1)(1 - 0.1q-1)(1 + 0.2q-1)

(33)

1 (1 - 0.8q-1)(1 - 0.1q-1)(1 + 0.2q-1)(1 + 0.1q-1)

After isolating the fault from the disturbance models, the (CD)11 and (CD)21 statistics are then used,

(CD)11 ) (CD)21

Figure 8. Experimental study of the controlled levels of the tank of the achievable benchmark (O) and the faulty condition (0) in Example 2.

and the corresponding controllers are

min σh12 ) 0.024 7 0.462 4 - 0.994 5q-1 0.000 284 and GC2 ) GC1 ) 1 - q-1 1 - q-1

54.1848 ) 45.9323 > F0.025 ) 1.1319 1.1797

2,r (CD)1,r 1 and (CD)1 indicate that the other possible faults may come from GP1 or GP2. On the basis of interaction of the analysis of the two loops, Table 3 shows that there are faults from GL1 and GL2, but there is no explicit indication for the fault from GP1 or GP2. To tell whether GP1 or GP2 has the model error, similar diagnostic procedures are repeated, but all calculations are based on the new estimated GL1 and GL2. The hypothesis tests of the primary and secondary outputs are conducted as

F0.975 ) 0.8807 < (CI)10 ) (31)

where the control system is performed using a PI controller for the primary loop and a P controller for the secondary loop. In this case, two leaking tanks are built, and the rates of the upstream flows in the level tank system (w1 and w2) are increased. This will constrict the outlet flow and degrade control performance if it goes undetected. Figure 8 shows the levels of both tanks under the achievable minimum variance condition whose output variance (y1) is 0.0247 and output variance (y2) is 0.0395, and under the fault condition whose output variance (y1) is 1.3470 and output variance (y2) is 0.7842. Even though the variance of the faulty condition is significantly larger than the benchmark, it is still difficult for operators to find out the root causes solely from the output data. To isolate the possible fault, the primary and secondary process delays (f1 and f2), which are used to classify the cascade-invariant and cascade-dependent terms, should be checked first. The process delays of the inner and outer loops are still un-

(34)

25.5210 ) 96.8876 > F0.025 ) 1.1319 ) 0.2141

2.3334 ) 2.4671 0.9458 < F0.025 ) 1.1240

54.1848 ) 45.9323 > F0.025 ) 1.1240 1.1797 (35) 2.0722 F0.975 ) 0.8807 < (CI)10 ) (CI)20 ) ) 2.1966 0.9436 < F0.025 ) 1.1240 25.5210 (CD)20 ) ) 99.1280 > F0.025 ) 1.1240 0.2575 (CD)10 )

& (CD)1,r and (CI)2,a & The null hypotheses are (CI)1,a 0 0 0 2,r (CD)0 . On the basis of the diagnosis trees of the system shown in Figure 10 and interaction of these two trees listed in Table 4, there are faults from either GP1 or GP2. From the above procedures, the possible faults (GL1, GL2, and GP1 or GP2) can be identified to determine deterioration in the control performance. As the procedures do not improve the current performance, the control performance can be enhanced at the last stage after correcting the fault sources. When all faults are removed, the conventional closed-loop model identification is applied directly so the process models and the disturbance

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7549

Figure 9. Fault detection diagnosis trees of (a) the primary loop and (b) the secondary loop whose bold line indicates the fault condition in Example 2.

Figure 10. Fault detection diagnosis trees of (a) the primary loop and (b) the secondary loop whose bold line indicates the fault condition in Example 2, where the disturbance models are updated.

models10 can be obtained. The new achievable benchmark is 0.4658 (the output variances of h1 and h2 are 0.3024 and 0.5430, respectively), and the corresponding controller parameters are

0.4634 - 0.7938q-1 1 - q-1 -0.000 278 4 GC2 ) 1 - q-1

GC1 )

The performance bound will be used as the new benchmark to keep monitoring and diagnosing the next operation in the cascade system. 7. Conclusions

(36)

A statistical inference method is developed for the measure of the current cascade loop status in this paper. It has the functions of detection and diagnosis of loop performance. The

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Table 3. Overall Diagnostic Results from Both Loops in Example 2

Acknowledgment This work is partly sponsored by the Ministry of Economic Affairs, R.O.C., and the Center-of-Excellence Program on Membrane Technology, the Ministry of Education, Taiwan, R.O.C. Literature Cited

Table 4. Overall Diagnostic Results from Both Loops in Example 2, Where the Disturbance Models are Updated

basis for the quantitative version of the loop status via a series of the hypothesis testing procedures has been described. It enables the fault symptoms to be categorized in diagnosis trees. Applying time-series analysis to the primary and secondary outputs in the impulse response form, both outputs (of SCC and PCC) can be decomposed into CI and CD terms, respectively. The CI terms are affected by one of the disturbances (GL1 or GL2) only, but CD would be changed when any fault (G/P2, G/P2, GL1, GL2) occurs. The possibility of applying the method presented here is conditional upon the specification of the isolation symptom of the effect of GL1 and GL2 on the primary and secondary CI terms individually. The precision form of the current disturbance models cannot be given a priori; its identification based on an approximate stochastic disturbance model realization is applied here. Since the primary and secondary control output variances consist of the CI and CD terms, a stepwise diagnostic reasoning tree is developed. As long as the value of the test statistics exceeds the threshold value, the effect of the just-identified fault is eliminated. After the outcomes of the diagnosis of the primary and secondary loops are gotten, the final decision of the fault localization and the loop status in the cascade system is made by merging the outcomes of the primary and secondary loops. This combinational structure can pinpoint the problem loop. Even if the multiple-faults situation occurs, the proposed method can still narrow down the possible root cause. The functioning of the proposed method is confirmed to be successful and robust by a simulation problem and a laboratoryscale experiment. Although the proposed method is developed for PID controllers, it should be also applicable for any type of controller. Besides, the models of the control valve, the controlled process, and the sensor-transmitter are combined in this paper. Thus, when the fault is found in any of them, the combined model should be changed. An extended research that assesses and diagnoses the types of faults will be further studied in the future.

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ReceiVed for reView October 24, 2005 ReVised manuscript receiVed July 25, 2006 Accepted July 31, 2006 IE051181M