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LICENTIATE T H E S I S

Luleå University of Technology

Department of Computer Science and Electrical Engineering 2006:47|: 402-757|: -c -- 06 ⁄47 -- 

2006:47

Fault Detection in Lambda-Tuned Control Loops

Magnus Berndtsson

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Fault detection in lambda-tuned control loops

Magnus Berndtsson

Department of Computer Science and Electrical Engineering Lule˚a University of Technology

Lule˚a, Sweden

Supervisor:

Dr. Andreas Johansson Assistant supervisor Professor Thomas Gustafsson

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To Sara

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Abstract

Poorly operating control loops cause loss in productivity in almost every industry world- wide. Therefore performance monitoring has been an active area of research for the past decades. In this work, a newly developed fault detection method is applied to the moni- toring of λ-tuned control loops. The λ-tuning method has, due to its simple use, become very popular in the pulp and paper industry and is now spreading to other industries.

A model-based fault detection algorithm assuming uncertain process parameters is used for detecting changes in the process. The algorithm consists of two parts, a resid- ual and a time-varying threshold. The a priori information obtained from λ-tuning is used to create an observer, which is used as residual generator. Known process inputs are used together with upper bounds for the uncertainties and upper bounds for disturbances when calculating the detection threshold. The observer may have integral action in order to make the threshold tight to the residual.

Upper bounds for the uncertainties in the process parameters and upper bounds for disturbances are tuning parameters in the algorithm. Two different methods for finding those parameter values are proposed. The first is an approach based on allowed loss in phase margin and the second consists of solving a nonlinear optimization problem to minimize the difference between the residual and the threshold.

The algorithm is tested in a simple water tank system. The threshold handles step changes in reference value without giving any false alarms. A fault is introduced by widening of the output in the water tank which simulates a change in the process para- meters. The residual then gets larger than the threshold and the fault is detected.

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Contents

Introduction 1

1 Introduction . . . . 1

2 Performance monitoring and diagnostics . . . . 1

3 Oscillation detection . . . . 3

4 Performance Monitoring . . . . 4

5 Fault detection . . . . 8

6 Summary of Contributions . . . . 13

Paper A Dynamic Threshold generator for supervision of λ-tuned control loops 21 1 Introduction . . . . 23

2 Preliminaries . . . . 24

3 Background . . . . 25

4 Main Result . . . . 26

5 Tuning of the uncertainty bounds . . . . 31

6 Experimental setup . . . . 32

7 Experimental Results . . . . 33

8 Conclusions . . . . 34

9 Future Work . . . . 34

Paper B Fault detection in λ-tuned control loops using an observer with integral action 39 1 Introduction . . . . 41

2 Preliminaries . . . . 42

3 Fault detection algorithm . . . . 43

4 Threshold parameter design . . . . 48

5 Experimental setup . . . . 50

6 Experimental results . . . . 50

7 Conclusions and future work . . . . 53

Paper C Optimization of a dynamic threshold generator for λ-tuned control loops 57 1 Introduction . . . . 59

2 λ-tuning . . . . 60

3 Preliminaries . . . . 61

4 The Dynamic Threshold Generator . . . . 62

5 Residual generation . . . . 64

6 Residual evaluation . . . . 67

7 Threshold parameter design . . . . 71

8 Experimental setup . . . . 75

9 Experimental results . . . . 75

10 Conclusions and future work . . . . 78

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11 Acknowledgements . . . . 78

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Acknowledgements

First I would like to thank my supervisor Andreas Johansson for all of his guidance and support. I would also like to thank my supervisor during the first years, co-supervisor and also the head of the research group Thomas Gustafsson. My other colleagues at the Con- trol Engineering Group, especially Michael Bask and Mikael Stocks, deserve my greatest gratitude for giving help and support both professionally and personally. I also would like to thank all of my colleagues, both at Systemteknik in Lule˚a and S2 at Chalmers, for the company and interesting discussions in the fika room. I especially would like to mention Martin for the company in the running tracks, Johan Carlson for making an excellent LATEXtemplate and Patrik, Kristina and Maja for sharing their rooms with me.

I would also like to thank David, Bj¨orn, Hans, Daniel and Per for giving me places to stay during my visits in Lule˚a.

Banverket and Process IT deserve acknowledgment for their funding during my stud- ies.

Finally I will give my thanks to my family and especially Sara for their love and support.

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Part I

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Introduction

1 Introduction

In a process industry facility there are hundreds maybe even thousands of controllers.

There are controllers for controlling flow, pressure, fluid levels, paper quality, ventilation, and so on. Many of the controllers has been manually tuned once, at the time they were installed. Some of the controllers are not tuned at all (they use the pre-set parameters that come with the controller) and in some controllers the parameters are changed on a daily basis. This cause problems and according to [21] as many as 60% of all industrial controllers have poor performance. It is not difficult to see that this problem is very expensive and therefore the field of closed-loop performance monitoring and diagnostics is an active area of research.

2 Performance monitoring and diagnostics

A typical control loop is the water tank system (see fig 1). The controller receives the water level in the tank from a level sensor. By comparing the signal from the sensor to a reference level it is possible to control the water flow from the pump to make the water level in the water tank be the same as the reference value.

Now imagine a small leak in the water tank. If the controller includes integration, which many controllers do since the far most common controller is the PI-controller.

Then there will be no problem for the controller to adjust the water flow from the pump to make the level in the water tank the same as the reference level. This way we will not have any problems with the leak and the rest of the process will work just fine. But what if the fluid in the tank is for example oil? It may then be of interest to be informed if there is a leakage in the tank even though the level of the fluid is right. This kind of error will be possible to find with some of the methods in performance monitoring and

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2 Introduction

Controller Pump

Figure 1: Water tank system

diagnostics.

It may not be obvious though why a method developed for monitoring the performance of a controller could be used to detect a leakage in a tank. The water flow in a water tank can be described with the following system equation

G(s) = k

1 + T se−Ls (1)

Suppose the output from the water tank is widened. This will affect the dynamics of the system, both by changing the time constant T and the gain k. If the time constant is increased with Tl and the system gain with kl, the new system will look like

Gl(s) = k + kl

1 + (T + Tl)se−Ls (2)

If the controller is tuned for the system G(s), the performance of the controller may decrease if the system changes to Gl(s).

Because many of the methods in performance monitoring and diagnostics not are developed for this purpose they may not be easy or even possible to use in this way. It would be of interest though to have a method that can detect faults in the processes and also detect a badly working controller. This is the main ideas behind the work in [5]

The closed-loop performance monitoring and diagnostics can be divided into two large areas, oscillation detection and performance monitoring. Several survey articles have been written in the subject e.g. [21] and [18].

An interesting investigation and discussion about the practical issues in implementing some of these methods can be read in [15].

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3. Oscillation detection 3

3 Oscillation detection

Many of the errors in control loops, e.g. stiction, badly tuned controllers and oscillating load disturbances occur as oscillations in the control loops. This can change the quality in the product or increase the cost of the production.

One interesting example of oscillation detection is described in [11]. The method consists of two steps. First there is a method to decide if there is a load disturbance.

If a load disturbance exists, the second step is for detecting if the load disturbance is an oscillation. For detecting if there is a load disturbance, H¨agglund uses the integrated absolute error (IAE) between successive zero crossings of the control error, i.e.

IAE =

 ti

ti−1

|e(t)|dt (3)

where ti−1and ti are two consecutive instances of zero crossings.

To detect an oscillation there must be an upper limit for IAE. H¨agglund suggests that the upper limit should be

IAElim 2a ω

where a is the amplitude of the oscillation and ω is the frequency. ω should be chosen as the ultimate frequency ωu. It will then be possible to detect oscillations up to this frequency. If ωu is unknown and if there exist a controller integral time, ω should be chosen as ωi = 2π/Ti, where Ti is the controller integral time. A reasonable choice for a is 1% which means that the method will detect oscillations with an amplitude higher than 2%.

When deciding if the load disturbance is an oscillation the number of load disturbances are monitored over a supervision time Tsup. If the number exceeds a certain limit nlim, it can be concluded that an oscillation is present. Tsupcan according to [11] be chosen as

Tsup≤ nlim

Tu 2

where Tuis the ultimate oscillation period. If Tu is unknown it is possible to use the controller integration time Tsup= 50Ti.

Some further developments of this method can be found in [20]. The method in [8]

also uses the IAE. There the IAE is combined with the time between zero crossings.

Another interesting method in oscillation detection can be found in [16].

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4 Introduction

4 Performance Monitoring

A basic problem in the second area of performance monitoring and diagnostics, perfor- mance monitoring, is to distinguish the difference between a well performing control loop and a badly performing one. A way to do this is proposed by Harris in [13]. There he introduced an index to measure the performance. It is then possible to use this index to decide which controller loops that needs further work. This index is what is later to be named as the ”Harris-index”. The idea behind the index is that if it exists a perfect control loop it would be possible to compare the existing loop with the perfect one and by that decide how good the loop is performing. Of course the perfect control loop does not exist in reality and the problem is then to find one which is optimal in the most important properties for solving this problem. A commonly used ”perfect” controller is the Minimum Variance (MV) controller. According to [19] the MV controller is optimal in the sense that it cancels disturbances as fast as possible. This makes it possible to use it as reference when assessing the performance of the control loop.

4.1 Minimum Variance Control

To describe the MV controller consider the time-series model y(k) = Bp(q−1)

Ap(q−1)q−du(k) + w(k) (4) where q−1 is the backward shift operator. The pole-zero excess is d. Like in [13] the dead time is denoted d. (sometimes the dead time is denoted as d-1).

The controller C(q−1) must be stable and is assumed to be one degree of freedom.

u(k) = C(q−1)(r(k)− y(k)) The disturbance w(k) is generated by white noise.

The controller error can be described as

e(k) = r(k)− y(k)

= Ap(q1)

Ap(q−1) + Bp(q−1)C(q−1)q−d(r(k)− w(k))

This description depends on a known process model, i.e. it assumes that Ap, Bpand d are known. If this is not the case it is possible to assume a simpler model for the controller error to make an identification easier. The proposed model is

e(k) =Be(q−1) Ae(q−1)a(k)

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4. Performance Monitoring 5 where a(k) is white noise. The impulse response gives a good description on how fast the controller reacts to disturbances. The impulse response for e(k) can be written in time series as

e(k) =

 j=0

fjq−ja(k)

where fjare the coefficients of the time series impulse response. If the first coefficient f0 is normalized to 1, the variance of e(k) can be described as

σ2e= var(e(k)) =

 j=0

fj2σ2a

Constraints on the impulse response coefficients can be related to common perfor- mance specifications i.e. closed loop settling time and decay rate [22]. Thereby it is possible to realize that the coefficients describe how fast the controller can react on disturbances. The pole excess d places a limit on how fast a controller can reject distur- bances i.e. the controller will not react on a disturbance earlier than d samples after the disturbance occured. Because an MV controller rejects disturbances as fast as possible the controller error e(k) under MV control is a moving average of the first d terms of the impulse response.

The variance of e(k) under MV control is

σM V2 = (1 + f12+ ... + fd−12 2a

To be able to achieve MV-control all the disturbances on the plant must be known.

It is thereby practically impossible to implement but what is considered a well function- ing loop in the process industry will frequently have variance well above the minimum variance.

4.2 Harris index

Since MV control is the best possible control and σ2M V is not affected by any feedback it is possible to use it as a reference when deciding how the current controller is perform- ing. Harris did this [13] and created an index which is the ratio between σ2M V and σe2. Evaluated as time series the index will be

IH=

d−1

j=0fj2



j=0fj2

Several authors along Harris himself has used and further developed he Harris-index, e.g. [2].

Another development is the normalized performance index as defined in [7]. It is a performance index closely related to the Harris index and is evaluated from a linear

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6 Introduction

predictor. The prediction error when predicting b steps in the future (b is called the prediction horizon) is denoted

˜

e(k)= e(k) − ˆe(k|k − b) The variance of the prediction error is

σ˜e= (1 + f12+ ... + fb−12 a The normalized performance index is then

η(b) = 1σ˜e σe

If b is the dead-time of the system d the predictor has similar properties as the minimum variance controller and the normalized performance index is then closely related to the Harris index. If b is larger than d the index can be written as

η(b) = 1

b−1

j=0fj2



j=0fj2 (5)

The problem with the normalized performance index is to choose the prediction hori- zon b. Different choices will give different results. In [17] it is suggested that selecting b is treated as an engineering criterion representing a demand made by the control engineer on the control loop.

Another suggestion how to choose the prediction horizon b can be found in [12]. The idea behind the choice is to use information obtained from the controller tuning. It is concentrated on a special kind of tuning, the λ-tuning.

4.3 λ-tuning

λ-tuning is a method for tuning of PI-controllers based on IMC-principle. It is a procedure in a practical point of view which holds for first order plants with time delay, i.e.

G(s) = k

1 + T se−Ls (6)

The tuning consists of an identification part where the parameters of the system (1) are identified. The method for identification is an open loop step response according to fig 1. The process gain is calculated as k =ΔOUTΔP V . T is the process time-constant which is the time it takes for the process to reach 63% of ΔP V . L is the time-delay of the process, i.e. the time it takes from the step input until the output responds.

The controller is a PI,

Cλ(s) = kc(1 + 1

Tis) (7)

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4. Performance Monitoring 7

140 160 180 200 220 240 260 280 300

0 5 10 15 20 25

ΔPV

OUT 63% of

L T Δ

ΔPV

Figure 2: A step response forλ tuning.

The controller parameters should be chosen as

Ti = T kc = Ti

k(λ + L)

The parameter λ is considered as the closed loop time constant, and is to be chosen in the interval λ∈ [T, 3T ]. If λ = T the tuning is considered aggressive and λ = 3T is considered robust.

4.4 λ-Monitoring

The method in [12] is called λ-monitoring. Unlike the Harris index the method is not supposed to give information about how close the current performance is to optimal per- formance. The index rather reflects how a loop is fulfilling the design specifications. The algorithm monitors that the loop has certain disturbance rejection capabilities and those are reflected by η(b).

The selection of the prediction horizon b in η(b) depends in addition to the tuning on the disturbance model. For λ-tuning the disturbance model is set equal to the one in the controller design i.e. a step disturbance

D(q−1) = 1 1− q−1

The set-point response from a Dahlin-controller for (4) is the same as Td(q−1) = 1− e−h/λ

1− e−h/λq−1q−d

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8 Introduction

if h is the sampling time and λ is the time constant given in continuous time [12].

The Dahlin-controller is in this case equivalent to a IMC-controller and thereby also to λ-tuning.

The control error is then

e(k) = [1− Td(q−1)]D(q−1)a(k)

where a(k) is a deterministic or stochastic disturbance. If α = e−h/λ the impulse response of e(k) is [12]

e(k) = (1 + q−1+ ... + q−d+ αq−d−1+ α2q−d−2+ ...)a(k) This impulse response is then used when calculating (5).

As the monitoring should reflect how well the controller is fulfilling the design spec- ifications, the main characteristics in η(b) should be captured. In [12] it is then recom- mended to choose the prediction horizon b to (λ + L)/h.

5 Fault detection

The main purpose of performance monitoring and diagnostics is to diagnose how good a control loop is performing. But sometimes it is more interesting to decide if the control loop is working or not. Suppose that there is a controller which controls the pressure in a gas-tank. It may not be critical that the pressure is exactly the same as the reference all the time. The time it takes to reach the right pressure is perhaps not very important either. But if the pressure gets to high the tank will explode. This is an example where it is more important that the controller is working than to have optimal performance. It is of course possible to use the performance monitoring algorithms for this purpose but there exist better ones. With a better algorithm it may be possible to detect smaller faults earlier by using detailed process knowledge, see for example [4]. There is a large research area which deals with this kinds of problems i.e. to decide if there are any faults present. This research area is called fault detection. A closely relating area, which has gotten a lot of attention lately, is the Abnormal Event Management (AEM) [23]. AEM deals with the detection, diagnosis and correction of abnormal conditions of faults in a process. The fault detection is just one part of the AEM. This thesis is concentrated on the fault detection part.

The faults which can be detected can be divided into three different categories de- pending on what part of the system they occur in

• Process faults, When the system parameters change and when the plant is subject to unknown input signals e.g. leakage, loads and oscillations. This type of faults can be divided in two different areas depending on how they affect the system

Additive faults, Unknown input signals are often described as additive faults.

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5. Fault detection 9 Multiplicative faults, Changes in the process parameters can be considered as

multiplicative faults.

• Sensor faults, Faults in the measurement sensors are called sensor faults. They are often described as additive faults but are sometimes better described as multi- plicative faults, e.g. complete failure.

• Actuator faults, these faults can be described in the same way as sensor faults.

A typical actuator fault is sticking valves.

The most important properties in fault detection algorithms are

• Few missed detections

• Low false alarm rate

The first point, Few missed detections is quite obvious. An example is a fault de- tection on a temperature sensor in an engine. A few missed detections and the engine could overheat and be permanently damaged. Another example is the earlier mentioned example with a oil tank. If the detection algorithm does not detect a leak then the oil could cause a lot of damage in the environment. But it may not be equally obvious that the other point, Low false alarm rate, actually causes more problems in the industry than the first one. Intuitively one may think that it is better to have one false alarm than a missed detection because if there is a false alarm then nothing wrong actually has happened. But there are some large problems with false alarms. Suppose there is a false alarm in the temperature sensor mentioned earlier. The alarm will probably cause some error code that limits the power from the engine until the problem is taken care of by an authorized workshop. This kind of behavior is not acceptable by a car owner. Another large problem is if there are a lot of false alarms. In a large process plant there may be as many as 1500 process variables observed every few seconds [3]. If every observed process variable in a plant get one false alarm every day there are thousands of false alarms in just one day. Then the risk of discarding a real alarm as a false is large.

The properties of Few missed detections and Low false alarm rate are contradictory and a trade off between them is required.

The first methods of detecting faults, mainly instrumental ones, used hardware re- dundancy. Hardware redundancy means that three or more sensors measure the same signal and a voting criterion is carried out to determine if any of the signals may be faulty. This method was introduced sometime in the late 60s. The drawback with the method is the need of several sensors which is expensive. Another common problem is the use of one single kind of sensors. If all of the sensors are the same kind, which is very common due to less costs, the risk that the same problems will occur in all of them within a short period of time is large. This will make the redundancy obsolete.

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10 Introduction

G

E

u y

r D Alarm

Figure 3: A general structure for fault detection systems.

Due to the problems with hardware redundancy, research was initiated on what now may be called analytical redundancy. Analytical redundancy is using different sensors and the analytical relations between them to generate a residual to evaluate. That way the expenses are reduced and the problem with similar sensors is solved. The drawback with analytical redundancy is that there must exist different sensors with a clear analyt- ical relation.

If there only is one sensor or if the analytical relation between the sensors is hard to find there is a possibility to use other model-based methods for creating the residual.

The fault detection algorithms can be divided into different types depending on the type of model description that is used. If it is possible to derive a quantitative process model then this can be used for residual generation. More about this method is described in the sequel. Another possibility is to design a knowledge-based fault detection algorithm.

For this kind of algorithm, a qualitative process model is required [24]. A third method is to use a history based method to create the residual. An advantage with this ap- proach is that there is no need for an a priori process model. The drawback is the large amount of measurement data that is required. This kind of methods can be found in [25].

The rest of the thesis is concentrated on the quantitative process model approach.

Figure 3 shows the general structure of a model-based fault detection system where G is the process, u is the process input and y is the process output. E is the process estimator, the residual generator with the residual r as output. D is the detection algorithm, the residual evaluator which has the decision, i.e. the alarm as output. The residual generator is a system with the known system inputs and the residual as output.

The residual is then evaluated in the residual evaluator and a decision is made if there is a detection or not.

5.1 Residual generation

The residual is a signal that ideally is zero when no fault is present and nonzero when there is a fault. Due to model uncertainties and unknown disturbances the residual nor- mally always is nonzero. Therefore the residual generation should be determined such that the dependence on disturbances is minimized, while the effect of faults is maximized.

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5. Fault detection 11

A common linear description of the process is

˙x(t) = Ax(t) + Bu(t) + Edd(t) + Eff (t) (8) y(t) = Cx(t) + Du(t) + Fdd(t) + Fff (t) (9) where the vectors x(t), u(t), d(t), y(t) and f (t) are the state, known input, unknown input, output and fault vector respectively.

If the process model is the approximation in equation (8) linear observers is a natural choice of estimator. One possibility is the Kalman filter. A drawback with using the Kalman filter in fault detection is that the feedback gain matrix is designed in a way that considers sensitivity with respect to disturbances and not to faults. A more common possibility is then the Luenberger observer. A method which uses the Luenberger observer and has been used for fault detection is the fault detection filter.

A linear observer for (8) that can be used as residual generator is

ˆ

x(t) = Aˆx(t) + Bu(t) + L(y(t)− ˆy(t)) ˆ

y(t) = C ˆx(t) + Du(t)

Define the estimation error as e(t)= x(t) −ˆx(t) and the residual as r(t) = V (y(t)−ˆy(t) then

˙e(t) = (A− LC)e(t) + Edd(t) + Eff (t)− L(Fdd(t) + Fff (t)) r(t) = V (Ce(t) + Fdd(t) + Fff (t))

where the feedback gain matrix L and the matrix V are the design parameters. A general observer-based residual generator is shown in figure 4, where Q is the observer and dp, dmare the process and measurement disturbances.

Other examples on estimators that can be used for residual generation for fault detec- tion are the unknown input observer, the parity space observer and the frequency domain observer. An early article with the use of unknown input observers in fault detection is [26]. The parity space approach is described in [6] and the frequency domain observer can be found in [9]. Another way to generate the residual is by using system identification [10].

5.2 Residual evaluation

Because the residual is affected both by the uncertainties in the process parameters and by the disturbances, the residual will generally not be 0 even if no fault is present. To

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12 Introduction

E G

+-

Q r

u y

y^

dm

dp

Figure 4: An observer-based residual generator.

determine if there is a fault it is then necessary to have some kind of residual evaluation.

A common way to do this is by using a threshold that stays larger than the residual as long as there is no fault and the uncertainties and disturbances remain within some preset bounds. If some of the characteristics of the faults are known it is possible to use an evaluation signal, which is a function of the residual, to compare to the threshold.

One possible choice of evaluation signal s is, as proposed in [9], the signal 2-norm of the residual, i.e.

s(t) =r(t)2=



0 |r(τ)|2

The advantage of using the 2-norm is that it is then straightforward to optimize the residual generator to minimize the influence from disturbances. A drawback with this choice is that the 2-norm can not be calculated until t =∞ and can therefore not be used for online fault detection. Instead one may use a windowed version of the 2-norm [9].

s(t) =

 1 β

 t

t−β

|r(τ)|2

Another common choice of evaluation signal is the absolute value of the residual s =|r(t)|.

In [14] an evaluation operator based on a weighted absolute integral is used

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13

s(t)=

 t

0

w(t− τ)|r(τ)|dτ (10)

The purpose of the weighting function w is to increase the influence from the most recent data, e.g. by windowing or exponential forgetting.

In [1] there are some early work on how to choose the threshold when using an evaluation signal.

In [14], the problem of choosing thresholds is analyzed when assuming bounded pa- rameter uncertainties in the process. The result is a dynamic threshold generator, i.e. a dynamic system generating the threshold based upon the measured signals.

6 Summary of Contributions

The three papers included in this thesis presents a new fault detection algorithm for detecting poorly operating λ-tuned control loops. The a priori information obtained in a λ-tuning is used to create a dynamic threshold generator for model-based fault detection of the process.

6.1 Paper A

The fault detection algorithm is presented. It is a Luenberger-observer based residual generator with a dynamic threshold generator. The maximum allowed loss in phase mar- gin for the closed loop has been chosen as the main tuning parameter when choosing the upper bounds for the uncertainties in the threshold generator.

6.2 Paper B

The observer is extended with integral action which makes it possible to obtain a fault detection threshold which is tight to the residual. A linear optimization approach is used for finding the upper bounds for the uncertainties in the threshold generator.

6.3 Paper C

A nonlinear optimization problem is solved for finding the upper bounds for the uncer- tainties and the measurement and process disturbances in the threshold generator. An evaluation operator based on a weighted absolute integral is used.

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[18] S. Joe Qin. Control performance monitoring, a review and assessment. Computers and Chemical Engineering, 23:173–186, 1998.

[19] K.J. ˚Astr¨om. Computer control of a paper machine - an application of linear sto- chastic control theory. IBM Journal of Research and Development, 11(4):389–405, 1967.

[20] N.F. Thornhill and T. H¨agglund. Detection and diagnosis of oscillations in control loops. Control Engineering Practice, 5(10):1342–1354, 1997.

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[22] Mathew L. Tyler and Manfred Morari. Performance monitoring of control systems using likelihood methods. American Control Conference, Seattle, Washington, USA, 1995.

[23] Kewen Yin Venkat Venkatasubramanian, Raghunathan Rengaswamy and Surya N.

Kavuri. A review of process fault detection and diagnosis part i: Quantitative model-based methods. Computers and Chemical Engineering, 27:293–311, 2003.

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[24] Raghunathan Rengaswamy Venkat Venkatasubramanian and Surya N. Kavuri. A review of process fault detection and diagnosis part ii: Qualitative models and search strategies. Computers and Chemical Engineering, 27:313–326, 2003.

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Part II

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Paper A Dynamic threshold generator for supervision of λ-tuned control loops

Authors:

Magnus Berndtsson and Andreas Johansson

Reformatted version of paper originally published in:

Proceedings of the 25th American Control Conference, 2006, Minneapolis, USA

2006, IEEE. Reprinted with permission.c

21

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22 Paper A

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Dynamic threshold generator for supervision of λ-tuned control loops

Magnus Berndtsson and Andreas Johansson

Abstract

A new approach for detecting poorly operating λ-tuned control loops is presented. The a priori information obtained in a λ-tuning is used to create a dynamic threshold generator for model-based fault detection of the process. The maximum allowed loss in phase margin for the closed loop has been chosen as the main tuning parameter. The proposed method is tested successfully on measurement data from a water tank process.

1 Introduction

Poorly operating control loops cause loss in productivity in almost every industry world- wide. Therefore performance monitoring has been an active area of research for the past decades. This work focuses on fault detection for a special kind of controller, the λ-tuned PI controller.

Common reasons for oscillations in control loops is not just bad controller tuning [6]

but also physical problems e.g. sticking valves. There are several commonly used meth- ods for detecting this kind of oscillations [3], [13], [11]. Other methods monitor how well the controller is performing, for example compared to the minimum variance controller [8].

To be able to distinguish the difference between good and bad performance of a control loop, it is necessary to know what kind of behavior that characterizes good performance.

This kind of a priori information, is time consuming and thus expensive to obtain. A solution to this is to use already known information, e.g. from the controller tuning. The swedish pulp and paper industry has developed a standard for tuning of PI controllers for first order systems with time-delay [12]. The tuning method is called λ-tuning and is based on the IMC(Internal model control)-principle. It is easy to apply and is intended to be used by ordinary control-room operators. The method has become very popular in the pulp and paper industry because of its simplicity and is now spreading to other industries.

A great advantage in using the same method for tuning a lot of controllers in the plant is that it is then possible to develop a controller supervision based on a priori information from this controller tuning.

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24 Paper A

An example of a monitoring tool that uses the information obtained from λ-tuning is presented in [7]. Ingimundarson has developed a method based on the Harris index [8]

but takes advantage of the known properties of the process. The Harris index, however, compares the performance of the control loop to that of an optimal controller. In a plant with hundreds, maybe even thousands of control loops, optimal performance may not be a relevant comparison. To make every controller behave close to the optimum will be an impossible task for the staff of the plant.

The idea behind this work is to use the known properties of the process to create a model based fault-detection algorithm. Unlike the Harris index the goal is not to compare the existing controller to an optimal one but to detect changes in the process parameters (the lambda-tuning parameters).

The main tuning parameter is the minimum allowed phase margin. This will make the method easy to use, and enable detection of process changes that may decrease sta- bility.

The idea behind model-based fault detection is to use the redundancy in the infor- mation obtained from the measurements in combination with a process model. If the measured output does not match the expected output produced by a process model, then the presence of a fault can be deduced. Provided an analytical process model, a fault detection algorithm essentially consists of two steps, the residual generation and residual evaluation. The purpose of the residual generation is to generate a signal which is nonzero when there is a fault and zero otherwise. However, due to disturbances and uncertain- ties, the residual is generally nonzero even if no fault is present. This fact necessitates the second step of the fault detection algorithm, the residual evaluation which consists of comparing the residual to a threshold. If the model describes the process perfectly then the residual will only be affected by the non measured disturbances, otherwise the residual will also depend on known input signals. A way to deal with the problem of inaccurate process parameters is to assume a process model with parameter uncertainties and to use these uncertainties when calculating the threshold. [10] [9] [2]

2 Preliminaries

In the following, | · | denotes element-wise absolute value when applied to a matrix.

Inequalities between matrices is also to be interpreted element-wise. The notation represents the Kronecker product, for which a useful property is

Property 1 Let x∈ Rn and y∈ Rm. Then x⊗ y = (x ⊗ Im)y = (In⊗ y)x

where Ipdenotes the identity matrix of dimension p. For functions|·| is to be interpreted pointwise, so that|F |(t)= |F (t)|. Inequalities between functions is also intended point- wise, i.e. F ≤ G means F (t) ≤ G(t) for all t. A linear operator defined by convolution

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by a weighting function is denoted by the symbol of the weighting function written in bold-face font, thus e.g. FG(t)= (F ∗ G)(t).

3 Background

3.1 Fault detection algorithm

The fault detection algorithm consists of two parts. The residual generator, which gener- ates the residual, and the residual evaluator, which compares the residual to a threshold in order to decide whether a fault has occurred. Due to the model uncertainties and dis- turbances the residual will always be nonzero and is affected by the known input signals.

To achieve a threshold that is as tight as possible to the residual, the threshold should also depend on the known input signals. According to [10] [9] a parametric uncertainty description may provide this property to the threshold. Consider the system,

˙x = Ax + P (π⊗ x) + Eπ + F + μ (1a)

y = Cx + ν (1b)

where A ∈ Rn×n is Hurwitz, P ∈ Rn×nm, E(t) ∈ Rn×m and π(t) ∈ Rm. Known input signals enter via the time-varying terms F (t) and E(t). π(t) are the parameter uncertainties and μ(t),ν(t) are the process and measurement disturbances. The system (1) is, under some conditions, obtained by a Taylor expansion of a nonlinear system with uncertain parameters [10].

A commonly used residual generator is a Luenberger observer. Applying a Luenberger observer to (1) gives an error system with the same structure as (1) where it is possible to choose the observer residual as output. The residual evaluation often involves taking the absolute value of that residual|r(t)|.

The dynamic threshold generator is a dynamic system whose inputs are the known process inputs and the output is an upper bound for|r(t)|, i.e. the threshold. The fol- lowing Theorem gives a dynamic threshold generator for an error system of the form (1) [10] [9].

Theorem 1 Assume that|π(t)| ≤ Π ∈ Rmfor all t and let G(t)= e At. Let H(t)∈ Rn×n be a function that satisfies H(t)≥ |G(t)P |(Π ⊗ In) for all t and let H be the operator defined by convolution with H. Let Γ∈ Rn×n be a function that satisfies Γ(t) ≥ |G(t)|

for all t and let Γ be the linear operator defined by convolution with Γ. Then, ifH < 1 for some induced operator norm  ·  then (I − H)−1 < ∞ and

|y| ≤ |C|(I − H)−1(Γ(|E|Π + |μ|) + Γ|x(0)|) + |ν| (2)

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26 Paper A

For details and proof see [10] [9].

Apart from known input signals and the system parameters A,B and C, upper bounds for the uncertainties π and upper bounds for the disturbances μ, ν are needed when determining an upper bound for|y| according to (2).

3.2 Lambda-tuning

In a document published by the Swedish Pulp and Paper Industries Engineering Co, the following method is recommended for tuning of PI controllers [12].

A first order plant with time delay is assumed, i.e.

Gλ(s) = ke−Ls

1 + T s (3)

The controller is a PI controller with the structure Cλ(s) = Kc(1 + 1

Tis) (4)

The controller parameters should be chosen as

Ti = T Kc = Ti

k(λ + L)

The parameter λ is chosen to be in the interval λ∈ [T, 3T ]. If λ = T the tuning is considered aggressive and λ = 3T is considered robust.

4 Main Result

The dynamic threshold generator (Theorem 1) will now be used for developing an al- gorithm to detect significant changes in a process controlled by a λ-tuned controller. A process description in state-space form suitable for λ-tuning is

˙x = 1 Tx + k

Tu + μ (5a)

y = x + ν (5b)

which is a state-space description of (3) with output y. The input u is defined as u(t)= w(t − L)

where w is assumed to be the input in (3). Also, a process disturbance μ and a mea- surement disturbance ν have been introduced. The parameters T and k are the time

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constant and static gain, respectively. It is assumed that these parameters are uncertain and time-varying with known nominal value T0 and k0, respectively, i.e.

T (t) = T0(1 + π1(t)) (6a)

k(t) = k0(1 + π2(t)) (6b)

Let π= [π 1π2]T and define upper bounds for the disturbances and parameter uncertain- ties in (5) as

Π = [Π1Π2]T ≥ |π(t)|

M (t) ≥ |μ(t)|

N (t) ≥ |ν(t)|

The process (5) may be expressed as

˙x = f (x, u, π) + μ y = x + ν

where

f (x, u, π) = 1

T0(1 + π1)x +k0(1 + π2) T0(1 + π1)u

A second order Taylor expansion of f is

f (x, u, π) = Axx + Buu + Bππ + Pxx(x⊗ x) + Puu(u⊗ u) + Pππ⊗ π) + Pxu(u⊗ x) + P⊗ x) + P⊗ u) + fh(x, u, π)

References

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