1997 European Control Conference (ECC) 1-4 July 1997, Brussels, Belgium
A QUALITATIVE/QUANTITATIVE REPRESENTATION OF SIGNALS FOR SUPERVISION OF CONTINUOUS SYSTEMS ¨
J. Colomer*, J. Melendez*, J. Ll. de la Rosa*, J. Aguilar-Martin* * LEA-SICA, IIiA, University of Girona Av. LLuís Santaló s/n E-17071, Girona (Catalonia) ¨ LEA-SICA, LAAS-CNRS 7 Av. du Colonel Roche 31077 Toulouse (France) e-mail:
[email protected] fax :+34 72 418098 Keywords : Qualitative analysis, Symbolic Interpretation, Supervisory Control, Intelligent Control
Thus, significant information could be supplied directly to ES after an abstraction procedure. In the following paragraphs this abstraction tool is widely described.
Abstract
2 Antecedents
Correct interpretation of measured process data is essential for the supervision of control systems. Qualitative representation of signals based on episodes is presented. A new classification of episodes based on the second derivative sign is performed. Qualitative information associated to each episode is completed with quantitative information. This representation is applied to linear systems responses as an example to distinguish the different dynamics of these systems. Finally, in an application with real signals, its utility and limitations are shown.
Symbolic representation of trends of signals for process supervision is not a new topic in the bibliography and it is used in different ways. For example [1] proposes a qualitative modelling method based on causal relations between qualitative variables and [2] describes a representation language called ALCMEN where indices are used to roughly describe signals and which is used to perform qualitative relations between variables. [3] uses a description of signals in episodes obtained from triangular and trapezoidal representations. This description is improved by applying wavelet transform to obtain a representation of signals from batch processes by different time and frequency scales [4]. Since the work, here presented, began after analysing triangular and trapezoidal representation, in the next paragraphs an introduction of the basis of these techniques is presented.
1 Introduction Interpretation of measured process data is an important task in Supervisory Control, specially for Fault Detection and Diagnosis. Since a great deal of the process data is available for the operators and supervision engineers, it is difficult to get outstanding data and extract the most significant information. Using Artificial Intelligence (AI) in these areas has become more significant. In supervisory applications Expert Systems (ES) are not successfully used because of interfacing difficulties between process signals and expert knowledge bases (KB) that is, they lack of good human-like interpretation from signals. Then, the aim of this work is another useful tool to interpret measured process signals from the point of view of process behaviour.
ISBN 978-3-9524269-0-6
2.1 Triangular and trapezoidal representation of process trends[3] This qualitative representation of time-records of process variables is based on a detection of changes in the sign of first or second derivative of variables ( represented by [¶x] and [¶¶x] ). Then, episodes are built between these change instants. A triangular episode is the triangular region constructed by
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intersecting lines of the initial slope, the final slope and the average slope through the boundary points (See Fig. 1). (tj , x( tj )) x
which the slope is equal to the slope of the line joining the boundary points of the episode.
[¶ ¶ x ] =
Fig. 1 Triangular representation
There are only seven basic types of triangular episodes (Fig. 2) under the constancy of [¶x] and [¶¶x].
[¶¶x] = [¶x] = +
[¶¶x] = -
[¶¶x] = + [¶x] = +
[¶¶x] = +
[¶x] = -
[¶¶x] = 0
[¶x] = +
[¶x] = -
5 4 3 2 1 0 -1 -2 -3
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t
2.2 Another trapezoidal representation
Trends of variables could be represented by a descriptive representation consisting of a series of episodes (Fig. 3).
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[¶¶x] = -
Trapezoidal episodes can be built using the triangular representation, grouping consecutive triangular episodes with the same [¶¶x]. Or can be built from another trapezoidal representation, grouping trapezoidal episodes and obtaining representations at different scales. This method, called qualitative scaling, is a local scaling and brings out the local character of trends with minimum distortion.
[¶x] = -
[¶¶x] = 0
40
(tj , x( tj ))
Fig. 5 Convexity point.
Fig. 2 Different episodes depending on the sign of [¶ ¶x] and [¶¶ ¶¶x].
20
x convexity point
(ti , x( ti ))
[¶¶x] = 0 [¶x] = 0
0
[¶ ¶ x ] = +
Fig. 4 Trapezoidal episodes
(ti , x( ti ))
-4
-
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Another trapezoidal representation is given by [5]. In this case triangular episodes are obtained following the previous description. Extrema and inflexion points are called distinguished time points (dtp) and episodes are built between these dtp. Then, trapezoidal episodes are defined by the constancy of one of the qualitative values [¶x] or [¶¶x] so, they are built grouping adjacent triangular episodes with the same [¶x] or [¶¶x]. Then, the representation can be scaled at different abstraction levels grouping episodes. The result is a qualitative representation of process variables.
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Fig. 3 Series of episodes offers descriptive representation of trends of signals.
This representation allows explicit information of significant information of a trend. Qualitative features like ‘increasing’, ‘spikes’, ‘oscillations’, etc. could be directly expressed by the type of matched triangular shape. At the same time quantitative information, like co-ordinates of extrema and inflexion points, are also available. The trapezoidal representation (Fig. 4) is based on trapezoidal episodes. A trapezoidal episode is an episode whose qualitative context is defined by the constancy of the qualitative value [¶¶x]. In addition it includes the value and the slopes at the boundary points and the convexity point of the episode. The boundary points of episodes are the second order zero crossings and convexity points are defined to be the point at
3 A qualitative representation based on convexity of signals. In the following paragraphs this notation will be used : x(n) : Sample of signal at instant n. dx(n) : Derivative of signal at instant n. ddx(n) : Second derivative of signal at instant n. The qualitative representation of signals described in the last paragraphs is improved by extending the set of episodes. In this way, the range of values of second derivative is split in three intervals (negative, zero and positive) and a qualitative value (or label) is associated to each one. Then, qualitative value ‘zero’ represents an interval not only a numerical value.
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Accordingly this partition of second derivative, distinguished time points (dtp) are detected only when there is a change in the qualitative value of ddx(n). Limits of zero interval must be chosen according to system dynamics. Doing the same with dx(n) or x(n) representation richness would be increased. But, at the same time, this makes its interpretation difficult. Only ddx(n) has been chosen because it is the most representative of the signal dynamics and determines the trapezoidal representation described in section 2.1. A more accurate representation could be done by dividing the range of values of ddx(n) in more than three intervals with its qualitative values. The representation would be more accurate but, at the same time, more complex and harder to use.
3.1 Qualitative information: Types of episodes In the previous paragraph, three possible qualitative values for [¶¶x] are described so, obviously, three basic types of episodes are done. A more accurate classification of episodes can be done by means of the slope of the line joining the boundary points (dtp) of the episode. This slope can be qualified by the same qualitative values: ‘positive’, ‘negative’ or ‘zero’, and conserve the meaning of [¶x] in the triangular representation. Then, this notation is also used here. [¶x] gives information about tendency of the current episode.
[¶ ¶x] 0
A
A
[¶¶ ¶¶x] < 0
B -
E
¶¶x] =0 [¶¶
D
-
D
C
¶¶x] > 0 [¶¶
Fig. 6 Types of episodes
So, according to this new parameter, nine types of basic episodes are given (3 values of [¶¶x] x 3 values of the [¶x]). Moreover, two subsets can be differentiated when both [¶x] and [¶¶x], are different from ‘zero’. Thus, thirteen types of episodes are obtained (See Fig. 6) and used to represent qualitative description of the evolution of the variables. This classification offers a more detailed description than the triangular and trapezoidal representations in the sense that episodes with [¶¶x]=0 are significative. In the trapezoidal representations described below, episodes with [¶¶x]=0 are not detected (ddx is zero only in one point) or are included in the other types, so never appear in the qualitative representation ; only episodes A,B,C,D,AB or CD compose the qualitative trend. But with the new classification method (with
and parameters r1 and r2 can be calculated from A. INCORPORERINCORPORER
3.4 Area For each episode, the area of the region limited by the signal and the line joining the boundary points can provide useful knowledge about the oscillation magnitude in the episode. Then, comparing areas of consecutive episodes we can know the evolution of oscillation. An approximation of this area can be calculated subtracting the Area2 from the Area1 (Fig. 8) approxim ation
n r, x ( nr) Area
l(n)=r1·n+r2
n l, x ( n l)
Fig. 7
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INCORPORER 1.5
A r e a 1
A r e a 2
1
Fig. 8 0.5
These two areas can be calculated from nr, nl and a1, a2 a3 and a4 parameters. INCORPORERINCORPORERNote that episodes with [¶¶x] = ‘zero’ (F, G and E types) have Area=0.
0
0
20
40
60
80
100
120
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D AB- CD¯ AB- CD¯
160
G
Fig. 10
4 Symbolic representation of linear systems
4.3 Higher order systems 1 0.9 0.8
In this section, some step responses of linear systems are analysed and its symbolic representation is obtained. First, second and higher order systems, with stable or unstable responses, have been studied. Symbolic representation, together with the quantitative information (areas), is used in order to distinguish the different dynamics of these systems. Qualitative values of [¶x] (-,¯ ¯ and =) are used in order to distinguish behaviours that cannot be distinguished before with other classifications . Also, new quantitative values, as is the area , are used in the same way.
Higher order systems present several dynamics. Oscillations and stability are deduced in the same way as for second order systems. In Fig. 11 an example is presented.
4.1 First order systems
4.4 Systems with an integrator (type 0 systems)
1
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
10
D
20
30
AB- CD -
40
50
AB-
60
70
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G
Fig. 11
The main characteristic of systems with an integrator is the constancy of [¶x] which shows that the system is always rising. There is a diminution of areas, so there is a relative stability but the last episode is E type so there is not absolute stability.
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
9
2.5
0.1 0
0
8
A
10
20
30
40
50
G
60
70
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2
7 6
1.5 5
Fig. 9
4
1
3 0.5
Only two episodes are done, the first one (A) shows that the signal, in this episode, is rising but its slope is decreasing. The second episode shows the stability of the system since the signal stays on level.
4.2 Second order systems In second order systems two different dynamics can be differentiated :with or without oscillations. The first one is similar to first order systems with a D type at the beginning while in the second one episodes AB and CD appears (Fig. 10). The stability is shown by the diminution of areas, the changes in [¶x] and finally by the appearance of episode G.
2 1
0
0
5
10
15
20
25
30
CD- AB- CD- AB-
35
40
0
0
5
10
15
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CD-
E
25
30
35
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E
Fig. 12
4.5 Unstable systems INCORPORERSEQARABEINCORPORERSEQARABE 200
100
0 -100
-200
-300 -400 0
50
100
CD-
150
AB¯
200
250
300
350
400
450
500
CD- AB¯ CD-
Fig. 13
Unstable systems are characterised by an increasing or permanent oscillation ; therefore, AB and CD types with a non
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diminution of the area are the main characteristics. Types E,F and G , which identify relative or absolute stability, never appear in unstable systems.
5. Application example : Fault detection in a laboratory plant The laboratory plant where fault situations are caused is composed by two coupled tanks with two pipes connecting them, as depicted in Fig. 14. The goal is to control the level in the second tank by pumping the fluid to the first tank while the liquid flows through valve 2 (V2). The control signal of this process is the pump voltage and the measure from the tank 2 level is also available. Correct operations are defined as follows: valves (V1 and V2) are open and the process is correctly controlled. When failures are introduced the expert diagnostic system must be able to detect and identify them. Therefore, the goal of the supervisory system is to track the process and detect situations that incite failures or process malfunctions, as well as to know when the process works in normal operating conditions.
The symbolic representation for the two signals in every situation is :
situation 1 2 3 4 (pos.) 4 (neg.)
Error
PID
Motorpump
-
G-D-A-(-E-A)-G G-D-A-(-E-A)-G G-B-(F-B)-(CD2-E-AB2)-... ...-C-(F-C)-G
A
G
B
C
9
D
8
G
7
G 6
5
0
200
400
600
800
1000
Fig. 16 Control signal in a set-point change 4.5
Tank 1 Level 1
G G
10
Set Point
+
level2
G G G-B-C-G G-D-A-G-B-C-G G-B-C-G-D-A-G
Table 1 Episodes in brackets appear only in some cases, depending on the set point. With these representations it can be distinguished when there is a set-point change and when valve 2 is closed.
4
Control
control
Tank 2 V1
4
Level2
V2
G
A
3.5 3 2.5 2
Fig. 14
1.5
In order to test the symbolic representation utility, the presented tool is applied to level2 and control signals The goal is to distinguish between four possible situations: 1. 2. 3. 4.
Normal situation Valve 1 closed Valve 2 closed Set-point change (positive or negative)
In normal conditions, noise and a small oscillation appear in the signals, so the zero interval for [¶x] and [¶¶x] has been chosen to detect only a g type in this case: [¶ ¶x] +/- 0.01 +/- 0.005
Control Level2
[¶ ¶¶x] +/- 0.06 +/- 0.00015
0.5
D
G
1
0
200
400
600
800
1000
1200
Fig. 17 level2 signal when v2 is closed
But is not possible to distinguish between the normal situation and the situation where v1 is closed. In these cases, behaviour differences cannot be appreciated by this method because of the limits chosen for the zero interval. This choice is determined for the noise presence. If the interval was reduced then noise would have been detected and a wrong representation for all situations (1-4) would have been the result. So, to distinguish between these two situations is not possible without any kind of noise filtering; in the next section a solution for this problem is proposed. 7.2
G
7 6.8 6.6
6.4
6.4 6.2
6.2
6 6
5.8 5.6
5.8
5.4 5.2
5.6
5.4
0
200
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800
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Fig. 15 Control signal in normal conditions
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0
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Fig. 18 Control signal when v1 is closed
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6. Conclusions The presented framework is based on the triangular and trapezoidal representations used and described in previous works as mentioned in the references and briefly described in this paper. The main goal is to obtain qualitative information that could be supplied to ES as an abstraction of process behaviour in order to facilitate the design of supervisory systems for continuous systems. This information is obtained dividing signals in time-episodes and a classification of these episodes. Also, simplified quantitative data is obtained to help and complete the qualitative information. First, the utility of the obtained information is contrasted by means of a representation of linear systems dynamics. The utility of using the new classification of episodes and new quantitative values is shown. New quantitative and qualitative values are used to distinguish behaviours that cannot be distinguished before with other classifications. Second, the application in real data shows that noise and not useful behaviours determine the choices of the interval where ddx and dx are considered zero. These elections are important for the detection and classification of episodes and according to them some interesting behaviours could be masked. One possible solution for this problem is the use of the wavelet transform and the multiresolution analysis of signals. The decomposition of a signal on a wavelet basis allows the extraction of features that are dominant at various scales. The application of the wavelet decomposition of signals before the qualitative representation described in this paper will allow a more accurate classification and representation [4].
It is possible to know on-line the type of the episode from the last dtp detection; but, if the current time is not any dtp, the obtained type could be wrong ; the correct type would be the one obtained if the signal is analysed until the next dtp. For example, in Fig. 19, at the instant t' the obtained type since the last detected dtp(t1) is C ,but this is not the correct type (CD ¯). The correct one only can be determined after the next dtp (t2). This problem could induce important errors in the consequent use of this information so it is necessary to find a method to correct these possible 'bad deductions'. One possible solution is to make a prediction of the episode type depending on the already known episode, detected at each time instant. CD¯ C t1
t'
t2
Fig. 19. Type detected at an intermediate point t' To make this prediction, the use of fuzzy reasoning is proposed, both in the prediction and in the classification of the episode between the last dtp and the current time.
8. Acknowledgements Research supported by the TAP96-1114-C03-03 project of CICYT program from the Spanish government.
7. Future work
9. References
This work has been developed under MATLAB using ‘.m’ files description. In order to be useful in designing supervisory systems these algorithms will be integrated with an ES, CEES [7] and qualitative reasoning tools (ALCMEN [2]) in Simulink. This integration will be done using object oriented approach in Simulink [8]. An immediate application of described qualitative representations of signals in episodes is in the analysis of data records. The triangular representation of trends has been applied in the evaluation and diagnosis of batch operations [6] and for the interpretation of biological processes [5]. In these two applications, the analysis in episodes is applied off-line using time-records of process variables. The final objective of this work is to expand its use to signals from continuous process, making the classification on-line, not only as an analysis tool but as an interfacing tool for ES in Supervisory tasks. However, some problems appear in the application of these techniques in on-line supervision of continuous processes or for event detection. The main one is that episodes only can be completely characterised when they have finished since it is necessary to know their two boundary points to classify.
[1] Feray-Beaumont S., Gentil S., Leyval L., “Declarative Modelling for Process Supervision”, Revue d’Intelligence Artificialle, vol.3, nº4, pp. 135-150. (1989). [2] Aguilar-Martin
J., “Qualitative control, diagnostic and supervision of complex processes”, Mathematics and Computers in Simulation, vol 36, pp 115-127. (1994)
[3] Cheung, J. T., Stephanopoulos G., “Representation of process
trends, parts I and II”. Computers Chemical Engineering, vol.14, pp. 495-540 (1990). [4] Bakshi, B. R. ; Stephanopoulos G. ; “Representation of process
trends, parts III and IV”. Computers Chemical Engineering, vol.18, N. 4, pp 267-302, (1994) [5] Ayrolles
L.; “Abstraction temporelle et interprétation quantitative/qualitative de processus à dynamiques multiples Aplication aux processus biologiques”, These de Doctorat Universite Paul Sabatier, Raport LAAS N. 96034, Toulouse, France (1996)
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[6] Bakshi B.R. et alt., “Analysis of operating data for
evaluation,diagnosis and control of batch operations”, Journal of Process Control, vol 4, No. 4, pp 179-194. (1994) [7] De la Rosa J.Ll., 1994., “®Heuristics for cooperation of expert
systems.Application to process control” , Doctoral Thesis, Universitat de Girona, Girona, (1994). [8] Meléndez J., Colomer J., de la Rosa J. LL., Aguilar-Martin J.,
Vehí J., "Embedding Objects into Matlab/Simulink for Process Supervision", Proceedings of the 1996 IEEE International Symposium on CACSD, pp 20-25,. (1996)
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