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SIMULACE TECHNOLOGICKÝCH PROCESŮ HYBRIDNÍ TECHNIKOU VYUŽÍVAJÍCÍ MATEMATICKO-FYZIKÁLNÍCH MODELŮ A UMĚLÝCH NEURONOVÝCH SÍTÍ SIMULATION OF TECHNOLOGICAL PROCESSES USING HYBRID TECHNIQUE EXPLORING MATHEMATICAL-PHYSICAL MODELS AND ARTIFICIAL NEURAL NETWORKS Milan HEGER a, Ivo ŠPIČKA b, Martin BOGAR c, Mária STRÁŇAVOVÁ d, Jiří FRANZ e a VŠB-TU Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Česká republika,
[email protected] b VŠB-TU Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Česká republika,
[email protected] c VŠB-TU Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Česká republika,
[email protected] d VŠB-TU Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Česká republika,
[email protected] e Tieto Czech s.r.o, Varenská 51, 702 00 Ostrava-Moravská Ostrava, Česká republika,
[email protected] Abstrakt Optimalizace řízení technologických procesů je obvykle svázána s využitím matematických modelů. Většina technických prostředků využitých pro řízení na úrovni vlastní technologie není přizpůsobena na řešení složitých matematických operací a navíc výpočet s kvalitními podrobnými matematickými modely dynamických systémů je časově velmi náročný a spolu s optimalizací v reálném čase prakticky neřešitelný. Na druhé straně matematický popis umělých neuronových sítí (UNS) je velmi jednoduchý a algoritmy naučené UNS jsou snadno aplikovatelné do stávajících technických prostředků řízení technologických procesů. Aby však mohly být modely na bázi UNS úspěšně použity, musí být UNS účelně naučena na datech, která zahrnují všechny možné varianty, které by mohly nastat v reálném procesu včetně poruchových a havarijních stavů. Taková data však prakticky není možné získat z reálného technologického procesu. Nabízí se však možnost naučit UNS off-line na datech získaných simulacemi s využitím přesných matematických modelů a získat tak model hybridní. Vhodnou organizací simulací je pak zajištěno, že UNS bude správně reagovat i na takové situace, které jsou v reálných podmínkách řízení zcela výjimečné. Cílem tohoto článku je pak na několika praktických procesech ukázat filozofii a možnosti použití těchto hybridních modelů. Abstract The optimization of the technological processes control is usually connected with mathematical models usage. Most of technical instruments for control on the level of own technology is not customized for the hard mathematical operations solving and in addition the computation with quality precisely models of the dynamic systems is very time consuming and together with the real time optimization is not really solvable. On the other hand the mathematical description of artificial neural networks (ANN) is very simple and the algorithms of the learned ANN are easily implemented into existing technological processes control means. For successful using of the models on the base of ANN, the ANN needs to be rationally learned on the data which occupy all eventual variants which could occur in the real process including malfunction and crash states. But such a data is not practically possible to get from real technological process. There is possibility of off-line ANN learning with using data given by simulations based on the high precision mathematical models and by this way to get the hybrid model. By the useful organization it is secured, that ANN will also react correctly to such situations which are highly exceptional in real control conditions. The goal of this paper is to present the philosophy and the possibilities of this hybrid models usage on several practical processes. Klíčová slova: Optimalizace, řízení, umělé neuronové sítě, hybridní modely. Key words: Optimization, control, artificial neural networks, hybrid models.
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Introduction Mass exploitation of artificial intelligence elements, most frequently artificial neural networks in the process of technical praxis solving has been encountered recently. Exploitation of artificial neural networks has its great value whenever it is difficult to obtain solution by usual analytic procedures. In order to have results obtained by artificial neural network effective, it is always necessary to carefully analyze the whole problems in their complexity and choose such methods to expect real results from neural network with high probability. For example, prediction of observed value time trends in the process of solving dynamic processes, which is often loaded with error, but from control point of view, predictions of certain parameters, such as material temperature at the end of heating process, are often sufficient. These predictions are more accurate and faster available. Another artificial neural network exploitation problem tends to be insufficiency and quality of data for the process of artificial neural network learning. The article should demonstrate hybrid models exploitation, so models based on mathematical-physic al bases and artificial neural network work basis. Application of these models could be expected wherever sufficiently precise mathematical models of systems are known, but where their exploitation in real time is due to complexity of their computation unreal. 1.
HYBRID MODEL PROBLEMS
Under hybrid model idea is mostly imagined a model, where part of the computation is executed based on mathematics-physics basis and other with artificial neural network exploitation. This model could be signed as parallel hybrid model. Mathematics-physics models seem to be advantageous for generating of large amount alternatives of real object behaviour and data acquired this way then use for artificial neural network learning. This approach could be named as serial hybrid model. The main reason for serial hybrid models application is effective advantage of computation exploitation on one side and effort for the elimination of their characteristic disadvantages on the other side. Originally, the idea of serial hybrid model has been invented in an effort to find an effective solution for online control of material complex heating and cooling processes intended for forming, it can as well be used for hybrid models creation in other industrial parts. 2.
HYBRID MODEL EXPLOITATION
The principle of creation and serial hybrid model exploitation will be explained on the following simple example. Two-phase RC capacitor charging element (Fig.1), which is connected as low-pass.
R(t)
i1(t)
i3(t) C(t)
u1(t)
From system point of view it is proportional dynamic system of 1st order, the mathematic description of which can be (provided that i3 = 0 and parameters R and C can change in time) expressed by the following linear differential equation (1),
u2(t)
i2(t)
Obr. 1. Zapojení RC členu Fig. 1. Scheme of RC element
T t u2' t u2 t u1 t
(1)
where u1(t) – RC input electric voltage actual value u2(t) – output electric voltage actual value T(t) – actual system time constant and given by expression T(t) = R(t) * C(t) R(t) – resistor, the value of which can change in time C(t) – capacity, the value of which can change in time
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Jump in voltage is used as input signal, the value of which will after some time be increased (see Fig.2.). This way is the process of RC capacitor charging divided in two steps that differ in the length Δt1 = t1 – t0 and Δt2 = t2 – t1 and also input voltage value
u11 t , which is constant for Δt1 a u12 t , which
is constant for Δt2.
t2
t1
Analytic differential equation solution describing both steps RC capacitor charging defined this way is simple and its mathematical relationship of voltage u 2 t on 1
Obr. 2. Časové průběhy nabíjení kondenzátoru RC členu Fig. 2. RC capacitor charging time courses
output RC element in time interval t0 ≤ t ≤ t1 may be expressed, where the expression (2) is acquired t T u t u t 1 e 1 1 2
1 1
(2)
and in time interval t1 ≤ t ≤ t2 which describes the expression (3) t t1 T 2 u t u t u t u t 1 e . 2 2
1 2
2 1
1 2
(3)
Time course of RC capacitor charging element for parameter values from Tab.1 is shown in Fig.2. Tab 1. Parametry RC členu Table 1. Parameters of RC element. R = 1 MΩ C = 1 µF
T1 = 1 s T2 = 1 s
t1 = 1 s t2 = 3 s
u11 = 10 V 2 u1 = 20 V
Δ t1 = 1s Δ t2 = 2s
From the following graph can be seen that some kind of simplified analogy between RC capacitor charging element and course of surface material temperature during heating could be found. In technical practice, due to various accidents or changes in control strategy, some deviations from ideal
course could be found. This state can be simulated by parameters change u1 t , u1 t , t1, t2 and value 1
change R(t) a C(t) respectively in single charging phases.
2
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Set of numbers of 10000 solutions differential equation (1), (2) and (3) respectively for basic parameters set according to Tab.1 and loaded with random value changes in the range ± 10% can be seen in Fig.3. Solutions with tabular values Final charging time and final voltage value on the RC element output at the end of charging process are considerably variable. Similar set of solutions could be found in the process of various heating conditions simulation, only the mathematical model including all physical-chemical processes in three dimensional heat diffusion would be substantially complicated and single solutions generating would Set of all solutions be rather time consuming. For optimal heating control, where multiple simulation repetition is t2 t1 required before control system executes competent decision, this kind of model is in praxis non applicable. Obr. 3. Množina 10000 náhodných řešení nabíjení On the other hand, a model based on artificial neural networks has its great advantage because of universal inner structure, which is generally independent of work physical principles of simulated object. The basis is effective interconnection of mathematical analogues of physical neuron. It proceeds from the following and generally known basic mathematical relationship describing neuron: Fig. 3. Set of 10000 random solving of charging
n z ij f neur wk xk k 0
(4)
Neural network for the example with RC capacitor element charging then could have a structure according to Fig.4. Even if the structure looks rather complicated at first sight, voltage definition at RC element output at the end of charging process - u2(t2) by means of learned artificial neural network for 1 2 given input values R1, R2, C, t1, t2 u1 a u1 is then effortless.
R1 R2 C t1
u2(t2)
t2
difference [V]
u22(t2)MM , u22(t2)ANN [V]
u11 Because the precision of the results acquired by artificial neural network 2 u1 exploitation 20 0,2 depends not Obr. 4. Struktura UNS 18 0,16 only on quality, 16 0,12 Fig. 4. ANN structure 14 0,08 but also on the 12 0,04 amount and data distribution, 70 curves with 10 0 random choice of input variables with ±10% 8 -0,04 6 -0,08 allowance (training set) were generated with aid 4 -0,12 of mathematical model for artificial neural 2 -0,16 network learning process. Voltage values at the 0 -0,2 1 6 11 16 21 26 ends of single charging u22(t2)MM were the Experiment No. outputs. Having been artificial neural network mathematical model model with ANN difference between the two models learned in "Neuronek" program [1], the functionality was verified at voltage prediction Obr. 5. Výsledky predikcí pomocí UNS 2 u2 (t2)ANN. For this reason, testing data sum about 27 experiments in volume was generated. Fig. 5. Prediction results by means of ANN Expected results and prediction deviations by
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means of artificial neural network are stated in figure 5. Maximum error was around ±0,1 V, which is very satisfactory result. 3.
HYBRID MODEL EXPLOITATION IN THE PROCESS OF STEEL SPECIMENS COOLING CURVES
In paper [2.] was shown that cooling curves for specimens, which vary in shape, have the same character and therefore is sufficient when artificial neural network would not predict cooling curve, but corresponding time transformation coefficient (TTC) only, by reciprocal value of which must be transformed physically measured (referential) cooling curve of know shape, therefore the time course of cooling predicted curve for specimens of required dimensions (see Fig. 6) is acquired. In real-time cooling process of two geometrically different steel specimens intended for rolling process, two cooling curves were acquired by means of maximum surface temperature measurement. For correct artificial neural network learning is the count insufficient. Therefore it is effective to use mathematical-physical three-dimensional model of heat diffusion spread in material, which results from Fourier partial differential equation [3.]:
c
Tmaterial Tmaterial t
(5)
and external heat transfer model at third category conditions results from equation [3.]:
Tsurface n
Tsurface Tmediums
(6)
Model parameters could be defined based on identification of both measured cooling curves, on which mathematical-physical models were also verified. To acquire more data for artificial neural network learning process, another five simulations for various geometric specimen shapes were carried out on these models (see Fig. 6). Artificial neural network was learned in "Neuronek" program and tested on another specimen with simulated cooling. The geometric dimensions of the specimens with the appropriate TTC for particular experiments are listed in the table. 2. From Fig.7 is obvious that prediction by means of artificial neural network exhibits a good relationship with mathematical model results (maximum error do not exceed 5%, which represents 2 s as the maximum error of determining the cooling down time). Tab 2. Parametry vzorků Table 2. The specimens' parameters
purpose measured
Sumulation for ANN learning
testing
Experiment No. 1 2 3 4 5 6 7 8
Wide of specimen 2,3 10,1 3,4 4,4 7,2 10 7,2 7,2
Height of specimen 20,3 40,2 20 30 30 30 20 40
TTC 1 0,267 0,46 0,295 0,425 0,26 0,412 0,248
1200
Specimens
1000
2.3x20.3
800
10.1x40.2 3.4x20.0
600
4.4x30.0 400
7.2x30.0
200
10.0x30.0
40
prediction with ANN 30
time [s]
Temperature [°C]
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20
10
simulation
7.2x20.0
0
0
30
60
90
120
150
7.2x40.0
time [s]
0 600
800
1000
1200
temperature [°C]
4.
Obr. 6. Křivky chladnutí
Obr. 7. Přesnost predikce
Fig. 6. Cooling down curves
Fig. 7. Accuracy of prediction
CONCLUSION
Even if two professionally different but relatively simple examples for hybrid model exploitations were used, it can be seen that satisfactory accuracy for technical praxis was achieved. For successful hybrid models application is fundamental:
to create truthful mathematical model of a real object,
to properly choose seeking output value or more values, which characterize the process,
to choose significant input variables and strategy of changes of their values, so that the simulation would cover the whole field of real object controlling methods,
to carry out sufficient number of simulations so that artificial neural network is learned with satisfactory accuracy.
By hybrid model application can control system acquire fast means not only for result prediction of actual control strategy, but also for results computation of various control options. It enables to correct control mechanism in time with goal to achieve optimal control. ACKNOWLEDGEMENTS The methodology described and results were obtained in the framework of the solution of Research Plan MSM 6198910015 (Ministry of Education of the Czech Republic) and project 105091366 (Czech Science Foundation). LITERATURE [1.]
HEGER, M.; DAVID, J. Neuronek – program pro výuku neuronových sítí. In Sborník semináře XXVI. ASŘ 2001: Instrumets and Control, Ostrava, 2001, ISBN 80-7078-890-9.
[2.]
HEGER, M.; FRANZ, J.; ŠPIČKA, I. Využití prvků umělé inteligence pro predikci času chladnutí kovových vzorků před tvářením. Hutnické listy. roč. LXI, 2008, Sv. č. 2
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[3.]
ŠPIČKA, I.; HEGER, M.; FRANZ, J. The Mathematical-Physical models and The Neural Network Exploitation for Time Prediction of Cooling Down Low Range Specimen. Archives of Matallurgy and Materials. 2010, Sv. vol. 55, 3/2010
