6837092896

Diagnostyka’ 4(48)/2008    11

KEKEZ, AMBROZIK, RADZISZEWSKI, Modeling of Cylinder Pressure in Compression Ignition Engine ...

°CA

to rcproduce. Genetic algorithm consists of several stcps: 1) initial population is generated. 2) value of "fitness function" for each individual is calculated. 3) selection of Solutions based on value of "fitness function" and randomness. 4) crossover (new chromosome is constructed from two others), 5) mutation (random change of smali part of chromosome). Steps 2-5 are repeated until a good enough solution is found. In case of GFSm system, each chromosome represents one fuzzy model. The "fitness function” reflects the accuracy of tliis model and, optionally. its transparency i.e. the number of “if-then” rules in the knowledge base.

In our e\periments. we used measuremcnt data for diesel oil to create fuzzy model (and to test its accuracy) by means of GFSm system. The acquired fuzzy model liad two inputs: xl (crankshaft angle), x₂ (time value. proportional to rotational speed of crankshaft) and one output y (cylinder pressure value). The model predicts value of cylinder pressure (y) for all possible crankshaft angle and rotational speed values (x_h x₂) in AD3.152UR engine fueled by diesel oil.

Relationships between input and output variables are stored in the form of so-called knowledge base, which consists of set of rules and set of membership functions of fuzzy sets used in rules. GFSm system builds the fuzzy model on the basis of training data. complying with settings for number of rules and number of fuzzy sets.

| Knowledge base |

Fig. 3. Mamdani-type fuzzy model with two input

variables x, and x₂. and one output variable y

The GFSm system created several models which liad different number of rules, depending on program settings. Training data. used in experiments. contained pressure curves for speeds 1000 and 1800 rpm of CA. The single model built by GFSm describes pressure curves for all allowable crankshaft speeds. With settings "ma.ximum number of fuzzy sets describing one input = 50" and "highest accuracy models preferred (number of rules is irrelevant)” the GFSm system created a model consisting of 40 rules. The acąuired model allows to predict the value of maximum pressure with error not exceeding 5%, and the value of inean indicated pressure with error in the rangę of 0.3% to 13.5% (depending on rotational speed of CA). Becausc 13.5% error is too high for praclical technical applications, a new model was created by means of GFSm system. The settings were identical like previously. but training data contained the pressure data for the rangę of 340° to 390° CA. The acąuired model, which liad 20 rules. w as morę precise in the above mentioned rangę, for all crankshaft speeds. However, in order to compute mean indicated pressure we necd a model describing pressure in the whole rangę of 180° to 540° CA. not only in its fragment. For tliis reason. the next (third) model was created by means of GFSm. The settings were: "maximum number of fuzzy sets describing one input = 50”, “models with smali number of rules preferred” and "each nile in knowledge base must have condition (premise) regarding x, yariable” (this variable describes tlie value of crankshaft angle). The acąuired model is morę generał (only 12 rules) and also describes pressure curves for all crankshaft speeds. Fig. 4 shows “model” curves (computed with use of tliis model).

a)

pc, MPa 9

b)

pc, MPa 9

- 2000

0

340    350    360    370    380    390

°CA

Fig. 4. Relationsliip between cylinder pressure in diesel engine (w orking in extenial speed cliaracteristic regime and fueled by diesel oil) and crankshaft angle. computed by the fuzzy system, for different crankshaft speeds in the rangę of: a) 180° - 540° CA, b) 340° - 390° CA