Identification

Air Heater : Model Complexity Estimation
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Figure 7.10. Complexity estimation for the air heater model based on the data file AERO.c




S=3 M=3 (O.E.E.M.P) A = 1 FILE:AERO.C NS = 128 DELAY D=0 COEFFICIENTS OF POLYNOMIAL A: A(1) = -0.6589 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.1724
B(2) = 0.0579 COEFFICIENTS OF POLYNOMIAL C: C(1) = -0.1248
VALIDATION TEST: Whiteness of the residual error
System variance: 0.00351 Model variance: 0.00346 Error variance R(0): 3.837E-05
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.:|RN(i)|  0.192, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.000000 RN(1) = 0.1303
RN(2) = 0.1029 RN(3) = 0.0192
RN(4) = 0.0326
The result of the validation is not acceptable. The results of the identification can be further improved by taking into account the fact that disturbances are present. A first approach is to choose structure S3 with the method M3, the output error with extended prediction model, which gives also an estimation of the model of disturbances. The following results are obtained (with decreasing adaptation gain):
The results of the validation are without doubt satisfactory.

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Air Heater : Step Responses

									OEEPM OEFC

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Time (t/T_s)

Figure 7.11. Step response for the models identified in the air heater example: OEEPM - output error with extended prediction model, variable forgetting factor, d = 0, n_A
= 1, n_{B =} 2; OEFC - output error with fixed compensator, d = 0, n_A = 1, n_B = 2

The step response for this model is shown in Figure 7.11 (OEEPM). Structure S2 can also be used, which may lead to an asymptotically unbiased estimated model without modeling the disturbances. The output error with fixed compensator (method M2) is chosen with n_D=0 (because n_A=1), and with an adaptation gain with forgetting factor ( ₁(0)  0.97 ). The results are summed up in the following

S=2 M=2 (OEFC) A=3 FILE: AERO.C NE=128 DELAY D=0 COEFFICIENTS OF POLYNOMIAL A: A(1) = -0.6837 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.1771 B(2) = 0.043
VALIDATION TEST: Error / prediction uncorrelation
System variance: 0.00351 Model variance: 0.00317 Error variance R(0): 9.77 E-05
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.:|RN(i)|  0.192, Pract. Val.: |RN(i)|  0.15 RN(0) = -0.2208 RN(1) = 0.1508
RN(2) = 0.0432 RN(3) = 0.0116
RN(4) = 0.0469
table:
The step response for this model, which has passed the validation test, is presented in Figure 7.11 (OEFC). The model obtained with the output error with fixed compensator has a static gain and a rise time slightly larger than the corresponding values for the previous model (a larger rise time means that the identified model is slower).

S=3 M=3 (OEEPM) A=1 FILE: AERO.C NS=128 DELAY D=0 COEFFICIENTS OF POLYNOMIAL A: A(1) = -0.6589 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.1724
B(2) = 0.0579
VALIDATION TEST: Error / prediction uncorrelation
System variance: 0.00351 Model variance: 0.00317 Error variance R(0): 1.02E-04
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.: |RN(i)|  0.192, Pract. Val.:|RN(i)|  0.15 RN(0) = - 0.2154 RN(1) = - 0.1465
RN(2) = - 0.1033 RN(3) = - 0.1094 RN(4) = - 0.1769
In this case, in order to compare further the quality of the two models, the model identified with the output error with extended prediction model should also be validated by the uncorrelation test. This test gives the following results:
Both models are validated and the results are very close.
As the models identified have the same quality, one of them will be used to compute the controller and the result of the design will be tested on both models.

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