Identification

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Figure 7.14. I/O data set for the distillation column QXD

Thus a new identification is performed with d = 1, n_B = 3, n_A = 2 and the results obtained are

S = 1 M = 1(RLS) A = 1 FILE: QXD NS = 256 DELAY D=1 COEFFICIENTS OF POLYNOMIAL A A(1) = -.23806
A(2) = -.18820 COEFFICIENTS OF POLYNOMIAL B B(1) = .07982
B(2) = .18303
B(3) = .13244
VALIDATION TEST: Whiteness of the residual error
System variance: 0.0542 Model variance: 0.0274 Error variance R(0): 0.0268
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.: |RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.0000 RN(1) = -0.1216
 RN(2) = -0.2278  RN(3) = 0.0112 RN(4) = 0.0718

It is observed that the validation is unsatisfactory since the residual error is not sufficiently white (|RN(2)|) > 0.15). We may consider that this situation arises from the fact that the disturbances are incorrectly modeled by the structure S1. A different structure (S3) and the output error method with extended estimation model (M3), still with decreasing adaptation gain, are then tried. In this way a disturbance model will be identified. The results obtained are

S = 3 M = 3 (OEEPM) A = 1 FILE:QXD NS = 256 DELAY D=1 COEFFICIENTS OF POLYNOMIAL A A(1) = -.5065
A(2) = -.1595 COEFFICIENTS OF POLYNOMIAL B B(1) = 0.0908
B(2) = .1612
B(3) = .0776 COEFFICIENTS OF POLYNOMIAL C C(1) = -.4973
C(2) = -.2593
VALIDATION TEST: Whiteness of the residual error
System variance: 0.0542 Model variance: 0.0308 Error variance R(0): 0.0231
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.: |RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.000000 RN(1) = -0.0241
RN(2) = -0.0454 RN(3) = 0.0518
RN(4) = 0.1014

The validation results obtained are very good. This is therefore a representative model. Figure 7.15 gives the step response for this model with n_A = 2 (gain normalized to 1). The analysis of this response shows that the rise time is t_R  0 T_s (with a delay d = 1). On the other hand, since the number of cells of the generator register of the PRBS is N = 8, the duration of the largest pulse is less than t_R. The validation will thus not be significant for the steady state gain (several models may be validated without their steady state gain being the same). A direct verification of the steady state gain shows that the value obtained (0.987) is correct.
Note also that the polynomial B(q^-1) is unstable, B(1) < B(2) revealing the presence of a fractional delay greater than 0.5T_s (which explains the presence of an unstable zero).


S = 3 M = 3 (OEEPM) A = 1 FILE:QXD NS = 256 DELAY D=1 COEFFICIENTS OF POLYNOMIAL A A(1) = -.7096 COEFFICIENTS OF POLYNOMIAL B B(1) = .0839
B(2) = 0.1415
B(3) = 0.0528 COEFFICIENTS OF POLYNOMIAL C C(1) = -.688
VALIDATION TEST: Whiteness of the residual error
System variance: 0.0542 Model variance: 0.02952 Error variance R(0): 0.0227
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.: |RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.0000 RN(1) = -0.0035
RN(2) = -0.0748 RN(3) = -0.0131
RN(4) = 0.0442
As the coefficient A(2) is small if compared to A(1) and the validation results are very good, one may think of identifying a new model with n_A=1. In this case the results are

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