Identification

Distillation Column : Step Responses

								OEEPM nA=2 OEEPM nA=1

0 2 4 6 8 10 12 14 16 18 20
Time (t/T_s)

Figure 7.15. FIle QXD: normalized step responses for the two models identified (static gain normalized to 1)

The model obtained is validated. The static gain of the model is 0.958. It will be necessary to scale the values of the coefficients of the polynomial B(q^-1) in order to obtain the static gain previously obtained (nevertheless the difference found between the two static gain can be neglected).

The normalized step responses of both models are presented in Figure 7.15.

3. 
   DC Motor
   

The identification of a DC motor model is examined in this section. The input of the system is the voltage applied to a power amplifier that feeds the motor, and the output is the speed measured by means of a tachometer. A short description of the global system is shown in Figure 7.16. From the identification point of view, let us consider the cascade power amplifier, motor, tachometer, filter on the measured output as the plant.

The file MOT3.c¹⁰ contains 256 centered I/O data obtained with a sampling period of 15 ms. The input is a PRBS generated by a shift register with seven cells and a clock frequency f_s/2 (sequence length: 254).
¹⁰ Available from the website: http://landau-bookic.lag.ensieg.inpg.fr.

Figure 7.16. Schematic representation of a DC motor and other external elements
Motor Speed
0.4

Amplitude (Volt)
0.2
0
-0.2

-0.4

50 100

150

200 250

Motor Input Voltage
0.4

Amplitude (Volt)
0.2
0
-0.2

-0.4

50 100 150 200 250
Samples

Figure 7.17. I/O data set for the DC motor (MOT3.c)

The magnitude of the PRBS is set to 0.3 V and applied when the operating point of the system is an input voltage of 3V (scales for u: 0-10V corresponding to a speed variation from 0 to +1500 rpm). This file is shown in Figure 7.17.

The result of the complexity estimation algorithm (instrumental variable with delayed inputs method using estorderiv.sci or estorderiv.m) is (see Figure 7.18)


n = max (n_A,n_B+d) = 2

The detailed complexity estimation gives the values n_A = 1, n_B = 2, d = 0. This complexity is coherent with the model obtained from physical equations that describe the DC motor, with at most two time constants, but the electrical time

DC Motor : Model Complexity Estimation
0.25

0.2

0.15

0.1

0.05

0
0 1 2 3 4

Complexity (order)


Figure 7.18. Model complexity estimation for the DC motor (input: voltage, output: speed) using the data file MOT3.c

constant in this case is very small if compared to the electro-mechanical one. The absence of a pure time delay is not surprising as well. The degree of the polynomial B , n_B = 2 reveals the presence of a fractional delay as a result of the filtering action on the measure.

S=1 M=1(RLS) A=1 FILE:MOT3.c NS=256 DELAY D=0 COEFFICIENTS OF POLYNOMIAL A A(1) = - 0.5402 COEFFICIENTS OF POLYNOMIAL B B(1) = 0.2629
B(2) = 0.2257
VALIDATION TEST: Whiteness of the residual error
System variance: 0.0412 Model variance: 0.041 Error variance R(0): 1.157 E-4
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.:|RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.0000  RN(1) = - 0.4529 
RN(2) = 0.3113 RN(3) = 0.0332 RN(4) = - 0.0297
A first identification is carried on with structure S1 and the recursive least squares (M1) with a decreasing adaptation gain (A1), the following results are obtained:
The model obtained is not validated. Next an identification with the structure S3 is performed by using the output error with extended prediction model method (M3) that simultaneously estimates the plant model and the disturbance model with the choice n_C = 1 (a decreasing adaptation gain is still used). In this case the results are

S=3 M=3 (OEEPM) A=1 FILE:MOT3.c NS=256 DELAY D=0 COEFFICIENTS OF POLYNOMIAL A A(1) = - 0.5372 COEFFICIENTS OF POLYNOMIAL B B(1) = 0.2628
B(2) = 0.2273 COEFFICIENTS OF POLYNOMIAL C C(1) = 0.3808
VALIDATION TEST: Whiteness of the residual error
System variance: 0.0412 Model variance: 0.041 Error variance R(0): 1.004 E-04
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.:|RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.0000 RN(1) = - 0.0144
--> RN(2) = 0.3414 <-- RN(3) = 0.0952 RN(4) = 0.0399

It can be observed that the model obtained is not validated too. If the choice n_C = 1 does not allow one to model suitably the effects of disturbances on the system (as RN(2) still has a large value), it is advisable to perform a new identification with n_C = 2.
The following results for the whitening test on the residual prediction errors and for the uncorrrelation test between the output error and the output prediction are found:

RN(0) = 1.0000	RN(1) = - 0.0432	RN(2) =	0.0404
RN(3) = 0.0453	RN(4) = 0.01179
VALIDATION TEST: Error / prediction uncorrelation System variance:0.0412 Model variance:0.0408 Error variance R(0): 2.71 E-04 NORMALIZED AUTOCORRELATION FUNCTIONS Validation Criterion:Theor. Val.:|RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15
RN(0) = 0.0119	RN(1) = 0.0156	RN(2) = 0.0284
RN(3) = 0.0356	RN(4) = 0.0186

S=2 M=4 (OEFC) A=1 FILE:MOT2C NS=256 DELAY D=0 COEFFICIENTS OF POLYNOMIAL A A(1) = - 0.5347 COEFFICIENTS OF POLYNOMIAL B B(1) = 0.2629
B(2) = 0.2266
VALIDATION TEST: Error / prediction uncorrelation
System variance: 0.0412 Model variance: 0.0406
Error variance R(0): 2.723 E -04
Both the whitening test and the uncorrelation test give good values. Thus the model identified is validated. Another structure and method can be tested nevertheless with the same model complexity. If the structure S2 and the output error with fixed compensator method (M4) are chosen, in order to obtain an asymptotically unbiased estimation without estimating the disturbance model, one gets (d =0)

NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion:Theor. Val.: |RN(i)|  0.136, Pract. Val.:|RN(i)|  0.15 RN(0) = 0.043 RN(1) = 0.0591 RN(2) = 0.0631
RN(3) = 0.0564 RN(4) = 0.031

Note that the results of the validation are very good, but slightly worse than those obtained with the output error with extended prediction model method. However, the parameters obtained in these two cases are very close. The step responses obtained using these models are extremely close each other and are shown in Figure 7.19.