Identification

Figure 7.20. View of the flexible transmission (Laboratoire d’Automatique de Grenoble INPG/CNRS/UJF)

MOTOR AXIS


LOAD

_m

AXIS POSITION

D A C
u(t)

y(t)

^ref

Figure 7.21. Control scheme for the flexible transmission

Plant Output
1.5

1

0.5

0

-0.5

-1

-1.5
50 100 150 200 250

Plant Input
0.15

0.1

0.05

0

-0.05

-0.1

-0.15
50 100 150 200 250
Samples


Figure 7.22. I/O data used for the identification of the flexible transmission
Flexible Transmission : Model Complexity Estimation
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 1 2 3 4 5 6 7 8
Complexity (order)


Figure 7.23. Complexity estimation for the flexible transmission model based on the file poulb01.c

S=1 M=1 (RLS) A=1 FILE: POULBO1.C NS=254 DELAY D=2 COEFFICIENTS OF POLYNOMIAL A: A(1) = -1.5748
A(2) = 1.8329
A(3) = -1.4784
A(4) = 0.8895 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.3010
B(2) = 0.4181
VALIDATION TEST: Whiteness of the residual error
System variance: 0.3317 Model variance: 0.1053 Error variance R(0):
0.0007
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion: Theor. Val.: |RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.0000  RN(1) = -0.5727 
 RN(2) = 0.2360  RN(3) = -0.0475 RN(4) = -0.0158
If a new identification is performed with d = 2, n_B = 2, n_A = 4 by using structure S1 and the recursive least squares methods, one gets the following results:

30

20

10



Magnitude (dB)
0

-10

-20

-30

-40
Flexible Transmission : Magnitude Bode Diagrams

									RLS OEEPM

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5
Frequency (f/f_s)

Figure 7.24. Frequency characteristics of the models identified for the flexible transmission (RLS – recursive least squares, OEEPM – output error with extended prediction model)

The validation is not satisfactory as RN(1), RN(2) are greater than 0.15. The frequency characteristics of this model is shown in Figure 7.25 (line RLS).

Thus one tries structure S3 which introduces a model for the disturbance (ARMAX model). The output error with extended prediction model is used (OEEPM) with a decreasing adaptation gain (A1). The results of identification and validation are given below:

S=3 M=3 (OEEPM) A=1 FILE:POULBO1.C NS=254 DELAY D=2 COEFFICIENTS OF POLYNOMIAL A: A(1) = -1.60955
A(2) = 1.87644
A(3) = -1.49879
A(4) = 0.88574 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.30530
B(2) = 0.39430 COEFFICIENTS OF POLYNOMIAL C: C(1) = -0.67530
C(2) = 0.2283
C(3) = -0.0653
C(4) = -0.0585
VALIDATION TEST: Whiteness of the residual error
System variance: 0.1061 Model variance: 0.1055 Error variance R(0): 0.0004
NORMALIZED AUTOCORRELATION FUNCTIONS
Validation Criterion: Theor. Val.: |RN(i)|  0.136, Pract. Val.: |RN(i)|  0.15 RN(0) = 1.0000 RN(1) = -0.0425
RN(2) = 0.0959 RN(3) = -0.0563
RN(4) = -0.0407

The validation is really satisfactory, as all the values RN(i) are smaller than 0.136 for i = 1,2,3,4. The frequency characteristics of this model is shown in Figure 7.24 (line OEEPM). The comparison of the frequency characteristics of the models identified shows that a good validation corresponds to the identification of a less damped second resonant mode.

5. Concluding Remarks

This chapter has shown how effectively an identification of a plant model has to be carried out. The different steps can be summarized as follows:

• 
  Input/output data acquisition using a PRBS (pseudo-random-binary sequence) as input
  
• 
  Conditioning of the acquired data (DC removing, data scaling, filtering);
  
• 
  selection or initial estimation of the system order n = max (n_A, n_B +d)
  
• 
  Selection or estimation of n_A, n_B, d either by order estimation techniques, or by inspection of the numerical values of the estimated parameters
  
• 
  Identification and validation using several structures plant + disturbance and identification methods with the objective of obtaining the best acceptable model with lowest n_A and n_B
  
• 
  Analysis of the model identified and validated both in the frequency and time domain
  

5. Notes and References

The proceedings of the IFAC Symposiums Identification and System Parameter Estimation, Pergamon Press, Oxford, are a good source of information about several types of system identification. See also:

Isermann R. (1980) Practical aspects of process identification, Automatica, vol.16, pp. 575-587.
Ljung L. (1999) System Identification - Theory for the User, 2nd edition, Prentice Hall, Englewood Cliffs.
For the identification of flexible structures see also:
Van den Bossche E., Dugard L., Landau I.D. (1986) Modelling and Identification of a Flexible Arm, Proceedings American Control Conference, Seattle, U.S.A.
Landau I.D., Langer J., Rey D., Barnier J. (1996) Robust control of a 360^o flexible arm using the combined pole placement (sensitivity function shaping method), IEEE Trans. On Control Systems Technology, vol. 4, no. 4, Juillet 1996, pp. 369-383.
Landau I.D. (2001b) Identification des systèmes. Les bases, in Identification des systèmes, (I.D. Landau, A. Besançon-Voda éd.), Hermes, Paris.
For the identification of DC motors see also:
Landau I.D., Rolland F. (1993) Identification and digital control of electrical drives, Control Engineer Practice, vol.1, n°3.