Identification

File T1 : Model Complexity Estimation
0.25

0.2

0.15

0.1

0.05

0
0 1 2 3 4 5 6 7 8

Complexity (order)


Figure 7.7. Estimation of the model complexity (n=max(n_A,n_B+d)) from the data collected in the file T1

5. Plant Identification Examples

1. 
   Air Heater
   

The diagram of the system is represented in Figure 7.8. The air is heated at the pipe input by means of an electrical resistor supplied by a power amplifier. The temperature of the air at the output is measured by a thermocouple. This section is concerned with identifying the dynamic model linking the power amplifier control to the temperature of the output air, around a certain temperature. The steady state characteristic of the air heater + power amplifier is very non-linear. This results in the appearance of a DC component at the output even for centered input signals of low magnitude.

temperature measurement
y(t)




power amplifier
u(t)
Figure 7.8. Schematic representation of the air heater (Laboratoire d’Automatique de Grenoble INPG/CNRS)

The data acquisition has been carried on around an operating point corresponding to an output temperature of 60^o and a DC input signal u₀=5V (y₀(t)

= 3.2V).
The file AERO.dat contains 128 raw input/output data obtained with a sampling period of 5 s. The input was a PRBS generated with a length register equal to 6 and a clock frequency f_s/2 (sequence length: 126 samples). The raw data file was centered. The I/O data set relative to the centered file AERO.c⁸ is shown in Figure 7.9.
A first identification is started up, using the S1 structure and the recursive least squares method M1 with decreasing adaptation gain. Since this is a thermal system, n_A = 2 can be a good initial choice. On the other hand d = 0 (no prior knowledge on the delay) and n_B = 2. The following results are obtained:

⁸ Available from the web site: http:/landau-bookic.lag.ensieg.inpg.fr

S = 1 M = 1(RLS) A = 1 FILE: AERO.C NS = 128 DELAY D = 0 COEFFICIENTS OF POLYNOMIAL A: A(1) = -0.66872
A(2) = 0.00115 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.17360
B(2) = 0.05424

The system clearly has a time delay less than 0.5 T_s (since |b₁|>|b₂| ), thus the choice d=0 was a good one. As the coefficient A(2) has a very small value (resulting from the product of poles) one can say that the value of one of the poles is very small and it can be neglected.