Identification

Distillation Column

The diagram of the binary column (water-methanol), which is the subject of the identification, is given in Figure 7.12a and a view of the plant is shown in Figure 7.12b.

The control variables are the heating power (QB) and the reflux rate (R). The flow rate (LF) and concentration (XF) of the input product are assumed constant (their variations produce disturbances in the system). The controlled variables are the flow rate of the top product (LD) and the concentration of the top product (XD).
(T) designate the temperature and (LB) the flow rate of the product at the bottom. The corresponding block diagram is given in Figure 7.13.
The input and output variations will be related around an operating point by a linear model characterized by a 2 × 2 transfer matrix operator


^{ LD } _ ^H₁₁ (q ^1) H₁₂ (q ^1)^{ R } _ ^{ w}₁ ^

 _XD
^H (q ^1)
H (q ^1)^QB
_w 

  ^_{ 21}
22 ^^
  2 

where w₁ and w₂ represent the unmodeled disturbances. Each element of the transfer matrix operator is of the form

H_ij (q


^1) 
q ^d b₁q ^1  b₂ q ^2  ... 1  a q ^1  a q ^2  ...

1 2

=

=

=



Figure 7.12a,b. Binary distillation column (Laboratoire d'Automatique de Grenoble, INPG/CNRS). a Functional diagram, b View
that corresponds to a model of the form:


A(q ^1) y(t)  q ^d B(q ^1)u(t)  w(t)
(reflux
_rate) (flow)

PLANT
_R L D

QB
(heating power)

X D
(concentr.)

Figure 7.13. Block diagram of the distillation column (inputs-outputs)

Henceforward we shall be concerned with the identification of the transfer between the heating power (QB) and the concentration of the top product (XD). The input/output file is named QXD⁹ file. It contains an I/O data set made of 256 samples. The input is a pseudo random binary sequence generated with a register with 8 cells (N = 8, L = 255). Figure 7.14 displays the inputs and outputs of the centered QXD file. The sampling period was 10 s.

S=1 M=1 (RLS) A=1 FILE: QXD NS=256 DELAY D=0 INSTANT K = 256
FORGETTING FACTOR = 1 COEFFICIENTS OF POLYNOMIAL A: A(1) = -0.23644
A(2) = -0.21889 COEFFICIENTS OF POLYNOMIAL B: B(1) = 0.01177
B(2) = 0.07982
B(3) = 0.18311
B(4) = 0.13271
The identification procedure starts after having centered the I/O sequences. As no prior knowledge of the system is available, first we choose structure S1, the recursive least squares method (M1) with a decreasing adaptation gain A1, delay d=0, degree for the polynomial B: n_B = 4 (in order to capture a possible time delay of the system) and degree for the polynomial A: n_A = 2 (since it is a chemical plant). The initial value of the parameters to be estimated is set to zero. The results obtained with this method are summarized below:
Note that B(1) < 0.15 B(2). This shows that we have to set the delay d equal to 1, as a first approximation.
⁹ Available from the book website http://landau-bookic.lag.ensieg.inpg.fr.