Example

 Closed Loop Identification Module - Examples

The disk provided with CLID contains a  file democlid.m. This file corresponds to a simulation example in closed  loop. The data have been generated  as follows :

The plant model is  :

A (q-1) y (t) = B (q-1) u (t) + C (q-1) e (t),

with

A = 1 - 1.5 q-1 + 0.7 q-2

B (q-1) = q-1 + 0.5 q-2,

C (q-1) of degree 2,

e (t) white noise.

The controller is stored in the file democlid.m. It has the following coefficients:

        R (q-1) = 0.8659 - 1.2763 q-1 + 0.5204 q-2

        S (q-1) = 1 - 0.6283 q-1 - 0.3717 q-2

        T (q-1) = 0.11

The external excitation signal is a PRBS applied on the reference. The PRBS is generated by a shift register with 7 cells and a frequency divider of 2.

Use of Closed-Loop Output Error Algorithms

Once you have run democlid.m in Matlab window, you can identify a plant model using four different algorithms as follows :

 

[B1,A1]=cloe(y,r,2,2,0,R,S,T,Fin,lam1,lam0)

B1 = 0    0.9527    0.4900

A1 = 1.0000   -1.4808    0.6716

For fcloe algorithm we can use the identified model by cloe algorithm as an initial model.

[B2,A2]=fcloe(y,r,2,2,0,R,S,T,B1,A1,Fin,lam1,lam0)

B2 = 0    0.9532    0.5074

A2 = 1.0000   -1.4750    0.6744

Using afcloe algorithm we obtaine :

[B3,A3]=afcloe(y,r,2,2,0,R,S,T,Fin,lam1,lam0)

B3 = 0    0.9684    0.4722

A3 = 1.0000   -1.4844    0.6821

Using xcloe algorithm we identify an ARMAX model for noise, so we need to choose the order of  the H polynomial (nh=4).

[B4,A4,H4] =xcloe(y,r,2,2,4,0,R,S,T,Fin,lam1,lam0)

B4 =  0    0.9461    0.4856

A4 =  1.0000   -1.4524    0.6465

H4 =  1.0000    1.1532   -0.6807   -0.5082   -0.1238

The following table summarizes the coefficients of the various identified models as well as those of the simulated model.

Method

A(1)

A(2)

B(1)

B(2)

cloe

-1.4808

0.6716

0.9527

0.4900

fcloe

-1.4750

0.6744

0.9532

0.5074

afcloe

-1.4844

0.6821

0.9684

0.4722

xcloe

-1.4524

0.6465

0.9461

0.4856

Simulated model

-1.5

0.7

1

0.5

 

 

 Validation of the Models identified in Closed-Loop

Statistical Test 

The results of the statistical validation test for the model identified using theclosed loop error with adaptive filter are given below. (for details see chapter 3 of this manual section clvalid function).

[yhat,lossf,Pcal,Piden]=clvalid(B3,A3,R,S,T,y,r,1);

 

r0=3.7055 e-004

rni= 0.0042 0.0061 0.0125 0.0169 0.0150

 

It can be observed (see also the figure) that all the cross-corelations are below  the validation level for 97% (2.136/(N)1/2  - where N is the number of data)

Poles Closeness Test

The computed closed loop poles based on the identified model are :

Pcal =   1.0000  -1.2742  0.4160  0.0245  -0.0078   

 

The identified closed loop poles are :

Piden =  1.0000   -1.3773    0.6312   -0.1607    0.0589

As it can be observed in the next figure, the computed poles and the identified poles are very close.