HOME WORK I:

SYSTEM IDENTIFICATION ASSIGNMENT

T.Arriessa Sukhairi

(108013235855)

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Problem: A cantilever beam is given but without knowledge of its properties (i.e. dimensions,

mass, flexural rigidity (EI), etc.). One is interested to do frequency analysis for this cantilever

beam using MATLAB\ System Identification Toolbox. In order to do so, a number of steps have

to be performed.

Step 1 - Data measurement

The beam is attached with 4 piezoelectric patches to act as actuators. Experimental equipments

are arranged as in Figure 1. Data acquisition equipment (DAQ) feeds 4 band-limited white

noise excitation signals to the 4 actuators on the beam through a power amplifier. Vibration

near tip of the beam is recorded by a velocity sensor (vibrometer).

Figure 1. Experiment setup for data collection

Plant: Single-sided clamped aluminum beam with 4 piezoelectric patches and a laser

vibrometer (MISO system).

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Plant-input: voltage signals to the four piezoelectric patches u[k].

Plant-output: deflection velocity near to the end point y[k].

Measurement was already done in 10 seconds with sampling period Ts = 0.001s. Four input

signals and one output signal are stored in Data_samplingfreq_1000_hz.mat file

which can be downloaded from blackboard in the directory ‘Kursunterlagen\Sytem

Identification_matlab files.zip’.

Step 2 - Model Identification: Indentify a linear state-space model for the plant using

the subspace state-space system identification method N4SID in MATLAB\ System

Identification Toolbox (using GUI or command lines). Perform this step with various

model orders, e.g. 8, 10, 12, 14, etc. Describe the identification process.

By using the system identification tool in MATLAB we process our data from

Data_samplingfreq_1000_hz.mat.

Figure 2. System Identification Toolbox

Then we input our data in left side of the toolbox by choosing ‘time domain data’

and then insert the value U from the Data_samplingfreq_1000_hz.mat file

as input variable and Y from the Data_samplingfreq_1000_hz.mat file as

output variable. Set the sampling interval as 0.001 s

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Figure 3. Import Data

Then we will get the data as shown in the figure 4. Select the range in which we

are going to do the analysis. Here I choose the range between ± 3 – 7.

Figure 4. Input and Output Signals

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Figure 5. Range ± 3 – 7

Figure 6. Combination the Input and Output and Range ± 3 – 7

Step 3 - Model Verification: Compare the simulated output using the models obtained from

task 2 with the measured output (use the command compare if you do system identification by

command lines). Comment on the percentage of fit when model order is increased and choose

simplest model that fits the measured data best.

For this task, we need to compare the percentage result that we get by increasing or

decreasing the model order, to find and choose the simplest model that fits the measured

data best.

We can change the value in Specify value to get the state space models (mathematical

model) that we are going to compare with the model from the experiment.

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Figure 7. Choosing Model Order

Here I choose the value 15, 18, 20, 19, 22.

Then we compare the value as can be seen in the figure 8

Figure 8. Model Output

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The percentage that I get :

15 (SS2) 18 (SS3) 20 (SS4) 19 (SS5) 22 (SS6)

91.07 % 92.51 % 91.35 % 94.16 % 90.26 %

Table 1. The percentage from comparing the simulated output using the models obtained from task 2

with the measured output

The best percentage that I get is when I specify the value as 19.

Figure 9. The Percentage at Specify Value 19

From the table 1 can be seen that the percentage value will increase until it reach the

peak (in this case is 19) then it will decrease.

Step 4 - Frequency analysis: From the best model selected in step 3, plot the frequency

response of the plant and record the first 4 resonant frequencies of the beam.

Plot the frequency response

Figure 10. Frequency Response

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From the plot of the frequency response we get the beam resonant frequencies

1 2 3 4

84.7881 rad/s 455.8061 rad/s 1242.594 rad/s 2465,7624 rad/s

Table 2. Resonant Frequencies

Figure 11. 1st Resonant Frequency

Step 5 – Use the identified model: From the identified state-space model of the beam, assign

those system properties to the given Simulink model ‘state_space_model_beam.mdl’ and plot

the model time response when the input for all 4 channels excitation is of the form

u(t) = 10*sin(2πf1t)

where f1 is the first resonant frequency.

After we get the best fit model for our system, then send the data into workspace. By using the

data that we have, then we store the value a, b, c, and k that we have from the data to create

variable A, B, C, and K in which will be used in Simulink. Then change the value f1 from the

formula with the value of 1 in this case 84.7881 rad/s in source block parameter (see figure

12). Then do the simulation and we can see the velocity as been shown in figure 13.

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Figure 12. Simulink Model

Figure 12. Source Block Parameter Input

Figure 13.Output Velocity