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2. Identification of one-dimensional linear dynamic objects

Widespread identification of linear dynamic objects in the time domain is the method based on the use of the Kalman filter. The implementation of the Kalman filter is found in many foods modeling and identification, including ItACS.
 
Initial data for the Kalman filter is a set of data "time-in-out" a priori information about the structure of the object model (order of the system), and the initial value is determined by the parameters of the dynamic link and the amount of dispersion.
 
Example 1. A linear dynamic object of the second order.
The block diagram of the known model of the object, which is fed to the input of the constant exposure is presented in Figure 1.

Figure 1. Block diagram of an example of the object 1
 
Transient dynamic object is shown in Figure 2 and fully completed in 2 seconds.

Figure 2: Dependency of the dynamic object output signal versus time for Example 1
 
Assuming that the only known dependence U(t) and Y(t), and the procedure for dynamic object (the numerator of the second order, the second order denominator), determine the unknown parameters of the model.
 
To determine the parameters of the Kalman filter is used, with the input and output signals are calculated sensitivity function of the second kind, the covariance matrix is ​​filled and consistently going evaluation of the unknown parameters.
 
The identification process is automated and is implemented in the product ItACS, in which you need to connect a text file containing each line set of "time-in-out" (export dependency U(t) and Y(t) in a single file), and set the order of the dynamic object . Dialog ItACS after performing the given action is shown in Figure 3.
 
Figure 3. The dialog box for ItACS Example 1
 
Thus, when using a Kalman filter in the time domain of the true values ​​of the greatest difference from the estimated amounts of 0.1867% with a standard deviation of the true output signal of the estimated 0.01058.
 
To identify a dynamic object in the frequency domain should have a set of baseline data "frequency, the real part, the imaginary part." Get the data by conducting an experiment on the computer in the product may MathCAD. For this it is necessary to describe the transfer function in the frequency domain and obtain its real Re and imaginary Im portions that display a file. An example of the implementation of this algorithm is shown in Figure 4.
Figure 4. Example of initial data for the product of Example 1 MathCAD
 
Then the program connects IfACS created file is selected order management, given initial approximate settings and identification, the results of which are shown in Figure 5.
 
Figure 5. The dialog box for IfACS Example 1
 
Thus, when using a Kalman filter in the frequency domain the greatest difference from the true value is estimated 0.8191% with a standard deviation of the true output signal of the estimated 0.00054.
 
There is another method of identification in the time domain, implemented in SimACS product. To use it you need to assemble the model with the object under study, set to link the unknown parameters and program them to connect to the model file input and output file, and add the real and the comparator receives the signal and identification unit.
 
To connect to the file SimACS used external impact, the magnitude of which signal corresponds to the input file name and is «$u.d», and the output signal «$y.d». The unknown parameters are recorded as «?P0», «?P1» etc. and programmed, respectively, as «R=P[0];», «R=P[1];» etc. Additional nonlinear function blocks CKO (written in Latin) and CID have a title called «$0» and «$1» respectively. Under the tab "Identification" specifies the number of unknown parameters, equal to five, and is indicated for the parameter identification algorithm, equal to 111. Thereafter, the identification, the results of which are shown in Figure 6.
 

Figure 6. Identification SimACS for Example 1
 
Thus, when used in product identification SimACS time domain true values do not differ from the estimates and the true standard deviation of the output signal of the estimated zero.
 
Example 2. A linear system of second order.
The block diagram of the known model of the object, which is fed to the input of the constant exposure is shown in Figure 7.
Figure 7. Block diagram of the system for Example 2
 
To identify the first dynamic unit should be removed to a separate file with the output of the adder in front of him and out of dynamic management.
 
Then the product is connected ItACS text file containing each line set of "time-in-out" and specifies the order of the transfer function (zero order of the numerator and denominator of the first order). Dialog ItACS after performing the given action is shown in Figure 8.
 
Figure 8. Dialog ItACS for Example 2
 
Thus, when using a Kalman filter in the time domain the greatest difference from the true value is 0.01% estimated with a standard deviation of the true output signal of the estimated 0.09473.
 
Implementation of identification of dynamic objects in the frequency domain is carried out in the product MathCAD and is presented in Figure 9.
Figure 9: Example of initial data for the product of Example 2 MathCAD
 
Then the program connects IfACS created file is selected order management, given initial approximate settings and identification, the results of which are shown in Figure 10.
 
Figure 10. Dialog IfACS for Example 2
 
Thus, when using a Kalman filter in the frequency domain the greatest difference from the true value is estimated 0.1439% with a standard deviation of the true output signal of the estimated 0.00135.
 
Results of the model and the results of identification in the time domain product SimACS shown in Figure 11.
 

Figure 11. Identification SimACS for Example 2
 
Thus, when used in product identification SimACS time domain true values do not differ from the estimates and the true standard deviation of the output signal of the estimated zero.

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