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4. Identification of the multidimensional linear dynamic objects

The identification of multidimensional linear dynamic objects is carried out by determining the transfer functions of their generalized coordinates in conducting several experiments.
 
For generalized coordinates for two-dimensional objects are carried out two experiments in which the block diagram is constant, and input actions are reversed. For convenience, the following performance index calculation input and output signals consist of two numbers: the physical number of the signal and the number of experience.
 
For example, a two-dimensional linear dynamic object for the two experiments is shown in Figure 12.
Figure 12. Block diagram of a two-dimensional object to the two tests
 
For a system block diagram of the recording system of equations for the output signals of two experiments with respect to other input signals or output signals. When writing equations, use the shortest path, moreover if we consider the traffic signal coincides with the direction of the link, the signal is multiplied by the transfer function of the unit, otherwise the signal is divided by the transfer function.
 
Thus, the signal Y11 equals the signal combiner, the input of which is fed a signal U11, passing through link W1, W3 and a signal unit, which operates on the input signal W4, the output of which is a known signal Y21. Then the input signal W4 can be defined as the passage of Y22 through the inverse function W4. As used in the calculation of the links and the signals are presented graphically in Figure 13.

Figure 13. Y11 signal on the block diagram
 
Further, the signal Y21 equals W4 signal level, the input of which is fed a signal from the adder, and an equal amount of U11 with signal level W2, the input signal is equal to Y11. As used in the calculation of the links and the signals are presented graphically in Figure 14.
Figure 14. Production of signal Y21 on the block diagram
 
As a result, the output signals of two experiments will be obtained equations (1):
 (1)

Further, the system of equations for the two experiments are referring to the matrix, with the transfer functions are carried in a separate column (2):
 (2)

After this we must find all expressions disposed in the third matrix. For convenience, all matrix calculations are divided into upper and lower parts, each of which has two lines. To determine the first row of the last matrix is necessary to substitute the top two lines of the first column of the matrix in the upper portion of the second array in the first column, find the resulting matrix the determinant of the second order and divided by the determinant of the matrix, but without substitution column. Thus, the expression for the W1 takes the form (3):
 (3)

Similarly, operations are performed for the second row, wherein the first matrix is substituted only in the second column, and the denominator remains the same as in the first row (4):
 (4)

Lines 3 and 4 are calculated by the bottom of the matrix and are (5) and (6):
 (5)
 (6)

After receiving all four expressions are necessary to obtain the transfer functions W1, W2, W3 and W4. The transfer functions W1 and W4 in the expressions (3) and (5) have already been determined, the transfer function W2 must be obtained from (6) by dividing by W4 (7) and the transfer function W3 can be obtained from (4) by multiplying by W4 (8):
 
 
 
(7)
 
 
 
(8)
 
For generalized coordinates is necessary to put a block diagram of the system blocks are calculated separately obtained by the numerator and denominator of the transfer function.
 
Implementation of generalized coordinates for the transfer function of the product W1 SimACS carried out in the following sequence. On the block diagram is placed non-linear element with the name «CBEPTKA4» (Latin script) and the title name "$0", the number of inputs is extended to four and are connected in series with the circuit signals Y11, Y22, Y12, Y21. To the output of a single non-linear element is connected with the output amplifying unit connected to the chart. The resulting function will implement the numerator W1 (3). Similarly, in the scheme implemented by the denominator of the transfer function, with a link to a non-linear signals are U11, U22, U12, U21, which is connected to the amplifying element with a single output on the same schedule. Transformed block diagram for generalized coordinates for the transfer function W1 is shown in Figure 15.
Figure 15. Block diagram for finding the generalized coordinates link W1
 
The output signals of individual amplifying units added to the numerator and denominator are the input and output of the test unit W1 and applied in product ItACS for unknown parameters W1 link the known input and output signals of all two-dimensional object, instead of the link. The dialog box in ItACS shown in Figure 16.
 
Figure 16. Dialog for ItACS link W1
 
Similar actions are performed for parts of W2 and W4, and to link W3 should use nonlinear links «CBEPTKA8» 8 inputs, in which the daisy U11, Y12, U12, Y11, Y21, Y12, Y22, Y11 for the calculation of the numerator and U11, Y22 , U12, Y21, U21, Y12, U22, Y11 denominator. Block diagram of the resulting system is shown in Figure 17.
Figure 17. Block diagram for finding the generalized coordinates link W3
 
Following the withdrawal of the generalized coordinates of the numerator and denominator W3 held in the product identification ItACS, a dialog box is shown in Figure 18.
 

Figure 18. Dialog for ItACS link W3
 
Identification of multivariate dynamic objects by means of product SimACS.
For identification SimACS you want to display all input U11, U21, U12, U22 and the output Y11, Y21, Y12, Y22 signals into separate files with the same name.
 
Then it is going to a model in which all input signals are connected from the files, and adds the output signals of the files that are compared with the corresponding outputs of the model. The block diagram model SimACS is shown in Figure 19.
Figure 19. Block diagram for identification SimACS
 
On the block diagram of the system is necessary to introduce an additional non-linear element «MAX» (Latin writing) with 4 inputs to select the highest value of the standard deviation of output from the known signals. The title name from the non-linear element is equal to "$1" and the title name from the non-linear element «CID» must be equal to "$2". Thereafter scheme specified unknown coefficients, e.g., «?P0», «?P1» and «?P2», which are programmed as «R=P[0];», «R=P[1];» and «R=P[2];», respectively, with the block diagram of the location of the two tests should be the same. After identification will get the results that are shown in Figure 20.
 
Figure 20. The results of the identification of the two-dimensional object in the product SimACS
 
Пример. Идентификация многомерных систем
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