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Structural matrix schemes and transfer matrices

Lecture



Matrix structural diagrams

Matrix structural schemes are essentially a compact graphical representation of the classical structural scheme of a multidimensional object or control system. They are based on the operator form of representation of equations, on the replacement of real signals by their images according to Laplace.

Using several levels of representation of schemes that differ in the degree of aggregation (association) of links and system elements.

Let us consider different levels of representation of matrix structural schemes on the example of some generalized structure of a multidimensional automatic control system.

I level

In fig. 3 are shown   Structural matrix schemes and transfer matrices - vectors of variable images,

  Structural matrix schemes and transfer matrices

Fig. one

  Structural matrix schemes and transfer matrices ,

  Structural matrix schemes and transfer matrices - transfer function matrices,

  Structural matrix schemes and transfer matrices ,

  Structural matrix schemes and transfer matrices ,

  Structural matrix schemes and transfer matrices .

On the shown in fig. 1 matrix block diagram summing elements, matrix links and branch points perform the same functions as in conventional block diagrams. Therefore, in accordance with the scheme, you can write the system of matrix equations -

  Structural matrix schemes and transfer matrices

Level II

Consider the second level, assuming for the matrix structural scheme   Structural matrix schemes and transfer matrices .

  Structural matrix schemes and transfer matrices

Fig. 2

Consider at this level the opamp described by the matrix   Structural matrix schemes and transfer matrices

  Structural matrix schemes and transfer matrices

(one)

Let's move from matrix equations to scalar -

  Structural matrix schemes and transfer matrices ,

  Structural matrix schemes and transfer matrices

(2)

The last expression shows that the transfer function   Structural matrix schemes and transfer matrices in accordance with the principle of superposition is the transfer function between   Structural matrix schemes and transfer matrices m entrance and   Structural matrix schemes and transfer matrices output, in the absence of signals on all inputs, except   Structural matrix schemes and transfer matrices th

Level III

Consider the third level of matrix structural only for the control object. On the basis of the system of operator equations (2) obtained for the object, it is possible to draw a block diagram of the control object.

  Structural matrix schemes and transfer matrices

Fig. 3

From the proposed example of the complexity levels of the matrix structural scheme, it can be seen that the representation of even not very complex multidimensional control systems in the form of level III schemes, that is, in the form of classical structural schemes, leads to a cumbersome graphical representation that does not reflect the characteristic connections and functional elements of the system.

Transfer matrices

The transfer or equivalent matrixes refer to models of the "input-output" type and represent the matrix connecting the input and output of a multidimensional system. In fig. 4 shows a multi-dimensional system.

  Structural matrix schemes and transfer matrices

Fig. four

The matrix operator equation describing the system has the form -

  Structural matrix schemes and transfer matrices ,

Where   Structural matrix schemes and transfer matrices - the transfer or matrix of the system, the components of which will be the transfer functions connecting the components of the input and output vectors of the system.

Analogues of equivalent matrices in one-dimensional systems are transfer functions that connect the input and output of an object or system. Matrices   Structural matrix schemes and transfer matrices from the above matrix structural scheme (see Fig. 1) are, in fact, the transfer matrices of the multidimensional functional elements of the system.

Equivalent matrices of multidimensional systems can be obtained in two ways.

  1. The transfer functions that connect the respective inputs and outputs of the system are determined. That is, the matrix is ​​determined by its components. Components are determined by known methods in accordance with the principle of superposition.

  2. The transfer matrices are determined as a result of equivalent transformations of the matrix structural schemes or by matrix operator equations. The transformation of the matrix structural schemes is carried out in accordance with the rules of equivalent transformations of conventional structural schemes; it is only necessary to take into account the specifics of operations with vectors and matrices (non-compliance with the commutative law, replacement of division by multiplication by the inverse matrix, the concept of unit and zero matrix, etc.).

As an example, we find the transfer matrix connecting the input and output of the system considered above (Fig. 1), while we believe in accordance with the principle of superposition that the disturbance signal is absent (   Structural matrix schemes and transfer matrices ). Then the structural diagram takes the form shown in Fig. five.

  Structural matrix schemes and transfer matrices

Fig. five

Define   Structural matrix schemes and transfer matrices satisfying the following matrix operator equation -

  Structural matrix schemes and transfer matrices .

For determining   Structural matrix schemes and transfer matrices let us use the transformation of matrix operator equations, which can be written in a matrix structural diagram.

  Structural matrix schemes and transfer matrices

(3)

  Structural matrix schemes and transfer matrices

(four)

  Structural matrix schemes and transfer matrices

(five)

Substitute   Structural matrix schemes and transfer matrices from (4) to (3)

  Structural matrix schemes and transfer matrices

(6)

Substitute   Structural matrix schemes and transfer matrices from (5) to (6)

  Structural matrix schemes and transfer matrices

(7)

Open the brackets in the right part (7)

  Structural matrix schemes and transfer matrices

(eight)

Transfer the item with   Structural matrix schemes and transfer matrices from the right side of expression (8) to the left side

  Structural matrix schemes and transfer matrices

(9)

We will take out   Structural matrix schemes and transfer matrices brackets to the right

  Structural matrix schemes and transfer matrices

(ten)

Where   Structural matrix schemes and transfer matrices - unit matrix   Structural matrix schemes and transfer matrices th order.

If a

  Structural matrix schemes and transfer matrices ,

then the matrix

  Structural matrix schemes and transfer matrices ,

is nondegenerate and can be obtained from the inverse matrix -

  Structural matrix schemes and transfer matrices

(eleven)

Multiply the left and right sides of equation (10) on the right by the inverse matrix (11), after simple transformations we get -

  Structural matrix schemes and transfer matrices

Then we finally get -

  Structural matrix schemes and transfer matrices .

Consequently

  Structural matrix schemes and transfer matrices

(12)

By expression (12), knowing the matrix expressions for the elements of the system, one can always determine the transfer matrix of the system as a whole.

Test questions and tasks

    1. Give the definition of a matrix structural diagram.

    2. In what forms can matrix structural schemes be presented?

    3. Determine the vector   Structural matrix schemes and transfer matrices if vector   Structural matrix schemes and transfer matrices has the form -

  Structural matrix schemes and transfer matrices ,

vectors are related by the equation -

  Structural matrix schemes and transfer matrices ,

Where

  Structural matrix schemes and transfer matrices ,

Answer:

  Structural matrix schemes and transfer matrices .

    1. According to the matrix equation

  Structural matrix schemes and transfer matrices ,

define   Structural matrix schemes and transfer matrices .

Answer:

  Structural matrix schemes and transfer matrices

    1. The control object is described by the transfer matrix -

  Structural matrix schemes and transfer matrices ,

which links vectors -

  Structural matrix schemes and transfer matrices

Draw a block diagram linking the components of the vectors   Structural matrix schemes and transfer matrices .

Answer:

  Structural matrix schemes and transfer matrices

    1. Give the definition of the transfer (equivalent) matrix.

    2. Define the components of the transfer matrix of the control object.

    3. What ways can be determined transfer matrices of multidimensional objects.

    4. For the multidimensional system shown in Fig. 1, determine the transfer matrix connecting the vectors   Structural matrix schemes and transfer matrices and   Structural matrix schemes and transfer matrices .

Answer:

  Structural matrix schemes and transfer matrices .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control