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Rules for the conversion of structural schemes. negative connections. problem solving examples

Lecture



A structural diagram of any complexity can be reduced to an equivalent single-loop (having one main feedback) or to an equivalent open-loop block diagram.

If there are a large number of elements in the system, a structural method is used, according to which the equation of the entire system is obtained on the basis of an algorithmic scheme using several simple rules.

The essence of the method is that by the known transfer functions of the individual elements of the system, which are the detecting links of unidirectional action, using the rules of the series, parallel and counter-parallel (with feedback) links of the links, they receive an equivalent transfer function or transfer function of the equivalent link. The result is a single-circuit closed chain consisting of several equivalent links.

Moreover, this chain can have multiple inputs and multiple outputs. Further, according to the single-circuit scheme, transfer functions between the open inputs and outputs of the system are recorded.

Structural transformations of the circuits are valid if they keep all the input and output signals of the system unchanged.

Schema conversion rules

  1. If the node is transferred against the direction of the signal, then the portable branch should include elements with those transfer functions that are encountered along the path of the old and new points of the node's location.
  2. If the node is transferred in the direction of the signal, then the portable branch should include elements with transfer functions inverse to the transfer functions of elements that occur along the path between the new and the old points of the node location.

Structural transformations of the circuits are valid if they keep all the input and output signals of the system unchanged.

The possibility of structural transformations of systems is provided by the principle of superposition and reversibility of signals:

Rules for the conversion of structural schemes.  negative connections.  problem solving examples (2.1) (2.1)

Rule 1: Permutation of the same type of elements (Table 2.1)

Table 2.1

Rearrange the node (adder or links)

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Neighboring elements of the same type can be rearranged in places without changing anything in the scheme itself.

Rule 2: Transfer node from the output to the input of the adder (through the adder against the signal). Such a transfer is accompanied by duplication of the element (adder or link) (Fig. 2.1).

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.1. Transfer node:

and - the transformed scheme; b - transformed circuit

Rule 3: Transferring a node through a link against the signal travel (Fig. 2.2).

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.2. Transfer node through link:

and - the transformed scheme; b - transformed circuit

Rule 4: Transfer node and link through the adder. With such a transfer, it is necessary to install a subtractive device in the node branch (Fig. 2.3).

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.3. Moving a node through the adder in the direction of the signal:

and - the transformed scheme; b - transformed circuit

Rule 5: Transfer the link through the adder in the direction of the signal (Fig. 2.4).

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.4. Link transfer through adder:

and - the transformed scheme; b - transformed circuit

In this case, a dynamic link is added to the branch:

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Rule 6: Transfer node through the link in the direction of the signal. With this transfer, an additional link with the reverse transfer function must be installed in the node branch (Fig. 2.5)

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.5. Transfer node through link:

and - the transformed scheme; b - transformed circuit

Rule 7: Transfer the link through the adder against the signal. In this case, the link with the transfer function must be added to the portable branch (Fig. 2.6).

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.6. Link transfer through adder:

and - the transformed scheme; b - transformed circuit

Block diagram of a closed two-loop system with negative feedback links (without cross-links)

Block diagram of a closed two-loop system with negative feedback links (without cross-links)

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.7. Structural scheme

Formulas for head-to-side connection can be generalized for an arbitrary number of contours nested into one another without cross-feedbacks.

For the structural scheme presented in Fig. 2.7, the transfer function of the system is:

Rules for the conversion of structural schemes.  negative connections.  problem solving examples (2.2)

Rules for the conversion of structural schemes.  negative connections.  problem solving examples - transfer function of the internal circuit of the system;

Rules for the conversion of structural schemes.  negative connections.  problem solving examples - transfer function of the external circuit.

The transfer function of a closed multi-loop system without cross feedbacks is equal to the ratio of the transfer function of the open loop system to the algebraic sum of the unit and the transfer functions of the elements of nested contours.

Examples of problem solving

Example 1. Simplifying the scheme shown in Fig. 2.8, determine which elementary dynamic link corresponds to the transfer function of the system.

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.8. Structural scheme

Decision:

  1. This block diagram shows a multi-loop closed automatic system with cross-links. To determine the transfer function, it is necessary to convert the block diagram presented in Fig. 2.8, using schema conversion rules.
  2. We get rid of cross-connections by swapping adders and nodes (Fig. 2.9)

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.9. Structural scheme

We receive the block diagram presented on fig. 2.10:

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.10. Structural scheme

3. Having added two signals going to adder 4 and two signals going to adder 5 (fig. 2.10), we get the following block diagram (fig. 2.11):

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.11. Structural scheme

Parallel connection of links with transfer functions w (p) = 4 / p and w (p) = 3 gives us a link with transfer function w (p) = 4 / p + 3.

The parallel connection of links with transfer functions w (p) = 0.1p and w (p) = 0.2 gives us a link with transfer function w (p) = 0.1p + 0.2.

4. Parallel connection of links in adder 3 will be equal to w (p) = 4 + 4p (Fig.2.12.)

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

Fig. 2.12. Structural scheme

The transfer function of two links connected in anti-parallel with negative feedback, using the formula (1.19), can be represented as the following expression: Rules for the conversion of structural schemes.  negative connections.  problem solving examples .

We write the transfer function of the entire system, presented in Figure 2.12

Rules for the conversion of structural schemes.  negative connections.  problem solving examples

This transfer function corresponds to the aperiodic link. Rules for the conversion of structural schemes.  negative connections.  problem solving examples with parameters k = 5, T = 3s.


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control