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Preparation of equations of automatic control system (ACS)

Lecture



Formulas used in solving

The transfer function of the control for an open system:

  Preparation of equations of automatic control system (ACS) (3.1)

Disturbing transfer function for an open-loop system:

  Preparation of equations of automatic control system (ACS) (3.2)

The transfer function of the control for a closed system:

  Preparation of equations of automatic control system (ACS) (3.3)

The transfer function of the disturbing effects for a closed system:

  Preparation of equations of automatic control system (ACS) (3.4)

The transfer function of the control for the error:

  Preparation of equations of automatic control system (ACS) (3.5)

Disturbance transfer function for error:

  Preparation of equations of automatic control system (ACS) (3.6)

A (p) - Polynomial open automatic system

B (p) - Polynomial control action of an open automatic system

C (p) - Polynomial disturbing effect of an open automatic system

  Preparation of equations of automatic control system (ACS) (3.7)

D ( p ) is the characteristic polynomial of a closed automatic system.

The equations for an open automatic system, for a closed automatic system and for an error:

  Preparation of equations of automatic control system (ACS) (3.8)

  Preparation of equations of automatic control system (ACS) (3.9)

  Preparation of equations of automatic control system (ACS) (3.10)

The characteristic polynomials of the open A (p) and closed D ( p ) systems, as well as the roots of the corresponding characteristic equations, play an important role in the study of the dynamic properties of speakers.

Transfer functions   Preparation of equations of automatic control system (ACS) they are called the main or main operators of the automatic system, since they completely determine the dynamic and accuracy characteristics of the system.

Transfer functions   Preparation of equations of automatic control system (ACS) They are not the main operators, as they depend on the point of application of the disturbing action.

In practical studies of the automatic system, you need to know the links between the main operators:

  Preparation of equations of automatic control system (ACS) (3.11)

  Preparation of equations of automatic control system (ACS) (3.12)

  Preparation of equations of automatic control system (ACS) (3.13)

Thus, it is enough to know one of the main operators and the transfer function of feedback   Preparation of equations of automatic control system (ACS) to define all the other main operators.

Examples of problem solving

Example 1. Make an equation of the automatic control system, the scheme of which is shown in Figure 3.1:

  Preparation of equations of automatic control system (ACS)

Fig. 3.1. Structural scheme

Decision:

Determine the transfer function W (p) of an open-loop system for a given influence and the transfer function of the system W f (p) for a disturbing effect:

  Preparation of equations of automatic control system (ACS)

Define the transfer function R (p) of the closed-loop system for the control action and the transfer function R f (p) of the closed-loop system for the disturbing influence:

  Preparation of equations of automatic control system (ACS)

  Preparation of equations of automatic control system (ACS)

Define the transfer function for the error on the main impact of E (p) and the transfer function of the error on the disturbing influence of E f (p):

  Preparation of equations of automatic control system (ACS)

Knowing the transfer functions, you can write any equation of the system:

a) Equation of a closed automatic system:

  Preparation of equations of automatic control system (ACS)

b) Equation of an open automatic system:

  Preparation of equations of automatic control system (ACS)

c) Error equation:

  Preparation of equations of automatic control system (ACS) .

The well-known analogy between transfer functions and transfer operators allows one to compose equations of automatic systems not only in Laplace images, but also differential equations.


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control