Topic: elementary dynamic links

SUMMARY OF LECTURE № 9

1. GENERAL PROVISIONS

Logarithmic characteristics of the open-loop systems are used in the synthesis of automatic control systems. The transfer function of the open-loop system is presented as a set of elementary dynamic links.

Classification of links is produced, by the form of the transfer function. For this, consider open-loop system of automatic control. We will use early adopted signs/

Here: – input value; – output value; – disturbance action.

Are known to the properties of any of the automatic control can be described using the static and dynamic characteristics.

Recall that the static characteristics of real objects are nonlinear.

In general, they have the form:

Fig. 9.1 – Static characteristics

To dynamic characteristics they include time and frequency characteristics.

а) reaction to a stepped controlling action

b) reaction to a stepped disturbance action

c) reaction to a impulse controlling action

d) reaction to impulse disturbance action

Fig. 9.2 – Time domain characteristics

Fig. 9.3 – Frequency response

Nonlinear characteristics are linearized. In this case, they use linearization attributes:

1. Accuracy – .

2. The operating point.

3. Range of the input signal variation .

4. Range of the output signal variation .

5. The structure of the equation.

6. Evaluative values of the coefficients –

7. The time scale – .

The structure of the linear mathematical model in a general form is as follows:

Important note: the initial conditions are nonzero:

Make changes, move on to the operator equation:

Introduce the signs:
;

;

Given this, we get:

Therefore:

Redenote:

и

We expand polynomials, using the theorem of Bezout (remember it from the school course!), get:

Here:

– zeros of a polynomial ;

– polynomial’s pole ;

– zeros of a polynomial ;

Let , then

In general, the transfer function can be represented as follows:

;

– the transient coefficient; – time constants; – degree of damping.

Using this expression, write down the transfer functions of the elementary links:

- lagless links;

– force links of the 1st order;

– force links of the 2nd order;

– integrating links;

– differentiator;

– aperiodic links;

– oscillatory links.

2. CALCULATION CHARACTERISTICS OF THE ELEMENTARY DYNAMIC LINKS

Consider the lagless element:

On the example of the two-stroke potentiometric sensor we design the calculation characteristics of the lagless element.

Fig. 9.4 – Scheme of a two-stroke potentiometric sensor

Let , then

Using this information, we can construct a static, time and frequency responses.

Fig. 9.5 – Static characteristic

Note: the model is fundamentally poorer then original!

а) reaction to a stepped b) reaction to impulse c) Nyquist diagram d) AFC

controlling action disturbnse action

Fig. 9.6 – Dynamic characteristics

Please, try to set the calculated static and dynamic

characteristics for the rest of the transfer functions of the elementary

links!

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