Course paper on the course "Management Theory" Completed: art gr. Accepted(a): tashkent-2022 bbedenie

1.2. Stability analysis of a closed system

To analyze the stability of the initial system by the Nyquist criterion, the amplitude-phase characteristic (AFC) of an open system is constructed. The construction of the AFC of an open system is carried out on a computer. For manual AFC calculation, the recommended sequence of actions is as follows:

By setting C with values , an AFC is constructed, and the stability of a closed system is determined from it.

AFH:

Fig. 2. AFC of an open system

The hodograph of an open system covers the point (-1, w0). Therefore, the system in the closed state is not stable.

1.3. Calculation of the linear system correction

Perform correction of the linear system. At the same time, it is necessary to provide the following quality indicators of the regulatory process in the adjusted system:

a) overshoot σ≤20%;
b) the duration of the transition process, not exceeding the value of t_reg=0.35 s;
c) accuracy is 2.5 times higher than that of the unadjusted system.

The desired logarithmic characteristic of an open system is constructed based on the requirements for the designed system: the required gain, the order of system astaticism, the permissible transition time, and the permissible value of overshoot.

The low-frequency part of the amplitude response is determined by the gain of the system in the open state and the order of its astaticism. The frequency interval of the low-frequency part of the characteristic lies in the range from the minimum initial frequencies to the frequency of the first conjugation of the approximated characteristic. The slope of the initial low-frequency segment of the characteristic is equal to -20 db / dec, where is the order of astaticism of the system.
The specified initial segment of the characteristic must pass through a point with an ordinate equal to 20lgK and an abscissa =1, where Kis the system's selected gain, i.e. this part must coincide with the low-frequency part of the original system to simplify correction tools.
The mid-frequency part of the amplitude characteristic is the most essential part of the characteristic, since its type mainly determines the quality of the transient process of the system.
At the cut-off frequency, the slope _cn is determined by the required transition time and the permissible overshoot: , where the coefficient is selected depending on the permissible overshoot value (Fig.3).
The mid-frequency section of the desired LFC is drawn to the left and right until the modules are equal to , . It is also selected и depending on the permissible overshoot (Fig. 3). corresponds _{to 2j}, and corresponds _{to 3j}. In this case, it should be borne in mind that the larger the values of the intervals _2j – _cf.w and _{cf. w} - _3j , the smaller the overshoot. Therefore, the final choice of these intervals should be consistent with the transition requirement.

Fig. 3. Dependence Лof L2of L2 anda0 on.

Fig. 4 .Dependence of stocks of L and ot.

The central LFC segment is paired with the low-frequency part in a straight line with a slope of -40 db / dec -60 db/dec.

When constructing the desired LACH, it is desirable that it differs as little as possible from the original LACH. This is necessary for simplifying corrective measures. When forming the desired LAX, you can increase the reserves modulo, if necessary for matching the asymptotes, and so on. Such an increase will really improve the quality of the system.
Recommended sequence diagram for constructing the desired LFC system:


.

Let's use the nomogram to determine the cutoff frequency for these quality indicators:

; ;

Fig. 5. Nomogram for determining the cutoff frequency of the desired LAX.

Adjusted system gain:

;
Transfer function of an open uncorrected system:
;



Fig. 6. LFC of the initial system.