You get a bonus - 1 coin for daily activity. Now you have 1 coin

20 Root Criteria for Transient Quality

Lecture



This group of criteria is based on an assessment of the quality of transients by the values ​​of the poles and zeros of the transfer function of the system between the inputs and outputs of interest to us.

As is known, the transient response of a system can be defined as follows: -

  20 Root Criteria for Transient Quality

(one)

Where   20 Root Criteria for Transient Quality - the roots of the characteristic equation of the system

  20 Root Criteria for Transient Quality .

It is obvious that the character of the transition process is also affected by the numerator   20 Root Criteria for Transient Quality and denominator   20 Root Criteria for Transient Quality transfer function. But, in most cases, when analyzing systems by response to a control action,   20 Root Criteria for Transient Quality has no roots, that is, the transfer function has no zeros. Then the nature of the transition process can be assessed only by the poles of the transfer function, thereby exposing the analysis of the roots of the characteristic equation of the system -

  20 Root Criteria for Transient Quality

(2)

In the case of an approximate quality assessment based on the roots of the characteristic equation on the complex plane, the area of ​​location of the roots is distinguished, the boundaries of which are set according to the quality requirements of the processes, as shown in Fig. one.

  20 Root Criteria for Transient Quality

Fig. one

Borders of the area shown in fig. 1, are set by the following parameters:

    •   20 Root Criteria for Transient Quality - criterion for the duration of the transition process,

    •   20 Root Criteria for Transient Quality - oscillation of the transition process, is determined by   20 Root Criteria for Transient Quality ,

    •   20 Root Criteria for Transient Quality - maximum root removal from the imaginary axis.

Consider these options.

Duration criterion   20 Root Criteria for Transient Quality defined as the distance from the imaginary axis to the nearest real root or the nearest pair of complex conjugate roots.

Find out whether this parameter characterizes the duration of the transition process? There are two cases of the location of the roots on the border area.

      1. Let the closest to the imaginary axis, that is, lying on the border of the region, be the real root -

  20 Root Criteria for Transient Quality ,

then the corresponding component of the transition process, in accordance with (1) will be -

  20 Root Criteria for Transient Quality

(3)

Where   20 Root Criteria for Transient Quality - decomposition coefficient (1).

      1. If the complex-conjugate pair of roots is closest to the imaginary axis,

  20 Root Criteria for Transient Quality ,

then the corresponding component of the transition process, in accordance with (1) will be -

  20 Root Criteria for Transient Quality

(four)

Where   20 Root Criteria for Transient Quality - oscillation frequency.

From (3) and (4) we see that the time attenuation components   20 Root Criteria for Transient Quality determines the factor -

  20 Root Criteria for Transient Quality ,

Where   20 Root Criteria for Transient Quality - the value of the minimum real root or minimum real part of the roots,   20 Root Criteria for Transient Quality - corresponding   20 Root Criteria for Transient Quality , the greatest time constant. Thus, we can assume that the transition process of the system is completed not earlier than the component   20 Root Criteria for Transient Quality . Consequently,   20 Root Criteria for Transient Quality determines the duration of the transition process, being a quantity inversely proportional to the time of regulation. Knowing   20 Root Criteria for Transient Quality , we can estimate the time of regulation or transition process by the following relation -

  20 Root Criteria for Transient Quality ,

Where   20 Root Criteria for Transient Quality - half the width of the area, when hit in which the transition process is complete. If a   20 Root Criteria for Transient Quality , and the extreme root is real, then we have -

  20 Root Criteria for Transient Quality .

Oscillation criterion   20 Root Criteria for Transient Quality determined by angle   20 Root Criteria for Transient Quality in the following way -

  20 Root Criteria for Transient Quality .

Where   20 Root Criteria for Transient Quality - respectively, the real and imaginary parts of the complex conjugate pair of roots located on the border of the region (see Fig. 1). By increasing   20 Root Criteria for Transient Quality the oscillation of the system increases.

Farther from the imaginary axis, the boundary of the region   20 Root Criteria for Transient Quality , determine the roots that have an extremely small impact on the transition process.

Other things being equal, the system requires an increase   20 Root Criteria for Transient Quality and declines   20 Root Criteria for Transient Quality .

As an example of the influence of the location of the roots on the nature of the transient processes, we will show the graphs presented in Fig. 2 and 3.

  20 Root Criteria for Transient Quality Fig. 2

  20 Root Criteria for Transient Quality Fig. 3

If the transfer function of the system has zeros, then the system quality assessment only by poles can give a significant error.

To clarify the nature of the influence of zeros on the quality of transient processes, let us present the system as follows, as shown in Fig. four.

  20 Root Criteria for Transient Quality

Fig. four

We specify the problem, let

  20 Root Criteria for Transient Quality ,

but   20 Root Criteria for Transient Quality has the form shown in fig. 5. In this case, we consider two options for the schedule:

  •   20 Root Criteria for Transient Quality ,

  •   20 Root Criteria for Transient Quality .

From consideration of fig. 5 it can be concluded that members   20 Root Criteria for Transient Quality with positive coefficients lead to increased oscillation and speed, and negative coefficients delay the transition process.

  20 Root Criteria for Transient Quality

Fig. five

In those cases when it is required to obtain the desired type of transition process, methods are used based on the relationship of the coefficients of the characteristic equation of the system or its roots with the type of the transition process with given dynamic indicators.

Consider the characteristic equation of the form -

  20 Root Criteria for Transient Quality

(five)

Convert (5)

  20 Root Criteria for Transient Quality

(6)

According to the formula Vieta   20 Root Criteria for Transient Quality defined as the sum of all the roots of the equation,   20 Root Criteria for Transient Quality - the sum of the products of all pairs of roots,   20 Root Criteria for Transient Quality - the sum of the products of all triples of roots, etc., and   20 Root Criteria for Transient Quality is defined as the product of all the roots of the equation -

  20 Root Criteria for Transient Quality .

Now, if we can specify the location of the roots on the complex plane, based on the requirements of the quality of the dynamics, then using the Vieta formulas, we can find the values ​​of the coefficients of the characteristic equation that are related to the parameters of the system.

Pay special attention to the coefficient   20 Root Criteria for Transient Quality the more   20 Root Criteria for Transient Quality then, ceteris paribus, more real parts of the roots, therefore, faster transient. If the roots are real and multiple, then -

  20 Root Criteria for Transient Quality .

Denote

  20 Root Criteria for Transient Quality

(7)

Where   20 Root Criteria for Transient Quality is called the geometric mean root of the characteristic equation.

Then equation (6) in view of (7) has the form -

  20 Root Criteria for Transient Quality

On the complex area of ​​the roots of the characteristic equation   20 Root Criteria for Transient Quality determines the point on the real axis - the geometric center of all the roots of the system, and the coefficients   20 Root Criteria for Transient Quality determine the relative position of the roots. It is easy to show that   20 Root Criteria for Transient Quality determine the transition curve in relative time   20 Root Criteria for Transient Quality and value   20 Root Criteria for Transient Quality determines the time scale for this process.

In practice, the above approach is used as follows:

    1. For a particular system, determine the desired type of transition.

    2. To ensure the specified requirements, the previously calculated coefficients of the characteristic equation are selected from the reference literature, thereby selecting the “desired” characteristic equation -

  20 Root Criteria for Transient Quality

(eight)

    1. Determine the characteristic equation for the structure and parameters of the system -

  20 Root Criteria for Transient Quality

(9)

Where   20 Root Criteria for Transient Quality - coefficients functionally related to system parameters.

    1. Get the system of algebraic equations, equating the coefficients of equations (8) and (9) with the same powers of the Laplace operator   20 Root Criteria for Transient Quality -

  20 Root Criteria for Transient Quality

(ten)

  1. The system (10) is solved with respect to the variable parameters of the system (parameters of the regulators), which makes it possible to determine the parameters providing the specified look and quality of the transition process.

The algorithm described above is often called the method of standard coefficients or the standard arrangement of the roots of the characteristic equation of the control system. Consider as an illustration two standard root arrangements that are most common in control systems for electromechanical drives of various installations.

Binomial root distribution

Binomial root distribution is used to ensure a given speed with the monotony of transients. The standard binomial characteristic equation has the form -

  20 Root Criteria for Transient Quality

In this case, we have   20 Root Criteria for Transient Quality multiple real roots with a negative real part equal to   20 Root Criteria for Transient Quality . View transients for   20 Root Criteria for Transient Quality from 1 to 4 is shown in fig. 6. The characteristic equations for these cases are of the form -

  20 Root Criteria for Transient Quality

Butterworth distribution

Correct is the comparison of the automatic control system and the ideal low-pass filter (LPF), when for the system bandwidth (LF) a maximum horizontal position of LAFC is required, which ensures transmission of control signals without distortion. For the high frequency range (HF), they require maximum signal rejection, since this is a range of interference signals. Fig. 7 illustrates the approximation of the desired system characteristic to the characteristic of an “ideal” low-pass filter.

The distribution of Butterworth roots provides a compromise between these requirements, achieving a high uniformity in the LF passband with an acceptable slope of the characteristic in the HF suppression band.

  20 Root Criteria for Transient Quality

Fig. 6

  20 Root Criteria for Transient Quality

Fig. 7

The roots of the characteristic equation are located on the complex plane, on a circle with a radius   20 Root Criteria for Transient Quality and the angular distance between the roots -   20 Root Criteria for Transient Quality , symmetrically about the real axis, as shown in Fig. eight.

  20 Root Criteria for Transient Quality

Fig. eight

View transients for   20 Root Criteria for Transient Quality from 1 to 4 is shown in fig. 9.

  20 Root Criteria for Transient Quality

Fig. 9

The characteristic equations and parameters of the transition process for these cases have the form -

  20 Root Criteria for Transient Quality

Comparison of transient characteristics shows that the Butterworth distribution provides a higher response than the binomial distribution, with low overshoot and oscillatory response.

Test questions and tasks

    1. How to explain the effect on transient roots of the characteristic equation?

    2. What component of the transition process gives the negative real root of the characteristic equation?

    3. What components of the transition process give the complex conjugate roots of the characteristic equation?

    4. What determines the roots of the characteristic equation closest to the imaginary axis of the complex plane?

    5. How is the mean geometric root of the characteristic equation related to the system speed?

    6. What effect does the transfer function zero have on the transient?

    7. When should I use the binomial distribution of the roots of the characteristic equation for system settings?

    8. When should I use the distribution of the roots of the characteristic Butterworth equation on the system settings?

    9. Determine the coefficients of the characteristic equation with a binomial root distribution for a third-order control system, if the required regulation time   20 Root Criteria for Transient Quality .

Answer :

The desired characteristic equation has the form -

  20 Root Criteria for Transient Quality .

    1. Determine the coefficients of the characteristic equation with the Butterworth root distribution for a fourth-order control system, if the required regulation time   20 Root Criteria for Transient Quality .

Answer :

The desired characteristic equation has the form -

  20 Root Criteria for Transient Quality .


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control