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18 Calculation of linear integral estimates

Lecture



Consider the problem of computing the integral of a linear integral estimate. You can first solve analytically differential equations describing the system, define the control error further, then substitute the expression for the error into the linear estimate integral and, taking it, obtain the expression for   18 Calculation of linear integral estimates .

But you can do otherwise.

Let the free motion of the system regulation error be described by the equation

  18 Calculation of linear integral estimates

(one)

Integrating this equation -

  18 Calculation of linear integral estimates

After integration we get -

  18 Calculation of linear integral estimates

(2)

Upper limit substitutions are given by members of the following form -

  18 Calculation of linear integral estimates

(3)

since all derived errors in steady state go to zero.

Substitution of the lower limit is given by members of the form -

  18 Calculation of linear integral estimates

(four)

which are the initial conditions of equation (1).

Substituting (3) and (4) in (2), we get

  18 Calculation of linear integral estimates

(five)

And since

  18 Calculation of linear integral estimates ,

finally get

  18 Calculation of linear integral estimates

(6)

Solving (6) relatively   18 Calculation of linear integral estimates , we obtain an expression for calculating the linear integral error -

  18 Calculation of linear integral estimates

(7)

Now we can define   18 Calculation of linear integral estimates on the coefficients of the characteristic equation of the system and the initial conditions of the transition process error.

For the synthesis of systems, determining the parameters of minimizing   18 Calculation of linear integral estimates , you should use the usual methods of research functions on the extremum. Therefore, if we want to define a system parameter, for example, the parameter   18 Calculation of linear integral estimates providing   18 Calculation of linear integral estimates , need to decide on the parameter   18 Calculation of linear integral estimates The following equation is

  18 Calculation of linear integral estimates .

Consider a few examples of using linear integral estimation.

Example

The system has a characteristic equation

  18 Calculation of linear integral estimates

(eight)

Define an expression for   18 Calculation of linear integral estimates if the initial conditions are of the form -

  18 Calculation of linear integral estimates .

Determine the value of the parameter   18 Calculation of linear integral estimates at which the integral estimate has a minimum.

Decision

Denote -

  18 Calculation of linear integral estimates .

Use to find   18 Calculation of linear integral estimates expression (7) -

  18 Calculation of linear integral estimates

(9)

From consideration (9) we obtain that   18 Calculation of linear integral estimates in this case it does not have an extremum, and we will receive a smaller value of the integral error with a smaller value   18 Calculation of linear integral estimates . Indeed, because equation (8) is the characteristic equation of an aperiodic link, the parameter   18 Calculation of linear integral estimates - this is the time constant. The transition process for two different time constants will have the form shown in Fig. one.

  18 Calculation of linear integral estimates

Fig. one

Example

The system has a characteristic equation

  18 Calculation of linear integral estimates .

Define an expression for   18 Calculation of linear integral estimates if the initial conditions are of the form -

  18 Calculation of linear integral estimates .

Determine the value of the parameter   18 Calculation of linear integral estimates at which the integral estimate has a minimum.

Decision

Denote -

  18 Calculation of linear integral estimates .

Use to find   18 Calculation of linear integral estimates expression (7) -

  18 Calculation of linear integral estimates .

If a   18 Calculation of linear integral estimates then monotonous processes   18 Calculation of linear integral estimates provided with the smallest   18 Calculation of linear integral estimates and   18 Calculation of linear integral estimates . If a   18 Calculation of linear integral estimates then a decrease in the attenuation coefficient decreases the linear integral estimate, but this leads to a deterioration of the transition process, an increase in its oscillatory behavior.

With oscillatory processes in systems, the linear integral estimate gives a significant error. At the same time, the minimum estimate can correspond to a process with a large number of oscillations with a significant amplitude, low speed, since, in fact, the evaluation involves the addition of positive and negative areas of the area under the integral curve. This is illustrated in Fig. 2 and 3, showing two processes that may have the same value of a linear integral estimate.

  18 Calculation of linear integral estimates

Fig. 2

  18 Calculation of linear integral estimates

Fig. 3

And since the form of the transition process in the analysis of the automatic control system is often unknown in advance, it is impractical to use linear integral estimates in practice.

You can try to use the integral from the error module of the following form -

  18 Calculation of linear integral estimates

(ten)

In fig. 4 shows an exemplary view of the error variation curves and its modulus. But the analytical calculation of the integral from the error module according to the mathematical model of the system turned out to be very cumbersome, therefore this estimate was not widely used.

  18 Calculation of linear integral estimates

Fig. four

Quadratic integral estimate

In most cases, with the possibility of an oscillatory transition process occurring in the system, a quadratic integral estimate is used, which has the following form -

  18 Calculation of linear integral estimates

(eleven)

Evaluation   18 Calculation of linear integral estimates does not depend on the sign of deviations of the error, and hence on the form of the transition process, it will be monotonous, aperiodic or oscillatory in nature. In fig. 5 and 6, an exemplary view of the error variation curves and the square error is shown.

  18 Calculation of linear integral estimates

Fig. five

  18 Calculation of linear integral estimates

Fig. 6

Consider the procedure for calculating the quadratic estimate of the mathematical model of the system. The control system is represented as shown in Fig. 7

  18 Calculation of linear integral estimates

Fig. 7

The image of the Laplace signal at the output of the system has the form -

  18 Calculation of linear integral estimates

(12)

Where   18 Calculation of linear integral estimates - Laplace image of a single step function - system input signal.

For an automatic control system, the mathematical model of which is reduced to the form (12), the integral quadratic error is determined by the following expression -

  18 Calculation of linear integral estimates

(13)

Where

  18 Calculation of linear integral estimates

(14)

at   18 Calculation of linear integral estimates all items with indices less than 0 and more   18 Calculation of linear integral estimates replaced by 0.

Determinants   18 Calculation of linear integral estimates in (13) where   18 Calculation of linear integral estimates are obtained by replacing in the determinant   18 Calculation of linear integral estimates (14) (   18 Calculation of linear integral estimates ) th column of the following column -

  18 Calculation of linear integral estimates .

Coefficients   18 Calculation of linear integral estimates in expression (13) are defined as follows:

  18 Calculation of linear integral estimates

(15)

in determining   18 Calculation of linear integral estimates coefficients whose indices are less than 0 and more   18 Calculation of linear integral estimates replaced by 0.

Test questions and tasks

    1. What parameters of the mathematical model of the object are required to calculate the linear integral evaluation?

    2. Why is it impossible to use a linear integral estimate in the case of the oscillatory nature of transients?

    3. What integral estimates should be used in the event that oscillatory transients are possible in the system?

    4. Give a definition of the quadratic integral estimate of the transition process.

    5. When minimizing the quadratic estimate, what kind of transition process tends?

    6. What parameters of the mathematical model of an object are required to calculate the quadratic integral estimate?

    7. The control object is described by the transfer function -

  18 Calculation of linear integral estimates .

Calculate the linear integral transient estimate for the initial error value   18 Calculation of linear integral estimates .

Answer :

Linear integral evaluation   18 Calculation of linear integral estimates .

  1. The control object is described by the transfer function -

  18 Calculation of linear integral estimates .

Calculate the linear integral transient estimate for the initial error value   18 Calculation of linear integral estimates .

Answer :

Linear integral evaluation   18 Calculation of linear integral estimates .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control