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19 Quadratic integral estimate taking into account the derivative

Lecture



The disadvantage of the quadratic integral estimate   19 Quadratic integral estimate taking into account the derivative As with previous assessments, the fact that minimizing estimates does not impose restrictions on the form of the transition process. For example, shown in Fig. 1 graphics   19 Quadratic integral estimate taking into account the derivative - (a, b, c) can have the same values   19 Quadratic integral estimate taking into account the derivative while significantly different in the form of the transition process.

  19 Quadratic integral estimate taking into account the derivative

Fig. one

In addition, it often turns out that selected by   19 Quadratic integral estimate taking into account the derivative system parameters lead to a substantially oscillatory process, large derivatives due to the desire to bring the process closer to an ideal jump.

Therefore, another type of integral quadratic estimate is used, in which the restriction is imposed not only on the deviation   19 Quadratic integral estimate taking into account the derivative but also at the rate of its change   19 Quadratic integral estimate taking into account the derivative . This estimate has the following form -

  19 Quadratic integral estimate taking into account the derivative

(one)

Where   19 Quadratic integral estimate taking into account the derivative - some time constant.

Difference between grades   19 Quadratic integral estimate taking into account the derivative and   19 Quadratic integral estimate taking into account the derivative can be represented graphically, as shown in Fig. 2

  19 Quadratic integral estimate taking into account the derivative

Fig. 2

That is optimized by   19 Quadratic integral estimate taking into account the derivative the transition process strives for a perfect leap, and optimized by   19 Quadratic integral estimate taking into account the derivative - to the curve of exponential type, which is described by the following expression -

  19 Quadratic integral estimate taking into account the derivative .

We prove the last statement. To do this, we analyze the expression (1).

  19 Quadratic integral estimate taking into account the derivative ,

given that

  19 Quadratic integral estimate taking into account the derivative ,

we get

  19 Quadratic integral estimate taking into account the derivative

(2)

Given the fact that the last term in (2) is a constant value -

  19 Quadratic integral estimate taking into account the derivative ,

square estimate   19 Quadratic integral estimate taking into account the derivative will have a minimum at

  19 Quadratic integral estimate taking into account the derivative

(3)

The solution of the differential equation (3) has the form -

  19 Quadratic integral estimate taking into account the derivative ,

and if we go from errors to output variables, we get -

  19 Quadratic integral estimate taking into account the derivative ,

Q.E.D.

Consequently, choosing system parameters by   19 Quadratic integral estimate taking into account the derivative , you can bring the transition process to the exponent with a given time constant   19 Quadratic integral estimate taking into account the derivative thus limiting the rate of increase of the output value   19 Quadratic integral estimate taking into account the derivative .

Method of determination   19 Quadratic integral estimate taking into account the derivative may be similar to the method of determining   19 Quadratic integral estimate taking into account the derivative , considered above, if we present the quadratic estimate with the derivative in the following form -

  19 Quadratic integral estimate taking into account the derivative ,

Where   19 Quadratic integral estimate taking into account the derivative is determined by the formulas for   19 Quadratic integral estimate taking into account the derivative but considering the order of the numerator   19 Quadratic integral estimate taking into account the derivative -   19 Quadratic integral estimate taking into account the derivative increases by 1.

In the theory of automatic control, quadratic estimates with higher order derivatives are used (up to   19 Quadratic integral estimate taking into account the derivative ) to more accurately specify the desired form of the transition process, it is natural that this also complicates the process of calculating estimates.

Calculation of quadratic integral estimates

Consider the calculation and use of quadratic errors in an example.

Example

In a control system with a transfer function -

  19 Quadratic integral estimate taking into account the derivative ,

let's set   19 Quadratic integral estimate taking into account the derivative :

  • from the condition   19 Quadratic integral estimate taking into account the derivative ,

  • from the condition   19 Quadratic integral estimate taking into account the derivative ,

and compare transients for these two cases.

Decision

Get the expression for   19 Quadratic integral estimate taking into account the derivative . To do this, convert the transfer function of the system to the specified form

  19 Quadratic integral estimate taking into account the derivative ,

then we get

  19 Quadratic integral estimate taking into account the derivative

(four)

Expression for   19 Quadratic integral estimate taking into account the derivative takes the form -

  19 Quadratic integral estimate taking into account the derivative

(five)

We define the components (5) according to the parameter of the transfer function of the system (4).

  19 Quadratic integral estimate taking into account the derivative

(6)

To find   19 Quadratic integral estimate taking into account the derivative define (   19 Quadratic integral estimate taking into account the derivative ), with   19 Quadratic integral estimate taking into account the derivative ,   19 Quadratic integral estimate taking into account the derivative

  19 Quadratic integral estimate taking into account the derivative ,

Replace in the expression (6) for   19 Quadratic integral estimate taking into account the derivative first column view column

  19 Quadratic integral estimate taking into account the derivative .

Then we get

  19 Quadratic integral estimate taking into account the derivative .

Define   19 Quadratic integral estimate taking into account the derivative -

  19 Quadratic integral estimate taking into account the derivative .

After substituting the obtained components into (5), we obtain the expression for the quadratic integral estimate.

  19 Quadratic integral estimate taking into account the derivative

(five)

Find the expression for the partial derivative with respect to   19 Quadratic integral estimate taking into account the derivative from the expression (5)

  19 Quadratic integral estimate taking into account the derivative ,

equating the resulting expression to zero, we obtain the equation for finding the optimal value   19 Quadratic integral estimate taking into account the derivative .

  19 Quadratic integral estimate taking into account the derivative .

As a result, we obtain the value optimized by the quadratic estimate.   19 Quadratic integral estimate taking into account the derivative -

  19 Quadratic integral estimate taking into account the derivative

(6)

The transfer function of the system   19 Quadratic integral estimate taking into account the derivative take the form -

  19 Quadratic integral estimate taking into account the derivative .

In fig. 3 we show the view of the transient process of the system with a single step effect and optimized by   19 Quadratic integral estimate taking into account the derivative by parameter.

  19 Quadratic integral estimate taking into account the derivative

Fig. 3

Thus, we have the following indicators of the quality of the transition process,

  19 Quadratic integral estimate taking into account the derivative

(7)

Define   19 Quadratic integral estimate taking into account the derivative according to the method developed above for   19 Quadratic integral estimate taking into account the derivative -

  19 Quadratic integral estimate taking into account the derivative ,

expression for   19 Quadratic integral estimate taking into account the derivative we take from the previous case -

  19 Quadratic integral estimate taking into account the derivative .

Define now   19 Quadratic integral estimate taking into account the derivative . The transfer function of the system for this case has the form -

  19 Quadratic integral estimate taking into account the derivative ,

then we get

  19 Quadratic integral estimate taking into account the derivative

(eight)

Expression for   19 Quadratic integral estimate taking into account the derivative takes the form -

  19 Quadratic integral estimate taking into account the derivative

(9)

We define the components (9) according to the parameter of the transfer function of the system (8).

  19 Quadratic integral estimate taking into account the derivative

(ten)

Determine the coefficients   19 Quadratic integral estimate taking into account the derivative -

  19 Quadratic integral estimate taking into account the derivative .

  19 Quadratic integral estimate taking into account the derivative we do not define, because   19 Quadratic integral estimate taking into account the derivative . To find   19 Quadratic integral estimate taking into account the derivative define (   19 Quadratic integral estimate taking into account the derivative ), with   19 Quadratic integral estimate taking into account the derivative ,   19 Quadratic integral estimate taking into account the derivative

  19 Quadratic integral estimate taking into account the derivative ,

Replace in expression (10) for   19 Quadratic integral estimate taking into account the derivative second column view column

  19 Quadratic integral estimate taking into account the derivative .

Then we get

  19 Quadratic integral estimate taking into account the derivative .

After substituting the obtained components into (9), we obtain the expression for the quadratic integral estimate.

  19 Quadratic integral estimate taking into account the derivative

(eleven)

Finally we get

  19 Quadratic integral estimate taking into account the derivative

(12)

Find the expression for the partial derivative with respect to   19 Quadratic integral estimate taking into account the derivative from expression (12)

  19 Quadratic integral estimate taking into account the derivative ,

equating the resulting expression to zero, we obtain the equation for finding the optimal value   19 Quadratic integral estimate taking into account the derivative .

  19 Quadratic integral estimate taking into account the derivative .

As a result, we obtain an optimized quadratic estimate with the derivative   19 Quadratic integral estimate taking into account the derivative -

  19 Quadratic integral estimate taking into account the derivative

(13)

We believe for definiteness   19 Quadratic integral estimate taking into account the derivative then

  19 Quadratic integral estimate taking into account the derivative .

The transfer function of the system when   19 Quadratic integral estimate taking into account the derivative take the form -

  19 Quadratic integral estimate taking into account the derivative .

In fig. 3 we show the view of the transient process of the system with a single step effect and optimized by   19 Quadratic integral estimate taking into account the derivative by parameter.

  19 Quadratic integral estimate taking into account the derivative

Fig. four

Thus, we have the following indicators of the quality of the transition process,

  19 Quadratic integral estimate taking into account the derivative

(14)

Comparing the transient processes, we see that when optimizing by a quadratic estimate taking into account the derivative (   19 Quadratic integral estimate taking into account the derivative ) received significantly smaller values ​​of overshoot and speed, with a smoother increase of the variable.

Test questions and tasks

    1. Give the definition of a quadratic integral estimate taking into account the derivative, explain its components.

    2. What kind of transient process should we strive for while minimizing the integral quadratic estimate with regard to the derivative?

    3. How to calculate the quadratic integral estimate taking into account the derivative?

    4. Calculate the integral quadratic estimate of the transition process in the system with the transfer function -

  19 Quadratic integral estimate taking into account the derivative ,

if the input to the system is a single step function.

Answer :

Integral quadratic estimate   19 Quadratic integral estimate taking into account the derivative .

    1. Calculate the integral quadratic taking into account the derivative estimate of the transition process in the system with the transfer function -

  19 Quadratic integral estimate taking into account the derivative ,

if the input system is a single step function, and the time constant for the evaluation   19 Quadratic integral estimate taking into account the derivative .

Answer :

Integral quadratic estimate   19 Quadratic integral estimate taking into account the derivative .

  1. Determine the regulator parameter of the control system that provides the minimum of the quadratic estimate.

  19 Quadratic integral estimate taking into account the derivative

Answer :

Parameter proportional-integral controller   19 Quadratic integral estimate taking into account the derivative


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control