**1. Introduction**

Digital automatic control systems (ACS) have won everywhere. Their advantages over analog are undeniable—these controllers implement control algorithms of almost any complexity, completely inaccessible to analog ACS.

They are very reliable and stable. Most often, their setup is simple and convenient, like working with mobile phones.

There are no problems with discreteness of output signals in terms of level and time for most ACS. The discreteness in time in fractions of milliseconds and in level in fractions of a percent for the overwhelming number of electromechanical ACS (the most complex of possible structures) with their working range of speed and effort changes is insignificant. Important impulse elements remain in these systems —power converters, which actively affect processes in power currents of engines.

*<sup>W</sup>* <sup>∗</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>≈</sup> <sup>X</sup>

*Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems…*

under consideration.

**Figure 1.**

**Figure 2.**

**329**

*Frequency characteristics of suppression link.*

*Structural diagrams of ACS.*

in the later book by Tsypkina [2].

*DOI: http://dx.doi.org/10.5772/intechopen.91467*

25.4 and 28.2—are described in the statement:

element." structures shown in **Figure 1** are equivalent.

*N*

D*ne*

�*jω<sup>n</sup>* (2)

<sup>2</sup>*WH*ð Þ *jω* (3)

*n*¼0

It is stipulated that the clock frequency is greater than the range of frequencies

Detailed mathematical calculations of approximately the same results are given

Discrete transformations with cumbersome results—paragraphs 25.3 and

... "with sufficiently small pulse repetition periods, the pulse system can be considered. as a continuous one containing the same continuous part and a delay

�*jω<sup>T</sup>*

*<sup>W</sup>* <sup>∗</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>≈</sup>*<sup>e</sup>*

For high-precision electromechanical systems (electric drives), the problems of discreteness of information signals and power currents remain important.

Indeed, discreteness in time and in the level of the processed signals inevitably breaks continuous ACSs and makes their behavior unpredictable. If in analog versions of the ACSs were important—the order of the differential equations describing the control object, the presence of nonlinear links and the requirements for the dynamics of the system, then in digital systems in the 70–80s of the twentieth century, time for calculating control signals became very important.

## **2. Statement of problems: question status**

A study of the fundamental works of leading scientists of the 80s showed the following. All discrete analysis methods, pulsed digital systems, in one way or another are connected with the use of a delay link and lattice functions. These are discrete transformations of continuous channels and transfer functions— Z-transforms, D-transforms, discrete Laplace transforms, and others. What they have in common, most importantly for working with real ACS, is that ALL elements of the control system are subjected to transformations—continuous, linear, with simple and complex transfer functions. It means, that all previous developments on ACS obtained for continuous ACS, that is, stability, accuracy, quality, performance, etc. must be forgotten and remade in the language of discrete transformations and transfer functions. In this case, despite seeming a very "serious" mathematical apparatus, all these transformations, along with cumbersomeness, retain many inaccurate assumptions and reservations.

For example, in the book by Meerov et al. ([1], p. 332), it is said about inverse Z transformations:

*"Transformation makes sense if the series converges" ...*

And on p. 350:

*"If only the function F \* (z,) is given, then ... in principle, there is no procedure for finding F (p)."*

Discrete transfer functions of the simplest links of ACS are very complex, cumbersome, and almost unacceptable for engineering calculations—in "Example 7.4" on p. 354 of the same book, discrete transfer function of an aperiodic link

$$\begin{split} W^\*(\mathbf{z}, \boldsymbol{\varepsilon}) &= \sum\_{i=1}^2 K\_i \boldsymbol{z}\_i^{\varepsilon} \frac{\mathbf{z}}{\mathbf{z} - \mathbf{z}\_i} = \frac{k}{a} \mathbf{z} \left[ \frac{\mathbf{1}}{\mathbf{z} - \mathbf{1}} - \frac{e^{-aT\_p \varepsilon}}{\mathbf{z} - e^{-aT\_p}} \right] \\ &= \frac{k}{a} \mathbf{z} \frac{(\mathbf{1} - e^{-aT\_p \varepsilon})\mathbf{z} + \left(e^{-aT\_p \varepsilon} - e^{-aT\_p}\right)}{(\mathbf{z} - \mathbf{1})(\mathbf{z} - e^{-aT\_p})} \end{split} \tag{1}$$

At the end of these calculations, simplifications are made in the same book, which lead to formulas 7.138 on p. 370 with the words: "you can limit yourself to a finite number of terms in equation (7.137)" and the frequency response of the sampling link is reduced to the response of the delay:

*Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems… DOI: http://dx.doi.org/10.5772/intechopen.91467*

$$\mathcal{W}^\*(\ j\omicron) \approx \sum\_{n=0}^N \omega r\_n e^{-j\overline{\alpha}n} \tag{2}$$

It is stipulated that the clock frequency is greater than the range of frequencies under consideration.

Detailed mathematical calculations of approximately the same results are given in the later book by Tsypkina [2].

Discrete transformations with cumbersome results—paragraphs 25.3 and 25.4 and 28.2—are described in the statement:

... "with sufficiently small pulse repetition periods, the pulse system can be considered. as a continuous one containing the same continuous part and a delay element." structures shown in **Figure 1** are equivalent.

$$W^\*\left(\begin{smallmatrix} jo \end{smallmatrix}\right) \approx e^{-j o \frac{\overline{\gamma}}{2}} W\_H(\begin{smallmatrix} jo \end{smallmatrix}) \tag{3}$$

#### **Figure 1.** *Structural diagrams of ACS.*

in fractions of a percent for the overwhelming number of electromechanical ACS (the most complex of possible structures) with their working range of speed and effort changes is insignificant. Important impulse elements remain in these systems —power converters, which actively affect processes in power currents of engines. For high-precision electromechanical systems (electric drives), the problems of

Indeed, discreteness in time and in the level of the processed signals inevitably

A study of the fundamental works of leading scientists of the 80s showed the following. All discrete analysis methods, pulsed digital systems, in one way or another are connected with the use of a delay link and lattice functions. These are

For example, in the book by Meerov et al. ([1], p. 332), it is said about inverse

*"If only the function F \* (z,) is given, then ... in principle, there is no procedure for*

on p. 354 of the same book, discrete transfer function of an aperiodic link

*Kiz<sup>ε</sup> i z z* � *zi*

Discrete transfer functions of the simplest links of ACS are very complex, cumbersome, and almost unacceptable for engineering calculations—in "Example 7.4"

> ¼ *k α z*

At the end of these calculations, simplifications are made in the same book, which lead to formulas 7.138 on p. 370 with the words: "you can limit yourself to a finite number of terms in equation (7.137)" and the frequency response of the

<sup>1</sup> � *<sup>e</sup>*�*αTp<sup>ε</sup>* � �*<sup>z</sup>* <sup>þ</sup> *<sup>e</sup>*�*αTp<sup>ε</sup>* � *<sup>e</sup>*�*αTp*

1 *z* � 1 � *<sup>e</sup>*�*αTp<sup>ε</sup> <sup>z</sup>* � *<sup>e</sup>*�*αTp*

ð Þ *<sup>z</sup>* � <sup>1</sup> *<sup>z</sup>* � *<sup>e</sup>*�*αTp* ð Þ*:* (1)

� �

� �

discrete transformations of continuous channels and transfer functions— Z-transforms, D-transforms, discrete Laplace transforms, and others. What they have in common, most importantly for working with real ACS, is that ALL elements of the control system are subjected to transformations—continuous, linear, with simple and complex transfer functions. It means, that all previous developments on ACS obtained for continuous ACS, that is, stability, accuracy, quality, performance, etc. must be forgotten and remade in the language of discrete transformations and transfer functions. In this case, despite seeming a very "serious" mathematical apparatus, all these transformations, along with cumbersomeness, retain many

discreteness of information signals and power currents remain important.

**2. Statement of problems: question status**

*Control Theory in Engineering*

inaccurate assumptions and reservations.

*"Transformation makes sense if the series converges" ...*

*<sup>W</sup>* <sup>∗</sup> ð Þ¼ *<sup>z</sup>*, *<sup>ε</sup>* <sup>X</sup>

2

*i*¼1

¼ *k α z*

sampling link is reduced to the response of the delay:

Z transformations:

And on p. 350:

**328**

*finding F (p)."*

breaks continuous ACSs and makes their behavior unpredictable. If in analog versions of the ACSs were important—the order of the differential equations describing the control object, the presence of nonlinear links and the requirements for the dynamics of the system, then in digital systems in the 70–80s of the twentieth century, time for calculating control signals became very important.

**Figure 2.** *Frequency characteristics of suppression link.*

At the same time, it is said that the sampling time is "small," although it is not specified how small it should be and how wrong if "not small."

or equal to the sampling frequency [11, 12]. This link breaks the connection at high frequencies without the ability to adjust this action sequentially connected link. "Included" in a closed-loop control system. it leads to instability if the cutoff

*<sup>W</sup>* <sup>¼</sup> *<sup>A</sup>*ð Þ *<sup>ω</sup> <sup>e</sup>jφ ω*ð Þ

<sup>1</sup> � *ωτ* , *if <sup>ω</sup>*<sup>≤</sup>

�∞, *if ω*>

*ωτ*�1, *if ω* ≤

0, *if ω* >

�∞, *if ω*>

*K*3 *ωτ* � 1

A graphical interpretation of the suppression link is shown in **Figure 2**. *A* of the formula (4)–(6) are phase- and amplitude-frequency characteristics. They differ from formula (3), especially in the frequency zone close to the clock frequency and show that in this frequency zone a signal is suppressed, which cannot be overcome by sequential correction, since no serial link can overcome the amplitude suppression by formula (3). The phase shift (2), at the lower frequencies, similar to the shift of the delay unit in the zone of the clock frequency, increases sharply and also

*<sup>K</sup>*<sup>2</sup> � *<sup>e</sup>* <sup>1</sup>

1 *τ*

1 *τ*

> 1 *τ*

> 1 *τ*

> > 1 *τ*

> > 1 *τ*

, *if ω*<

(4)

(5)

(6)

� *<sup>K</sup>*<sup>1</sup> � ð Þ *τω*

8 >><

>>:

8 >><

>>:

**Figure 3** shows the logarithmic characteristics of the suppression link—

value (� ∞), no sequential correction can overcome this limitation, unlike the phase characteristics of the links proposed in the sources [1, 2], which can theoret-

As follows from the formulas and frequency characteristics of the proposed suppression link, for any sequential correction at a frequency below the quantization frequency, the phase shift will reach a critical value of �180° and lead to instability of the closed loop. Depending on other parts of the system, how far from

It should be noted that the negative phase shift increases much faster than the suppression of the amplitude coefficient. So, at a frequency three times lower than

According to these characteristics, the features of the proposed suppression link

At the clock frequency and higher, in the ACS "after" the suppression link, no sequential correction and feedback of the system will work. Disturbances at these

Since at a frequency equal to the clock frequency, the phase takes the conditional

frequency of the system becomes close to the sampling frequency.

**suppression link**

cannot be seriously corrected.

are very clearly visible.

ically be corrected.

**331**

amplitude and phase characteristics.

The desired formula may look like this:

*DOI: http://dx.doi.org/10.5772/intechopen.91467*

*φ ω*ð Þ¼

8 >><

>>:

*A*ð Þ¼ *ω*

*Lg A*½ �¼ ð Þ *ω*

frequencies will also not be worked out by the regulators.

the quantization frequency this will happen?

**4. Formula of transfer function and frequency characteristics of**

*Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems…*

For engineers, this condition sounds something like this:

"The cutoff frequency of the system must be less than the quantization frequency by at least 10 times, otherwise nothing can be guaranteed." Moreover, the delay link does not change the amplitude frequency response, that is, when the condition of "smallness" of the quantization interval is satisfied, one can completely forget about it.

In this case, the phase characteristic of the delay link shown in **Figure 2** allows a formal possibility of its correction by successive links, but this is only in the case if the correction frequencies are far from the quantization frequency, and since there is a "veto" for their "rapprochement"—the prohibition of the original methodology—this possible correction is simply excluded by the method itself.

Over the past years, naturally, a lot of works on these topics have been written and published. But, practically, in almost all approaches remained the same. All methods are based on discrete Laplace transforms. The operator in these transformations is replaced by the exponential function of the delay unit, and the sampling time is included in these transformations by a parameter. Frequencies close to the clock frequency are not considered [3–10].

Thus, the "traditional" ACS theory offers two fundamental approaches:


It does not take a lot of imagination to understand that the second approach is chosen more often in engineering calculations and studies.

One of the most commonly used devices in electromechanical systems is pulsed power amplifiers—frequency converters for asynchronous drives and voltage converters for DC drives. The switching frequency of power elements is usually in the range from 4 to 16 kHz. Mechanical processes in these systems range from 0 to 20 Hz. That is, the condition of "smallness" of the switching period of pulse elements is fulfilled. The frequency of clocking of control signals in microprocessors is most often not mentioned even in advertising materials for converters.
