**7. Correction of processes in an asynchronous electric drive**

Traditionally, it is customary to be considered among electric drive engineers that the discreteness of control signals only affects the controllability of electromechanical systems, because it always "breaks" continuous connections. However, the interpretation of discretization by suppression links shows that sampling allows you to "clear" the frequency characteristics of corrective devices from "side" effects. An example is the PID controller discussed above. Under the conditions of the controller, only the differential channel "works" in the high-frequency zone. In the continuous controller, all channels are rumbled, although the integral and proportional channels are greatly weakened. The use of discrete elements at the output of each channel allows them to be completely filtered out, which cannot be done in a continuous controller and it is difficult to come to such a decision without using the concept of suppression link. A system with nonlinear dynamics is asynchronous electric drives with frequency control.

As shown in [18, 19], the traditionally applied methods and control algorithms ("transvector control") do not always provide the necessary dynamic characteristics of asynchronous electric drives.

In the same works, an alternative control algorithm is described—a dynamic positive relationship with the effective value of the stator current ("DOS+"). This connection allows you to compensate for changes in rotational speed under static and low-frequency loads [19]. In order for the communication to correct only static modes and the low-frequency region, the devices use dynamic links—low-pass filters [18]. As experiments and modeling show, these tasks are performed.

**Figure 16** shows a diagram of the model of an asynchronous electric drive with corrective connections for the stator current (**Figure 16a**) and rotation speed (**Figure 16b**), and **Figures 17** and **18** show the processes of acceleration and load surges with several versions of dynamic links including a discrete element with a low sampling frequency. Static modes are well compensated. With current correction (**Figure 17**), the high-frequency oscillation in currents and speeds at different speeds is preserved by slightly changing its parameters at different speeds of rotation.

**Figure 18** shows the processes in the model with additional correction for rotation speed (connection in **Figure 16b**) in which the signal passes through two parallel links—proportional with low-frequency sampling ("Zero-Order Hold1"— 0.1 s) and differential with high-frequency sampling ("Zero-Order Hold2"— 0.005 s).

As follows from **Figure 18**, the oscillation is completely eliminated at all speeds. In this system, there are three different discrete links: "Zero-Order Hold"—0.3 s, "Zero-Order Hold1"—0.1 s; and "Zero-Order Hold2"—0.005 s (**Figure 16b**); and the system as a whole is significantly superior to all known options for the frequency regulation of induction motors. Moreover, the system is quite easily implemented in industrial frequency converters, since it does not require high accuracy in measuring direct coordinates or in perfect processing.

begin to link individual objects and phenomena into continuous chains. No wonder continuous mathematics, created in the seventeenth and eighteenth centuries, became one of the crowning results of almost three thousand years of our civilization. Without this mathematics, Aristotle and Archimedes created their teachings, the whole of Ancient Rome and the millennial Byzantium created their own civilizations. But voluntarily or involuntarily, the philosophers of antiquity turned to the concepts of the continuous and discrete and received very interesting paradoxes

*Diagrams of processes in asynchronous electric drive model with stator current correction and discrete elements.*

*Scheme of the asynchronous electric drive model and the correction of current (a) and speed (b) with discrete*

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

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

and statements.

**341**

**Figure 17.**

**Figure 16.**

*elements.*

Discreteness is one of the fundamental principles in science. Needless to say, the initial concepts in human thinking are discrete. We perceive the world around us as separate phenomena and objects. Only after the transition to abstract thinking, we

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

**Figure 16.**

means, that fast movements are imperfect, which matches the model. This confirms the validity of the previously derived criteria for sliding along the frequency response and the effectiveness of the proposed frequency response suppression links for

Traditionally, it is customary to be considered among electric drive engineers that the discreteness of control signals only affects the controllability of electromechanical systems, because it always "breaks" continuous connections. However, the interpretation of discretization by suppression links shows that sampling allows you to "clear" the frequency characteristics of corrective devices from "side" effects. An example is the PID controller discussed above. Under the conditions of the controller, only the differential channel "works" in the high-frequency zone. In the continuous controller, all channels are rumbled, although the integral and proportional channels are greatly weakened. The use of discrete elements at the output of each channel allows them to be completely filtered out, which cannot be done in a continuous controller and it is difficult to come to such a decision without using the concept of suppression link. A system with nonlinear dynamics is asynchronous

As shown in [18, 19], the traditionally applied methods and control algorithms ("transvector control") do not always provide the necessary dynamic characteris-

In the same works, an alternative control algorithm is described—a dynamic positive relationship with the effective value of the stator current ("DOS+"). This connection allows you to compensate for changes in rotational speed under static and low-frequency loads [19]. In order for the communication to correct only static modes and the low-frequency region, the devices use dynamic links—low-pass filters [18]. As experiments and modeling show, these tasks are performed.

**Figure 16** shows a diagram of the model of an asynchronous electric drive with

**Figure 18** shows the processes in the model with additional correction for rota-

As follows from **Figure 18**, the oscillation is completely eliminated at all speeds. In this system, there are three different discrete links: "Zero-Order Hold"—0.3 s, "Zero-Order Hold1"—0.1 s; and "Zero-Order Hold2"—0.005 s (**Figure 16b**); and the system as a whole is significantly superior to all known options for the frequency regulation of induction motors. Moreover, the system is quite easily implemented in industrial frequency converters, since it does not require high

Discreteness is one of the fundamental principles in science. Needless to say, the initial concepts in human thinking are discrete. We perceive the world around us as separate phenomena and objects. Only after the transition to abstract thinking, we

tion speed (connection in **Figure 16b**) in which the signal passes through two parallel links—proportional with low-frequency sampling ("Zero-Order Hold1"— 0.1 s) and differential with high-frequency sampling ("Zero-Order Hold2"—

accuracy in measuring direct coordinates or in perfect processing.

corrective connections for the stator current (**Figure 16a**) and rotation speed (**Figure 16b**), and **Figures 17** and **18** show the processes of acceleration and load surges with several versions of dynamic links including a discrete element with a low sampling frequency. Static modes are well compensated. With current correction (**Figure 17**), the high-frequency oscillation in currents and speeds at different speeds is preserved by slightly changing its parameters at different speeds of

assessing the dynamics of even complex nonlinear control systems.

electric drives with frequency control.

*Control Theory in Engineering*

tics of asynchronous electric drives.

rotation.

0.005 s).

**340**

**7. Correction of processes in an asynchronous electric drive**

*Scheme of the asynchronous electric drive model and the correction of current (a) and speed (b) with discrete elements.*

**Figure 17.** *Diagrams of processes in asynchronous electric drive model with stator current correction and discrete elements.*

begin to link individual objects and phenomena into continuous chains. No wonder continuous mathematics, created in the seventeenth and eighteenth centuries, became one of the crowning results of almost three thousand years of our civilization.

Without this mathematics, Aristotle and Archimedes created their teachings, the whole of Ancient Rome and the millennial Byzantium created their own civilizations. But voluntarily or involuntarily, the philosophers of antiquity turned to the concepts of the continuous and discrete and received very interesting paradoxes and statements.

And this is the trick of Zeno, because the number of intervals is not a length of time especially if the intervals are infinitesimal. The paradox turns into Sophism. We do not know knowingly did it Zeno ... Hardly. Otherwise, he would have created a theory of infinitesimal quantities 2000 years earlier than Descartes and Leibniz, who created higher mathematics in which discreteness and, especially, its infinitesimal values play a fundamental role. Judging by Zeno's other aporias—for example, "On the Arrow," he felt a "discrepancy" between ordinary discrete thinking, based on observations and practical experience and continuity, which scientists spoke about in his time. And he showed this problem in every way in the Achilles

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

One of the founders of modern science is Pierre Fermat, the author of many important decisions and discoveries. But he is best known for 400 years thanks to the paradox or "Fermat's theorem," which is a very vivid illustration of the possibilities of discretization of variables of mathematical quantities, since it is precisely the discreteness of four independent variables in Fermat's theorem that leads one

Fermat's theorem states that there are no positive integers that would be a

If any positive values of X, Y, or Z (or at least one of them) were allowed, then an equation with three unknowns for any degree would have an infinite number of

But here is what happens if discreteness is introduced into an indisputable and understandable statement. It turns out that with such discreteness it is impossible to find at least one combination of three numbers and a degree corresponding to the

Let us try to formulate; the theorem is a paradox with an "emphasis" on the

The sum of the natural degrees of two natural numbers is unequal to the same

For the first degree, this condition is not fulfilled, that is, for any two positive

For the second degree, there are solutions to the equation but not for any pair of

But for the third degree is no longer. Rather, there are, but some very large ones that mathematicians find once every hundred years. It is very difficult to check if

About 30 years proof of Fermat's theorem was found [20]. Only reasonably good specialists, mathematicians, can understand it. And for all other people, this is not a solution to the original paradox: the simplest mathematical paradox connecting the

It is as if helicopter pilots would win in mountaineering competitions. No one

Moreover, natural numbers are what most people see in their practical lives. And all the others are fractions. Complex vectors were for many years a "fabrication of

What is the essence of Fermat's paradox? It may be that the discreteness of numbers turns an expression with several degrees of freedom (one equation with three unknowns) into a practically unsolvable expression. In other words, only a rigid definition of the variables involved in this condition turns excessive freedom

paradox—irresponsibly changing the discreteness of time.

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

equation to four unknowns for a condition that cannot be fulfilled.

solution to the equation Xn + Yn = Z<sup>n</sup> for n greater than 2.

solutions. This is undeniable and understandable.

degree, starting from the third, no natural number.

simplest expressions to the simplest numbers.

integers there is a third for the equality to be fulfilled.

there are such numbers yet. Even if there are very good computers.

argues with the proof, or almost no one ... But questions remained.

solution of the Fermat equation.

discreteness of variables:

numbers.

into nonexistence.

**343**

**9. Fermat's paradox**

#### **Figure 18.**

*Diagrams of processes in asynchronous electric drive model with speed correction and discrete elements.*

Consider two points that people have been pondering over for centuries, unaware that the whole thing is in very small detail.
