**2. Problem statement**

100 years, they have been used in mechanisms with virtually no control over the speed of rotation or the mechanical moment developed. Only in the 1980s of the twentieth century, that is, 100 years after the invention, with the start of production of powerful controlled semiconductor switches, it became possible to effectively control the stator frequency and, along with it, almost any AED coordinates. There was a very interesting situation. The principles of frequency control were developed at the beginning of the twentieth century, but not having a wide practical application,

"transvector" control were immediately recognized as "reducing" the AED to a DC drive, that is, practically to a linear stationary system. The use of AED with frequency converters in a wide practice has encountered a number of problems. It turned out that AEDs are a substantially nonlinear link and the existing control methods (scalar control (SC), space vector control (SVC)) do not remedy this situation too much. At present, the following situation is "generally accepted": scalar control preserves the nonlinearity of the AED, but is not intended for dynamic mechanisms and does not require dynamic analysis, and vector control reduces the AED to a DC drive; therefore, complex nonlinear methods are not applied to anything. Everything is complicated by the fact that both linear and nonlinear equations of electromechanical

Reality is more complicated than any mathematical model. At the same time, strict solutions exist only in the area of linear stationary systems (LSS), which are described by linear differential equations with constant coefficients in the "left" part of the equations—the one that describes the control object itself—in our case, the electric

• LSS equations have strict solutions, the qualities of which - stability, static and dynamic errors and the time of transient processes do not depend on the "right" part of the equations, which, as a rule, describe external perturbations

• The most important of the estimates is the assessment of sustainability which is

• If in the "right" part of the equation, that is, at any input of the system, there is a harmonic signal of a certain frequency, then all blocks will have harmonic input and output signals of only this frequency, different in amplitude about

Naturally, real electric drives can be identified by LSS with only very large approximations. In direct-current drives with independent excitation, these approximations are insignificant; they mostly relate to mechanical structures with stiffness and gaps. In asynchronous electric drives, the reduction to LSS is associated with much larger errors. The equations of a generalized AC motor, even with significant assumptions, can be reduced to linear equations with variable coefficients or to nonlinear control systems. For engineering calculations, the differences between these systems are very conditional. If variable coefficients depend on the same coordinates of the electric drive (rotational speed, stator current, etc.), the

they largely remained a theory. Developed in the 1970s, the principles of

complexes are only an approximation to real technical systems.

or job signals.

*Control Theory in Engineering*

the phase.

**170**

**1.2 About mathematical models and linear stationary systems**

drive. For such systems, you can highlight a number of important features:

made by sufficiently accurate, for example, criteria Nyquist.

• Systems and individual blocks are identified by transfer functions and frequency characteristics strictly defined by differential equations.

#### **2.1 Asynchronous electric drives and linear stationary systems**

Linear stationary systems, which we will further call simply linear, have one big advantage over reality. These equations have exact solutions. But nonlinear equations are either very difficult to solve or not at all. But this is not the biggest problem in the interaction of linear and nonlinear systems. Linear cybernetic systems, and electric drives in particular, have processes whose quality—stability, transient time, and the magnitude of the static and dynamic errors of the drive—do not depend on the input "master" signals and on external influences and disturbances, since they are determined only by parameters of the control system itself. Thus, such system is predictable. To identify it, it is not necessary to test it with signals of different magnitudes and rates; it should not allow unexpected operating modes and all the more emergency ones. In addition, it is quite simply adjusted by regulators and feedbacks. For nonlinear control systems, all these are just dreams and desires. Systems behave differently at different speeds and under different loads, stable at nominal speed, and they become oscillatory at low speeds, etc. But probably the biggest problem is that they cannot be adjusted by the usual methods—PI and PID regulators behave completely unpredictable. Paraphrasing the classic, one can say: "All linear systems are the same, and nonlinear ones are each nonlinear in their own way." From the foregoing, it is clear that any engineer would prefer to deal with linear control systems and electric drives, or at least with the systems closer to linear with those tasks and disturbances that this system is experiencing. Nonlinear components can be very different—inevitable "imperfections," restrictions, dead zones, backlashes, etc. But there are also "fundamental" nonlinearities in electric drives; this is the *moment formation*—the operation of multiplying two variable functions current and magnetic flux. In asynchronous motors, these are periodic functions; as a result, this drive even has a mechanical characteristic that is strictly nonlinear. It is obvious and follows from the Kloss formula and the equivalent scheme, in which there is an element, dependent on slip. Those nonlinearities of asynchronous electric drives are known, but adjusting them with simple means (*IR*, compensation) does not work. The electric drives becomes ineffective. To overcome this, a special "vector" control is applied, which also turns out to lead to new nonlinearities and problems, including unexpected ones. In this way, to bring an automatic system closer to a linear one is to make it predictable, adjustable, reliable, and efficient. In the proposed paper, some methods of such an approximation are given. We called them linearization methods. Usually, this term is called the simplification of the original nonlinear equations of the system. But we left this term unchanged. In our opinion, this term reflects too well the goals and results of this work to replace it with another.

In modern high-tech industrial mechanisms, electric actuators play a very important role. The quality of technical complexes and their competitiveness depend on their ability to "fend off" disturbances, for example, change in air

temperature, "dips" and "surge" of power voltage, wear of mechanical parts, and, most importantly, external loads.

AC drives with asynchronous and synchronous electric motors have significantly better "robustness" properties, the latter, as a rule, differing from brushless DC motor only by the control method. But control problems are the reason why AC drives are still rarely used in complex technological mechanisms. These problems are connected with essentially nonlinear equations, which describe the processes of formation of a mechanical torque in an asynchronous electric motor. Neglecting them leads to the fact that the simplified formulas do not at all reflect the processes occurring in AC drives. Accounting for these nonlinearities leads to equations that are very difficult to solve. Even for assessing sustainability, it is difficult to choose the appropriate mathematical apparatus. This state of affairs requires the creation of a new engineering calculation method and the synthesis of AED control system.

Engineering calculation is a solution that determines the technical and economic development; it is limited not only by the conditions of the task itself. It is necessary to solve it so that the result could be effective and useful. These are significant limitations that theoretical science sometimes does not have.

2.The transfer functions of the rotor circuit and its models must be exactly the same. This is also not always the case. Moreover, the structure of the model does not take into account the initial conditions, which are determined by external disturbances. Therefore, processes at zero initial conditions (drive acceleration) usually occur "correctly" but react to the external load much

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

3.The basic equations of the processes for the vector control of the stator current projections are obtained by representing the operator image of the product of functions by the product of a function by the reference image of another factor, that is, Parseval's integral is simplified significantly, but the errors of

4.All equations allow sinusoidal processes: currents, EMF, and voltages. Really, after all, "vector control" controls the current or voltage vector, and what if the signals are not harmonic? Who controls other harmonics? What are the

5. In the calculations it is assumed that the rotor flow does not change, which of course does not correspond to reality. But the price of this assumption is also

6.And the last question. In the scheme there is no link reflecting the dynamics of the converter itself, that is, systems with a switch, delay links, and filters. This

To evaluate all these errors, especially to show the modes in which they will be significant, no engineering technique is trying to do. Moreover, "as time passes," the assumptions are forgotten, and on the basis of these "new" ideas, vector control is constructed, which inverse operators "linearize" the drive, turning it into a similarity to a DC drive. These algorithms fundamentally contain errors, which are especially pronounced in dynamic modes and in the countering of external torque perturbations. In this case, as mentioned above, in the dynamic links of a real electric motor appear initial conditions, which cannot be in the links opposite to the dynamic links of the electric motor. These links become different. In conventional control schemes, external disturbances lead to a shift in the equilibrium state. In the scheme with direct linearization (**Figure 1**), it destroys that. The same is true for other assumptions. Analytically it is very difficult to take into account their

worse and in accordance with other dynamic characteristics.

projections of other harmonics? The question is rhetorical.

link can be very conditionally considered an inertia-free link.

this simplification are not specified in any work.

not known.

**173**

**Figure 1.**

*Block diagram of space vector control.*

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

In this regard, when developing new electric drives based on an asynchronous motor and frequency converter (FC), great attention should be paid to the methodology of experimental studies. The authors have been working on these problems for about 15 years.

The first few years were devoted to theoretical studies, namely, the analysis of the dynamics (primarily stability) of systems with variable carrying signals [1].

A major project was the project of introducing frequency-controlled drives to self-propelled wagons for the mining industry in 2008–2010 [2, 3]. As a result of the modernized control algorithm introduction, it was possible to increase the loading capacity of the car by 1.5 times.

In subsequent years, numerous experimental studies were carried out, the results of which are reflected in publications [4–8] and patents [9–11], introduced in industry and energy. The algorithms of sensorless correction with frequency control were significantly refined, the existing algorithms were investigated in detail, and the problems of these algorithms were identified. This work was carried out in collaboration with representatives of *Schneider Electric* in Russia, who provided equipment for the experiments. The authors thank the company. In this regard, the starting materials for research were experiments. Their goal was to clarify the advantages of vector control over scalar ones and find a convenient way to identify the dynamics of such drives. However, the results of the experiments forced to significantly adjust the research plan.

#### **2.2 On the existing control methodologies and assumptions**

A lot of books have been written about the problems of AED vector control. The original structures of the model built into the regulator contain many mathematical inaccuracies. To assess these inaccuracies, we consider the block diagram of a current-controlled drive, given in the monograph by Usoltsev (**Figure 1**).

The desired linearization occurs, if several conditions are met:

1.Exactly coincides the rotor rotation speed and the rotation speed of the vectors of the stator current. In fact, in the sensorless circuit, this condition cannot always be satisfied, especially in dynamic modes. As a result, the connection between the model and the motor becomes a very complex link with floating frequency characteristics; as a result the control inevitably falls apart.

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction… DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 1.** *Block diagram of space vector control.*

temperature, "dips" and "surge" of power voltage, wear of mechanical parts, and,

AC drives with asynchronous and synchronous electric motors have significantly better "robustness" properties, the latter, as a rule, differing from brushless DC motor only by the control method. But control problems are the reason why AC drives are still rarely used in complex technological mechanisms. These problems are connected with essentially nonlinear equations, which describe the processes of formation of a mechanical torque in an asynchronous electric motor. Neglecting them leads to the fact that the simplified formulas do not at all reflect the processes occurring in AC drives. Accounting for these nonlinearities leads to equations that are very difficult to solve. Even for assessing sustainability, it is difficult to choose the appropriate mathematical apparatus. This state of affairs requires the creation of a new engineering calculation method and the synthesis of AED control system. Engineering calculation is a solution that determines the technical and economic development; it is limited not only by the conditions of the task itself. It is necessary to solve it so that the result could be effective and useful. These are significant

In this regard, when developing new electric drives based on an asynchronous motor and frequency converter (FC), great attention should be paid to the methodology of experimental studies. The authors have been working on these problems

The first few years were devoted to theoretical studies, namely, the analysis of the dynamics (primarily stability) of systems with variable carrying signals [1]. A major project was the project of introducing frequency-controlled drives to self-propelled wagons for the mining industry in 2008–2010 [2, 3]. As a result of the modernized control algorithm introduction, it was possible to increase the loading

In subsequent years, numerous experimental studies were carried out, the results of which are reflected in publications [4–8] and patents [9–11], introduced in industry and energy. The algorithms of sensorless correction with frequency control were significantly refined, the existing algorithms were investigated in detail, and the problems of these algorithms were identified. This work was carried out in collaboration with representatives of *Schneider Electric* in Russia, who provided equipment for the experiments. The authors thank the company. In this regard, the starting materials for research were experiments. Their goal was to clarify the advantages of vector control over scalar ones and find a convenient way to identify the dynamics of such drives. However, the results of the experiments

A lot of books have been written about the problems of AED vector control. The original structures of the model built into the regulator contain many mathematical inaccuracies. To assess these inaccuracies, we consider the block diagram of a current-controlled drive, given in the monograph by Usoltsev (**Figure 1**). The desired linearization occurs, if several conditions are met:

1.Exactly coincides the rotor rotation speed and the rotation speed of the vectors of the stator current. In fact, in the sensorless circuit, this condition cannot always be satisfied, especially in dynamic modes. As a result, the connection between the model and the motor becomes a very complex link with floating

frequency characteristics; as a result the control inevitably falls apart.

limitations that theoretical science sometimes does not have.

most importantly, external loads.

*Control Theory in Engineering*

for about 15 years.

**172**

capacity of the car by 1.5 times.

forced to significantly adjust the research plan.

**2.2 On the existing control methodologies and assumptions**


To evaluate all these errors, especially to show the modes in which they will be significant, no engineering technique is trying to do. Moreover, "as time passes," the assumptions are forgotten, and on the basis of these "new" ideas, vector control is constructed, which inverse operators "linearize" the drive, turning it into a similarity to a DC drive. These algorithms fundamentally contain errors, which are especially pronounced in dynamic modes and in the countering of external torque perturbations. In this case, as mentioned above, in the dynamic links of a real electric motor appear initial conditions, which cannot be in the links opposite to the dynamic links of the electric motor. These links become different. In conventional control schemes, external disturbances lead to a shift in the equilibrium state. In the scheme with direct linearization (**Figure 1**), it destroys that. The same is true for other assumptions. Analytically it is very difficult to take into account their

influence. Therefore, an engineering technique is proposed that combines a qualitative analysis of the equations, transfer functions, and frequency characteristics with modeling and experiments. In contrast to the usual starting and braking modes at one average speed, the conditions for them are formed, taking into account all the above features, namely, at different frequencies, load surges, and accelerations.

Thus, the nonlinearity of the equations of AED is well known and does not cause doubts. But there are no techniques that would allow to evaluate the effect of these unaccounted nonlinearities on the drive final characteristics. One can only assume a violation of stability under complex torque loads and complex reference signals. Since nonlinearity is preserved for all known most widely used control methods (scalar, vector, direct torque control), it is advisable to identify the dynamics of adjustable electric drives with detailed modeling and experiments in a variety of modes: start-up brakes and modes of different torque loads. So, vector control, in order to reduce AED to DC systems, tries to linearize AD with nonlinear transformations. With these inaccuracies nonlinearity only increases.

#### **2.3 Problems of AED practice**

As mentioned above, the research was based on experiments and modeling. A model diagram is shown in **Figure 2**. The electric drive was subsequently accelerated to speeds of 30, 60, 90, 120, and 150 rad/s corresponding to the frequencies of the supply voltage of 10 – 20 – 30 – 40 – 50 Hz, and a load was drawn at each speed of rotation (**Figure 3**). The load was set by stepwise action equal to the nominal value of the motor torque. The oscilloscope displays the speed, electromagnetic torque, and other necessary variables of the drive.

A stand was made for experiments (**Figure 4**). It consists of two asynchronous motors (M1, with a squirrel-cage rotor; M2, with a wound rotor) with a rated power of 370 W, synchronous speed of rotation 1500 rpm, nominal speed of rotation of 1370 rpm, and rated voltage of 380 V, controlled by two FC ATV32 (UZ1) and ATV71 (UZ2) by *Schneider Electric*. Interconnected motor shafts are connected to a speed sensor (encoder), information from which is transmitted to the FC UZ2. Used in the stand frequency converters belong to the middle technical-economic class. They have a relatively low cost, which allows them to be widely used at enterprises

of various levels (including small businesses), and at the same time, they have a broad functionality, including all standard, well-developed control algorithms, which makes them universal in terms of application in various technical systems. The first results of the experiments were the operation of the drive with load surges. These results were poorly explained from the point of view of the absolute advantages of vector control and the proximity of this drive to DC drives. The parallel movement of the mechanical characteristics under the frequency control of the drive should lead to the same absolute speed drops with the same loading torques. Instead, the "dip" speeds turned out to be different both in absolute and relative values, both for scalar and vector control. If different processes were expected for scalar control, the results for the vector were unexpected (**Figures 5** and **6**). The dynamics of acceleration processes up to speeds of 10, 20 rad/s, and so on in a scalar control are somewhat different, as in the vector one. In case of load surges, the processes differ both in dynamics (transient time) and in static speed dip. Moreover, at speeds of 60 rad/s (i.e., at a frequency of 10 Hz) and below, both

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

*Schematic circuit of the stand for the study of the dynamic characteristics of the drive.*

**Figure 3.**

**Figure 4.**

**175**

*Modeling processes in an asynchronous electric drive.*

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 2.** *Diagram of AED model.*

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction… DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 3.**

influence. Therefore, an engineering technique is proposed that combines a qualitative analysis of the equations, transfer functions, and frequency characteristics with modeling and experiments. In contrast to the usual starting and braking modes at one average speed, the conditions for them are formed, taking into account all the above features, namely, at different frequencies, load surges, and accelerations. Thus, the nonlinearity of the equations of AED is well known and does not cause doubts. But there are no techniques that would allow to evaluate the effect of these unaccounted nonlinearities on the drive final characteristics. One can only assume a violation of stability under complex torque loads and complex reference signals. Since nonlinearity is preserved for all known most widely used control methods (scalar, vector, direct torque control), it is advisable to identify the dynamics of adjustable electric drives with detailed modeling and experiments in a variety of modes: start-up brakes and modes of different torque loads. So, vector control, in order to reduce AED to DC systems, tries to linearize AD with nonlinear trans-

As mentioned above, the research was based on experiments and modeling. A model diagram is shown in **Figure 2**. The electric drive was subsequently accelerated to speeds of 30, 60, 90, 120, and 150 rad/s corresponding to the frequencies of the supply voltage of 10 – 20 – 30 – 40 – 50 Hz, and a load was drawn at each speed of rotation (**Figure 3**). The load was set by stepwise action equal to the nominal value of the motor torque. The oscilloscope displays the speed, electromagnetic

A stand was made for experiments (**Figure 4**). It consists of two asynchronous motors (M1, with a squirrel-cage rotor; M2, with a wound rotor) with a rated power of 370 W, synchronous speed of rotation 1500 rpm, nominal speed of rotation of 1370 rpm, and rated voltage of 380 V, controlled by two FC ATV32 (UZ1) and ATV71 (UZ2) by *Schneider Electric*. Interconnected motor shafts are connected to a speed sensor (encoder), information from which is transmitted to the FC UZ2. Used in the stand frequency converters belong to the middle technical-economic class. They have a relatively low cost, which allows them to be widely used at enterprises

formations. With these inaccuracies nonlinearity only increases.

torque, and other necessary variables of the drive.

**2.3 Problems of AED practice**

*Control Theory in Engineering*

**Figure 2.**

**174**

*Diagram of AED model.*

*Modeling processes in an asynchronous electric drive.*

**Figure 4.** *Schematic circuit of the stand for the study of the dynamic characteristics of the drive.*

of various levels (including small businesses), and at the same time, they have a broad functionality, including all standard, well-developed control algorithms, which makes them universal in terms of application in various technical systems.

The first results of the experiments were the operation of the drive with load surges. These results were poorly explained from the point of view of the absolute advantages of vector control and the proximity of this drive to DC drives. The parallel movement of the mechanical characteristics under the frequency control of the drive should lead to the same absolute speed drops with the same loading torques. Instead, the "dip" speeds turned out to be different both in absolute and relative values, both for scalar and vector control. If different processes were expected for scalar control, the results for the vector were unexpected (**Figures 5** and **6**). The dynamics of acceleration processes up to speeds of 10, 20 rad/s, and so on in a scalar control are somewhat different, as in the vector one. In case of load surges, the processes differ both in dynamics (transient time) and in static speed dip. Moreover, at speeds of 60 rad/s (i.e., at a frequency of 10 Hz) and below, both

inputs of the FC of the test drive (**Figure 8**). In this case, the higher the amplitude of the oscillation, the more effective the drive. The same sine-wave reference signals were applied to the input of the load drive converter (**Figure 9**). The ability of the drive to maintain a given speed was investigated. The efficiency of the drive is higher in the case in which the amplitude of the speed oscillations is lower.

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

*Step loading response of AED with vector control with speed feedback (ω1 = ω2 = 30 rad/s; PI-regulator*

As follows from **Table 1**, the drive efficiency with vector control is not the best, including and with speed feedback. These results required a different theoretical

approach to the problem.

*Speed diagrams with harmonic (sinusoidal) speed reference signal.*

*Speed diagrams for harmonic torque perturbation testing.*

*settings: Кt1 = 40%; Кt2 = 20%).*

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 7.**

**Figure 8.**

**Figure 9.**

**177**

scalar and vector controls behave almost exactly the same. It should be noted that in the drive equations, there are no prerequisites for this state of affairs.

It should be noted that maintaining the speed under shock torque perturbations is one of the most difficult tasks of automated electric drives. Even DC motors, closed in speed or angular position, with very good control characteristics cannot avoid significant dynamic failures and complex speed recovery processes during load gain. But the processes of working out such loads are the most important characteristics of complex control systems.

When the speed sensor is installed and the PI regulator is turned on, the processes in the drive with vector control also do not match the expected ones. In the AED with the PI speed regulator, the two-time changes in the regulator parameters do not completely change the process (**Figure 7**). Attention should be paid to the duration of the process, which is substantially longer than the process time in an open-ended drive. Similar results were obtained in the simulation.

Experiments with periodically varying loads were carried out using a similar technique. Since various properties of the drive are evident at different frequencies of the stator voltage, the drive under study accelerates to different speeds, corresponding to the frequencies of the stator voltage—10, 20, 30, 40, and 50 Hz, respectively. A constant reference signal was summed up with a periodic sinusoidal signal with varying amplitude and frequency. This sum signal was fed to the analog *Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction… DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 7.**

*Step loading response of AED with vector control with speed feedback (ω1 = ω2 = 30 rad/s; PI-regulator settings: Кt1 = 40%; Кt2 = 20%).*

inputs of the FC of the test drive (**Figure 8**). In this case, the higher the amplitude of the oscillation, the more effective the drive. The same sine-wave reference signals were applied to the input of the load drive converter (**Figure 9**). The ability of the drive to maintain a given speed was investigated. The efficiency of the drive is higher in the case in which the amplitude of the speed oscillations is lower.

As follows from **Table 1**, the drive efficiency with vector control is not the best, including and with speed feedback. These results required a different theoretical approach to the problem.

**Figure 8.** *Speed diagrams with harmonic (sinusoidal) speed reference signal.*

**Figure 9.** *Speed diagrams for harmonic torque perturbation testing.*

scalar and vector controls behave almost exactly the same. It should be noted that in

When the speed sensor is installed and the PI regulator is turned on, the processes in the drive with vector control also do not match the expected ones. In the AED with the PI speed regulator, the two-time changes in the regulator parameters do not completely change the process (**Figure 7**). Attention should be paid to the duration of the process, which is substantially longer than the process time in an

Experiments with periodically varying loads were carried out using a similar technique. Since various properties of the drive are evident at different frequencies

corresponding to the frequencies of the stator voltage—10, 20, 30, 40, and 50 Hz, respectively. A constant reference signal was summed up with a periodic sinusoidal signal with varying amplitude and frequency. This sum signal was fed to the analog

of the stator voltage, the drive under study accelerates to different speeds,

It should be noted that maintaining the speed under shock torque perturbations is one of the most difficult tasks of automated electric drives. Even DC motors, closed in speed or angular position, with very good control characteristics cannot avoid significant dynamic failures and complex speed recovery processes during load gain. But the processes of working out such loads are the most important

the drive equations, there are no prerequisites for this state of affairs.

*Step loading response of AED with vector control (ω1 = 30 rad/s; ω2 = 90 rad/s; ω3 = 150 rad/s).*

*Step loading response of AED with scalar control (ω1 = 30 rad/s; ω2 = 90 rad/s; ω3 = 150 rad/s).*

open-ended drive. Similar results were obtained in the simulation.

characteristics of complex control systems.

**Figure 5.**

*Control Theory in Engineering*

**Figure 6.**

**176**


the frequency of the output signal occurred. The process of disruption is also accompanied by a significant increase in current. The point of control failures did not change when the parameters of the controller built into the inverter were changed, the mechanical part of the electric drive changed, and the load changed. It was found that the frequency of disruptions depends only on the amplitude of the input signal. The dependence of the disruption frequency on the amplitude is determined experimentally. It can be assumed that the cause of failures is the disregard of the initial conditions by the motor model; the parameters of the sine

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

It should be noted that in AED studies with later models of converters (*ATV71* and *ATV32*) with open vector control, these failures are not present, but they occur under other conditions. It is important to note that the possibility of such control failures under certain external conditions is not mentioned at all by any documents. And the dangers of resonance processes in mechanisms with such drives are very

**2.5 Performance identification of the asynchronous electric drives by the**

signals contain harmonics of frequencies other than the main one.

As mentioned above, an asynchronous electric drive is a nonlinear control system, moreover, "on a carrier" harmonic signal. The equations describing it, most often, are reduced to vector interpretations of all signals—rotor and stator currents, EMF, voltages, etc. Both vector and scalar controls are constructed using these equations. At the same time, everyone is well aware that the real variables of the coordinates and signals contain the harmonics of other frequencies, the presence of which "collapses" most of the provisions of these theories. For example, widely used transformations of coordinates in *d* and *q* will not make sense if the current

It remains to find out the "share" of high (or other) harmonics in the signals of an asynchronous electric drive. During the experiments, it was decided to analyze the spectra of rotor currents in asynchronous electric drives with different control methods—scalar (**Figure 11**), vector sensorless (**Figure 12**), and vector speed feedback (**Figure 13**). These experiments are described in detail in a number of articles. This paper presents the results of modeling and experiments and new comments on

As noted in [6, 8] in the case of vector control, especially when the speed loop is

*The diagram of the speed and current of the rotor of an asynchronous drive with vector control. Spectrum of the*

closed, the proportion of other harmonics, as compared with the main one, is

wave determine these conditions.

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

**spectrum of rotor currents**

these results.

**Figure 11.**

**179**

*rotor current signal.*

high and can have very serious consequences.

**Table 1.**

*Results of experiments for periodic signals.*

#### **2.4 Instability systems AED with vector control**

Dynamic properties of the electric drive are best evaluated by frequency characteristics. For their registration, we carried out the experiments in which a constant reference signal and a periodic sinusoidal signal with varying amplitude and frequency were fed to the analog inputs of the FC. The effective value of the stator current was selected as an output signal. This signal is selected as the most reliable of the signals computed by the inverter, since the models for calculating the speed and torque of the motor are not known. Most often, the stator current signals are close to harmonic and indicate a "correct" system response to input signals (**Figure 10**).

At certain values of the amplitude (A) and frequency (f), "disruptions" of control characterized by a mismatch between the frequency of the input signal and

#### **Figure 10.**

*The stator current diagrams during the development of variable tasks 1 and 8 Hz (a), during the "control failure" (b) and the dependence of the reference signal frequency on the speed on the same signal amplitude, at which the "breakdown" of control in the FC occurs (c).*

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction… DOI: http://dx.doi.org/10.5772/intechopen.88223*

the frequency of the output signal occurred. The process of disruption is also accompanied by a significant increase in current. The point of control failures did not change when the parameters of the controller built into the inverter were changed, the mechanical part of the electric drive changed, and the load changed. It was found that the frequency of disruptions depends only on the amplitude of the input signal. The dependence of the disruption frequency on the amplitude is determined experimentally. It can be assumed that the cause of failures is the disregard of the initial conditions by the motor model; the parameters of the sine wave determine these conditions.

It should be noted that in AED studies with later models of converters (*ATV71* and *ATV32*) with open vector control, these failures are not present, but they occur under other conditions. It is important to note that the possibility of such control failures under certain external conditions is not mentioned at all by any documents. And the dangers of resonance processes in mechanisms with such drives are very high and can have very serious consequences.

### **2.5 Performance identification of the asynchronous electric drives by the spectrum of rotor currents**

As mentioned above, an asynchronous electric drive is a nonlinear control system, moreover, "on a carrier" harmonic signal. The equations describing it, most often, are reduced to vector interpretations of all signals—rotor and stator currents, EMF, voltages, etc. Both vector and scalar controls are constructed using these equations. At the same time, everyone is well aware that the real variables of the coordinates and signals contain the harmonics of other frequencies, the presence of which "collapses" most of the provisions of these theories. For example, widely used transformations of coordinates in *d* and *q* will not make sense if the current signals contain harmonics of frequencies other than the main one.

It remains to find out the "share" of high (or other) harmonics in the signals of an asynchronous electric drive. During the experiments, it was decided to analyze the spectra of rotor currents in asynchronous electric drives with different control methods—scalar (**Figure 11**), vector sensorless (**Figure 12**), and vector speed feedback (**Figure 13**). These experiments are described in detail in a number of articles. This paper presents the results of modeling and experiments and new comments on these results.

#### **Figure 11.**

*The diagram of the speed and current of the rotor of an asynchronous drive with vector control. Spectrum of the rotor current signal.*

**2.4 Instability systems AED with vector control**

*Results of experiments for periodic signals.*

*Control Theory in Engineering*

**Drive control system Test the periodic reference**

(**Figure 10**).

**Figure 10.**

**178**

*which the "breakdown" of control in the FC occurs (c).*

**Table 1.**

Dynamic properties of the electric drive are best evaluated by frequency characteristics. For their registration, we carried out the experiments in which a constant reference signal and a periodic sinusoidal signal with varying amplitude and frequency were fed to the analog inputs of the FC. The effective value of the stator current was selected as an output signal. This signal is selected as the most reliable of the signals computed by the inverter, since the models for calculating the speed and torque of the motor are not known. Most often, the stator current signals are close to harmonic and indicate a "correct" system response to input signals

**signal**

Open system (scalar control) 5.0 87 2.38 245 Open system (vector control) 5.03 84 2.19 270 Speed feedback system 4.99 84 3.31 230

**Test of periodic torque perturbation**

**Δω, rad/s Δφ, el. deg. Δω, rad/s Δφ, el. deg.**

At certain values of the amplitude (A) and frequency (f), "disruptions" of control characterized by a mismatch between the frequency of the input signal and

*The stator current diagrams during the development of variable tasks 1 and 8 Hz (a), during the "control failure" (b) and the dependence of the reference signal frequency on the speed on the same signal amplitude, at*

#### **Figure 12.**

*The diagram of the speed and current of the rotor of an asynchronous drive with closed-loop vector control. Spectrum of the rotor current signal.*

#### **Figure 13.**

*Diagram of the speed and current of the rotor of an asynchronous drive with scalar control. Spectrum of the rotor current signal.*

significantly higher than with scalar control. At the same time, the frequencies of rotor currents with the same load are the highest for a system with a speed loop. Since, in an asynchronous electric drive, the mechanical moment is formed when the rotor rotates at the speed of rotation of the electromagnetic field, it is natural to evaluate the efficiency of the formation of the moment by this error rate or, as it is called in the electric drive, by slip. The greater the slip for the formation of the moment, the less effective way of its formation is applied. As it is known, the frequency of the rotor current in an asynchronous electric drive is rigidly connected with the slip. Experiments show that with vector control the efficiency of the formation of the moment, at least in the converters of the frequency of the middle technical-economic class, is lower than with scalar control. This is also confirmed by the simulation of asynchronous electric drive systems (**Figures 14** and **15**).

The frequency of the rotor current under load (at a speed of 90 rps) with vector control is 10.6 Hz, scalar without feedback—2.72 Hz.

Another conclusion from these experiments is that the components of the signals of other harmonics are very significant, which makes the errors of the generalized equations, expansions on the *d* and *q* axes, and other transformations very essential. Since nonlinear operations are carried out in vector control in models of a control unit of frequency converters, these errors increase, as evidenced by higher frequencies of rotor currents under identical loads compared to scalar control.

**Control system No-load Under load** Vector control 2.1 Hz 6.25 Hz Vector speed feedback control 2.1 Hz 8.75 Hz Scalar control without feedback 1.69 Hz 4.75 Hz

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

Thus, multiple nonlinear operations are the reason for the appearance of nonfundamental harmonics in the electric motor currents and, as a result, the ineffi-

ciency of control methods.

**Figure 14.**

**Figure 15.**

**Table 2.**

**181**

*Modeling processes in an asynchronous electric drive with vector control system.*

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

*Modeling processes in an asynchronous electric drive with scalar control.*

*Frequency of the fundamental harmonic for various control algorithms.*

Analysis of the spectra of rotor currents convincingly shows that the formation of the necessary torque in a vector-controlled electric drive, even when the speed loop is closed, is not the most effective; the reason for this is probably the presence of significant harmonics at a frequency of 3–8 Hz (**Figures 12** and **13**). **Table 2** shows the experimental values of the main frequencies of the rotor currents, which cast doubt on the generally accepted opinion about the high efficiency of the formation of the torque in vector control.

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction… DOI: http://dx.doi.org/10.5772/intechopen.88223*

#### **Figure 14.**

*Modeling processes in an asynchronous electric drive with vector control system.*

#### **Figure 15.**

significantly higher than with scalar control. At the same time, the frequencies of rotor currents with the same load are the highest for a system with a speed loop. Since, in an asynchronous electric drive, the mechanical moment is formed when the rotor rotates at the speed of rotation of the electromagnetic field, it is natural to evaluate the efficiency of the formation of the moment by this error rate or, as it is called in the electric drive, by slip. The greater the slip for the formation of the moment, the less effective way of its formation is applied. As it is known, the frequency of the rotor current in an asynchronous electric drive is rigidly connected with the slip. Experiments show that with vector control the efficiency of the formation of the moment, at least in the converters of the frequency of the middle technical-economic class, is lower than with scalar control. This is also confirmed by

*Diagram of the speed and current of the rotor of an asynchronous drive with scalar control. Spectrum of the*

*The diagram of the speed and current of the rotor of an asynchronous drive with closed-loop vector control.*

the simulation of asynchronous electric drive systems (**Figures 14** and **15**).

control is 10.6 Hz, scalar without feedback—2.72 Hz.

formation of the torque in vector control.

**Figure 12.**

**Figure 13.**

**180**

*rotor current signal.*

*Spectrum of the rotor current signal.*

*Control Theory in Engineering*

The frequency of the rotor current under load (at a speed of 90 rps) with vector

Analysis of the spectra of rotor currents convincingly shows that the formation of the necessary torque in a vector-controlled electric drive, even when the speed loop is closed, is not the most effective; the reason for this is probably the presence of significant harmonics at a frequency of 3–8 Hz (**Figures 12** and **13**). **Table 2** shows the experimental values of the main frequencies of the rotor currents, which cast doubt on the generally accepted opinion about the high efficiency of the

*Modeling processes in an asynchronous electric drive with scalar control.*


#### **Table 2.**

*Frequency of the fundamental harmonic for various control algorithms.*

Another conclusion from these experiments is that the components of the signals of other harmonics are very significant, which makes the errors of the generalized equations, expansions on the *d* and *q* axes, and other transformations very essential. Since nonlinear operations are carried out in vector control in models of a control unit of frequency converters, these errors increase, as evidenced by higher frequencies of rotor currents under identical loads compared to scalar control.

Thus, multiple nonlinear operations are the reason for the appearance of nonfundamental harmonics in the electric motor currents and, as a result, the inefficiency of control methods.

## **2.6 AED nonlinearity is the main cause of "complex"dynamics**

The main nonlinear operation in the drive is the multiplication operation, which is very difficult to transfer to the Laplace transform domain and frequency transformations. In AC drives, the multiplication operation is performed, usually with harmonic or close to them variables. It is very important to consider that multiplying the original harmonic signal by a harmonic signal with a "carrier" frequency shifts the frequency of the original signal to the carrier frequency. In AED, this multiplication occurs twice.

The system can thus be represented as a symmetrical three-phase system with modulating and demodulating links (**Figure 16**).

As a result of the transformations, the transfer function of the torque shaping circuit will take the form

$$W\_{\hat{f}t} = \frac{3}{2} ReW(p + \dot{y}f) \tag{1}$$

for *W p*ð Þ¼ <sup>1</sup> ð Þ *<sup>T</sup>*Σ*p*þ<sup>1</sup> :

$$W\_{ft} = \frac{3}{2} \frac{T\_{\Sigma}p + 1}{\left(T\_{\Sigma}p + \mathbf{1}\right)^2 + \left(T\_{\Sigma}f\right)^2} \tag{2}$$

stability in a system "prone" to oscillation, as can be seen from the frequency

*Amplitude and phase frequency characteristics of the AED at different frequencies of the supply voltage (W1*

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

nous electric drives and explain the differences in drive dynamics at different speeds and complexity when closing the speed loop, described above. It should be noted that the representation of the torque shaping loop in the drive of the AC drive by a family of frequency characteristics, each of which corresponds to its carrier frequency, is not quite a strict solution, but other methods are even more complicated and also contain errors. **Figure 17** shows the frequency characteristics of the structure corresponding to the scheme shown in **Figure 16** at frequencies of 5 and 50 Hz. The differences are very significant, as well as the problematic synthesis of the control system, which should stabilize the acceleration process from 15.7 to 157.08 rad/s with changes in the frequency characteristic of the torque-forming

Formulas (1) and (2) describe families of frequency characteristics of asynchro-

The above results lead to the need to form a method for identifying AED dynamics. Another mathematical operation that allows to obtain transfer functions of such a structure can be a multidimensional Laplace transform with transitions to one variable using the method described in [12, 13]. The transfer function of the equivalent link after two multiplication operations is as follows, which is very similar to

All this allows to proceed to the following mathematical transformations.

Both process modeling (**Figure 3**) and experimental research (**Figures 5**–**9**) show that at different speeds and with different loads on the drive, the processes are

**3. Identification of AED dynamics by frequency characteristics**

The possibilities of effective correction will be discussed below.

characteristics of the structure.

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

*(f1 = 50 Hz), W2 (f2 = 5 Hz)).*

unit.

**183**

**Figure 17.**

the formula (2).

**3.1 Modeling of processes in AED**

High-frequency signals obtained as a result of modulation and demodulation form a symmetrical system upon addition and do not form a high-frequency component in the electromagnetic and mechanical torques, and the shift from the carrier frequency remains in the low-frequency components. This shift in the frequency response of the dynamic link characterizing the electromagnetic processes —in the stator and rotor under frequency control—is variable and largely affects the dynamics of the drive.

From this transfer function, it can be assumed how these transfer functions, which vary with frequency *ω* and slip *β*, "work" in vector and scalar controls.

With vector control, the modulating units are energized, the amplitude and phase of which are "modeled" in the control unit so as to linearize this transfer function, and such linearization potentially contains many errors. If there is no-load measurement (*β*) in the inverter, then the linearization procedure will be the same for any loads, which naturally leads to regulation errors that we observe in experiments.

With scalar control and the effect of IR and S compensations, local positive feedback is included in the structure. If this connection is "hard," it breaks the

**Figure 16.** *Structural scheme for the analysis of nonlinearities of AED.*

#### **Figure 17.**

**2.6 AED nonlinearity is the main cause of "complex"dynamics**

multiplication occurs twice.

*Control Theory in Engineering*

circuit will take the form

for *W p*ð Þ¼ <sup>1</sup>

the dynamics of the drive.

**Figure 16.**

**182**

*Structural scheme for the analysis of nonlinearities of AED.*

ð Þ *<sup>T</sup>*Σ*p*þ<sup>1</sup> :

modulating and demodulating links (**Figure 16**).

The main nonlinear operation in the drive is the multiplication operation, which is very difficult to transfer to the Laplace transform domain and frequency transformations. In AC drives, the multiplication operation is performed, usually with harmonic or close to them variables. It is very important to consider that multiplying the original harmonic signal by a harmonic signal with a "carrier" frequency shifts the frequency of the original signal to the carrier frequency. In AED, this

The system can thus be represented as a symmetrical three-phase system with

As a result of the transformations, the transfer function of the torque shaping

*T*Σ*p* þ 1

High-frequency signals obtained as a result of modulation and demodulation form a symmetrical system upon addition and do not form a high-frequency component in the electromagnetic and mechanical torques, and the shift from the carrier frequency remains in the low-frequency components. This shift in the frequency response of the dynamic link characterizing the electromagnetic processes —in the stator and rotor under frequency control—is variable and largely affects

From this transfer function, it can be assumed how these transfer functions, which vary with frequency *ω* and slip *β*, "work" in vector and scalar controls.

With vector control, the modulating units are energized, the amplitude and phase of which are "modeled" in the control unit so as to linearize this transfer function, and such linearization potentially contains many errors. If there is no-load measurement (*β*) in the inverter, then the linearization procedure will be the same for any loads, which naturally leads to regulation errors that we observe in experiments. With scalar control and the effect of IR and S compensations, local positive feedback is included in the structure. If this connection is "hard," it breaks the

*ReW p*ð Þ þ *jf* (1)

ð Þ *<sup>T</sup>*Σ*<sup>p</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>T</sup>*Σ*<sup>f</sup>* <sup>2</sup> (2)

*Wft* <sup>¼</sup> <sup>3</sup> 2

*Wft* <sup>¼</sup> <sup>3</sup> 2

*Amplitude and phase frequency characteristics of the AED at different frequencies of the supply voltage (W1 (f1 = 50 Hz), W2 (f2 = 5 Hz)).*

stability in a system "prone" to oscillation, as can be seen from the frequency characteristics of the structure.

The possibilities of effective correction will be discussed below.

Formulas (1) and (2) describe families of frequency characteristics of asynchronous electric drives and explain the differences in drive dynamics at different speeds and complexity when closing the speed loop, described above. It should be noted that the representation of the torque shaping loop in the drive of the AC drive by a family of frequency characteristics, each of which corresponds to its carrier frequency, is not quite a strict solution, but other methods are even more complicated and also contain errors. **Figure 17** shows the frequency characteristics of the structure corresponding to the scheme shown in **Figure 16** at frequencies of 5 and 50 Hz. The differences are very significant, as well as the problematic synthesis of the control system, which should stabilize the acceleration process from 15.7 to 157.08 rad/s with changes in the frequency characteristic of the torque-forming unit.

The above results lead to the need to form a method for identifying AED dynamics.

Another mathematical operation that allows to obtain transfer functions of such a structure can be a multidimensional Laplace transform with transitions to one variable using the method described in [12, 13]. The transfer function of the equivalent link after two multiplication operations is as follows, which is very similar to the formula (2).

All this allows to proceed to the following mathematical transformations.

### **3. Identification of AED dynamics by frequency characteristics**

#### **3.1 Modeling of processes in AED**

Both process modeling (**Figure 3**) and experimental research (**Figures 5**–**9**) show that at different speeds and with different loads on the drive, the processes are qualitatively different. Therefore, they should be described by different frequency characteristics. The task that should be solved first of all is the formation of the frequency characteristics of the asynchronous drive for each specific mode.

#### *3.1.1 The proposed solution*

The basis for choosing the method for calculating the dynamic mechanical characteristic, propose in the same monograph by Usoltsev ([14], p. 135).

The resulting formula connects the electromagnetic torque with the critical slip and the current slip; all of these values depend on the frequency of the stator voltage:

$$m = \frac{2M\_k}{\left(1 + T\_2/p\right)\left[\frac{S\_k}{\beta}\left(1 + T\_2'p\right)\right] + \frac{\beta}{S\_k}}\ ,\tag{3}$$

The transfer function of the torque driver changes as the stator voltage and slip

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

It should be noted that at β = 0, the transfer function, as well as the structural diagram, exactly coincides with the linear transfer function and structural diagram

*Amplitude and phase frequency characteristics of the AED at different frequencies of the supply voltage (10 Hz*

*(a) and 50 Hz (b)) and slips corresponding to small (ω1) and nominal (ω2) loads.*

frequency changes, that is, it is essentially nonlinear.

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 19.**

**185**

where *T*<sup>0</sup> <sup>2</sup> <sup>¼</sup> *Lk <sup>R</sup>*<sup>2</sup> is the rotary time constant and *<sup>β</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>ω</sup>*<sup>1</sup> is the relative slip.

Usoltsev calls this formula a dynamic mechanical characteristic and simplifies it to a first-order dynamic link that cannot be described by the processes presented in **Figure 3**.

The refinement of the linearization conditions allows us to obtain a different formula for the dynamic link connecting the torque developed by the induction motor with the rotational speed (Kloss dynamic formula), while some of the coefficients of the formula depend on the frequency of the supply voltage and slip:

$$m = \frac{2\mathcal{M}\_k \left(T\_2' p + \mathbf{1}\right)}{\left(1 + T\_2/p\right)^2 \frac{S\_k}{\beta} + \frac{\beta}{S\_k}}\tag{4}$$

Then, the equation of the connecting torque (m), the relative slip (*β*), and the motor parameters (*T*<sup>0</sup> <sup>2</sup>*, Mk, Sk*) will take the form:

$$m = \frac{2\mathbf{M}\_k \left(T\_2' p + \mathbf{1}\right) \mathbf{S}\_k \boldsymbol{\beta}}{\left(\mathbf{1} + T\_2 \mathbf{1}' p\right)^2 \mathbf{S}\_k^2 + \boldsymbol{\beta}^2},\tag{5}$$

and the transfer function connecting the absolute slip and the torque will take the form:

$$\mathbf{W}(\mathbf{p}) = \frac{2\mathbf{M}\_{\mathbf{k}} \left(\mathbf{T}\_{2}^{\prime}\mathbf{p} + \mathbf{1}\right)\mathbf{S}\_{\mathbf{k}}}{\mathbf{o}\_{1}\left[\left(\mathbf{1} + \mathbf{T}\_{2}\mathbf{'}\mathbf{p}\right)^{2}\mathbf{S}\_{\mathbf{k}}^{2} + \emptyset^{2}\right]} \tag{6}$$

where *ω*<sup>1</sup> is the frequency of the stator voltage.

The block diagram of the drive in the work area will take the form shown in **Figure 18**.

**Figure 18.**

*Structural diagram of an asynchronous motor in the working area of mechanical characteristics.*

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction… DOI: http://dx.doi.org/10.5772/intechopen.88223*

The transfer function of the torque driver changes as the stator voltage and slip frequency changes, that is, it is essentially nonlinear.

It should be noted that at β = 0, the transfer function, as well as the structural diagram, exactly coincides with the linear transfer function and structural diagram

#### **Figure 19.**

*Amplitude and phase frequency characteristics of the AED at different frequencies of the supply voltage (10 Hz (a) and 50 Hz (b)) and slips corresponding to small (ω1) and nominal (ω2) loads.*

qualitatively different. Therefore, they should be described by different frequency characteristics. The task that should be solved first of all is the formation of the frequency characteristics of the asynchronous drive for each specific mode.

The basis for choosing the method for calculating the dynamic mechanical characteristic, propose in the same monograph by Usoltsev ([14], p. 135).

> *<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*Mk* 1 þ *T*<sup>2</sup> <sup>0</sup> ð Þ *<sup>p</sup> Sk*

*<sup>R</sup>*<sup>2</sup> is the rotary time constant and *<sup>β</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

The resulting formula connects the electromagnetic torque with the critical slip and the current slip; all of these values depend on the frequency of the stator voltage:

> *<sup>β</sup>* 1 þ *T*<sup>0</sup> <sup>2</sup>*<sup>p</sup>* � � h i

Usoltsev calls this formula a dynamic mechanical characteristic and simplifies it to a first-order dynamic link that cannot be described by the processes presented in

The refinement of the linearization conditions allows us to obtain a different formula for the dynamic link connecting the torque developed by the induction motor with the rotational speed (Kloss dynamic formula), while some of the coefficients of the formula depend on the frequency of the supply voltage and slip:

<sup>2</sup>*<sup>p</sup>* <sup>þ</sup> <sup>1</sup> � �

<sup>2</sup>*<sup>p</sup>* <sup>þ</sup> <sup>1</sup> � �*Sk<sup>β</sup>*

2p <sup>þ</sup> <sup>1</sup> � �Sk

2 S2 <sup>k</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup>

2 *S*2

2 *Sk <sup>β</sup>* <sup>þ</sup> *<sup>β</sup> Sk*

*<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*Mk <sup>T</sup>*<sup>0</sup>

<sup>2</sup>*, Mk, Sk*) will take the form:

*<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*Mk <sup>T</sup>*<sup>0</sup>

W pð Þ¼ 2Mk <sup>T</sup><sup>0</sup>

*Structural diagram of an asynchronous motor in the working area of mechanical characteristics.*

where *ω*<sup>1</sup> is the frequency of the stator voltage.

1 þ *T*<sup>2</sup> <sup>0</sup> ð Þ *p*

1 þ *T*<sup>2</sup> <sup>0</sup> ð Þ *p*

Then, the equation of the connecting torque (m), the relative slip (*β*), and the

and the transfer function connecting the absolute slip and the torque will take

ω<sup>1</sup> 1 þ T2 <sup>0</sup> ð Þ p

The block diagram of the drive in the work area will take the form shown in

þ *β Sk* *,* (3)

(4)

*<sup>ω</sup>*<sup>1</sup> is the relative slip.

*<sup>k</sup>* <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> *,* (5)

h i (6)

*3.1.1 The proposed solution*

*Control Theory in Engineering*

where *T*<sup>0</sup>

motor parameters (*T*<sup>0</sup>

**Figure 3**.

the form:

**Figure 18**.

**Figure 18.**

**184**

<sup>2</sup> <sup>¼</sup> *Lk*

for the asynchronous drive, given in the monograph by Usoltsev [14]. In the proposed nonlinear interpretation, formula (5) and the block diagram (**Figure 18**) explain some of the problems of an asynchronous electric drive. To this end, it is proposed to consider the transfer functions and the corresponding frequency characteristics at "frozen" (fixed) but different values of the frequency of the stator voltage and slip. In this case, instead of the traditional characteristics of the control object, it will be necessary to consider "families" grouped by varying stator voltage (its frequency) or slip.

The frequency characteristics of an asynchronous electric drive with frequency control based on an asynchronous motor with a squirrel-cage induction motor used in the research stand (**Figure 4**) are shown in **Figure 19**. They are built using the *Matlab Simulink*© application.

There can be distinguished linear element with frequency characteristic WLF and nonlinear element (NE) that has an upper bound—static, for static nonlinear-

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

Popov obtained stability criteria by frequency characteristics of linear elements and boundary characteristics of nonlinear element. However, the practical application of the criterion for electric drives remained only limited. Plotting modified hodographs needed for the criterion was not very convenient. It was difficult to distinguish "weak" elements and suggest their adjustment. Real electric drives can hardly reduce to structures shown in **Figure 20** due to multiple cross couples, so the

ð*t*

0

Nyquist criterion is still used even if results are not sufficiently accurate.

There are several known formulations of this criterion.

should cross it even a number of times (**Figure 21**).

It is commonly believed that modern electric drives do not have a stability problem. All conventional systems use Pc, PI, and PID controllers as it is assumed that the whole system is close to a second-order linear system where these controllers are the most efficient. As a result, the wider application is being found by methods for building automatic systems based on stability criterion for linear

1.For the closed-loop system, it is necessary and sufficient that for frequencies where a Bode magnitude plot is positive (i.e., L(ω) > 0), the phase frequency characteristic of the open-loop system should not cross the axis �180° or

In practice, most often this variant of the criterion is formulated as a limitation of a phase shift of the logarithmic frequency characteristic of the open-loop system at the cutoff frequency (i.e., at L(ω) = 0) with a lower bound value (�180°).

Let us consider one of the most important features of the criterion—when it is used, only a certain range of frequency is taken into account, namely, the cutoff frequency of the system or some region around it. This results in a large number of practical consequences—criteria of negligibility of elasticity of servo drive gears, requirements for parameters of actuating motors and information systems, methods for separate, etc. Phase shift at the cutoff frequency may be used for assessment of stability "margin" of the control system (the difference between the phase shift and the critical value �180°). Along with that, results of experiments are often gravely inconsistent with a theory, but it is normally assumed that this inconsistence is

*X Y*ð Þ≤*KY* (7)

*X t*ð Þ*dt* ≤*KY* (8)

ities (7), or integral (8).

*Nonlinear automatic control system block diagram.*

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

**Figure 20.**

systems—the Nyquist criterion.

within tolerable limits.

**187**

The amplitude and phase frequency characteristics of an electric motor at a stator voltage frequency of 10 Hz and slip corresponding to small and nominal loads are shown in **Figure 19a**. **Figure 19b** shows similar characteristics for a stator voltage frequency of 50 Hz.

The given frequency characteristics well explain some of the problems of AED. When operating at low frequencies of the stator voltage, the phase shifts with changing load (and slip) change significantly, which leads to instability and inefficient operation at low speeds. It is necessary to pay attention to the change in frequency characteristics when changing the frequency of the stator voltage; this affects the acceleration processes. Thus, the nonlinearity of the transfer functions of the torque driver requires linearization to improve the efficiency of the electric drive. One of the widely used methods of linearization are various types of so-called "transvector" control. With this control, the dynamic links of the reverse dynamic links of the motor are formed in the control device. These links are adapted to different motor operation modes.

It should be noted that in a real drive, a perfect adaptation is impossible. The transfer functions incorporated in the software of the frequency converter and the real asynchronous motor may vary for several reasons (some parameters are difficult to measure, the structure of the real electric motor is much more complicated than the model, and some parameters may change during operation). Dynamic links are quite complex. This leads to the fact that the equivalent transfer functions of AED may in some modes contain resonant links that lead to control failures, high-frequency harmonics, and differences in dynamics at different speeds noted during the experiments [15, 16]. The stability analysis of such systems presents a known complexity. Moreover, the classical stability criteria for nonlinear systems do not apply to systems with dynamic nonlinearities. It is advisable to consider some of the "offshoots" of one of these criteria—Popov's criterion.
