**4. Analysis of stability of electric drives as nonlinear systems**

#### **4.1 Popov's criterion for nonlinear systems**

Stability theory is a modern mathematical field which is probably the most widely used in the modern engineering for the last 100 years. Moreover, multiple research works on this theory were inspired or conditioned by practical problems of cybernetics and electromechanical systems. The similar mathematical field is definitely the theory of stability of nonlinear systems conceived by Romanian mathematician Popov [17]. This theory, once known as absolute stability theory and later as hyperstability theory, describes conditions of stability for automatic control systems which may reduce to the simplest structure given in **Figure 20**.

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

**Figure 20.** *Nonlinear automatic control system block diagram.*

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

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

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

The given frequency characteristics well explain some of the problems of AED.

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.

**4.1 Popov's criterion for nonlinear systems**

**186**

**4. Analysis of stability of electric drives as nonlinear systems**

Stability theory is a modern mathematical field which is probably the most widely used in the modern engineering for the last 100 years. Moreover, multiple research works on this theory were inspired or conditioned by practical problems of cybernetics and electromechanical systems. The similar mathematical field is definitely the theory of stability of nonlinear systems conceived by Romanian mathematician Popov [17]. This theory, once known as absolute stability theory and later as hyperstability theory, describes conditions of stability for automatic control systems which may reduce to the simplest structure given in **Figure 20**.

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

(its frequency) or slip.

*Control Theory in Engineering*

*Matlab Simulink*© application.

voltage frequency of 50 Hz.

different motor operation modes.

There can be distinguished linear element with frequency characteristic WLF and nonlinear element (NE) that has an upper bound—static, for static nonlinearities (7), or integral (8).

$$X(Y) \le KY \tag{7}$$

$$\int\_{0}^{t} X(t)dt \le KY \tag{8}$$

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

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 systems—the Nyquist criterion.

There are several known formulations of this criterion.

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

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 within tolerable limits.

a particular link. According to the hodograph, such an analysis is possible only if this feature is singled out in a separate link. The calculation of the frequency locus of the entire system is a rather cumbersome process, and its automation makes it difficult to solve synthesis problems. The second is that in a frequency drive, it is impossible without simplification to divide the system into a purely linear link and a nonlinear structure, as is necessary in the Popov criterion. Hence, the first task is to formulate the Popov criterion in the categories of amplitude and phase frequency

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

**4.2 Popov's sustainability criterion for systems with nonlinear dynamics**

Re 1ð Þ þ *jωq WLF* þ

Re 1ð Þ þ *jωq WLF* þ

This condition is met if the phase shift of the combination of elements given in

1.linear element with frequency characteristic WLF of the linear part of the

3.instantaneous element where К is an upper bound of the nonlinear element

When this equivalent circuit is introduced into consideration, the Popov crite-

This condition may be easily checked by frequency characteristics and analyzed using methods close to synthesis methods according to the Nyquist criterion. In contrast with this latter, the whole frequency range should be analyzed like in the Popov criterion, and not only the cutoff frequency like in the Nyquist criterion. The phase shift may be assessed using methods for building equivalent frequency characteristics. Here, two elements—linear and arbitrary lead—are series-connected, and the instantaneous element is paralleled. The analysis may be based on the rule

It may be assumed that this approach is also applicable for control systems where the dynamic element may be assessed by the family of frequency characteristics, that is, for all systems for which, without a serious error, it is impossible to distinguish linear dynamic element from nonlinear one having a static upper bound

rion may be formulated as follows: **nonlinear system reduced to the circuit presented in Figure 20 is stable if the phase shift of the equivalent circuit given in Figure 22 is** �**90° minimum, which is equivalent to positivity of the real part**

Analytically, the Popov criterion is as follows: the closed-loop automatic control system as in **Figure 20** is stable if there is a real positive *q* such that for linear and

1

1 *K*

<sup>&</sup>gt;<sup>0</sup> (10)

*<sup>K</sup>* <sup>&</sup>gt;<sup>0</sup> (9)

characteristics, to make it look like the Nyquist criterion.

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

nonlinear elements of the system, the following condition is met:

This condition is equivalent to the following one (as K is real):

Let us consider the elements of this diagram:

2.arbitrary lead element determined by *q*

like in the Popov criterion for the initial system

**of the frequency characteristic of this equivalent circuit.**

of "positive limiting" of paralleled elements (**Figure 22**).

(**Figure 20**). Here, the condition will be analytical:

**Figure 22** is over (�90°).

initial system

**189**

**Figure 21.** *Nyquist criterion by the Bode magnitude plot (1, instable system; 2, stable system).*

In any case, attempts to question the very possibility of constructing electric drives, as something close to the desired linear systems, were extremely rare. Despite the fact that the presence of highly essential nonlinearities in them is not disputed by anyone. But their influence is not considered to be sufficient to abandon the usual methods of building electric drives.

The peak of research in this theory fell on the 1970s–1980s of the twentieth century and, at present, is considered irrelevant. This is due to many reasons. Let us discuss some of them more thoroughly. The first obvious reason is that many problems have already been solved over the years. The second is that due to technological progress in related areas of technology (in electronics, in engineering, especially in computing), the problem of sustainability has become "routine" for many cases. Since the characteristics of almost all elements of the systems become such that they do not affect the stability with modern requirements on accuracy and speed. However, in recent years, a class of systems has appeared, or rather "manifested," in which in the near future, a stability analysis will become extremely important. At the same time, the nonlinearity of these systems is obvious, and modern means and methods with which these nonlinearities are trying to compensate, according to many researchers, can lead not only to inefficient modes of operation but also to the emergence of critical and even emergency situations.

The authors conducted a whole range of experiments with these systems, and they concluded that it was necessary to analyze control systems from the point of view of stability and take into account their essential nonlinearities. These nonlinearities are such that linearization is quite difficult to carry out, and simplifications lead to the "emasculation" of any complexity in these tasks, while in practice all the difficulties remain. So Usoltsev [14] forcedly reduces the nonlinear asynchronous electric drive to a linear system II or even I order.

We assume that the widespread introduction of frequency control systems is a sufficient reason to return to the formulation and solution of problems of analyzing the nonlinear system stability. In our opinion, there are two main reasons for the rare use of Popov criterion in electric drives. One of them, purely technical, is that the criterion operates with frequency loci. Engineers are accustomed to working with amplitude and phase frequency characteristics, on which the influence of each link and any of its features is very clear. According to these characteristics, it is convenient to apply the Nyquist criterion and evaluate the effect on the stability of

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

a particular link. According to the hodograph, such an analysis is possible only if this feature is singled out in a separate link. The calculation of the frequency locus of the entire system is a rather cumbersome process, and its automation makes it difficult to solve synthesis problems. The second is that in a frequency drive, it is impossible without simplification to divide the system into a purely linear link and a nonlinear structure, as is necessary in the Popov criterion. Hence, the first task is to formulate the Popov criterion in the categories of amplitude and phase frequency characteristics, to make it look like the Nyquist criterion.

#### **4.2 Popov's sustainability criterion for systems with nonlinear dynamics**

Analytically, the Popov criterion is as follows: the closed-loop automatic control system as in **Figure 20** is stable if there is a real positive *q* such that for linear and nonlinear elements of the system, the following condition is met:

$$(\operatorname{Re}(\mathbf{1} + jaq)\mathcal{W}\_{LF} + \frac{\mathbf{1}}{K} > \mathbf{0}) \tag{9}$$

This condition is equivalent to the following one (as K is real):

$$\operatorname{Re}\left[ (\mathbf{1} + j\alpha q) \mathcal{W}\_{LF} + \frac{\mathbf{1}}{K} \right] > \mathbf{0} \tag{10}$$

This condition is met if the phase shift of the combination of elements given in **Figure 22** is over (�90°).

Let us consider the elements of this diagram:


When this equivalent circuit is introduced into consideration, the Popov criterion may be formulated as follows: **nonlinear system reduced to the circuit presented in Figure 20 is stable if the phase shift of the equivalent circuit given in Figure 22 is** �**90° minimum, which is equivalent to positivity of the real part of the frequency characteristic of this equivalent circuit.**

This condition may be easily checked by frequency characteristics and analyzed using methods close to synthesis methods according to the Nyquist criterion. In contrast with this latter, the whole frequency range should be analyzed like in the Popov criterion, and not only the cutoff frequency like in the Nyquist criterion. The phase shift may be assessed using methods for building equivalent frequency characteristics. Here, two elements—linear and arbitrary lead—are series-connected, and the instantaneous element is paralleled. The analysis may be based on the rule of "positive limiting" of paralleled elements (**Figure 22**).

It may be assumed that this approach is also applicable for control systems where the dynamic element may be assessed by the family of frequency characteristics, that is, for all systems for which, without a serious error, it is impossible to distinguish linear dynamic element from nonlinear one having a static upper bound (**Figure 20**). Here, the condition will be analytical:

In any case, attempts to question the very possibility of constructing electric drives, as something close to the desired linear systems, were extremely rare. Despite the fact that the presence of highly essential nonlinearities in them is not disputed by anyone. But their influence is not considered to be sufficient to aban-

The peak of research in this theory fell on the 1970s–1980s of the twentieth century and, at present, is considered irrelevant. This is due to many reasons. Let us discuss some of them more thoroughly. The first obvious reason is that many problems have already been solved over the years. The second is that due to technological progress in related areas of technology (in electronics, in engineering, especially in computing), the problem of sustainability has become "routine" for many cases. Since the characteristics of almost all elements of the systems become such that they do not affect the stability with modern requirements on accuracy and speed. However, in recent years, a class of systems has appeared, or rather "manifested," in which in the near future, a stability analysis will become extremely important. At the same time, the nonlinearity of these systems is obvious, and modern means and methods with which these nonlinearities are trying to compensate, according to many researchers, can lead not only to inefficient modes of operation but also to the emergence of critical and even emergency

The authors conducted a whole range of experiments with these systems, and they concluded that it was necessary to analyze control systems from the point of view of stability and take into account their essential nonlinearities. These nonlinearities are such that linearization is quite difficult to carry out, and simplifications lead to the "emasculation" of any complexity in these tasks, while in practice all the difficulties remain. So Usoltsev [14] forcedly reduces the nonlinear asyn-

We assume that the widespread introduction of frequency control systems is a sufficient reason to return to the formulation and solution of problems of analyzing the nonlinear system stability. In our opinion, there are two main reasons for the rare use of Popov criterion in electric drives. One of them, purely technical, is that the criterion operates with frequency loci. Engineers are accustomed to working with amplitude and phase frequency characteristics, on which the influence of each link and any of its features is very clear. According to these characteristics, it is convenient to apply the Nyquist criterion and evaluate the effect on the stability of

chronous electric drive to a linear system II or even I order.

don the usual methods of building electric drives.

*Nyquist criterion by the Bode magnitude plot (1, instable system; 2, stable system).*

situations.

**188**

**Figure 21.**

*Control Theory in Engineering*

**Figure 22.** *Equivalent circuit of nonlinear automatic control system and its frequency responses.*

$$\operatorname{Re}\left[ (\mathbf{1} + jaq)\mathcal{W}\_{\sim} + \frac{\mathbf{1}}{K} \right] > \mathbf{0} \tag{11}$$

**4.3 Correction for servo drive with elastic element**

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

tural correction before the generally accepted variant.

*Stability assessment for servo drive with elastic element according to the Nyquist criterion (satisfied).*

(**Figures 24** and **25**).

from 0 to ∞ Hz.

**Figure 24.**

**191**

The difference between the suggested criterion (the suggested formulation of the Popov criterion, to be exact) and the Nyquist criterion is well seen considering the example of stability of a servo drive with a finite rigidity of gears. According to the Nyquist criterion, it is enough to "move the cutoff frequency away" from the frequency of elastic oscillations to "forget" about it. However, in practice it does not work like that. Stability testing by the means of suggested method proves it

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

As can be seen from **Figures 19** and **20**, the link describing the gearbox, taking into account backlash and friction, can only be represented by a family of highorder dynamic links that will not allow an acceptable phase shift of an equivalent combination of links to satisfy the stability condition in the entire frequency range

Effect of stabilization (or a fine stability margin) may be reached via introduction of actuating motor rate feedback which allows the adjustment of the combination of elements (**Figure 26**) and fulfillment of stability conditions for the whole system (or sufficient stability margin) (**Figure 27**). At the same time, it is necessary to pay special attention to the fact that this connection "works" in the entire frequency range—from 0 to +∞ Hz. The use of differentiating channels in PID regulators is always limited to the cutoff zone or slightly larger frequency range, because at high frequencies this channel enhances the influence of interference. And for the stability of the drive when the "shifted" frequency response of the elastic link, D-channel of regulator is not enough. Thus, the proposed interpretation of the stability criterion makes it possible to substantiate the advantages of struc-

or according to the equivalent circuit, for the phase shift:

$$
\rho \left[ (\mathbf{1} + jaq) \mathcal{W}\_{\sim} + \frac{\mathbf{1}}{K} \right] > -90^{\circ} \tag{12}
$$

Stability requires the positivity of a real part of the complex transfer function of the equivalent circuit (**Figure 23**) or the limitation of the phase shift of the same complex of elements at a level of �90° for all possible frequency characteristics. Phase margin may be estimated by the difference between the actual phase and �90°. Values of frequencies where the phase is below �90° allow the assessment of frequency of self-oscillations in the system.

**Figure 23.** *Conditions of stability for control systems with dynamic nonlinear elements.*

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

## **4.3 Correction for servo drive with elastic element**

The difference between the suggested criterion (the suggested formulation of the Popov criterion, to be exact) and the Nyquist criterion is well seen considering the example of stability of a servo drive with a finite rigidity of gears. According to the Nyquist criterion, it is enough to "move the cutoff frequency away" from the frequency of elastic oscillations to "forget" about it. However, in practice it does not work like that. Stability testing by the means of suggested method proves it (**Figures 24** and **25**).

As can be seen from **Figures 19** and **20**, the link describing the gearbox, taking into account backlash and friction, can only be represented by a family of highorder dynamic links that will not allow an acceptable phase shift of an equivalent combination of links to satisfy the stability condition in the entire frequency range from 0 to ∞ Hz.

Effect of stabilization (or a fine stability margin) may be reached via introduction of actuating motor rate feedback which allows the adjustment of the combination of elements (**Figure 26**) and fulfillment of stability conditions for the whole system (or sufficient stability margin) (**Figure 27**). At the same time, it is necessary to pay special attention to the fact that this connection "works" in the entire frequency range—from 0 to +∞ Hz. The use of differentiating channels in PID regulators is always limited to the cutoff zone or slightly larger frequency range, because at high frequencies this channel enhances the influence of interference. And for the stability of the drive when the "shifted" frequency response of the elastic link, D-channel of regulator is not enough. Thus, the proposed interpretation of the stability criterion makes it possible to substantiate the advantages of structural correction before the generally accepted variant.

**Figure 24.** *Stability assessment for servo drive with elastic element according to the Nyquist criterion (satisfied).*

Re 1ð Þ þ *jωq W*� þ

*φ* ð Þ 1 þ *jωq W*� þ

or according to the equivalent circuit, for the phase shift:

*Equivalent circuit of nonlinear automatic control system and its frequency responses.*

frequency of self-oscillations in the system.

*Conditions of stability for control systems with dynamic nonlinear elements.*

**Figure 22.**

*Control Theory in Engineering*

**Figure 23.**

**190**

1 *K*

1 *K*

Stability requires the positivity of a real part of the complex transfer function of the equivalent circuit (**Figure 23**) or the limitation of the phase shift of the same complex of elements at a level of �90° for all possible frequency characteristics. Phase margin may be estimated by the difference between the actual phase and �90°. Values of frequencies where the phase is below �90° allow the assessment of

>0 (11)

> � 90° (12)

**4.4 Corrections of asynchronous electric drive**

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

**Figure 28.**

**Figure 29.**

**193**

*Popov criterion.*

It is interesting to analyze the stability of AED, the block diagram of which contains several nonlinear dynamic links described above. As shown in **Figure 1**, vector control "linearizes" the drive converting it to a linear structure. As was shown above, this linearization is very conditional and not too accurate mathematically. However, even if we make assumptions about Laplace transformations with variable coefficients, it is impossible to avoid discrepancies in the parameters (and structure) of the model and the real motor. This will lead to the fact that their serial connection will produce a disproportionate linear link and a floating dynamic link that will be close to the resonant link. Even if we assume that this link is close to linear, for stability it is necessary to apply the stability criterion for nonlinear systems (a variant of the Popov criterion) and, therefore, to consider the entire frequency range, and not just the cutoff frequency. **Figures 28** and **29** show how applying the Nyquist criterion gives an incorrect result (and the Popov criterion is not satisfied) for a vector-controlled drive closed in speed with a PID controller. As the experiments showed, the applied algorithms are not very effective, and the hard positive feedback on the stator current breaks the stability. In works [6, 15, 18] the positive dynamic connection on current is described. Without compromising

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

*Assessment of stability in the AC drive according to the Popov criterion (not satisfied).*

*Stabilization of the drive by positive feedback of the stator current with a first-order filter. Verification by*

**Figure 25.** *Stability assessment for servo drive according to the Popov criterion (not satisfied).*

**Figure 26.** *Block diagram of servo drive with stabilizing rate feedback.*

**Figure 27.** *Stability assessment for servo drive with stabilizing rate feedback according to the Popov criterion (satisfied).*

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