How the inertial Nyquist hodograph is constructed. Amplitude-phase characteristic (Nyquist hodograph). The use of l.a.ch.h. and phase frequency characteristics for system stability analysis

This is the locus of points that describes the end of the frequency transfer function vector as the frequency changes from -∞ to +∞. The value of the segment from the origin to each point of the hodograph shows how many times at a given frequency the output signal is greater than the input, and the phase shift between the signals is determined by the angle to the mentioned segment.

All other frequency dependences are generated from the AFC:

• U(w) - even (for closed ACS P(w));
• V(w) - odd;
• A(w) - even (frequency response);
• j(w) - odd (PFC);
• LACHH & LPCHH - are used most often.

Logarithmic frequency characteristics.

Logarithmic frequency responses (LFC) include logarithmic amplitude response (LAFC) and logarithmic phase response (LPCH) built separately on the same plane. The construction of LAFC & LPFC is made according to the expressions:

L(w) = 20 lg | W(j w)| = 20 lg A(w), [dB];

j(w) = arg( W(j w)), [rad].

Value L(w) is expressed in decibels . Bel is a logarithmic unit corresponding to a tenfold increase in power. One Bel corresponds to an increase in power by 10 times, 2 Bels - by 100 times, 3 Bels - by 1000 times, etc. A decibel is equal to one tenth of a Bel.

Examples of AFC, AFC, PFC, LAFC and LPFC for typical dynamic links are shown in Table 2.

Table 2. Frequency characteristics of typical dynamic links.

Principles of automatic control

According to the principle of control, ACS can be divided into three groups:

1. With regulation by external influence - the Poncelet principle (used in open ACS).
2. With regulation by deviation - the Polzunov-Watt principle (used in closed ACS).
3. With combined regulation. In this case, the ACS contains closed and open control loops.

The principle of control by external disturbance

Disturbance sensors are required in the structure. The system is described by the transfer function of an open system: x(t) = g(t) - f(t).

Advantages:

• It is possible to achieve complete invariance to certain perturbations.
• The problem of system stability does not arise, since no OS.

Flaws:

• A large number of disturbances requires an appropriate number of compensation channels.
• Changes in the parameters of the regulated object lead to errors in the control.
• Can only be applied to objects whose characteristics are clearly known.

Deviation control principle

The system is described by the transfer function of an open system and the closure equation: x(t) = g(t) - y(t) W oc ( t). The algorithm of the system is concluded in an effort to reduce the error x(t) to zero.

Advantages:

• Environmental protection leads to a decrease in the error, regardless of the factors that caused it (changes in the parameters of the regulated object or external conditions).

Flaws:

• OS systems have a stability problem.
• It is fundamentally impossible to achieve absolute invariance to perturbations in systems. The desire to achieve partial invariance (not the first OS) leads to the complication of the system and the deterioration of stability.

Combined control

Combined control consists in a combination of two principles of control by deviation and external disturbance. Those. the control signal to the object is formed by two channels. The first channel is sensitive to the deviation of the controlled value from the reference. The second one forms the control action directly from the setting or disturbing signal.

x(t) = g(t) - f(t) - y(t)Woc(t)

Advantages:

• The presence of environmental protection makes the system less sensitive to changes in the parameters of the regulated object.
• Adding a reference or perturbation sensitive channel(s) does not affect the stability of the feedback loop.

Flaws:

• Channels that are sensitive to a task or to a disturbance usually contain differentiating links. Their practical implementation is difficult.
• Not all objects allow forcing.

ATS sustainability analysis

The concept of stability of the regulatory system is associated with its ability to return to a state of equilibrium after the disappearance of external forces that brought it out of this state. Stability is one of the main requirements for automatic systems.

The concept of stability can also be extended to the case of ACS movement:

• undisturbed movement,
• indignant movement.

The movement of any control system is described using a differential equation, which generally describes 2 operating modes of the system:

Steady state mode

Driving mode

In this case, the general solution in any system can be written as:

forced the component is determined by the input action on the CS input. The system reaches this state at the end of transient processes.

Transitional the component is determined by the solution of a homogeneous differential equation of the form:

The coefficients a 0 ,a 1 ,…a n include the system parameters => changing any coefficient of the differential equation leads to a change in a number of system parameters.

Solution of a homogeneous differential equation

where are the constants of integration, and are the roots of the characteristic equation of the following form:

The characteristic equation is the denominator of the transfer function set to zero.

The roots of the characteristic equation can be real, complex conjugate and complex, which is determined by the parameters of the system.

To assess the stability of systems, a number of sustainability criteria

All sustainability criteria are divided into 3 groups:

root

- algebraic

An important theorem from the theory of functions of a complex variable establishes: let the function be single-valued inside a simply connected contour C and, moreover, be single-valued and analytic on this contour. If it is not equal to zero on C and if there can be only a finite number of singular points (poles) inside the contour C, then

where is the number of zeros, and is the number of poles inside C, each of which is taken into account according to its multiplicity.

This theorem follows directly from Cauchy's residue theorem, which states that

We replace by and note that the singularities are preserved both at zeros and at the poles. Then the residues found at these singular points will be equal to the multiplicities of the singular points with a positive sign at zeros and a negative sign at the poles. The theorem formulated above is now obvious.

Relation (11.2-1) can also be written as

Since on the contour C will generally have both real and imaginary parts, its logarithm will be written in the form

Provided that C does not vanish anywhere on the boundary, integration in (II.2-3) gives directly

where denote an arbitrary beginning and end of a closed contour C. Therefore,

Combining the results (II.2-1) and (II.2-7), we find that the product of the total change in angle (complete reversal around the origin) when running around contour C is equal to the difference between zeros and poles inside contour C.

If is the total number of revolutions around the origin with C running around, then we can write

moreover, the contour C is traversed in the direction corresponding to the increase in the positive angle, and the revolution is called positive if it also occurs in the direction corresponding to the increase in the positive angle.

Rice. II.2-1. A closed contour enclosing the end part of the right half-plane.

Now these results can be applied directly to the problem of determining stability. We want to know if the denominator of the transfer function has zeros in the right half-plane.

Therefore, contour C is chosen so as to completely cover the right half-plane. This circuit is shown in Fig. where the large semicircle enclosing the right half-plane is given by the relations

as it tends to infinity in the limit.

Suppose it is written as

where is an entire function of and that have no common factors. Let us construct a diagram in the complex plane, changing the values ​​along the contour C. This diagram will give us some closed contour. In the general case, it will be an entire function of polynomial form, which, obviously, has no poles in the finite part of the plane. If transcendental, then the number P of poles in the final part of the right half-plane is to be determined. Knowing P and determining from the diagram when C runs through, we can now determine, according to equation (II.2-8), the number of zeros in the right half-plane

Rice. II.2-2. Simple single-loop control system.

For the system to be stable, it must be equal to zero. Therefore, the application of this criterion involves two stages: the first is the determination of the poles in the right half-plane, and the second is the construction of the diagram when C runs through. The first stage is usually quite simple. The second one can present considerable difficulties, especially if it is of the third or higher order and if it contains transcendent members.

For the feedback control system shown in general view in fig. The complexity of plotting can be greatly reduced by using an open-loop transfer function. The transfer function of a closed system is related to the transfer function of an open system by the relation

where they can have both poles and zeros. In the stability problem, it is desirable to know whether has poles in the right half-plane. This is equivalent to having the zeros of the function in the right half-plane, or having the zeros of the function in the right half-plane, shifted by -1. expressions (II.2-12) in the form where K is the open loop gain. Now the poles are identical to zeros with respect to

To apply the Nyquist criterion, we first draw the contour C, which covers

the entire right half-plane. After that, we calculate the total number of revolutions with the same movement around the point. The change in the gain K changes only the position of the point and does not affect the location more difficult calculation if it has a polynomial or transcendental form. The stability of the system is then determined by the direct application of equation (II.2-8), which establishes

Therefore, the system is stable only if it is equal to zero, where now the number of zeros of the denominator (II.2-12) in

Rice. II.2-3. Two possible modifications of circuits with bypassing the poles on the imaginary axis.

When applying the criterion in this form, attention should be paid to the choice of contour C enclosing the right half-plane. Relationship (11.2-1), and hence (11.2-13) require the absence of singularities of the displayed function on the contour C. There are frequent cases when it has a pole at the origin or even several pairs of complex conjugate poles on the imaginary axis. To consider these special cases, congur C is modified by going around each of the singularities in very small semicircles, as shown in Fig. II.2-3. If the features are poles, then the modified contour C can pass either to the right or to the left of them, as shown in Fig. II.2-3,a and II.2-3,b, respectively. If the singularity is not a pole, then the contour must always pass to the right of it, since relation (II.2-1) allows only such singularities as poles inside the contour C. Those poles on the imaginary axis that go around to the left lie inside the contour C and, therefore, must be taken into account in P. In this case, the contour C in the nearest neighborhood of the singular point is usually chosen in the form

where the angle varies from to tends to zero in the limit.

The hodograph when running through the contour C, consists mainly of four parts. Hodograph at

excluding neighborhoods of singularities on the imaginary axis, is simply the frequency response of an open system. Therefore, the hodograph at can be obtained by displaying it at relative to the real axis. When an infinite semicircle runs through, the value for all physically feasible systems is zero, or at most a finite constant. Finally, the hodograph when running through small semicircles in the vicinity of the poles on the imaginary axis is determined by direct substitution of the expression (II.2-14) into this function. Thus, the mapping of the contour C onto the function plane is completed.

When applying the criterion in this form, the nature of the restrictions imposed on it becomes apparent. First, it can have only a finite number of pole-type singularities in the right half-plane. Second, it can have only a finite number of singularities (poles or branch points) on the imaginary axis. The class of functions can be extended to include functions that have branch points only if the branch points lie in the left half-plane and if the principal value of the function is used. Third, the essential features of the form in the numerator are permissible, since the absolute value of this function, when varied within the right half-plane, is between and 0.

It is advisable to show the application of the Nyquist criterion by an example. Let the controlled system with feedback be defined by the relations

The transfer function of the given elements corresponds to a two-phase induction motor operating at a frequency from a half-wave magnetic amplifier. The presence of negative attenuation is associated with low rotor resistance. The first question arises: is it possible to stabilize the given elements only due to the gain factor? Let us therefore

The transfer function of an open system takes the form

We see, firstly, that it has only one pole in the right half-plane and this pole is located at the point. II.2-4, a, is shown in fig. II.2-4, b and shows that with the selected gain around the point there is one positive turn.

Rice. II.2-4. Examples of Nyquist diagrams.

Therefore, using the Nyquist criterion, expressed by equation (II.2-13), we arrive at the result

An increase in K creates the possibility of more positive revolutions due to the helical nature of the multiplier part of the diagram, we can therefore conclude that the system is unstable for all positive values ​​of K.

For negative values ​​of K, we can either rotate our diagram about the origin and consider rotations around a point, or use an existing diagram and consider rotations around a point. The latter method is easier; it directly shows that, at least, there are no positive turns around. This gives at least one zero in the right half-plane for negative values ​​of K. Therefore, we conclude that the system is unstable for all values ​​of K, both positive and negative, and therefore some correction is required to make the system stable.

The Nyquist criterion can also be applied when the frequency response of an open system is built from experimental data. The transfer function of an open system must then be stable and, therefore, cannot have poles in the right half-plane, i.e. . In order to construct the Nyquist hodograph correctly, care must be taken to accurately determine the behavior of the system at very low frequencies.

When applying the Nyquist criterion to multi-loop systems, the construction starts from the innermost loop and continues to the outer loops, carefully counting the number of poles in the SPP from each individual loop. The work involved in this method can often be reduced by destroying some of the contours as a result of the block diagram transformation. The choice of the hodograph construction sequence for multiloop systems depends on the block diagram, as well as on the location of the given and corrective elements in the loops.

Job condition.

Using the Mikhailov stability criterion, Nyquist determine the stability of a single-loop control system that has a transfer function of the form in the open state

in the formula put down the values ​​of K, a, b and c according to the option.

W(s) = , (1)

Construct Mikhailov and Nyquist hodographs. Determine the cutoff frequency of the system.

Determine the critical value of the system gain.

Solution.

The tasks of analysis and synthesis of control systems are solved with the help of such a powerful mathematical apparatus as the operational calculus (transformation) of Laplace. The tasks of analysis and synthesis of control systems are solved with the help of such a powerful mathematical apparatus as the operational calculus (transformation) of Laplace. The general solution of the operator equation is the sum of the terms determined by the values ​​of the roots of the characteristic polynomial (polynomial):

D(s) =  ds n d n ) .

Construction of Mikhailov's hodograph.

A) We write out the characteristic polynomial for a closed system described by equation (1)

D(s) = 50 + (25s+1)(0.1s+1)(0.01s+1) = 50+(625+50s+1)(0.001+0.11s+1) =0.625+68.85 +630.501+50.11s+51.

Polynomial roots D(s) may be: null; real (negative, positive); imaginary (always paired, conjugate) and complex conjugate.

B) Transform to the form s→ ωj

D()=0.625+68.85+630.501+50.11+51=0.625ω-68.85jω- 630.501ω+50.11jω+51

ω – signal frequency, j = (1) 1/2 – imaginary unit. J 4 = (-1) 4/2 =1, J 3 = (-1) 3/2 = -(1) 1/2 = - j, J 2 = (-1) 2/2 = -1, J \u003d (-1) 1/2 \u003d j,

C) Separate the real and imaginary parts.

D= U()+jV(), where U() is the real part and V() is the imaginary part.

U(ω) =0.625ω-630.501ω+51

V(ω) =ω(50.11-68.85ω)

D) We construct the Mikhailov hodograph.

We construct the Mikhailov hodograph near and passed from zero, for this we construct D(jw) when changing w from 0 to +∞. Let's find the intersection points U(w) and V(w) with axles. Let's solve the problem using Microsoft Excel.

We set the values ​​of w in the range from 0 to 0.0001 to 0.1, calculate in Table. excel values U(ω) and V(ω), D(ω); find intersection points U(w) and V(w) with axles,

We set the values ​​of w in the range from 0.1 to 20, calculate in Table. excel values U(w) and V(w), D; find intersection points U(w) and V(w) with axles.

Table 2.1 - Definition of the real and imaginary parts and the polynomial itself D() using Microsoft Excel

Rice. A, B, ….. Dependencies U(ω) and V(ω), D(ω) from ω

According to fig. A, B, ... .. find the intersection points U(w) and V(w) with axles:

at ω = 0 U(ω)= …. And V(ω)= ……

Fig.1. Mikhailov's hodograph at ω = 0:000.1:0.1.

Fig.2. Mikhailov's hodograph at ω = 0.1:20

E) Conclusions about the stability of the system according to the hodograph.

The stability (as a concept) of any dynamic system is determined by its behavior after the removal of external influence, i.e. its free movement under the influence of initial conditions. A system is stable if it returns to its original state of equilibrium after the signal (perturbation) that brought it out of this state ceases to act on the system. An unstable system does not return to its original state, but continuously moves away from it with time. To assess the stability of the system, it is necessary to investigate the free component of the solution to the equation of dynamics, i.e. the solution to the equation:.

D(s) =  ds n d n )= 0.

Check the stability of the system using the Mikhailov criterion :

Mikhailov criterion: For a stable ASR, it is necessary and sufficient that the Mikhailov hodograph (see Fig. 1 and Fig. 2), starting at w = 0 on the positive real semiaxis, sequentially goes around in the positive direction (counterclockwise) as w increases from 0 to ∞ n quadrants, where n is the degree of the characteristic polynomial.

It can be seen from the solution (see Fig. 1 and Fig. 2) that the hodograph satisfies the following conditions of the criterion: It starts on the positive real semiaxis at w = 0. The hodograph does not satisfy the following conditions of the criterion: it does not bypass all 4 quadrants in the positive direction (the degree of the polynomial n=4) at ω.

We conclude that this open-loop system is not stable. .

Construction of the Nyquist hodograph.

A) Let's make a change in the formula (1) s → ωj

W(s) = =,

B) Open the brackets and select the real and imaginary parts in the denominator

C) Multiply by the conjugate and select the real and imaginary parts

,

where U() is the real part and V() is the imaginary part.

D) Let's construct the Nyquist hodograph: - dependence of W() on .

Fig.3. Nyquist hodograph.

E) Check the stability of the system using the Nyquist criterion:

Nyquist criterion: In order for the system, which was stable in the open state, to be stable in the closed state, it is necessary that the Nyquist hodograph, when the frequency changes from zero to infinity, does not cover the point with coordinates (-1; j0).

It can be seen from the solution (see Fig. 3) that the hodograph satisfies all the conditions of the criterion:

Hodograph changes its direction clockwise

The hodograph does not cover the point (-1; j0)

We conclude that this open-loop system is stable .

Determination of the critical value of the system gain.

A) In paragraph 2, the real and imaginary parts have already been distinguished

B) In order to find the critical value of the system gain, it is necessary to equate the imaginary part to zero, and the real part to -1

C) Find from the second (2) equation

The numerator must be 0.

We accept that then

C) We substitute into the first (1) equation and find

Critical value of the system gain.

Literature:

1.Methods of classical and modern theory of automatic control. Volume 1

Analysis and statistical dynamics of automatic control systems. M: Ed. MSTU named after Bauman. 2000

2. Voronov A.A. Theory of automatic control. T. 1-3, M., Nauka, 1992

The Nyquist stability criterion was formulated and substantiated in 1932 by the American physicist H. Nyquist. The Nyquist stability criterion is most widely used in engineering practice for the following reasons:

- the stability of the system in the closed state is studied by the frequency transfer function of its open part W p (jw), and this function, most often, consists of simple factors. The coefficients are the real parameters of the system, which allows you to choose them from the conditions of stability;

- to study stability, it is possible to use the experimentally obtained frequency characteristics of the most complex elements of the system (control object, executive bodies), which increases the accuracy of the results obtained;

- the stability of the system can be investigated by logarithmic frequency characteristics, the construction of which is not difficult;

- the stability margins of the system are quite simply determined;

- it is convenient to use for assessing the stability of ACS with a delay.

The Nyquist stability criterion makes it possible to evaluate the stability of the ACS by the AFC of its open-loop part. There are three cases of applying the Nyquist criterion.

1. The open part of the ACS is stable.For the stability of a closed system, it is necessary and sufficient that the AFC of the open part of the system (the Nyquist hodograph) when changing frequencies w from 0 to +¥ did not cover the point with coordinates [-1, j 0]. On fig. 4.6 shows the main possible situations:

1. - a closed system is absolutely stable;

2. - ATS is conditionally stable, i.e. stable only in a certain range of change in the transmission coefficient k;

3. - ATS is on the borderline of sustainability;

4. - ATS is unstable.

Rice. 4.6. Nyquist hodographs when the open part of the ACS is stable

2. The open part of the ACS is on the border of stability.In this case, the characteristic equation has zero or purely imaginary roots, while the other roots have negative real parts.

For the stability of a closed system, if the open part of the system is on the stability boundary, it is necessary and sufficient that the AFC of the open part of the system when changing w from 0 to +¥, supplemented by an arc of infinitely large radius in the section of the discontinuity, did not cover the point with coordinates [-1, j 0]. In the presence of ν zero roots of the AFC of the open part of the system at w=0 by an arc of infinitely large radius moves from the positive real semiaxis by an angle of degrees clockwise, as shown in Fig. 4.7.

Rice. 4.7. Nyquist hodographs with zero roots

If there is a pair of purely imaginary roots w i =, then the AFC at a frequency w i an arc of infinitely large radius moves clockwise through an angle of 180°, which is shown in Fig. 4.8.

Rice. 4.8. Nyquist hodograph in the presence of a pair of purely imaginary roots

3. The open part of the system is unstable, i.e. characteristic equation has l roots with a positive real part. In this case, for the stability of a closed system, it is necessary and sufficient that when the frequency changes w from 0 to +¥ AFC of the open part of the ACS covered the point

[-1, j 0) l/2 times in the positive direction (counterclockwise).

With a complex shape of the Nyquist hodograph, it is more convenient to use another formulation of the Nyquist criterion, proposed by Ya.Z. Tsypkin using the transition rules. The transition of the AFC of the open-loop part of the system with increasing w the segment of the real axis from -1 to -¥ from top to bottom is considered positive (Fig. 4.9), and from bottom to top negative. If the AFC starts on this segment at w=0 or ends at w=¥ , then it is considered that the AFC makes the transition half.

Rice. 4.9. The transitions of the Nyquist hodograph through the segment P( w) from -¥ to -1

Closed system is stable, if the difference between the number of positive and negative transitions of the Nyquist hodograph through the segment of the real axis from -1 to -¥ is equal to l/2, where l is the number of roots of the characteristic equation with a positive real part.

Construction of Nyquist hodographs from the transfer function of an open-loop system given as a polynomial

The frequency Nyquist criterion in the study of the stability of automatic systems is based on the amplitude-phase frequency response of an open system and can be formulated as follows:

if the characteristic equation of an open-loop system of the n-th order has k roots with a positive real part (k = 0, 1, ..... n) and n-k roots with a negative real part, then for the stability of the closed system it is necessary and sufficient that the amplitude-phase hodograph frequency response of an open system (Nyquist hodograph) covered the point (-1, j0) of the complex plane at an angle k p, or, which is the same, covered the point (-1, j0) in the positive direction, i.e. counterclockwise, k times.

For a particular case, when the characteristic equation of an open system has no roots with a positive real part (k = 0), i.e. when it is stable in the open state, the Nyquist criterion is formulated as follows:

the automatic control system is stable in the closed state, if the amplitude-phase frequency response of the open system when the frequency changes from 0 to? does not cover the point of the complex plane with coordinates (-1, j0).

The Nyquist stability criterion is convenient to apply to feedback systems, especially high-order systems.

To build the Nyquist hodograph, we will use the transfer function of an open-loop system in symbolic form from Practical Lesson No. 5

We write it in symbolic-digital form for the given parameters of all elements of the system, except for the transfer coefficient of the magnetic amplifier:

Let us write the equation of the amplitude-phase frequency response, select the real and imaginary frequency responses, and construct a family of Nyquist hodographs as a function of the frequency and transfer coefficient of the magnetic amplifier.

Building a graph of the amplitude-phase frequency response in MathСad

Fig.3. A family of Nyquist hodograph curves constructed for an open-loop transfer function as a function of k mu .

Figure 3 shows that one of the Nyquist hodographs passes through a point with coordinates (j0, -1) . Consequently, in a given range of change in the transfer coefficient of the magnetic amplifier, there is also its critical value. To determine it, we use the following relations:

Therefore, the critical gain of the magnetic amplifier is:

k mukr =11.186981170416560078

Let's make sure this is true. To do this, we construct the Nyquist hodograph curves for three values ​​of the magnetic amplifier transfer coefficient: k mu = 0.6 k mukr ; k mu = k mukr ; k mu =1.2k mukr

Fig.4.

k mu = 0.6 k mucr; k mu = k mukr; k mu =1.2 k mucr

The curves in Fig. 4 confirm that the critical transfer coefficient of the magnetic amplifier has been found correctly.

The use of l.a.ch.h. and phase frequency characteristics for system stability analysis

The stability criterion for the system in terms of the logarithmic amplitude frequency response (l.a.h..x) and the phase frequency response can be formulated as follows:

The automatic control system, unstable in the open state, is stable in the closed state, if the difference between the numbers of positive transitions (transition of the phase frequency response from bottom to top through the line u(u) = -180 ° ) and the numbers of negative transitions (transition of the phase frequency response from top to bottom through the line u(u) = -180 ° ) phase frequency response u(u) through the line u(u) = -180 ° is equal to zero in the frequency range at which L.a.h..x (L(u)> 0) .

To build a phase frequency response, it is desirable to represent the transfer function in the form of typical dynamic links.

and build a phase characteristic using the expression:

«+» - corresponds to the typical dynamic links of the numerator of the transfer function;

«-« - corresponds to the typical dynamic links of the denominator of the transfer function.

To construct an asymptotic l.a.ch.ch. we use the transfer function of an open system, presented in the form of typical dynamic links:

To do this, we use a transfer function of the form:

We represent this transfer function in the form of typical dynamic links:

The parameters of typical dynamic links are defined as shown below:

The phase characteristic equation will look like:

Let us determine the frequency at which the phase frequency response intersects the axis c(u) = -180 °

To construct an L.A.Ch. let's use the expression:

Figure 5 shows the graphs of the L.A.Ch. for two values ​​of the transfer coefficient of the magnetic amplifier k mu = 10 and k mu = 80 .

Fig.5.

Analysis of l.a.h.h. and phase frequency response show that with an increase in the transfer coefficient of the magnetic amplifier from 8 to 80 the system goes from stable to unstable. Let us determine the critical transfer coefficient of the magnetic amplifier.

If there are no additional requirements for stability margins to the system, then it is recommended to take them equal:

DL(u) = -12db Dc(u) = 35°h 45

Let us determine at what transfer coefficient of the magnetic amplifier this condition is satisfied.

This is also confirmed by the graphs shown in Figure 6.