Harmonic linearization. Method of harmonic linearization: Guidelines for laboratory work Method of harmonic linearization of self-oscillation matlab

Purpose of the harmonic linearization method.

The idea of the harmonic linearization method was proposed in 1934. N. M. Krylov and N. N. Bogolyubov. In relation to automatic control systems, this method was developed by L. S. Goldfarb and E. P. Popov. Other names for this method and its modifications are the harmonic balance method, the method of describing functions, and the method of equivalent linearization.

The harmonic linearization method is a method for studying self-oscillations. It allows you to determine the conditions of existence and parameters of possible self-oscillations in nonlinear systems.

Knowledge of the parameters of self-oscillations allows us to present a picture of possible processes in the system and, in particular, determine the conditions of stability. Suppose, for example, that as a result of studying self-oscillations in some nonlinear system, we obtained the dependence of the amplitude of these self-oscillations A from transmission coefficient k linear part of the system shown in Fig. 12.1, and we know that self-oscillations are stable.

From the graph it follows that with a large value of the transmission coefficient k, When k > k kr, there are self-oscillations in the system. Their amplitude decreases to zero as the transmission coefficient decreases k before k cr. In Fig. 12.1, the arrows conventionally show the nature of transient processes at different values k: at k > k kr the transient process caused by the initial deviation contracts to self-oscillations. From the figure it is clear that when k< k cr, the system turns out to be stable. Thus, k kr is the critical value of the transmission coefficient according to the stability condition. Exceeding it leads to the fact that the initial mode of the system becomes unstable and self-oscillations arise in it. Consequently, knowledge of the conditions for the existence of self-oscillations in the system allows us to determine the conditions of stability.

The idea of harmonic linearization.

Let's consider a nonlinear system, the diagram of which is shown in Fig. 12.2, and . The system consists of a linear part with a transfer function W l ( s) and nonlinear link NL with a specific characteristic . A link with a coefficient of - 1 shows that the feedback in the system is negative. We believe that there are self-oscillations in the system, the amplitude and frequency of which we want to find. In the considered mode, the input quantity X nonlinear link and output Y are periodic functions of time.

The harmonic linearization method is based on the assumption that the oscillations at the input of the nonlinear link are sinusoidal, i.e. e. that

, (12.1)

WhereA– amplitude and is the frequency of these self-oscillations, and is a possible constant component in the general case when the self-oscillations are asymmetrical.

In reality, self-oscillations in nonlinear systems are always nonsinusoidal due to the distortion of their shape by the nonlinear element. Therefore, the specified initial assumption means that the harmonic linearization method is fundamentally close and the scope of its application is limited to cases where self-oscillations at the input of a nonlinear link are quite close to sinusoidal. In order for this to take place, the linear part of the system must not allow higher harmonics of self-oscillations to pass through, i.e., be low pass filter. The latter is illustrated in Fig. 12.2, b . If, for example, the frequency of self-oscillations is equal to , then the linear part shown in Fig. 12.2, b The frequency response will play the role of a low-pass filter for these oscillations, since the second harmonic, whose frequency is equal to 2, will practically not pass to the input of the nonlinear link. Therefore, in this case the harmonic linearization method is applicable.

If the frequency of self-oscillations is equal to , the linear part will freely pass the second, third and other harmonics of self-oscillations. In this case, it cannot be said that the oscillations at the input of the nonlinear link will be quite close to sinusoidal, i.e. the prerequisite necessary for applying the harmonic linearization method is not met.

In order to determine whether the linear part of the system is a low-pass filter and thereby determine the applicability of the harmonic linearization method, it is necessary to know the frequency of self-oscillations. However, it can only be known by using this method. Thus, The applicability of the harmonic linearization method must be determined at the end of the study as a test.

Let us note that if, as a result of this test, the hypothesis that the linear part of the system plays the role of a low-pass filter is not confirmed, this does not mean that the results obtained are incorrect, although, of course, it casts doubt on them and requires additional verification in some way. another method.

So, assuming that the linear part of the system is a low-pass filter, we assume that the self-oscillations at the input of the nonlinear link are sinusoidal, that is, they have the form (12.1). The oscillations at the output of this link will no longer be sinusoidal due to their distortion by nonlinearity. As an example in Fig. 12.3, a curve is plotted at the output of the nonlinear link for a certain amplitude of the input purely sinusoidal signal according to the link characteristic given there.

Fig. 12.3. Passage of a harmonic oscillation through a nonlinear link.

However, since we believe that the linear part of the system passes only the fundamental harmonic of self-oscillations, it makes sense to be interested only in this harmonic at the output of the nonlinear section. Therefore, we will expand the output oscillations into a Fourier series and discard the higher harmonics. As a result we get:

;

; (12.3)

;

Let us rewrite expression (12.2) in a form more convenient for subsequent use, substituting into it the following expressions for and obtained from (12.1):

Substituting these expressions into (12.2), we will have:

(12.4)

. (12.5)

The following notations are introduced here:

. (12.6)

Differential equation (12.5) is valid for a sinusoidal input signal (12.1) and determines the output signal of the nonlinear link without taking into account higher harmonics.

The coefficients in accordance with expressions (12.3) for the Fourier coefficients are functions of the constant component, amplitude A and the frequency of self-oscillations at the input of the nonlinear link. At fixed A, and equation (12.5) is linear. Thus, if we discard higher harmonics, then for a fixed harmonic signal the original nonlinear link can be replaced by an equivalent linear one, described by equation (12.5). This replacement is called harmonic linearization .

In Fig. Figure 12.4 conventionally shows a diagram of this link, consisting of two parallel links.

Rice. 12.4. Equivalent linear element obtained as a result of harmonic linearization.

One link () passes the constant component, and the other - only the sinusoidal component of self-oscillations.

The coefficients are called harmonic linearization coefficients or harmonic transfer coefficients: - transmission coefficient of the constant component, and - two transmission coefficients of the sinusoidal component of self-oscillations. These coefficients are determined by nonlinearity and values and according to formulas (12.3). There are ready-made expressions defined using these formulas for a number of typical nonlinear links. For these and, in general, all inertia-free nonlinear links, the quantities do not depend on and are functions only of the amplitude A And .

Ministry of Education and Science of the Russian Federation

Saratov State Technical University

Balakovo Institute of Engineering, Technology and Management

Harmonic linearization method

Guidelines for laboratory work in the course “Theory of Automatic Control” for students of specialty 210100

Approved

editorial and publishing council

Balakovo Institute of Technology,

technology and management

Balakovo 2004

Purpose of the work: Study of nonlinear systems using the method of harmonic linearization (harmonic balance), determination of harmonic linearization coefficients for various nonlinear links. Gaining skills in finding the parameters of symmetrical oscillations of constant amplitude and frequency (self-oscillations), using algebraic, frequency methods, and also using the Mikhailov criterion.

BASIC INFORMATION

The harmonic linearization method refers to approximate methods for studying nonlinear systems. It allows one to quite simply and with acceptable accuracy assess the stability of nonlinear systems and determine the frequency and amplitude of oscillations established in the system.

It is assumed that the nonlinear ACS under study can be represented in the following form

and the nonlinear part must have one nonlinearity

This nonlinearity can be either continuous or relay, single-valued or hysteretic.

Any function or signal can be expanded into a series according to a system of linearly independent, in a particular case, orthonormal functions. The Fourier series can be used as such an orthogonal series.

Let us expand the output signal of the nonlinear part of the system into a Fourier series

, (2)

here are the Fourier coefficients,

. (3)

Thus, the signal according to (2) can be represented as an infinite sum of harmonics with increasing frequencies etc. This signal is supplied to the input of the linear part of the nonlinear system.

Let us denote the transfer function of the linear part

, (4)

and the degree of the numerator polynomial must be less than the degree of the denominator polynomial. In this case, the frequency response of the linear part has the form

where 1 - has no poles, 2 - has a pole or poles.

For the frequency response it is fair to write

Thus, the linear part of a nonlinear system is a high-pass filter. In this case, the linear part will transmit only low frequencies without attenuation, while high frequencies will be significantly attenuated as the frequency increases.

In the harmonic linearization method, the assumption is made that the linear part of the system will pass only the DC component of the signal and the first harmonic. Then the signal at the output of the linear part will have the form

This signal passes through the entire closed circuit of the system Fig. 1 and at the output of the nonlinear element without taking into account higher harmonics, according to (2) we have

. (7)

When studying nonlinear systems using the harmonic linearization method, cases of symmetrical and asymmetrical oscillations are possible. Let us consider the case of symmetric oscillations. Here and.

Let us introduce the following notation

Substituting them into (7), we get . (8)

Considering that

. (9)

According to (3) and (8) when

. (10)

Expression (9) is a harmonic linearization of nonlinearity; it establishes a linear relationship between the input variable and the output variable at . The quantities are called harmonic linearization coefficients.

It should be noted that equation (9) is linear for specific quantities and (the amplitude and frequency of harmonic oscillations in the system). But in general, it retains nonlinear properties, since the coefficients are different for different and . This feature allows us to study the properties of nonlinear systems using the harmonic linearization method [Popov E.P.].

In the case of asymmetric oscillations, harmonic linearization of the nonlinearity leads to the linear equation

. (12)

Just like equation (9), linearized equation (11) preserves the properties of a nonlinear element, since the harmonic linearization coefficients , , as well as the constant component depend on both the displacement and the amplitude of harmonic oscillations.

Equations (9) and (11) allow us to obtain the transfer functions of harmonically linearized nonlinear elements. So for symmetrical vibrations

, (13)

in this case the frequency transfer function

depends only on the amplitude and does not depend on the frequency of oscillations in the system.

It should be noted that if the odd-symmetric nonlinearity is unambiguous, then in the case of symmetric oscillations in accordance with (9) and (10) we obtain that , (15)

(16)

and the linearized nonlinearity has the form

For ambiguous nonlinearities (with hysteresis), the integral in expression (16) is not equal to zero, due to the difference in the behavior of the curve when increasing and decreasing , therefore the full expression (9) is valid.

Let's find the harmonic linearization coefficients for some nonlinear characteristics. Let the nonlinear characteristic have the form of a relay characteristic with hysteresis and dead zone. Let's consider how harmonic oscillations pass through a nonlinear element with such a characteristic.



If the condition is met, that is, if the amplitude of the input signal is less than the dead zone, then there is no signal at the output of the nonlinear element. If the amplitude is , then the relay switches at points A, B, C and D. Let us denote and .

. (18)

When calculating the harmonic linearization coefficients, it should be borne in mind that with symmetrical nonlinear characteristics, the integrals in expressions (10) are at half-cycle (0, ) with a subsequent doubling of the result. Thus

. (19)

For a nonlinear element with a relay characteristic and a dead zone

For a nonlinear element having a relay characteristic with hysteresis

Harmonic linearization coefficients for other nonlinear characteristics can be obtained similarly.

Let's consider two ways to determine symmetrical oscillations of constant amplitude and frequency (self-oscillations) and the stability of linearized systems: algebraic and frequency. Let's look at the algebraic method first. For the closed system Fig. 1, the transfer function of the linear part is equal to

Let us write the harmonically linearized transfer function of the nonlinear part

The characteristic equation of a closed-loop system has the form

. (22)

If self-oscillations occur in the system under study, this indicates the presence of two purely imaginary roots in its characteristic equation. Therefore, let us substitute the value of the root into the characteristic equation (22).

. (23)

Let's imagine

We obtain two equations that determine the desired amplitude and frequency

. (24)

If real positive values of amplitude and frequency are possible in the solution, then self-oscillations may occur in the system. If the amplitude and frequency do not have positive values, then self-oscillations in the system are impossible.

Let's consider example 1. Let the nonlinear system under study have the form

In this example, the nonlinear element is a sensing element with a relay characteristic, for which the harmonic linearization coefficients

The actuator has a transfer function of the form

The transfer function of the regulated object is equal to

. (27)

Transfer function of the linear part of the system

, (28)

Based on (22), (25) and (28), we write down the characteristic equation of the closed system

, (29)

Let 1/sec, sec, sec, v.

In this case, the parameters of the periodic motion are equal

7,071 ,

Let's consider a method for determining the parameters of self-oscillations in a linearized automatic control system using the Mikhailov criterion. The method is based on the fact that when self-oscillations occur, the system will be on the stability boundary and Mikhailov’s hodograph in this case will pass through the origin of coordinates.

In example 2, we will find the parameters of self-oscillations under the condition that the nonlinear element in the system Fig. 4 is a sensitive element having a relay characteristic with hysteresis, for which the harmonic linearization coefficients

The linear part remained unchanged.

Let us write down the characteristic equation of the closed system

Mikhailov's hodograph is obtained by replacement.

The task is to select such an amplitude of oscillations at which the hodograph will pass through the origin of coordinates. It should be noted that in this case the current frequency is , since it is in this case that the curve will pass through the origin.

Calculations carried out in MATHCAD 7 at 1/sec, sec, sec, v and v gave the following results. In Fig. 5, Mikhailov’s hodograph passes through the origin of coordinates. To increase the accuracy of calculations, we will enlarge the required fragment of the graph. Figure 6 shows a fragment of the hodograph, enlarged in the vicinity of the origin. The curve passes through the origin at c.

Fig.5. Fig.6.

The oscillation frequency can be found from the condition that the modulus is equal to zero. For frequencies

module values are tabulated

Thus, the oscillation frequency is 6.38. It should be noted that the accuracy of calculations can easily be increased.

The resulting periodic solution, determined by the amplitude and frequency values, must be examined for stability. If the solution is stable, then a self-oscillatory process takes place in the system (stable limit cycle). Otherwise the limit cycle will be unstable.

The easiest way to study the stability of a periodic solution is to use the Mikhailov stability criterion in graphical form. It was found that at , the Mikhailov curve passes through the origin of coordinates. If you give a small increment, then the curve will take a position either above zero or below. So in the last example we will give an increment in, that is, and . The position of the Mikhailov curves is shown in Fig. 7.

When the curve passes above zero, which indicates the stability of the system and a damped transition process. When the Mikhailov curve passes below zero, the system is unstable and the transition process is divergent. Thus, a periodic solution with an amplitude in and an oscillation frequency of 6.38 is stable.

To study the stability of a periodic solution, an analytical criterion obtained from the graphical Mikhailov criterion can also be used. Indeed, to find out whether the Mikhailov curve will go when above zero, it is enough to look where the point of the Mikhailov curve, which is located at the origin of coordinates, will move.

If we expand the displacement of this point along the X and Y coordinate axes, then for the stability of a periodic solution, the vector determined by the projections onto the coordinate axes

should be located to the right of the tangent MN to the Mikhailov curve, if looking along the curve in the direction of increasing, the direction of which is determined by the projections

We write the analytical stability condition in the following form

In this expression, partial derivatives are taken with respect to the current parameter of the Mikhailov curve

It should be noted that the analytical expression of the stability criterion (31) is valid only for systems not higher than the fourth order, since, for example, for a fifth-order system at the origin of coordinates, condition (31) can be satisfied, and the system will be unstable

Let us apply criterion (31) to study the stability of the periodic solution obtained in Example 1.

, ,

Introduction

Relay systems have become widespread in the practice of automatic control. The advantage of relay systems is their simplicity of design, reliability, ease of maintenance and configuration. Relay systems represent a special class of nonlinear automatic control systems.

Unlike continuous ones in relay systems, the regulatory action changes abruptly whenever the relay control signal (most often this is a control error) passes through some fixed (threshold) values, for example, through zero.

Relay systems, as a rule, have high performance due to the fact that the control action in them changes almost instantly, and the actuator is exposed to a piecewise constant signal of maximum amplitude. At the same time, self-oscillations often occur in relay systems, which in many cases is a disadvantage. In this paper, a relay system with four different control laws is studied.

Structure of the system under study

The system under study (Fig.) 1 includes a comparison element ES, a relay element RE, an actuator (ideal integrator with gain = 1), a control object (an aperiodic link with three time constants , , and gain). The values of the system parameters are given in table. 1 Appendix A.

Static characteristics (input-output characteristics) of the relay elements under study are shown in Fig. 2.

In Fig. 2a shows the characteristics of an ideal two-position relay, Fig. 2b characteristic of a three-position relay with a dead zone. In Fig. 2,c and 2,d show the characteristics of a two-position relay with positive and negative hysteresis, respectively.

The investigated ASR can be modeled using well-known modeling packages, for example, SIAM or VisSim.

Comment. In some simulation packages, the output value

the relay signal can only take values ±1 instead of ±B, where B is an arbitrary number. In such cases, it is necessary to take the integrator gain equal to .

Work order

To complete the work, each student receives a version of the initial data from the teacher (see section 2).

The work is carried out in two stages.

The first stage is computational and research (can be performed outside the laboratory).

The second stage is experimental (carried out in the laboratory). At this stage, using one of the packages, the transient processes in the system under study are simulated for the modes calculated at the first stage, and the accuracy of the theoretical methods is checked.

The necessary theoretical material is presented in section 4; Section 5 contains test questions.

3.1. Calculation and research part

1. Obtain expressions for the amplitude-frequency and phase-frequency, real and imaginary characteristics of the linear part of the system.

2. Calculate and construct the amplitude-phase characteristic of the linear part of the system. For calculations, use programs from the TAU package. Necessarily print real and imaginary frequency response values(10 – 15 points corresponding third and second quadrants).

4. Using Goldfarb’s graphic-analytical method, determine the amplitude and frequency of self-oscillations and their stability for all four relays. The parameters of self-oscillations can also be calculated analytically. Qualitatively depict the phase portrait of the system for each case.

5. For a three-position relay, determine one value of the gain of the linear part at which there are no self-oscillations, and the boundary value at which the self-oscillations fail.

experimental part

1. Using one of the available modeling packages, assemble a modeling scheme for the ASR under study. With the permission of the teacher, you can use a ready-made diagram. Configure the circuit parameters in accordance with the task.

2. Investigate the transient process in a system with an ideal relay (print it), applying a stepwise action x(t)=40*1(t) to the input. Measure the amplitude and frequency of self-oscillations, comparing them with the calculated values. Repeat the experiment, setting non-zero initial conditions (for example, y(0)=10, y(1) (0)=-5).

3. Investigate the transient process in a system with a three-position relay for two different values of the input signal amplitude x(t)= 40*1(t) and x(t)=15*1(t). Print transient processes, measure the amplitude and frequency of self-oscillations (if they exist), compare them with calculated values, and draw conclusions.

4. Investigate transient processes in a system with a three-position relay for other values of the gain of the linear part (see paragraph 5, section 3.1).

5. Investigate transient processes in a system with two-position relays with hysteresis under zero and non-zero initial conditions and x(t)=40*1(t). Print transient processes, measure the amplitude and frequency of self-oscillations (if they exist), compare them with calculated values, and draw conclusions.

Theoretical part

A widely used method for calculating nonlinear systems is the method of harmonic linearization (describing functions).

The method makes it possible to determine the parameters of self-oscillations (amplitude and frequency), the stability of self-oscillations, and the stability of the equilibrium position of a nonlinear ASR. Based on the harmonic linearization method, methods for constructing transient processes, analysis and synthesis of nonlinear ASR have been developed.

Harmonic linearization method

As already noted, in nonlinear and especially relay ASRs, stable periodic oscillations constant amplitude and frequency, the so-called self-oscillations. Moreover, self-oscillations can persist even with significant changes in system parameters. Practice has shown that in many cases the oscillations of the controlled variable (Fig. 3) are close to harmonic.

The proximity of self-oscillations to harmonic ones allows us to use the harmonic linearization method to determine their parameters - amplitude A and frequency w 0. The method is based on the assumption that the linear part of the system is a low-pass filter (filter hypothesis). Let us determine the conditions under which self-oscillations in the system can be close to harmonic. Let us limit ourselves to systems that, as in Fig. 3 can be reduced to a series connection of a nonlinear element and a linear part. Let us assume that the reference signal is a constant value; for simplicity, we will take it equal to zero. And the error signal (Figure 3) is harmonic:

(1)

The output signal of a nonlinear element, like any periodic signal - in Figure 3 these are rectangular oscillations - can be represented as the sum of the harmonics of the Fourier series.

Let us assume that the linear part of the system is a low-pass filter (Fig. 4) and passes only the first harmonic with frequency w 0. The second with a frequency of 2w 0 and higher harmonics are filtered by the linear part. In this case, on linear output parts will exist practically only first harmonic , and the influence of higher harmonics can be neglected

Thus, if the linear part of the system is a low-pass filter, and the frequency of self-oscillations w 0 satisfies the conditions

, (4)

The assumption that the linear part of the system is a low-pass filter is called filter hypothesis . The filter hypothesis is always satisfied if the difference in the degrees of the polynomials of the denominator and numerator of the transfer function of the linear part

(5)

at least two

Condition (6) is satisfied for many real systems. An example is a second-order aperiodic link and a real integrating

. (7)

When studying self-oscillations close to harmonic, only the first harmonic of periodic oscillations at the output of a nonlinear element is taken into account, since higher harmonics are still practically filtered out by the linear part. In the self-oscillation mode, it is carried out harmonic linearization nonlinear element. The nonlinear element is replaced by an equivalent linear one with complex gain (describing function) depending on the amplitude of the input harmonic signal:

where and are the real and imaginary parts,

– argument,

– module.

In the general case, it depends on both the amplitude and frequency of self-oscillations and the constant component. Physically complex gain of a nonlinear element, more often called harmonic linearization coefficient , There is complex gain of a nonlinear element at the first harmonic. Modulus of the harmonic linearization coefficient

(9)

is numerically equal to the ratio of the amplitude of the first harmonic at the output of the nonlinear element to the amplitude of the input harmonic signal.

Argument

(10)

characterizes the phase shift between the first harmonic of the output oscillations and the input harmonic signal. For unambiguous nonlinearities, such as, for example, in Fig. 2,a and 2,b, real expression and

For ambiguous nonlinearities, Fig. 2,c, 2,d, determined by the formula

where S is the area of the hysteresis loop. The area S is taken with a plus sign if the hysteresis loop is bypassed in the positive direction (Fig. 2, c) and with a minus sign otherwise (Fig. 2, d).

In the general case and are calculated using the formulas

, (12)

where , is a nonlinear function (characteristic of a nonlinear element).

Taking into account the above, when studying self-oscillations close to harmonic, the nonlinear ASR (Fig. 3) is replaced by an equivalent one with a harmonic linearization coefficient instead of a nonlinear element (Fig. 5). The output signal of the nonlinear element in Fig. 5 is designated as , this is

emphasizes that the nonlinear element generates only

the first harmonic of oscillations. Formulas for harmonic linearization coefficients for typical nonlinearities can be found in the literature, for example, in. Appendix Table B shows the characteristics of the relay elements under study, formulas for and their hodographs. Formulas and hodographs for the inverse harmonic linearization coefficient, defined by the expression

, (13)

where are both the real and imaginary parts. Hodographs and are constructed in coordinates , and , respectively.

Let us now write down the conditions for the existence of self-oscillations. The system in Fig. 5 is equivalent to linear. In a linear system, undamped oscillations exist if it is on the stability boundary. Let us use the condition of the stability boundary according to the Nyquist criterion:

. (14)

Equation (14) There is condition for the existence of self-oscillations, close to harmonic. If there are real positive solutions A and w 0 of equation (14), then in the nonlinear ASR there are self-oscillations close to harmonic. Otherwise, self-oscillations are absent or not harmonic. Equation (14) splits into two – with respect to the real and imaginary parts:

;

Dividing both sides of equation (14) by and taking into account formula (13), we obtain the condition for the existence of self-oscillations in the form of L.S. Goldfarb:

. (17)

Equation (17) also splits into two:

(18)

and in some cases it is more convenient to use them to determine the parameters of self-oscillations.

Goldfarb proposed a graphic-analytical method for solving system (17) and determining the stability of self-oscillations.

In coordinates , and , hodographs and are constructed (Fig. 6, a). If the hodographs intersect, then self-oscillations exist. The parameters of self-oscillations - A and w 0 are determined at the intersection points - frequency w 0 according to the hodograph, amplitude according to the hodograph. In Fig. 6,a – two points of intersection, which indicates the presence of two limit cycles.

To determine the stability of self-oscillations, according to Goldfarb, the left side of the AFC of the linear part is shaded when moving along the AFC in the direction of increasing frequency (Fig. 6).

Self-oscillations are stable if, at the intersection point, the hodograph of the nonlinear element passes from the unshaded area to the shaded area when moving in the direction of increasing amplitude A.

If the transition occurs from a shaded area to an unshaded area, then the self-oscillations are not stable.

In Fig. Figure 6b qualitatively depicts the phase portrait corresponding to two limit cycles in Fig. 6, a. The point of intersection with the parameters and in Fig. 6a corresponds to the unstable limit cycle in Fig. 6b, point with parameters and and to achieve disruption of self-oscillations, in this case hodographs and do not intersect. The same effect can be achieved by increasing the dead zone d or reducing the amplitude of the output signal of relay B. There is a certain limiting value K l at which the AFC of the linear part touches Error! Communication error. wherein , and the amplitude value is . Naturally, this leads to a qualitative change in the phase portrait of the system.

Let us illustrate the calculation of harmonic linearization coefficients with several examples: first for symmetrical vibrations, and then for asymmetrical ones. Let us first note that if the odd-symmetric nonlinearity F(x) is single-valued, then, according to (4.11) and (4.10), we obtain

and when calculating q(4.11) we can limit ourselves to integration over a quarter period, quadrupling the result, namely

For the loop nonlinearity F(x) (odd-symmetric), the full expression (4.10) will hold

and you can use the formulas

i.e., doubling the result of integration over a half-cycle.

Example 1. Let's study cubic nonlinearity (Fig. 4.4, i):

Addiction q(a) shown in Fig. 4.4, b. From Fig. 4.4, A it is clear that for a given amplitude I am straight q(a)x averages the curvilinear dependence F(x) on a given

plot -a£ X£ . A. Naturally, it's cool q(a) the slope of this averaging straight line q(a)x increases with amplitude A(for a cubic characteristic this increase occurs according to a quadratic law).

Example 2. Let's study the loop relay characteristic (Fig. 4.5, a). In Fig. 4.5,6 the integrand function F(a sin y) for formulas (4.21) is presented. Relay switching takes place at ½ X½= b , Therefore, at the moment of switching, the value y1 is determined by the expression sin y1= b /A. Using formulas (4.21) we obtain (for a³b)

In Fig. 4.5, b shows graphs of q(a) and q"(a). The first of them shows the change in the slope of the averaging straight line q( A)x s change A(see Fig. 4.5, a). Naturally, q( a)à0 at аа¥ at, since the output signal remains constant (F( x)=c)for any unlimited increase in the input signal X. From physical considerations it is also clear why q" <0. Это коэффициент при производной в формуле (4.20). Положительный знак давал бы опережение сигнала на выходе, в то время как гистерезисная петля дает запаздывание. Поэтому естественно, что q" < 0. Абсолютное значение q" decreases with increasing amplitude a, since it is clear that the loop will occupy the smaller part of the “working section” of the characteristic F( x), the greater the amplitude of oscillations of the variable X.

The amplitude-phase characteristic of such nonlinearity (Fig. 4.5, a), according to (4.13). presented in the form

Moreover, the amplitude and phase of the first harmonic at the nonlinearity output have the form, respectively

Where q And q" defined above (Fig. 4.5, b). Consequently, harmonic linearization transforms the nonlinear coordinate lag (hysteresis loop) into an equivalent phase lag, characteristic of linear systems, but with a significant difference - the dependence of the phase shift on the amplitude of input oscillations, which is not present in linear systems.

Example 3. We study unambiguous relay characteristics (Fig. 4.6, a, V). Similar to the previous one, we obtain, respectively

what is shown in Fig. 4.6, b, a.

Example 4. Let's study a characteristic with a dead zone, a linear section and saturation (Fig. 4.7, a). Here q"= 0, and the coefficient q(a) has two variants of values in accordance with Fig. 4.7, b, where F (a sin y) is constructed for them:

1) for b1 £ a £ b2, according to (4.19), we have

that taking into account the ratio a sin y1 = b 1 gives

2) for a ³ b2

which, taking into account the relation a sin y2 = b2 gives

The result is presented graphically in Fig. 4.7, a.

Example 5. As special cases, the corresponding coefficients q(a) for two characteristics (Fig. 4.8, a, b) are equal

which is shown graphically in Fig. 4.8, b, d. Moreover, for the characteristic with saturation (Fig. 4.8, a) we have q= k at 0 £ a£ b.

Let us now show examples of calculating harmonic linearization coefficients for asymmetrical vibrations with the same nonlinearities.

Example 6. For the case of cubic nonlinearity F( x) =kx 3 according to formula (4.16) we have

and according to formulas (4.17)

Example 7. For a loop relay characteristic (Fig. 4.5, A) using the same formulas we have

Example 8. For a characteristic with a dead zone (Fig. 4.1:1), the same expressions will apply F° And q. Their graphs are presented in Fig. 4.9, a, b. Wherein q"== 0. For an ideal relay characteristic (Fig. 4.10) we obtain

what is shown in Fig. 4.10, a and b.

Example 9. For a characteristic with a linear section q saturation (Fig. 4.11, a) for a ³ b+½ x 0 ½ we have

These dependencies are presented in the form of graphs in Fig. 4.11, b, V.

Example 10. For an asymmetrical characteristic

(Fig. 4. 12, a) using formula (4.l6) we find

and according to formulas (4.17)

The results are shown graphically in Fig. 4.12, b And V.

The expressions and graphs of the harmonic linearization coefficients obtained in these examples will be used below when solving research problems

self-oscillations, forced oscillations and control processes.

Based on the filter property of the linear part of the system (Lecture 12), we look for a periodic solution of the nonlinear system (Fig. 4.21) at the input of the nonlinear element approximately in the form

x = a sin w t (4.50)

with unknown people A and w. The form of nonlinearity is specified = F( x) and the transfer function of the linear part

Harmonic linearization of nonlinearity is performed

which leads to the transfer function

The amplitude-phase frequency response of the open circuit system takes the form

A periodic solution to the linearized system (4.50) is obtained if there is a pair of purely imaginary roots in the characteristic equation of the closed system.

And according to the Nyquist criterion, this corresponds to the passage W(j w) through point -1. Consequently, the periodic solution (4.50) is determined by the equality

Equation (4.51) determines the required amplitude A and frequency w of the periodic solution. This equation can be solved graphically as follows. On the complex plane (U, V), the amplitude-phase frequency response of the linear part Wl( j w) (Fig. 4.22), as well as the inverse amplitude-phase characteristic of nonlinearity with the opposite sign -1 / Wн( a). Dot IN their intersection (Fig. 4.22) and determines the values A and w, and the value A counted along curve -1 / Wн (a) , and the value of w is according to the curve Wл (jw).

Instead, we can use two scalar equations that follow from (4.51) and (4.52):

which also determine the two sought quantities A and w.

It is more convenient to use the last two equations on a logarithmic scale, using logarithmic

frequency characteristics of the linear part. Then instead of (4.53) and (4.54) we will have the following two equations:

In Fig. 4.23 on the left are graphs of the left-hand sides of equations (4.55) and (4.56), and on the right are the right-hand sides of these equations. In this case, along the abscissa axis on the left, the frequency w is plotted, as usual, on a logarithmic scale, and on the right is the amplitude A in natural scale. The solution to these equations will be the following values A and w, so that both equalities (4.55) and (4.56) are simultaneously observed. This solution is shown in Fig. 4.23 with thin lines in the form of a rectangle.

Obviously, it will not be possible to guess this solution right away. Therefore, attempts are made, shown in dashed lines. The last points of these trial rectangles M1 and M2 do not fall on the phase characteristic of nonlinearity. But if they are located on both sides of the characteristic, as in Fig. 4.23, then the solution is found by interpolation - by drawing straight line MM1 .

Finding a periodic solution is simplified in the case of unambiguous nonlinearity F( X). Then q"= 0 and equations (4.55) and (4.56) take the form

The solution is shown in Fig. 4.24.

Rice . 4.24.

After determining a periodic solution, it is necessary to investigate its stability. As already mentioned, a periodic solution occurs in the case when the amplitude-phase characteristic of the open circuit

passes through point -1. Let's give the amplitude a deviation D A. The system will return to a periodic solution if at D A> 0 oscillations die out, and at D A < 0 - расходятся. Следовательно, при DA> 0 characteristic W(jw, A) must be deformed (Fig. 4.25) so that at D A> 0 the Nyquist stability criterion was met, and for D A < 0 - нарушался.

So it is required that at a given frequency w there be

It follows that in Fig. 4.22 positive amplitude reading A along the curve -1/Wн ( A) must be directed from the inside to the outside through the curve Wл (jw) , as shown by the arrow. Otherwise, the periodic solution is unstable.

Let's look at examples.

Let the amplifier in the tracking system (Fig. 4.13, a) have relay characteristic(Fig. 4.17, A). Pa fig. 4.17, b a graph of the harmonic linearization coefficient q( A), and q’( A) =0. To determine the periodic solution using the frequency method, according to Fig. 4.22, we need to examine the expression

From formula (4.24) we obtain for this nonlinearity

The graph of this function is shown in Fig. 4.26.

The transfer function of the linear part has the form

The amplitude-phase characteristic for it is shown in Fig. 4.27. Function -1 / Wн ( A), being real in this case (Fig. 4.26), fits entirely on the negative part of the real axis (Fig. 4.27). In this case, in the area of amplitude change b £ a£ b amplitude is measured from the left from the outside into the curve Wл(jw), and in the section A>b - reversed. Therefore, the first intersection point ( A 1) gives an unstable periodic solution, and the second ( A 2) - stable (self-oscillations). This is consistent with the previous solution (example 2 lecture 15, 16).

Let us also consider the case loop relay characteristics(Fig. 4.28, a) in the same tracking system (Fig. 4.13, a). The amplitude-phase frequency response of the linear part is the same (Fig. 4.28, b). The expression for the curve –1/Wн( A), according to (4.52) and (4.23), takes the form

This is a straight line parallel to the abscissa axis (Fig. 4.28, b), with amplitude reading A from right to left. The intersection will give a stable periodic solution (self-oscillations). To obtain graphs of amplitude and frequency

from k l , presented in Fig. 4.20, needed in Fig. 4.28 construct a series of curves Wл(jw) for each value k l and find at their points of intersection with the line –1/Wн( A) corresponding values A and w.

The output signal of a nonlinear element, like any periodic signal - in Figure 3 these are rectangular oscillations - can be represented as the sum of the harmonics of the Fourier series.

Thus, if the linear part of the system is a low-pass filter, and the frequency of self-oscillations w 0 satisfies the conditions

, (4)

at least two

Condition (6) is satisfied for many real systems. An example is a second-order aperiodic link and a real integrating

where and are the real and imaginary parts,

– argument,

– module.

is numerically equal to the ratio of the amplitude of the first harmonic at the output of the nonlinear element to the amplitude of the input harmonic signal.

Argument

For ambiguous nonlinearities, Fig. 2,c, 2,d, determined by the formula

where S is the area of the hysteresis loop. The area S is taken with a plus sign if the hysteresis loop is bypassed in the positive direction (Fig. 2, c) and with a minus sign otherwise (Fig. 2, d).

In the general case and are calculated using the formulas

where , is a nonlinear function (characteristic of a nonlinear element).

Emphasizes that a nonlinear element only generates

where are both the real and imaginary parts. Hodographs and are constructed in coordinates , and , respectively.