Method of harmonic linearization. Calculation of harmonic linearization coefficients. Work order

The idea of ​​the harmonic linearization method belongs to N.M. Krylov and N.N. Bogolyubov and is based on replacing a nonlinear element of the system with a linear link, the parameters of which are determined under a harmonic input action from the condition of equality of the amplitudes of the first harmonics at the output of the nonlinear element and its equivalent linear link. This method can be used when the linear part of the system is a low-pass filter, i.e. filters out all harmonic components arising at the output of the non-linear element, except for the first harmonic.

Harmonic Linearization Coefficients and Equivalent Complex Gains of Nonlinear Elements. In a nonlinear system (Fig. 2.1), the parameters of the linear part and the nonlinear element are chosen in such a way that symmetrical periodic oscillations with a frequency w exist.

At the heart of the method of harmonic linearization of nonlinearities (Fig. 2.10), described by the equation

y n = F(x), (2.17)

there is an assumption that a harmonic action with a frequency w and an amplitude is applied to the input of a nonlinear element a, i.e.

x= a sin y, where y = wt, (2.18)

and only the first harmonic is distinguished from the entire spectrum of the output signal

y n 1 = a n 1 sin(y + y n 1), (2.19)

Where a n 1 - amplitude and y n 1 - phase shift;

in this case, higher harmonics are discarded and a connection is established between the first harmonic of the output signal and the input harmonic effect of the nonlinear element.

Rice. 2.10. Characteristics of a non-linear element

In the case of non-linear system insensitivity to higher harmonics, the non-linear element can be, in the first approximation, replaced by some element with an equivalent gain, which determines the first harmonic of periodic oscillations at the output depending on the frequency and amplitude of sinusoidal oscillations at the input.

For nonlinear elements with characteristic (2.17), as a result of expanding the periodic function F(x) into a Fourier series with sinusoidal oscillations at the input (2.18), we obtain an expression for the first harmonic of the output signal

y n 1 = b 1F siny + a 1F cozy, (2.20)

where b 1F , a 1F - expansion coefficients in a Fourier series, which determine the amplitudes of the in-phase and quadrature components of the first harmonic, respectively, which are determined by the formulas:

px= a w cos y, where p = d/dt,

then the relationship between the first harmonic of periodic oscillations at the output of the nonlinear element and sinusoidal oscillations at its input can be written as

y н 1 = x, (2.21)

where q = b 1F / a, q¢ = a 1F / a.

The last equation is called harmonic linearization equation, and the coefficients q and q¢ - harmonic linearization coefficients.

Thus, a nonlinear element, when exposed to a harmonic signal, is described by equation (2.21), which is linear, up to higher harmonics. This equation of a non-linear element differs from the equation of a linear link in that its coefficients q and q¢ change with a change in amplitude a and frequency w of oscillations at the input. This is the fundamental difference between harmonic linearization and conventional linearization, the coefficients of which do not depend on the input signal, but are determined only by the type of characteristic of the nonlinear element.

For various types of nonlinear characteristics, the harmonic linearization coefficients are summarized in the table. In the general case, the harmonic linearization coefficients q( a, w) and q¢( a, w) depend on the amplitude a and frequency w of oscillations at the input of the nonlinear element. However, for static nonlinearities these coefficients q( a) and q¢( a) are only a function of the amplitude a input harmonic signal, and for static single-valued nonlinearities, the coefficient q¢( a) = 0.

Subjecting Eq. (2.21) to the Laplace transformation under zero initial conditions and then replacing the operator s with jw (s = jw), we obtain equivalent complex gain non-linear element

W E (jw, a) = q + jq¢ = A e (w, a) e j y e (w , a) , (2.22)

where the modulus and argument of the equivalent complex gain are related to the harmonic linearization coefficients by the expressions

A E (w, a) = mod W E (jw, a) =

y E (w, a) = arg W E (jw, A) = arctg.

The equivalent complex transfer coefficient of a non-linear element makes it possible to determine the amplitude and phase shift of the first harmonic (2.19) at the output of the non-linear element under harmonic action (2.18) at its input, i.e.

a n 1 = a´A E (w, a); y n 1 \u003d y E (w, a).

Study of symmetric periodic regimes in nonlinear systems. In the study of nonlinear systems based on the method of harmonic linearization, first of all, the question of the existence and stability of periodic modes is solved. If the periodic regime is stable, then there are self-oscillations in the system with frequency w 0 and amplitude a 0 .

Consider a nonlinear system (Fig. 2.5), which includes a linear part with a transfer function

and a non-linear element with an equivalent complex gain

W E (jw, a) = q(w, a) + jq¢(w, a) = A E (w, a) e j y e (w , a) . (2.24)

Taking expression (2.21) into account, we can write the equation of the nonlinear system

(A(p) + B(p)´)x = 0. (2.25)

If self-oscillations occur in a closed nonlinear system

x= a 0 sin w 0 t

with a constant amplitude and frequency, then the harmonic linearization coefficients turn out to be constant, and the entire system is stationary. To assess the possibility of self-oscillations in a nonlinear system using the harmonic linearization method, it is necessary to find the conditions for the stability boundary, as was done in the analysis of the stability of linear systems. A periodic solution exists if a = a 0 and w = w 0 characteristic equation of a harmonically linearized system

A(p) + B(p)´ = 0 (2.26)

has a pair of imaginary roots l i = jw 0 and l i +1 = -jw 0 . The stability of the solution needs to be evaluated additionally.

Depending on the methods for solving the characteristic equation, methods for studying nonlinear systems are distinguished.

Analytical Method. To estimate the possibility of self-oscillations in a nonlinear system, jw is substituted into the harmonically linearized characteristic polynomial of the system instead of p

D(jw, a) = A(jw) + B(jw)´. (2.27)

The result is the equation D(jw, a) = 0, whose coefficients depend on the amplitude and frequency of the assumed self-oscillatory regime. Separating the real and imaginary parts

Re D(jw, a) = X(w, a);

Im D(jw, a) = Y(w, a),

we get the equation

X(w, a) + jY(w, a) = 0. (2.28)

If for real values a 0 and w 0 expression (2.28) is satisfied, then a self-oscillatory mode is possible in the system, the parameters of which are calculated according to the following system of equations:

From expressions (2.29), one can find the dependence of the amplitude and frequency of self-oscillations on the parameters of the system, for example, on the transfer coefficient k of the linear part of the system. To do this, it is necessary in equations (2.29) to consider the transfer coefficient k as a variable, i.e. write these equations in the form:

According to charts a 0 = f(k), w 0 = f(k), you can choose the transfer coefficient k, at which the amplitude and frequency of possible self-oscillations have acceptable values ​​or are completely absent.

frequency method. In accordance with the Nyquist stability criterion, undamped oscillations in a linear system arise when the amplitude-phase characteristic of an open-loop system passes through a point with coordinates [-1, j0]. This condition is also a condition for the existence of self-oscillations in a harmonically linearized nonlinear system, i.e.

W n (jw, a) = -1. (2.31)

Since the linear and nonlinear parts of the system are connected in series, the frequency response of an open-loop nonlinear system has the form

W n (jw, a) = W lch (jw)´W E (jw, a). (2.32)

Then, in the case of a static characteristic of a nonlinear element, condition (2.31) takes the form

W lch (jw) = - . (2.33)

The solution of equation (2.33) with respect to the frequency and amplitude of self-oscillations can be obtained graphically as the intersection point of the hodograph of the frequency response of the linear part of the system W lch (jw) and the hodograph of the inverse characteristic of the non-linear part, taken with the opposite sign (Fig. 2.11). If these hodographs do not intersect, then the regime of self-oscillations does not exist in the system under study.

Rice. 2.11. Hodographs of the linear and non-linear parts of the system

For the stability of the self-oscillatory regime with frequency w 0 and amplitude a 0 it is required that the point on the hodograph of the non-linear part - , corresponding to the increased amplitude a 0+D a compared with the value at the point of intersection of the hodographs, was not covered by the hodograph of the frequency response of the linear part of the system and the point corresponding to the reduced amplitude was covered a 0-D a.

On fig. 2.11 gives an example of the location of hodographs for the case when stable self-oscillations exist in a nonlinear system, since a 3 < a 0 < a 4 .

Study on logarithmic frequency responses.

When studying nonlinear systems by logarithmic frequency characteristics, condition (2.31) is rewritten separately for the modulus and argument of the equivalent complex gain of an open-loop nonlinear system

mod W lch (jw)W e (jw, a) = 1;

arg W lch (jw)W e (jw, a) = - (2k+1)p, for k=0, 1, 2, ...

with subsequent transition to logarithmic amplitude and phase characteristics

L h (w) + L e (w, a) = 0; (2.34)

y lch (w) + y e (w, a) = - (2k+1)p, for k=0, 1, 2, ... (2.35)

Conditions (2.34) and (2.35) allow us to determine the amplitude a 0 and frequency w 0 of the periodic solution of equation (2.25) according to the logarithmic characteristics of the linear part of the system L lch (w), y lch (w) and the nonlinear element L e (w, a), y e (w, a).

Self-oscillations with frequency w 0 and amplitude a 0 will exist in a nonlinear system if the periodic solution of Eq. (2.25) is stable. An approximate method for studying the stability of a periodic solution is to study the behavior of the system at a frequency w = w 0 and amplitude values a =a 0+D a And a =a 0-D a, where D a> 0 - small amplitude increment. When studying the stability of a periodic solution for a 0+D a And a 0-D a according to logarithmic characteristics, the Nyquist stability criterion is used.

In nonlinear systems with single-valued static characteristics of a nonlinear element, the harmonic linearization coefficient q¢( a) is equal to zero, and therefore, equal to zero and the phase shift y e ( a) contributed by the element. In this case, the periodic solution of the equation of the system

x = 0 (2.36)

exists if the following conditions are met:

L h (w) \u003d - L e ( a); (2.37)

y lch (w) = - (2k+1)p, for k=0, 1, 2, ... (2.38)

Equation (2.38) allows us to determine the frequency w \u003d w 0 of a periodic solution, and equation (2.37) - its amplitude a =a 0 .

With a relatively simple linear part, solutions to these equations can be obtained analytically. However, in most cases it is advisable to solve them graphically (Fig. 2.12).

When studying the stability of a periodic solution of equation (2.36), i.e. when determining the existence of self-oscillations in a nonlinear system with a single-valued nonlinear static characteristic, one uses Nyquist criterion: periodic solution with frequency w = w 0 and amplitude a =a 0 is stable if, as the frequency changes from zero to infinity and a positive increment of the amplitude D a> 0 the difference between the number of positive (from top to bottom) and negative (from bottom to top) transitions of the phase characteristic of the linear part of the system y lch (w) through the -p line is zero in the frequency range where L lch (w)³-L e (w 0 , a 0+D a), and is not equal to zero in the frequency range, where L h (w)³-L e (w 0, a 0-D a).

On fig. 2.12 shows an example of determining periodic solutions in a nonlinear system with a constraint. In such a system, there are three periodic solutions with frequencies w 01 , w 02 and w 03 , determined at the points of intersection of the phase characteristic y lch (w) with the line -180 0 . Periodic Solution Amplitudes a 01 , a 02 and a 03 are determined from the condition (2.37) by the logarithmic amplitude characteristics of the nonlinear element -L e (w 01 , a), -L e (w 02, a) and -L e (w 03, a).

Rice. 2.12. Logarithmic amplitude and phase characteristics

Of the three solutions defined in Fig. 2.12, two are stable. Solution with frequency w = w 01 and amplitude a =a 01 is stable, since in the frequency range 1, where L lch (w)³-L e (w 01, a 01+D a), the phase characteristic y lch (w) does not cross the line -180 0, but in the frequency range 2, where L lch (w)³-L e (w 01, a 01-D a), the phase characteristic y lch (w) once crosses the line -180 0 . Solution with frequency w = w 02 and amplitude a =a 02 is unstable, since in the frequency range where L h (w)³-L e (w 02, a 02+D a), the phase characteristic y lch (w) once crosses the line -180 0 . High-frequency periodic solution with frequency w = w 03 and amplitude a =a 03 is stable, because in the frequency range, where L h (w)³-L e (w 03, a 03+D a), there is one positive and one negative transition of the phase characteristic y lch (w) through the line -180 0, and in the frequency range where L lch (w)³-L e (w 03, a 03-D a), there are two positive and one negative transition of the phase characteristic y lch (w) through the line -180 0 .

In the considered system, with small perturbations, high-frequency self-oscillations with frequency w 03 and amplitude a 03 , and for large perturbations - low-frequency self-oscillations with frequency w 01 and amplitude a 01 .

Example. Investigate self-oscillating modes in a nonlinear system, the linear part of which has the following transfer function

where k=200 s -1 ; T 1 =1.5 s; T 2 \u003d 0.015 s,

and as a non-linear element, a relay with a dead zone is used (Fig. 2.4, b) at c=10 V, b=2 V.

Solution. According to the table for a relay with a dead zone, we find the coefficients of harmonic linearization:

At a³ b, q¢( a) = 0.

When constructing the characteristics of a nonlinear element, it is advisable to use the relative value of the amplitude of the input harmonic effect m = a/b. Let us rewrite the expression for the harmonic linearization coefficient in the form

where is the transmission coefficient of the relay;

Relative amplitude.

The relay transfer coefficient k n is related to the linear part of the system and we obtain the normalized harmonic linearization coefficients

and the normalized logarithmic amplitude characteristic of the relay element with the opposite sign

If m ® 1, then -L e (m) ® ¥; and when m >> 1 -L e (m) = 20 lg m. Thus, the asymptotes of the normalized logarithmic amplitude characteristic with the opposite sign are the vertical straight line and the straight line with a slope of +20 dB/dek, which pass through the point with coordinates L = 0, m = 1 (Fig. 2.13).

Rice. 2.13. Determining a Periodic Solution in a Relay System

with dead zone

a 0 = b´m 1 = = 58 V.

To solve the question of the existence of self-oscillations in accordance with the normalized logarithmic amplitude characteristic with the opposite sign of the nonlinear element and the transfer function of the linear part of the system

in fig. 2.13 plotted the logarithmic characteristics of L ch (w), -L e (m) and y ch (w).

The frequency of the periodic solution w 0 = 4.3 s -1 is determined at the point of intersection of the phase characteristic y lch (w) and the line -180 0 . The amplitudes of the periodic solutions m 1 = 29 and m 2 = 1.08 are found according to the characteristics L h (w) and -L e (m). A periodic solution with a small amplitude m 2 is unstable, while a periodic solution with a large amplitude m 1 is stable.

Thus, in the studied relay system, there is a self-oscillatory mode with a frequency w 0 = 4.3 s -1 and an amplitude a 0 = b´m 1 = = 58 V.

This chapter will present the method of harmonic linearization for the approximate determination of periodic solutions (self-oscillations) and the stability of nonlinear systems of any order, which, in theory, is close to the method of equivalent linearization or the harmonic balance method of N. M. Krylov and N. N. Bogolyubov, and according to the results - also to the method of a small parameter of BV Bulgakov.

The considered approximate method is a powerful tool for studying nonlinear automatic systems in the sense of simplicity and rather great universality of its apparatus in application to a wide variety of nonlinearities. However, it should be kept in mind that it solves the problem approximately. There are certain limitations to its applicability, which will be discussed below. These restrictions are usually well observed in problems of automatic control theory. Practical calculations and experiment show the acceptability of this method for many types of nonlinear systems.

Let some non-linear expression of the form

Expanding the function on the right side of expression (18.1) into a Fourier series, we obtain

which means that there is no constant component in this expansion. In this chapter, we will everywhere assume that the condition for the absence of a constant component (18.5) is satisfied. Subsequently (Chapter 19) a method will be given for studying self-oscillations in the presence of a constant component, i.e., in the case of non-fulfillment of condition (18.5).

If we take into account that from (18.2) and (18.3)

then formula (18.4) under condition (18.5) can be written as

where q - coefficients of harmonic linearization, determined by the formulas:

Thus, the non-linear expression (18.1) at is replaced by the expression (18.6), which, up to higher harmonics, is similar to the linear one. This operation is called harmonic linearization. The coefficients are constant at constant values, i.e. in the case of a periodic process. In a transient oscillatory process, with a change in a and co, the coefficients q and change (see Chap. 20). For different amplitudes and frequencies of periodic processes, the coefficients of expression (18.6) will be different in magnitude. This circumstance, which is very important for what follows, is a significant difference between harmonic linearization and the usual linearization method (§ 3.1), which leads to purely linear expressions that were used in the previous sections of the book. The specified circumstance will allow, by applying linear research methods to expression (18.6), to analyze the main properties of nonlinear systems that cannot be detected with ordinary linearization.

We also give harmonic linearization formulas for a simpler nonlinearity:

Two options are possible here: 1) the curve has a hysteresis loop (for example, Fig. 16.18, c, Fig. 16.22, d, e), and 2) the curve does not have a hysteresis loop (Fig. 16.8, b, Fig. 16.22, a and etc.).

In the presence of a hysteresis loop, when in fact there is a dependence on the sign of the derivative, the nonlinear function after harmonic linearization is replaced by the following expression (for

in the absence of a constant component:

If the curve does not have a hysteresis loop, then since at will

(with a hysteresis loop, this integral was not zero due to the difference in the shape of the curve with increasing and decreasing

Consequently, in the absence of a hysteresis loop, the nonlinear expression (18.8) is replaced by a simpler one:

i.e., a curvilinear or broken characteristic, up to higher harmonics, is replaced by a rectilinear one, the tangent of the slope of which q depends on the size of the oscillation amplitude a. In other words, a non-linear link is likened to a “linear” link with a gear ratio (gain) that depends on the amplitude a of the oscillations of the input value x.

The hysteresis loop introduces, according to (18.9), in addition, a derivative that gives the phase lag, since Thus, the non-linear coordinate lag in the form of a hysteresis loop turns into an equivalent linear phase lag during harmonic linearization.

It is possible to create a special non-linear link with a leading loop, which will be equivalent to a linear phase advance with the introduction of a derivative, but with the difference that the amount of phase advance will depend on the size of the oscillation amplitude, which is not the case in linear systems.

In cases where a nonlinear link is described by a complex equation that includes the sum of various linear and nonlinear expressions, each of the nonlinear terms is subjected to harmonic linearization separately. The product of nonlinearities is necessarily considered as a whole as one complex nonlinearity. In this case, non-linear functions of a different nature may occur.

For example, in the case of harmonic linearization of the second of equations (16.3), one will have to deal with the function at . In this case, we get

given that

If the function or function is the only non-linear function in the equation of the non-linear link, then with a harmonic

linearization can be put and

similarly to the former formulas (18.6) and (18.7). But in this case, the value of a in all calculations will be the amplitude of the velocity oscillations and not the coordinate x itself. The latter will then have an amplitude

When calculating the coefficients of harmonic linearization using formulas (18.10), it must be borne in mind that with symmetric nonlinear characteristics, the integral can be obtained by doubling the integral, i.e.

and for hysteresis-free characteristics that are symmetric with respect to the origin, when calculating, one can write

We give expressions for the coefficients of some of the simplest nonlinear links. Then they can be directly used in solving various specific problems.

Coefficients of harmonic linearization of relay links. Let's find the coefficients and equations of the most typical relay links using formulas (18.10). Let's take a general view of the characteristics of the relay link depicted by the graph in fig. 18.1, a, where there is any fractional number in the interval

Equations of other types of relay links will be obtained as special cases.

If the fluctuations of the input value have an amplitude, then according to Fig. 18.1, and there will be no movement in the system. If the amplitude is then relay switching occurs at points A, B, C, D (Fig. 18.1, b), in which we have

Therefore, after using the properties, each of the integrals (18.10) is divided into three terms:

and the first and third of them, according to Fig. 18.1, a and will be zeros. Therefore, expressions (18.10) take the form

and the relay link equation with a characteristic of the form fig. 18.1, but will have the form (18.9) with the values ​​obtained here and .

Let's consider special cases.

For a relay link with a characteristic without a hysteresis loop, but with a dead zone (Fig. 18.1, a), assuming from the above formulas, we obtain

For a relay characteristic with a hysteresis loop like fig. assuming we have

Finally, for an ideal relay link (Fig. 18.1, e), assuming we find

In the last example, it is easy to see the meaning of the harmonic linearization of the relay characteristic. The written expression for q means replacing the broken line characteristic with a straight line (Fig. 18.1, e) with such a slope that this line approximately replaces the section of the broken line that is covered by a given amplitude a. From here, the inversely proportional dependence on a given by formula (18.18) becomes quite understandable, since the greater the amplitude a of the fluctuations of the input value, the flatter the straight line approximately replacing the broken line should be.

The situation is similar with the relay characteristic in Fig. 18.1, r for which the slope of the line replacing it is given by formula (18.16). Consequently, any hysteresis-free relay link in the oscillatory process is equivalent to such a "linear" link, the gear ratio (gain) of which decreases with increasing amplitude of the input variable, starting from

As for the relay link with a hysteresis loop, according to (18.9) and (18.17) it is replaced by a linear link with a similar former gain , but, in addition, with the introduction of a negative derivative on the right side of the equation. The introduction of a negative derivative as opposed to a positive one (see § 10.2) introduces a phase lag in the response of the link to the input action. This serves as a "linear equivalent" replacing the hysteresis loop effect of the non-linearity. Wherein

the coefficient at the derivative according to (18.17) also decreases with an increase in the amplitude a of the oscillations of the input value, which is understandable, since the effect of the influence of the hysteresis loop on the process of oscillations in the relay link should be the smaller, the greater the amplitude of the oscillations compared to the width of the hysteresis loop.

Coefficients of harmonic linearization of other simple non-linear links. Consider a nonlinear link with a dead zone and saturation (Fig. 18.2, a). According to fig. 18.2, b, where

integral (18.10) on the section is divided into five terms, and two of them are equal to zero. That's why

whence with the replacement we get

where are determined by formulas (18.19). Since there is no hysteresis loop here

So, the equation of a nonlinear link with a characteristic of the form fig. 18.2, and will be where is determined by the expression (18.20).

As a special case, this gives a value for a link with a dead zone without saturation (Fig. 18.2, c). To do this, in the previous solution we need to put and, therefore, Then

As you can see, a link with a dead zone is likened here to a linear link with a gain reduced due to it. This decrease in the gain is significant at small amplitudes and small at large ones, and at

Purpose of the harmonic linearization method.

The idea of ​​the harmonic linearization method was proposed in 1934. N. M. Krylov and N. N. Bogolyubov. As applied 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 method of harmonic balance, the method of describing functions, the method of equivalent linearization.

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

Knowing the parameters of self-oscillations makes it possible to present a picture of possible processes in the system and, in particular, to determine the stability conditions. Suppose, for example, that as a result of the study of self-oscillations in some nonlinear system, we obtained the dependence of the amplitude of these self-oscillations A from the transfer 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 transfer coefficient k, When k>k cr, 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, arrows conditionally show the nature of transient processes at different values k: at k>k kr the transient process caused by the initial deviation shrinks to self-oscillations. It can be seen from the figure that at k< k cr, the system is stable. Thus, k kr is the critical value of the transmission coefficient according to the stability condition. Its excess leads to the fact that the initial mode of the system becomes unstable and self-oscillations occur 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.

Consider a nonlinear system, the scheme of which is shown in Fig. 12.2, and . The system consists of a linear part with a transfer function W l ( s) and a nonlinear link NL with a specific specification . A link with a coefficient - 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 mode under consideration, the input value X nonlinear link and output Y are periodic functions of time.

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

, (12.1)

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

In fact, self-oscillations in nonlinear systems are always non-sinusoidal due to the distortion of their shape by a nonlinear link. Therefore, this initial assumption means that the harmonic linearization method is fundamentally approximate and the scope of its application is limited to cases where self-oscillations at the input of a nonlinear link are sufficiently close to sinusoidal. In order for this to take place, the linear part of the system must not pass the higher harmonics of self-oscillations, i.e., be low pass filter. The latter is illustrated in Fig. 12.2, b . If, for example, the frequency of self-oscillations is , then the linear part c 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 method of harmonic linearization 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 argued that the oscillations at the input of the nonlinear link will be sufficiently close to sinusoidal, i.e. the prerequisite for applying the harmonic linearization method is not met.

In order to establish 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 as a result of using this method. Thus, the applicability of the harmonic linearization method has to be determined already at the end of the study as a test.

It should be noted that if, as a result of this verification, 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 by some by another method.

So, assuming that the linear part of the system is a low-pass filter, we assume that self-oscillations at the input of the nonlinear link are sinusoidal, i.e. they have the form (12.1). In this case, the oscillations at the output of this link will already be non-sinusoidal due to their distortion by the nonlinearity. As an example, in fig. 12.3, a curve is plotted at the output of a non-linear link for a certain amplitude of an input purely sinusoidal signal according to the link characteristic given in the same place.

Fig.12.3. The 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 link. Therefore, we expand the output oscillations in a Fourier series and discard the higher harmonics. As a result, we get:

;

; (12.3)

;

.

Let us rewrite expression (12.2) in a more convenient form 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)

Here are the notations:

. (12.6)

The differential equation (12.5) is valid for a sinusoidal input signal (12.1) and determines the output signal of a non-linear 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 higher harmonics are discarded, then for a fixed harmonic signal, the original non-linear link can be replaced by an equivalent linear one described by equation (12.5). This replacement is called harmonic linearization .

On fig. 12.4 schematically shows the diagram of this link, consisting of two parallel links.

Rice. 12.4. Equivalent linear link resulting from 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 gains: - transfer coefficient of the constant component, and - two transfer coefficients of the sinusoidal component of self-oscillations. These coefficients are determined by the nonlinearity and the values ​​of and by formulas (12.3). There are ready-made expressions defined by these formulas for a number of typical non-linear links. For these and in general for all inertial non-linear links, the quantities do not depend on and are functions only of the amplitude A And .

Let us illustrate the calculation of the harmonic linearization coefficients with several examples: first for symmetrical oscillations, and then for asymmetric ones. We 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 restrict ourselves to integration over a quarter of the period, quadrupling the result, namely

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

and you can use the formulas

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

Example 1. We investigate cubic non-linearity (Fig. 4.4, i):

Addiction q(a) shown in fig. 4.4, b. From fig. 4.4, A it can be seen that for a given amplitude i is straight q(a)x averages the curvilinear dependence F(x) on a given

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

Example 2. We investigate the loop relay characteristic (Fig. 4.5, a). On fig. 4.5.6 shows the integrand F(a sin y) for formulas (4.21). 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. By formulas (4.21) we obtain (for a³b)

On fig. 4.5, b shows the graphs q (a) and q"(a). The first of them shows the change in the steepness of the slope of the averaging straight line q( A)x s change A(see Fig. 4.5, a). Naturally, q( a)à0 at aॠat, 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 an increase in the amplitude a, since it is clear that the loop will occupy the smaller part of the “working area” of the characteristic F( x), the greater the amplitude of fluctuations of the variable X.

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

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

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

Example 3 V). Similarly to the previous one, we obtain, respectively,

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

Example 4. We examine 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 built 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

Graphically, the result is shown 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, g. At the same time, for a characteristic with saturation (Fig. 4.8, a) we have q=k at 0 £ a£ b.

Let us now show examples of calculating the harmonic linearization coefficients for asymmetrical oscillations with the same non-linearities.

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

and by formulas (4.17)

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

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

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

Example 9 x 0 ½ we have

These dependences are presented in the form of graphs in Figs. 4.11, b, V.

Example 10. For a non-symmetrical characteristic

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

and by 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 are looking 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 A and w. The form of the nonlinearity is given = F( x) and the transfer function of the linear part

Harmonic linearization of the nonlinearity is performed

which leads to the transfer function

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

The periodic solution of 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. Therefore, the periodic solution (4.50) is defined by the equality

Equation (4.51) determines the desired amplitude A and the frequency w of the periodic solution. This equation is 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 the non-linearity 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 measured along the curve -1 / Wn (a) , and the value of w - along the curve Wl (jw).

Instead, two scalar equations following from (4.51) and (4.52) can be used:

which also determine the two required quantities A and w.

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

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

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

It is obvious that it will not be possible to guess this solution right away. Therefore, attempts are made, shown by dashed lines. The last points of these trial rectangles M1 and M2 do not fall on the phase characteristic of the 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 a straight line MM1 .

Finding a periodic solution is simplified in the case of a single-valued 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 the periodic solution, it is necessary to investigate its stability. As already mentioned, a periodic solution takes place in the case when the amplitude-phase characteristic of an open circuit

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

So it is required that at a given frequency w be

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

Consider examples.

Let in the servo system (Fig. 4.13, a) the amplifier has relay characteristic(Figure 4.17, A). Pa fig. 4.17, b for it is shown the graph of the coefficient of harmonic linearization q( A), and q'( A)=0. To determine the periodic solution by the frequency method, according to Fig. 4.22, it is necessary to investigate the expression

From formula (4.24) we obtain for the given 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 same -1 / Wn ( A), being real in this case (Fig. 4.26), fits all on the negative part of the real axis (Fig. 4.27). At the same time, in the section of the change in the amplitude b £ a£ b the amplitude is counted from the left from the outside inside the curve Wl(jw), and on 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 lectures 15, 16).

Consider also the case relay loop 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 x-axis (Fig. 4.28, b), with amplitude reading A from right to left. The intersection will give a stable periodic solution (self-oscillations). To get amplitude versus frequency graphs

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

As already noted, in nonlinear and, in particular, relay ACPs, there are often observed stable periodic oscillations constant amplitude and frequency, the so-called self-oscillations. Moreover, self-oscillations can persist even with significant changes in the system parameters. Practice has shown that in many cases the fluctuations of the regulated value (Fig. 3) are close to harmonic.

The closeness of self-oscillations to harmonic ones makes it possible to use the method of harmonic linearization 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 ones. We restrict ourselves to systems that, as in Fig. 3 can be reduced to a series connection of a non-linear element and a linear part. We 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:

The output signal of a non-linear 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 out 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 self-oscillation frequency 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 between the degrees of the polynomials of the denominator and numerator of the transfer function of the linear part

at least two

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

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

where and are the real and imaginary parts of ,

- argument,

- module.

In the general case, it depends on both the amplitude and the frequency of self-oscillations and the constant component . Physically complex non-linear element gain, more commonly referred to as harmonic linearization coefficient , There is complex gain of the nonlinear element in the first harmonic. Harmonic linearization coefficient modulus

numerically equal to the ratio of the amplitude of the first harmonic at the output of the non-linear element to the amplitude of the input harmonic signal.

Argument

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

For ambiguous nonlinearities, fig. 2, c, 2, d, is 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. 2c) and with a minus sign otherwise (Fig. 2d).

In the general case, and are calculated by the formulas

where , is a non-linear function (characteristic of a non-linear element).

In view of the foregoing, 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 non-linear element in fig. 5 is marked as , it is

Emphasizes that the non-linear element generates only

the first harmonic of the vibrations. Formulas for the harmonic linearization coefficients for typical nonlinearities can be found in the literature, for example, in. The table in Appendix B shows the characteristics of the studied relay elements, formulas for and their hodographs. There are also formulas and hodographs for the reciprocal coefficient of harmonic linearization, defined by the expression

where are the real and imaginary parts of . The hodographs and are plotted in the coordinates , and , respectively.

Let us now write the conditions for the existence of self-oscillations. The system in fig. 5 is equivalent to linear. In a linear system, there are undamped oscillations if it is on the boundary of stability. Let us use the condition of the stability boundary according to the Nyquist criterion: . On fig. 6a – two intersection points, which indicates the presence of two limit cycles.