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 his apparatus in application to the most diverse 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 a dependence on the sign of the derivative is actually observed, 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” one 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
on condition
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 oscillations, 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 hysteresis loop on the process of oscillations in the relay link should be the smaller, the greater the oscillation amplitude 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
The idea of the harmonic linearization method belongs to N.M. Krylov and N.N. Bogolyubov and is based on the replacement of 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 non-linear 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 that appear at the output of the non-linear element, except for the first harmonic.
Harmonic Linearization Coefficients and Equivalent Complex Gains of Nonlinear Elements. In not linear system(Fig. 2.1) the parameters of the linear part and the non-linear element are chosen in such a way that there are symmetrical periodic oscillations with a frequency w.
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 non-linear element.
Rice. 2.10. Characteristics of a non-linear element
In case of insensitivity nonlinear system to higher harmonics, the nonlinear element can be replaced in the first approximation by some element with an equivalent gain, which determines the first harmonic of the periodic oscillations at the output depending on the frequency and amplitude of the 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 ordinary 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, the question of the existence and stability of periodic modes is first of all 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.
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 on the course "Theory of automatic control" for students of the specialty 210100
Approved
editorial and publishing council
Balakovo Institute of Technology,
technology and management
Balakovo 2004
The purpose of the work: The study of nonlinear systems using the method of harmonic linearization (harmonic balance), the determination of the coefficients of harmonic linearization for various nonlinear links. Obtaining skills in finding the parameters of symmetrical oscillations of constant amplitude and frequency (self-oscillations), using algebraic, frequency methods, as well as using the Mikhailov criterion.
BASIC INFORMATION
The method of harmonic linearization refers to approximate methods for studying nonlinear systems. It makes it possible to assess the stability of nonlinear systems quite simply and with acceptable accuracy, and to determine the frequency and amplitude of the oscillations established in the system.
It is assumed that the investigated nonlinear ACS can be represented in the following form
moreover, the non-linear part must have one non-linearity
. (1)
This non-linearity can be either continuous or relay, unambiguous 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. 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 input to 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 the nonlinear system is a high pass filter. In this case, the linear part will pass only low frequencies without attenuation, while high frequencies will be significantly attenuated as the frequency increases.
The harmonic linearization method assumes 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 look like
This signal passes through the entire closed loop of the system Fig.1 and at the output of the non-linear element without taking into account higher harmonics, according to (2) we have
.
(7)
In the study of nonlinear systems using the method of harmonic linearization, cases of symmetric and asymmetric oscillations are possible. Let us consider the case of symmetric oscillations. Here and.
We introduce the following notation
,
.
Substituting them into (7), we obtain . (eight)
Taking into account the fact that
,
, where ,
. (9)
According to (3) and (8) at
,
. (10)
Expression (9) is a harmonic linearization of the nonlinearity establishes a linear relationship between the input variable and the output variable at . The quantities and are called harmonic linearization coefficients.
It should be noted that equation (9) is linear for specific values and (amplitudes and frequencies 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 explore the properties of nonlinear systems using the method of harmonic linearization [Popov E.P.].
In the case of asymmetric oscillations, the harmonic linearization of the nonlinearity leads to the linear equation
,
,
. (12)
Just like equation (9), the linearized equation (11) retains 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 one to obtain the transfer functions of harmonically linearized nonlinear elements. So for symmetrical vibrations
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 on the course "Theory of automatic control" for students of the specialty 210100
Approved
editorial and publishing council
Balakovo Institute of Technology,
technology and management
Balakovo 2004
The purpose of the work: The study of nonlinear systems using the method of harmonic linearization (harmonic balance), the determination of the coefficients of harmonic linearization for various nonlinear links. Obtaining skills in finding the parameters of symmetrical oscillations of constant amplitude and frequency (self-oscillations), using algebraic, frequency methods, as well as using the Mikhailov criterion.
BASIC INFORMATION
The method of harmonic linearization refers to approximate methods for studying nonlinear systems. It makes it possible to assess the stability of nonlinear systems quite simply and with acceptable accuracy, and to determine the frequency and amplitude of the oscillations established in the system.
It is assumed that the investigated nonlinear ACS can be represented in the following form
moreover, the non-linear part must have one non-linearity
This non-linearity can be either continuous or relay, unambiguous 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. 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 input to 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 the nonlinear system is a high pass filter. In this case, the linear part will pass only low frequencies without attenuation, while high frequencies will be significantly attenuated as the frequency increases.
The harmonic linearization method assumes 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 look like
This signal passes through the entire closed loop of the system Fig.1 and at the output of the non-linear element without taking into account higher harmonics, according to (2) we have
.
(7)
In the study of nonlinear systems using the method of harmonic linearization, cases of symmetric and asymmetric oscillations are possible. Let us consider the case of symmetric oscillations. Here and.
We introduce the following notation
Substituting them into (7), we obtain . (eight)
Taking into account the fact that
.
(9)
According to (3) and (8) at
,
.
(10)
Expression (9) is a harmonic linearization of the nonlinearity and establishes a linear relationship between the input variable and the output variable at . The quantities and are called harmonic linearization coefficients.
It should be noted that equation (9) is linear for specific values and (amplitudes and frequencies 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 explore the properties of nonlinear systems using the method of harmonic linearization [Popov E.P.].
In the case of asymmetric oscillations, the harmonic linearization of the nonlinearity leads to the linear equation
,
,
.
(12)
Just like equation (9), the linearized equation (11) retains 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 one to obtain the transfer functions of harmonically linearized nonlinear elements. So for symmetrical vibrations
,
(13)
while 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 single-valued, 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 with increasing and decreasing , therefore, the full expression (9) is valid.
Let's find harmonic linearization coefficients for some non-linear characteristics. Let the non-linear characteristic take the form of a relay characteristic with hysteresis and a dead zone. Consider how harmonic oscillations pass through a nonlinear element with such a characteristic.
When 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 non-linear element. If the amplitude is , then the relay switches at points A, B, C and D. Denote and .
,
.
(18)
When calculating the coefficients of harmonic linearization, it should be borne in mind that with symmetric nonlinear characteristics, the integrals in expressions (10) are on the half-cycle (0, ) with a subsequent increase in the result by a factor of two. In this way
,
.
(19)
For a non-linear element with a relay characteristic and a dead zone
,
For a non-linear element having a relay characteristic with hysteresis
,
Harmonic linearization coefficients for other non-linear characteristics can be obtained similarly.
Let us consider two methods for determining symmetric oscillations of constant amplitude and frequency (self-oscillations) and stability of linearized systems: algebraic and frequency. Let's look at the algebraic way first. For a closed system Fig.1, the transfer function of the linear part is equal to
.
We write the harmonically linearized transfer function of the nonlinear part
.
The characteristic equation of a closed system has the form
.
(22)
If self-oscillations occur in the system under study, then this indicates the presence of two purely imaginary roots in its characteristic equation. Therefore, we substitute into the characteristic equation (22) the value of the root .
.
(23)
Imagine
We obtain two equations that determine the desired amplitude and frequency
,
.
(24)
If real positive values of the amplitude and frequency are possible in the solution, then self-oscillations can occur in the system. If the amplitude and frequency do not have positive values, then self-oscillations in the system are impossible.
Consider Example 1. Let the nonlinear system under study have the form
In this example, the non-linear 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 the characteristic equation of a closed system
,
(29)
,
Let 1/sec, sec, sec, c.
In this case, the parameters of the periodic motion are equal to
7,071
,
Let us consider a method for determining the parameters of self-oscillations in a linearized ACS using the Mikhailov criterion. The method is based on the fact that when self-oscillations occur, the system will be at the stability boundary and the Mikhailov hodograph in this case will pass through the origin.
In example 2, we find the parameters of self-oscillations, provided that the nonlinear element in the system Fig. 4 is a sensitive element that has a relay characteristic with hysteresis, for which the harmonic linearization coefficients
,
The linear part remained unchanged.
We write the characteristic equation of a closed system
The Mikhailov hodograph is obtained by replacing .
The task is to choose such an amplitude of oscillations at which the hodograph passes 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, in and in, gave the following results. In Fig.5 Mikhailov's hodograph passes through the origin. To improve the accuracy of calculations, we will increase the desired 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 of coordinates at .
Fig.5. Fig.6.
In this case, 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 value of the amplitude and frequency , must be investigated for stability. If the solution is stable, then a self-oscillating process (stable limit cycle) takes place in the system. 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, let's increment in, that is, and . The position of the Mikhailov curves is shown in Fig.7.
At , the curve passes above zero, which indicates the stability of the system and the damped transient process. When the Mikhailov curve passes below zero, the system is unstable and the transient is divergent. Thus, a periodic solution with an amplitude of 6 and an oscillation frequency of 6.38 is stable.
To study the stability of a periodic solution, an analytical criterion obtained from the Mikhailov graphical criterion can also be used. Indeed, in order to find out whether the Mikhailov curve will go at above zero, it is enough to look where the point of the Mikhailov curve will move, which at is located at the origin of coordinates.
If we expand the displacement of this point along the X and Y coordinate axes, then for the stability of the 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, when viewed along the curve in the direction of increase, the direction of which is determined by the projections
Let us 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, condition (31) can be satisfied, and the system will be unstable
We apply criterion (31) to study the stability of the periodic solution obtained in Example 1.
,
,
,
,
Introduction
Relay systems are widely used in the practice of automatic control. The advantage of relay systems is the simplicity of design, reliability, ease of maintenance and configuration. Relay systems are a special class of non-linear ACPs.
Unlike continuous in relay systems, the control action changes abruptly whenever the control signal of the relay (most often this is a control error) passes through some fixed (threshold) values, for example, through zero.
Relay systems, as a rule, have high speed due to the fact that the control action in them changes almost instantly, and a piecewise constant signal of maximum amplitude acts on the actuator. At the same time, self-oscillations often occur in relay systems, which in many cases is a disadvantage. In this paper, we study a relay system with four different control laws.
The structure of the system under study
The system under study (Fig.) 1 includes an ES comparison element, a RE relay element, an actuator (an ideal integrator with a gain = 1), a control object (an aperiodic link with three time constants , , and a gain ). The values of the system parameters are given in Table. 1 appendix A.
Static characteristics (input-output characteristics) of the studied relay elements are shown in fig. 2.
On fig. 2,a shows the characteristics of an ideal two-position relay, in fig. 2b characteristic of a three-position relay with a dead zone. On fig. 2c and 2d show the characteristics of a two-position relay with positive and negative hysteresis, respectively.
The ACP under study can be modeled using known modeling packages such as SIAM or VisSim.
Comment. In some simulation packages, the value of the output
relay signal can only take the values ±1 instead of ±B, where B is an arbitrary number. In such cases, it is necessary to take the gain of the integrator 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 research (can be performed outside the laboratory).
The second stage is experimental (conducted in the laboratory). At this stage, with the help of one of the packages, transient processes are simulated in the system under study for the modes calculated at the first stage, and the accuracy of theoretical methods is checked.
The necessary theoretical material is presented in Section 4; section 5 contains control questions.
3.1. Settlement - research part
1. Get expressions for the amplitude-frequency and phase-frequency, real and imaginary characteristics of the linear part of the system.
2. Calculate and build the amplitude-phase characteristic of the linear part of the system. For the calculation, use the programs from the TAU package. Necessarily print real and imaginary frequency response values(10 - 15 points corresponding to third and second quadrants).
4. Using the graphic-analytical method of Goldfarb, determine the amplitude and frequency of self-oscillations and their stability for all four relays. The self-oscillation parameters can also be calculated analytically. Qualitatively depict the phase portrait of the system for each of the cases.
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 self-oscillations are stalled.
experimental part
1. Using one of the available simulation packages, assemble the simulation circuit of the ACP under study. With the permission of the teacher, you can use the finished scheme. Set the scheme parameters according to the task.
2. Investigate the transient process in a system with an ideal relay (print out), applying a jump 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 by 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 the 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 clause 5, section 3.1).
5. Investigate transient processes in a system with two-position relays with hysteresis at 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 the 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 ACP. On the basis of the method of harmonic linearization, methods have been developed for constructing transient processes, analysis and synthesis of nonlinear ASR.
Harmonic linearization method
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:
(1)
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
(5)
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
,
. (7)
In the study of 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 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
(9)
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
(10)
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
,
, (12)
where , is a non-linear function (characteristic of a non-linear element).
In view of the foregoing, in the study of 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
, (13)
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. We use the condition of the stability boundary according to the Nyquist criterion:
. (14)
Equation (14) there is the 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 ones. Otherwise, self-oscillations are absent or are not harmonic. Equation (14) splits into two - with respect to the real and imaginary parts:
;
;
Dividing both parts of equation (14) by and taking into account formula (13), we obtain the condition for the existence of self-oscillations in the form of Goldfarb L.S.:
. (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. 6a). If the hodographs intersect, then self-oscillations exist. The parameters of self-oscillations - A and w 0 are determined at the intersection points - the frequency w 0 according to the hodograph, the amplitude according to the hodograph. On fig. 6a – two intersection points, 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 hatched when moving along the AFC in the direction of increasing frequency (Fig. 6).
Self-oscillations are stable if, at the point of intersection, 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 the shaded region to the unshaded region, then self-oscillations are not stable.
On fig. 6b qualitatively depicts the phase portrait corresponding to the 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. 6, b, to a point with parameters and and to achieve a breakdown of self-oscillations , in this case, the 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 limit value K l at which the AFC of the linear part touches Error! Communication error. wherein , and the value of the amplitude is . Naturally, this leads to a qualitative change in the phase portrait of the system.