home Heating systems Method of harmonic linearization. Harmonic Linearization Method: Guidelines for Laboratory Work Analysis of a Nonlinear System by Harmonic Linearization Method

Method of harmonic linearization. Harmonic Linearization Method: Guidelines for Laboratory Work Analysis of a Nonlinear System by Harmonic Linearization Method

Let us illustrate the calculation of the harmonic linearization coefficients with several examples: first for symmetrical oscillations, and then for asymmetric ones. Let us preliminarily 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, but 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£ . but. Naturally, the steepness q(a) slope of this averaging line q(a)x increases with amplitude but(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 /but. 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( but)x s change but(see Fig. 4.5, a). Naturally, q( a)à0 at aà¥ at, since the output signal remains constant (F( x)=c) for any unlimited increase in the input signal X. From physical considerations it is also clear why q" <0. Это коэффициент при производной в формуле (4.20). Положительный знак давал бы опережение сигнала на выходе, в то время как гистерезисная петля дает запаздывание. Поэтому естественно, что q" < 0. Абсолютное значение q" decreases with 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 in). 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, but) 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, in.

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 in.

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 but 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 but 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 but and w, and the value but 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 but 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 but in natural scale. The solution of these equations will be such values but 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 but. The system will return to a periodic solution if, for D but> 0 oscillations are damped, and at D but < 0 - расходятся. Следовательно, при Dbut> 0 characteristic W(jw, but) should be deformed (Fig. 4.25) so that at D but> 0, the Nyquist stability criterion was observed, and for D but < 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 but along the curve -1/Wn ( but) 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, but). Pa fig. 4.17, b for it is shown the graph of the coefficient of harmonic linearization q( but), and q'( but)=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 ( but), 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 but>b - reversed. Therefore, the first intersection point ( but 1) gives an unstable periodic solution, and the second ( but 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( but), 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 but 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( but) corresponding values but and w.

The method of harmonic linearization (harmonic balance) allows you to determine the conditions for the existence and parameters of possible self-oscillations in non-linear automatic control systems. Self-oscillations are determined by limit cycles in the phase space of systems. Limit cycles divide space (generally - multidimensional) on the domains of damped and divergent processes. As a result of calculating the parameters of self-oscillations, one can draw a conclusion about their admissibility for a given system or about the need to change the parameters of the system.

The method allows:

Determine the conditions for the stability of a nonlinear system;

Find the frequency and amplitude of free oscillations of the system;

Synthesize corrective circuits to ensure the required parameters of self-oscillations;

Investigate forced oscillations and evaluate the quality of transient processes in non-linear automatic control systems.

Conditions for applicability of the harmonic linearization method.

1) When using the method, it is assumed that linear part of the system is stable or neutral.

2) The signal at the input of the non-linear link is close in shape to the harmonic signal. This provision needs some explanation.

Figure 1 shows the block diagrams of the non-linear ACS. The circuit consists of series-connected links: a non-linear link y=F(x) and a linear

th, which is described by the differential equation

For y = F(g - x) = g - x we obtain the equation of motion of a linear system.

Consider free movement, i.e. for g(t) º 0. Then,

In the case when there are self-oscillations in the system, the free motion of the system is periodic. Non-periodic movement over time ends with the system stopping to some final position (usually, on a specially provided limiter).

With any form of a periodic signal at the input of a non-linear element, the signal at its output will contain, in addition to the fundamental frequency, higher harmonics. The assumption that the signal at the input of the nonlinear part of the system can be considered harmonic, i.e., that

x(t)@a×sin(wt),

where w=1/T, T is the period of free oscillations of the system, is equivalent to the assumption that the linear part of the system effectively filters higher harmonics of the signal y(t) = F(x (t)).

In the general case, when a nonlinear element of a harmonic signal x(t) acts at the input, the output signal can be Fourier transformed:

Fourier series coefficients

To simplify the calculations, we set C 0 =0, i.e., that the function F(x) is symmetric with respect to the origin. Such a limitation is not necessary and is done by analysis. The appearance of the coefficients C k ¹ 0 means that, in the general case, the nonlinear transformation of the signal is accompanied by phase shifts of the converted signal. In particular, this takes place in non-linearities with ambiguous characteristics (with various kinds of hysteresis loops), both delay and, in some cases, phase advance.

The assumption of effective filtering means that the amplitudes of higher harmonics at the output of the linear part of the system are small, that is,

The fulfillment of this condition is facilitated by the fact that in many cases the amplitudes of the harmonics already directly at the output of the nonlinearity turn out to be significantly less than the amplitude of the first harmonic. For example, at the output of an ideal relay with a harmonic signal at the input

y(t)=F(с×sin(wt))=a×sign(sin(wt))

there are no even harmonics, and the amplitude of the third harmonic in three times less than the amplitude of the first harmonic

Let's do assessment of the degree of suppression higher harmonics of the signal in the linear part of the ACS. To do this, we make a number of assumptions.

1) Frequency of free oscillations of ACS approximately equal to the cutoff frequency its linear part. Note that the frequency of free oscillations of a nonlinear automatic control system can differ significantly from the frequency of free oscillations of a linear system, so that this assumption is not always correct.

2) We take the ACS oscillation index equal to M=1.1.

3) LAH in the vicinity of the cutoff frequency (w s) has a slope of -20 dB/dec. The boundaries of this section of the LAH are related to the oscillation index by the relations

4) The frequency w max is conjugating with the LPH section, so that when w > w max the LAH slope is at least minus 40 dB/dec.

5) Non-linearity - an ideal relay with characteristic y = sgn(x) so that only odd harmonics will be present at its non-linearity output.

The frequencies of the third harmonic w 3 \u003d 3w c, the fifth w 5 \u003d 5w c,

lgw 3 = 0.48+lgw c ,

lgw 5 = 0.7+lgw c .

Frequency w max = 1.91w s, lgw max = 0.28+lgw s. The corner frequency is 0.28 decades away from the cutoff frequency.

The decrease in the amplitudes of the higher harmonics of the signal as they pass through the linear part of the system will be for the third harmonic

L 3 \u003d -0.28 × 20-(0.48-0.28) × 40 \u003d -13.6 dB, that is, 4.8 times,

for the fifth - L 5 \u003d -0.28 × 20-(0.7-0.28) × 40 \u003d -22.4 dB, that is, 13 times.

Consequently, the signal at the output of the linear part will be close to harmonic

This is equivalent to assuming that the system is a low pass filter.

When a harmonic signal is applied to the input of a linear system

a harmonic signal is also set at the output of the system, but with a different amplitude and shifted in phase with respect to the input. If a sinusoidal signal is applied to the input of a non-linear element, then periodic oscillations are formed at its output, but in form they differ significantly from sinusoidal ones. As an example, in fig. 8.17 shows the nature of the change in the output variable of a non-linear element with a relay characteristic (8.14) when sinusoidal oscillations (8.18) enter its input.

Expanding the periodic signal at the output of a nonlinear element into a Fourier series, we represent it as the sum of a constant component and an infinite set of harmonic components:

, (8.19)

where – constant coefficients of the Fourier series; – oscillation frequency of the first harmonic (fundamental frequency), equal to the frequency of input sinusoidal oscillations; T - the period of oscillation of the first harmonic, equal to the period of the input sinusoidal oscillations.

The output signal of the non-linear element is fed to the input of the linear part of the ACS (see Fig. 8.1), which, as a rule, has a significant inertia. In this case, the high-frequency components of the signal (8.19) practically do not pass to the output of the system, i.e. the linear part is a filter in relation to the high-frequency harmonic components. In this regard, and also taking into account that the amplitudes of the harmonic components in decrease with increasing harmonic frequency, for an approximate estimate of the output value of a nonlinear element, in a large number of cases it is sufficient to take into account only the first harmonic component in .

Therefore, in the absence of a constant component in the output oscillations, expression (8.19) can be approximately written as:

Expressing from formula (8.20) the function , and from the derivative - function , we transform the expression (8.20) as follows:

. (8.21)

Thus, the non-linear dependence of the output value on the input value in a non-linear element is approximately replaced by a linear dependence described by expression (8.21).

Having performed the Laplace transformation in expression (8.21), we obtain:

As for continuous links, we introduce into consideration transfer function of a nonlinear harmonically linearized element , as the ratio of the image of the output quantity to the image of the input quantity:

. (8.22)

Table 8.1

Coefficients of harmonic linearization of typical nonlinearities

Static characteristic of a non-linear element
Linear response with deadband
Linear characteristic with limitation
Linear response with deadband and clipping
Characteristic "backlash"
Ideal relay characteristic
Unambiguous relay characteristic with deadband
Ambiguous relay response with deadband
Cubic parabola:
Characteristic "hysteresis loop"

The transfer function of a non-linear element has a significant difference from the transfer function of a linear system, which lies in the fact that it depends on the amplitude and frequency of the input signal.

Expression (8.22) can be written as:

q(A) + q 1 (A), (8.23)

where q(A),q 1 (A) are the coefficients of harmonic linearization, defined as the ratio of the coefficients of the Fourier series for the first harmonic of the output oscillations to the amplitude of the input oscillations:

q(A) = q 1 (A) = . (8.24)

Replacing in expression (8.23) R on , we obtain an expression for complex gain of the non-linear element :

q(A) +j q 1 (A), (8.25)

which is an analogue of the AFC for a linear link.

As an example, let's define an expression for the complex transfer coefficient of a non-linear element with a relay static characteristic (8.14). Fourier series coefficients A 1 And B 1 for the indicated nonlinearity are:

B 1 .

It is obvious that the coefficient B 1 will be equal to zero for any non-linear element with odd-symmetric static non-linearity.

where - transfer function of the linear part of the system; - transfer function of a non-linear element after its linearization.

If , then expression (8.26) can be written as:

Replacing in expression (8.27) R on , we obtain a complex expression in which it is necessary to separate the real and imaginary parts:

[ q(A) +j q 1 (A) ] . (8.28)

In this case, we write the condition for the occurrence of periodic oscillations in the system with frequency and amplitude:

(8.29)

If the solutions of system (8.29) are complex or negative, the mode of self-oscillations in the system is impossible. The presence of positive real solutions for and indicates the presence of self-oscillations in the system, which must be checked for stability.

As an example, let's find the conditions for the occurrence of self-oscillations in the ACS, if the transfer function of its linear part is equal to:

(8.30)

and a non-linear element of the "hysteresis loop" type.

The transfer function of a harmonically linearized non-linear element (see Table 8.1) is:

. (8.31)

Substituting expressions (8.30) and (8.31) into expression (8.26) and replacing R on , find the expression for :

From here, in accordance with expression (8.29), we obtain the following conditions for the occurrence of self-oscillations in the system:

The solution of the system of equations (8.29) is usually difficult, since the harmonic linearization coefficients have a complex dependence on the amplitude of the input signal. In addition, in addition to determining the amplitude and frequency , it is necessary to evaluate the stability of self-oscillations in the system.

The conditions for the occurrence of self-oscillations in a nonlinear system and the parameters of limit cycles can be investigated using frequency stability criteria, for example, the Nyquist stability criterion. According to this criterion, in the presence of auto-oscillations, the amplitude-phase characteristic of an open-loop harmonically linearized system is equal to

passes through the point (-1, j0). Therefore, for and the following equality holds:

. (8.32)

The solution of equation (8.32) with respect to the frequency and amplitude of self-oscillations can be obtained graphically. To do this, on the complex plane, it is necessary, by changing the frequency from 0 to , to construct the AFC hodograph of the linear part of the system and, by changing the amplitude BUT from 0 to , build a hodograph of the inverse characteristic of the non-linear part , taken with a minus sign. If these hodographs do not intersect, then the mode of self-oscillations in the system under study does not exist (Fig. 8.18, b).

When the hodographs intersect (Fig. 8.18, a), self-oscillations arise in the system, the frequency and amplitude of which are determined by the values and at the intersection point.

If and - intersect at several points (Fig. 8.18, a), then this indicates the presence of several limit cycles in the system. In this case, oscillations in the system can be stable and unstable.

The stability of the self-oscillatory regime is estimated as follows. The self-oscillation mode is stable if the point on the hodograph of the non-linear part , corresponding to an amplitude greater than the value at the point of intersection of the hodographs, is not covered by the hodograph of the frequency response of the linear part of the system. Otherwise, the self-oscillatory regime is unstable.

On fig. 8.18, and the hodographs intersect at points 1 and 2. Point 1 determines the unstable mode of self-oscillations, since the hodograph point corresponding to the increased amplitude is covered by the hodograph of the frequency response of the linear part of the system. Point 2 corresponds to a stable mode of self-oscillations, the amplitude of which is determined by the hodograph and the frequency - by the hodograph.

As an example, let us estimate the stability of self-oscillations in two nonlinear systems. We will assume that the transfer functions of the linear parts of these systems coincide and are equal:

but their non-linear elements included in them are different. Let the first system include a non-linear element "ideal relay", described by system (8.14), and the second one - a non-linear element with a static characteristic "cubic parabola". Using the data in Table 8.1, we get:

On fig. 8.19 shows the hodographs of these systems together with the AFC hodograph of the linear part of the system. Based on the foregoing, it can be argued that stable self-oscillations with frequency and amplitude occur in the first system, and unstable self-oscillations occur in the second system.

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 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 replaced in the first approximation 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 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, 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 solution methods characteristic equation distinguish between methods for studying nonlinear systems.

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 .

Research on logarithmic frequency characteristics .

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-oscillatory 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

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 . (8)

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 under the condition 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