1.4 Analysis of methods for studying nonlinear systems

Basics of the phase trajectory method

Harmonic linearization method

2.7.3.1. Exact methods for studying nonlinear systems

1. Direct Lyapunov method. It is based on Lyapunov's theorem on the stability of nonlinear systems. The Lyapunov function is used as a research apparatus, which is a sign-definite function of the coordinates of the system, which also has a sign-definite derivative in time. The application of the method is limited by its complexity.

2. Popov’s method (Romanian scientist) is simpler, but suitable only for some special cases.

3. Method based on piecewise linear approximation. The characteristics of individual nonlinear links are divided into a number of linear sections, within which the problem turns out to be linear and can be solved quite simply.

The method can be used if the number of sections into which the nonlinear characteristic is divided is small (relay characteristics). With a large number of areas it is difficult. The solution is only possible with the help of a computer.

4. Phase space method. Allows you to study systems with nonlinearities of arbitrary type, as well as with several nonlinearities. At the same time, in the phase space, a so-called phase portrait of the processes occurring in linear system. By the appearance of the phase portrait, one can judge the stability, the possibility of self-oscillations, and the accuracy in a steady state. However, the dimension of the phase space is equal to the order of the differential equation of the nonlinear system. Application for systems higher than second order is practically impossible.

5. To analyze random processes you can use mathematical apparatus theory of Markov random processes. However, the complexity of the method and the ability to solve the Fokker-Planck equation, which is required in the analysis only for first- and in some cases second-order equations, limits its use.

Thus, although exact methods for analyzing nonlinear systems allow one to obtain accurate, correct results, they are very complex, which limits them practical use. These methods are important from a purely scientific, cognitive, research point of view, and therefore they can be classified as purely academic methods, the practical application of which to real complex systems doesn't make sense.

2.7.3.2. Approximate methods for studying nonlinear systems

The complexity and limitations of the practical application of exact methods for analyzing nonlinear systems have led to the need to develop approximate, simpler methods for studying these systems. Approximate methods make it possible in many practical cases to quite simply obtain transparent and easily visible results of the analysis of nonlinear systems. Approximate methods include:

1. The method of harmonic linearization, based on replacing a nonlinear element with its linear equivalent, and equivalence is achieved for some motion of the system that is close to harmonic. This makes it possible to quite simply investigate the possibility of self-oscillations occurring in the control system. However, the method can also be applied to study transient processes of nonlinear systems.

2. The method of statistical linearization is also based on replacing a nonlinear element with its linear equivalent, but when the system moves under the influence of random disturbances. The method makes it possible to relatively simply study the behavior of a nonlinear system under random influences and find some of its statistical characteristics.

Harmonic linearization method

Let us apply to nonlinear systems described by a differential equation of any order. Let us consider it only in relation to the calculation of self-oscillations in an automatic control system. Let us divide the closed-loop control system into linear and nonlinear parts (Fig. 7.2) with transfer functions and, respectively.

For a linear link:

A nonlinear link can have nonlinear dependencies of the form:

etc. Let us limit ourselves to a dependence of the form:

Rice. 7.2. Towards the harmonic linearization method

Let us pose the problem of studying self-oscillations in this nonlinear system. Strictly speaking, self-oscillations will be non-sinusoidal, but we will assume that for the variable x they are close to the harmonic function. This is justified by the fact that the linear part (7.1), as a rule, is a low-pass filter (LPF). Therefore, the linear part will delay the higher harmonics contained in the variable y. This assumption is called the filter hypothesis. Otherwise, if the linear part is a filter high frequencies(HPF), then the harmonic linearization method may give erroneous results.

Let Substituting into (7.2), we expand (7.2) into a Fourier series:

Let us assume that there is no constant component in the desired oscillations, i.e.

This condition is always met when the nonlinear characteristic is symmetrical with respect to the origin of coordinates and there is no external influence applied to the nonlinear link.

We accepted that, then.

In the written expansion, we will make a replacement and discard all the higher harmonics of the series, considering that they are filtered out. Then for the nonlinear link we obtain the approximate formula

where and are the harmonic linearization coefficients determined by the Fourier series expansion formulas:

Thus, the nonlinear equation (7.2) is replaced by an approximate equation for the first harmonic (7.3), similar to the linear equation. Its peculiarity is that the coefficients of the equation depend on the desired amplitude of self-oscillations. In the general case, with a more complex dependence (7.2), these coefficients will depend on both amplitude and frequency.

The performed operation of replacing a nonlinear equation with an approximate linear one is called harmonic linearization, and coefficients (7.4), (7.5) are called harmonic transmission coefficients of the nonlinear link.

From (7.3) it follows that for the system under consideration the transfer function of the nonlinear link is:

Taking into account (7.1) and (7.3), we obtain the transfer function of the open-loop system:

and characteristic equation closed system:

Substituting into (7.6), we find the frequency transfer function of the open-loop system:

Does not depend on [see (7.8)].

The module of the equivalent transfer function of a nonlinear link is determined by the formula:

and is equal to the ratio of the amplitude of the first harmonic at its output to the amplitude of the input value. The argument of the frequency transfer function of the nonlinear link is equal to:

It can be shown that for nonlinear links with unambiguous and symmetrical relative to the origin of coordinates characteristics that do not have hysteresis loops, therefore - purely real, and

The inverse of the equivalent transfer function of a nonlinear link is often used:

called the equivalent impedance of the nonlinear link. Its use is convenient when calculating self-oscillations using the Nyquist criterion. As an example of using the harmonic linearization method, consider the relay characteristic of a three-position relay without a hysteresis loop (Fig. 7.3). As can be seen from Fig. 7.3, the static characteristic is symmetrical with respect to the origin of coordinates, therefore, . Therefore, it is only necessary to find the coefficient using formula (7.4). To do this, we apply a sinusoidal function to the input of the link and construct y(t) (Fig. 7.4).

Rice. 7.3. Static characteristic of three-position

relay without hysteresis loop

As can be seen from Fig. 7.4, with

The phase angle corresponding to x 1 = b is equal to arcsin (b/a) (Fig. 7.4).

Taking into account the symmetry of the integrand and in accordance with (7.4), we have:

Because , then we finally have:

In a similar way, it is possible to perform harmonic linearization of other nonlinear links. The linearization results are given in , .

As noted above, the harmonic linearization method is convenient for analyzing the possibility of the appearance of a self-oscillation regime in a nonlinear system and determining its parameters. To calculate self-oscillations, various stability criteria are used. The simplest and most obvious way is to use the Nyquist criterion. It is especially convenient to use the Nyquist criterion in the case where there is a nonlinear dependence of the form and the equivalent transfer function of the nonlinear link depends only on the amplitude of the input signal.

Rice. 7.4. Example of linearization of a relay characteristic

Conditions for the occurrence of self-oscillations: the appearance of a pair in solution (7.7) is purely imaginary roots, and all other roots lie in the left half-plane (connection with the point –1,j0).

Let’s equate (7.7) to minus one:

To solve (7.12), we set different values of , and construct the AFC. At some a = A, the AFC will pass through the point (-1,j0), which corresponds to the absence of stability reserves.

The frequency and correspond to the frequency and amplitude of the desired harmonic oscillation: (Fig. 7.5).

In a similar way, it is possible to find a periodic solution for nonlinear dependencies of any type, leading, in particular, to the fact that the equivalent transfer function of a nonlinear element depends not only on amplitude, but also on frequency. If we limit ourselves to considering a nonlinear dependence of the form , then the process of finding the periodic regime can be simplified.

Rice. 7.5. Condition for the occurrence of self-oscillations

Let us write equation (7.12) in the form:

See (7.11). (7.13)

Equation (7.13) can be easily solved graphically. For this purpose, it is necessary to separately construct the AFC and the inverse AFC taken with the opposite sign. The intersection point of two AFCs determines the solution (7.13). We find the frequency of the periodic mode by the frequency marks on the graph, and the amplitude by the amplitude marks on the graph (Fig. 7.6).

However, the found periodic regime corresponds to self-oscillations only when it is stable in the sense that this regime can exist in the system for an indefinitely long time. The stability of the periodic mode can be determined as follows.

Let us assume that the linear part of the system in the open state is stable or neutral. Let's give the amplitude A some positive increment A. Then it will increase, therefore it will decrease. As a result, it decreases and therefore moves even further away from the point (-1,j0). A decreases and will tend to 0. Similarly, if A received a negative increment - A. Then it will decrease, therefore, it will increase, it will increase, and, therefore, the amplitude will increase, because AFC will approach the point (-1,j0) (decrease in stability margins).

Rice. 7.6. The condition for the occurrence of self-oscillations during nonlinear

dependencies of the type

Consequently, any random deviation of A changes the system in such a way that the amplitude restores its value. This corresponds to the stability of the periodic mode, which corresponds to self-oscillations.

The stability criterion for the periodic mode here comes down to the fact that the part of the curve corresponding to smaller amplitudes is covered by the AFC of the linear part of the system, which corresponds to the presence of one point of intersection of the characteristic with the negative part of the axis of real values (see Fig. 7.6).

When the AFC of an open-loop system crosses the negative part of the axis of real values twice, it is possible for the AFC to pass through the point (-1,j0) for two values of and (Fig. 7.7).

The two intersection points correspond to two possible periodic solutions with parameters and . Similar to what was done above, you can make sure that the first point corresponds to an unstable mode of periodic oscillations, and the second to a stable one, i.e. self-oscillations (Fig. 7.8).

In more complex cases, when, say, it is unstable, it is possible to determine the stability of the resulting periodic mode by considering the location of the AFC of the open-loop system. What remains common here is that in order to obtain stability of the periodic regime, it is necessary that a positive increase in amplitude leads to convergent processes in the system, and a negative one to divergent ones.

In the absence of possible periodic modes close to harmonic in the system, which is revealed by the above calculation, there are many different options for the behavior of the system. However, in systems whose linear part has the property of suppressing higher harmonics, especially in such systems where for some parameters there is a periodic solution, but for others not, there is reason to believe that in the absence of a periodic solution the system will be stable relative to the equilibrium state. In this case, the stability of the equilibrium state can be assessed by the requirement that when the linear part is stable or neutral in the open state, its AFC does not cover the hodograph

Method for statistical linearization of nonlinear characteristics

To evaluate the statistical characteristics of nonlinear systems, you can use the method of statistical linearization, based on replacing the nonlinear characteristic with a linear one, which in a certain sense of statistics is equivalent to the original nonlinear characteristic.

Replacing a nonlinear transformation with a linear one is approximate and can be fair only in some respects. Therefore, the concept of statistical equivalence, on the basis of which such a replacement is made, is not unambiguous, and it is possible to formulate various criteria for the statistical equivalence of the nonlinear and the linear transformations replacing it.

In the case when a nonlinear inertia-free dependence of the form (7.2) is subjected to linearization, the following statistical equivalence criteria are usually applied:

The first requires equality of mathematical expectations and variances of processes and , where is the output value of the equivalent linearized link, and is the output value of the nonlinear link;

The second requires minimizing the mean square of the difference between the processes at the output of the nonlinear and linearized elements.

Let us consider linearization for the case of applying the first criterion. Let us replace the nonlinear dependence (7.2) with a linear characteristic (7.14), which has the same mathematical expectations and dispersion as those available at the output of the nonlinear link with characteristic (7.2). For this purpose, we present (7.14) in the form: , where is a centered random function.

According to the selected criterion, the coefficients and must satisfy the following relationships:

From (7.15) it follows that statistical equivalence occurs if

Moreover, the sign must coincide with the sign of the derivative of the nonlinear characteristic F( x).

The quantities are called statistical linearization coefficients. To calculate them, you need to know the signal at the output of the nonlinear link:

where is the probability density of the distribution of a random signal at the input of the nonlinear link.

For the second criterion, the statistical linearization coefficients are selected in such a way as to ensure a minimum of the mean square difference between the processes at the output of the nonlinear and linearized link, i.e. ensure equality

The coefficients of statistical linearization, as follows from (7.16), (7.17) and (7.18), depend not only on the characteristics of the nonlinear link, but also on the distribution law of the signal at its input. In many practical cases, the distribution law of this random variable can be assumed to be Gaussian (normal), described by the expression

This is explained by the fact that nonlinear links in control systems are connected in series with linear inertial elements, the distribution laws of which output signals are close to Gaussian for any distribution laws of their input signals. The more inertial the system, the closer the distribution law of the output signal is to Gaussian, i.e. inertial devices of the system lead to the restoration of the Gaussian distribution, violated by nonlinear links. In addition, changes in the distribution law within a wide small range affect the statistical linearization coefficients. Therefore, it is believed that the signals at the input of nonlinear elements are distributed according to the Gaussian law.

In this case, the coefficients and depend only on the signal at the input of the nonlinear link, therefore, for typical nonlinear characteristics, the coefficients and can be calculated in advance, which significantly simplifies the calculations of systems using the method of statistical linearization. For normal law distributions and typical nonlinear links when calculating nonlinear systems, you can use the data given in.

Application of statistical linearization method for analysis

stationary modes and failure of tracking

Possibility of replacing the characteristics of nonlinear links linear dependencies allows you to use methods developed for linear systems when analyzing nonlinear systems. Let us apply the method of statistical linearization to analyze stationary modes in the system shown in Fig. 7.9,

where F(e) is the static characteristic of the nonlinear element (discriminator);

W(p) – transfer function of the linear part of the system.

The task of the analysis is to assess the influence of the discriminator characteristics on the accuracy of the system and determine the conditions under which the normal operation of the system is disrupted and tracking fails.

When analyzing the accuracy of operation with respect to the non-random component of the signal g(t), the nonlinear element F(e) in accordance with the method of statistical linearization is replaced by a linear link with a transmission coefficient . The dynamic error, as shown earlier, is found by the formula:

An example of finding and , as well as determining the condition for failure of tracking, is given in.

Self-test questions

1. Name approximate methods for analyzing nonlinear systems.

2. What is the essence of the harmonic linearization method?

3. What is the essence of the statistical linearization method?

4. For which nonlinear links does q¢ (a) = 0?

5. What criteria for statistical equivalence do you know?

The presence of nonlinearities in control systems leads to the description of such a system by nonlinear differential equations, often quite high orders. As is known, most groups do not linear equations not resolved in general view, and we can only talk about special cases of the solution, therefore, in the study of nonlinear systems, various approximate methods play an important role.

Using approximate methods for studying nonlinear systems, it is usually impossible to obtain a sufficiently complete understanding of all the dynamic properties of the system. However, with their help it is possible to answer a number of individual essential questions, such as the question of stability, the presence of self-oscillations, the nature of any particular modes, etc.

Currently exists big number various analytical and graph-analytical methods for studying nonlinear systems, among which we can highlight the methods of phase plane, fitting, point transformations, harmonic linearization, Lyapunov’s direct method, frequency methods for studying Popov’s absolute stability, methods for studying nonlinear systems using electronic models and computers.

a brief description of some of the listed methods.

The phase plane method is accurate, but has limited application, since it is practically inapplicable for control systems, the description of which cannot be reduced to second-order controls.

The harmonic linearization method is an approximate method; it has no restrictions on the order of differential equations. When applying this method, it is assumed that there are harmonic oscillations at the output of the system, and the linear part of the control system is a high-pass filter. In the case of weak filtering of signals by the linear part of the system, when using the harmonic linearization method, it is necessary to take into account higher harmonics. At the same time, the analysis of stability and quality of control processes of nonlinear systems becomes more complicated.

The second Lyapunov method allows one to obtain only sufficient conditions for stability. And if on its basis the instability of the control system is determined, then in a number of cases, to check the correctness of the obtained result, it is necessary to replace the Lyapunov function with another one and perform stability analysis again. Moreover, there is no common methods definition of the Lyapunov function, which complicates the practical application of this method.

The absolute stability criterion allows you to analyze the stability of nonlinear systems using frequency characteristics, which is a great advantage this method, since it combines the mathematical apparatus of linear and nonlinear systems into a single whole. The disadvantages of this method include the complexity of calculations when analyzing the stability of systems with an unstable linear part. Therefore, to obtain the correct result on the stability of nonlinear systems, one has to use various methods. And only the coincidence of various results will allow us to avoid erroneous judgments about the stability or instability of the designed automatic control system.

The characteristic shown in Figure 1.5 b is a three-position relay, in which an additional position is due to insensitivity. The equation of such a characteristic

x out		x in	< a ,

x out	B siqn(xin)	x in	>a.

The characteristic shown in Figure 1.5c is a two-position relay with hysteresis. It is also called a “relay with memory”. It “remembers” its previous state and within x input< a сохраняет это своё значение. Уравне-

definition of such a characteristic

xout = b siqn(x − a)	xin > 0,
xout = b siqn(x + a)
xout = b siqn(x + a)	x in< 0 ,

x out = + b	xin > − a ;	x&in< 0,
x out = − b	xin< a;	xin > 0,

The characteristic shown in Figure 1.5 d is a three-position relay with hysteresis, in which an additional position is due to the dead zone. The equation of such a characteristic

From the above equations it is clear that in the absence of a hysteresis loop, the output action of the relay depends only on the value of xin or xout = f (xin).

In the presence of a hysteresis loop, the value of x out also depends on the derivative with respect to x in or x out = f (x in ,x & in), where x & in characterizes the presence of “memory” in the relay.

To solve problems of analysis and synthesis of a nonlinear system, it is first necessary to construct its mathematical model, which characterizes the connection between the output signals of the system and the signals reflecting the influences applied to the system. As a result, we obtain a nonlinear differential equation high order, sometimes with a number of logical relationships. Modern Computer Engineering allows you to solve any nonlinear equations and will need to be solved incredibly a large number of these nonlinear differential equations. Then choose the best one. But at the same time, one cannot be sure that the chosen solution is truly optimal and it is not known how to improve the chosen solution. Therefore, one of the problems of control theory is as follows.

Creation of control system design methods that allow you to determine the best structure and optimal ratios of system parameters.

To complete this task you need the following calculation methods that

allow enough in simple form determine mathematical connections between the parameters of a nonlinear system and the dynamic indicators of the control process

leniya. And without finding a solution to a nonlinear differential equation. To solve the problem, the nonlinear characteristics of real elements of the system are replaced by some idealized approximate characteristics. Calculation of nonlinear systems using such characteristics gives approximate results, but the main thing is that the obtained dependencies make it possible to relate the structure and parameters of the system with its dynamic properties.

In the simplest cases and mainly for a second-order nonlinear system, it is used phase path method, which allows you to clearly show the dynamics of motion of a nonlinear system at various types nonlinear link taking into account the initial conditions. However, it is difficult to take into account various external influences using this method.

For a high order system it is used harmonic linearization method. With conventional linearization, a nonlinear characteristic is treated as linear and loses some properties. With harmonic linearization, the specific properties of the nonlinear link are preserved. But this method is approximate. It is used when a number of conditions are met, which will be shown when calculating a nonlinear system using this method. Important property This method is that it directly connects the system parameters with the dynamic indicators of the regulation process.

To determine the statistical error of regulation under random influences, use statistical linearization method. The essence of this method is that the nonlinear element is replaced by an equivalent linear element, which transforms the first two statistical moments in the same way as the nonlinear element random function: expectation (mean) and variance (or standard deviation). There are other methods for analyzing nonlinear systems. For example, small parameter method in the form of B.V. Bulgakov. Asymptotic method N.M. Krylov and N.N. Bogolyubova to analyze a process in time near a periodic solution. Grapho-analytical The method allows a nonlinear problem to be reduced to a linear one. Harmonic balance method, which was used by L.S. Goldfarb for analyzing the stability of nonlinear systems using the Nyquist criterion. Graphic-analytical methods, among which the most widely used method is D.A. Bashkirova. Of the variety of research methods in this textbook will be considered: the method of phase trajectories, the method of point transformations, the method of harmonic linearization E.P. Popov, graphic-analytical method L.S. Goldfarb, criterion of absolute stability by V.M. Popov, method of statistical linearization.

"Theory of automatic control"

"Methods for studying nonlinear systems"

1. Method of differential equations

The differential equation of a closed nonlinear system of nth order (Fig. 1) can be transformed to the system n-differential equations first order in the form:

where: – variables characterizing the behavior of the system (one of them may be a controlled variable); – nonlinear functions; u – setting influence.

Typically, these equations are written in finite differences:

where are the initial conditions.

If the deviations are not large, then this system can be solved as a system of algebraic equations. The solution can be represented graphically.

2. Phase space method

Let us consider the case when the external influence is zero (U = 0).

The movement of the system is determined by a change in its coordinates - as a function of time. The values at any time characterize the state (phase) of the system and determine the coordinates of the system having n-axes and can be represented as the coordinates of some (representing) point M (Fig. 2).

Phase space is the coordinate space of the system.

As time t changes, point M moves along a trajectory called the phase trajectory. If we change the initial conditions, we get a family of phase trajectories called a phase portrait. The phase portrait determines the nature of the transition process in a nonlinear system. The phase portrait has special points to which the phase trajectories of the system tend or move away (there may be several of them).

The phase portrait may contain closed phase trajectories, which are called limit cycles. Limit cycles characterize self-oscillations in the system. The phase trajectories do not intersect anywhere, except for special points characterizing the equilibrium states of the system. Limit cycles and equilibrium states can be stable or unstable.

The phase portrait completely characterizes the nonlinear system. Characteristic feature Nonlinear systems are the presence of various types of motions, several equilibrium states, and the presence of limit cycles.

The phase space method is a fundamental method for studying nonlinear systems. It is much easier and more convenient to study nonlinear systems on the phase plane than by plotting transient processes in the time domain.

Geometric constructions in space are less visual than constructions on a plane, when the system is of second order, and the phase plane method is used.

Application of the phase plane method for linear systems

Let us analyze the relationship between the nature of the transition process and the curves of phase trajectories. Phase trajectories can be obtained either by integrating the phase trajectory equation or by solving the original 2nd order differential equation.

Let the system be given (Fig. 3).

Let us consider the free movement of the system. In this case: U(t)=0, e(t)=– x(t)

In general, the differential equation has the form

Where (1)

This is a homogeneous differential equation of the 2nd order; its characteristic equation is equal to

. (2)

The roots of the characteristic equation are determined from the relations

(3)

Let us represent a 2nd order differential equation in the form of a system

1st order equations:

(4)

where is the rate of change of the controlled variable.

In the linear system under consideration, the variables x and y represent the phase coordinates. We construct the phase portrait in the space of coordinates x and y, i.e. on the phase plane.

If we exclude time from equation (1), we obtain the equation of integral curves or phase trajectories.

. (5)

This is a separable equation

Let's consider several cases

The files GB_prog.m and GB_mod.mdl, and the analysis of the spectral composition of the periodic mode at the output of the linear part - using the files GB_prog.m and R_Fourie.mdl. Contents of the file GB_prog.m: % Study of nonlinear systems by the harmonic balance method % Files used: GB_prog.m, GB_mod.mdl and R_Fourie.mdl. % Designations used: NE - nonlinear element, LP - linear part. %Clearing all...

Inertia-free in the permissible (limited from above) frequency range, beyond which it becomes inertial. Depending on the type of characteristics, nonlinear elements with symmetrical and asymmetrical characteristics are distinguished. A characteristic that does not depend on the direction of the quantities that determine it is called symmetric, i.e. having symmetry relative to the origin of the system...

There are exact and approximate methods for studying nonlinear systems; exact methods include the methods of phase trajectories, point transformations, Popov's frequency method, the method of sections of the parameter space, the fitting method; approximate methods include the harmonic linearization method.

The method of phase trajectories is that the behavior of the nonlinear system under study is considered and described not in the time domain (in the form of equations of processes in the system), but in the phase space of the system (in the form of phase trajectories).

The state of a nonlinear automatic control system is characterized using the phase coordinates of the system

defining the state vector of the system in the phase space of the system

Y (y1, y2, y3,...yn).

When introducing phase coordinates into consideration, a nonlinear differential equation of order n for a free process in a nonlinear system

transforms to a system of n first order differential equations

During the process in the system, the phase coordinates yi change and the system state vector Y describes a hodograph in the n-dimensional phase space of the system (Fig. 56). The hodograph of the state vector (trajectory of movement of the representing point M corresponding to the end of the vector) is the phase trajectory of the system. The type of phase trajectory is uniquely related to the nature of the process in the system. Therefore, the properties of a nonlinear system can be judged by its phase trajectories.

The phase trajectory equation can be obtained from the above system of first-order equations relating phase coordinates and taking into account the properties of the system by eliminating time. The phase trajectory does not reflect the time of processes in the system.

The connection between the phase trajectory y(x) and the process x(t) is illustrated in Fig. 57. The phase trajectory is constructed in phase coordinates 0XY, where x is the output value of the system, y is the rate of change of the output value (the first derivative of x’). The transient process x(t) is plotted in x–t coordinates (output value – time).

Method of point transformations of surfaces allows you to determine all kinds of movement ( free vibrations) nonlinear dynamic systems after any initial deviations. The method has been developed for the analysis and synthesis of motions of systems described by differential equations of low order (second, third), as well as for a system with relay control taking into account delay.

The replacement is carried out in sections, for each of which the nonlinear part of the characteristic is represented by a linear segment. This makes it possible to obtain an integrable linear differential equation that approximately reflects the process within a given section. For a system described by a second-order differential equation, the progress of the calculation can be shown on the phase plane, along the axes of which the variable under study l and its time derivative y are plotted. The solution of the dynamic problem comes down to the study of the point transformation of the coordinate semi-axis into itself.

Fig. 10.7. Point transformation method

Frequency method Romanian scientist V.M. Popov, proposed in 1960, solves the problem of absolute stability of a system with one single-valued nonlinearity, specified by the limiting value of the transfer coefficient k of the nonlinear element. If the control system has only one unambiguous nonlinearity z=f(x), then by combining all the other links of the system into a linear part, one can obtain its transfer function Wlch(p), i.e. obtain the design diagram Fig. 7.1.
There are no restrictions on the order of the linear part, i.e. the linear part can be anything. The outline of the nonlinearity may be unknown, but it must be unambiguous. It is only necessary to know within what angle arctg k (Fig. 7.2) it is located, where k is the maximum (maximum) transmission coefficient of the nonlinear element.

Fig.7.2. Characteristics of a nonlinear element

The graphical interpretation of V.M. Popov’s criterion is associated with the construction of the a.f.h. modified frequency response of the linear part of the system W*(jω), which is defined as follows:
W*(jω) = Re WLC(jω) + Im WLC(jω),
where Re WLC(jω) and Im WLC(jω) are the real and imaginary parts of the linear system, respectively.
V.M. Popov’s criterion can be presented either in algebraic or frequency form, as well as for the cases of stable and unstable linear parts. The frequency form is most often used.
Formulation of V.M. Popov’s criterion in the case of a stable linear part: to establish the absolute stability of a nonlinear system, it is sufficient to select a straight line on the complex plane W*(jω) passing through the point (, j0) so that the entire curve W*(jω) lies on the right from this straight line. The conditions for fulfilling the theorem are shown in Fig. 7.3.

Rice. 7.3. Graphic interpretation of the criterion by V.M. Popov for an absolutely stable nonlinear system

In Fig. 7.3 shows the case of absolute stability of a nonlinear system for any form of unambiguous nonlinearity. Thus, to determine the absolute stability of a nonlinear system using the method of V.M. Popov, it is necessary to construct a modified frequency characteristic of the linear part of the system W*(jω), determine the limiting value of the transmission coefficient k of the nonlinear element from the condition and draw a straight line through the point (-) on the real axis of the complex plane so that the characteristic W*(jω) lies on the right from this straight line. If such a straight line cannot be drawn, then this means that absolute stability for a given system is impossible. The outline of the nonlinearity may be unknown. It is advisable to use the criterion in cases where the nonlinearity may change during the operation of the ACS, or its mathematical description is unknown.

Fitting method has found its application in constructing phase portraits of nonlinear systems, which can be represented in the form of linear and nonlinear parts (Fig. 11.10), with the linear part being a second-order system, and the nonlinear part being characterized by a piecewise linear static characteristic.

linear part

nonlinear part

Rice. 11.10 Block diagram of a nonlinear system

According to this method, the phase trajectory is constructed in parts, each of which corresponds to a linear section of the static characteristic. In such a section under consideration, the system is linear and its solution can be found by directly integrating the equation for the phase trajectory of this section. Integration of the equation when constructing a phase trajectory is carried out until the latter reaches the boundary of the next section. The values of the phase coordinates at the end of each section of the phase trajectory are the initial conditions for solving the equation in the next section. In this case, they say that the initial conditions are adjusted, i.e. the end of the previous section of the phase trajectory is the beginning of the next. The boundary between sections is called a switch line.

Thus, the construction of a phase portrait using the fitting method is carried out in the following sequence:

initial conditions are selected or specified;

a system of linear equations is integrated for the linear section where the initial conditions fall until the moment of reaching the boundary of the next section;

the initial conditions are adjusted.

[ siqn(x − a2

) + siqn(x + a1 )]

[ siqn(x + a2

) + siqn(x − а1 )]

There are no general universal methods for studying nonlinear systems - the variety of nonlinearities is too great. However, for individual species nonlinear systems developed effective methods analysis and synthesis.

The harmonic linearization method is intended to represent the nonlinear part of the system with some equivalent transfer function if the signals in the system can be considered harmonic.

This method can be effectively used to study periodic oscillations in automatic systems, including the conditions of the absence of these oscillations as harmful.

Characteristic of the harmonic linearization method is the consideration one and only nonlinear element. NE can be divided to static And dynamic. Dynamic NE are described by nonlinear differential equations and are much more complex. Static NE are described by the function F(x).