Analysis of nonlinear systems. Harmonic linearization method

"Theory of automatic control"

"Methods for studying nonlinear systems"

1. Method of differential equations

The differential equation of a closed nonlinear system of the nth order (Fig. 1) can be transformed to a system of n-differential equations of the 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. A characteristic feature of nonlinear systems is 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...

Chapter7

Analysis of nonlinear systems

The control system consists of individual functional elements, for the mathematical description of which standard elementary links are used (see Section 1.4). Among the typical elementary links there is one inertia-free (reinforcing) link. The static characteristic of such a link connecting the input x and day off y quantities, linear: y=Kx. Real functional elements of the control system have a nonlinear static characteristic y=f(x). Type of nonlinear dependence f(∙) can be varied:

Functions with variable slope (functions with the “saturation” effect, trigonometric functions, etc.);

Piecewise linear functions;

Relay functions.

Most often, it is necessary to take into account the nonlinearity of the static characteristic of the sensitive element of the control system, i.e. nonlinearity of the discriminatory characteristic. Usually they strive to ensure the operation of the control system in the linear section of the discriminatory characteristic (if the type of function allows this f(∙)) and use the linear model y=Kx. Sometimes this cannot be achieved due to large values of the dynamic and fluctuation components of the control system error, or due to the so-called significant nonlinearity of the function f(∙), inherent, for example, in relay functions. Then it is necessary to perform an analysis of the control system taking into account links that have a nonlinear static characteristic, i.e. analyze a nonlinear system.

7.1. Features of nonlinear systems

Processes in nonlinear systems are much more diverse than processes in linear systems. Let us note some features of nonlinear systems and processes in them.

1. The principle of superposition does not hold: the reaction of a nonlinear system is not equal to the sum of reactions to individual influences. For example, an independent calculation of the dynamic and fluctuation components of the tracking error, performed for linear systems (see Section 3), is impossible for nonlinear systems.

2. The property of commutativity is not applicable to the structural diagram of a nonlinear system (linear and nonlinear links cannot be interchanged).

3. In nonlinear systems, the conditions of stability and the very concept of stability change. The behavior of nonlinear systems, from the point of view of their stability, depends on the impact and initial conditions. In addition, in a nonlinear system, a new type of steady-state process is possible - self-oscillations with constant amplitude and frequency. Such self-oscillations, depending on their amplitude and frequency, may not disrupt the performance of the nonlinear control system. Therefore, nonlinear systems are no longer divided into two classes (stable and unstable), like linear systems, but are divided into a larger number of classes.

For nonlinear systems, Russian mathematician A.M. Lyapunov in 1892 introduced the concepts of stability “in the small” and “in the large”: a system is stable “in the small” if, with some (sufficiently small) deviation from the point of stable equilibrium, it remains in a given (limited) region ε, and the system is stable “large” if it remains in the region ε for any deviation from the stable equilibrium point. Note that the region ε can be set as small as desired near the stable equilibrium point, therefore, what is given in Sect. 2, the definition of stability of linear systems remains in force and is equivalent to the definition of asymptotic stability according to Lyapunov. At the same time, the previously discussed criteria for the stability of linear systems for real nonlinear systems should be perceived as criteria for stability “in the small.”

4. In nonlinear systems, transient processes change qualitatively. For example, in the case of the function f(∙) with a variable slope in a 1st order nonlinear system, the transient process is described by an exponential with a changing parameter T.

5. The limited aperture of the discriminatory characteristic of a nonlinear system is the cause of tracking failure (the system is stable “in the small”). In this case, it is necessary to search for a signal and enter the system into tracking mode (the concept of a search-and-track meter is given in Section 1.1). In synchronization systems with a periodic discrimination characteristic, jumps in the output value are possible.

The presence of the considered features of nonlinear systems leads to the need to use special methods for analyzing such systems. The following are considered:

A method based on solving a nonlinear differential equation and allowing, in particular, to determine the error in a steady state, as well as the capture and hold bands of a nonlinear PLL system;

Methods of harmonic and statistical linearization, convenient for analyzing systems with a significantly nonlinear element;

Methods of analysis and optimization of nonlinear systems based on the results of the theory of Markov processes.

7.2. Analysis of regular processes in a nonlinear PLL system

Item:

"Theory of automatic control"

Subject:

"Methods for studying nonlinear systems"

1. Method of differential equations

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

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 called the coordinate space of the system.

As time t changes, point M moves along a trajectory called phase trajectory. If we change the initial conditions we get a family of phase trajectories called 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.

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 the system be given (Fig. 3).

Let us consider the free movement of the system. Moreover: 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) 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

. (6)

Let's consider several cases

1. Let the roots of the characteristic equation (3) have the form

(those.

). (7)

In this case, the transition process is described by the equations

x = A sin (wt+j), (8)

y = Aw cos (wt+j),

those. represents undamped oscillations with constant amplitude A and initial phase – j.

On the phase plane (Fig. 4), these equations are parametric equations of an ellipse with semi-axes A and wA (where A is the integration constant).

If we designate

The ellipse equation can be obtained by solving the equation of phase trajectories

(9)

The equilibrium state is determined from the condition

in this case x 0 = y 0 = 0.

The singular point is called the “center” and corresponds to stable equilibrium, since the phase trajectories do not move away from it.

2. Let the roots of the characteristic equation (3) have the form

(10)

In this case, the transition process is described by the equations:

From the equation of phase trajectories

we get the equation

This is an equation of a family of hyperbolas when A changes (Fig. 5).

The presence of nonlinearities in control systems leads to the description of such a system by nonlinear differential equations, often of quite high orders. As is known, most groups of nonlinear equations cannot be solved in a general form, and one can only talk about special cases of 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, there are a large number of different analytical and graph-analytical methods for studying nonlinear systems, among which we can highlight the methods of the phase plane, fitting, point transformations, harmonic linearization, Lyapunov’s direct method, frequency methods for studying the absolute stability of Popov, methods for studying nonlinear systems on electronic models and computers.

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. In addition, there are no general methods for determining the Lyapunov function, which makes the practical application of this method difficult.

The absolute stability criterion allows you to analyze the stability of nonlinear systems using frequency characteristics, which is a great advantage of 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, it is necessary 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.

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, a so-called phase portrait of the processes occurring in the nonlinear system is constructed in phase space. 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 the mathematical apparatus of the 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 precise methods for analyzing nonlinear systems allow one to obtain accurate, correct results, they are very complex, which limits their practical application. 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 does not 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 high-pass filter (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 the characteristic equation of the 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 in solution (7.7) of a pair of 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 regime, 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 the normal distribution law 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

The ability to replace the characteristics of nonlinear links with linear dependencies allows the use of 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?