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

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

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.

1.4 Analysis of methods for studying nonlinear systems

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 high-order nonlinear differential equation, sometimes with a number of logical relations. Modern computer technology makes it possible to solve any nonlinear equations, and an incredibly large number of these nonlinear differential equations will need to be solved. 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 one to determine in a fairly simple form the 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 for various types of 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. An important property of 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, in the same way as the nonlinear element, transforms the first two statistical moments of a random function: the mathematical expectation (average value) and dispersion (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, this textbook will consider: 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.

A system is considered nonlinear if its order is >2 (n>2).

The study of high-order linear systems involves overcoming significant mathematical difficulties, since there are no general methods for solving nonlinear equations. When analyzing the motion of nonlinear systems, numerical and graphical integration methods are used, which allow one to obtain only one particular solution.

Research methods are divided into two groups. The first group is methods based on searching for exact solutions of nonlinear differential equations. The second group is approximate methods.

The development of exact methods is important both from the point of view of obtaining direct results and for studying various special modes and forms of dynamic processes of nonlinear systems that cannot be identified and analyzed by approximate methods. Precise methods include:

1. Direct Lyapunov method

2. Phase plane methods

3. Fitting method

4. Point transformation method

5. Parameter space cutting method

6. Frequency method for determining absolute stability

To solve many theoretical and practical problems, discrete and analog computing technology is used, which makes it possible to use mathematical modeling methods in combination with semi-natural and full-scale modeling. In this case, computer technology is interfaced with real elements of control systems, with all their inherent nonlinearities.

Approximate methods include analytical and graph-analytical methods that make it possible to replace a nonlinear system with an equivalent linear model, followed by the use of methods of the linear theory of dynamic systems to study it.

There are two groups of approximate methods.

The first group is based on the assumption that the nonlinear system under study is close in its properties to the linear one. These are small parameter methods, when the motion of the system is described using power series with respect to some small parameter that is present in the equations of the system, or which is introduced artificially into these equations.

The second group of methods is aimed at studying the natural periodic oscillations of the system. It is based on the assumption that the desired oscillations of the system are close to harmonic. These are methods of harmonic balance or harmonic linearization. When using them, a nonlinear element under the influence of a harmonic input signal is conditionally replaced by equivalent linear elements. The analytical justification for harmonic linearization is based on the principle of equality of frequency, amplitude and phase output variables, the equivalent linear element and the first harmonic of the output variable of a real nonlinear element.

The greatest effect is achieved by a reasonable combination of approximate and exact methods.

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.

Basics of the phase trajectory 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 motion (free vibrations) of 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.

Harmonic linearization method

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

• 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).

Strictly speaking, linear systems do not exist in nature; all real systems are nonlinear. Various sensors, detectors, discriminators, amplifiers, analog-to-digital and digital-to-analog converters, control devices and actuators have nonlinear characteristics.

There is no general theory for the analysis of nonlinear systems. Scientists have developed various methods for analyzing nonlinear systems that allow solving analysis problems under certain conditions and restrictions.

Let us characterize the most common methods of analyzing nonlinear systems.

Phase plane method. This method is also called the method of phase portraits or phase spaces. This method allows you to visually analyze, using graphical constructions, the behavior of nonlinear systems described by nonlinear differential equations of no higher than the second (third) order.

Piecewise linear approximation method. This method uses a piecewise linear approximation of the characteristic of a nonlinear element, analyzes the system as linear for various signal values, and then stitches the analysis results together. The method is characterized by high labor intensity of analysis and low accuracy of results, especially at the “crosslinking” points.

Harmonic linearization method. This method is used in cases where a linear low-pass filter is connected after the nonlinear element, and the input effect is harmonic.

Statistical linearization method. This method is used in cases where a stationary random process acts as an input signal. In this method, the real nonlinear element is replaced by a linear element whose output mathematical expectation and variance of the process are the same as the output of the real nonlinear element. Methods for determining the parameters of an equivalent linear element may be different.

Markov process method. This method is used for non-stationary random input signals, but an analytical solution can only be found for systems no higher than second order.

Computer simulation method. This method claims to be universal; it has no fundamental restrictions on the nature of nonlinearity and the order of the system. Currently, this is the most common method for analyzing nonlinear systems; the only drawback of the method is the absence of any analytical results of the analysis (in the form of formulas).

• Method of harmonic linearization in the design of nonlinear automatic control systems.[Djv-10.7M] Edited by Yu.I. Topcheeva. Team of authors.
  (Moscow: Mashinostroenie Publishing House, 1970. - Series “Nonlinear Automatic Control Systems”)
  Scan: AAW, processing, Djv format: Ilya Sytnikov, 2014
• BRIEF CONTENTS:
  Preface (5).
  Chapter I. Theoretical foundations of the harmonic linearization method (E.P. Popov) (13).
  Chapter II. A new form of harmonic linearization for control systems with nonlinear hysteresis characteristics (E.I. Khlypalo) (58).
  Chapter III. Harmonic linearization method based on assessing the sensitivity of a periodic solution to higher harmonics and small parameters (A.A. Vavilov) (88).
  Chapter IV. Determination of amplitude and phase frequency characteristics of nonlinear systems (Yu.I. Topcheev) (117).
  Chapter V. Approximate frequency methods for analyzing the quality of nonlinear control systems (Yu.I. Topcheev) (171).
  Chapter VI. Improving the accuracy of the harmonic linearization method (V.V. Pavlov) (186).
  Chapter VII. Application of the harmonic linearization method to discrete nonlinear control systems (S.M. Fedorov) (219).
  Chapter VIII. Application of the asymptotic method of N.M. Krylov and N.N. Bogolyubov in the analysis of nonlinear control systems (A.D. Maksimov) (236).
  Chapter IX. Application of harmonic linearization to nonlinear self-tuning control systems (Yu.M. Kozlov, S.I. Markov) (276).
  Chapter X. Application of the harmonic linearization method to nonlinear automatic systems with finite state machines (M.V. Starikova) (306).
  Chapter XI. An approximate method for studying oscillatory processes and sliding modes in automatic systems with variable structure (M.V. Starikova) (390).
  Chapter XII. An approximate study of a pulse-relay control system (M.V. Starikova) (419).
  Chapter XIII. Determination of oscillatory processes in complex nonlinear systems with various initial deviations (M.V. Starikova) (419).
  Chapter XIV. Application of the harmonic linearization method to systems with periodic nonlinearities (L.I. Semenko) (444).
  Chapter XV. Application of the harmonic linearization method to systems with two nonlinearities (V.M. Khlyamov) (467).
  Chapter XVI. Amplitude-phase characteristics of relay mechanisms with DC and AC motors, obtained using the harmonic linearization method (V.V. Tsvetkov) (485).
  Applications (518).
  Literature (550).
  Alphabetical index (565).

Publisher's abstract: This book is part of a series of monographs devoted to nonlinear automatic control systems.
It systematically, quite comprehensively, sets out the theory of nonlinear automatic control systems, based on the method of harmonic linearization. The main attention is paid to the theoretical foundations of the harmonic linearization method and its practical applications to continuous, discrete, self-tuning systems, as well as systems with finite state machines and tunable structure. Ways to improve the accuracy of the harmonic linearization method by taking into account the influence of higher harmonics are considered. The proposed methods are illustrated with numerous examples.
The book is intended for scientists, engineers, teachers and graduate students of higher educational institutions dealing with automatic control issues.