Approximate methods for studying nonlinear systems. Analysis of nonlinear systems. Method of harmonic linearization. Application of the theory of Markov processes for the analysis of nonlinear systems. Relay type non-linearity

A system is considered non-linear if its order >2 (n>2).

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

Research methods are divided into two groups. The first group is methods based on finding exact solutions to 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 regimes and forms of dynamic processes of nonlinear systems that cannot be identified and analyzed by approximate methods. The exact methods are:

1. Direct Lyapunov method

2. Phase plane methods

3. Fitting method

4. Method of point transformations

5. Method of sections of the space of parameters

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 joined with the real elements of control systems, with all their inherent non-linearities.

Approximate methods include analytical and graph-analytical methods that allow replacing a nonlinear system with an equivalent linear model, followed by the use of methods of the linear theory of dynamical systems for its study.

There are two groups of approximate methods.

The first group is based on the assumption that the studied nonlinear system is close to the linear one in terms of its properties. These are methods of a small parameter, 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 into these equations artificially.

The second group of methods is aimed at studying natural periodic oscillations of the system. It is based on the assumption that the desired oscillations of the system are close to harmonic ones. These are methods of harmonic balance or harmonic linearization. When they are used, a conditional replacement of a non-linear element, which is under the action of a harmonic input signal, is performed with equivalent linear elements. The analytical substantiation of 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 non-linear element.

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

"Theory of automatic control"

"Methods of research of nonlinear systems"

1. Method of differential equations

The differential equation of a closed non-linear system of the nth order (Fig. 1) can be converted 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 can be a controlled value); are non-linear functions; u is the driving force.

Usually, 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 action is equal to zero (U = 0).

The motion of the system is determined by the change in its coordinates - as a function of time. The values ​​at any moment of 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).

The phase space is the space of coordinates of the system.

With a change in time t, the point M moves along a trajectory called the phase trajectory. If we change the initial conditions, we obtain a family of phase trajectories called the phase portrait. The phase portrait determines the nature of the transient process in a nonlinear system. The phase portrait has singular points that the phase trajectories of the system tend to or leave from (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. Phase trajectories do not intersect anywhere, except for singular points characterizing the equilibrium states of the system. Limit cycles and equilibrium states may or may not be stable.

The phase portrait completely characterizes the nonlinear system. A characteristic feature of nonlinear systems is the presence of various types of motions, several states of equilibrium, 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 transients in the time domain.

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

Applying the Phase Plane Method to Linear Systems

Let us analyze the relationship between the nature of the transient process and the curves of the 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).

Consider the free motion 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 2nd order homogeneous differential equation; its characteristic equation is

. (2)

The roots of the characteristic equation are determined from the relations

(3)

Let us represent the 2nd order differential equation as 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 are phase coordinates. The phase portrait is built in the space of coordinates x and y, i.e. on the phase plane.

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

. (5)

This is a separable equation

Let's consider several cases

Files GB_prog.m and GB_mod.mdl, and 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 GB_prog.m file: %Investigation of nonlinear systems by the harmonic balance method %Used files: GB_prog.m, GB_mod.mdl and R_Fourie.mdl. % Notation used: NE - non-linear element, LP - linear part. %Clear all...

Inertial in the permissible (limited from above) frequency range, beyond which it passes into the category of inertial ones. Depending on the type of characteristics, nonlinear elements with symmetrical and asymmetric characteristics are distinguished. Symmetric is a characteristic that does not depend on the direction of the quantities that determine it, i.e. having symmetry with respect to the beginning of the system...

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Novosibirsk State Technical University

Department of Electric Drive and Automation of Industrial Installations

COURSE WORK

in the discipline "Theory of automatic control"

Analysis of nonlinear automatic control systems

Student: Tishinov Yu.S.

Ema-71 Group

Coursework Supervisor

TASK FOR COURSE WORK:

1. Investigate ACS with a given block diagram, type of nonlinearity and numerical parameters using the phase plane method.

1.1 Verify the results of the calculations in paragraph 1 using structural modeling.

1.2 Investigate the influence of the input action and nonlinearity parameters on the system dynamics.

2. Investigate ACS with a given block diagram, type of nonlinearity and numerical parameters using the harmonic linearization method.

2.1 Verify the results of the calculations in paragraph 2 using structural modeling.

2.2 Investigate the influence of the input action and nonlinearity parameters on the system dynamics

1. We investigate the ACS with a given block diagram, the type of nonlinearity and numerical parameters using the phase plane method.

Option number 4-1-a

Initial data.

1) Structural diagram of a non-linear ACS:

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A system in which work and control operations are performed by technical devices is called automatic control system (ACS).

Structural diagram is called a graphic representation of the mathematical description of the system.

The link on the structural diagram is depicted as a rectangle indicating external influences and the transfer function is written inside it.

The set of links, together with the communication lines that characterize their interaction, forms a block diagram.

2) Block diagram parameters:

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Phase plane method

The behavior of a nonlinear system at any time is determined by the controlled variable and its (n? 1) derivative, if these quantities are plotted along the coordinate axes, then the resulting n? dimensional space will be called the phase space. The state of the system at each moment of time will be determined in the phase space by the representing point. During the transition process, the representative point moves in the phase space. The trajectory of its movement is called the phase trajectory. In steady state, the representative point is at rest and is called a singular point. The set of phase trajectories for various initial conditions, together with singular points and trajectories, is called the phase portrait of the system.

When studying a nonlinear system by this method, it is necessary to convert the block diagram (Fig. 1.1) to the form:

The minus sign indicates that the feedback is negative.

where X 1 and X 2 - output and input values ​​of the linear part of the system, respectively.

Let's find the differential equation of the system:

Let's make a replacement, then

We solve this equation with respect to the highest derivative:

Let's assume that:

We divide equation (1.2) by equation (1.1) and obtain a nonlinear differential equation for the phase trajectory:

where x 2 \u003d f (x 1).

If this DE is solved by the isocline method, then it is possible to construct a phase portrait of the system for various initial conditions.

An isocline is the locus of points in the phase plane that the phase trajectory intersects at the same angle.

In this method, the nonlinear characteristic is divided into linear sections and for each of them a linear DE is recorded.

To obtain the isocline equation, the right side of equation (1.3) is equated to a constant value N and solved relatively.

Taking into account the nonlinearity, we get:

Given values ​​of N in the range from to, a family of isoclines is constructed. On each isocline, an auxiliary straight line is drawn at an angle to the x-axis

where m X - scale factor along the x-axis;

m Y - scale factor along the y-axis.

Choose m X = 0.2 units/cm, m Y = 40 units/cm;

Final formula for angle:

We calculate the family of isoclines and the angle for the site, we summarize the calculation in Table 1:

Table 1

We calculate the family of isoclines and the angle for the site, we summarize the calculation in Table 2:

table 2

We calculate the family of isoclines and the angle for the site, we summarize the calculation in Table 3:

Table 3

Let's construct a phase trajectory

To do this, the initial conditions are selected on one of the isoclines (point A), two straight lines are drawn from point A to the intersection with the next isocline at angles b 1, b 2, where b 1, b 2? respectively, the angles of the first and second isoclines. The segment cut off by these lines is divided in half. From the obtained point, the middle of the segment, two lines are again drawn at angles b 2, b 3, and again the segment is divided in half, etc. The resulting points are connected by a smooth curve.

Families of isoclines are built for each linear section of the nonlinear characteristic and are separated from each other by switching lines.

It can be seen from the phase trajectory that a singular point of the stable focus type has been obtained. It can be concluded that there are no self-oscillations in the system, and the transient process is stable.

1.1 Check the results of calculations using structural modeling in the MathLab program

Structural scheme:

Phase portrait:

The transient process at the input action equal to 2:

Xout.max = 1.6

1.2 We study the influence of the input action and nonlinearity parameters on the system dynamics

Let's increase the input signal to 10:

Xout.max = 14.3

Treg = 0.055

X out. max=103

T reg = 0.18

Let's increase the sensitivity zone to 15:

Xout.max = 0.81

Reduce the sensitivity zone to 1:

Xout.max = 3.2

The simulation results confirmed the calculation results: Figure 1.7 shows that the process is convergent, there are no self-oscillations in the system. The phase portrait of the simulated system is similar to the calculated one.

Having studied the influence of the input action and nonlinearity parameters on the dynamics of the system, we can draw the following conclusions:

1) with an increase in the input action, the level of the steady state increases, the number of oscillations does not change, the control time increases.

2) with an increase in the dead zone, the level of the steady state increases, the number of oscillations also remains unchanged, the control time increases.

2. We investigate the ACS with a given block diagram, the type of nonlinearity and numerical parameters using the method of harmonic linearization.

Option #5-20-c

Initial data.

1) Block diagram:

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2) Parameter values:

3) Type and parameters of non-linearity:

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The most widely used for the study of high-order nonlinear automatic control systems (n > 2) is the approximate method of harmonic linearization using frequency representations developed in the theory of linear systems.

The main idea of ​​the method is as follows. Let a closed autonomous (without external influences) nonlinear system consist of a series-connected nonlinear inertialess NC and a stable or neutral linear part of the LP (Figure 2.3, a)

u=0 x z X=X m sinwt z y

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y \u003d Y m 1 sin (wt +)

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To judge the possibility of the existence of monoharmonic undamped oscillations in this system, it is assumed that a harmonic sinusoidal signal x(t) = X m sinwt acts at the input of the nonlinear link (Fig. 2.3,b). In this case, the signal at the output of the nonlinear link z(t) = z contains a spectrum of harmonic components with amplitudes Z m 1 , Z m 2 , Z m 3 , etc. and frequencies w, 2w, 3w, etc. It is assumed that this signal z(t), passing through the linear part W l (jw), is filtered by it to such an extent that in the signal at the output of the linear part y(t) all higher harmonics Y m 2 , Y m 3 and etc. and assume that

y(t)Y m 1 sin(wt +)

The last assumption is called the filter hypothesis, and the fulfillment of this hypothesis is a necessary condition for harmonic linearization.

The equivalence condition for the circuits shown in fig. 2.3, a and b, can be formulated as an equality

x(t) + y(t) = 0(1)

When the filter hypothesis y(t) = Y m 1 sin(wt +) is fulfilled, equation (1) splits into two

Equations (2) and (3) are called harmonic balance equations; the first of them expresses the balance of amplitudes, and the second - the balance of the phases of harmonic oscillations.

Thus, in order for undamped harmonic oscillations to exist in the system under consideration, conditions (2) and (3) must be satisfied if the filter hypothesis is satisfied

Let us use the Goldfarb method for the graph-analytical solution of the characteristic equation of the form

W LCH (p) W NO (A) +1 = 0

W LCH (jw) W NO (A) = -1

For an approximate determination of self-oscillations, the AFC of the linear part of the system and the inverse negative characteristic of the nonlinear element are constructed.

To build the AFC of the linear part, we transform the block diagram to the form of Fig. 2.4:

As a result of the transformation, we obtain the scheme of Fig. 2.5:

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Find the transfer function of the linear part of the system:

Let's get rid of the irrationality in the denominator by multiplying the numerator and denominator by the conjugate to the denominator, we get:

Let's break it down into imaginary and real parts:

To construct the inverse negative characteristic of a nonlinear element, we use the formula:

Nonlinearity parameters:

A is the amplitude, provided that.

The AFC of the linear part of the system and the inverse negative characteristic of the non-linear element are shown in fig. 2.6:

To determine the stability of self-oscillations, we use the following formulation: if the point corresponding to the increased amplitude compared to the intersection point is not covered by the frequency response of the linear part of the system, then the self-oscillations are stable. As can be seen from Figure 2.6, the solution is stable, therefore, self-oscillations are established in the system.

2.1 Let's check the calculation results using structural modeling in the MathLab program.

Figure 2.7: Structural diagram

The transient process with an input action equal to 1 (Fig. 2.8):

automatic control non-linear harmonic

As can be seen from the graph, self-oscillations are established. Let us check the influence of nonlinearity on the stability of the system.

2.2 Let's investigate the influence of the input action and nonlinearity parameters on the system dynamics.

Let's increase the input signal to 100:

Let's increase the input signal to 270

Let's reduce the input signal to 50:

Let's increase the saturation to 200:

Reduce saturation to 25:

Reduce saturation to 10:

The simulation results did not unequivocally confirm the calculation results:

1) Self-oscillations occur in the system, and a change in saturation affects the amplitude of the oscillations.

2) With an increase in the input action, the value of the output signal changes and the system tends to a stable state.

LIST OF SOURCES USED:

1. Collection of problems on the theory of automatic regulation and control. Ed. V.A. Besekersky, fifth edition, revised. - M.: Nauka, 1978. - 512 p.

2. Theory of automatic control. Part II. Theory of nonlinear and special systems of automatic control. Ed. A.A. Voronova. Proc. allowance for universities. - M.: Higher. school, 1977. - 288 p.

3. Topcheev Yu.I. Atlas for the design of automatic control systems: textbook. allowance. ? M.: Mashinostroenie, 1989. ? 752 p.

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The presence of nonlinearities in control systems leads to the description of such a system by nonlinear differential equations, often of sufficiently high orders. As is known, most groups of nonlinear equations cannot be solved in a general form, and one can only talk about particular cases of solution, therefore, in the study of nonlinear systems, various approximate methods play an important role.

By means of approximate methods for studying nonlinear systems, it is impossible, as a rule, to obtain a sufficiently complete idea of ​​all the dynamic properties of the system. However, they can be used to answer a number of separate essential questions, such as the question of stability, the presence of self-oscillations, the nature of any particular regimes, etc.

Currently, there are a large number of different analytical and graph-analytical methods for studying nonlinear systems, among which are 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 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 method of harmonic linearization refers to approximate methods, 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, higher harmonics must be taken into account. This complicates the analysis of the stability and quality of the control processes of nonlinear systems.

The second Lyapunov method allows one to obtain only sufficient stability conditions. And if on its basis the instability of the control system is determined, then in some cases, to verify the correctness of the result obtained, it is necessary to replace the Lyapunov function with another one and perform the stability analysis again. In addition, there are no general methods for determining the Lyapunov function, which makes it difficult to apply this method in practice.

The absolute stability criterion allows one 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 complication of calculations in the analysis of 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 different results will make it possible to avoid erroneous judgments about the stability or instability of the designed automatic control system.

Chapter7

Analysis of Nonlinear Systems

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

Functions with variable slope (functions with the effect of "saturation", trigonometric functions, etc.);

Piecewise linear functions;

relay functions.

Most often, one has to take into account the nonlinearity of the static characteristic of the sensing element of the control system, i.e. non-linearity of the discrimination characteristic. Usually, they strive to ensure the operation of the control system in the linear section of the discriminatory characteristic (if the form of the function allows it) f(∙)) and use the linear model y=Kx. Sometimes this cannot be ensured due to large values ​​of the dynamic and fluctuation components of the CS error, or due to the so-called significant non-linearity 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 the links that have a non-linear static characteristic, i.e. to analyze the 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 is not fulfilled: the response of a non-linear system is not equal to the sum of responses 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 inapplicable to the block diagram of a nonlinear system (linear and non-linear links cannot be interchanged).

3. In nonlinear systems, the stability conditions 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, a new type of steady process is possible in a nonlinear system - 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), as linear systems, but are divided into more classes.

For nonlinear systems, the Russian mathematician A.M. Lyapunov in 1892 introduced the concepts of stability “in the small” and “in the large”: the system is stable “in the small” if, for 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 of ε for any deviation from the point of stable equilibrium. Note that the region ε can be set arbitrarily small near the point of stable equilibrium; therefore, the given in Sec. 2, the definition of the stability of linear systems remains valid and is equivalent to the definition of asymptotic stability in the sense of Lyapunov. At the same time, the stability criteria for linear systems considered earlier for real nonlinear systems should be taken as stability criteria “in the small”.

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

5. The limited aperture of the discriminatory characteristic of the nonlinear system is the cause of the disruption of tracking (the system is stable "in the small"). In this case, it is necessary to search for a signal and enter the system into the tracking mode (the concept of a search-tracking 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 the analysis of such systems. The following are considered:

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

Methods of harmonic and statistical linearization, convenient in the analysis of systems with an essentially non-linear 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