General linearization method
In most cases, it is possible to linearize non-linear dependencies using the method of small deviations or variations. To consider ᴇᴦο, let's turn to some link in the automatic control system (Fig. 2.2). The input and output quantities are denoted by X1 and X2, and the external perturbation is denoted by F(t).
Let us assume that the link is described by some non-linear differential equation of the form
To compile such an equation, you need to use the appropriate branch of technical sciences (for example, electrical engineering, mechanics, hydraulics, etc.) that studies this particular type of device.
The basis for linearization is the assumption that the deviations of all variables included in the link dynamics equation are sufficiently small, since it is precisely on a sufficiently small section that the curvilinear characteristic can be replaced by a straight line segment. The deviations of the variables are measured in this case from their values in the steady process or in a certain equilibrium state of the system. Let, for example, the steady process is characterized by a constant value of the variable X1, which we denote as X10. In the process of regulation (Fig. 2.3), the variable X1 will have the values where denotes the deviation of the variable X 1 from the steady value X10.
Similar relationships are introduced for other variables. For the case under consideration, we have ˸ and also .
All deviations are assumed to be sufficiently small. This mathematical assumption does not contradict the physical meaning of the problem, since the very idea of automatic control requires that all deviations of the controlled variable during the control process be sufficiently small.
The steady state of the link is determined by the values X10, X20 and F0. Then equation (2.1) should be written for the steady state in the form
Let us expand the left side of equation (2.1) in the Taylor series
where D are higher order terms. Index 0 for partial derivatives means that after taking the derivative, the steady value of all variables must be substituted into its expression.
The higher order terms in formula (2.3) include higher partial derivatives multiplied by squares, cubes and higher degrees of deviations, as well as products of deviations. They will be small of a higher order compared to the deviations themselves, which are small of the first order.
Equation (2.3) is a link dynamics equation, just like (2.1), but written in a different form. Let us discard the higher-order smalls in this equation, after which we subtract the steady-state equations (2.2) from Eq. (2.3). As a result, we obtain the following approximate equation of the link dynamics in small deviations˸
In this equation, all variables and their derivatives enter linearly, that is, to the first degree. All partial derivatives are some constant coefficients in the event that a system with constant parameters is being investigated. If the system has variable parameters, then equation (2.4) will have variable coefficients. Let us consider only the case of constant coefficients.
General linearization method - concept and types. Classification and features of the category "General linearization method" 2015, 2017-2018.
The method of harmonic linearization (harmonic balance) allows you to determine the conditions for the existence and parameters of possible self-oscillations in non-linear automatic control systems. Self-oscillations are determined by limit cycles in the phase space of systems. Limit cycles divide space (generally - multidimensional) on the domains of damped and divergent processes. As a result of calculating the parameters of self-oscillations, one can conclude that they are admissible for a given system or that it is necessary to change the system parameters.
The method allows:
Determine the conditions for the stability of a nonlinear system;
Find the frequency and amplitude of free oscillations of the system;
Synthesize corrective circuits to ensure the required parameters of self-oscillations;
Investigate forced oscillations and evaluate the quality of transient processes in non-linear automatic control systems.
Conditions for applicability of the harmonic linearization method.
1) When using the method, it is assumed that linear part of the system is stable or neutral.
2) The signal at the input of the non-linear link is close in shape to the harmonic signal. This provision needs some explanation.
Figure 1 shows the block diagrams of the non-linear ACS. The circuit consists of series-connected links: a non-linear link y=F(x) and a linear
th, which is described by the differential equation
For y = F(g - x) = g - x we obtain the equation of motion of a linear system.
Consider free movement, i.e. for g(t) º 0. Then,
In the case when there are self-oscillations in the system, the free motion of the system is periodic. Non-periodic movement over time ends with the system stopping to some final position (usually, on a specially provided limiter).
With any form of a periodic signal at the input of a non-linear element, the signal at its output will contain, in addition to the fundamental frequency, higher harmonics. The assumption that the signal at the input of the nonlinear part of the system can be considered harmonic, i.e., that
x(t)@a×sin(wt),
where w=1/T, T is the period of free oscillations of the system, is equivalent to the assumption that the linear part of the system effectively filters higher harmonics of the signal y(t) = F(x (t)).
In the general case, when a nonlinear element of a harmonic signal x(t) acts at the input, the output signal can be Fourier transformed:
Fourier series coefficients
To simplify the calculations, we set C 0 =0, i.e., that the function F(x) is symmetric with respect to the origin. Such a limitation is not necessary and is done by analysis. The appearance of the coefficients C k ¹ 0 means that, in the general case, the nonlinear transformation of the signal is accompanied by phase shifts of the converted signal. In particular, this takes place in nonlinearities with ambiguous characteristics (with various kinds of hysteresis loops), both delay and, in some cases, phase advance.
The assumption of effective filtering means that the amplitudes of higher harmonics at the output of the linear part of the system are small, that is,
The fulfillment of this condition is facilitated by the fact that in many cases the amplitudes of the harmonics already directly at the output of the nonlinearity turn out to be significantly less than the amplitude of the first harmonic. For example, at the output of an ideal relay with a harmonic signal at the input
y(t)=F(с×sin(wt))=a×sign(sin(wt))
there are no even harmonics, and the amplitude of the third harmonic in three times less than the amplitude of the first harmonic
Let's do assessment of the degree of suppression higher harmonics of the signal in the linear part of the ACS. To do this, we make a number of assumptions.
1) Frequency of free oscillations of ACS approximately equal to the cutoff frequency its linear part. Note that the frequency of free oscillations of a nonlinear automatic control system can differ significantly from the frequency of free oscillations of a linear system, so that this assumption is not always correct.
2) We take the ACS oscillation index equal to M=1.1.
3) LAH in the vicinity of the cutoff frequency (w s) has a slope of -20 dB/dec. The boundaries of this section of the LAH are related to the oscillation index by the relations
4) The frequency w max is conjugating with the LPH section, so that when w > w max the LAH slope is at least minus 40 dB/dec.
5) Non-linearity - an ideal relay with characteristic y = sgn(x) so that only odd harmonics will be present at its non-linearity output.
The frequencies of the third harmonic w 3 \u003d 3w c, the fifth w 5 \u003d 5w c,
lgw 3 = 0.48+lgw c ,
lgw 5 = 0.7+lgw c .
Frequency w max = 1.91w s, lgw max = 0.28+lgw s. The corner frequency is 0.28 decades away from the cutoff frequency.
The decrease in the amplitudes of the higher harmonics of the signal as they pass through the linear part of the system will be for the third harmonic
L 3 \u003d -0.28 × 20-(0.48-0.28) × 40 \u003d -13.6 dB, that is, 4.8 times,
for the fifth - L 5 \u003d -0.28 × 20-(0.7-0.28) × 40 \u003d -22.4 dB, that is, 13 times.
Consequently, the signal at the output of the linear part will be close to harmonic
This is equivalent to assuming that the system is a low pass filter.
With regard to the function Z \u003d cp (X, X 2, ..., XJ, nonlinear with respect to the system of its arguments, the solution of the problem in the formulation formulated above can, as a rule, be obtained only approximately on the basis of the linearization method. The essence of the linearization method is that a non-linear function is replaced by some linear one and then, according to already known rules, the numerical characteristics of this linear function are found, considering them approximately equal to the numerical characteristics of the non-linear function.
Let's consider the essence of this method using the example of a function of one random argument.
If the random variable Z is a given function
random argument X, then its possible values z associated with the possible values of the argument X a function of the same kind, i.e.
(for example, if Z = sin X, then z= sinX).
We expand the function (3.20) in a Taylor series in a neighborhood of the point X= m , limiting ourselves only to the first two terms of the expansion, and we will assume that 
The value of the derivative of the function (3.20) with respect to the argument X at X = t x.
This assumption is equivalent to replacing the given function (3.19) by the linear function
On the basis of theorems on mathematical expectations and variances, we obtain calculation formulas for determining the numerical characteristics mz i in the form
Note that in the case under consideration, the standard deviation a r should be calculated by the formula
(The modulus of the derivative is taken here because it
may be negative.)
Application of the linearization method to find the numerical characteristics of a nonlinear function
an arbitrary number of random arguments leads to calculation formulas for determining its mathematical expectation, which have the form 
x 2, ..., x n) by arguments X. and X. respectively, calculated taking into account the signs at the point w x, m^, t Xp, i.e. by replacing all their arguments x v x 2, ..., x n their mathematical expectations.
Along with formula (3.26) for determining the dispersion D? you can use the calculation formula of the form
where g x x - correlation coefficient of random arguments X.
As applied to a nonlinear function of independent (or at least uncorrelated) random arguments, formulas (3.26) and (3.27) have the form
Formulas based on the linearization of non-linear functions of random arguments make it possible to determine their numerical characteristics only approximately. The accuracy of the calculation is less, the more the given functions differ from linear ones and the greater the dispersion of the arguments. It is not always possible to estimate the possible error in each specific case.
To refine the results obtained by this method, a technique based on preserving in the expansion of a nonlinear function not only linear, but also some subsequent terms of the expansion (usually quadratic) can be used.
In addition, the numerical characteristics of a nonlinear function of random arguments can be determined on the basis of a preliminary search for the law of its distribution for a given distribution of the system of arguments. However, it should be borne in mind that the analytical solution of such a problem is often too complicated. Therefore, to find the numerical characteristics of nonlinear functions of random arguments, the method of statistical modeling is widely used.
The basis of the method is the simulation of a series of tests, in each of which a certain set of x i, x 2i , ..., xni random argument values x v x 2 ,..., x n from the set corresponding to their joint distribution. The obtained values with the help of the given relation (3.24) are transformed into the corresponding values z. of the investigated function Z. According to the results z v z 2 , ..., z., ..., zk all to such tests, the desired numerical characteristics are calculated by methods of mathematical statistics.
Example 3.2. Based on the linearization method, determine the mathematical expectation and standard deviation of a random variable 
1. By formula (3.20) we obtain
2. Using the table of derivatives of elementary functions, we find
and calculate the value of this derivative at the point 
:
3. By formula (3.23) we obtain 
Example 3.3. Based on the linearization method, determine the mathematical expectation and standard deviation of a random variable
1. By formula (3.25) we obtain
2. Let us write formula (3.27) for the function of two random arguments 
3. Find the partial derivatives of the Z function with respect to the arguments X 1 and X 2:
and calculate their values at the point (m Xi ,t x2):
4. Substituting the data obtained into the formula for calculating the Z variance, we obtain Dz= 1. Therefore, u r = 1.
Differential equations can be linearized by the following methods:
1. The non-linear function of the working area is expanded into a Taylor series.
2. Nonlinear functions given in the form of graphs are linearized in the working plane by straight lines.
3. Instead of directly determining partial derivatives, variables are introduced into the original nonlinear equations.
,
.
(33)
4. This method is based on the determination of coefficients by the least squares method.
,
(34)
where 
- time constant of the pneumatic actuator;
- gear ratio of the pneumatic actuator;
- damping coefficient of the pneumatic actuator.
The internal structure of the ACS elements is most simply determined using the block diagrams of graphs. Unlike well-known block diagrams in graphs, variables are indicated in the form of time, and arcs denote either parameters or transfer functions of typical links. There is an even relationship between them.
mm non-linear elements
The linearization methods considered in the first chapter are applicable when the nonlinearity included in the LSA object is at least once differentiable or approximated by a tangent with a small error of some neighborhood close to the operating point. There is a whole class of nonlinearities for which both conditions are not satisfied. Usually these are significant non-linearities. These include: step, piecewise linear and multi-valued functions with discontinuity points of the first kind, as well as power and transtendental functions. The use of CCMs that provide the execution of logical-algebraic operations in systems has led to new types of linearities, which are represented through continuous variables using special logic.
For the mathematical description of such nonlinearities, equivalent transfer functions are used, depending on the linearization coefficients, which are obtained by minimizing the mean square of the reproduction error of a given input signal. The shape of the input signals coming to the input of the nonlinearities can be arbitrary. In practice, harmonic and random types of input signals and their temporal combinations are most widely used. Accordingly, the linearization methods are called harmonic and static.
General method for describing equivalent transfer functions ne
The entire class of essential nonlinearities is divided into two groups. The first group includes single-valued nonlinearities, in which the connection between the input 
and weekends 
vector signals depends only on the form of the static characteristic of the nonlinearity 
.
.
In this case, with a certain form of input signals:
.
Using the linearization matrix 
you can find the approximate value of the output signals:
.
From (42) it follows that the matrix of linearization coefficients of single-valued nonlinearities are real quantities and their equivalent transfer functions:
.
The second group includes two-valued (multi-valued) nonlinearities, in which the relationship between the input and output signals depends not only on the shape of the static characteristic, but is also determined by the history of the input signal. In this case, expression (42) will be written as:
.
To take into account the influence of the prehistory of the input periodic signal, we will take into account not only the signal itself 
, but also the rate of its change, the differential 
.
For input signals:
the approximate value of the input signal will be:
where 
and 
- coefficients of harmonic linearization of two-valued nonlinearities;
- oscillation period on the right harmonic;
- harmonic function.
Equivalent transfer function:
There are nonlinearities of a more general form:
,
,
where 
and 
- coefficients of harmonic linearization;
is the harmonic number.
Periodic linearization coefficient matrices 
. With this in mind, the transfer function of two two-valued nonlinearities can be represented by analogy with the transfer function
Using, we define a generalized formula for calculating the transfer function of single-valued and two-valued nonlinearities.
In the case of single-valued nonlinearity, the matrix of linearization coefficients 
, depending on the parameters of the vector 
, we choose in such a way as to linearize the mean value of the squared difference between the exact 
and approximate 
input signals:
After transformations, simplifications, tricks and increased vigilance, we get the equivalent transfer function in the form of a system of matrices: 
,
.
,
at 
,
.
.
Determine the linearization coefficient for single-valued non-linearity. When the first harmonic of a sinusoidal signal arrives at its input:
where 
.
.
Equation (56) is the first harmonic linearization factor for single-valued non-linearity, it defines the equivalent transfer function 
.
In the future, a comparison of the formula for determining the linearization coefficients of the simplest nonlinearities when periodic signals are applied to their input: sinusoidal, triangular, we will show the expediency of using the resulting equivalent transfer functions.
The linearization coefficient is determined 
,
.
,
.
Example. Determine the linearization coefficient of a two-valued nonlinearity when the first harmonic of a sinusoidal signal enters its input and has one input. From the system of matrices (60), we obtain:
,
.
In this example, we write the input signal as:
,
.
When for a two-valued nonlinearity the general equivalent function is:
.
.
AT
Rice. 2.2. ATS link
Let us assume that the link is described by some non-linear differential equation of the form
To compile such an equation, you need to use the appropriate branch of technical sciences (for example, electrical engineering, mechanics, hydraulics, etc.) that studies this particular type of device.
The basis for linearization is the assumption that the deviations of all variables included in the link dynamics equation are sufficiently small, since it is precisely on a sufficiently small section that the curvilinear characteristic can be replaced by a straight line segment. The deviations of the variables are measured in this case from their values in the steady process or in a certain equilibrium state of the system. Let, for example, a steady process is characterized by a constant value of the variable X 1 , which we denote as X 10 . In the process of regulation (Fig. 2.3), the variable X 1 will have the values where 
denotes the deviation of the variable X 1 from the steady value of X 10 .
BUT
Rice. 2.3. Link regulation process
.
Next, you can write: 
;
and 
, because 
and 
All deviations are assumed to be sufficiently small. This mathematical assumption does not contradict the physical meaning of the problem, since the very idea of automatic control requires that all deviations of the controlled variable during the control process be sufficiently small.
The steady state of the link is determined by the values of X 10 , X 20 and F 0 . Then equation (2.1) can be written for the steady state in the form
Let us expand the left side of equation (2.1) in the Taylor series
where are higher order terms. The index 0 for partial derivatives means that after taking the derivative, the steady value of all variables must be substituted into its expression 
.
The higher order terms in formula (2.3) include higher partial derivatives multiplied by squares, cubes and higher degrees of deviations, as well as products of deviations. They will be small of a higher order compared to the deviations themselves, which are small of the first order.
Equation (2.3) is a link dynamics equation, just like (2.1), but written in a different form. Let us discard the higher order smalls in this equation, after which we subtract the steady state equations (2.2) from Eq. (2.3). As a result, we obtain the following approximate link dynamics equation in small deviations:
In this equation, all variables and their derivatives enter linearly, that is, to the first degree. All partial derivatives are some constant coefficients in the event that a system with constant parameters is being investigated. If the system has variable parameters, then equation (2.4) will have variable coefficients. Let us consider only the case of constant coefficients.
Obtaining equation (2.4) is the goal of the linearization done. In the theory of automatic control, it is customary to write the equations of all links so that the output value is on the left side of the equation, and all other terms are transferred to the right side. In this case, all terms of the equation are divided by the coefficient at the output value. As a result, equation (2.4) takes the form
where the following notation is introduced
.
(2.6)
In addition, for convenience, it is customary to write all differential equations in operator form with the notation
Then the differential equation (2.5) can be written in the form
This record will be called the standard form of the link dynamics equation.
The coefficients T 1 and T 2 have the dimension of time - seconds. This follows from the fact that all the terms in equation (2.8) must have the same dimension, and for example, the dimension 
(or px 2) differs from the dimension of x 2 per second to the minus first power ( 
). Therefore, the coefficients T 1 and T 2 are called time constants
.
The coefficient k 1 has the dimension of the output value divided by the dimension of the input. It is called transmission ratio link. For links whose output and input values have the same dimension, the following terms are also used: gain - for a link that is an amplifier or has an amplifier in its composition; gear ratio - for gearboxes, voltage dividers, scaling devices, etc.
The transfer coefficient characterizes the static properties of the link, since in the steady state 
. Therefore, it determines the steepness of the static characteristic at small deviations. If we depict the entire real static characteristic of the link 
, then the linearization gives 
or 
. The transmission coefficient k 1 will be the tangent of the slope 
tangent at that point C (see Fig. 2.3), from which small deviations x 1 and x 2 are measured.
It can be seen from the figure that the above linearization of the equation is valid for control processes that capture such a section of the AB characteristic, on which the tangent differs little from the curve itself.
In addition, another, graphical method of linearization follows from this. If the static characteristic and point C are known, which determines the steady state around which the regulation process takes place, then the transfer coefficient in the link equation is determined graphically from the drawing according to the dependence k 1 = tg 
taking into account the scale of the drawing and dimensions x 2. In many cases graphical linearization method
turns out to be more convenient and leads to the goal faster.
The dimension of the coefficient k 2 is equal to the dimension of the gain k 1 times the time. Therefore, equation (2.8) is often written in the form
where 
is the time constant.
P
Rice. 2.4. Independent excitation motor
The factor k 3 is the gain for external perturbation.
As an example of linearization, consider an electric motor controlled from the side of the excitation circuit (Fig. 2.4).
To find a differential equation that relates the speed increment to the voltage increment on the excitation winding, we write the law of equilibrium of electromotive forces (emf) in the excitation circuit, the law of equilibrium of emf in the armature circuit and the law of equilibrium of moments on the motor shaft:
;
.
In the second equation, for simplicity, the term corresponding to the self-induction emf in the armature circuit is omitted.
In these formulas, R B and R I are the resistances of the excitation circuit and the armature circuit; І В and І Я - currents in these circuits; U V and U I are the voltages applied to these circuits; V is the number of turns of the excitation winding; Ф – magnetic flux; Ω is the angular speed of rotation of the motor shaft; M is the moment of resistance from external forces; J is the reduced moment of inertia of the engine; C E and C M - coefficients of proportionality.
Let us assume that before the appearance of an increment in the voltage applied to the excitation winding, there was a steady state, for which equations (2.10) will be written as follows:
(2.11)
If now the excitation voltage will receive an increment U B = U B0 + ΔU B, then all variables that determine the state of the system will also receive increments. As a result, we will have: І В = І В0 + ΔІ В; Ф = Ф 0 + ΔФ; I I \u003d I I0 + ΔІ I; Ω = Ω0 + ΔΩ.
We substitute these values into (2.10), discard the higher-order small ones and get:
(2.12)
Subtracting equations (2.11) from equations (2.12), we obtain a system of equations for deviations:
(2.13)
AT
Rice. 2.5. Magnetization curve
determined from the magnetization curve of the electric motor (Fig. 2.5).
The joint solution of system (2.13) gives
where is the transfer coefficient, 
,
;
(2.15)
electromagnetic time constant of the excitation circuit, s,
(2.16)
where L B = a B is the dynamic coefficient of self-induction of the excitation circuit; electromagnetic time constant of the engine, s,
.
(2.17)
From expressions (2.15) - (2.17) it can be seen that the system under consideration is essentially non-linear, since the transfer coefficient and time "constant" are, in fact, not constant. They can be considered constant only approximately for a certain mode, provided that the deviations of all variables from the steady-state values are small.
An interesting is the special case when in the steady state U B0 = 0; I B0 = 0; Ф 0 = 0 and Ω 0 = 0. Then formula (2.14) becomes
.
(2.18)
In this case, the static characteristic will relate the increase in engine acceleration 
and voltage increment in the excitation circuit.
