The Laplace transformation of the DE makes it possible to introduce a convenient concept of a transfer function that characterizes the dynamic properties of the system.
For example, the operator equation
3s 2 Y(s) + 4sY(s) + Y(s) = 2sX(s) + 4X(s)
can be transformed by taking X(s) and Y(s) out of brackets and dividing by each other:
Y(s)*(3s 2 + 4s + 1) = X(s)*(2s + 4)
The resulting expression is called the transfer function.
Transfer function is called the ratio of the image of the output effect Y(s) to the image of the input X(s) under zero initial conditions.
(2.4)
The transfer function is a fractional rational function of a complex variable:
,
where B(s) = b 0 + b 1 s + b 2 s 2 + … + b m s m - numerator polynomial,
A(s) = a 0 + a 1 s + a 2 s 2 + … + a n s n - denominator polynomial.
The transfer function has an order that is determined by the order of the denominator polynomial (n).
From (2.4) it follows that the image of the output signal can be found as
Y(s) = W(s)*X(s).
Since the transfer function of the system completely determines its dynamic properties, the initial task of calculating the ASR is reduced to determining its transfer function.
2.6.2 Examples of typical links
A link of a system is an element of a system that has certain dynamic properties. The links of control systems can have a different physical nature (electrical, pneumatic, mechanical, etc. links), but are described by the same remote control, and the ratio of input and output signals in the links is described by the same transfer functions.
In TAU, a group of simplest units is distinguished, which are usually called typical. The static and dynamic characteristics of typical links have been studied quite fully. Standard links are widely used in determining the dynamic characteristics of control objects. For example, knowing the transient response constructed using a recording device, it is often possible to determine what type of links the control object belongs to, and therefore its transfer function, differential equation, etc., i.e. object model. Typical links Any complex link can be represented as a connection of simpler links.
The simplest typical links include:
amplification,
inertial (1st order aperiodic),
integrating (real and ideal),
differentiating (real and ideal),
aperiodic 2nd order,
oscillatory,
delayed.
1) Reinforcing link.
The link amplifies the input signal by K times. The link equation y = K*x, transfer function W(s) = K. The parameter K is called gain .
The output signal of such a link exactly repeats the input signal, amplified by K times (see Figure 1.18).
With stepwise action h(t) = K.
Examples of such links are: mechanical transmissions, sensors, inertia-free amplifiers, etc.
2) Integrating.
2.1) Ideal integrating.
The output value of the ideal integrating link is proportional to the integral of the input value:
; W(s) = 
When a step action link x(t) = 1 is applied to the input, the output signal constantly increases (see Figure 1.19):
This link is astatic, i.e. does not have a steady state.
An example of such a link is a container filled with liquid. The input parameter is the flow rate of the incoming liquid, the output parameter is the level. Initially, the container is empty and in the absence of flow the level is zero, but if you turn on the liquid supply, the level begins to increase evenly.
2.2) Real integrating.
P 
The transfer function of this link has the form
W(s) = 
.
The transition response, unlike an ideal link, is a curve (see Fig. 1.20):
h(t) = K . (t – T) + K . T. e - t / T .
An example of an integrating link is a DC motor with independent excitation, if the stator supply voltage is taken as the input effect, and the rotor rotation angle is taken as the output effect. If voltage is not supplied to the motor, then the rotor does not move and its angle of rotation can be taken equal to zero. When voltage is applied, the rotor begins to spin, and its angle of rotation is first slowly due to inertia, and then increases faster until a certain rotation speed is reached.
3) Differentiating.
3.1) Ideal differentiator.
The output quantity is proportional to the time derivative of the input:
; W(s) = K*s
With a step input signal, the output signal is an impulse (-function): h(t) = K. (t).
3.2) Real differentiating.
Ideal differentiating links are not physically realizable. Most objects that represent differentiating links belong to real differentiating links, the transfer functions of which have the form
W(s) = 
.
Step response: 
.
Example of a link: electric generator. The input parameter is the angle of rotation of the rotor, the output parameter is voltage. If the rotor is rotated at a certain angle, voltage will appear at the terminals, but if the rotor is not rotated further, the voltage will drop to zero. It cannot drop sharply due to the presence of inductance in the winding.
4) Aperiodic (inertial).
This link corresponds to DE and PF of the form
; W(s) = 
.
Let us determine the nature of the change in the output value of this link when a stepwise effect of the value x 0 is applied to the input.
Image of step effect: X(s) = 
. Then the image of the output quantity is:
Y(s) = W(s) X(s) = 
= K x 0 
.
Let's break down the fraction into prime ones:
=
+
=
=
-
=
-
The original of the first fraction according to the table: L -1 () = 1, the second:
L -1 ( 
}
=
.
Then we finally get
y(t) = K x 0 (1 - 
).
The constant T is called time constant.
Most thermal objects are aperiodic links. For example, when voltage is applied to the input of an electric furnace, its temperature will change according to a similar law (see Figure 1.22).
5) Second order links
The links have remote control and PF of the form
,
W(s) = 
.
When a step effect with amplitude x 0 is applied to the input, the transition curve will have one of two types: aperiodic (at T 1 2T 2) or oscillatory (at T 1< 2Т 2).
In this regard, second-order links are distinguished:
aperiodic 2nd order (T 1 2T 2),
inertial (T 1< 2Т 2),
conservative (T 1 = 0).
6) Delayed.
If, when a certain signal is applied to the input of an object, it does not react to this signal immediately, but after some time, then the object is said to have a delay.
Lag– this is the time interval from the moment the input signal changes until the output signal begins to change.
A lagging link is a link in which the output value y exactly repeats the input value x with some delay :
y(t) = x(t - ).
Link transfer function:
W(s) = e - s .
Examples of delays: the movement of liquid along a pipeline (how much liquid was pumped at the beginning of the pipeline, so much of it will come out at the end, but after some time while the liquid moves through the pipe), the movement of cargo along a conveyor (the delay is determined by the length of the conveyor and the speed of the belt), etc. .d.
The Laplace transformation of the DE makes it possible to introduce a convenient concept of a transfer function that characterizes the dynamic properties of the system.
For example, the operator equation
3s 2 Y(s) + 4sY(s) + Y(s) = 2sX(s) + 4X(s)
can be transformed by taking X(s) and Y(s) out of brackets and dividing by each other:
Y(s)*(3s 2 + 4s + 1) = X(s)*(2s + 4)
The resulting expression is called the transfer function.
Transfer function is called the ratio of the image of the output effect Y(s) to the image of the input X(s) under zero initial conditions.
(2.4)
The transfer function is a fractional rational function of a complex variable:
,
where B(s) = b 0 + b 1 s + b 2 s 2 + … + b m s m - numerator polynomial,
A(s) = a 0 + a 1 s + a 2 s 2 + … + a n s n - denominator polynomial.
The transfer function has an order that is determined by the order of the denominator polynomial (n).
From (2.4) it follows that the image of the output signal can be found as
Y(s) = W(s)*X(s).
Since the transfer function of the system completely determines its dynamic properties, the initial task of calculating the ASR is reduced to determining its transfer function.
Examples of typical links
A link of a system is an element of a system that has certain dynamic properties. The links of control systems can have a different physical nature (electrical, pneumatic, mechanical, etc. links), but are described by the same remote control, and the ratio of input and output signals in the links is described by the same transfer functions.
In TAU, a group of simplest units is distinguished, which are usually called typical. The static and dynamic characteristics of typical links have been studied quite fully. Standard links are widely used in determining the dynamic characteristics of control objects. For example, knowing the transient response constructed using a recording device, it is often possible to determine what type of links the control object belongs to, and therefore its transfer function, differential equation, etc., i.e. object model. Typical links Any complex link can be represented as a connection of simpler links.
The simplest typical links include:
· intensifying,
· inertial (1st order aperiodic),
integrating (real and ideal),
differentiating (real and ideal),
· aperiodic 2nd order,
· oscillatory,
· delayed.
1) Reinforcing link.
The link amplifies the input signal by K times. The link equation y = K*x, transfer function W(s) = K. The parameter K is called gain
.
The output signal of such a link exactly repeats the input signal, amplified by K times (see Figure 1.18).
With stepwise action h(t) = K.
Examples of such links are: mechanical transmissions, sensors, inertia-free amplifiers, etc.
2) Integrating.
2.1) Ideal integrating.
The output value of the ideal integrating link is proportional to the integral of the input value:
; W(s) =
When a step action link x(t) = 1 is applied to the input, the output signal constantly increases (see Figure 1.19):
This link is astatic, i.e. does not have a steady state.
An example of such a link is a container filled with liquid. The input parameter is the flow rate of the incoming liquid, the output parameter is the level. Initially, the container is empty and in the absence of flow the level is zero, but if you turn on the liquid supply, the level begins to increase evenly.
2.2) Real integrating.
The transfer function of this link has the form
The transition response, unlike an ideal link, is a curve (see Fig. 1.20):
h(t) = K . (t – T) + K . T. e - t / T .
An example of an integrating link is a DC motor with independent excitation, if the stator supply voltage is taken as the input effect, and the rotor rotation angle is taken as the output effect. If voltage is not supplied to the motor, then the rotor does not move and its angle of rotation can be taken equal to zero. When voltage is applied, the rotor begins to spin, and its angle of rotation is first slowly due to inertia, and then increases faster until a certain rotation speed is reached.
3) Differentiating.
3.1) Ideal differentiator.
The output quantity is proportional to the time derivative of the input:
With a step input signal, the output signal is a pulse (d-function): h(t) = K. d(t).
3.2) Real differentiating.
Ideal differentiating links are not physically realizable. Most objects that represent differentiating links belong to real differentiating links, the transfer functions of which have the form
Transition characteristic: .
Example of a link: electric generator. The input parameter is the angle of rotation of the rotor, the output parameter is voltage. If the rotor is rotated at a certain angle, voltage will appear at the terminals, but if the rotor is not rotated further, the voltage will drop to zero. It cannot drop sharply due to the presence of inductance in the winding.
4) Aperiodic (inertial).
This link corresponds to DE and PF of the form
; W(s) = .
Let us determine the nature of the change in the output value of this link when a stepwise effect of the value x 0 is applied to the input.
Image of step effect: X(s) = . Then the image of the output quantity is:
Y(s) = W(s) X(s) = = K x 0 .
Let's break down the fraction into prime ones:
= + = 
= - = -
The original of the first fraction according to the table: L -1 ( ) = 1, the second:
Then we finally get
y(t) = K x 0 (1 - ).
The constant T is called time constant.
Most thermal objects are aperiodic links. For example, when voltage is applied to the input of an electric furnace, its temperature will change according to a similar law (see Figure 1.22).
5) Second order links
The links have remote control and PF of the form
,
W(s) = 
.
When a step effect with amplitude x 0 is applied to the input, the transition curve will have one of two types: aperiodic (at T 1 ³ 2T 2) or oscillatory (at T 1< 2Т 2).
In this regard, second-order links are distinguished:
· aperiodic 2nd order (T 1 ³ 2T 2),
· inertial (T 1< 2Т 2),
· conservative (T 1 = 0).
6) Delayed.
If, when a certain signal is applied to the input of an object, it does not react to this signal immediately, but after some time, then the object is said to have a delay.
Lag– this is the time interval from the moment the input signal changes until the output signal begins to change.
A lagging link is a link in which the output value y exactly repeats the input value x with some delay t:
y(t) = x(t - t).
Link transfer function:
W(s) = e - t s .
Examples of delays: the movement of liquid along a pipeline (how much liquid was pumped at the beginning of the pipeline, so much of it will come out at the end, but after some time while the liquid moves through the pipe), the movement of cargo along a conveyor (the delay is determined by the length of the conveyor and the speed of the belt), etc. .d.
Link connections
Since the object under study, in order to simplify the analysis of its functioning, is divided into links, then after determining the transfer functions for each link, the task arises of combining them into one transfer function of the object. The type of transfer function of the object depends on the sequence of connections of the links:
1) Serial connection.
W rev = W 1. W2. W 3...
When links are connected in series, their transfer functions multiply.
2) Parallel connection.
W rev = W 1 + W 2 + W 3 + …
When links are connected in parallel, their transfer functions fold up.
3) Feedback
Transfer function by reference (x):
“+” corresponds to negative OS,
"-" - positive.
To determine the transfer functions of objects with more complex connections of links, either sequential enlargement of the circuit is used, or they are converted using the Meson formula.
Transfer functions of ASR
For research and calculation, the structural diagram of the ASR through equivalent transformations is brought to the simplest standard form “object - controller” (see Figure 1.27). Almost all engineering methods for calculating and determining the settings of regulators are applied to such a standard structure.
In the general case, any one-dimensional ASR with main feedback can be brought to this form by gradually enlarging the links.
If the output of the system y is not fed to its input, then an open-loop control system is obtained, the transfer function of which is defined as the product:
W ¥ = W p . W y
(W p - PF of the regulator, W y - PF of the control object).
|
|
|
This transfer function Фз(s) determines the dependence of y on x and is called the transfer function of a closed-loop system along the channel of the reference action (by reference).
For ASR there are also transfer functions through other channels:
Ф e (s) = = - by mistake,
Ф in (s) = = - by disturbance,
where W (s) – transfer function of the control object through the disturbance transmission channel.
With regard to taking into account the disturbance, two options are possible:
The disturbance has an additive effect on the control action (see Figure 1.29a);
The disturbance affects the measurements of the controlled parameter (see Figure 1.29b).
An example of the first option could be the influence of voltage fluctuations in the network on the voltage supplied by the regulator to the heating element of the object. An example of the second option: errors in measuring a controlled parameter due to changes in ambient temperature. W u.v. – model of the influence of the environment on measurements.
Figure 1.30
Parameters K0 = 1, K1 = 3, K2 = 1.5, K4 = 2, K5 = 0.5.
In the block diagram of the ASR, the links corresponding to the control device stand in front of the links of the control object and generate a control action on the object u. The diagram shows that the regulator circuit includes links 1, 2 and 3, and the object circuit includes links 4 and 5.
Considering that links 1, 2 and 3 are connected in parallel, we obtain the transfer function of the controller as the sum of the transfer functions of the links:
Links 4 and 5 are connected in series, therefore the transfer function of the control object is defined as the product of the transfer functions of the links:
Open-loop transfer function:
from which it is clear that the numerator B(s) = 1.5. s 2 + 3 . s + 1, denominator (also the characteristic polynomial of an open-loop system) A(s) = 2. s 3 + 3 . s 2 + s. Then the characteristic polynomial of the closed system is equal to:
D(s) = A(s) + B(s) = 2 . s 3 + 3 . s 2 + s + 1.5. s 2 + 3 . s + 1 = 2. s 3 + 4.5. s 2 + 4 . s+1.
Closed-loop system transfer functions:
on assignment 
,
by mistake 
.
When determining the transfer function from a disturbance, W a.v. is taken. = W ou. Then
. ¨
The ultimate goal of ACS analysis is to solve (if possible) or study the differential equation of the system as a whole. Usually the equations of the individual links that make up the ACS are known, and the intermediate task of obtaining the differential equation of the system from the known DEs of its links arises. In the classical form of representing DEs, this task is fraught with significant difficulties. Using the concept of a transfer function greatly simplifies it.
Let some system be described by a differential equation of the form.
By introducing the notation = p, where p is called the operator, or symbol, of differentiation, and now treating this symbol as an ordinary algebraic number, after taking x out and x in out of brackets, we obtain the differential equation of this system in operator form:
(a n p n +a n-1 p n-1 +…+a 1 p +a 0)x out = (b m p m +b m-1 p m-1 +…+b 1 p+b 0)x in. (3.38)
The polynomial in p at the output value is
D(p)=a n p n +a n -1 p n -1 +…+a 1 p+a 0 (3.39)
is called the eigenoperator, and the polynomial at the input value is called the influence operator
K(p) = b m p m +b m-1 p m-1 +…+b 1 p+b 0 . (3.40)
The transfer function is the ratio of the influence operator to its own operator:
W(p) = K(p)/D(p) = x out / x in. (3.41)
In what follows, we will almost everywhere use the operator form of writing differential equations.
Types of connections of links and algebra of transfer functions.
Obtaining the transfer function of an automatic control system requires knowledge of the rules for finding the transfer functions of groups of links in which the links are interconnected in a certain way. There are three types of connections.
1. Sequential, in which the output of the previous link is the input for the next one (Fig. 3.12):
| x out |
Rice. 3.14. Back-to-back - parallel connection.
Depending on whether the feedback signal x is added to the input signal xin or subtracted from it, positive and negative feedback are distinguished.
Still based on the property of the transfer function, we can write
W 1 (p) =x out /(x in ±x); W 2 (p) = x/x out; W c =x out /x in. (3.44)
Eliminating the internal coordinate x from the first two equations, we obtain the transfer function for such a connection:
W c (p) = W 1 (p)/ . (3.45)
It should be kept in mind that in the last expression the plus sign corresponds to negative feedback.
In the case when a link has several inputs (such as, for example, a control object), several transfer functions of this link are considered, corresponding to each of the inputs, for example, if the link equation has the form
D(p)y = K x (p)x + K z (p)z (3.46)
where K x (p) and K z (p) are operators of influences on inputs x and z, respectively, then this link has transfer functions on inputs x and z:
W x (p) = K x (p)/D(p); W z (p) = K z (p)/D(p). (3.47)
In the future, in order to reduce entries in the expressions of transfer functions and corresponding operators, we will omit the “p” argument.
From a joint consideration of expressions (3.46) and (3.47) it follows that
y = W x x+W z z, (3.48)
that is, in the general case, the output value of any link with several inputs is equal to the sum of the products of the input values and the transfer functions for the corresponding inputs.
Transfer function of ACS based on disturbance.
The usual form of the ACS structure, operating on the deviation of a controlled variable, is as follows:
| W o z =K z /D object W o x =K x /D |
| W p y |
| z |
| y |
| -x |
Fig.3.15. Closed ATS.
Let us pay attention to the fact that the regulatory influence is applied to the object with a changed sign. The connection between the output of an object and its input through the regulator is called the main feedback (as opposed to possible additional feedback in the regulator itself). According to the very philosophical meaning of regulation, the action of the regulator is aimed at reduction of deviation controlled variable, and therefore the main feedback is always negative. In Fig. 3.15:
W o z - transfer function of the object by disturbance;
W o x - transfer function of the object according to the regulatory influence;
W p y - transfer function of the controller according to the deviation y.
The differential equations of the plant and the controller look like this:
y=W o x x +W o z z
x = - W p y y. (3.49)
Substituting x from the second equation into the first and performing grouping, we obtain the ATS equation:
(1+W o x W p y)y = W o z z . (3.50)
Hence the transfer function of the ACS for disturbance
W c z = y/z =W o z /(1+W o x W p y) . (3.51)
In a similar way, you can obtain the transfer function of the ACS for the control action:
W c u = W o x W p u /(1+W o x W p y) , (3.52)
where W p u is the transfer function of the controller according to the control action.
3.4 Forced oscillations and frequency characteristics of ACS.
In real operating conditions, the ACS is often exposed to periodic disturbing forces, which is accompanied by periodic changes in controlled quantities and regulatory influences. These are, for example, vibrations of the ship when sailing in rough seas, fluctuations in the speed of rotation of the propeller and other quantities. In some cases, the amplitudes of oscillations of the system's output quantities can reach unacceptably large values, and this corresponds to the phenomenon of resonance. The consequences of resonance are often disastrous for the system experiencing it, for example, capsizing a ship, destroying an engine. In control systems, such phenomena are possible when the properties of elements change due to wear, replacement, reconfiguration, or failures. Then there is a need to either determine safe ranges of operating conditions or properly configure the ATS. These issues will be considered here as they apply to linear systems.
Let some system have the structure shown below:
| x=A x sinωt |
| y=A y sin(ωt+φ) |
Fig.3.16. ACS in forced oscillation mode.
If the system is subject to a periodic influence x with amplitude A x and circular frequency w, then after the end of the transition process, oscillations of the same frequency with amplitude A y and shifted relative to the input oscillations by a phase angle j will be established at the output. The output oscillation parameters (amplitude and phase shift) depend on the frequency of the driving force. The task is to determine the parameters of output oscillations from the known parameters of oscillations at the input.
In accordance with the ACS transfer function shown in Fig. 3.14, its differential equation has the form
(a n p n +a n-1 p n-1 +…+a 1 p+a 0)y=(b m p m +b m-1 p m-1 +…+b 1 p+b 0)x. (3.53)
Let us substitute into (3.53) the expressions for x and y shown in Fig. 3.14:
(a n p n +a n-1 p n-1 +…+a 1 p+a 0)A y sin(wt+j)=
=(b m p m +b m-1 p m-1 +…+b 1 p+b 0)A x sinwt. (3.54)
If we consider the oscillation pattern shifted by a quarter of the period, then in equation (3.54) the sine functions will be replaced by cosine functions:
(a n p n +a n-1 p n-1 +…+a 1 p+a 0)A y cos(wt+j)=
=(b m p m +b m-1 p m-1 +…+b 1 p+b 0)A x coswt. (3.55)
Let's multiply equation (3.54) by i = and add the result with (3.55):
(a n p n +a n -1 p n -1 +…+a 1 p+a 0)A y =
= (b m p m +b m-1 p m-1 +…+b 1 p+b 0)A x (coswt+isinwt). (3.56)
Using Euler's formula
exp(±ibt)=cosbt±isinbt,
Let us reduce equation (3.56) to the form
(a n p n +a n-1 p n-1 +…+a 1 p+a 0)A y exp=
= (b m p m +b m-1 p m-1 +…+b 1 p+b 0)A x exp(iwt). (3.57)
Let us perform the operation of differentiation with respect to time provided by the operator p=d/dt:
A y exp=
A x exp(iwt). (3.58)
After simple transformations related to reduction by exp(iwt), we obtain
The right side of expression (3.59) is similar to the expression of the ACS transfer function and can be obtained from it by replacing p=iw. By analogy, it is called the complex transfer function W(iw), or the amplitude-phase characteristic (APC). The term frequency response is also often used. It is clear that this fraction is a function of a complex argument and can also be represented in this form:
W(iw) = M(w) +iN(w), (3.60)
where M(w) and N(w) are real and imaginary frequency characteristics, respectively.
The ratio A y / A x is the AFC modulus and is a function of frequency:
A y / A x = R (w)
and is called the amplitude-frequency response (AFC). Phase
the shift j =j (w) is also a function of frequency and is called the phase frequency response (PFC). By calculating R(w) and j(w) for the frequency range (0…¥), it is possible to construct an AFC graph on the complex plane in coordinates M(w) and iN(w) (Fig. 3.17).
| ω |
| R(ω) |
| ω cp |
| ω res |
Fig.3.18. Amplitude-frequency characteristics.
The frequency response of system 1 shows a resonant peak corresponding to the largest amplitude of forced oscillations. Work in the area near the resonant frequency can be disastrous and is often completely unacceptable by the operating rules of a particular regulated object. Frequency response type 2 does not have a resonant peak and is more preferable for mechanical systems. It can also be seen that as the frequency increases, the amplitude of the output oscillations decreases. Physically, this is easily explained: any system, due to its inherent inertial properties, is more easily subject to swinging by low frequencies than by high frequencies. Starting at a certain frequency, the output oscillation becomes negligible, and this frequency is called the cutoff frequency, and the range of frequencies below the cutoff frequency is called the bandwidth. In the theory of automatic control, the cutoff frequency is taken to be one at which the frequency response value is 10 times less than at zero frequency. The property of a system to dampen high-frequency vibrations is called the property of a low-pass filter.
Let us consider the method of calculating the frequency response using the example of a second-order link, the differential equation of which
(T 2 2 p 2 + T 1 p + 1)y = kx. (3.62)
In forced oscillation problems, a more visual form of the equation is often used
(p 2 +2xw 0 p + w 0 2)y = kw 0 2 x, (3.63)
where is called the natural frequency of oscillations in the absence of damping, x =T 1 w 0 /2 is the damping coefficient.
The transfer function looks like this:
By replacing p = iw we obtain the amplitude-phase characteristic
Using the rule for dividing complex numbers, we obtain the expression for the frequency response:
Let us determine the resonant frequency at which the frequency response has a maximum. This corresponds to the minimum denominator of expression (3.66). Equating the derivative of the denominator with respect to frequency w to zero, we have:
2(w 0 2 - w 2)(-2w) +4x 2 w 0 2 *2w = 0, (3.67)
from where we obtain the value of the resonant frequency, which is not equal to zero:
w res = w 0 Ö 1 - 2x 2 . (3.68)
Let's analyze this expression, for which we consider individual cases that correspond to different values of the attenuation coefficient.
1. x = 0. The resonant frequency is equal to the natural frequency, and the magnitude of the frequency response turns to infinity. This is a case of so-called mathematical resonance.
2. . Since the frequency is expressed as a positive number, and from (68) for this case either zero or an imaginary number is obtained, it follows that at such values of the attenuation coefficient the frequency response does not have a resonant peak (curve 2 in Fig. 3.18).
3. . The frequency response has a resonant peak, and with a decrease in the attenuation coefficient, the resonant frequency approaches its own and the resonant peak becomes higher and sharper.
We will assume that the processes taking place in the ACS are described by linear differential equations with constant coefficients. Thus, we will limit ourselves to considering linear ACS with constant parameters, i.e. parameters that do not depend on either time or the state of the system.
Let for a dynamic system (see figure)
the differential equation is written in operator form
where D(P) and M(P) are polynomials in P.
P – differentiation operator;
x(t) – output coordinate of the system;
g(t) – input influence.
Let us transform (1) according to Laplace, assuming zero initial conditions.
Let us introduce the notation
;
,
we get, taking into account that
We use the notation
, (5)
then equation (3) will take the form:
. (6)
Equation (6) connects the image X (S) of the output coordinate of the system with the image G(S) of the input action. Function Ф(S) characterizes the dynamic properties of the system. As follows from (4) and (5), this function does not depend on the impact applied to the system, but depends only on the parameters of the system. Taking into account (6) the function F(S) can be written as follows
Function Ф(S) is called the transfer function of the system. From (7) it is clear that the transfer function is the ratio of the Laplace image of the input coordinate of the system to the Laplace image of the input action under zero initial conditions.
Knowing the transfer function of the system Ф(S) Having determined the image G(S) of the influence g(t) applied to the system, one can find from (6) the image X(S) of the output coordinate of the system x (t), then, moving from the image X(S) to the original x(t), obtain the process of changing the output coordinate of a system when an input influence is applied to this system.
The polynomial in the denominator of the transfer function is called the characteristic polynomial, and the equation
characteristic equation.
For a system described by an nth-order equation, the characteristic equation is an algebraic equation of the nth degree and has n roots, S 1 S 2 ... S n , among which there can be both real and complex conjugate.
The root of the polynomial in the denominator of the transfer function is called the poles of this transfer function, and in the numerator - zeros.
Let's represent the polynomials in the form:
Therefore the transfer function
. (11)
It follows that specifying zeros and poles determines the transfer function up to a constant factor 
.
In the case when the real parts of all poles of the transfer function are negative, i.e.
, k=1,2…n, the system is called stable. In it, the transition component of the output quantity (proper motion) fades over time.
System frequency characteristics
Conversion of a harmonic input signal by a linear system
The transfer function of the automatic system with respect to the control action g(t) is
(1)
Let the impact
g(t) = A 1 sin ω 1 t,
And it is required to determine the change in X(t) in a steady process, i.e. Find a particular solution to equation (1), discussed earlier.
Note that as a result of the application of an influence, a transient process occurs in the system, which tends to 0 over time, because the system is assumed to be stable. We are not considering it. Such a transition allows us to consider the action g(t) as specified on the entire time axis (the initial moment of application of the control action to the system is not considered) and use the previously obtained expression for the spectral characteristic of the sinusoid.
To determine x(t) in a steady state, we transform both sides of the differential equation (1) according to Fourier. By this we mean that
;
,
Note that
transfer function in which S 
Besides
Then the spectral characteristic of forced oscillations of the controlled quantity is determined from (3) in the form
In (4) the functional multiplier Ф(jω) takes into account the change in the spectral characteristic when the influence g(t) passes through a linear dynamic system.
Let's imagine a complex function Ф(jω) in demonstrative form
and find x(t) using the inverse Fourier transform formula:
using the filtering properties of the delta function, and taking into account (5), we will have
Because 
,,
(6)
It follows that in steady state the response x(t) of a linear automatic system to sinusoidal influences is also a sinusoid. The angular frequencies of the input and output signals are the same. The amplitude at the system output is A 1 │ Ф(jω)│, and the initial phase is arg Ф(jω).
If the input of a linear system receives a periodic influence in the form
,
then, using the principle of superposition, which is valid for a linear system, we find that in this case the forced steady motion of the system
(7)
Moreover, the value of ω here should be given discrete values, i.e. assume ω=kω 1
Knowing the frequency spectra of the input signal, you can easily determine the frequency spectra of the signal at the system input. If, for example, the amplitude frequency spectrum A k of the input signal g(t) is known, then the amplitude frequency spectrum of the output signal is A k │ Ф(jkω 1 ) │.
In the expressions under consideration, the function Ф(jω) characterizes the dynamic properties of the automatic system itself and does not depend on the nature of the influences applied to the system. It can easily be obtained from the transfer function by formally replacing S with jω
Function Ф(jω) from the continuous argument ω is called the amplitude-phase characteristic of the AFC system in relation to the control action g(t) applied to the system.
Based on (3), AFC can also be defined as the ratio of the spectral characteristics of the signal at its input. AF module Ф(j ) characterizes the change in the amplitude of a harmonic signal as it passes through the system, and its argument is the phase shift of the signal.
Function Ф(j ) received the name amplitude-frequency response (AFC), and the function arg Ф(j ) – phase-frequency response (PFC).
Let the influence g(t) applied to the automatic system be a complex harmonic with frequency 1, i.e.
The response of the system to such an impact in a steady state is determined by the equality
Or using Euler's formula
and also that
;
We will find the integral on the right side of the equality using the filtering properties of the delta function.
determines in complex form the steady-state response of the system to influence in the form of a complex harmonic with frequency 1.
AFC can be used not only to analyze steady-state oscillations at the output of an automatic system, but also to determine the control process as a whole. In the latter case, it is convenient to consider the moment of time t 0 of application to the control system as the zero moment of time and use the formulas of the one-sided Fourier transform. Having determined the spectral characteristic 
and finding the spectral characteristic of the controlled variable using the formula
The change in the controlled variable x(t) after applying the influence g(t) is found using the inverse Fourier transform formula.
