In the study of control systems, they are usually presented as an interconnected set of individual elements - dynamic links. dynamic link refers to a device of any physical form and design, having an input and an output as shown in Figure 2.1, and for which an equation (usually differential) is given that relates the signals at the input and output.
Figure 2.1 - Diagram of a dynamic link
The classification of dynamic links is made according to the form of the differential equation. Devices of any type (electrical, electromechanical, hydraulic, thermal, etc.) can be described by the same differential equations, which makes it possible to use the same approaches for designing various devices.
If the equation relating the signals , linearly, then they speak of a linear dynamic link
The equation of a linear dynamic link has the following form:
where are constant coefficients; .
However, the form of the differential equation is not the only feature by which dynamic links are compared.
The main characteristics of the links are:
Differential equations of motion;
transfer functions;
Time characteristics (transient function, impulse (weight) function;
Frequency characteristics (amplitude-frequency characteristics, amplitude-phase frequency characteristics, logarithmic frequency characteristics).
transfer function The link is the ratio of the images of the output and input signals under zero initial conditions. Let us subject equation (2.1) to the Laplace transformation, assuming the initial conditions to be zero and replacing the original signals with their images:
From here we get
Relation (2.2) does not depend on the signal images and is determined only by the parameters of the dynamic link itself , , has the form of a fractional-rational function.
Type equation
is called the characteristic equation of the dynamic link, since the denominator of the transfer function is the characteristic polynomial of the differential equation describing the dynamic link.
Timing determine the dynamic properties of the link. They are determined at the link output when standard signals are applied to the input.
transition function or the transient response is a transient process at the output of the link that occurs when a jump action is applied to its input with a jump value equal to one (Figure 2.2). Such an action is called a unit step function and is denoted
The step function is a common type of input action in the ACS. This type of impact can include an instantaneous change in the load of the electric generator, an increase in the torque on the motor shaft, an instantaneous change in the setting for the engine speed, an instantaneous rotation of the command axis of the servo system.
Figure 2.2 - Single step (a) and transition (b) functions
The Laplace image of the unit step function is defined as
To determine the image of the transition function with a known transfer function of the link, you must perform the following operation:
The original is found using the inverse Laplace transform (Appendix B) applied to (1.5).
impulse transition function or weight function is the response of the link to the unit impulse function. The unit impulse function, or - function, is the derivative of the unit step function:
The delta function is defined by the expression
The main property of the delta function is that
that is, it has a unit area. This function can be described as a short but powerful impulse. The delta function is also a common input in automatic systems. For example, short-term load shock on the motor shaft, short-term short-circuit current of the generator, switched off by fuses, etc.
It is easy to establish that the image of the -function is defined
The image of the weight function is the transfer function:
Therefore, to find the original of the impulse transition function, it is necessary to apply the inverse Laplace transform to the transfer function of the link (system).
The delta function and the weight function of a certain link are shown in Figure 2.3
Figure 2.3 - Delta function (a) and weight function (b)
The transition and impulse functions are related by the relations
frequency response dynamic link is called the function complex argument, obtained by formal replacement with in the expression of the transfer function. Frequency characteristics are obtained by considering the movement of a link (system) when a harmonic effect is applied to its input.
The function , which is obtained from the transfer function (2.2):
is called the frequency transfer function.
The frequency transfer function, as a function of the complex argument, can be represented as
where is the real (real) part of ; is the imaginary part of ; – module (amplitude) ; – argument (phase).
Amplitude, phase, real and imaginary parts of the function are functions of frequency, so the frequency transfer function is used and represented as amplitude-phase, real, imaginary, amplitude and phase frequency responses.
Thus, the following frequency characteristics of dynamic links are considered in TAU:
1. Amplitude-frequency response (AFC) -
2. Phase response (PFC) -
3. Real frequency response (VCH) -
5. Amplitude-phase frequency response (APFC), which is defined as the hodograph of a vector (the curve described by the end of this vector), built on complex plane when changing the frequency from 0 to .
physical meaning frequency characteristics can be determined as follows. At harmonic influence in stable systems after the end of the transient, the output value also changes according to the harmonic law, but with a different amplitude and phase. In this case, the ratio of the amplitudes of the output and input values is equal to the module, and the phase shift is equal to the argument of the frequency transfer function. And, therefore, the amplitude frequency response shows a change in the ratio of amplitudes, and the phase frequency response shows the phase shift of the output value relative to the input depending on the frequency of the input harmonic effect.
A general view of the frequency characteristics is shown in Figure 2.4.
Figure 2.4 - Frequency characteristics:
amplitude-phase (a), amplitude-frequency (b), phase-frequency (c), real frequency (d), imaginary frequency (e) characteristics
Logarithmic frequency characteristics (LFC).Logarithmic amplitude frequency response(LAFC) of a dynamic link is such a representation of the amplitude frequency response (AFC), in which the modulus (amplitude) of the frequency response is expressed in decibels, and the frequency is in a logarithmic scale:
Logarithmic phase frequency response(LPCH) of a dynamic link is called a graph of the dependence of the phase-frequency characteristic (PFC) on the logarithm of the frequency. When constructing logarithmic characteristics, the frequency is plotted on the abscissa axis on a logarithmic scale, and the value itself is written at the mark corresponding to the value. Quite often, LACH and LPCH are built on the same graph to give a complete picture of the properties of the object.
The unit is the decibel, and the unit of the logarithm of frequency in LFC is the decade. decade The interval over which the frequency changes by a factor of 10 is called. When the frequency changes by 10 times, it is said that it has changed by one decade.
When constructing the LPFC, the angles are counted along the y-axis on the usual scale in degrees or radians.
When constructing the LFC, the ordinate axis is drawn through an arbitrary point, and not through a point (the frequency corresponds to an infinitely distant point: at ). Since , then the origin of coordinates is most often taken at the point .
8. Integrating link with deceleration
Here, is the link gain, and is the time constant, s.
1.3.1 Features of the classification of ACS links The main task of the TAU automatic control theory is to develop methods by which it would be possible to find or evaluate the quality indicators of dynamic processes in ACS. In other words, not all physical properties elements of the system, but only those that influence, are associated with the type of dynamic process. The structural design of the element, its overall dimensions, the way of summing up are not considered.
energy, design features, range of materials used, etc. However, parameters such as mass, moment of inertia, heat capacity, combinations of RC, LC, etc., which directly determine the type of dynamic process, will be important. Features of the physical performance of the element are important only to the extent that they will affect its dynamic performance. Thus, only one selected property of an element is considered - the nature of its dynamic process. This allows us to reduce the consideration of a physical element to its dynamic model in the form of a mathematical model. Model solution, i.e. differential equation describing the behavior of the element, gives a dynamic process that is subject to a qualitative assessment.
The classification of ACS elements is based not on the design features or features of their functional purpose (control object, comparison element, regulatory body, etc.), but on the type of mathematical model, i.e. mathematical equations of connection between the output and input variables of the element. Moreover, this connection can be specified both in the form of a differential equation and in another transformed form, for example, using transfer functions (PF). The differential equation provides comprehensive information about the properties of the link. Having solved it, with one or another given law of the input value, we get a reaction, by the form of which we evaluate the properties of the element.
The introduction of the concept of a transfer function makes it possible to obtain a connection between the output and input quantities in operator form and, at the same time, use some properties of the transfer function, which make it possible to significantly simplify the mathematical representation of the system and use some of their properties. To explain the concept of PF, consider some properties of the Laplace transform.
1.3.2 Some properties of the Laplace transform The solution of the models of the dynamic links of the ACS gives a change in the variables in the time plane. We are dealing with functions. X(t). However, using the Laplace transform, they can be transformed into functions [X(p)] with a different argument p and new properties.
The Laplace transform is a special case of type matching: one function is associated with another function. Both functions are interconnected by a certain dependence. Correspondence resembles a mirror, reflecting in a different way, depending on the form, the object in front of it. The type of display (correspondence) can be chosen arbitrarily, depending on the problem being solved. You can, for example, look for a correspondence between a set of numbers, the meaning of which boils down to how, according to the chosen number at from the area Y find number X from the area x. Such a relationship can be specified analytically, in the form of a table, graph, rule, etc.
Similarly, a correspondence between groups of functions can be established (Fig. 3.1 a), for example, in the form:
As a correspondence between the functions x(t) and x(p) (Fig. 3.1 b), the Laplace integral can be used:
subject to the conditions: x(t)= 0 at and at t.
In ACS, not absolute changes in variables are investigated, but their deviations from steady-state values. Hence, x(t) - a class of functions that describe the deviations of variables in the automatic control system and both conditions of the Laplace transformation are satisfied for them: the first - since there is no change in variables before the application of the perturbation, the second - since over time any deviation in a workable system tends to zero.
These are the conditions for the existence of the Laplace integral. Let's get, as an example, images of the simplest functions but to Laplace.
Rice. 3.1. Function display types
So, if the unit function x(t) = 1 is given, then
For the exponential function x(t) = e -α t, the image by
Laplace will look like:
Finally:
The resulting functions are no more complicated than the original ones. The function x(t) is called the original, and x(p)- her image. Conditionally direct and inverse Laplace transform can be represented as:
L=x(p),L -1<=x(t).
In this case, there is an unambiguous relationship between the original and the image, and vice versa, the only image of the function corresponds to the original. Consider some properties of the Laplace transform.
Image of the function differential. Let the function x(t) correspond to the image x(p): x(t)-> x(p)- It is necessary to find the image of its derivative x(t):
Thus
Under zero initial conditions
For the image of the derivative of the nth order:
Thus, the image of the derivative of a function is the image of the function itself, multiplied by the operator p to the extent n, where P is the order of differentiation.
Elementary dynamic link (EDZ) is called a mathematical model of an element in the form of a differential equation that is not subject to further simplification.
1.3.3 Inertial aperiodic link of the first order
Such a link is described by a first-order differential equation relating the input and output quantities:
An example of such a link, in addition to a thermocouple, a DC motor, an RL chain, can be a passive RC- chain (Fig. 3.2 d).
Using the basic laws of description electrical circuits we obtain a mathematical model of an aperiodic link in differential form:
Let's get the relationship between the input and output values of the link in the form of the Laplace transform:
Rice. 3.2. Examples of aperiodic links
The ratio of the output value to the input value gives an operator of the form.
What is a dynamic link? In previous lessons, we considered the individual parts of the automatic control system and called them elements automatic control systems. Elements can have a different physical appearance and design. The main thing is that some input x( t ) , and as a response to this input signal, the element of the control system forms some output signal y( t ) . Next, we found that the relationship between the output and input signals is determined by dynamic properties control, which can be represented as transfer function W(s). So here it is a dynamic link is any element of an automatic control system that has a certain mathematical description, i.e. for which the transfer function is known.
Rice. 3.4. Element (a) and dynamic link (b) ACS.
Typical dynamic links is the minimum required set of links to describe an arbitrary type of control system. Typical links include:
proportional link;
aperiodic link of the 1st order;
aperiodic link of the second order;
oscillatory link;
integrating link;
ideal differentiating link;
forcing link of the 1st order;
forcing link of the second order;
link with pure delay.
proportional link
The proportional link is also called inertialess .
1. Transfer function.
The transfer function of the proportional link has the form:
W(s) = K where K is the amplification factor.
The proportional link is described by the algebraic equation:
y(t) = K· X(t)
Examples of such proportional links are a lever mechanism, a rigid mechanical transmission, a gearbox, an electronic signal amplifier at low frequencies, a voltage divider, etc.
4. Transition function .
The transition function of the proportional link has the form:
h(t) = L -1 = L -1 = K· 1(t)
5. Weight function.
The weight function of the proportional link is:
w(t) = L -1 = Kδ(t)
Rice. 3.5. Transition function, weight function, phase response and proportional response .
6. Frequency characteristics .
Let's find the AFC, AFC, PFC and LAH of the proportional link:
W(jω ) = K = K +0j
A(ω
)
=
= K
φ(ω) = arctg(0/K) = 0
L(ω) = 20 log = 20 log(K)
As follows from the presented results, the amplitude of the output signal does not depend on the frequency. In reality, no link is able to uniformly pass all frequencies from 0 to ¥, as a rule, at high frequencies, the gain becomes smaller and tends to zero as ω → ∞. Thus, the mathematical model of a proportional link is some idealization of real links .
Aperiodic link I th order
Aperiodic links are also called inertial .
1. Transfer function.
The transfer function of the aperiodic link of the 1st order has the form:
W(s) = K/(T· s + 1)
where K is the amplification factor; T is the time constant characterizing the inertia of the system, i.e. the duration of the transition process in it. Insofar as time constant characterizes some time interval , then its value must always be positive, i.e. (T > 0).
2. Mathematical description of the link.
Aperiodic link of the 1st order is described by a first order differential equation:
T· dy(t)/ dt+ y(t) = K·X(t)
3. Physical implementation of the link.
Examples of an aperiodic link of the 1st order are: electric RC filter; thermoelectric converter; compressed gas tank, etc.
4. Transition function .
The transition function of the aperiodic link of the 1st order has the form:
h(t) = L -1 = L -1 = K – K e -t/T = K (1 – e -t/T )
Rice. 3.6. Transient response of aperiodic link of the 1st order.
The transient process of the aperiodic link of the first order has an exponential form. The steady value is: h set = K. The tangent at the point t = 0 crosses the line of the steady value at the point t = T. At the time t = T, the transition function takes the value: h(T) ≈ 0.632 K, over time T, the transient response gains only about 63% of the steady-state value.
Let's define regulation time T at for an aperiodic link of the 1st order. As is known from the previous lecture, the regulation time is the time after which the difference between the current and steady-state values will not exceed some given small value Δ. (Typically, ∆ is given as 5% of steady state).
h(T y) \u003d (1 - Δ) h set \u003d (1 - Δ) K \u003d K (1 - e - T y / T), hence e - T y / T \u003d Δ, then T y / T \u003d -ln (Δ), As a result, we get T y \u003d [-ln (Δ)] T.
At Δ = 0.05 T y = - ln(0.05) T ≈ 3 T.
In other words, the time of the transient process of the first order aperiodic link is approximately 3 times the time constant.
An ACS link is a mathematical model of an element or connection of elements of any part of the system. Links, like systems, can be described by high-order differential equations and, in the general case, their transfer functions can be represented as
But they can be represented as compounds of typical or elementary links, the order of differential equations of which is not higher than the second.
From the course of algebra, on the basis of Bezout's theorem, it is known that a polynomial of arbitrary order can be decomposed into prime factors of the form
,
. (4.64)
Therefore, the transfer function (4.63) can be represented as the product of prime factors of the form (4.64) and simple fractions of the form
,
,
.
(4.65)
Links, the transfer functions of which have the form of simple factors (4.63) or simple fractions (4.64), are called typical or elementary links.
Before proceeding to the study of elementary links, let us recall the formulas for the modulus and argument of a complex number. Let a complex number be represented as a ratio of two products of complex numbers
As 
,
, then for the modulus and argument of a complex number we have
,
.
Thus, the following rule of modules and arguments of complex numbers is valid: module complex number, represented as the ratio of two products of complex numbers, is equal to the ratio of the product of the modules of the numerator factors to the product of the modules of the denominator factors, and its argument is the difference between the sum of the arguments of the factors of the numerator and the sum of the arguments of the factors of the denominator.
proportional link. A link is called proportional, which is described by the equation 
or transfer function 
.
The frequency and time functions of this typical Even have the form:
,
,
,
,
,
,
.
Ha fig. 4.5 shows some of the characteristics of the proportional link: the amplitude-phase frequency response (4.5 a) is the point To on the real axis; phase frequency
jVa)L(w) b)h(t) in)
20 lgK K
K U w t
Fig.4.5 Characteristics of the proportional link
the characteristic (or AFC) coincides with the positive frequency axis; logarithmic amplitude frequency response (Fig. 4.56) is parallel to the frequency axis and passes on the level .
The transient response (Fig. 4.5c) is parallel to the time axis and passes at the level 
.
integrating link. An integrating link is a link that is described by the equation 
or transfer function 
. Frequency transfer function 
.
The remaining frequency and time functions have the form:
,
,
,
,
,
,
.
The AFC (Fig. 4.6a) of the integrating link coincides with the negative imaginary semiaxis. LPCH (Fig. 4.66) is parallel to the frequency axis and passes at the level: the phase shift does not depend on frequency and is equal to 
.
LACHH (Fig. 4.6b) - an inclined straight line passing through a point with coordinates 
and 
.
As can be seen from the equation, with an increase in frequency by the I decade, the ordinate 
,
decreases by 20 dB. Therefore, the slope of the LAFC is -20 dB / dec (read: minus twenty decibels per decade).
The transient response is a straight line passing through the origin with a slope equal to k. (Fig. 4.6c).
a B C)
jV
U
L(w)
(w)h(t)
0.1 1.0 w arctgK
-
/2
t
Fig 4.6 Characteristics of the integrating link
differentiating link. A differentiating link is a link that is described by the equation 
or transfer function 
.
The frequency and time functions of this link have the form
,
,
,
,
,
,
,
.
jVa)L(w)
(w) b)
+
/2
0,1 1,0 10
Fig.4.7 Characteristics of the differentiating link
The AFC (Figure 4.7a) coincides with the positive imaginary semiaxis. LPCH (Figure 4.7b) is parallel to the frequency axis and passes at the level 
, that is, the phase shift does not depend on frequency and is equal to 
/2.
LACH is a straight line passing through a point with coordinates 
=1,
and having a slope of 20 dB / dec (read: plus twenty decibels per decade): 
increases by 20 dB for each decade of frequency increase.
Aperiodic link. An aperiodic even of the first order is a link that is described by the equation
(4.66)
or transfer function
.
(4.67)
This link is also called the inertial link of the first order. The aperiodic link, in contrast to the links considered above, is characterized by two parameters: the time constant T and the transfer coefficient k.
.
(4.68)
Multiplying the numerator and denominator by the complex conjugate of the denominator, we get
,
.
(4.69)
The amplitude and phase frequency functions can be determined using the rule of modules and arguments.
Since the modulus of the numerator of the frequency transfer function (4.68) is equal to k, and the modulus of the denominator 
,then
(4.70)
Numerator argument 
is zero and the denominator argument 
. So
Having solved the differential equation (4.66) with 
and zero initial condition 
, we get the transient response 
. Weight function or impulse response
.
The AFC of an aperiodic Even (Fig. 4.8a) is a semicircle, which is not difficult to verify by excluding the AFC frequency from the parametric equations (4.69) 
.
LACH is shown in Figure 4.8b. In practice, they are usually limited to constructing the so-called asymptotic LACH (broken line in the same figure 4.86). In critical cases, when a small error can affect the conclusions about the state of the system under study, the exact LAFC is considered. However, the exact LAFC can be easily constructed from the asymptotic LAFC if we use the following relationship (
L
- the difference between the asymptotic and exact LACH):
T 
=
0,10
0,25 0,40 0,50 1,0 2,0 2,5 4,0 10,0
L=
0,04
0,25 0,62 0,96 3,0 0,96 0,62 0,25 0,04
Frequency 
, at which the asymptotes intersect, is called the conjugate frequency. Exact and asymptotic LAFC
Rio.4.8 Non-periodic characteristics
differ most strongly at the corner frequency; the deviation at this frequency is about 3 dB.
The asymptotic LAFC equation has the form:
It is obtained from equation (4.71) if in it under the root at 
ignore the first term, and 
-
the second term.
According to the resulting equation, the asymptotic LAFC can be constructed as follows: at the level 
frequencies 
draw a straight line parallel to the frequency axis, and then through a point with coordinates 
and 
- straight at an angle - -20 dB/dec.
By AFC or LAFC it is easy to determine the parameters T and k aperiodic link (Fig. 4.86).
LPCH is shown in fig. 4.86. This characteristic tends asymptotically to zero as 
and to 
at 
. At 
phase-frequency function takes the value - 
, i.e 
. LPCH
all aperiodic links have the same shape and can be obtained on the basis of one characteristic by a parallel shift along the frequency axis to the left or right, depending on the value of the time constant T. Therefore, to construct the LPFC of the aperiodic link, you can use the template shown in Fig. 4.8d.
The transient response of an aperiodic link (Fig. 4.8c) is an exponential curve that can be used to determine the parameters of this link: transfer coefficient k
determined by the established value 
; time constant T is equal to the value of t corresponding to the point of intersection of the tangent, built on the transient response at the origin, with its asymptote (Figure 4.8c).
Forcing link. A forcing link or a first-order forcing link is a link that is described by the equation
,
or transfer function
.
This link, like the aperiodic one, is characterized by two parameters: the time constant T and gear ratio k.
Frequency transfer function
.
The remaining frequency and time functions have the form:
,
,
,
,
,
,
.
The AFC is a straight line parallel to the imaginary axis and intersecting the real axis at the point U=
k.(Fig. 4.9a). As in the case of an aperiodic link, in practice one confines oneself to constructing an asymptotic LAFC. Frequency 
, corresponding to the break point of this characteristic, is called the corner frequency. Asymptotic LAFC at 
parallel to the frequency axis and intersects the y-axis at 
, and when 
has a slope of +20dB/dec.
The LFC of the boosting link can be obtained by mirroring the LFC of the aperiodic link with respect to the frequency axis, and to construct it, you can use the same template and nomogram that are used to construct the latter.
Oscillatory, conservative and aperiodic second order links. A link that can be described by the equation
(4.72)
or in another form
where, 
,
.
The transfer function of this link
(4.74)
This link is oscillatory if 
;-conservative if 
; - an aperiodic link of the second order, if 
.
Coefficient 
called the damping factor.
oscillating link
. The frequency transfer function of this link
.
Multiplying the numerator and denominator by the complex conjugate expression, we obtain the real and imaginary frequency functions of the oscillatory link:
,
The phase frequency function, as can be seen from the AFC (Figure 4.10b), changes monotonously from 0 to - 
and is expressed by the formula
(4.75)
LPCH (Fig. 410b) at 
asymptotically tends to the frequency axis, and at 
to a straight line 
. It can be built using a template. But for this it is necessary to have a set of templates corresponding to different values of the damping coefficient.
Amplitude frequency function
and logarithmic amplitude-frequency function
The asymptotic LPCH equation has the form
(4.75)
where 
- corner frequency. Asymptotic LACH (Fig. 4.106) for 
parallel to axis
frequencies, and at 
has a slope of -40 dB/dec.
Rice. 4.10. Characteristics of the oscillatory link
Should have in
mind that the asymptotic LAFC (Fig. 4.10b) for small values of the damping coefficient is quite different from the exact LAFC. The exact LAFC can be constructed from the asymptotic LAFC, using the deviation curves of the exact LAFC from the asymptotic ones (Fig. 4.10d). Having solved the differential equation (4.72) of the oscillatory link at 
and zero initial conditions 
find the transition function.
,
,
,
.
weight function
.
According to the transient response (Fig. 4.10c), the parameters of the oscillatory link can be determined as follows.
At the first stage of ACS design, the problems of system synthesis are solved on the basis of data on the purpose of the system and the design features of the control object. When forming the ACS structure at this stage, the functionally necessary elements of the systems are used, the so-called ACS links (value sensors, signal converters, regulators, actuators, etc.).
The second stage of the ACS design is the analysis of the conformity of the quality characteristics of the designed system with the required ones. To carry out all types of analysis of the ACS considered in section 3, it is necessary to have its model in the form of a differential equation of the form (1) or a transfer function of the form (2).
To obtain ACS models, the concept is introduced typical elementary link. A typical elementary link is understood as a set of ACS elements, the dynamic processes in which are described by a linear differential equation of the form (1) not higher than the second order ( n£2). The introduction of elementary links makes it possible to reduce the whole variety of technical devices to a small number of typical links, which makes it possible to use common methods analysis for any ACS. Types of elementary units of ACS are given in application 1.
Reinforcing inertialess link
The links of this type include any element of the ACS, which at each moment of time has a proportional relationship between the output value y(t) and input action x(t), i.e. this link, not only in statics, but also in dynamics, is described by an algebraic equation of the form:
y(t) = k× x(t),
where k– static conversion factor (gain factor) of the link.
Strictly speaking, the amplifying link is not dynamic, since the change y(t) occurs instantly, immediately after the change x(t). The link differential equation is said to have null order. The transfer function of the link has the form W(p) = k.
When applying to the input of a single step x(t) = 1(t) the output will instantly receive the same signal, amplified in k times (Fig. 35).
| Rice. 35 |
It is clear that no real technical device can instantly transform the input action, however, the speed of some elements of the ACS is so high (the duration of the transient process is less than a second) that they can be considered links of this type. Examples of such elements are a potentiometer, a lever, an electronic amplifier. In the first approximation, without taking into account the phenomenon of twisting and backlash, a gearbox can be considered an inertialess amplifying link.
In the literature, there are other names for the amplifying inertialess link: amplifier, ideal amplifier or proportional link .
Aperiodic link of the first order
This type of link (cf. Appendix 1) is described by a first-order differential equation:
,
where k– static conversion factor (gain factor) of the link; T is some constant having the dimension of time (link time constant).
On fig. 36 shows the transient characteristics of aperiodic links of the first order with k= 10 and different time constants T. It can be seen that when increasing T link output value y(t k, i.e. time constant T characterizes inertia link, and determines the time of the transient process tp. In practical calculations tp for an aperiodic link of the first order, they are taken approximately equal to 3× T.
| Rice. 36 |
.
Aperiodic links of the first order are such ACS devices as electric RL- and RC- circuits (used as ACS corrective devices), DC electric generator (used as an ACS control device), a temperature sensor - a thermocouple, a flow tank with liquid or gas (control objects in chemical-technological ACS) and much more.
Get the dynamics model RC-circuit in a theoretical way: we write the equations of the input and output circuits (Fig. 37) according to the Kirchhoff law:
| Rice. 37 |
U in(t) and output - U out(t) variables RC i(t
,
i(t) into the input circuit equation:
.
The resulting equation corresponds to the differential equation of the aperiodic link of the first order, for which the time constant T = R× C, i.e. determined by the resistor and capacitor values used in RC-contour; k = 1; y(t) = U out(t); x(t) = U in(t).
In the literature, there are other names for the aperiodic link of the first order: inertial link of the first order or relaxation link .
4.3. Aperiodic link of the second order and oscillatory
sustainable
The second-order aperiodic link and the oscillatory stable link have the general form of a differential equation (see Fig. Appendix 1):
,
but aperiodic of the second order a link with such an equation is called under the condition 
, a oscillatory- given that 
.
General view of the transfer function for both links:
.
Note that under the condition 
the equation 
will have a positive discriminant and, accordingly, real roots. This allows decomposing the denominator of the second-order aperiodic link transfer function into factors of the form:
where 
.
If we take into account that when the links are connected in series, their transfer functions are multiplied, then it turns out that the second-order aperiodic link is equivalent to two first-order aperiodic links connected in series one after another, with a common static conversion coefficient k and time constants T 3 and T 4 .
On fig. 38 shows the transient characteristics of two aperiodic links of the second order with k= 5 and different time constants T 1 and T 2. It can be seen that when increasing T 1 and T 2 link output value y(t) more slowly reaches a steady-state value equal to k, i.e. the time constants and for this link determine the time of the transient process.
Important! Pay attention: despite the visual similarity of the transient characteristics of the aperiodic links of the first and second orders, they have fundamental differences. The 2nd order characteristic has an inflection point: at time zero, the rate of change y(t) is minimal, then it increases up to the inflection point, and after it decreases. The initial section of the transient characteristics of the second-order links (for t 0 to 0.5 seconds) is shown in fig. 38 in the selected enlarged fragment. In the same place, for comparison, a similar section of the characteristics of the first-order links shown in Fig. 36. It can be seen that for them the rate of change y(t) is maximum at time t= 0. Further, during the time t p rate of change y(t) decreases to zero (see Fig. 36).
The time interval up to the inflection point of the transient response of the second-order aperiodic link is calculated by the formula:
.
Given that , i.e. for oscillatory stable link, denominator of the transfer function will have a negative discriminant and, accordingly, complex conjugate roots. It is known from the theory of differential equations that the free movement of such a system contains harmonic components (sine, cosine), which gives fluctuations in the output value when the input signal changes.
The transfer function of an oscillatory link is usually written as:
where T is the time constant of the oscillatory link; x is the attenuation coefficient (for an oscillatory stable link 0< x < 1). Чем больше x, тем быстрее затухают колебания переходной характеристики звена. При x = 0 получается oscillatory harmonic a link that gives undamped oscillations at the output (see. Appendix one). For x ³ 1, we have an aperiodic link of the second order.
On fig. 39 shows the transient characteristics of two oscillatory links with the same k= 8 and time constant T= 1, and different attenuation coefficients x. It can be seen that the oscillatory characteristics of the transient and overshoot for the link with x = 0.25 is greater than for the link with x = 0.5.
On fig. 40 shows the transient characteristics of two oscillatory links with the same values of the static conversion factor k= 8 and damping factor x = 0.3, and different time constant values T. It can be seen that the time of the transient process at the link with T= 2 is greater than that of the link with T = 1.
| Rice. 39 |
| Rice. 40 |
Oscillatory or aperiodic links of the second order (depending on the values specifications, which determine the ratio of the time constants T 1 and T 2) are such ACS devices as electric RLC-circuit; DC motor (see dynamic model derivation in section 2.3.1), elastic mechanical transmissions, e.g. for transmission rotary motion with elasticity, moment of inertia and coefficient of high-speed friction, differential pressure gauge (sensor for measuring differential pressure) and other devices.
Get the dynamics model RLC-circuit in a theoretical way: we write the equations of the input and output circuits (Fig. 41) according to the Kirchhoff law:
| Rice. 41 |
The purpose of modeling is to obtain a differential equation of the form (1) connecting the input - U in(t) and output - U out(t) variables RC-contour. To do this, in the equations of the input and output circuits, get rid of the intermediate internal variable of the circuit - current i(t). Differentiate the output circuit equation:
,
and substitute the result of the expression i(t) into the input circuit equation:
T 1 = R× C and 
, i.e. determined by the values of the resistor, capacitor and inductor used in RLC-contour; k = 1; y(t) = U out(t); x(t) = U in(t). The specific type of link - second-order aperiodic or oscillatory - depends on the ratio of time constants T 1 and T 2 (or respectively), i.e. ultimately determined by the denominations R, L and C. Transient response examples RLC-contours are shown in fig. 42.
| Rice. 42 |
Let's get a model of the dynamics of a mechanical system with linear movement, the parameters of the mechanical elements of which are mass, damping (friction) and elasticity (Fig. 43). Note that in the system under consideration, movement occurs only in one direction, movement in the transverse direction is not allowed.
Consider the action of an external force F(t) on isolated mechanical elements separately. For mass M according to Newton's second law:
,
where v(t) - speed; a(t) is the acceleration, and s(t) is the output linear displacement (see Fig. 43).
The speed of movement of the damper piston under the action of force F(t) is defined as follows:
,
where G is the coefficient of resistance (damping).
| Rice. 43 |
For an elastic spring, in accordance with Hooke's law, the equation of motion has the form:
,
where H- coefficient of elasticity of the spring.
In the system as a whole (see Fig. 43) on a body of mass M three forces acting - external force F(t), friction force and elastic force, therefore, for the sum of forces it is true:
The resulting equation of dynamics is of the second order, however, to reduce it to the form of a standard differential equation of an oscillatory or aperiodic link of the second order (see. Appendix 1) constant coefficient of the term s(t) on the left side should be equal to 1. Let us bring the equation of dynamics to a typical form by dividing the left and right sides by the coefficient of elasticity of the spring H:
The resulting equation corresponds to a differential equation for which the time constants T 1 = G/ H and 
, i.e. determined by the mass, as well as by the quantities G and H; k = 1 / H; y(t) = s(t); x(t) = F(t).
Thus, we have shown that a mechanical system of the form shown in fig. 43 is also an oscillatory or aperiodic link of the second order. The specific type of link depends on the ratio of the time constants T 1 and T 2 (or respectively), i.e. is ultimately determined by the quantities M, G and H. The considered mechanical system can be used, for example, as a link in the model of the braking system of a car per one wheel (in addition to the considered link, such a model requires taking into account the mass of the car and the elasticity of the tire).
From the considered examples, it can be seen that, despite the difference in ACS devices and their purpose, their mathematical models have the form of the same second-order differential equation. The types of links considered in the literature are sometimes called inertial links of the second order .
Integrating links
An ideal integrating link is a link whose output value is proportional to the time integral of the input value (see Fig. Appendix 1):
,
where k– static conversion coefficient (gain) of an ideal integrating link, equal to the ratio of the rate of change of the output value to the input.
The transfer function of the link has the form:
.
The transient response of an ideal integrating link has the form of an inclined straight line, since the integral is geometrically the area limited by the graph of the stepped input action x(t), which increases with time. t. The solution of the differential equation of an ideal integrating link has the form:
,
whence for a unit step ( x(t) = 1 for t³ 0) with zero initial conditions y(0) = 0 we obtain a linearly increasing transient response y(t) = k× t. On fig. 44 shows the transient characteristics of ideal integrators with different values k.
| Rice. 44 |
The simplest household example of an ideal integrating link is a bathtub into which water is drawn. Input action x(t) for this object is the inflow (flow) of water through the tap, and the output value y(t) is the water level in the bath. When water enters, the level rises, i.e. the system "accumulates" (integrates) the input signal.
Examples of ideal integrating links are such ACS devices as an operational amplifier used in the integration mode (Fig. 45– a) and hydraulic damper (Fig. 45– b).
The equation for the operational amplifier used in the integration mode is:
,
which corresponds to the equation of an ideal integrating link, for which k = 1/R× C, U in = x(t), U out = y(t).
| Rice. 45 |
| a) |
| b) |
For a hydraulic damper, the input force is the force F acting on the piston, and the output value is the displacement of the piston s. Since the speed of the piston is proportional to the applied force:
,
where G- coefficient of resistance (damping), then the movement of the piston will be proportional to the integral of the applied force:
.
The resulting equation corresponds to the equation of an ideal integrating link, for which k = 1/G, F(t) = x(t), s(t) = y(t).
The considered type of integrating links is called ideal, because its equation does not take into account the inertia of the ACS device described by the link. In the literature, this type of link is sometimes called astatic link.
All real devices introduce some slowdown, so a more accurate model of real integrators is deceleration integrator
,
those. is the product of the transfer functions of an ideal integrating link and a first-order aperiodic link. Thus, an integrator with deceleration can be represented by a series connection of these two varieties of typical links. An engine can be described by such a link if the output value is not the angular velocity, but the angle of rotation, which is an integral of angular velocity, as well as the damper, if we consider its equation of motion more precisely.
Differentiators
The ideal differentiator gives the output a value proportional to the derivative of the input signal, i.e. the rate of change of the input action (see. Appendix 1):
,
where k is the static conversion factor (gain) of the ideal differentiating link. The transfer function of the link has the form: 
.
The differentiating link reacts not to a change in the input value itself, but to a change in its derivative, that is, to the trend in the development of events. Therefore, they say that the differentiating link has a proactive, predictive action. With it, you can speed up the response of the ACS to changing inputs.
Let us analyze the shape of the transient characteristic of an ideal differentiating link (see Fig. Appendix one). When applying to the input of a link of a single step x(t) = 0 for t< 0 и x(t) = 1 for t > 0. The derivative of the constant value is equal to zero, therefore, y(t) = 0 for t< 0 и для t >0. And only at the moment of direct change of the input action from zero to one, i.e. at time t = 0, the derivative of the input signal dx(t)/dt not equal to zero:
As a result, the transient response of an ideal differentiating link at time t = 0 theoretically has the form of an impulse with an infinitely large amplitude and an infinitely short duration. It is clear that such a transient response cannot be obtained using a real ACS device. Therefore, an ideal differentiating link, as well as links of this type of the first and second orders (see Fig. Appendix 1) are model and refer to physically unrealizable links.
Approximately, an operational amplifier switched on in the differentiation mode can be considered as an ideal differentiating link (Fig. 46– a), and a DC tachogenerator, if we consider the angle of rotation of its rotor a( t), and as the output - armature voltage u i(t) (Fig. 46– b).
In a DC tachogenerator with a constant excitation flux, the armature voltage can be considered proportional to the angular velocity of rotation. In turn, the rotation speed is a derivative of the angle of rotation:
,
which corresponds to the differential equation of an ideal differentiating link with a static transformation coefficient k, y(t) = u i(t); x(t) = a( t).
In practice, ACS differentiating devices introduce some slowdown in the work (have inertia), therefore, a more accurate model of real devices is differential link with deceleration, whose transfer function has the form:
,
those. is the product of the transfer functions of an ideal differentiating link and a first-order aperiodic link. Thus, a differentiating link with deceleration can be represented by a serial connection of these two types of typical links. Examples of a differentiating link with deceleration are a transformer, a capacitive differentiating circuit (Fig. 47– a) and a mechanical differentiating device consisting of a spring and a damper (Fig. 47– b).
| Rice. 47 |
| a) |
| b) |
We obtain a model of the dynamics of a capacitive differentiating circuit (see Fig. 47– a). We write the equations of the input and output circuits according to the Kirchhoff law:
Differentiate the input circuit equation:
,
and put a current into it i(t), expressing it from the output circuit equation:
We derive the transfer function of the capacitive differentiating circuit:
Received W(p k = T = R× C.
We obtain a model of the dynamics of a mechanical differentiating device (see Fig. 47– b) for y(t) = s out(t); x(t) = s in(t) under the assumption that the friction element (damper) and elasticity (spring) have zero mass. The damper motion equation for this case has the form:
,
where G is the coefficient of resistance (damping). For a spring with an elastic coefficient H the equation of motion is:
,
so after substitution:
We derive the transfer function of a mechanical differentiating device:
Received W(p) corresponds to the transfer function of the differentiating link with deceleration, for which k = T = G/H.