The Laplace transformation of the DE makes it possible to introduce a convenient concept of the transfer function characterizing 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 converted 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 the ratio of the image of the output action 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,
А(s) = a 0 + a 1 s + a 2 s 2 + … + a n s n is the denominator polynomial.
The transfer function has an order, which 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
The link of the system is its element, which has certain properties in a dynamic sense. The links of control systems can have a different physical nature (electrical, pneumatic, mechanical, etc. links), but they can be described by the same control, and the ratio of input and output signals in the links can be described by the same transfer functions.
In TAU, a group of the simplest links is distinguished, which are usually called typical. The static and dynamic characteristics of standard links have been studied quite fully. Typical links are widely used in determining the dynamic characteristics of control objects. For example, knowing the transient response built using a recording device, it is often possible to determine what type of links the control object belongs to, and, consequently, its transfer function, differential equation, etc., i.e. object model. Typical links Any complex link can be represented as a combination of the simplest links.
The simplest typical links include:
amplifying,
inertial (aperiodic of the 1st order),
integrating (real and ideal),
differentiating (real and ideal),
aperiodic 2nd order,
oscillatory,
lagging.
1) Reinforcing link.
The link amplifies the input signal by K times. The link equation y \u003d K * x, the transfer function W (s) \u003d 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).
Under step action h(t) = K.
Examples of such links are: mechanical transmissions, sensors, inertialess amplifiers, etc.
2) Integrating.
2.1) Ideal integrator.
The output value of an ideal integrator is proportional to the integral of the input value:
; W(s) = 
When a stepped 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 integrator.
P 
the transfer function of this link has the form
W(s) = 
.
The transient response, in contrast to the 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 action, and the rotor rotation angle is taken as the output action. If voltage is not applied 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 up, and the angle of its rotation at first slowly due to inertia, and then increase rapidly until a certain rotation speed is reached.
3) Differentiating.
3.1) The ideal differentiator.
The output value is proportional to the time derivative of the input:
; W(s) = K*s
With a stepped 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 of the objects that are differentiating links refer to real differentiating links, the transfer functions of which have the form
W(s) = 
.
Transient response: 
.
Link example: electric generator. The input parameter is the angle of rotation of the rotor, the output parameter is voltage. If the rotor is rotated a certain angle, voltage will appear on the terminals, but if the rotor is not rotated further, the voltage will drop to zero. It cannot fall 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's determine the nature of the change in the output value of this link when a step action of the value x 0 is applied to the input.
Step action image: X(s) = 
. Then the image of the output quantity:
Y(s) = W(s) X(s) = 
= K x 0 
.
Let's decompose the fraction into simple 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) Links of the second order
The links have DU and PF of the form
,
W(s) = 
.
When a stepped action 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, the links of the second order are distinguished:
aperiodic 2nd order (T 1 2T 2),
inertial (T 1< 2Т 2),
conservative (T 1 \u003d 0).
6) Delayed.
If, when a certain signal is applied to the input of an object, it does not respond to this signal immediately, but after some time, then the object is said to have a delay.
Lag is the time interval from the moment the input signal changes to the start of the output signal change.
A lagging link is a link whose output value y exactly repeats the input value x with some delay :
y(t) = x(t - ).
Link transfer function:
W(s) \u003d e - s.
Examples of delays: the movement of liquid through the pipeline (how much liquid was pumped at the beginning of the pipeline, so much will be released at the end, but after a while, while the liquid moves through the pipe), the movement of cargo along the conveyor (the delay is determined by the length of the conveyor and the speed of the belt), etc. .d.
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 individual links that are part of the ACS are known, and an intermediate problem arises of obtaining a differential equation of the system from the known DE of its links. With the classical form of representation of DE, this problem is associated with significant difficulties. Using the concept of transfer function greatly simplifies it.
Let some system be described by a DE 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 get 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)
Polynomial in p, standing at the output value,
D(p)=a n p n +a n -1 p n -1 +…+a 1 p+a 0 (3.39)
is called an eigenoperator, and the polynomial at the input value is called the action 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 action 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 link connections and algebra of transfer functions.
Obtaining the transfer function of the ACS 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. Counter-parallel connection.
Depending on whether the feedback signal x is added to the input signal x in or subtracted from it, positive and negative feedbacks are distinguished.
Still based on the property of the transfer function, we can write
W 1 (p) \u003d x out / (x in ± x); W 2 (p) \u003d x / x out; W c \u003d 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)
Note that in the last expression, the plus sign corresponds to negative feedback.
In the case when some link has several inputs (as, for example, an object of regulation), several transfer functions of this link corresponding to each of the inputs are considered, 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 the action operators for inputs x and z, respectively, then this link has transfer functions for inputs x and z:
Wx(p) = Kx(p)/D(p); W z (p) = K z (p)/D(p). (3.47)
In the future, in order to reduce the entries in the expressions of transfer functions and the corresponding operators, we will omit the “p” argument.
It follows from the joint consideration of expressions (3.46) and (3.47) 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 by perturbation.
The usual form of the ACS structure, working on the deviation of the controlled value, 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 SAR.
Let's pay attention to the circumstance that the regulating action arrives at the object with a changed sign. The connection between the object's output and its input through the controller is called the main feedback (in contrast to possible additional feedbacks in the controller itself). According to the very philosophical meaning of regulation, the action of the regulator is aimed at deviation reduction adjustable value, and therefore the main feedback is always negative. On fig. 3.15:
W o z - transfer function of the object in perturbation;
W o x - transfer function of the object on the regulatory action;
W p y - transfer function of the controller by 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 grouping, we get the CAP equation:
(1+W o x W p y)y = W o z z . (3.50)
Hence the transfer function of the ACS with respect to the perturbation
W c z \u003d y / z \u003d W o z / (1 + W o x W p y) . (3.51)
In a similar way, you can get 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 for the control action.
3.4 Forced vibrations and frequency characteristics of the ACS.
In real operating conditions, the automatic control system is often subjected to the action of periodic disturbing forces, which is accompanied by periodic changes in the controlled values and control actions. Such, for example, are the oscillations of the vessel during the course of the waves, oscillations in the frequency of rotation of the propeller, and other quantities. In some cases, the oscillation amplitudes of the output values of the system can reach unacceptably large values, and this corresponds to the resonance phenomenon. The consequences of resonance are often detrimental to 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, and failures. The need then arises either to define safe operating ranges or to properly tune the ACS. These questions will be considered here as applied to linear systems.
Let some system have the following structure:
| x=A x sinωt |
| y=A y sin(ωt+φ) |
Fig.3.16. ACS in the mode of forced oscillations.
If the system is affected by a periodic action x with amplitude A x and circular frequency w, then after the end of the transient process, oscillations of the same frequency with amplitude A y and shifted relative to the input oscillations by the phase angle j will be established at the output. The parameters of the output oscillations (amplitude and phase shift) depend on the frequency of the driving force. The task is to determine the parameters of the output oscillations from the known parameters of the oscillations at the input.
In accordance with the transfer function of the ACS 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)
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)
Applying the Euler formula
exp(±ibt)=cosbt±isinbt,
we bring 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's perform the time differentiation operation provided by the p=d/dt operator:
A y exp=
Axexp(iwt). (3.58)
After simple transformations related to the reduction by exp(iwt), we get
The right side of expression (3.59) is similar to the CAP transfer function expression 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 (AFC). Often the term frequency response is also used. It is clear that this fraction is a function of the 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 the real and imaginary frequency responses, respectively.
The ratio A y / A x is the AFC module and is a function of frequency:
A y / A x \u003d R (w)
and is called the amplitude-frequency characteristic (AFC). phase
shift j =j (w) is also a function of frequency and is called the phase frequency response (PFC). Calculating R(w) and j(w) for the frequency range (0…¥), it is possible to plot the AFC graph on the complex plane in coordinates M(w) and iN(w) (Fig. 3.17).
| ω |
| R(ω) |
| ωcp |
| 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 zone near the resonant frequency can be disastrous and often generally unacceptable by the rules of operation of a particular object of regulation. Frequency response of type 2 does not have a resonant peak and is more preferable for mechanical systems. It can also be seen that with increasing frequency, the amplitude of the output oscillations decreases. Physically, this is easily explained: any system, due to its inherent inertial properties, is more easily subjected to swinging by low frequencies than by high ones. Starting from a certain frequency, the fluctuations in the output become insignificant, and this frequency is called the cutoff frequency, and the frequency range 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 the system to dampen high-frequency oscillations is called the property of the low-pass filter.
Consider the methodology for 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 problems of forced oscillations, a more illustrative 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 then looks like this:
By replacing p = iw we get the amplitude-phase characteristic
Using the rule for dividing complex numbers, we obtain an expression for the frequency response:
Let us determine the resonant frequency at which the frequency response has a maximum. This corresponds to the minimum of the denominator of expression (3.66). Equating to zero the derivative of the denominator with respect to the frequency w, we have:
2(w 0 2 - w 2)(-2w) +4x 2 w 0 2 *2w = 0, (3.67)
whence we obtain the value of the resonant frequency, which is not equal to zero:
w cut \u003d w 0 Ö 1 - 2x 2. (3.68)
Let us analyze this expression, for which we consider individual cases, which correspond to different values of the attenuation coefficient.
1. x = 0. The resonant frequency is equal to its own, and the frequency response modulus goes 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 for such values of the damping 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.
After simple transformations, we get
(3.54)
Rule: transfer function of the system with negative feedback is equal to a fraction, the numerator of which is the transfer function of the direct channel , and the denominator is the sum of unity and the product of the transfer functions of the forward and reverse channels of the system.
When positive feedback formula (3.54) takes the form
(3.55)
In practice, there are usually systems with negative feedback, for which the transfer function is found by relation (3.54).
3.3.4. Transfer rule
In some cases, to obtain the overall transfer function of the system using structural transformations, it would be more convenient to move the signal application point through the link closer to the output or input. With such a transformation of the structural scheme, one should adhere to regulations: the transfer function of the system must remain unchanged.
Consider the situation when the point of application of the signal is transferred through the link closer to the output. The initial structure of the system is shown in fig. 3.31. Let us define for it the resulting transfer function
Let's transfer the point of application of the signal through the link with the transfer function by adding some transfer function to this channel.
 |
Rice. 3.32. Block diagram of the converted system.
For it, the transfer function has the form
Since when the system structure is transformed, its transfer function should not change, by equating the right parts of expressions (3.56) and (3.57), we determine the desired transfer function
Thus, when moving the signal application point closer to the system output, the transfer function of the link through which the signal is transferred should be added to the channel.
Similar rule can be formulated to move the point of application of the signal closer to the input of the system: the inverse transfer function of the link through which the signal is transferred should be added to the corresponding channel.
Example 3.1
Determine the overall transfer function of the system, the block diagram of which is shown in fig. 3.33.
Let us preliminarily determine the transfer functions of typical link connections: transfer function of a parallel connection of links
and the transfer function of series-connected links
 |
Rice. 3.33. Structural diagram of the system
Taking into account the introduced notation, the structure of the system can be reduced to the form shown in Fig. 3.34.
Using structural transformations, we write the general transfer function of the system
Substituting for and their values, we finally get
Example 3.2
Determine the transfer function of the automatic target tracking system of the radar station , the block diagram of which is shown in fig. 3.35.
Rice. 3.35. Structural diagram of the automatic target tracking system
Here is the transfer function of the system receiver; - transfer function of the phase detector; - transfer function of the power amplifier; - transfer function of the engine; - transfer function of the reducer; - transfer function of the antenna speed sensor; - transfer function of the corrective device.
Using the rules of structural transformations, we write
transfer function
Define the transfer function of the inner loop
and direct channel system
Let us define the total transfer function of the system
Substituting instead of intermediate transfer functions , the initial values, we finally get
3.4. Block diagrams corresponding to differential equations
The second way to draw up a block diagram is based on the use of differential equations. Consider it first for an object whose behavior is described by vector-matrix equations (2.1), (2.2):
(3.59)
Let us integrate the equation of state in (3.59) with respect to time and define the state and output variables as
(3.60)
Equations (3.60) are basic for charting.
 |
Rice. 3.36. Structural diagram corresponding to the equations
object state
The block diagram corresponding to equations (3.60) is more convenient to depict, starting with the output variables y, and it is desirable to place the input and output variables of the object on the same horizontal line (Fig. 3.36).
For a single-channel object, a block diagram can be drawn up according to equation (2.3), resolving it with respect to the highest derivative
Having integrated (3.61) n times, we get
(3.62)
The system of equations (3.62) corresponds to the block diagram shown in fig. 3.37.
Rice. 3.37. Block diagram corresponding to equation (3.61)
As you can see, a single-channel control object, whose behavior is described by equation (3.61), can always be structurally represented as a chain of n series-connected feedback integrators.
Example 3.3
Draw a block diagram of an object whose model is given by the following system of differential equations:
Let us preliminarily integrate the equations of state
 |
Rice. 3.38. Structural charting illustration
according to the equations of state
In accordance with the integral equations in Figs. 3.38 we will depict the block diagram of the system.
3.5. Transition from transfer function to canonical description
Let's discuss the most well-known ways of transforming the mathematical model of an object in the form of an arbitrary transfer function to a description in state variables. For this purpose, we use the appropriate block diagrams. Note that this problem is ambiguous, since state variables for an object can be chosen in different ways (see Sec. 2.2).
Consider two options for the transition to the description in state variables from the transfer function of the object
(3.63)
where Let us first represent (3.63) as a product of two transfer functions:
Each of these representations (3.63) has its own simple model in state variables, which is called canonical form.
3.5.1. First canonical form
Consider the transformation of the mathematical model of the system with the transfer function (3.64). Its block diagram can be represented as two series-connected links
(Fig. 3.39).
Rice. 3.39. Structural representation of the system (3.64)
For each link of the system, we write the corresponding operator equation
(3.66)
Let us determine from the first equation (3.66) the highest derivative of the variable z, which corresponds to the value in the operator form
The resulting expression allows us to represent the first equation (3.66) as a chain of n feedback integrators (see Sec. 3.5), and the output variable y is formed in accordance with the second equation (3.66) as the sum of the variable z and her m derivatives (Fig. 3.40).
Rice. 3.40. Scheme corresponding to equations (3.66)
Using structural transformations, we obtain the block diagram of the system shown in fig. 3.41.
Rice. 3.41. Block diagram corresponding to the canonical form
Note that the block diagram corresponding to the transfer function (3.64) consists of the chain n integrators, where n- the order of the system. Moreover, the coefficients of the denominator of the original transfer function (coefficients of the characteristic polynomial) are in feedback, and the coefficients of the polynomial of its numerator are in direct connection.
It is not difficult to pass from the obtained block diagram to the model of the system in state variables. For this purpose, we take the output of each integrator as a state variable
which allows us to write the differential equations of state and the system output equation (3.63) in the form
(3.67)
The system of equations (3.67) can be represented in vector-matrix form (2.1) with the following matrices:
The system model in state variables (3.67) will be called first canonical form.
3.5.2. Second canonical form
Let us consider the second way of transition from the transfer function (3.63) to the description in state variables, for which we schematically represent the structure of the system (3.65) in fig. 3.42.
Rice. 3.42. Structural representation of the transfer function (3.65)
Its operator equations have the form
(3.68)
Similarly to the previous case, we represent the first equation in (3.68) as a chain of n integrators with feedback, and the input action z form in accordance with the second equation (3.68) in the form of the sum of the control u and m its derivatives (Fig. 3.43).
As a result of structural transformations, we obtain a block diagram of the system shown in fig. 3.44. As we can see, in this case, too, the block diagram corresponding to the transfer function (3.65) consists of the chain n integrators. The coefficients of the characteristic polynomial are also in feedback, and the coefficients of the polynomial of its numerator are in direct connection.
Rice. 3.43. Scheme corresponding to equations (3.68)
Rice. 3.44. Block diagram corresponding to the transfer function (3.65)
Again, we choose the output values of the integrators as state variables and write down the differential equations of state and the output equation with respect to them
(3.69)
Using equations (3.69), we define the matrices
The system model in state variables of type (3.69) will be called second canonical form.
Note that the matrix A is unchanged for the first or second canonical forms and contains the coefficients of the denominator of the original transfer function (3.63). The transfer function numerator coefficients (3.63) contain the matrix C(in the case of the first canonical form) or matrix B(in the case of the second canonical form). Therefore, the equations of state corresponding to two canonical representations of the system can be written directly in terms of the transfer function (3.63) without going to the block diagrams shown in Figs. 3.40 and 3.43.
As you can see, the transition from the transfer function to the description in state variables is an ambiguous task. We have considered the options for the transition to the canonical description, which are most often used in the theory of automatic control.
Example 3.4
Get two versions of the canonical description and the corresponding block diagrams for the system, the model of which has the form
We use the representation of the transfer function in the form (3.64) and write down the operator equations for it
from which we proceed to the block diagram shown in Fig. 3.45.
Rice. 3.45. Block diagram corresponding to the first canonical form
Based on this structural scheme, we write the equations of the first canonical form in the form
To pass to the second canonical form, we represent the transfer function of the system in the form (3.65) and write the following operator equations for it:
which corresponds to the block diagram shown in fig. 3.46.
Rice. 3.46. Block diagram corresponding to the second canonical form
We now write the system model in the form of the second canonical form
3.6. Scope of the structural method
The structural method is convenient in the calculation of linear automatic systems, but has its limitations. The method involves the use of transfer functions, so it can be applied, as a rule, under zero initial conditions.
When using the structural method, you must adhere to the following regulations: for any transformation of the system, its order should not decrease, i.e., it is unacceptable to reduce the same factors in the numerator and denominator of the transfer function. By reducing the same multipliers, we thereby throw out the really existing links from the system. Let's illustrate this statement with an example.
Example 3.5
Consider a system consisting of integrating and differentiating links, which are connected in series.
The first version of the connection of the links is shown in Fig. 3.47.
Using structural transformations, we find the general transfer function
This implies the conclusion that such a connection of links is equivalent to an inertialess link, i.e., the signal at the output of the system repeats the signal at its input. Let us show this by considering the equations of individual links. The output signal of the integrator is determined by the relation
where is the initial condition on the integrator. The signal at the output of the differentiating link, and hence the entire system, has the form
which corresponds to the conclusion made on the basis of the analysis of the overall transfer function of the links.
The second variant of connecting the links is shown in Fig. 3.48, i.e. the links have been swapped. The transfer function of the system is the same as in the first case,
However, now the output of the system does not repeat the input signal. This can be seen by considering the equations of the links. The signal at the output of the differentiating link corresponds to the equation
and at the output of the system is determined by the relation
As you can see, in the second case, the output signal differs from the signal at the output of the first system by the value of the initial value, despite the fact that both systems have the same transfer function.
Conclusion
This section considers the dynamic characteristics of typical links that make up control systems of arbitrary configuration. The features of block diagrams constructed on the basis of transfer functions and differential equations are discussed. Two ways of transition from the transfer function of the system through block diagrams to its models in the form of state variables corresponding to various canonical forms are given.
It should be noted that the representation of the system in the form of a block diagram allows in some cases to evaluate its statics and dynamics and, in essence, gives a structural portrait of the system.
3.1. Draw a block diagram of the system, the differential equation of which has the form:
a) 
in) 
3.2. Draw a block diagram of the system, the model of which is presented in state variables:
a) 
b) 
in) 
G) 
3.3. Determine the transfer functions of systems if their block diagrams have the form shown in fig. 3.49.
Rice. 3.49. Block diagrams for task 3.3
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3.4. Block diagrams of the system are known (Fig. 3.50). Record their models in state variables.
Rice. 3.50. Block diagrams for task 3.4
3.5. The block diagram of the system is known (Fig. 3.51).
Rice. 3.51.
1. Determine the transfer function under the assumption that
2. Determine the transfer function assuming
3. Write the system model in state variables.
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4. Repeat paragraphs. 1 and 2 for the system whose block diagram is shown in fig. 3.52.
Rice. 3.52. Block diagram for task 3.5
3.6 .
3.7. Draw a block diagram corresponding to the first canonical form of describing a system that has a transfer function
1. Write down the first canonical form.
2. Draw a block diagram corresponding to the second canonical form of the system description.
3. Write down the second canonical form.
3.8. Draw a block diagram corresponding to the first canonical form of describing a system that has a transfer function
1. Write down the first canonical form.
2. Draw a block diagram corresponding to the second canonical form of the system description.
3. Write down the second canonical form.
Literature
1. Andreev Yu.N. Control of finite-dimensional linear objects. - M.: Nauka, 1978.
2. Besekersky V.A..,Popov E.P.. Theory of automatic control. - M.: Nauka, 1974.
3. Erofeev A. A. Theory of automatic control. - St. Petersburg: Poly-technics, 1998.
4. Ivashchenko N.N. Automatic regulation. - M.: Mashinostroenie, 1978.
5. Pervozvansky A.A. Course of the theory of automatic control. - M.: Higher. school, 1986.
6. Popov E.P. Theory of linear systems of automatic regulation and control. - M.: Higher. school, 1989.
7. Konovalov G.F. Radioautomatics. - M.: Higher. school, 1990.
8. Philips Ch.,Harbor R. Feedback control systems. - M.: Basic Knowledge Laboratory, 2001.
LINEAR SYSTEMS
AUTOMATIC CONTROL
Publishing house OmSTU
Ministry of Education and Science of the Russian Federation
State educational institution
higher professional education
"Omsk State Technical University"
LINEAR SYSTEMS
AUTOMATIC CONTROL
Methodical instructions for practical work
Publishing house OmSTU
Compiler E. V. Shendaleva, cand. tech. Sciences
The publication contains guidelines for practical work on the theory of automatic control.
It is intended for students of the specialty 200503, "Standardization and Certification", studying the discipline "Fundamentals of Automatic Control".
Published by decision of the editorial and publishing council
Omsk State Technical University
© GOU VPO "Omsk State
Technical University", 2011
The need to use the methodology of management theory for standardization and certification specialists arises when determining:
1) quantitative and (or) qualitative characteristics of the properties of the test object as a result of the influence on it during its operation, when modeling the object and (or) influences, the law of change of which must be provided with the help of an automatic control system;
2) dynamic properties of the object of measurements and tests;
3) the influence of the dynamic properties of measuring instruments on the results of measurements and tests of the object.
Methods for studying objects are considered in practical works.
Practical work 1
Dynamic features
Exercise 1.1
Find weight function w(t) by the known transition function
h(t) = 2(1–e –0.2 t).
Decision
w(t)=h¢( t), so when differentiating the original expression
w(t)=0.4e –0.2 t .
Exercise 1.2
Find the transfer function of the system from the differential equation 4 y¢¢( t) + 2y¢( t) + 10y(t) = 5x(t). The initial conditions are zero.
Decision
The differential equation is converted into standard form by dividing by the coefficient with the term y(t)
0,4y¢¢( t) + 0,2y¢( t) + y(t) = 0,5x(t).
The resulting equation is transformed according to Laplace
0,4s 2 y(s) + 0,2sy(s) + y(s) = 0,5x(s)
and then written as a transfer function:
where s= a + i w is the Laplace operator.
Exercise 1.3
Find Transfer Function W(s) of the system with respect to the known weight function w(t)=5–t.
Decision
Laplace transform
. (1.1)
Using the relationship between transfer function and weight function W(s) = w(s), we get
.
The Laplace transform can be obtained by calculation (1.1), using Laplace transform tables or using the Matlab software package. The program in Matlab is given below.
syms s t
x=5-t% time function
y=laplace(x)% is a Laplace-transformed function.
Exercise 1.4
Using the transfer function of the system, find its response to a single step action (transition function)
.
Decision
Inverse Laplace Transform
, (1.2)
where c is the abscissa of convergence x(s).
According to the principle of superposition, valid for linear systems
h(t)=h 1 (t)+h 2 (t),
where h(t) is the transition function of the entire system;
h 1 (t) is the transition function of the integrating link
;
h 2 (t) is the transient function of the amplifying link
.
It is known that h 1 (t)=k 1× t, h 2 (t)=k 2×δ( t), then h(t)=k 1× t+k 2×δ( t).
The inverse Laplace transform can be obtained by calculation (1.2), using the Laplace transform tables or using the Matlab software package. The program in Matlab is given below.
syms s k1 k2% notation for symbolic variables
y=k1/s+k2% Laplace-transformed function
x=ilaplace(y)% is a temporary function.
Exercise 1.5
Find the amplitude-frequency and phase-frequency characteristics from the known transfer function of the system
.
Decision
To determine the amplitude-frequency (AFC) and phase-frequency characteristics (PFC), it is necessary to move from the transfer function to the amplitude-phase characteristic W(i w) why change the argument s → i w
.
Then represent the AFC in the form W(i w)= P(w)+ iQ(w), where P(w) is the real part, Q(w) is the imaginary part of the AFC. To obtain the real and imaginary parts of the AFC, it is necessary to multiply the numerator and denominator by a complex number conjugate to the expression in the denominator:
AFC and PFC are determined, respectively, by the formulas
, 
;
, 
Amplitude-phase characteristic W(j w) can be represented as
.
Exercise 1.6
Define Signal y(t) at the output of the system according to the known input signal and the transfer function of the system
x(t)=2sin10 t; 
.
It is known that when exposed to the input signal x(t)=B sinw t per system output signal y(t) will also be harmonic, but will differ from the input amplitude and phase
y(t) = B× A(w)sin,
where A(w) – frequency response of the system; j(w) - PFC of the system.
By the transfer function, we determine the frequency response and phase response
j(w)=- arctg0,1w.
At frequency w = 10s –1 A(10) = 4/ = 2 and j(10) = –arctg1=–0.25p.
Then y(t) = 2×2 sin(10 t-0.25p) = 4 sin(10 t-0.25p).
test questions:
1. Define the concept of a weight function.
2. Define the concept of a transition function.
3. What is the purpose of using the Laplace transform when describing dynamic links?
4. What equations are called linear differential?
5. For what purpose, when passing to an equation in operator form, is the original differential equation converted into a standard form?
6. How is the expression with an imaginary number eliminated from the denominator of the amplitude-phase characteristic?
7. Specify the direct Laplace transform command in the Matlab software package.
8. Specify the inverse Laplace transform command in the Matlab software package.
Practical work 2
Transfer functions
Exercise 2.1
Find the transfer function of the system according to its block diagram.
Decision
The main ways of connecting links in block diagrams are: parallel, serial and connection of links with feedback (typical sections of links).
The transfer function of a system of parallel connected links is equal to the sum of the transfer functions of individual links (Fig. 2.1)
. (2.1)
Rice. 2.1. Parallel connection of links
The transfer function of a system of series-connected links is equal to the product of the transfer functions of individual links (Fig. 2.2)
(2.2)
Rice. 2.2. Serial connection of links
Feedback is the transfer of a signal from the output of a link to its input, where the feedback signal is algebraically summed with an external signal (Fig. 2.3).
Rice. 2.3 Connection with feedback: a) positive, b) negative
Transfer function of positive feedback connection
, (2.3)
negative feedback connection transfer function
. (2.4)
The transfer function of a complex control system is determined step by step. To do this, select sections containing serial, parallel connections and connections with feedback (typical sections of links) (Fig. 2.4)
W 34 (s)=W 3 (s)+W 4 (s); 
.
Rice. 2.4. Structural diagram of the control system
Then the selected typical section of the links is replaced by one link with the calculated transfer function and the calculation procedure is repeated (Fig. 2.5 - 2.7).
Rice. 2.5. Replacing the Parallel Connection and the Feedback Connection with a Single Link
Rice. 2.6. Replacing a feedback connection with a single link
Rice. 2.7. Replacing a serial connection with a single link
(2.5)
Exercise 2.2
Determine the transfer function if the transfer functions of the links included in it:
Decision
When substituting into (2.5) the transfer functions of the links
The transformation of the block diagram with respect to the input control action (Fig. 2.7, 2.11) can be obtained by calculation (2.5) or using the Matlab software package. The program in Matlab is given below.
W1=tf(,)% Transmission function W 1
W2=tf(,)% Transmission function W 2
W3=tf(,)% Transmission function W 3
W4=tf(,)% Transmission function W 4
W5=tf(,)% Transmission function W 5
W34=parallel(W3,W4)% parallel connection ( W 3 + W 4)
W25=feedback(W2,W5)
W134=feedback(W1,W34)% negative feedback
W12345=series(W134,W25)% serial connection ( W 134× W 25)
W=feedback(W12345,1)
Exercise 2.3.
Find the transfer function of a closed system by the perturbing action
Decision
In order to determine the transfer function of a complex system by a disturbing action, it is necessary to simplify it and consider it relative to the disturbing input action (Fig. 2.8 - 2.12).
Fig.2.8. The initial block diagram of the automatic system
Rice. 2.9. Block diagram simplification
Rice. 2.10. Simplified block diagram
Rice. 2.11. Structural diagram relative to the input control action
Rice. 2.12. Structural diagram of the system with respect to the disturbing action
After bringing the block diagram to a single-loop transfer function for the perturbing action f(t)
(2.6)
The transformation of the block diagram with respect to the perturbing action (Fig. 2.12) can be obtained by calculation (2.6) or using the Matlab software package.
W1=tf(,)% Transmission function W 1
W2=tf(,)% Transmission function W 2
W3=tf(,)% Transmission function W 3
W4=tf(,)% Transmission function W 4
W5=tf(,)% Transmission function W 5
W34=parallel(W3,W4)% parallel connection
W25=feedback(W2,W5)% negative feedback
W134=feedback(W1,W34)% negative feedback
Wf=feedback(W25,W134)% negative feedback.
Exercise 2. 4
Determine the closed-loop transfer function for the error.
Decision
A block diagram for determining the transfer function of a closed system for a control error is shown in fig. 2.13.
Rice. 2.13. Structural diagram of the system in relation to control error
Closed-loop transfer function for error
(2.7)
When substituting numerical values
The transformation of the block diagram with respect to the control error signal (Fig. 2.13) can be obtained by calculation (2.7) or using the Matlab software package.
W1=tf(,)% Transmission function W 1
W2=tf(,)% Transmission function W 2
W3=tf(,)% Transmission function W 3
W4=tf(,)% Transmission function W 4
W5=tf(,)% Transmission function W 5
W34=parallel(W3,W4)% parallel connection)
W25=feedback(W2,W5)% negative feedback
W134=feedback(W1,W34)% negative feedback
We=feedback(1,W134*W25)% negative feedback
test questions:
1. List the main ways of connecting links in block diagrams.
2. Determine the transfer function of the system of parallel connected links.
3. Determine the transfer function of the system of series-connected links.
4. Determine the transfer function with positive feedback.
5. Determine the negative feedback transfer function.
6. Determine the transfer function of the communication line.
7. Which Matlab command is used to determine the transfer function of two parallel connected links?
8. Which Matlab command is used to determine the transfer function of two serially connected links?
9. Which Matlab command is used to determine the transfer function of a link covered by feedback?
10. Draw a block diagram of the system to determine the transfer function for the control action.
11. Write the transfer function for the control action.
12. Draw a block diagram of the system for determining the transfer function from the perturbing parameter.
13. Write the transfer function for the perturbing parameter.
14. Draw a block diagram of the system for determining the transfer function for control error.
15. Write the transfer function for the control error.
Practical work 3
Complex Transfer Function Decomposition
The Laplace transformation of the DE makes it possible to introduce a convenient concept of the transfer function characterizing 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 converted 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 the ratio of the image of the output action 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,
А(s) = a 0 + a 1 s + a 2 s 2 + … + a n s n is the denominator polynomial.
The transfer function has an order, which 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
The link of the system is its element, which has certain properties in a dynamic sense. The links of control systems can have a different physical nature (electrical, pneumatic, mechanical, etc. links), but they can be described by the same control, and the ratio of input and output signals in the links can be described by the same transfer functions.
In TAU, a group of the simplest links is distinguished, which are usually called typical. The static and dynamic characteristics of standard links have been studied quite fully. Typical links are widely used in determining the dynamic characteristics of control objects. For example, knowing the transient response built using a recording device, it is often possible to determine what type of links the control object belongs to, and, consequently, its transfer function, differential equation, etc., i.e. object model. Typical links Any complex link can be represented as a combination of the simplest links.
The simplest typical links include:
amplifying,
inertial (aperiodic of the 1st order),
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 \u003d K * x, the transfer function W (s) \u003d 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).
Under step action h(t) = K.
Examples of such links are: mechanical transmissions, sensors, inertialess amplifiers, etc.
2) Integrating.
2.1) Ideal integrator.
The output value of an ideal integrator is proportional to the integral of the input value:
; W(s) =
When a stepped 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 integrator.
The transfer function of this link has the form
The transient response, in contrast to the 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 action, and the rotor rotation angle is taken as the output action. If voltage is not applied 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 up, and the angle of its rotation at first slowly due to inertia, and then increase rapidly until a certain rotation speed is reached.
3) Differentiating.
3.1) The ideal differentiator.
The output value is proportional to the time derivative of the input:
With a stepped input, the output is a pulse (d-function): h(t) = K . d(t).
3.2) Real differentiating.
Ideal differentiating links are not physically realizable. Most of the objects that are differentiating links refer to real differentiating links, the transfer functions of which have the form
Transient response: .
Link example: electric generator. The input parameter is the angle of rotation of the rotor, the output parameter is voltage. If the rotor is rotated a certain angle, voltage will appear on the terminals, but if the rotor is not rotated further, the voltage will drop to zero. It cannot fall 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's determine the nature of the change in the output value of this link when a step action of the value x 0 is applied to the input.
Step action image: X(s) = . Then the image of the output quantity:
Y(s) = W(s) X(s) = = K x 0 .
Let's decompose the fraction into simple 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) Links of the second order
The links have DU and PF of the form
,
W(s) = 
.
When a stepped action 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, the links of the second order are distinguished:
aperiodic 2nd order (T 1 ³ 2T 2),
inertial (T 1< 2Т 2),
conservative (T 1 \u003d 0).
6) Delayed.
If, when a certain signal is applied to the input of an object, it does not respond to this signal immediately, but after some time, then the object is said to have a delay.
Lag is the time interval from the moment the input signal changes to the start of the output signal change.
A lagging link is a link whose 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 through the pipeline (how much liquid was pumped at the beginning of the pipeline, so much will be released at the end, but after a while, while the liquid moves through the pipe), the movement of cargo along the 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 is divided into links in order to simplify the analysis of functioning, after determining the transfer functions for each link, the task arises of combining them into one transfer function of the object. The type of the transfer function of the object depends on the sequence of connecting the links:
1) Serial connection.
W about \u003d W 1. W2. W 3 ...
When the links are connected in series, their transfer functions multiply.
2) Parallel connection.
W about \u003d W 1 + W 2 + W 3 + ...
When the links are connected in parallel, their transfer functions add up.
3) Feedback
Transfer function according to the task (x):
"+" corresponds to negative OS,
"-" - positive.
To determine the transfer functions of objects that have more complex connections of links, either sequential enlargement of the circuit is used, or they are converted according to the Meson formula.
Transfer functions of ASR
For research and calculation, the structural diagram of the ASR is brought to the simplest standard form “object - controller” by means of equivalent transformations (see Figure 1.27). Almost all engineering methods for calculating and determining the parameters of regulator settings are applied for such a standard structure.
In the general case, any one-dimensional ACP with main feedback can be reduced to this form by gradually increasing the links.
If the output of the system y is not applied 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 controller, W y - PF of the control object).
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This transfer function Ф з (s) determines the dependence of y on x and is called the transfer function of a closed system along the channel of the master influence (by assignment).
For ACP, there are also transfer functions for other channels:
Ф e (s) = = - by mistake,
Ф in (s) = = - by perturbation,
where W s.v. (s) is the transfer function of the control object over the perturbation transmission channel.
There are two options for taking into account perturbations:
The perturbation has an additive effect on the control action (see Figure 1.29, a);
The disturbance affects the measurements of the controlled parameter (see Figure 1.29, b).
An example of the first option can be the influence of voltage fluctuations in the network on the voltage supplied by the regulator to the heating element of the object. Example of the second option: errors in the measurements of the regulated parameter due to changes in the ambient temperature. W – model of the influence of the environment on measurements.
Figure 1.30
Parameters K 0 = 1, K 1 = 3, K 2 = 1.5, K 4 = 2, K 5 = 0.5.
In the block diagram of the ACP, 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 links 1, 2 and 3 belong to the regulator circuit, and links 4 and 5 belong to the object circuit.
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, so the transfer function of the control object is defined as the product of the transfer functions of the links:
Transfer function of an open system:
whence it can be seen that the numerator B(s) = 1.5. s 2 + 3 . s + 1, the denominator (aka the characteristic polynomial of an open system) A(s) = 2 . s 3 + 3 . s2 + s. Then the characteristic polynomial of the closed system is equal to:
D(s) = A(s) + B(s) = 2 . s 3 + 3 . s2 + s + 1.5. s 2 + 3 . s + 1 = 2 . s 3 + 4.5. s 2 + 4 . s + 1.
Transfer functions of a closed system:
on assignment 
,
by mistake 
.
When determining the transfer function from the perturbation, W r.v. = W oy. Then
. ¨
