Typical units of self-propelled guns and their characteristics. Typical units of self-propelled guns
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 → ∞. In this way, 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.
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, such parameters 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 region Y find number X from the region 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. Consequently, 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, only the unique 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):
In this way
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 for describing 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.
OTP BISN (KSN)
Purpose of work– the acquisition by students of practical skills in the use of methods for designing on-board integrated (complex) surveillance systems.
Laboratory work is carried out in a computer class.
Programming environment: MATLAB.
Airborne integrated (complex) surveillance systems are designed to solve the problems of search, detection, recognition, determining the coordinates of search objects, etc.
One of the main directions for increasing the efficiency of solving the set targets is the rational management of search resources.
In particular, if the carriers of the IOS are unmanned aerial vehicles (UAVs), then the management of search resources consists in planning the trajectories and controlling the flight of the UAV, as well as controlling the line of sight of the IOS, etc.
The solution of these problems is based on the theory of automatic control.
Typical links of the automatic control system (ACS)
Transmission function
In the theory of automatic control (TAU), the operator form of writing differential equations is often used. In this case, the concept of a differential operator is introduced p = d/dt so, dy/dt = py , but p n = d n /dt n . This is just another notation for the operation of differentiation.
The integration operation inverse to differentiation is written as 1/p . In operator form, the original differential equation is written as an algebraic one:
aop (n) y + a 1 p (n-1) y + ... + any = (aop (n) + a 1 p (n-1) + ... + an)y = (bop (m) + b 1 p (m-1) + ... + bm)u
This form of notation should not be confused with operational calculus, if only because time functions are directly used here y(t), u(t) (originals), not their Images Y(p), U(p) , obtained from the originals using the Laplace transform formula. At the same time, under zero initial conditions, up to notation, the entries are indeed very similar. This similarity lies in the nature of differential equations. Therefore, some rules of operational calculus are applicable to the operator form of the equation of dynamics. So operator p can be considered as a factor without the right to permutation, that is py yp. It can be taken out of brackets, etc.
Therefore, the equation of dynamics can also be written in the form:
Differential operator W(p) called transfer function. It determines the ratio of the output value of the link to the input at each moment of time: W(p) = y(t)/u(t) , that's why it's also called dynamic gain.
in steady state d/dt = 0, i.e p = 0, so the transfer function turns into the link transfer coefficient K = b m / a n .
Transfer function denominator D(p) = a o p n + a 1 p n - 1 + a 2 p n - 2 + ... + a n called characteristic polynomial. Its roots, i.e. the values of p for which the denominator D(p) goes to zero and W(p) tends to infinity is called transfer function poles.
Numerator K(p) = b o p m + b 1 p m - 1 + ... + b m called operator gain. Its roots, which K(p) = 0 And W(p) = 0, called transfer function zeros.
An ACS link with a known transfer function is called dynamic link. It is represented by a rectangle, inside which the expression of the transfer function is written. That is, this is an ordinary functional link, the function of which is given by the mathematical dependence of the output value on the input value in dynamic mode. For a link with two inputs and one output, two transfer functions must be written for each of the inputs. The transfer function is the main characteristic of the link in dynamic mode, from which all other characteristics can be obtained. It is determined only by the system parameters and does not depend on the input and output values. For example, one of the dynamic links is the integrator. Its transfer function W and (p) = 1/p. The ACS scheme, composed of dynamic links, is called structural.
Differentiator link
There are ideal and real differentiating links. Dynamic equation of an ideal link:
y(t) = k(du/dt), or y=kpu .
Here, the output quantity is proportional to the rate of change of the input quantity. Transmission function: W(p) = kp . At k = 1 the link performs a pure differentiation W(p) = p . Transient response: h(t) = k 1’(t) = d(t) .
An ideal differentiating link cannot be realized, since the magnitude of the surge in the output value when a single step action is applied to the input is always limited. In practice, real differentiating links are used that perform approximate differentiation of the input signal.
His equation: Tpy + y = kTpu .
Transmission function: W(p) = k(Tp/Tp + 1).
When a single step action is applied to the input, the output value is limited in magnitude and stretched in time (Fig. 5).
According to the transient response, which has the form of an exponential, it is possible to determine the transfer coefficient k and time constant T. Examples of such links can be a four-terminal network of resistance and capacitance or resistance and inductance, a damper, etc. Differentiating links are the main tool used to improve the dynamic properties of ACS.
In addition to those considered, there are a number of links, which we will not dwell on in detail. These include the ideal forcing link ( W(p) = Tp + 1 , practically unrealizable), a real forcing link (W(p) = (T 1 p + 1)/(T 2 p + 1) , at T1 >> T2 ), retarded link ( W(p) = e - pT ), reproducing input action with time delay and others.
Inertialess link
Transmission function:
AFC: W(j) = k.
Real frequency response (VCH): P() = k.
Imaginary frequency response (MFH): Q() = 0.
Amplitude-frequency characteristic (AFC): A() = k.
Phase frequency response (PFC): () = 0.
Logarithmic frequency response (LAFC): L() = 20lgk.
Some frequency responses are shown in Fig.7.
The link passes all frequencies equally with an increase in amplitude by k times and without phase shift.
Integrating link
Transmission function:
Consider the special case when k = 1, i.e.
AFC: W(j) = 
.
VCH: P() = 0.
MCH: Q() = - 1/ .
Frequency response: A() = 1/ .
PFC: () = - /2.
LAF: L() = 20lg(1/ ) = - 20lg().
The frequency response is shown in Fig. 8.
The link passes all frequencies with a phase delay of 90 degrees. The amplitude of the output signal increases with decreasing frequency, and decreases to zero with increasing frequency (the link "fills up" high frequencies). LAFC is a straight line passing through the point L() = 0 at = 1. With an increase in frequency per decade, the ordinate decreases by 20lg10 = 20 dB, that is, the slope of the LAFC is - 20 dB / dec (decibel per decade).
Aperiodic link
For k = 1, we obtain the following FH expressions:
W(p) = 1/(Tp + 1);
;
;
;
() = 1 - 2 = - arctg( T);
;
L() = 20lg(A()) = - 10lg(1 + ( T)2).
Here A1 and A2 are the amplitudes of the numerator and denominator of the LPFC; 1 and 2 are the numerator and denominator arguments. LPCH:
The frequency response is shown in Fig.9.
The AFC is a semicircle with a radius of 1/2 centered at the point P = 1/2. When constructing an asymptotic LAFC, it is considered that when< 1 = 1/T можно пренебречь ( T) 2 выражении для L(), то есть L() - 10lg1 = 0.. При >1 neglect the unit in the expression in brackets, that is, L(ω) - 20lg(ω T). Therefore, the LAFC passes along the abscissa to the corner frequency, then - at an angle - 20 dB / dec. The frequency ω 1 is called the corner frequency. The maximum difference between the real LAFC and the asymptotic ones does not exceed 3 dB at = 1 .
LPCH asymptotically tends to zero as ω decreases to zero (the lower the frequency, the less phase distortion of the signal) and to - /2 as it increases to infinity. Inflection point = 1 at () = - /4. The LPFC of all aperiodic links have the same shape and can be constructed from a typical curve with a parallel shift along the frequency axis.
Reporting Form
The electronic report must include:
1. Group, full name student
2. Name of laboratory work, topic, task option;
3. Schemes of typical links;
4. Results of calculations: transients, LAFC, for various parameters of links, graphics;
5. Conclusions on the results of calculations.
Laboratory work 2.
Compensation principle
If the disturbing factor distorts the output value to unacceptable limits, then apply compensation principle(Fig. 6, KU - corrective device).
Let be y about- the value of the output quantity, which is required to be provided according to the program. In fact, due to the perturbation f, the output registers the value y. Value e \u003d y o - y called deviation from the set value. If somehow it is possible to measure the value f, then the control action can be corrected u at the input of the op-amp, summing the CU signal with a corrective action proportional to the disturbance f and offset its effect.
Examples of compensation systems: a bimetallic pendulum in a clock, a compensation winding of a DC machine, etc. In Fig. 4, there is a thermal resistance in the heating element (NE) circuit R t , the value of which varies depending on temperature fluctuations environment, adjusting the voltage on the NO.
The virtue of the principle of compensation: quick response to disturbances. It is more accurate than the open loop principle. Flaw: the impossibility of taking into account all possible perturbations in this way.
Feedback principle
The most widely used in technology feedback principle(Fig.5).
Here, the control variable is corrected depending on the output value y(t). And it doesn't matter what perturbations act on the OS. If the value y(t) deviates from the required, then the signal is corrected u(t) to reduce this deviation. The connection between the output of an op-amp and its input is called main feedback (OS).
In a particular case (Fig. 6), the memory generates the required value of the output value y o (t), which is compared with the actual value at the output of the ACS y(t).
Deviation e = y o -y from the output of the comparing device is fed to the input regulator R, which combines UU, UO, CHE.
If e 0, then the controller generates the control action u(t), acting until equality is ensured e = 0, or y = y o. Since the difference of signals is applied to the regulator, such feedback is called negative, Unlike positive feedback when the signals are added.
Such a control in the deviation function is called regulation, and such an ACS is called automatic control system(SAR).
The disadvantage of the inverse principle connection is the inertia of the system. Therefore, it is often used combination of this principle with the principle of compensation, which allows you to combine the advantages of both principles: the speed of response to a disturbance of the compensation principle and the accuracy of regulation, regardless of the nature of the disturbances of the feedback principle.
The main types of ACS
Depending on the principle and law of the functioning of the memory, which sets the program for changing the output value, the main types of ACS are distinguished: stabilization systems, software, tracking And self-tuning systems, among which are extreme, optimal And adaptive systems.
IN stabilization systems a constant value of the controlled variable is ensured for all types of disturbances, i.e. y(t) = const. The memory generates a reference signal with which the output value is compared. The memory, as a rule, allows setting the reference signal, which allows you to change the value of the output quantity at will.
IN software systems a change in the controlled value is provided in accordance with the program generated by the memory. A cam mechanism, a punched tape or magnetic tape reader, etc. can be used as a memory. Clockwork toys, tape recorders, players, etc. can be attributed to this type of self-propelled guns. Distinguish systems with time program providing y = f(t), And systems with a spatial program, in which y = f(x), used where it is important to obtain the required trajectory in space at the output of the ACS, for example, in a copy machine (Fig. 7), the law of motion in time does not play a role here.
tracking systems differ from software programs only in that the program y = f(t) or y = f(x) unknown in advance. A device that monitors the change of some external parameter acts as a memory. These changes will determine the changes in the output value of the ACS. For example, a robot hand that mimics the movements of a human hand.
All three considered types of ACS can be built according to any of the three fundamental principles of control. They are characterized by the requirement that the output value coincide with some prescribed value at the ACS input, which itself can change. That is, at any moment in time, the required value of the output quantity is uniquely determined.
IN self-tuning systems The memory is looking for such a value of the controlled variable, which in some sense is optimal.
So in extreme systems(Fig. 8) it is required that the output value always takes an extreme value from all possible ones, which is not predetermined and can change unpredictably.
To find it, the system performs small trial movements and analyzes the response of the output value to these trials. After that, a control action is generated that brings the output value closer to the extreme value. The process is repeated continuously. Since the ACS data continuously evaluates the output parameter, they are performed only in accordance with the third control principle: the feedback principle.
Optimal systems are a more complex version of extremal systems. Here, as a rule, complex processing of information about the nature of the change in output values and disturbances, about the nature of the influence of control actions on output values occurs, theoretical information, information of a heuristic nature, etc. can be involved. Therefore, the main difference between extreme systems is the presence of computers. These systems can operate according to any of the three fundamental principles of control.
IN adaptive systems the possibility of automatic reconfiguration of parameters or changes in the ACS circuit diagram in order to adapt to changing external conditions is provided. Accordingly, there are self-tuning And self-organizing adaptive systems.
All types of ACS ensure that the output value matches the required value. The only difference is in the program for changing the required value. Therefore, the foundations of TAU are built on the analysis of the simplest systems: stabilization systems. Having learned to analyze the dynamic properties of ACS, we will take into account all the features of more complex types of ACS.
Static characteristics
The ACS operating mode, in which the controlled variable and all intermediate values do not change in time, is called established, or static mode. Any link and ACS as a whole in this mode is described equations of statics kind y = F(u,f) in which there is no time t. The corresponding graphs are called static characteristics. The static characteristic of a link with one input u can be represented by a curve y = F(u)(Fig. 9). If the link has a second perturbation input f, then the static characteristic is given by the family of curves y = F(u) at different values f, or y = F(f) at various u.
So an example of one of the functional links of the control system is a conventional lever (Fig. 10). The equation of statics for it has the form y = Ku. It can be represented as a link whose function is to amplify (or attenuate) the input signal in K once. Coefficient K = y/u, equal to the ratio of the output value to the input is called gain link. When the input and output quantities are of a different nature, it is called transmission ratio.
The static characteristic of this link has the form of a straight line segment with a slope a = arctg(L 2 /L 1) = arctg(K)(Fig. 11). Links with linear static characteristics are called linear. The static characteristics of real links are, as a rule, non-linear. Such links are called non-linear. They are characterized by the dependence of the transmission coefficient on the magnitude of the input signal: K = y/ u const.
For example, the static characteristic of a saturated DC generator is shown in Fig. 12. Usually, a non-linear characteristic cannot be expressed by any mathematical relationship and it has to be specified in a table or graph.
Knowing the static characteristics of individual links, it is possible to construct a static characteristic of the ACS (Fig. 13, 14). If all links of the ACS are linear, then the ACS has a linear static characteristic and is called linear. If at least one link is non-linear, then ACS nonlinear.
Links for which you can set a static characteristic in the form of a rigid functional dependence of the output value on the input are called static. If there is no such connection and each value of the input value corresponds to a set of values of the output value, then such a link is called astatic. Depicting its static characteristics is meaningless. An example of an astatic link is a motor whose input value is
voltage U, and the output - the angle of rotation of the shaft, the value of which at U = const can take any value.
The output value of the astatic link, even in steady state, is a function of time.
Lab 3
Dynamic mode of ACS
Equation of dynamics
The steady state is not typical for ACS. Usually, the controlled process is affected by various perturbations that deviate the controlled parameter from a given value. The process of establishing the desired value of the controlled variable is called regulation. Due to the inertia of the links, regulation cannot be carried out instantly.
Let us consider an automatic control system, which is in a steady state, characterized by the value of the output quantity y=yo. Let at the moment t = 0 any disturbing factor acted on the object, deviating the value of the controlled variable. After some time, the regulator will return the ACS to its original state (taking into account the static accuracy) (Fig. 1).
If the regulated value changes in time according to an aperiodic law, then the regulation process is called aperiodic.
With sharp disturbances, it is possible oscillatory damped process (Fig. 2a). There is also such a possibility that after some time T p undamped oscillations of the regulated value will be established in the system - undamped oscillatory process (Fig. 2b). The last view - divergent oscillatory process (Fig. 2c).
Thus, the main mode of operation of the ACS is considered dynamic mode, characterized by the flow in it transients. That's why the second main task in the development of ACS is the analysis of the dynamic modes of operation of ACS.
The behavior of the ACS or any of its links in dynamic modes is described dynamics equation y(t) = F(u,f,t), which describes the change in values over time. As a rule, this is a differential equation or a system of differential equations. That's why the main method for studying ACS in dynamic modes is the method of solving differential equations. The order of differential equations can be quite high, that is, both the input and output quantities themselves are dependent on the dependence u(t), f(t), y(t), and the rate of their change, acceleration, etc. Therefore, the equation of dynamics in general form can be written as follows:
F(y, y', y”,..., y (n) , u, u', u”,..., u (m) , f, f ', f ”,..., f ( k)) = 0.
To a linearized ACS, you can apply superposition principle: the reaction of the system to several simultaneously acting input actions is equal to the sum of the reactions to each action separately. This allows a link with two inputs u And f decompose into two links, each of which has one input and one output (Fig. 3).
Therefore, in the future, we will restrict ourselves to studying the behavior of systems and links with one input, the equation of dynamics of which has the form:
a o y (n) + a 1 y (n-1) + ... + a n - 1 y' + a n y = b o u (m) + ... + b m - 1u' + b m u.
This equation describes the ACS in the dynamic mode only approximately with the accuracy given by the linearization. However, it should be remembered that linearization is possible only with sufficiently small deviations of the values and in the absence of discontinuities in the function F in the vicinity of the point of interest to us, which can be created by various switches, relays, etc.
Usually n m, because at n< m ACS is technically unrealizable.
Structural diagrams of ACS
Equivalent transformations of block diagrams
The block diagram of the ACS in the simplest case is built from elementary dynamic links. But several elementary links can be replaced by one link with a complex transfer function. For this, there are rules for the equivalent transformation of block diagrams. Consider possible ways transformations.
1. serial connection(Fig. 4) - the output value of the previous link is fed to the input of the next one. In this case, you can write:
y 1 = W 1 y o ; y 2 \u003d W 2 y 1; ...; y n = W n y n - 1 =>
y n \u003d W 1 W 2 ..... W n .y o \u003d W eq y o,
where 
.
That is, a chain of serially connected links is converted into an equivalent link with a transfer function equal to the product of the transfer functions of individual links.
2. Parallel - consonant compound(Fig. 5) - the same signal is applied to the input of each link, and the output signals are added. Then:
y \u003d y 1 + y 2 + ... + y n \u003d (W 1 + W 2 + ... + W3) y o \u003d W eq y o,
where 
.
That is, a chain of links connected in parallel - according to, is converted into a link with a transfer function equal to the sum of the transfer functions of individual links.
3. Parallel - counter connection(Fig. 6a) - the link is covered by positive or negative feedback. The section of the circuit along which the signal goes in the opposite direction with respect to the system as a whole (that is, from output to input) is called feedback loop with transfer function W os. In this case, for a negative OS:
y = W p u; y 1 = W os y; u = y o - y 1 ,
Consequently
y = W p y o - W p y 1 = W p y o - W p W oc y = >
y(1 + W p W oc) = W p y o = > y = W eq y o ,
where 
.
Similarly: 
- for positive OS.
If Woc = 1, then the feedback is called unit (Fig. 6b), then W equiv \u003d W p / (1 ± W p).
A closed system is called single-loop if, when it is opened at any point, a chain of series-connected elements is obtained (Fig. 7a).
The section of the circuit, consisting of series-connected links, connecting the point of application of the input signal with the point of removal of the output signal is called straight circuit (Fig. 7b, transfer function of the direct circuit W p \u003d Wo W 1 W 2). A chain of series-connected links included in a closed circuit is called open circuit(Fig. 7c, open circuit transfer function W p = W 1 W 2 W 3 W 4). Based on the above methods of equivalent transformation of block diagrams, a single-loop system can be represented by one link with a transfer function: W equiv \u003d W p / (1 ± W p)- the transfer function of a single-circuit closed system with negative feedback is equal to the transfer function of the forward circuit divided by one plus the transfer function of the open circuit. For a positive OS, the denominator has a minus sign. If you change the point of removal of the output signal, then the form of the direct circuit changes. So, if we consider the output signal y 1 at the link output W 1, then W p = Wo W 1. The expression for the open circuit transfer function is independent of the point at which the output signal is taken.
Closed systems are single-loop And multiloop(Fig. 8). To find the equivalent transfer function for a given circuit, you must first transform individual sections.
If a multi-loop system has cross links(Fig. 9), then to calculate the equivalent transfer function, you need additional rules:
4. When transferring the adder through a link along the signal path, it is necessary to add a link with the transfer function of the link through which the adder is transferred. If the adder is transferred against the signal path, then a link with a transfer function is added, the inverse transfer function of the link through which we transfer the adder (Fig. 10).
So, the signal is taken from the output of the system in Fig. 10a
y 2 = (f + y o W 1)W 2 .
The same signal should be taken from the outputs of the systems in Fig. 10b:
y 2 \u003d fW 2 + y o W 1 W 2 \u003d (f + y o W 1)W 2,
and in Fig.10c:
y 2 = (f(1/W 1) + y o)W 1 W 2 = (f + y o W 1)W 2 .
With such transformations, nonequivalent sections of the communication line may appear (they are shaded in the figures).
5. When transferring a node through a link along the signal path, a link is added with a transfer function, the inverse transfer function of the link through which we transfer the node. If the node is transferred against the signal path, then a link is added with the transfer function of the link through which the node is transferred (Fig. 11). So, the signal is taken from the output of the system in Fig. 11a
y 1 = y o W 1 .
The same signal is taken from the outputs of Fig. 11b:
y 1 \u003d y o W 1 W 2 / W 2 \u003d y o W 1
y 1 = y o W 1 .
6. Mutual permutations of nodes and adders are possible: nodes can be interchanged (Fig. 12a); adders can also be interchanged (Fig. 12b); when transferring the node through the adder, it is necessary to add a comparing element (Fig. 12c: y \u003d y 1 + f 1 \u003d\u003e y 1 \u003d y - f 1) or adder (Fig. 12d: y = y1 + f1).
In all cases of transfer of elements of the block diagram, there are non-equivalent regions communication lines, so you need to be careful in places where the output signal is picked up.
With equivalent transformations of the same block diagram, different transfer functions of the system can be obtained for different inputs and outputs.
Lab 4
Laws of regulation
Let some ACS be given (Fig. 3).
The law of regulation is a mathematical dependence, according to which the control action on the object would be produced by a non-inertial regulator.
The simplest of them is proportional law of regulation, at which
u(t) = Ke(t)(Fig. 4a),
where u(t) is the control action generated by the regulator, e(t)- deviation of the controlled value from the required value, K- coefficient of proportionality of the regulator Р.
That is, to create a control action, it is necessary to have a control error and that the value of this error be proportional to the disturbing effect f(t). In other words, the ACS as a whole should be static.
These regulators are called P-regulators.
Since when a disturbance affects the control object, the controlled variable deviates from the required value at a finite speed (Fig. 4b), at the initial moment a very small value e is applied to the controller input, causing weak control actions u. To increase the speed of the system, it is desirable to force the control process.
To do this, links are introduced into the controller that form at the output a signal proportional to the derivative of the input value, that is, differentiating or forcing links.
Such a regulation is called about
STRUCTURAL SCHEMES OF LINE ACS
Typical links of linear ACS
Any complex ACS can be represented as a set of more simple elements(remember functional And block diagrams). Therefore, to simplify the study of processes in real systems they are presented as a set idealized schemes, which are exactly described mathematically and approximately characterize real links systems in a certain range of signal frequencies.
When compiling block diagrams some typical elementary links(simple, further indivisible), characterized only by their transfer functions, regardless of their design, purpose and principle of operation. Classify them by type equations describing their work. In the case of linear ACS, the following are distinguished link types:
1. Described by linear algebraic equations with respect to the output signal:
but) proportional(static, inertialess);
b) delayed.
2. Described by differential equations of the first order with constant coefficients:
but) differentiating;
b) inertial-differentiating(real differentiating);
in) inertial(aperiodic);
G) integrating(astatic);
e) integro-differentiating(elastic).
3. Described by second-order differential equations with constant coefficients:
but) inertial link of the second order(aperiodic link of the second order, oscillatory).
Using the mathematical apparatus outlined above, consider transfer functions, transitional And pulse transient(by weight) characteristics, as well as frequency characteristics these links.
Here are the formulas that will be used for this purpose.
1. Transmission function: .
2. Step response: 
.
3. : or 
.
4. KCHH: .
5. Amplitude frequency response: ,
where 
, 
.
6. Phase frequency response: 
.
According to this scheme, we study typical links.
Note that although for some typical links n(order of derivative output parameter on the left side of the equation) equals m(order of derivative input parameter on the right side of the equation), not more m, as mentioned earlier, however, when constructing real ACS from these links, the condition m
proportional(static , inertialess ) link . This is the simplest link, output signal which is directly proportional input signal:
where k- coefficient of proportionality or link transfer.
Examples of such a link are: a) valves with linearized characteristics (when the change fluid flow in proportion to the degree of change stem position) in the above examples of control systems; b) voltage divider; c) leverage, etc.
Passing in (3.1) to images, we have:
1. Transmission function: .
2. Step response: , Consequently .
3. impulse response: 
.
4. KCHH: .
6. PFC: 
.
Accepted description of the relationship between entrance And way out only valid for perfect link and corresponds real links only when low frequencies, . When in real links, the transfer coefficient k begins to depend on the frequency and high frequencies drops to zero.
lagging link. This link is described by the equation
where is the delay time.
An example lagging link serve: a) long electric lines without losses; b) long pipeline, etc.
Transmission function, transitional and pulse transient characteristic, CFC, as well as frequency response and phase response of this link:
2. means: .
Figure 3.1 shows: a) KCHH hodograph lagging link; b) AFC and PFC of the retarded link. Note that when increasing, the end of the vector describes an ever-increasing angle in a clockwise direction.
Fig.3.1. Hodograph (a) and AFC, PFC (b) of the retarded link.
Integrating link. This link is described by the equation
where is the link transfer coefficient.
Examples of real elements whose equivalent circuits are reduced to integrator, are: a) an electric capacitor, if we consider input signal current, and weekend- voltage on the capacitor: 
; b) a rotating shaft, if you count input signal angular velocity of rotation, and the output - the angle of rotation of the shaft: 
; etc.
Let's define the characteristics of this link:
2. 
.
We use the Laplace transform table 3.1, we get:
.
We multiply by since the function at .
3. 
.
4. 
.
Figure 3.2 shows: a) hodograph of the CFC of the integrating link; b) frequency response and phase response of the link; c) transient response of the link.
Fig.3.2. Hodograph (a), frequency response and phase response (b), transient response (c) of the integrating link.
Differentiator link. This link is described by the equation
where is the link transfer coefficient.
Let's find the characteristics of the link:
2. 
, considering that , we find: .
3. 
.
4. 
.
Figure 3.3 shows: a) link hodograph; b) frequency response and phase response of the link.
but) b)
Rice. 3.3. Hodograph (a), frequency response and phase response (b) of the differentiating link.
An example differentiating link are ideal capacitor And inductance. This follows from the fact that the voltage u and current i tied for capacitor FROM and inductance L according to the following relations:
Note that real capacity has a small capacitive inductance, real inductance It has interturn capacitance(which are especially pronounced at high frequencies), which brings the above formulas to the following form:
, 
.
In this way, differentiator can not be technically implemented, because order the right side of his equation (3.4) is greater than the order of the left side. And we know that the condition must be satisfied n>m or, at the very least, n=m.
However, one can approach this equation given link, using inertial-differentiating(real differentiating)link.
Inertial-differentiating(real differentiating ) link is described by the equation:
where k- link transfer coefficient, T- time constant.
Transmission function, transitional And impulse response, CFC, AFC and PFC of this link are determined by the formulas:
We use the property of the Laplace transform - image shift(3.20), according to which: if , then .
From here: 
.
3. 
.
5. 
.
6. 
.
Figure 3.4 shows: a) CFC graph; b) frequency response and phase response of the link.
but) b)
Fig.3.4. Hodograph (a), frequency response and phase response of a real differentiating link.
In order for the properties real differentiator close to properties ideal, it is necessary to simultaneously increase the transmission coefficient k and decrease the time constant T so that their product remains constant:
kT= k d,
where k e is the transfer coefficient of the differentiating link.
This shows that in the dimension of the transmission coefficient k d differentiating link included time.
Inertial link of the first order(aperiodic link ) is one of the most common links ACS. It is described by the equation:
where k– link transfer coefficient, T is the time constant.
The characteristics of this link are determined by the formulas:
2. 
.
Using properties integration of the original And image shift we have:
.
3. 
, because 
at , then on the entire time axis this function is equal to 0 (at ).
5. 
.
6. 
.
Figure 3.5 shows: a) CFC graph; b) frequency response and phase response of the link.
Fig.3.5. Hodograph (a), frequency response and phase response of the inertial link of the first order.
Integro-differentiating link. This link is described by a first-order differential equation in the most general form:
where k- link transfer coefficient, T 1 And T 2- time constants.
Let's introduce the notation:
Depending on the value t the link will have different properties. If , then link its properties will approach integrating And inertial links. If , then the given link properties will be closer to differentiating And inertial-differentiating.
Let's define the characteristics integro-differentiating link:
1. 
.
2. 
, this implies:
Because 
at t® 0, then:
.
6. 
.
In Fig.3.6. given: a) CFC chart; b) frequency response; c) PFC; d) transient response of the link.
but) b)
in) G)
Fig.3.6. Hodograph (a), frequency response (b), phase response (c), transient response (d) of the integro-differentiating link.
Inertial link of the second order. This link is described by a second-order differential equation:
where (kapa) is the damping constant; T- time constant, k- link transfer coefficient.
The response of the system described by equation (3.8) to a single step action at is damped harmonic oscillations, in this case the link is also called oscillatory . When vibrations do not occur, and link described by equation (3.8) is called aperiodic link of the second order . If , then the oscillations will be undamped with frequency.
An example of the constructive implementation of this link can serve as: a) an electric oscillatory circuit containing capacity, inductance and ohmic resistance; b) weight suspended on spring and having damping device, etc.
Let's define the characteristics inertial link of the second order:
1. 
.
2. 
.
Roots characteristic equation standing in the denominator are determined:
.
Obviously, there are three possible cases here:
1) for the roots of the characteristic equation negative real miscellaneous and , then the transient response is determined by:
;
2) for the roots of the characteristic equation negative reals are the same 
:
3) at , the roots of the characteristic equation of the link are complex-conjugated 
, and
the transient response is determined by the formula:
,
i.e., as noted above, it acquires oscillatory character.
3. We also have three cases:
1) 
,
because 
at ;
2) , because at ;
3) 
, because 
at .
5. 
.
Typical dynamic links and their characteristics
dynamic link is called an element of the system that has certain dynamic properties.
Any system can be represented as a limited set of typical elementary links, which can be of any nature, design and purpose. The transfer function of any system can be represented as a fractional-rational function:
(1)Thus, the transfer function of any system can be represented as a product of prime factors and simple fractions. Links, the transfer functions of which are in the form of simple factors or simple fractions, are called typical or elementary links. Typical links differ in the form of their transfer function, which determines their static and dynamic properties.
As can be seen from the decomposition, the following links can be distinguished:
1. Amplifying (inertialess).
2. Differentiating.
3. Forcing link of the 1st order.
4. Forcing link of the 2nd order.
5. Integrating.
6. Aperiodic (inertial).
7. Vibrational.
8. Delayed.
When studying automatic control systems, it is presented as a set of elements not according to their functional purpose or physical nature, but according to their dynamic properties. To build control systems, it is necessary to know the characteristics of typical links. The main characteristics of the links are the differential equation and the transfer function.
Consider the main links and their characteristics.
Reinforcing link(inertialess, proportional). An amplifying link is called, which is described by the equation:
or transfer function:
(3)In this case, the transient function of the amplifying link (Fig. 1a) and its weight function (Fig. 1b), respectively, have the form:
The frequency characteristics of the link (Fig. 2) can be obtained from its transfer function, while the AFC, AFC and PFC are determined by the following relationships:
.
The logarithmic frequency response of the amplifying link (Fig. 3) is determined by the relation
.Link examples:
1. Amplifiers, for example, direct current (Fig. 4a).
2. Potentiometer (Fig. 4b).
3. Reducer (Fig. 5).
The logarithmic frequency characteristics of the link (Fig. 8) are determined by the formula
These are asymptotic logarithmic characteristics, the true characteristic coincides with it in the region of high and low frequencies, and the maximum error will be at the point corresponding to the associated frequency, and is equal to about 3 dB. In practice, asymptotic characteristics are usually used. Their main advantage is that when changing the system parameters ( k And T) characteristics move parallel to themselves.
Link examples:
1. An aperiodic link can be implemented on operational amplifiers(Fig. 9).
ÆÆ
