Typical dynamic links of automatic control systems. Typical links of self-propelled guns Elementary dynamic links
What is a dynamic link? In previous lessons, we looked at individual parts of the automatic control system and called them elements automatic control systems. Elements may have different physical appearance and design. The main thing is that such elements are supplied with some input signal x( t ) , and as a response to this input signal, the control system element generates some output signal y( t ) . We further established that the relationship between the output and input signals is determined by dynamic properties control elements, which can be represented as transfer function W(s). So, 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) of the self-propelled gun.
Typical dynamic links– this is the minimum required set of links to describe a control system of any type. Typical links include:
proportional link;
aperiodic link of the first order;
aperiodic link of the second order;
oscillatory link;
integrating link;
ideal differentiating link;
1st order forcing link;
second-order forcing link;
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 gain.
The proportional link is described by the algebraic equation:
y(t) = K· X(t)
Examples of such proportional links include 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 weighting function of the proportional link is equal to:
w(t) = L -1 = K·δ(t)
Rice. 3.5. Transition function, weight function, AFC and proportional frequency response .
6. Frequency characteristics .
Let's find the AFC, AFC, PFC and LAC of the proportional link:
W(jω ) = K = K +0·j
A(ω
)
=
= K
φ(ω) = arctan(0/K) = 0
L(ω) = 20 lg = 20 lg(K)
As follows from the presented results, the amplitude of the output signal does not depend on frequency. In reality, not a single link is able to uniformly pass all frequencies from 0 to ¥; as a rule, at high frequencies, the gain becomes smaller and tends to zero as ω → ∞. Thus, the mathematical model of the 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 first order has the form:
W(s) = K/(T· s + 1)
where K is the gain; T – time constant characterizing the inertia of the system, i.e. the duration of the transition process in it. Because the the time constant characterizes a certain time interval , then its value must always be positive, i.e. (T > 0).
2. Mathematical description of the link.
An aperiodic link of the first 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 first order can be: an electric RC filter; thermoelectric converter; compressed gas tank, etc.
4. Transition function .
The transition function of the first-order aperiodic link has the form:
h(t) = L -1 = L -1 = K – K e -t/T = K·(1 – e -t/T )
Rice. 3.6. Transition characteristic of an aperiodic link of the 1st order.
The transition process of the aperiodic link of the first order has an exponential form. The steady-state value is: h set = K. The tangent at the point t = 0 intersects the line of the steady-state value at the point t = T. At the time t = T, the transition function takes the value: h(T) ≈ 0.632·K, i.e. during 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 first order. As is known from the previous lecture, the control time is the time after which the difference between the current and steady values will not exceed a certain specified small value Δ. (Typically, Δ is set to 5% of the steady-state value.)
h(T y) = (1 – Δ) h mouth = (1 – Δ) K = K (1 – e - T y/ T), hence e - T y/ T = Δ, then T y / T = -ln(Δ), As a result, we obtain T y = [-ln(Δ)]·T.
At Δ = 0.05 T y = - ln(0.05) T ≈ 3 T.
In other words, the time of the transition process of the aperiodic link of the first order is approximately 3 times the time constant.
Typical dynamic links and their characteristics
Dynamic link An element of a system that has certain dynamic properties is called.
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 the product of prime factors and simple fractions. Links whose transfer functions are in the form of simple factors or simple fractions are called standard or elementary links. Typical links differ in the type 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. Reinforcing (inertia-free).
2. Differentiating.
3. Forcing link of the 1st order.
4. Forcing link of the 2nd order.
5. Integrating.
6. Aperiodic (inertial).
7. Oscillatory.
8. Lagging.
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, you need to know the characteristics of typical units. The main characteristics of the links are the differential equation and the transfer function.
Let's consider the main links and their characteristics.
Reinforcing link(inertia-free, proportional). A reinforcing link is a link that is described by the equation:
or transfer function:
(3)In this case, the transition function of the amplifying link (Fig. 1a) and its weight function (Fig. 1b) respectively have the form:
The frequency characteristics of a 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 amplifier section (Fig. 3) is determined by the relation
.Link examples:
1. Amplifiers, for example, DC (Fig. 4a).
2. Potentiometer (Fig. 4b).
3. Gearbox (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 conjugate frequency and is equal to about 3 dB. In practice, asymptotic characteristics are usually used. Their main advantage is that when changing system parameters ( k And T) characteristics move parallel to themselves.
Link examples:
1. The aperiodic link can be implemented using operational amplifiers (Fig. 9).
ÆÆ
OTP BISN (KSN)
Purpose of work– students acquire practical skills in using methods for designing on-board integrated (complex) surveillance systems.
Laboratory work is carried out in a computer lab.
Programming environment: MATLAB.
Onboard integrated (complex) surveillance systems are designed to solve problems of search, detection, recognition, determining the coordinates of search objects, etc.
One of the main directions for increasing the efficiency of solving set target tasks is the rational management of search resources.
In particular, if the carriers of the SPV are unmanned aerial vehicles (UAVs), then the management of search resources consists of planning trajectories and controlling the flight of the UAV, as well as controlling the line of sight of the SPV, etc.
The solution to these problems is based on the theory of automatic control.
Lab 1
Typical links of an automatic control system (ACS)
Transmission function
In the theory of automatic control (ACT), the operator form of writing differential equations is often used. At the same time, the concept of a differential operator is introduced p = d/dt So, dy/dt = py , A pn=dn/dtn . This is just another designation for the operation of differentiation.
The inverse integration operation of differentiation is written as 1/p . In operator form, the original differential equation is written as algebraic:
a o p (n) y + a 1 p (n-1) y + ... + a n y = (a o p (n) + a 1 p (n-1) + ... + a n)y = (b o p (m) + b 1 p (m-1) + ... + bm)u
This form of notation should not be confused with operational calculus, if only because functions of time are used directly here y(t), u(t) (originals), and not them 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 records 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 writing 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 dynamics equation can also be written as:
Differential operator W(p) called transfer function. It determines the ratio of the output value of the link to the input value 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, that is p = 0, therefore the transfer function turns into the link transmission 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, that is, the values of p at which the denominator D(p) goes to zero, and W(p) tends to infinity are called poles of the transfer function.
Numerator K(p) = b o p m + b 1 p m - 1 + ... + b m called operator gain. Its roots, at which K(p) = 0 And W(p) = 0, are called zeros of the transfer function.
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 specified 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 a 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 quantities. For example, one of the dynamic links is the integrator. Its transfer function W and (p) = 1/p. An ACS diagram composed of dynamic links is called structural.
Differentiating link
There are ideal and real differentiating links. Equation of dynamics 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 carries out pure differentiation W(p) = p . Step response: h(t) = k 1’(t) = d(t) .
It is impossible to implement an ideal differentiating link, 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 extended in time (Fig. 5).
From the transient response, which has the form of an exponential, the transfer coefficient can be determined 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 means used to improve the dynamic properties of self-propelled guns.
In addition to those discussed, there are a number of other links that we will not dwell on in detail. These include the ideal forcing link ( W(p) = Tp + 1 , practically impossible), a real forcing link (W(p) = (T 1 p + 1)/(T 2 p + 1) , at T 1 >> T 2 ), lagging link ( W(p) = e - pT ), reproducing input influence with a time delay and others.
Inertia-free link
Transmission function:
AFC: W(j) = k.
Real frequency response (RFC): P() = k.
Imaginary frequency response (IFC): Q() = 0.
Amplitude-frequency response (AFC): A() = k.
Phase frequency response (PFC): () = 0.
Logarithmic amplitude-frequency response (LAFC): L() = 20lgk.
Some frequency characteristics are shown in Fig. 7.
The link transmits all frequencies equally with an increase in amplitude by k times and without a phase shift.
Integrating link
Transmission function:
Let's consider the special case when k = 1, that is
AFC: W(j) = 
.
VChH: P() = 0.
MCH: Q() = - 1/ .
Frequency response: A() = 1/ .
Phase response: () = - /2.
LACHH: L() = 20lg(1/ ) = - 20lg().
Frequency characteristics are shown in Fig. 8.
The link passes all frequencies with a phase delay of 90 o. The amplitude of the output signal increases as the frequency decreases, and decreases to zero as the frequency increases (the link “overwhelms” high frequencies). The LFC is a straight line passing through the point L() = 0 at = 1. As the frequency increases by a decade, the ordinate decreases by 20lg10 = 20 dB, that is, the slope of the LFC is - 20 dB/dec (decibels per decade).
Aperiodic link
For k = 1 we obtain the following expressions for frequency response:
W(p) = 1/(Tp + 1);
;
;
;
() = 1 - 2 = - arctan( 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. LFCHH:
Frequency characteristics are shown in Fig.9.
The AFC is a semicircle of radius 1/2 with a center at point P = 1/2. When constructing the asymptotic LFC it is considered that when< 1 = 1/T можно пренебречь ( T) 2 выражении для L(), то есть L() - 10lg1 = 0.. При >1 neglect unity in the expression in brackets, that is, L(ω) - 20log(ω T). Therefore, the LFC runs along the abscissa axis to the mating frequency, then at an angle of 20 dB/dec. The frequency ω 1 is called the corner frequency. The maximum difference between real LFCs and asymptotic ones does not exceed 3 dB at = 1.
The LFFC 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 LFFCs of all aperiodic links have the same shape and can be constructed using a standard curve with a parallel shift along the frequency axis.
Reporting form
The electronic report must indicate:
1. Group, full name student;
2. Name of laboratory work, topic, assignment option;
3. Diagrams of typical links;
4. Calculation results: transient processes, LAPFC, for various parameters of links, graphics;
5. Conclusions based on the calculation results.
Laboratory work 2.
Compensation principle
If a disturbing factor distorts the output value to unacceptable limits, then apply principle of compensation(Fig.6, KU - correction device).
Let y o- the value of the output quantity that is required to be provided according to the program. In fact, due to the disturbance f, the value is recorded at the output y. Magnitude e = y o - y called deviation from the specified value. If somehow it is possible to measure the value f, then the control action can be adjusted u at the op-amp input, summing the op-amp signal with a corrective action proportional to the disturbance f and compensating for its influence.
Examples of compensation systems: a bimetallic pendulum in a clock, a compensation winding of a DC machine, etc. In Fig. 4, in the circuit of the heating element (HE) there is a thermal resistance R t, the value of which varies depending on temperature fluctuations environment, adjusting the voltage on the NE.
The merits of the principle of compensation: speed of response to disturbances. It is more accurate than the open-loop control principle. Flaw: the impossibility of taking into account all possible disturbances in this way.
Feedback principle
The most widespread in technology is feedback principle(Fig. 5).
Here the control action is adjusted depending on the output value y(t). And it no longer matters what disturbances act on the op-amp. If the value y(t) deviates from the required one, the signal is adjusted u(t) in order 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 output value y o (t), which is compared with real value at the ACS output y(t).
Deviation e = y o -y from the output of the comparing device is supplied to the input regulator R, which combines UU, UO, CHE.
If e 0, then the regulator generates a control action u(t), valid until equality is achieved e = 0, or y = y o. Since a signal difference is supplied to the controller, such feedback is called negative, Unlike positive feedback, when the signals add up.
Such control in the deviation function is called regulation, and such a self-propelled gun is called automatic control system(SAR).
The disadvantage of the inverse principle communication 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 disturbances of the compensation principle and the accuracy of regulation, regardless of the nature of the disturbances of the feedback principle.
Main types of self-propelled guns
Depending on the principle and law of operation of the memory, which sets the program for changing the output value, the main types of automatic control systems are distinguished: stabilization systems, software, tracking And self-adjusting systems, among which we can highlight extreme, optimal And adaptive systems.
IN stabilization systems a constant value of the controlled quantity is ensured under 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 adjustment of 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 ensured 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. This type of self-propelled guns includes wind-up toys, tape recorders, record players, etc. Distinguish systems with time program, providing y = f(t), And systems with 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 copying 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. The memory is a device that monitors changes in some external parameter. These changes will determine changes in the output value of the ACS. For example, a robot's hand repeating the movements of a human hand.
All three considered types of self-propelled guns 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 a certain prescribed value at the input of the ACS, 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 a value of the controlled quantity that is in some sense optimal.
So in extreme systems(Fig. 8) it is required that the output value always takes the extreme value of all possible, which is not determined in advance and can change unpredictably.
To search for it, the system performs small test movements and analyzes the response of the output value to these tests. After this, 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, there is complex processing of information about the nature of changes in output quantities and disturbances, about the nature of the influence of control actions on output quantities; theoretical information, information of a heuristic nature, etc. can be involved. Therefore, the main difference between extreme systems is the presence of a computer. These systems can operate according to any of the three fundamental management principles.
IN adaptive systems it is possible to automatically reconfigure parameters or change the circuit diagram of the ACS in order to adapt to changing external conditions. In accordance with this, they distinguish self-adjusting 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 self-propelled guns, we will take into account all the features of more complex species Self-propelled guns.
Static characteristics
The operating mode of the ACS, in which the controlled quantity and all intermediate quantities do not change over time, is called established, or static mode. Any link and self-propelled guns as a whole are described in this mode 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 disturbance input f, then the static characteristic is given by a family of curves y = F(u) at different values f, or y = F(f) at different u.
So, an example of one of the functional links of the control system is an ordinary lever (Fig. 10). The static equation for it has the form y = Ku. It can be depicted 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 quantity to the input quantity is called gain link When the input and output quantities are of different nature, it is called transmission coefficient.
The static characteristic of this link has the form of a straight line segment with a slope a = arctan(L 2 /L 1) = arctan(K)(Fig. 11). Links with linear static characteristics are called linear. The static characteristics of real links are, as a rule, nonlinear. Such links are called nonlinear. 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. Typically, a nonlinear characteristic cannot be expressed by any mathematical relationship and must be specified tabularly or graphically.
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 nonlinear, then the self-propelled gun nonlinear.
Links for which a static characteristic can be specified in the form of a rigid functional dependence of the output value on the input value are called static. If there is no such connection and each value of the input quantity corresponds to a set of values of the output quantity, then such a link is called astatic. It is pointless to depict its static characteristics. An example of an astatic link is a motor, the input quantity of which is
voltage U, and the output is 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 self-propelled guns
Dynamic equation
The steady state is not typical for self-propelled guns. Typically, the controlled process is affected by various disturbances that deviate the controlled parameter from the specified value. The process of establishing the required value of the controlled quantity is called regulation. Due to the inertia of the links, regulation cannot be carried out instantly.
Let us consider an automatic control system that is in a steady state, characterized by the value of the output quantity y = y o. Let in the moment t = 0 the object was affected by some disturbing factor, deviating the value of the controlled quantity. After some time, the regulator will return the ACS to its original state (taking into account static accuracy) (Fig. 1).
If the controlled quantity changes over time according to an aperiodic law, then the control process is called aperiodic.
In case of sudden disturbances it is possible oscillatory damped process (Fig. 2a). There is also a possibility that after some time T r undamped oscillations of the controlled quantity will be established in the system - undamped oscillatory process (Fig. 2b). 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 transient processes. That's why the second main task in the development of ACS is the analysis of the dynamic operating modes of the ACS.
The behavior of the self-propelled guns or any of its links in dynamic modes is described dynamics equation y(t) = F(u,f,t), describing the change in quantities 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 related by dependence u(t), f(t), y(t), as well as their rate of change, acceleration, etc. Therefore, the dynamics equation 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.
You can apply to a linearized ACS superposition principle: the system’s response to several simultaneously acting input influences is equal to the sum of the reactions to each influence separately. This allows a link with two inputs u And f decomposed into two links, each of which has one input and one output (Fig. 3).
Therefore, in the future we will limit ourselves to studying the behavior of systems and links with one input, the dynamics equation 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 dynamic mode only approximately with the accuracy that linearization gives. 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, since when n< m Self-propelled guns are technically unrealizable.
Structural diagrams of self-propelled guns
Equivalent transformations of block diagrams
The structural diagram of an 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 purpose, there are rules for equivalent transformation of block diagrams. Let's consider possible methods of transformation.
1. Serial connection(Fig. 4) - the output value of the previous link is fed to the input of the subsequent one. In this case, you can write:
y 1 = W 1 y o ; y 2 = W 2 y 1 ; ...; y n = W n y n - 1 = >
y n = W 1 W 2 .....W n .y o = W eq y o ,
Where 
.
That is, a chain of links connected in series is transformed into an equivalent link with a transfer function equal to the product of the transfer functions of individual links.
2. Parallel - consonant connection(Fig. 5) - the same signal is supplied to the input of each link, and the output signals are added. Then:
y = y 1 + y 2 + ... + y n = (W 1 + W 2 + ... + W3)y o = W eq y o ,
Where 
.
That is, a chain of links connected in parallel - accordingly, is transformed into a link with a transfer function, equal to the amount 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 through which the signal goes in the opposite direction relative to the system as a whole (that is, from output to input) is called feedback circuit with transfer function W os. Moreover, for a negative OS:
y = W p u; y 1 = W os y; u = y o - y 1 ,
hence
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 
.
Likewise: 
- for positive OS.
If W oc = 1, then the feedback is called single (Fig. 6b), then W eq = W p /(1 ± W p).
A closed system is called single-circuit, if when it is opened at any point, a chain of series-connected elements is obtained (Fig. 7a).
A section of a circuit consisting of links connected in series, connecting the point of application of the input signal to the point of collection of the output signal is called straight chain (Fig. 7b, transfer function of the direct chain W p = 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-circuit system can be represented by one link with a transfer function: W eq = W p /(1 ± W p)- the transfer function of a single-circuit closed-loop 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 at which the output signal is taken, the appearance of the straight circuit changes. So, if we consider the output signal y 1 at the link output W 1, That W p = Wo W 1. The expression for the open-circuit transfer function does not depend on the point at which the output signal is taken.
There are closed systems single-circuit And multi-circuit(Fig. 8). To find the equivalent transfer function for a given circuit, you must first transform individual sections.
If a multi-circuit system has crossing connections(Fig. 9), then to calculate the equivalent transfer function additional rules are needed:
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 direction of the signal, then a link is added with a transfer function inverse to the transfer function of the link through which the adder is transferred (Fig. 10).
So the signal is removed from the system output in Fig. 10a
y 2 = (f + y o W 1)W 2 .
The same signal should be removed from the outputs of the systems in Fig. 10b:
y 2 = fW 2 + y o W 1 W 2 = (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 .
During such transformations, non-equivalent sections of the communication line may arise (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 inverse to the transfer function of the link through which the node is transferred. If a node is transferred against the direction of the signal, then a link is added with the transfer function of the link through which the node is transferred (Fig. 11). So the signal is removed from the system output in Fig. 11a
y 1 = y o W 1 .
The same signal is removed from the outputs of Fig. 11b:
y 1 = y o W 1 W 2 /W 2 = y o W 1
y 1 = y o W 1 .
6. Mutual rearrangements of nodes and adders are possible: nodes can be swapped (Fig. 12a); adders can also be swapped (Fig. 12b); when transferring a node through an adder, it is necessary to add a comparing element (Fig. 12c: y = y 1 + f 1 => y 1 = y - f 1) or adder (Fig. 12d: y = y 1 + f 1).
In all cases of transferring elements of a structural diagram, problems arise non-equivalent areas communication lines, so you need to be careful 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
Regulatory laws
Let some kind of ACS be given (Fig. 3).
The control law is a mathematical relationship according to which the control action on an object would be generated by an inertia-free regulator.
The simplest of them is proportional control law, at which
u(t) = Ke(t)(Fig. 4a),
Where u(t)- this is the control action generated by the regulator, e(t)- deviation of the controlled value from the required value, K- proportionality coefficient of the regulator R.
That is, to create a control action, it is necessary that there is a control error and that the magnitude of this error is proportional to the disturbing influence f(t). In other words, the self-propelled guns as a whole must be static.
Such regulators are called P-regulators.
Since when a disturbance influences the control object, the deviation of the controlled quantity from the required value occurs at a finite speed (Fig. 4b), then at the initial moment a very small value e is supplied to the controller input, causing weak control actions u. To increase the speed of the system, it is desirable to speed up the control process.
To do this, links are introduced into the controller that generate an output signal proportional to the derivative of the input value, that is, differentiating or forcing links.
This regulation law is called about
BLOCK DIAGRAMS OF LINEAR self-propelled guns
Typical links of linear self-propelled guns
Any complex self-propelled guns 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 collection idealized schemes, which are accurately described mathematically and approximately characterize real links systems in a certain range of signal frequencies.
When compiling block diagrams some typical elementary units(simple, no longer divisible), characterized only by their transfer functions, regardless of their design, purpose and principle of operation. They are classified by type equations describing their work. In the case of linear self-propelled guns, the following are distinguished: types of links:
1.Described by linear algebraic equations regarding the output signal:
A) proportional(static, inertia-free);
b) lagging.
2. Described by first order differential equations with constant coefficients:
A) differentiating;
b) inertial-differentiating(real differentiating);
V) inertial(aperiodic);
G) integrating(astatic);
d) integro-differentiating(elastic).
3.Described by second order differential equations with constant coefficients:
A) second order inertial link(second order aperiodic link, oscillatory).
Using the mathematical apparatus outlined above, consider transfer functions, transitional And pulse transient(weight) characteristics, and frequency characteristics these links.
We present 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: 
.
Using this scheme, we study typical links.
Note that although for some typical links n(derivative order output parameter on the left side of the equation) equals m(derivative order input parameter on the right side of the equation), and no more m, as mentioned earlier, however, when constructing real self-propelled guns 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 transmission of the link.
Examples of such a link are: a) valves with linearized characteristics (when change fluid flow proportional to the degree of change rod position) in the examples of regulatory systems discussed above; b) voltage divider; c) lever transmission, etc.
Passing to images in (3.1), we have:
1. Transmission function: .
2. Step response: , hence .
3. Impulse transient response: 
.
4. KCHH: .
6. FCHH: 
.
The accepted description of the relationship between entrance And exit valid only for ideal link and corresponds real links only when low frequencies, . When in real links the transmission coefficient k begins to depend on frequency and at high frequencies drops to zero.
Lagging link. This link is described by the equation
where is the delay time.
Example lagging link serve: a) long electrical lines without losses; b) long pipeline, etc.
Transmission function, transitional and pulse transient characteristic, frequency response, as well as frequency response and phase response of this link:
2. means: .
Figure 3.1 shows: a) hodograph CFC lagging link; b) AFC and phase response of the lagging link. Note that as we increase, the end of the vector describes a clockwise, ever-increasing angle.
Fig.3.1. Hodograph (a) and frequency response, phase response (b) of the lagging link.
Integrating link. This link is described by the equation
where is the link transmission coefficient.
Examples of real elements whose equivalent circuits are reduced to integrating unit, are: a) an electric capacitor, if we consider input signal current, and on days off– voltage on the capacitor: 
; b) a rotating shaft, if we count input signal angular speed of rotation, and the output – angle of rotation of the shaft: 
; etc.
Let us determine the characteristics of this link:
2. 
.
Using 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.
Differentiating link. This link is described by the equation
where is the link transmission coefficient.
Let's find the characteristics of the link:
2. 
, taking into account that , we find: .
3. 
.
4. 
.
Figure 3.3 shows: a) link hodograph; b) frequency response and phase response of the link.
A) b)
Rice. 3.3. Hodograph (a), frequency response and phase response (b) of the differentiating link.
Example differentiating link are ideal capacitor And inductance. This follows from the fact that the voltage u and current i connected for capacitor WITH 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 leads the above formulas to the following form:
, 
.
Thus, differentiating link can't 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 met n>m or, as a last resort, n = m.
However, it is possible to get closer to this equation given link, using inertial-differentiating(real differentiator)link.
Inertial-differentiating(real differentiator ) link described by the equation:
Where k- link transmission coefficient, T- time constant.
Transmission function, transitional And impulse transient response, frequency response, frequency response and phase response of this link are determined by the formulas:
We use the property of the Laplace transform - image offset(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.
A) b)
Fig.3.4. Hodograph (a), frequency response and phase response of a real differentiating link.
In order for the properties real differentiating link approached the 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 d – transmission coefficient of the differentiating link.
From this it can be seen that in the dimension of the transmission coefficient k d differentiating link included time.
First order inertial link(aperiodic link ) one of the most common links Self-propelled guns. It is described by the equation:
Where k– link transmission coefficient, T– time constant.
The characteristics of this link are determined by the formulas:
2. 
.
Taking advantage of the properties integration of the original And image shift we have:
.
3. 
, because 
at , then on the entire time axis this function equals 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 first-order inertial link.
Integro-differentiating link. This link is described differential equation first order in the most general form:
Where k- link transmission coefficient, T 1 And T 2- time constants.
Let us introduce the notation:
Depending on the value t the link will have different properties. If , then link its properties will be close to integrating And inertial links If , then given link properties will be closer to differentiating And inertial-differentiating.
Let's define the characteristics integrative link:
1. 
.
2. 
, this implies:
Because 
at t® 0, then:
.
6. 
.
In Fig. 3.6. are given: a) CFC graph; b) frequency response; c) FCHH; d) transient response of the link.
A) b)
V) G)
Fig.3.6. Hodograph (a), frequency response (b), phase response (c), transient response (d) of the integrative link.
Second order inertial link. This link is described by a second order differential equation:
where (kapa) is the attenuation constant; T- time constant, k- link transmission coefficient.
The response of the system described by equation (3.8) to a single stepwise action at is damped harmonic oscillations, in this case the link is also called oscillatory . When vibrations will not occur, and link, described by equation (3.8) is called aperiodic second order link . If , then there will be oscillations undamped with frequency.
An example of 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 second order inertial link:
1. 
.
2. 
.
The roots of the characteristic equation in the denominator are determined:
.
Obviously, there are three possible cases here:
1) when the roots of the characteristic equation negative real different and , then the transient response is determined:
;
2) when the roots of the characteristic equation negative reals are the same 
:
3) when the roots of the characteristic equation of the link are comprehensively-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. 
.
1.3.1 Features of the classification of ACS units The main task of the theory of automatic control of TAU is to develop methods with the help of which it would be possible to find or evaluate quality indicators of dynamic processes in ACS. In other words, not all are considered physical properties elements of the system, and only those that influence are associated with the type of dynamic process. The design of the element, its overall dimensions, and method of connection are not considered.
energy, design features, range of materials used, etc. However, parameters such as mass, moment of inertia, heat capacity, combinations of RC, LC, etc., which directly determine the type of dynamic process, will be important. Features of the physical execution of an element are important only to the extent that they will affect its dynamic performance. Thus, only one selected property of the 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. The model solution, i.e. The differential equation describing the behavior of the element gives a dynamic process that is subject to qualitative assessment.
The classification of ACS elements is based not on the design features or the features of their functional purpose (control object, comparison element, regulatory body, etc.), but on the type of mathematical model, i.e. mathematical equations for the relationship between the output and input variables of an element. Moreover, this relationship 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, for one or another given law of the input quantity, we obtain a reaction, by the type of which we evaluate the properties of the element.
The introduction of the concept of a transfer function allows us to obtain the connection between the output and input quantities in operator form and at the same time take advantage of some properties of the transfer function, which make it possible to significantly simplify the mathematical representation of the system and take advantage of 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 Solving models of dynamic links of an automatic control system gives a change in 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 that reflects the object in front of it in different ways, depending on its shape. 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 selected number at from the region Y find the 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 following conditions: x(t)= 0 at and at t.
In ACS, not absolute changes in variables are studied, but their deviations from steady-state values. Hence, x(t) - a class of functions that describe deviations of variables in an automatic control system and for them both conditions of the Laplace transform are satisfied: the first - since before the disturbance is applied, the variables do not change, the second - since over time any deviation in a working system tends to zero.
These are the conditions for the existence of the Laplace integral. Let us obtain, as an example, images of the simplest Laplace functions.
Rice. 3.1. Types of function display
So, if a unit function x(t) = 1 is given, then
For an exponential function x(t) = e -α t image by
Laplace will have the form:
Finally:
The resulting functions are no more complex than the original ones. The function x(t) is called the original, and x(p)- her image. Conventionally, the direct and inverse Laplace transform can be represented as:
L=x(p),L -1<=x(t).
In this case, there is an unambiguous connection between the original and the image, and vice versa, only a single image of the function corresponds to the original. Let's consider some properties of the Laplace transform.
Picture of differential function. Let the function x(t) correspond to the image x(p): x(t)-> x(p)- It is necessary to find an image of its derivative x(t):
Thus
At zero initial conditions
To depict the nth order derivative:
Thus, the image of the derivative of a function is the image of the function itself, multiplied by the operator p to a degree n, Where P- order of differentiation.
An 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 connecting the input and output quantities:
An example of such a link, in addition to a thermocouple, a DC electric motor, or an RL chain, can be a passive RC- chain (Fig. 3.2 d).
Using the basic laws for describing electrical circuits, we obtain mathematical model aperiodic link in differential form:
Let us obtain the connection between the input and output quantities of the link in the form of the Laplace transform:
Rice. 3.2. Examples of aperiodic links
The ratio of the output quantity to the input quantity is given by an operator of the form.
