Purpose of the work

Calculation by frequency method of a correcting device for linear system(Fig. 4.1) .

Fig.4.1. Block diagram of the original system

Basics

The first stage of the frequency synthesis method is the construction of a logarithmic amplitude-frequency response (LAFC) of an open-loop system. Then, according to the requirements for the quality of the transient process ( t p And s%) build the mid-frequency section of the desired LFC, which has a slope - 20 db/dec and intersects the x-axis at point ( lgw c >0), - Where w c- cutoff frequency, w c =(0.6 - 0.9)·w n, w n - positivity rate. Based on the given overshoot s%, using nomograms (Fig. 4.2) determine the stability margin modulo D.L., limiting the mid-frequency section of the LFC, and w p =Np/t p, Where N- proportionality coefficient corresponding to the found value Pmax.

For example, when s=25% we get P max =1.22, N=4.

Fig.4.2. Nomograms for determining the parameters of the desired LFC

In the region of high and low frequencies, the desired characteristic is matched with the LFC of the original system. By subtracting the characteristic of the open-loop system from the desired LFC, we obtain the LFC of the correcting link, from which its transfer function is determined. The block diagram of the system, taking into account the corrective link, is shown in Fig. 4.3.

Guidelines

To perform laboratory work, it is necessary to calculate the parameters of the corrective link in accordance with the requirements for the quality of processes in a closed system. The work is done using one of the packages application programs for research of self-propelled guns ( COMPAS, SIMNON, MATLAB) .

Fig.4.3. Block diagram of the adjusted system

Work order

4.1. Enter a model of the system under study (Fig. 4.1), the parameters of which are given in the table. Draw process graphs y(t), D(t).

4.2. Based on the requirements for the quality of transient processes in the system, calculate the parameters of the corrective link.

Table 4.1

4.3. Create a model of the corrective link and include it in the system. Remove the transient process in the adjusted system and make sure that the quality indicators correspond to the specified ones.

4.4. Change the parameters of the corrective link, record the transition process, determine the quality indicators of the process, compare them with the results of paragraph 4.3.

5.1. Purpose of the work.

5.2. Structural diagrams of the system without correction and with correction.

5.3. LFC of the original system, desired LFC of the open-loop system and the correcting link.

5.4. Transfer function of the correcting link.

5.5. Transient processes according to clauses 4.1, 4.3, 4.4.

6.Security questions

6.1. Which part of the LFC determines the properties of the system in static mode?

6.2. Which part of the LFC determines the properties of the system in dynamics?

6.3. How to construct its asymptotic LFC from the transfer function of a system?

6.4. How are external disturbances taken into account when designing a controller?

6.5. How are the quality indicators of a closed-loop system related to the type of desired LFC?

6.6. How to restore its transfer function based on the LFC of the correcting link?

Laboratory work No. 5

Investigating the Properties of State Observers

Purpose of the work

Explore methods for constructing and properties of state observers for dynamic objects.

Basics

We consider linear stationary objects whose behavior is described by the transfer function

W(p) = =(5.1)

U T2p2+2dTp+1

There are a number of methods for synthesizing control systems (methods for the analytical design of optimal controllers, the modal synthesis method), the use of which involves the use of system state variables in the control law. However, in practice, only the output variable of the system is usually available for measurement y(t), therefore, the problem arises of obtaining an estimate of the state vector x(t).

To estimate state variables, a special technical system- state assessment filter (state observer). The laboratory work examines such methods of constructing state observers as the parallel model method and the Kalman filter. The parallel model method can be used for stable linear stationary objects (5.1). In this case, the state observer equation has the form

T 2 ÿ+2dTý+y=KU(5.2)

The corresponding block diagram of the object (5.1) with a state observer is shown in Fig. 5.1.

In the case when the control object (5.1) is unstable or it is necessary to speed up the process of estimating state variables, a Kalman filter is usually used, which, in addition to the parallel model, contains a stabilizing additive L(p). The block diagram of the system is shown in Fig. 5.2.

Transfer function connecting variables Δ And U, has the form:

W (p) = = - .(5.3)

U T 2 p 2 +2dTp+1+KL(p)

The characteristic equation of the observer is as follows

T 2 p 2 +2dTp+1+KL(p)=0. (5.4)

Selection of stabilizing additive coefficients L(p) is carried out based on the requirements for the quality of transient processes in the observer. In this case, the desired characteristic equation is formed, the coefficients of which are equal to the coefficients of equation (5.4).

Fig.5.1. Block diagram of an object with an observer

as a parallel model

Fig.5.2. Block diagram of an object with an observer

in the form of a Kalman filter

Guidelines

3.1. Calculate the stabilizing additive L(p)=K З, based on the process in the observer.

τ 2 p+1

transient processes in the observer, where t p- desired transition time ; σ% - the amount of permissible overshoot.

3.3. Items marked with * are carried out on the recommendation of the teacher.

Work order

4.1. Assemble a scheme for modeling the system (5.1) with a state observer using the parallel model method (Fig. 5.1) in accordance with the option number.

Table 5.1

4.2. Draw transient graphs for the object and observer state variables, as well as the error Δ(t),

4.3. Carry out the simulation in the same way as in paragraph 4.2, applying a unit value to the input of the system under study step effect under different initial conditions for the object and the observer.

4.4. Change value T in the object 2 times and repeat step 4.3.

4.5. Assess impact K on the properties of the system, successively increasing and decreasing its value for the object by 2 times relative to the nominal value and repeating clause 4.3.

4.6. Assemble a model of the system with a Kalman filter (Fig. 5.2) and a stabilizing additive L(p)=k Z Δ(t), by applying a single step effect to the input of the system under study at zero initial conditions.

4.7. Carry out the simulation in the same way as in paragraph 4.6, applying a single step effect to the input of the system under study under different initial conditions for the object and the observer.

4.8. Explore the impact K, successively increasing and decreasing its value by half relative to the calculated value and repeat paragraphs 4.6 and 4.7.

4.9*. Change value T in the object 2 times and repeat step 4.7.

4.10*. Assess impact K on the properties of the system, successively increasing and decreasing its value for the object by 2 times relative to the nominal value and repeating clause 4.7.

4.11. Assemble a model of the system with a Kalman filter and a stabilizing additive L(p)=K(τ 1 p+1)/(τ 2 p+1) and sketch transient graphs for the output variables of the object and observer, as well as the error Δ(t), by applying a single step effect to the input of the system under study at zero initial conditions.

4.12. Carry out the simulation in the same way as in paragraph 4.11, applying a single step effect to the input of the system under study under different initial conditions for the object and the observer.

4.13. Change value T in the object 2 times and repeat paragraph 4.12, compare with the results of paragraph. 4.4 and 4.9.

4.14. Assess impact K on the properties of the system, successively increasing and decreasing its value for the object by 2 times relative to the nominal value and repeating clause 4.12. compare with the results obtained in paragraphs 4.5 and 4.10.

5.1. Purpose of the work.

5.2. Block diagrams of the studied systems.

5.3. Calculation of stabilizing additive parameters L(p).

5.4. Graphs of simulation results.

5.5. Conclusions from the work.

6. Security questions

6.1. What is the scope of application of the parallel model method?

6.2. How does changing the parameters of an object affect the error in estimating state variables using the parallel model method?

6.3. How to choose the parameters of a stabilizing additive L(p)?

6.4. What are the applications of Kalman filters?

6.5. How does changing object parameters affect the error in estimating state variables using the Kalman filter?

6.6. Is it possible to change the rate at which state variables are estimated using a parallel model observer?

6.7. How are state variables estimated if the object and observer have different initial conditions?

Laboratory work No. 6

Clarification of the structural diagram of the control system for the selection and calculation of its elements and parameters. Experimental study of a system or its individual parts in laboratory conditions and making appropriate corrections to its circuit and design. Design and production of control system. Setting up the system in real operating conditions and trial operation.

If this work does not suit you, at the bottom of the page there is a list of similar works. You can also use the search button

Lecture No. 6 Synthesis of automatic control systems

SYNTHESIS OF ACS selection of structure and parameters of ACS, initial conditions and input influences in accordance with the required quality indicators and operating conditions.

Designing an ACS involves performing the following steps:

1. Study of the regulated object: drawing up a mathematical model, determining the parameters, characteristics and operating conditions of the object.
2. Formulation of requirements for ATS.
3. Choice of control principle; determination of the functional structure (technical synthesis).
4. Selection of elements of the control scheme taking into account static, dynamic, energy, operational and other requirements and their coordination with each other according to static and energy characteristics (the procedure is not formalized - engineering creativity).
5. The determination of the algorithmic structure (theoretical synthesis) is carried out using mathematical methods and on the basis of requirements written in a clear mathematical form. Determination of control laws and calculation of corrective devices that provide specified requirements.
6. Clarification of the structural diagram of the control system, selection and calculation of its elements and parameters.
7. Experimental study of a system (or its individual parts) in laboratory conditions and making appropriate corrections to its circuit and design.
8. Design and production of control system.
9. Setting up the system in real operating conditions (trial operation).

The design of an ACS begins with the selection of a control object and the main functional elements (amplifiers, actuators, etc.), that is, the power part of the system is developed.

The specified static and dynamic characteristics of the system are ensured by the appropriate choice of the structure and parameters of the power unit, special corrective devices and the entire ACS as a whole.

Purpose of corrective devices: ensure the required accuracy of the system operation and obtain an acceptable nature of the transient process.

Corrective links are introduced into the system in various ways: sequentially, local OOS, direct parallel connection, external (outside the control loop) compensating devices, coverage of the entire ACS stabilizing OOS, non-unit main feedback.

Types of DC electrical correcting devices: active and passive DC quadripoles, differentiating transformers, DC tachogenerators, tachometer bridges, etc.

By purpose correction devices are classified:

1. STABILIZING ensure the stability of self-propelled guns and improve their static and dynamic characteristics;
2. COMPENSATING reduce static and dynamic errors when building self-propelled guns using a combined principle;
3. FILTERING increasing the noise immunity of systems, for example, filtering higher harmonics when demodulating a direct channel signal;
4. SPECIALIZED to give the system special properties that improve the quality of the system.

ACS can be built according to the following structural diagrams:

1. With series correction circuit.

Amplifier U must have a high input impedance so as not to bypass the output of the correction circuit.

It is used in the case of slowly changing input influences, since with large mismatches saturation occurs in real nonlinear elements, the cutoff frequency goes to the left and the system slowly leaves the saturation state.

Fig.1.

Sequential correction is often used in stabilization systems or for loop correction with corrective feedback.

Decreases.

1. With anti-parallel correction circuit.

Fig.2.

It enters the input as a difference and deep saturation does not occur.

1. With series-parallel correction circuit.

Fig.3.

1. With combined correction chains.

The synthesis of ACS of subordinate control with two or more circuits is carried out by sequential optimization of the circuits, starting with the internal one.

The calculation of systems is divided into 2 stages: static and dynamic.

Static calculationconsists in selecting the main links of the system included in its main circuit, drawing up a block diagram of the latter and determining the parameters of the main elements of the system (gain factors that ensure the required accuracy, time constants of all elements, gear ratios, transfer functions of individual links, engine power). In addition, this includes the calculation and design of magnetic and semiconductor amplifiers and the selection of transistor or thyristor converters, motors, sensitive elements and other auxiliary devices of the systems, as well as the calculation of the accuracy in steady state operation and the sensitivity of the system.

Dynamic calculationincludes a large set of issues related to the stability and quality of the transient process (speed, performance characteristics and dynamic accuracy of the system). During the calculation process, corrective circuits are selected, the places where they are connected, and the parameters of the latter are determined. The transient curve is also calculated or the system is modeled in order to clarify the obtained quality indicators and take into account some nonlinearities.

Platforms on which stabilizing algorithms are built:

1. Classical (differential equations - time and frequency methods);
2. Fuzzy logic;
3. Neural networks;
4. Genetic and ant algorithms.

Regulator synthesis methods:

1. Classic scheme;
2. PID regulators;
3. Pole placement method;
4. LCH method;
5. Combined control;
6. Many stabilizing regulators.

Classic regulator synthesis

The classic block diagram of object control is shown in Fig. 1. Usually the regulator is turned on in front of the object.

Rice. 1. Classic block diagram of object management

The task of the control system is to suppress the action of external disturbances and ensure high-quality transient processes. These objectives are often contradictory. In fact, we need to stabilize the system so that it has the required transfer functions for the master action and the disturbance channel:

, .

For this we can use only one controller, which is why such a system is called a single-degree-of-freedom system.

These two transfer functions are related by the equality

Therefore, by changing one of the transfer functions, we automatically change the second one. Thus, they cannot be formed independently and the solution will always be some kind of compromise.

Let's see if such a system can provide zero error, that is, absolutely accurate tracking of the input signal. The transfer function by error is equal to

To make a mistake Always was zero, it is required that this transfer function be equal to zero. Since its numerator is not zero, we immediately see that the denominator must go to infinity. We can only influence the regulator, so we get. Thus,to reduce the error you need

increase the regulator gain.

However, the gain cannot be increased indefinitely. First of all, everything real devices have maximum permissible values ​​of input and output signals. Secondly, with a large gain of the circuit, the quality of transient processes deteriorates, the influence of disturbances and noise increases, and the system may lose stability. Therefore, in a circuit with one degree of freedom, it is impossible to ensure zero tracking error.

Let's look at the problem from the point of view of frequency characteristics. On the one hand, for high-quality tracking of the master signal, it is desirable that the frequency response be approximately equal to 1 (in this case). On the other hand, from the point of view of robust stability, it is necessary to ensure high frequencies ah, where the modeling error is large. In addition, the transfer function for disturbances should be such that these disturbances are suppressed, which we should ideally ensure.

When choosing a compromise solution, you usually do the following:

● at low frequencies, the condition is met, which ensures good tracking of low-frequency signals; in this case, that is, low-frequency disturbances are suppressed;

● at high frequencies are sought to ensure robust stability and suppression of measurement noise; in this case, that is, the system actually operates as an open circuit, the regulator does not respond to high-frequency interference.

Calculation of linear continuous automatic control systems for a given accuracy

In steady state

One of the main requirements that the ACS must satisfy is to ensure the necessary accuracy of reproduction of the master (control) signal in steady state.

The order of astatism and the transfer coefficient of the system are found based on the requirements for accuracy in steady state.If the transfer coefficient of the system, determined by the required value of statism and quality factor (in the case of an astatic automatic control system), turns out to be so large that it significantly complicates even simple stabilization of the system, it is advisable to increase the order of astatism and thereby reduce the given steady-state error to zero, regardless of the value of the system transfer coefficient . As a result, it becomes possible to choose the value of this coefficient based only on considerations of stability and quality of transient processes.

Let the structural diagram of the ACS be reduced to the form

Then, in the quasi-steady operating mode of the automatic control system, the mismatch can be represented in the form of a convergent series

where they act as weight constants.

Obviously, such a process can only take place if the function is slowly changing and fairly smooth.

If we imagine the transfer function of an open-loop system in the form

then at r =0

at r =1

at r =2

at r =3

The low-frequency part of the logarithmic amplitude frequency characteristics determines the accuracy of the system when processing slowly changing control signals in a steady state and is determined by the error rates. Error rates no longer have a significant impact on the accuracy of the ACS, and they can be ignored in practical calculations.

1. Calculation of the steady-state operating mode of the ACS based on the specified mismatch (error) coefficients

The accuracy of the system in steady state is determined by the value of the transfer coefficient of the open-loop system, which is determined depending on the form of specifying the requirements for the accuracy of the system.

The calculation is carried out as follows.

1. STATIC SAR. Here you set the value of the positional error coefficient, which is used to determine: .

dB

20 lgk pc

ω, s -1

1. ASTATIC SYSTEMS 1st order.

In this case, a coefficient is specified by which it is determined

If the coefficients and are specified, then, which determines the position of the low-frequency asymptote of the LFC of the open-loop system with a slope of -20 dB/dec, and the second asymptote has a slope of -40 dB/dec at the coupling frequency (Fig. 1).

Fig.1.

1. ASTATIC SYSTEMS of the 2nd order.

Using the given coefficient, we determine kpc:

dB

ω, s -1

2. Calculation of the steady-state operating mode of the ACS based on the specified maximum value of the mismatch (error) of the system

Based on the permissible value of the steady-state error and the type of control action, the parameters of the low-frequency part of the LFC system are selected.

1. Let the permissible maximum error under harmonic influence with amplitude and frequency and the order of astatism of the system be given.

Then the low-frequency asymptote of the LFC of the system should pass no lower than control point with coordinates:

(1)

and have a slope of -20 r dB/dec. Dependence (1) is valid when.

1. Let the permissible maximum error at maximum speed and the maximum acceleration of the input influence and the order of astatism be given r systems.

It is often convenient to use the method of equivalent sinusoidal action proposed by Ya.E. Gukaylo.

In this case, a mode is determined in which the amplitudes of velocity and acceleration are equal to the maximum specified values. Let the input influence change in accordance with a given law

. (2)

Equating the amplitude values ​​of velocity and acceleration obtained by differentiating expression (2) to the given values ​​and, we obtain

where, . Using these values, you can build a control

point B with coordinates and

At unit negative feedback,

With non-unit feedback.

If the input signal speed is maximum and the acceleration is decreasing, then the test point will move in a straight line with a slope of -20 dB/dec in the frequency range. If the acceleration is equal to the maximum value, and the speed decreases, then the control point moves in a straight line with a slope of -40 dB/dec in the frequency range.

The area located below control point B and two straight lines with slopes of -20 dB/dec and -40 dB/dec is a forbidden area for the LFC of the tracking system. Since the exact LFC passes below the intersection point of the two asymptotes by 3 dB, the desired characteristic at should be raised up by this amount, i.e.

In this case, the required value of the quality factor for speed, and the frequency at the point of intersection of the second asymptote with the frequency axis (Fig. 2)

In the case when the control action is characterized only by the maximum speed, the quality factor of the system in terms of speed at a given error value:

If only the maximum acceleration of the signal and the magnitude of the error are specified, then the acceleration quality factor is:

Fig.2.

1. Let the maximum static error along the control channel be specified (the input action is stepwise, the system is static along the control channel).

Fig.3.

Then the value is determined from the expression. Static accuracy automatic system can be determined from the equation:

where is the static accuracy of the closed-loop system,

deviation of the controlled variable in an open-loop system,

open-loop transmission coefficient required to ensure a given accuracy.

1. Let the maximum permissible static error along the disturbance channel be specified (the disturbance is stepwise, the system is static along the disturbance channel, Fig. 3).

Then the value is determined from the expression:

where is the transfer coefficient of the open-loop system along the disturbance channel,

where error of the system without a regulator.

In static control systems, the steady-state error caused by a constant disturbance is reduced by 1+ compared to an open-loop system. At the same time, the transfer coefficient of the closed-loop system also decreases by 1+ times.

1. Let the permissible speed error from the control action be given (the input action changes at a constant speed, the system is astatic of the first order).

Servo systems are usually designed as astatic first order ones. They operate under variable control input. For such systems in a steady state, the most characteristic change in the input effect is linear.

Then the quality factor of the system in terms of speed is determined from the expression:

Since the steady-state error is determined by the low-frequency part of the LFC, the low-frequency asymptote of the desired LFC can be constructed from the calculated value of the transfer coefficient.

3. Calculation of the steady-state operating mode of the ACS for a given maximum permissible error of a system with non-unit feedback

Let a priori information about the input signal be minimized:

1. Maximum absolute value of the first derivative of the input action (maximum tracking speed) ;
2. Maximum absolute value of the second derivative of the input action (maximum tracking acceleration) ;
3. The input can be a deterministic or random signal with any spectral density.

It is required to limit the maximum permissible error of the control system when reproducing a useful signal in a steady state of operation by the value.

The requirement for reproduction accuracy is most simply formulated for a harmonic input effect equivalent to the real input signal:

under the assumption that the amplitude and frequency are given, and the initial phase has an arbitrary value.

Let us establish a connection between the permissible error in reproducing the input effect and the parameters of the system and the input signal.

Let the block diagram of a continuous automatic control system be reduced to the form (Fig. 4).

Fig.4.

The error at the system output in the time domain is determined by the expression:

where is the reference (error-free) output function.

It can be shown that due to restrictions on speed and accelerationthe output function is different from the step function.

Let us map the last expression into the space of Laplace transforms:

Let us map into the space of Fourier transforms:

In the low frequency region (time constants of the feedback circuit), then

the maximum error amplitude is determined by the expression:

IN real systems usually at low frequencies, because the requirement must be fulfilled; mathematical expression to determineis converted at the control frequency () to the form

and in order for the output function to be reproduced with a maximum error no more than specified, the LFC of the designed system should not pass below the control point with coordinates and

4. Calculation of the steady state operation of a static automatic control system using the method of limit transitions

Statement

Let a generalized block diagram of a static ACS be given:

where, here the polynomials of numerators and denominators do not contain a factor p (their free terms are equal to one),

regulator transfer coefficient,

transfer coefficient of the object via the control channel,

feedback transfer coefficient,

transfer coefficient of the object along the disturbance channel,

Moreover, in a first approximation, the static and dynamic transfer coefficients of the links are assumed to be equal, the nominal input action corresponds to the nominal value of the output function along the control channel, and let the magnitude of the step disturbance action and the permissible static error along the disturbance channel in % of the nominal value of the output function be specified.

Then the transfer coefficients of the system along the control and disturbance channels in steady state are equal to the static transfer coefficients of the closed system and are determined by the formulas:

(1)

The static equations for the control and disturbance channels have the form

(2)

The transfer coefficients of the regulator and the feedback circuit are determined by the expressions:

(3)

Ways to increase the static accuracy of self-propelled guns

1. Increasing the transfer coefficient of an open-loop system in static systems.

Where, .

However, the stability conditions worsen as the value increases, that is, the errors in the dynamics increase.

1. Introduction to the Integral Component Controller.

2.1. Application of I-regulator: .

In this case, the system becomes astatic along the control and disturbance channels, and the static error becomes equal to zero. The LFC of the system will be much steeper than the original one, and the phase shift will increase by 90 degrees. The system may be unstable.

2.2. Installation of the PI controller: .

Here the static error is zero, and the stability conditions are better than those of a system with an I-controller.

2.3. Using the PID controller: .

The static error of the system is zero, and the stability conditions are better than in a system with a PI controller.

1. Introduction of non-unit feedback into the system if accurate reproduction of the information level of the input signal is required.

We believe that and are static links. , you need to choose something like this,

To; .

1. Input scaling

impact.

Here.

The output function will be equal to the information level of the input influence, if, hence, where.

1. Application of the principle of compensation through control and disturbance channels.

The calculation of compensating devices is described in the section “Calculation of combined control systems”.

Calculation of the dynamics of self-propelled guns

Synthesis of ACS using LFC

Currently, a large number of methods for the synthesis of correction devices have been developed, which are divided into:

• analytical synthesis methods, which use analytical expressions that connect system quality indicators with the parameters of corrective devices;
• graphic-analytical.

The most convenient of graphic-analytical synthesis methods classic universal method logarithmic frequency characteristics.

Essence of the method is as follows. First, the asymptotic LFC of the original system is constructed, then the desired LFC of the open-loop system is constructed; The LFC of the correcting device must change the shape of the LFC of the original system so that the LFC of the corrected system.

The most difficult and critical stage in the synthesis is the construction of the desired LFC. When constructing, it is assumed that the synthesized system has a single negative feedback and is a minimum-phase system. A quantitative relationship between the quality indicators of the transition function of minimum-phase systems with single OOS and the LFC of an open-loop system is established on the basis of nomograms by Chestnut-Mayer, V.V. Solodovnikov, A.V. Fateev, V.A. Besekersky.

The desired LFC is conventionally divided into three parts: low-frequency, mid-frequency and high-frequency. The low-frequency part is determined by the static accuracy of the system the accuracy of the ACS in steady state. In a static system, the low-frequency asymptote is parallel to the frequency axis; in astatic systems, the slope of the low-frequency asymptote is 20 * dB/dec, where  - order of astatism ( =1, 2, 3,…). The mid-frequency part is the most important, since it mainly determines the dynamics of processes in the system. The main parameters of the mid-frequency asymptote are its slope and cutoff frequency. The greater the slope of the mid-frequency asymptote, the more difficult it is to ensure good dynamic properties of the system. Therefore, a slope of 20 dB/dec is appropriate and extremely rarely exceeds 40 dB/dec. The cutoff frequency determines the performance of the system. The more, the higher the performance (the less). The high-frequency part of the desired LFC has little effect on the dynamic properties of the system. Generally speaking, it is better to have the greatest possible slope of its asymptote, which reduces the required power of the actuator and the influence of high-frequency interference.

The desired LFC is built on the basis of system requirements: requirements for static properties are specified in the form of astatism order and the transfer coefficient of the open-loop system; dynamic properties are most often set by the maximum permissible overshoot value and regulation time; sometimes a limitation is set in the form of the maximum permissible acceleration of the controlled variable at the initial mismatch.

Methods for constructing the desired LFC: construction according to V.V. Solodovnikov, using standard LFCs and nomograms for them, construction according to E.A. Sankovsky G.G. Sigalov, simplified construction, construction according to V.A. Besekersky, according to the method of A. V. Fateeva and other methods.

Advantages of frequency methods:

● Frequency characteristics reflecting mathematical model object, can be obtained relatively simply experimentally;

● Calculations based on frequency characteristics are reduced to simple and visual graphical-analytical constructions;

● Frequency methods combine simplicity and clarity in solving problems, regardless of the order of the system, the presence of transcendental or irrational links in the transfer function.

Synthesis of the desired LFC

Theoretical and experimental studies have established that the LFC of an open-loop control system, stable in a closed state, almost always intersects the frequency axis with a section having a slope of 20 dB/dec. Intersection of the frequency axis by a section of the LFC with a slope of 40 dB/dec or 60 dB/dec is possible, but is rarely used, because such a system is stable with a very low transfer coefficient.

The most rational form of the LFC of an open-loop system, stable in a closed state, has the slopes:

• low-frequency asymptote 0, -20, -40 dB/dec (determined by the order of astatism of the system);
• the asymptote connecting the low-frequency and mid-frequency asymptotes can have slopes of 20, -40, -60 dB/dec;
• mid-frequency asymptote 20 dB/dec;
• the asymptote connecting the mid-frequency with the high-frequency portion of the LFC, as a rule, has a slope of -40 dB/dec;
• the high-frequency section of the LFC is built parallel to the asymptotes of the high-frequency section of the LFC of the original open-loop system.

When constructing the desired LFCs, we proceed from the following requirements:

1. The adjusted system must satisfy the specified quality indicators ( acceptable error in steady state, the required stability margin, speed, overshoot and other indicators of the quality of transient processes).
2. The shape of the desired LFC should differ as little as possible from the LFC of the uncorrected system to simplify the stabilizing device.
3. You should strive to ensure that at high frequencies the LFC of the uncorrected system does not pass more than 20-25 dB.
4. The low-frequency part of the desired LFC must coincide with the LFC of the uncorrected system, since the transfer coefficient of the open-loop dynamically uncorrected system is selected taking into account the required accuracy in steady state.

The construction of the desired LFCs can be considered complete if all requirements for the quality of the system are satisfied. IN otherwise you should return to the calculation of the steady state of operation and change the parameters of the elements of the main circuit (select a motor of a different power or less inertial, use an amplifier with a lower time constant, enable strict negative feedback covering the most inertial elements of the system, etc.).

Algorithm for constructing the desired LFCs

1. Selecting the cutoff frequency Lf(w).

If the overshoot and decay time of the transient process are specified, then the nomograms of V.V. Solodovnikov or A.V. Fateev are used; if the oscillation index M is specified, then the calculation is carried out according to the method of V.A. Besekersky.

The construction of quality nomograms by V.V. Solodovnikov was based on the typical real frequency response of a closed-loop automatic control system (Fig. 2). For static systems ( =0), for astatic systems ( =1, 2,…) .

This method assumes that the ratio is maintained.

Dynamic quality indicators and are taken as initial ones, which are related to the parameters of the real frequency response of a closed ACS by the V.V. quality diagram. Solodovnikov (Fig. 3). Based on the given curve (Fig. 3), the corresponding value is determined. Then, using the curve, a value is determined that is equal to the specified value, we obtain where is the value of the cutoff frequency at which the control time will not exceed the specified value.

On the other hand, it is limited by the permissible acceleration of the controlled coordinate. Recommended where initial mismatch.

The regulation time can be approximately determined using an empirical formula, where the numerator coefficient is taken to be equal to 2 at, 3 at, 4 at.

It is always desirable to design a system with the highest possible performance.

As a rule, it does not exceed more than ½ decade. This is due to the complication of corrective devices, the need to introduce differentiating links into the system, which reduces reliability and noise immunity, and also due to restrictions on the maximum permissible acceleration of the controlled coordinate.

The cutoff frequency can only be increased by increasing it. In this case, the static accuracy increases, but the stability conditions worsen.

The selection decision must have sufficient justification.

1. Constructing a mid-frequency asymptote.

1. We pair the mid-frequency asymptote with the low-frequency asymptoteso that in the frequency range in which there is an excess of phase. The excess phase and excess module are determined using the nomogram (Fig. 4). The conjugate asymptote has a slope of 20, -40 or 60 dB/dec at =0 ( - order of astatism of the system); -40, -60 dB/dec at =1 and -60 dB/dec at  =2.

If the phase excess turns out to be smaller, then the conjugate asymptote should be shifted to the left or its slope reduced. If the phase excess is greater than permissible, then the conjugating asymptote is shifted to the right or its slope is increased.

The initial coupling frequency is determined from the expression.

1. We pair the mid-frequency asymptote with the high-frequency partso that in the frequency range where there is an excess of phase. The coupling frequency is determined by the ratio.

If at the coupling frequency<, то сопрягающую асимптоту смещают вправо или уменьшают ее наклон.

If >, then the conjugating asymptote is shifted to the left or its slope is increased. The recommended difference should be a few degrees. Right conjugate frequency of the conjugate asymptote.

Typically, the slope of this asymptote is -40 dB/dec, and the acceptable difference is. The test is performed at a frequency at which.

1. The high-frequency part is designed in parallel or combined with it.

This part of the characteristic affects the smooth operation of the system.

So, at the first stage of construction, the frequencies at which the mid-frequency asymptote is conjugated with the conjugating asymptotes are found from the conditions. At the second stage, the values ​​of the conjugate frequencies are specified taking into account phase excesses. At the third stage, all mating frequencies are adjusted according to the condition of their proximity to the mating frequency of the original system, i.e., if these frequencies do not differ significantly from each other.

Synthesis of a sequential type correction circuit

In the diagram in Fig. 1, the parameters of the correcting circuit can be obtained from here:

Let's move on to logarithmic frequency characteristics: ,

At high frequencies, the LFC of the regulator “by default” should not exceed 20 dB according to noise protection conditions. The fundamental principle of structural-parametric optimization of automatic control systems with feedback: the controller must contain a dynamic link with a transfer function equal to or close to the inverse transfer function of the controlled object.

Let's look at the example of calculating a sequential correction circuit.

Suppose it is necessary to adjust the static system. Let's assume that we built them too. We believe that the system has minimum phase links, so we do not build a phase-frequency characteristic (Fig. 2).

Now it is easy to reproduce the parameters of the correction circuit. The most commonly used correction devices are active and passive ones. R.C. -chains. Based on physical concepts, we build the circuit shown in Fig. 3.

Attenuating the signal with a divider R 1- R 2 at high frequencies corresponds to signal attenuation * by.

Where,

At high frequencies there is no distortion positive factor. We can shift the cutoff frequency to the left using a correction circuit and ensure the required stability and quality of system operation.

Advantages of sequential CG:

1. Simplicity of the correction device (in many cases implemented in the form of simple passive RC circuits);
2. Easy to turn on.

Flaws:

1. The effect of sequential correction decreases during operation when parameters change (gain coefficients, time constants), therefore, with sequential correction, increased demands are placed on the stability of element parameters, which is achieved by using more expensive elements;
2. Differentiating phase advancing R.C. -circuits (algorithms in microcontrollers) are sensitive to high-frequency interference;
3. Sequential integrating R.C. -circuits contain more bulky capacitors (the implementation of large time constants is required) than the circuits in the feedback circuit.

Usually used in low-power systems. This is explained, on the one hand, by the simplicity of sequential corrective devices, and on the other hand, by the inexpediency of using in these systems bulky parallel corrective devices commensurate with the size of the actuator, such as a tachogenerator.

It should be borne in mind that due to the saturation of amplifiers, it is not always advisable to form the desired LFC in the low and medium frequency range by sequentially incorporating integrating and integrating circuits or some other elements with similar characteristics into the system. Therefore, feedback is often used for shaping in the low and medium frequency range.

Synthesis of back-to-back corrective circuits

When choosing the location for connecting the correction circuit, you should be guided by the following rules:

1. Those links that significantly negatively affect the type of desired LFC should be covered.
2. The slope of the LFC of the links not covered by feedback is chosen to be close to the slope in the mid-frequency range. Fulfilling this condition allows you to have a simple correction circuit.
3. Corrective feedback should cover as many links with nonlinear characteristics as possible. In the limit, it is necessary to strive to ensure that among the links not covered by feedback there are no elements with nonlinear characteristics. This inclusion of feedback can significantly reduce the influence of the nonlinearity of the characteristics of the elements covered by the feedback on the operation of the system.
4. Feedback should cover links with a large gear ratio. Only in this case will the feedback be effective.
5. The signal to the feedback input must be taken from an element with sufficient power so that turning on the feedback does not load it. The signal from the feedback output should, as a rule, be applied to the input of system elements that have a high input resistance.
6. When choosing where to turn on the feedback inside the loop with corrective feedback, it is desirable that the slope of the LFC in the frequency range is 0 or 20 dB/dec. Fulfilling this condition allows you to have a simple correction circuit.

Often they cover the amplification path of the system or cover the power part of the system. Corrective feedback is usually used in powerful systems.

Advantages of CEP:

1. The dependence of the system quality indicators on changes in the parameters of the elements of the unchangeable part of the system is reduced, since in a significant frequency range the transfer function of the system section covered by the feedback is determined by the inverse value of the transfer function of the back-to-back correcting device. Therefore, the requirements for the elements of the original system are less stringent than for sequential correction.
2. The nonlinear characteristics of the elements covered by the feedback are linearized, since the transfer properties of the covered section of the system are determined by the parameters of the loop in the feedback circuit.
3. Powering back-to-back corrective devices, even when it requires high power, does not cause difficulties, since feedback usually starts from the terminal links of the system with a powerful output.
4. Back-to-back corrective devices operate with a lower level of interference than serial ones, since the signal entering them passes through the entire system, which is a low-pass filter. Due to this, the effectiveness of back-to-back corrective devices when interference is applied to the error signal decreases less than that of serial corrective devices.
5. In contrast to a sequential correcting device, feedback makes it possible to realize the largest time constant of the desired LFC with relatively small values ​​of its own time constants.

Flaws:

1. Back-to-back CPs often contain expensive or bulky elements (for example, tachogenerators, differentiating transformers).
2. The summation of the feedback signal and the error signal should be implemented so that the feedback does not shunt the amplifier input.
3. The loop formed by corrective feedback may be unstable. Reducing stability margins in internal circuits worsens the reliability of the system as a whole.

Determination methods:

1. Analytical;
2. Graphic-analytical;
3. Model-experimental.

After calculating the anti-parallel correction circuit, the stability of the internal circuit should be checked. If the main feedback is open and the internal loop is unstable, then the elements of the system may fail. If the internal circuit is unstable, then its stability is ensured by a series correction circuit.

An approximate method for constructing the LFC of corrective negative feedback

Let the block diagram of the projected

The system is reduced to the form shown

In Fig. 1.

corrective feedback;

transmission

open-loop source function (uncorrected)

systems.

For such a structural diagram, the transfer function of the adjusted open-loop system.

In the frequency range where,the equation will be written like this

Those.

Selection condition ; (1)

- selection equation (in the low and high frequency ranges) (2)

In the frequency range where,

Selection condition ; (3)

we get,

i.e.,

where - selection equation(in the mid frequency range). (4)

Then the construction algorithm is as follows:

1. We are building.
2. We are building.
3. We construct and determine the frequency range where this characteristic is greater than zero (selection condition (3)).
4. Based on the specific technical implementation of the system, it is determined, i.e. places of entry and exit of corrective feedback.
5. We are building.
6. In the selected frequency range, we construct the logarithmic frequency response of the correcting link, subtracting it from the selection equation (4).
7. In the low-frequency region, where (selection condition (1)), we choose such that the selection equation (2) is satisfied: .
8. In the high-frequency region, inequality (2) is usually satisfied with an asymptote slope of 0 dB/dec.
9. The slope and length of the conjugating asymptotes are chosen based on the simplicity of the circuit implementation of the correction device.
10. We determine and design according to LFC schematic diagram corrective unit.

Example. Let and be given. The links covered by the feedback are determined. Needs to be built. The construction is made in Fig. 2. The initial system is minimum-phase. After construction, the calculated contour should be checked for stability.

An exact method for constructing the LFC of the corrective feedback link

If it is required to strictly maintain specified quality indicators, then it is necessary to calculate the exact values ​​of the frequency characteristics of the correction circuit.

Initial block diagram of uncorrected self-propelled guns

Converted block diagram

Adjusted ACS Equivalent block diagram

Let us introduce the following notation: , (1)

Then.

This allows you to use closure nomograms and find i.

Let's assume that they are known. We use the closure nomogram in reverse order:

, => , .

Then from the expression

LFC of an anti-parallel correction circuit:

To select the parameters of the correcting circuit, it is necessary to represent the LFC in an asymptotic form.

Construction of the LFC of a direct parallel corrective link

Let us transform the block diagram of the designed system into the form of Fig. 1.

In this case, it is advisable to consider the transfer function.

Frequency characteristics and are determined similarly to the frequency characteristics of a serial correction circuit.

In the frequency range where, characteristics

those. the correcting circuit does not affect the operation of the system, but in the frequency range where the characteristics

and the behavior of the system is determined by the parameters of the direct parallel circuit.

In the frequency range, where, when determining the LFC, it is advisable to present parallel-connected links in the form, where, .

We will build the LFC of the sequential correction device as before. Using the closure nomogram, we find and and, finally, .

Design of a correction device

CU quality criteria:

1. Reliability;
2. Low cost;
3. Simplicity of circuit implementation;
4. Sustainability;
5. Noise immunity;
6. Low power consumption;
7. Ease of production and operation.

Restrictions:

1. It is not recommended to install capacitors or resistors in one correction section, the values ​​of which differ by two to three orders of magnitude.
2. The LFC of the correcting links can have a frequency extension of no more than 2-3 decades, and an amplitude attenuation of no more than 20-30 dB.
3. The transfer coefficient of a passive two-port network should not be designed less than 0.05-0.1.
4. Resistor values ​​in the active correction links:

a) in the feedback circuit no more than 1-1.5 MOhms and no less than tens of kOhms;

b) in a direct channel circuit from tens of kOhms to 1 MOhm.

1. Capacitor ratings: units of microfarads hundreds of pcFarads.

Types of corrective links

1. Passive quadripoles ( R - L - C chains).

If, then the influence of the load on information processes can be neglected. .

The output signal in these circuits is weaker (or equal in level) to the input.

Example. Passive integro-differentiating link.

Where.

The predominance of the differentiating effect is ensured if the magnitude of the attenuation k<0.5 или иначе.

Since the resistance is the greatest, it is advisable to start calculating the elements of the correcting circuit with the condition given.

Let us denote from where;

let's define an intermediate parameter =>

hence, k = D.

The input impedance of the link is DC,

on alternating current

When matching resistance, a sufficient condition for direct current is the fulfillment of the relation,

on alternating current.

1. Active quadripoles.

If the amplifier gain >>1.

Example . Active real differentiating link of the first order.

Moreover, .

is selected during setup (setting the amplifier to zero).

on alternating current, and on direct current the input resistance is equal.

The output resistance of operational amplifiers is tens of ohms and is determined mainly by the values ​​of resistors in the collector circuits of the output transistors.

The circuit does not provide advance in the entire frequency range, but only in a certain band near the cutoff frequency of the system, usually located in the low and medium frequency range of the original ACS. The ideal link strongly emphasizes high frequencies, in the region of which there is a spectrum of interference superimposed on the useful signal, while the real circuit transmits them without significant amplification.

1. Differentiation transformer.

Transformer primary winding circuit resistance.

transformer transformation ratio.

Transfer function of the stabilizing transformer at

looks like

Where, inductance of the transformer in idle mode; .

1. Passive four-terminal AC circuits.

In AC circuits, DC correction circuits can be used.

The circuit diagram for connecting the corrective circuits is as follows:

Coordination of elementary corrective links

Produced:

1. For loads of active links (load currents of amplifiers should not exceed the maximum permissible values);
2. According to resistance, output input (at direct current and the upper frequency of the system operating range).

The load values ​​of operational amplifiers are specified in the technical conditions of their use and are usually more than 1 kOhm.

Note. Sign<< означает меньше как минимум в 10 раз.

Requirements for operational amplifiers:

1. Voltage gain.
2. Small zero drift.
3. High input impedance (100 kOhm 3 MOhm).
4. Low output resistance (tens of ohms).
5. Frequency range of operation (bandwidth).
6. Power supply voltage +5V, but not less than 10V.
7. Design (number of amplifiers in one housing).

Typical regulators

Types of regulators:

1. P-regulator (Greek. statos standing; the static regulator forms a proportional regulation law);

With increasing k p The steady-state error decreases, but the measurement noise increases, which leads to an increase in the activity of the actuators (they operate in jerks), the mechanical part wears out and the service life of the equipment is significantly reduced.

Flaws:

● inevitable deviation of the controlled variable from the specified value if the object is static;

● slow response of the regulator to disturbances at the beginning of the transition process.

1. I-regulator (integral);
2. PD controller (proportional-derivative);
3. PI controller (proportional-integral);
4. PID controller (proportional-integral-derivative);

1. Relay regulator.

A type D regulator is used in feedback, but a DI regulator is not used.

These regulators in many cases can provideacceptable management, easy to set up and cheap in mass production.

PD regulator

Block diagram:

forcing link.

real transfer function of the PD controller.

regulation law.

(1) without regulator;

(2) P-regulator;

(3) PD controller.

Advantages of the PD controller:

1. The stability margin increases;
2. Quality improves significantly

regulation (oscillation decreases

And time of transition

process).

Disadvantages of the PD controller:

1. Low control accuracy (static operation

the original system does not change when k p =1);

1. Interference at high frequencies increases and

system operation is disrupted due to saturation

amplifiers;

1. Difficult to implement in practice.

Implementation of a PD controller

The input and feedback signals are simply summed up.

If you change the signs of the input influence and feedback, then an inverter should be connected to the controller output.

The Zener diodes in the op-amp feedback are designed to limit the output signal level to a specified value.

In input circuits and are turned on as needed. It is advisable that. If excluded, the amplifier may enter saturation mode due to interference. Selectable (value up to 20 kOhm).

Transfer function of the controller via the control channel:

PI controller

(Greek isos smooth, dromos running; isodromic regulator)

At low frequencies the integrating effect predominates (there is no static error), and at high frequencies the effect from (the quality of the transient process is better than with the I-law of regulation).

regulation law.

1. lack of a regulator;
2. P-regulator;
3. PI controller.

Advantages:

1. Ease of implementation;
2. Significantly improves control accuracy in static conditions:

The steady-state error with a constant input action is zero;

This error is not sensitive to changes to object parameters.

Flaws : the astatism of the system increases by one and, as a consequence, the stability reserves decrease, the oscillation of the transition process increases, and increases.

Implementation of a PI controller

PID controller

At low frequencies the integrating effect predominates, and at high frequencies the differentiating effect prevails.

regulation law.

When installing a PID controller, a static system becomes astatic (static error is zero), however, in dynamics, astaticism is removed due to the action of the differentiating component, i.e. the quality of the transient process improves.

Advantages:

1. High static accuracy;
2. High performance;
3. Large margin of stability.

Flaws:

1. Applicable to systems described

differential equations of low

order, when an object has one or two poles

or can be approximated by a second model

order.

1. Management quality requirements are average.

Implementation of a PID controller

where, and.

We determine by the LFC of the operational amplifier. Then the transfer function of the real controller has the form:

The systems most often use a PID controller.

1. For objects with delay, the inertial part of which is close to the first-order link, it is advisable to use a PI regulator;
2. For objects with a delay, the inertial part of which is of order, the best controller is a PID controller;
3. PID controllers are effective in terms of reducing steady-state error and improving performance transient response, when the control object has one or two poles (or can be approximated by a second-order model);
4. When the control process is highly dynamic, as, for example, in a flow or pressure control system, a differentiating component is not used to avoid the phenomenon of self-excitation.

Calculation of combined control systems

Combinedsuch control in an automatic system when, along with a closed control loop for deviation, an external compensating device for reference or disturbing influences is used.

Invariance principlethe principle of compensation for dynamic and static errors, regardless of the form of the input action through the control channel or compensation of the disturbing influence.

invariant with respect to

disturbing influence, if after completion of the transition process,

determined by the initial conditions, the controlled variable and the system error are not

depend on this influence.

The automatic control system isinvariant with respect to

setting influence, if after completion of the transition process determined

initial conditions, the system error does not depend on this influence.

1. Calculation of compensating devices along the disturbance channel

Let the block diagram of the original system be transformed to the form shown

in Fig. 1.

Let us transfer the point of application of the disturbance to the system input (Fig. 2).

Let's write the equation for the output coordinate: .

Influence on the output function from the disturbance f will be absent if the condition is metabsolute invariancesystems to disturbing influence:

Condition for complete compensation of disturbance.

External controllers are used to obtain invariance over the disturbance channel with an accuracy of , since the order of the denominator is usually higher than the order of the numerator.

Example . Let the object and the controller behave as aperiodic links. The largest time constant usually belongs to the object.

Then

Graphs in Fig. 3.

The compensating circuit must have differentiating properties, and active differentiating properties at high frequencies (since the characteristic is partly located above the frequency axis).

Achieving absolute invariance is impossible, but the compensation effect can be significant even with a simple compensating circuit that provides implementation in a limited frequency range (in Fig. 3).

It is technically difficult and not always possible to measure disturbances, therefore, when designing systems, indirect methods for measuring disturbances are often used.

2. Calculation of systems with error compensation via the control channel

For this system, the block diagram of which is shown in Fig. 4, the following relations are valid:

transfer function by error.

We can achieve the condition of complete error compensation if we select a compensating circuit with the following parameters:

(1) condition of absolute invariance of the system to an error along the control channel.

Servo systems are implemented as astatic ones. Let's consider an example for such systems (Fig. 5).

At high frequencies, second-order differentiation in the compensating circuit leads to saturation of the amplifiers at high noise levels. Therefore, an approximate implementation is carried out, which gives a tangible regulatory effect.

Astatic systems are characterized by a quality factor transfer coefficient k determined at =1 and  = k.

If k =10, then the error is 10%, since

Low quality system (Fig. 6).

Let us introduce a compensating circuit with a transfer function

A tachogenerator can serve as such a circuit if

The entrance is mechanical. Implementation of a low-Q system

Simple.

Let us obtain from condition (1).

Then, having a system with astatism of the 1st order, we obtain a system with

astatism of the second order (Fig. 7).

Always Y lags behind the control signal; By entering, we reduce the error. The compensating circuit does not affect stability.

As a rule, the compensating link must have differentiating properties and be implemented using active elements. Exact fulfillment of the condition of absolute invariance is impossible due to the technical inexpediency of obtaining a derivative higher than the second order (a high level of interference is introduced into the control loop, the complexity of the compensating device increases) and the inertia of real technical devices. The number of aperiodic links in the compensating device is designed equal to the number of elementary forcing links. The time constants of aperiodic links are calculated based on the operating conditions of the links in a significant frequency range, i.e.

The principle of constructing a multi-circuit automatic control system with cascade connection of regulators is calledthe principle of subordinate regulation.

The synthesis of ACS of subordinate control with two or more circuits is carried out by sequential optimization of the circuits, starting with the internal one.

∆θ ,

hail

∆L,

dB

W and (p)

W A1 (p)

1/T p

1/T 0

The LFC method is one of the most common methods for synthesizing automatic control, since the construction of LFC, as a rule, can be carried out practically without computational work. It is especially convenient to use asymptotic “ideal” LFCs.

The synthesis process usually includes the following operations;

1. Construction of the LFC of the unchangeable part of the system.

The unchangeable part of the control system contains a control object and an actuator, as well as the main feedback element and comparison element. The LFC of the unchangeable part is built according to the transfer function of the open-loop unchangeable part of the system.

2. Construction of the desired part of the LFC.

The schedule of the desired LFC is made based on the requirements for the designed control system. The desired LFC L can be conditionally divided into three parts: low-frequency, mid-frequency and high-frequency.

2.1 The low-frequency part determines the static accuracy of the system, the accuracy in steady states. In a static system, the low-frequency asymptote is parallel to the x-axis. In an astatic system, the slope of this asymptote is –20 mdB/dec, where is the order of astatism ( = 1.2). The ordinate of the low-frequency part Lz is determined by the value of the transfer coefficient K of the open-loop system. The wider the low-frequency part of Lz, the more high frequencies are reproduced by the system without closed-loop attenuation.

2.2 The mid-frequency part is the most important, since it determines the stability, stability margin and, consequently, the quality of transient processes, usually assessed by indicators of the quality of the transient response. The main parameters of the mid-frequency asymptote are its slope and cutoff frequency cp (the frequency at which Lz crosses the abscissa axis). The greater the slope of the mid-frequency asymptote, the more difficult it is to ensure good dynamic properties of the system. Therefore, the most appropriate slope is -20 dB/dec and extremely rarely it exceeds -40 dB/dec. The cutoff frequency cf determines the performance of the system and the value of the overshoot value. The greater the cp, the higher the speed, the shorter the regulation time Tpp of the transient response, the greater the overshoot.

2.3 The high-frequency part of the LFC has little effect on the dynamic properties of the system. It is better to have the slope of its asymptote as large as possible, which reduces the required power of the executive organ and the influence of high-frequency interference. Sometimes, when calculating, the high-frequency LFC is not taken into account.

where is a coefficient depending on the amount of overshoot,

Must be selected according to the schedule shown in Figure 1.

Figure 18 - Graph for determining the permissible overshoot of the coefficient.

The ordinate of the low-frequency asymptote is determined according to the coefficient

the gain and the slope of the high-frequency asymptote of the transient, open-loop CAP.

3. Determination of the parameters of the correction device.

3.1 The graph of the LFC of the correcting device is obtained by subtracting the values ​​of the unchangeable graph from the graph value of the desired LFC, after which its transfer function is determined from the LFC of the correcting device.

3.2 Based on the transfer function of the controller, select electrical diagram to implement a corrective device and the values ​​of its parameters are calculated. The regulator circuit can be based on passive or active elements.

3.3 The transfer function of the correcting device, obtained in paragraph 3.1, is included in the generalized block diagram of the ACS. Using the generalized block diagram of the corrected ACS, with the help of a computer, graphs of transient processes are constructed, which should be no worse than the given ones.

Example:

6. Synthesis of an automatic control system using the method of logarithmic frequency characteristics.

Synthesis of linear self-propelled guns

Basic concepts of control system synthesis

All mathematical problems solved in the theory of automatic control can be combined into two large classes - problems of analysis and problems of synthesis of automatic systems.

In analysis problems, the structure of the system is completely known, all (as a rule) parameters of the system are specified, and it is necessary to estimate some of its statistical or dynamic properties. The analysis tasks include calculating accuracy in steady-state conditions, determining stability, and assessing the quality of the system.

Synthesis problems can be considered as the inverse of analysis problems: they require determining the structure and parameters of the system according to given quality indicators. The simplest synthesis problems are, for example, the problem of determining the transfer coefficient of an open loop based on a given error or the condition for the minimum of the integral estimate.

The synthesis of linear automatic control systems is understood as the choice of such a structural diagram, its parameters, and characteristics that meet, on the one hand, the specified indicators of quality and ease of technical implementation and reliability, on the other hand.

Features of synthesis

The ACS includes a control object and corrective devices (these are devices whose structure and parameters change in accordance with the synthesis task).

The setting of quality indicators is defined as the upper limit of acceptable quality indicators, i.e. the specified quality indicators determine the area of ​​decision-making. Therefore, during synthesis, an optimization criterion is selected that allows one to determine an unambiguous choice of the structure and parameters of the ACS.

For modern self-propelled guns, the synthesis procedure determines the approximate characteristics of the self-propelled guns, so the final result is obtained as a result of analysis (tuning, modeling) of the synthesized self-propelled guns.

Stages of ACS synthesis

The control object is analyzed, the static and dynamic characteristics of the object are determined.

An optimization criterion is determined based on the specified quality indicators of the ACS.

A structural diagram of the ACS is constructed, and technical means for its implementation are selected.

Synthesis of optimal dynamic characteristics.

Approximation of the optimal dynamic regime, i.e. selection of dynamic characteristics (desired) that meet specified quality indicators and simplicity of technical implementation of corrective devices.

Determination of the dynamic characteristics of correction devices that provide the desired dynamic characteristics of the entire system.

Selection of a scheme and method of technical implementation of correcting devices according to a given dynamic characteristic of the correcting device.

Analysis of synthesized self-propelled guns.

Synthesis of systems using the LFC method

There are two ways to enable correction devices:

Consistently to the control object.

Here W 0 (p) is the transfer function of the object, and W core (p)– transfer function of the correcting device.

Dignity sequential switching circuit is the simplicity of technical implementation.

Flaws: high sensitivity of this circuit to interference; strong dependence on changes in object parameters.

Parallel to some part of the object.

D

Flaws: The correcting device of this circuit is implemented by expensive circuits, in contrast to circuit (1).

As the dynamic characteristics by which the ACS is synthesized, the LFC of the object’s open-loop system is selected, because it is quite easy to determine the parameters of the object.

Desired LAC

When constructing the desired LFC, three frequency ranges are distinguished:

Low frequencies ( With). This frequency range reflects static characteristics.

Mid range ( With). Determines the dynamic characteristics of an object under stepwise input influence.

Treble Range ( With). This frequency range does not affect statics, but determines the dynamic characteristics of the object under rapidly changing input influence.

Modal regulator.

It is a method of root synthesis, namely, according to the desired location of the roots of the characteristic equation on the complex plane, a modal controller is constructed, which represents the negative feedback coefficients for each dynamic variable.

The object description is given:

We set the type of the desired polynomial D zhel (p) - in accordance with the given (desired) quality indicators.

Let's introduce feedback like:

Where
- characteristic equation of a system with a regulator.

Example: Given a system of equations

n 1 U x 1 x 2 x 3

We need to consider the controllability matrix:

The system is controllable since the rank is equal to the order of the system

We choose the desired polynomial of the same degree as the system:

D yellow (p)=(p+w 0 ) 3 =p 3 +3p 2 w 0 +3pw 0+ w 0 3

- quality assessment, where - transition time

When selected
we get:

K oc1 = 2; K oc2 = -1; K oc3 =5;

Controllability and observability.

A system is called controllable if, by changing any of the input signals, it is possible to achieve the desired value at the system output in a finite time.

without it the system will be uncontrollable, and with it -

controlled.

Controllability criterion.

In order for the system to be controllable, it is necessary and sufficient that the rank of the controllability matrix be equal to n (the order of the object).

In general, the controllability matrix is ​​rectangular. If the system has one input, then the matrix has the dimension
.

Observability.

A system is called observable if the state variables X can be reconstructed from the output signals Y.

Observability, in contrast to measurability, involves not only measuring state variables X, but also calculating non-measurable variables X from measured ones.

Measurability is the case when any variable can be directly measured.

Observability criterion.

In order for the system to be observable, it is necessary and sufficient that the rank of the observability matrix be equal to n (the order of the object).

The method of logarithmic frequency characteristics is used to determine the frequency transfer functions of corrective devices that bring the dynamic performance closer to the desired one. This method is most effectively used for the synthesis of systems with linear or digital correction devices, since in such systems the frequency characteristics of the links do not depend on the amplitude of the input signals. Synthesis of ACS using the method of logarithmic frequency characteristics includes the following operations:

At the first stage, using the known transfer function of the unchangeable part of the ACS, its logarithmic frequency response is constructed. In most cases, it is sufficient to use asymptotic frequency characteristics.

At the second stage, the desired logarithmic frequency response of the ACS is constructed, which would satisfy the requirements. The type of desired LFC is determined based on the purpose of the system, the time of the transition process, overshoot and error rates. In this case, typical frequency characteristics are often used for systems with different orders of astatism. When constructing the desired LFC, you must be sure that the type of amplitude response completely determines the nature of the transient processes, and there is no need to introduce the phase frequency response into consideration. The latter is true in the case of minimum-phase systems, which are characterized by the absence of zeros and poles located in the right half-plane. When choosing the desired logarithmic amplitude and phase characteristics, it is important that the latter provides the required stability margin at the system cutoff frequency. For this purpose, special nomograms are used, the appearance of which is shown in Fig. 1.

Figure 16‑1 Curves for selecting the stability margin in amplitude (a) and phase (b) depending on the amount of overshoot

Satisfactory quality indicators of the ACS in dynamic modes are achieved when the amplitude characteristic of the abscissa axis intersects with a slope of –20 dB/dec.

Figure 16‑2 Determination of PCU characteristics

At the last stage, from a comparison of the frequency characteristics of the uncorrected system and the desired frequency characteristics, the frequency properties of the correcting device are determined. When using linear correction means, the logarithmic frequency response of a sequential correction device (SCD) can be found by subtracting the LFC of the uncorrected system from the desired LFC of the ACS, that is

Hence

It should be noted that from the transfer function of a sequential correcting device it is easy to determine the transfer functions of the links in the direct or feedback circuit, with the help of which the dynamic performance of the automatic control system is corrected.

The next step is to determine the implementation method, circuit and parameters of the correcting device.

The last stage of synthesis of the correction device is the verification calculation of the ACS, which consists of constructing graphs of transient processes for the system with the selected correction device. At this stage it is advisable to use funds computer technology and modeling software systems VinSim, WorkBench, CircuitMaker, MathCAD.