Refinement of the structural scheme of the selection of the selection and calculation of its elements and parameters. Experimental study of the system or its individual parts in the laboratory and making appropriate corrections in its scheme and design. Design and production of the regulatory system. Adjusting the system in real conditions of work. Experienced operation.

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Lecture number 6. Synthesis of automatic control systems

SAU synthesis is the choice of the structure and parameters of the Sau, the initial conditions and input effects in accordance with the required quality indicators and the conditions of operation.

SAU design assumes the following steps:

1. Study of the regulatory object: drawing up a mathematical model, determination of parameters, characteristics and working conditions of the object.
2. Formulating claims for SAR.
3. Choosing a management principle; Defining a functional structure (technical synthesis).
4. The choice of elements of the regulatory scheme, taking into account static, dynamic, energy, operational, and other requirements and coordination of them among themselves according to static and energy characteristics (the procedure is not formalized - engineering creativity).
5. The definition of the algorithmic structure (theoretical synthesis) is made with the help of mathematical methods and on the basis of the requirements recorded in a clear mathematical form. Definition of regulation laws and calculation of corrective devices providing specified requirements.
6. Clarification of the structural scheme of the regulation system, selecting and calculating its elements and parameters.
7. Experimental study of the system (or individual parts) in the laboratory and making appropriate fixes in its scheme and design.
8. Design and production of the regulatory system.
9. Adjusting the system in real working conditions (experienced operation).

SAU design starts from the selection of the control object and the main functional elements (amplifiers, executive devices, etc.), that is, we develop the power of the system.

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

Appointment of corrective devices: ensure the desired accuracy of the system and obtain an acceptable nature of the transition process.

Corrective links are introduced into the system in various ways: sequentially, local OOS, direct parallel inclusion, external (outside control circuit) compensating devices, coverage of the entire SAU stabilizing OOS, inadequate main feedback.

Types of electrical corrective DC devices: active and passive four-pole DC, differentiating transformers, DC tachogenerators, tachometric bridges, etc.

By destination Corrective devices are classified:

1. Stabilizing - to ensure the stability of the Sau and improve their static and dynamic characteristics;
2. Compensating - reduce static and dynamic errors when building a saau by a combined principle;
3. Filtering - increasing systemism of systems, such as filtering higher harmonics when demodulating a forward channel signal;
4. Specialized - to give a system of special properties, allowing to improve system quality indicators.

SAU can be built according to the following structural schemes:

1. With a consistent corrective chain.

The amplifier should have a large input resistance so as not to shunt the output of the correction chain.

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

Fig.1.

Sequential correction is often used in stabilization systems or to correct the contour with corrective feedback.

Decreases.

1. With a counter-parallel corrective chain.

Fig.2.

Enters the entrance as a difference and deep saturation does not occur.

1. With a sequential-parallel corrective chain.

Fig.3.

1. With combined correction chains.

Synthesis of Sau subordinate control with two or more circuits is carried out by successively optimizing the contours, starting with the internal one.

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

Static calculation It is to choose the main links of the system included in its main chain, drawing up a structural scheme of the latter and determining the parameters of the main elements of the system (gain coefficients that provide the required accuracy, constant time of all elements, gear ratios, gear ratios of individual units, engine power). In addition, this includes calculation and design of magnetic and semiconductor amplifiers and the selection of transistor or thyristor converters, engines, sensitive elements and other auxiliary systems of systems, as well as the calculation of accuracy in the steady mode of operation and sensitivity of the system.

Dynamic calculationincludes a large range of issues related to the stability and quality of the transition process (speed, characteristics of the development and dynamic accuracy of the system). In the calculation process, adjustment chains are selected, the places of switching on and the parameters of the latter are determined. The calculation of the transition curve or simulation of the system to clarify the qualitative indicators obtained and accounts for some non-linearities.

Platforms on which stabilizing algorithms are built:

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

Methods of regulators synthesis:

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

Classic synthesis of regulators

The classic structural control scheme of the object is shown in Fig. 1. Typically, the regulator is included in front of the object.

Fig. 1. Classic structural object management scheme

The control system task is to suppress the effect of external perturbation and provide high-quality transient processes. These tasks are often contradictory. In fact, we need to stabilize the system so that it has the required gear ratios according to the specifying effect and on the perturbation channel:

, .

To do this, we can use only one regulator, so such a system is called the system with one degree of freedom.

These two gear ratios are associated with equality.

Therefore, changing one of the gear ratios, automatically change and second. Thus, it is impossible to form independently and the solution will always be some compromise.

Let's see if it is possible to provide a zero error in such a system, that is, absolutely accurate tracking of the input signal. Error gear ratio is equal

In order to make a mistakealways It was zero, it is required that this gear ratio is zero. Since its numerator is not zero, we immediately get that the denominator must contact infinity. We can only affect the regulator, so we get. In this way,to reduce the error you need

increase the regulator gain.

However, it is impossible to increase the amplification to infinity. First, all real devices have the maximum valid values \u200b\u200bof the input and output signals. Secondly, with a large strengthening of the circuit, the quality of transition processes deteriorates, the effect of perturbations and noise increases, the system can lose stability. Therefore, in a scheme with one degree of freedom, it is impossible to provide a zero tracking error.

Let's look at the task in terms of frequency characteristics. On the one hand, for high-quality tracking of the specifying signal, it is desirable that the frequency response is approximately equal to 1 (in this case). On the other hand, from the point of view of robust stability, it is necessary to ensure at high frequencies where the modeling error is large. In addition, the transmission function for indignation should be such that these perturbations suppress, ideally, we must ensure.

Choosing a compromise solution, usually come as follows:

● at low frequencies achieve the implementation of the condition, which provides good tracking of low-frequency signals; At the same time, that is, low-frequency perturbations are suppressed;

● At high frequencies tend to do to ensure robust stability and suppression of measurement noise; At the same time, that is, the system actually works as open, the regulator does not respond to high-frequency interference.

Calculation of linear continuous SAU for a given accuracy

In the steady mode of operation

One of the basic requirements that Sau must meet is to ensure the necessary accuracy of playing the specifying (control) signal in the steady operation mode.

The procedure for apostation and the transmission coefficient of the system are based on the requirements for accuracy in the steady mode. If the transmission coefficient of the system, determined by the required value of statism and Quality (in the case of astatic SAU), is so large, which significantly makes it even simple to stabilize the system, it is advisable to increase the order of apostatam and it is reduced to zero, the set-down error regardless of the value of the system coefficient . As a result, it becomes possible to choose this coefficient, proceeding only for considerations of the stability and quality of transient processes.

Let the SAR structural scheme shown to the form

Then in the quasi-secret mode of the work of SAR, the mismatch represents in the form of a convergent row

where the role of weight constants perform.

Obviously, such a process can only take place if it is changed and enough smooth function.

If you present the transfer function of the open system in the form of

then at r \u003d 0

at r \u003d 1

at r \u003d 2

at r \u003d 3

The low-frequency part of the logarithmic amplitude frequency characteristics determines the accuracy of the system operation when working out slowly changing control signals in the steady state and is determined by error coefficients. Error coefficients no longer have a significant effect on the accuracy of SAU, and they can not be taken into account in practical calculations.

1. Calculation of the established mode of operation of SAR on specified mismatch coefficients (errors)

The accuracy of the system in the steady mode is determined by the magnitude of the transmission coefficient of the opening system, which is determined depending on the form of setting the requirements for the accuracy of the system.

The calculation is carried out as follows.

1. Static SAR. Here is the magnitude of the factor of the positional error, which is determined by :.

dB

20 LGK PC.

ω, s -1

1. Astatic systems of 1st order.

In this case, the coefficient for which is determined

If coefficients are specified and, which determines the position of the low-frequency asymptotation of the lated open system with an inclination -20 dB / dec, and the second asymptota has a slope -40 dB / deck with a conjugating frequency (Fig. 1).

Fig.1.

1. Astatic 2nd order systems.

According to the specified coefficient, we definek PC:

dB

ω, s -1

2. Calculation of the established mode of operation of the SAR for a given maximum value of the mismatch (error) of the system

Based on the permissible value of the established error and the type of control exposure, the parameters of the low-frequency part of the strip system are selected.

1. Let the allowable maximum error in the harmonic effects with the amplitude and frequency and the order of the system.

Then the low-frequency asymptota of the strip system should pass at no reference point with coordinates:

(1)

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

1. Let the allowable maximum error at maximum speed and maximum acceleration of the input effect and the order ofr system.

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

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

. (2)

Equating the amplitude values \u200b\u200bof the speed and acceleration obtained by the differentiation of the expression (2) specified values \u200b\u200band, we obtain

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

point in coordinates and

With single negative feedback,

With inadequate feedback.

If the speed of the signal at the input is maximum, and the acceleration decreases, the control point will move in a straight line with the tilt -20 dB / dec in the frequency range. If the acceleration is equal to the maximum value, and the speed decreases, the checkpoint moves in a straight line with the inclination of -40db / dec. in the frequency range.

The area below the control point in and two directs with inclons -20DB / Dec and -40DB / Dec is a benchmark for the lacc of the tracking system. Since accurate lacch passes below the intersection point of two asymptotes by 3 dB, then the desired characteristic when it should be raised up to this value, i.e.

In this case, the desired value of the speed of speed, and the frequency at the intersection point of the second asymptotes with the frequency axis (Fig.2)

In the event that the control action is characterized only by maximum speed, the speed of the speed system at a given error value:

If only the maximum acceleration of the signal and the magnitude of the error is set, then the goodness at acceleration:

Fig.2.

1. Let the maximum static error on the control channel (step input exposure, static system via control channel).

Fig.3.

Then the value is determined from the expression. The static accuracy of the automatic system can be determined from the equation:

where is the static accuracy of the closed system,

- deviation of the adjustable value in the open system,

- Transmission coefficient of opening system required to ensure a given accuracy.

1. Let the maximum allowable static error along the perturbation channel (perturbing exposure stepped, static system over the perturbation channel, Fig.3).

Then the value is determined from the expression:

where - the transmission coefficient of the open system on the perturbation channel,

where - the error of the system without a regulator.

In static control systems, the established error caused by a permanent perturbing effect decreases compared to the open system in 1+. At the same time, the transmission coefficient of the closed system is also reduced in 1+ times.

1. Let a permissible high-speed error from the control exposure (the input effect changes at a constant speed, a system of astatic first order).

The tracking systems are usually astatic first order. They work with a variable control exposure. For such systems in the steady mode, the most characteristic is the change in the input effect on the linear law.

Then the quality of the speed system is determined from the expression:

Since the established error is determined by the low-frequency part of the LACH, the low-frequency asymptota of the desired lacc can be built according to the calculated value of the transfer coefficient.

3. Calculation of the established mode of operation of the SAR on a given maximum permissible error of the system with inadequate feedback

Let a priori information about the input signal minimize:

1. Maximum module value of the first derivative of input effects (maximum tracking speed) -;
2. Maximum module value of the second derivative of the input effect (maximum speeding tracking) -;
3. The input effect 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 playing a useful signal in the steady operation mode.

Requirement of reproduction accuracy is most simply formulated for the harmonic input effect of the equivalent to the real input signal:

in the assumption that the amplitude and frequency are specified, and the initial phase has an arbitrary value.

Set the connection between the valid error of playing the input and system parameters and the input signal.

Let the structural scheme continuous SAU 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 (unmistakable) output function.

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

Display the last expression into Laplace transformations:

Display Fourier transformations in the space:

In the low frequency area (, -table time of the feedback chain), then

the maximum amplitude of the error is determined by expression:

In real systems at low frequencies, usuallybecause the requirement should be required; Mathematical expression to determine converted to the control frequency () to mind

and so that the output function is reproduced with the maximum error of the no more specified, the lacc of the designed system should not be held below the control point with coordinates and

4. Calculation of the established mode of operation of static SAU by limiting transitions

Statement

Let the generalized structural scheme of static SAR:

where, here the polynomials of numerals and denominators do not contain a multiplierp. (free members of them are equal to one),

- transmission coefficient of the regulator,

- transmitting factor of the object by control channel,

- transmission coefficient of feedback,

- transmitting factor of the object by the perturbation channel,

moreover, in the first approximation, static and dynamic gear ratios are taken equal to the nominal input effect correspond to the nominal value of the output function via the control channel, and let the value of the step of perturbing effect and the permissible static error over the perturbation channel in% of the nominal value of the output function.

Then the transmission coefficients of the system through control channels and perturbations in the steady mode are equal to static transfer coefficients of a closed system and are determined by formulas:

(1)

The equations of statics over control channels and perturbations are

(2)

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

(3)

Methods for increasing the static accuracy of SAU

1. Increase the transmission coefficient of opening system in staticsystems.

Where,.

However, the stability conditions deteriorate with increasing, that is, the errors are increasing in dynamics.

1. Introduction to the integrated component regulator.

2.1. Application and regulator :.

In this case, the system becomes anstic over control channels and perturbations, and the static error becomes zero. The strip system will go much steeper from the source, and the phase shift increases by - 90 degrees. The system may be unstable.

2.2. Installing a PI regulator :.

Here the static error is zero, and the stability conditions are better than the system with and the regulator.

2.3. Using the PID regulator :.

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

1. Introduction to the non-baking feedback system if accurate playback of the information level of the input signal is required.

We believe that - static links. , it is necessary to choose such

To; .

1. Scaling input

impact.

Here.

The output function will be equal to the information level of the input effect, if, from here, where.

1. Apply the principle of compensation over control channels and indignation.

The calculation of compensating devices is set out in the section "Calculation of combined control systems".

Calculation of the dynamics of saau

Synthesis SAU by lchh

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

• analytical synthesis methods that use analytical expressions that connect the quality of the system with the parameters of corrective devices;
• grafo-analytical.

The most convenient of the graph-analytical methods of synthesis is a classic universal method of logarithmic frequency characteristics.

Essence of the method is as follows. First, the asymptotic lacc of the initial system is built, then the desired lacked open system is built; The stroke corrective device must change the shape of the lacc of the source system so that the progress of the adjusted system.

The most difficult and responsible stage in the synthesis is the construction of the desired lach. When constructing, it is assumed that the synthesized system has a single negative feedback and is the minimum phase system. The quantitative relationship between the quality indicators of the transition function of minimal-phase systems with single OOS and the lated open system is mounted on the basis of the nomograms of Cebria-Mayer, V.Volodovnikova, A.V. Fateeva, V.A. Besseker.

The desired lacch 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 SAU operation in the steady mode. In the static system, the low-frequency asymptota parallel to the frequency axis, in the astatic systems, the slope of low-frequency asymptotes is -20 * dB / dec, where  - order of apostamia ( \u003d 1, 2, 3, ...). The mid-frequency part is the most important, since it basically determines the dynamics of processes in the system. The main parameters of the mid-frequency asymptotes are its slope and the cutoff frequency. The more the slope of the mid-frequency asymptotes, the harder it is to provide good dynamic properties of the system. Therefore, it is expedient to the tilt -20 dB / deck and extremely rare it exceeds -40 dB / dec. The cut frequency determines the speed of the system. The more, the higher the speed (the less). The high-frequency part of the desired lacc slightly affects the dynamic properties of the system. Generally speaking, it is better to have a greater inclination of its asymptotes, which reduces the required power of the executive body and the effect of high-frequency interference.

The desired lacch is based on the requirements for the system: the requirements for static properties are set as ordacy and the transmission coefficient of the open system; Dynamic properties are most often set in the maximum allowable value of the overalling and regulatory time; Sometimes they define a limit in the form of the maximum permissible acceleration of the adjustable value during the initial mismatch.

Methods of building the desired lach: Building in V.Solodovnikov, the use of typical lach and nomograms for them, building according to E.A. Sankovsky - G.G.Sigalov, a simplified construction, building according to V.A. Besseker, according to the method A. V.Pateeva et al. Methods.

The advantages of frequency methods:

● Frequency characteristics reflecting a mathematical model of an object can be relatively simply obtained by experimental way;

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

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

Synthesis of the desired lach

Theoretical and experimental studies have been established that the lacc of open control system is stable in a closed state, almost always crosses the frequency axis by a plot that has a slope -20 dB / dec. The inverting axis of the frequency area with a slope -40 dB / dec register or -60 dB / deck is possible, but it is rarely used, because such a system is stable at a very low gear ratio.

The most rational form of the lacc of open system, stable in a closed state, has slopes:

• low-frequency asymptotta 0, -20, -40 dB / deck (determined by the order of the system of system);
• asymptotta mating low-frequency with mid-frequency asymptotes may have slopes -20, -40, -60 dB / deck;
• mid-frequency asymptota -20 dB/ Dec;
• asymptota, mating mid-frequency with a high-frequency area of \u200b\u200bLachh, as a rule, has a slope -40 dB / deck;
• the high-frequency area of \u200b\u200bLachh is built parallel to the asymptoms of the high-frequency site of the lacc of the initial open system.

When constructing the desired lchedh, they proceed from the following requirements:

1. The adjusted system must satisfy the specified quality indicators (permissible error in the steady mode, the required supply of stability, speed, overshooting and other indicators of the quality of transient processes).
2. The form of the desired LFH should, if possible, differ little from the lchedh uncorrected system to simplify the stabilizing device.
3. It should be strive to ensure that at high frequencies it does not take higher than a non-incurred system by more than 20-25 dB.
4. The low-frequency part of the desired lacc should coincide with the lacc of the uncorrected system, since the transmission coefficient of open uncorrected system is chosen taking into account the required accuracy in the steady mode.

The construction of the desired LFH can be considered complete if all the requirements for the quality of the system are satisfied. Otherwise, it is necessary to return to the calculation of the steady operation mode and change the parameters of the main chain elements (select the engine of other power or less inertial, use the amplifier with a lower time constant, turn on the rigid negative feedback, covering the most inertial elements of the system, etc.) .

Algorithm for building the desired lchh

1. Selection of frequency cutL w (w).

If the transition proactically, V.Solodovnikova or A.V.Fateeva is used; If the oscillatory indicator is set, then the calculation is carried out according to the method of V.A. Besseker.

The basis for the construction of the name of the quality of V.Volodovnikov, a typical real frequency characteristic of a closed saau is based (Fig. 2). For static systems ( \u003d 0), for the astatic systems ( =1, 2,…) .

This method assumes that the ratio is observed.

Dynamic quality indicators and, which are associated with the parameters of the real frequency response, a closed SAU of the quality diagram V.V. Solodovnikova (Fig. 3). According to the curve (Fig. 3), the corresponding value is determined. Then, the via and curve is determined by the value that is equal to the specified, we obtain, where - the value of the cutting frequency at which the control time does not exceed the specified value.

On the other hand, it is limited to the permissible acceleration of the adjustable coordinate. Recommended where - initial mismatch.

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

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

As a rule, does not exceed more than ½ decades. This is due to the complication of corrective devices, the need to introduce into a system of differentiating links, which reduces reliability and noise immunity, as well as due to the limit on the maximum acceleration of the regulated coordinate.

Cutting frequency can be enhanced only by magnification. Static accuracy increases, but the conditions of sustainability deteriorate.

The decision to choose from the choice should have a sufficient justification.

1. We build mid-frequency asymptot.

1. Medium-grade asymptotype Matter with low-frequency asymptota So that in the frequency interval in which, have an excess phase. Excess phase and excess module Determine the nomogram (Fig. 4). Mattering asymptota has a slope -20, -40 or -60 dB / dec =0 ( - the procedure for the system of system); -40, -60 dB / dec \u003d 1 and -60 dB / dec for  \u003d 2.

If the excess phase is less, the mating asymptotus should be shifted to left or reduce its slope. If an excess phase is more permissible, the mating asymptotus is shifted to the right or increase its tilt.

The initial conjugating frequency is determined from the expression.

1. Medium-grade asymptotype Matter with a high-frequency part So that in the frequency interval, where, the excess phase was. The mating frequency is determined by the ratio.

If on the mating frequency<, то сопрягающую асимптоту смещают вправо или уменьшают ее наклон.

If\u003e, then the mating asymptot is shifted to the left or increase its tilt. The recommended difference should be several degrees. Right mating frequency of mating asymptotes.

As a rule, the slope of this asymptotype is -40 dB / dec, and permissible difference. Check is performed at the frequency at which.

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

This part of the characteristics affects the smoothness of the system.

So, at the first stage of the construction of the frequency on which the mid-frequency asymptota with mating asymptotes is conjugated, are from the conditions. At the second stage, the values \u200b\u200bof the conjugating frequencies, taking into account the excess phase, are specified. In the third stage, all mating frequencies are corrected by the condition of their proximity to the mating frequency of the original system, i.e., if these frequencies are slightly different from each other.

Synthesis of the corrective chain of a sequential type

In Scheme Fig. 1, the parameters of the correction chain can be obtained from here:

Let us turn to the logarithmic frequency characteristics :,

At the high frequencies of the LACH regulator "By default" should not exceed 20 dB under the condition of noiselessness. The fundamental principle of the structural and parametric optimization of the feedback SAU: The regulator must contain a dynamic link with a gear ratio equal to or close to the return gear function of the control object.

Consider the example of the calculation of the serial correction chain.

Let it be necessary to adjust the static system. Suppose that we are built. We believe that the system with minimal phase links, therefore, the phase-frequency response is not systemic (Fig. 2).

Now it is easy to reproduce the parameters of the correction chain. Most often active corrective devices and passiveRC -Spi. Based on physical representations, we build a circuit shown in Fig. 3.

Weakened the signal of the dividerR 1- R. 2 at high frequencies corresponds to a wavelength * on.

Where,

At high frequencies does not make distortion - a positive factor. Circulation frequency We have the ability to move to the left using a corrective chain and provide the required stability and quality of the system.

The advantages of serial ku:

1. The simplicity of the correction device (in many cases is implemented in the form of simple passiveRC -Conturov);
2. Easy to turn on.

Disadvantages:

1. The effect of sequential correction is reduced during operation when the parameters change (gain coefficients, constant time), so with consistent correction to the stability of the parameters of the elements, elevated requirements are made, which is achieved by the use of more expensive elements;
2. Differentiating phase-recoveryRC -Contacities (algorithms in microcontrollers) are sensitive to high-frequency interference;
3. Consecutive integratingRC - Constures contain more cumbersome capacitors (the implementation of large time constants is required) than contours in the feedback circuit.

Usually used in low-power systems. This is explained, on the one hand, the simplicity of consecutive corrective devices, and on the other hand, the inexpediency of use in these systems of bulky, commensurate with the size of the actuator engine of such parallel corrective devices as a tach generator.

It should be borne in mind that due to the saturation of the amplifiers, it is not always necessary to form the formation of the desired LACH in the low and medium-sized band due to the sequential inclusion in the system of integrating and integration circuits or any other elements with similar characteristics. Therefore, often for forming in the low and mid-frequency range apply feedback.

Synthesis of corrective chains of a counter-parallel type

When choosing a place of inclusion of the correction circuit, follow the following rules:

1. It is followed by those links that significantly adversely affect the appearance of the desired lach.
2. The slope of the string of links that are not covered by feedback, choose close to the inclination in the range of medium frequencies. Performing this condition allows you to have a simple correction chain.
3. Corrective feedback should cover as many links as possible with nonlinear characteristics. The limit needs to strive to ensure that there are no elements with nonlinear characteristics among links that are not covered by feedback. Such an inclusion of feedback can significantly reduce the influence of nonlinearity of the characteristics of the elements covered by feedback, to work the system.
4. Feedback should cover units with a large gear ratio. Only in this case the feedback will be effective.
5. The feedback signal must be removed from the element with sufficient power so that the feedback does not load it. The feedback signal should, as a rule, are supplied to the input of the system elements that have a large input resistance.
6. When choosing an inclusion of feedback inside the contour with corrective feedback, it is desirable that the slope of the lach in the frequency range was 0 or 20 dB / dec. Performing this condition allows you to have a simple correction chain.

It is often covered by the enhancement path of the system or the coverage of the system's power. Corrective feedbacks are usually used in powerful systems.

Advantages of CEP:

1. The dependence of the quality indicators of the system from changes in the parameters of the elements of an unchangeable part of the system is reduced, since in a substantial frequency range, the transfer function of the system covered by feedback is determined by the reverse magnitude of the transfer function of the counter-parallel corrective device. Therefore, the requirements for the elements of the source system are less rigid than when consistent correction.
2. The nonlinear characteristics of the elements covered by feedback are linearized, since the transfer properties of the system covered sections are determined by the circuit parameters in the feedback circuit.
3. Power supply of counter-parallel corrective devices Even when it requires high power, it does not cause difficulties, since the feedback usually begins on the terminal links of the system with a powerful output.
4. Meet-parallel corrective devices work at a lower interference level than successive, since the signal coming on them passes through the entire system that is a low-frequency filter. Due to this, the effectiveness of the action of the counter-parallel corrective devices when overlapping an error signal decreases less than consecutive corrective devices.
5. In contrast to the serial corrective device, feedback allows you to realize the largest time constant of the desired lacc with relatively small values \u200b\u200bof our own time constants.

Disadvantages:

1. Meet-parallel ki often contain expensive or bulky elements (for example, taogenerators, differentiating transformers).
2. The summation of the feedback signal and the error signal should be implemented so that the feedback does not shut the input of the amplifier.
3. The contour formed by corrective feedback may be unstable. Reducing stability reserves in internal circuits worsens the reliability of the functioning of the system as a whole.

Definition methods:

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

After calculating the counter-parallel corrective circuit, check the stability of the internal circuit. If you open the main feedback, and the inner contour is unstable, then the elements of the system can fail. If the inner contour is unstable, then its stability is provided by a sequential correction chain.

Approximate method for constructing a lfh corrective negative feedback

Let the structural scheme of the projected

Systems are shown to the form depicted

In Fig.1.

- corrective feedback;

- gear

function open source (non-uncorrected)

systems.

For such a block diagram, the transfer function of the adjusted open system.

In the frequency range, where, The equation will be written so

Those.

Selection condition; (one)

- choice equation (in low and high frequency bands) (2)

In the frequency range, where,

Selection condition; (3)

we get

i.e.,

from where - choice equation(in the middle frequency range). (4)

Then the construct algorithm is as follows:

1. We build.
2. We build.
3. We build and determine the frequency range, where this characteristic is greater than zero (the selection condition (3)).
4. Based on the specific technical implementation of the system, it is determined, i.e. Entry and corrective feedback output.
5. We build.
6. In the dedicated frequency range, we build a logarithmic frequency characteristic of the corrective link, deducted from the equation (4).
7. In the low-frequency region, where (the selection condition (1)), choose such that the choice equation (2) was performed :.
8. In the high-frequency region, inequality (2) is usually performed when the asymptotes of 0 dB / dec.
9. The slope and the length of the mating asymptotes are chosen based on the simplicity of the circuit implementation of the corrective device.
10. By lachm, we define and design a fundamental scheme of the corrective link.

Example. Let them and. The links covered by feedback are defined. Required to build. The construction is made in Fig. 2. The source system is minimally phase. After construction, check the calculated contour for stability.

Accurate method for constructing lchhl corrective feedback

If it is necessary to strictly withstand the specified quality indicators, then you need to calculate the exact values \u200b\u200bof the frequency characteristics of the corrective chain.

Source structural scheme Non-uncorrected SAU

Transformed structural scheme

Corrected SAU Equivalent Structural Scheme

We introduce notation:, (1)

then.

This allows you to use the nomograms of closures and find and.

Suppose that you are known. We use the short-circuit nomogram:

, => , .

Then from the expression

Lfh counter-parallel corrective chain:

To select the adjusting chain parameters, it is necessary to present in asymptotic form.

Building a lchh direct parallel corrective link

The structural scheme of the projected system is converting the form of Fig.1.

In this case, it is advisable to consider the gear ratio.

Frequency characteristics are determined similar to the frequency characteristics of the serial corrective chain.

In the frequency range, where, characteristics

those. Corrective chain does not affect the operation of the system, and in the frequency range, where, characteristics

and the behavior of the system is determined by the parameters of the straight parallel chain.

In the frequency range, where it is advisable to determine the lched and present the links parallel to the form in the form where ,.

Lchhi serial corrective device and construct as before. Using the closure nomogram, we will find and finally.

Designing a corrective device

Quality Criteria QU:

1. Reliability;
2. Low cost;
3. Simplicity of circuit implementation;
4. Stability;
5. Noise immunity;
6. Small energy consumption;
7. Easy production and operation.

Restrictions:

1. Installation is not recommended in one corrective link of condensers or resistors, which differ in two or three orders.
2. Lacha corrective links can have a frequency length of not more than 2-3 decades, weakening over the amplitude of not more than 20-30 dB.
3. The transmission coefficient of the passive quadrupole should not be designed less than 0.05-0.1.
4. Nominal resistors in active corrective links:

a) in the feedback circuit - no more than 1-1.5 mΩ and at least tens of com;

b) in the circuit of the direct channel - from dozens com by 1 mΩ.

1. Capacitator ratings: MKF units - hundreds of PCFrades.

Species of corrective links

1. Passive four-pole (R - L - C - Tspi).

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

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

Example. Passive integro differentiating link.

where.

The predominance of the differentiating effect is ensured if the amount of weakeningk.<0.5 или иначе.

Since resistance is the greatest, then the calculation of the elements of the corrective chain is advisable to start with the conditions, setting.

Denote, from where;

determine the intermediate parameter \u003d\u003e

hence, k \u003d d.

Permanent current input resistance,

on alternating current

When agreeing on resistance, a sufficient condition on constant current is the fulfillment of the relationship,

on alternating current.

1. Active four-pole.

If the transmitter coefficient of amplifier \u003e\u003e 1.

Example . Active actual first-order differentiating link.

Moreover,

- Selected when adjusting (setting the zero amplifier).

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

The output resistance of the operating amplifiers is tens of OM and is determined mainly, the amounts of resistors in the collector circuits of the output transistors.

The scheme provides ahead not in the entire frequency domain, but only in a certain band near the slice frequency of the system, located usually in the range of low and medium frequencies of the original Sau. The perfect link strongly emphasizes high frequencies, in the area of \u200b\u200bwhich there is a spectrum of interference with the useful signal, while the actual circuit transmits them without substantial gain.

1. Differentiation transformer.

Resistance to the transformer primary winding circuit.

- transformer transformation coefficient.

The transfer function of the stabilizing transformer when

has the form

Where, - the inductance of the transformer in idle mode; .

1. Passive four-solids of alternating current.

Corrective DC circuits can be used in the AC circuits.

The adjustment circuit circuit is as follows:

Coordination of elementary corrective links

Produced:

1. On the loads of active links (loading currents of amplifiers should not exceed the maximum permissible values);
2. By resistance, the output is the input (on the constant current and the upper frequency of the system of the system).

The values \u200b\u200bof the loads of operating amplifiers are set in the technical conditions of their use and usually make up more than 1 com.

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

Requirements for operating amplifiers:

1. Voltage gain.
2. Small scratch drift.
3. Great input resistance (100 com - 3mΩ).
4. Small output resistance (dozens).
5. Frequency range of operation (bandwidth).
6. Power supply voltage + 5V, but not less than 10V.
7. Constructive design (the number of amplifiers in one case).

Typical regulators

Types of regulators:

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

With increasing K p The steady error decreases, but measurement noise increases, which leads to an increase in the activity of the actuators (work by jerks), the mechanical part is wearing and significantly reduces the service life of the equipment.

Disadvantages:

● the inevitable deviation of the adjustable value from the specified value if the static object;

● Slowed regulator reaction to perturbing effects at the beginning of the transition process.

1. - and regulator (integral);
2. - PD-regulator (proportional and differential);
3. - PI regulator (proportional to integral);
4. - PID controller (proportional to integral-differential);

1. Relay regulator.

Type D control is used in feedback, and di does not apply.

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

PD-regulator

Structural scheme:

forcing link.

- Real PD-Regulator Personal Reception.

- the law of regulation.

(1) - without a regulator;

(2) - P-regulator;

(3) - PD regulator.

Advantages of PD regulator:

1. Increases stability;
2. The quality is significantly improved

regulation (oscillating decreases

And time of transition

process).

Disadvantages of PD regulator:

1. Low adjustment accuracy (work status

the source system does not change whenk n \u003d 1);

1. Interference at high frequencies amplifies and

violates the operation of the system due to saturation

amplifiers;

1. It is difficult to implement in practice.

Implementation of PD regulator

Input and feedback signals are simply summed up.

If you change the signs of the input and feedback, the inverter should be connected to the regulator output.

Stabilians in the feedback of the operating amplifier are designed to limit the level of the output signal of the specified value.

In the input circuits and are included as needed. It is desirable to. If you exclude, the amplifier due to the action of the interference can enter the saturation mode. Separate (value up to 20 com).

Transmission controller on control channel:

PI regulator

(Greek. Isos - Smooth, Dromos - Running; Isodromic regulator)

At low frequencies, an integrating effect prevails (there is no static error), and at high frequencies - the effect of the transition process is better than with the and-law of the regulation).

- the law of regulation.

1. - lack of a regulator;
2. - P-regulator;
3. - Pi-regulator.

Advantages:

1. Simplicity;
2. Significantly improves the control accuracy in statics:

The steady error at a constant input effect is zero;

This error is insensitive to changes in the object parameters.

disadvantages : Instation of the system per unit increases and, as a result, reduction of stability reserves, the oscillativity of the transition process increases, increases.

Realization of the PI Regulator

PID regulator

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

- the law of regulation.

The static system when installing the PID controller becomes an astatic (static error is zero), however, in dynamics, apostatam is removed due to the action of the differentiating component, i.e. the quality of the transition process is improved.

Advantages:

1. High static accuracy;
2. High speed;
3. Large stability stock.

Disadvantages:

1. Applicable for systems described

differential equations are low

order when the object has one or two poles

or can be approximated by the second model

order.

1. Requirements for quality management average.

Implementation of the PID Regulator

where, and.

By the lacht operating amplifier determine. Then the transfer function of the real regulator is viewed.

In systems, the PID is most often used.

1. For objects with delay, the inertial part of which is close to the first order link, it is advisable to apply Pi - regulator;
2. For objects with delay, the inertial part of which has an order, the best regulator is the PID - regulator;
3. PID - regulators are effective from the point of view of reducing the established error and improving the type of transitional characteristic when the control object has one or two poles (or it can be approximated by the second order model);
4. When the regulatory process is characterized by high dynamism, as, for example, in a stream or pressure system, the differentiating component does not apply to avoid the phenomenon of self-excitation.

Calculation of combined control systems

Combined - Such management in the automatic system, when, along with a closed contour circuit, an external compensating device is used according to a definition or disturbing effects.

The principle of invariance - The principle of compensation for dynamic and static errors, regardless of the form of input effects on the control channel or compensation of indignant effects.

invariant with respect to

indignant effectif after completing the transition process,

determined by the initial conditions, the adjustable value and the error of the system is not

depend on this impact.

Automatic regulation system isinvariant with respect to

specifying exposureif after completing the transition process determined by

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

1. Calculation of compensating devices on the perturbation channel

Let the structural scheme of the source system transformed into the form shown

in Fig.1.

We transfer to the input system the point of the indignation application (Fig. 2).

We write the equation for the output coordinates :.

Impact on the output function from the perturbationf. will be absent if the condition is satisfiedabsolute invariance Systems to indignant influence:

The condition for full compensation of indignation.

External regulators are used to obtain invariance over the perturbation 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 regulator behave as aperiodic links. The greatest time constant, as a rule, belongs to the object.

Then

Graphics in fig. 3.

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

The achievement of absolute invariance is impossible, but the compensation effect may be significant even with a simple compensating chain that provides realization in a limited frequency range (in Fig. 3).

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

2. Calculation of systems with error compensation for control channel

For this system, the structural circuit of which is depicted in Fig. 4, Fair the following ratios:

- Error gear ratio.

We can achieve the Terms of Compensation of the error if you select the compensating circuit with the parameters:

(1) - the condition of the absolute invariance of the system to error over the control channel.

The tracking systems are implemented by astatic. Consider an example for such systems (Fig. 5).

In the high frequency area, the second order differentiation in the compensation circuit leads to saturation of amplifiers at a high level of interference. Therefore, an approximate implementation is carried out, which gives a tangible control effect.

Astatic systems are characterized by good quality - transmission coefficientk. determined by \u003d 1 and  \u003d k.

If K. \u003d 10, then a 10% error, since

Low quality system (Fig. 6).

We introduce a compensation circuit with a gear ratio

Such a chain can serve as a tachogenerator if

Entry mechanical. Implementation of a low quality system

Simple.

Let it be from condition (1) we will get.

Then, having a system with 1-order apostaism, we get the system with

aestamism of the second order (Fig. 7).

Always y. lags behind the control signal; Entering, reducing the error. The compensating chain does not affect stability.

As a rule, a compensating link must have differentiating properties and implement using active elements. The exact execution of the condition of absolute invariance is impossible in view of the technical inappropriateness of obtaining the derivative above the second order (a high level of interference is introduced into the control circuit, the complexity of the compensating device) and the inertia of real technical devices increase. The number of aperiodic links in the compensating device is designed to be equal to the number of elementary forsing links. The constants of the aperiodic links are calculated by the condition of operation of the links in a substantial field of frequencies, i.e.

The principle of constructing a multicontural saau with a cascade inclusion of regulators is calledprinciple of subordinate regulation.

Synthesis of Sau subordinate control with two or more circuits is carried out by successively optimizing the contours, starting with the internal one.

∆θ ,

grad.

Δ l,

dB

W and (P)

W A1 (P)

1 / t p

1 / t 0

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The synthesis of corrective devices by the LACH method is based on the knowledge of the desired lacc-designed SAU in the open state. The LFCH is not considered, since the system is assumed to be minimally-phase and with a known lated phase characteristic is given.

The desired lacent is called such a lachh, which corresponds to the system with the required quality indicators (the time of control T p, overshooting S%, the established error E set). The task of the synthesis of the correction device is to choose its structure and parameters so as to maximize the lated adjusted system to the desired.

As desired, the so-called optimal characteristics are often chosen, which are the best in any sense. Systems with such characteristics are called optimal.

Transmission function and frequency response of the optimal system.

When constructing the desired strip open system, the concept of the optimal system is used. For each SAU, you can choose your optimality conditions. Here we call the control process with a stepwise exposure to optimal, if it is monotonous and the control time T p is minimal with a limited second derivative of the input value x (t).

Denote.

The transition time of the optimal system is denoted by T min.

The regulatory process will be optimal if the acceleration G has the maximum value G M and changes the sign when, i.e.

Then at (127)

at (128)

x 0 (t) - a controlled value in the optimal process.

When and then can be written in the form

Combining (1) - (3) using single step functions, we get

From dependence (130) you can get

Depending on the magnitude of the input impact, we will change

Let be .

this is the minimum time to test the step signal G 0 with acceleration of a controlled value that does not exceed G m.

Find the gear ratio of a closed optimal system

Considering (130), (131), we get

We define the transmission function of the open system. Have

and then from (132) and (133) we find

The resulting gear ratio is a transcendental function p. This means that the adopted form of the optimal process of regulation, determined by the expression (130), cannot be accurately implemented by a linear stationary Sau. However, it defines the limit to which the processes in the linear system with constant parameters should be brought.

The dependence (134) allows you to determine the lacc optimal saau.

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Ministry of Education and Science of the Russian Federation

FGBOU to Ivanovsky State Chemical Technology University of Technical Cybernetics and Automation.

COURSE WORK

By discipline: Automatic control theory

Subject: synthesis of automatic control systems

Ivanovo 2016.

Transitional Control Function

Table 1. Transitional control function.

annotation

In this course, the object of the study is a stationary inertial object with a delay submitted by the transitional function, as well as the control system.

The research methods are elements of the theory of automatic control, mathematical and simulation.

Using identification methods, approximation and graphical method, models of objects were obtained in the form of gear ratios, a model was set, which most accurately describes the specified object.

After selecting the object model, the parameters of the regulator settings were performed by the cygler-nicols methods and extended frequency characteristics.

To determine the method in which the best settings of the closed control system of the automatic control system were found, its simulation was carried out in the MATLAB environment using the Simulink package. According to the modeling results, a method was selected, with which the adjustment settings were calculated, which best satisfy the specified quality criterion.

The synthesis of the control system of the multidimensional object was also produced: a cascade control system, a combined control system, an autonomous control system. The parameters setting the PI regulators, compensators were calculated, responses to typical exposure were obtained. Automatic control regulator frequency

List of keywords:

Control object, control, settings, control system.

Volume information:

Scope of work- pages

Number of tables

Number of illustrations - 32.

Number of sources used-3

Introduction

In this course, the source data is the transitional function of the control object according to one of the dynamic channels. It is necessary to perform parametric identification of an object specified by the transition function by the graphic method, approximation and identification methods.

Based on the data obtained, set which model more accurately describes the specified object. The solution of this problem is a fairly relevant problem, since we often have a mathematical model itself, but only its acceleration curve.

After selecting the object model, we make calculating the parameters of the PI regulator. Calculation We produce with the help of cygler-nicols and extended frequency characteristics. In order to determine which method, the best adjustment settings were found, we use the degree of damping of the process as a quality criterion.

In this paper, the synthesis of the control system of the multidimensional object of three types was carried out: autonomous, cascade, combined. The settings of the regulators are calculated, system responses are studied according to various channels for typical exposure.

This term paper is a training. The skills obtained during its implementation can be used during the implementation of the course work on modeling management and final qualifying systems.

1. Sidentification of the control object

1.1 Identification using the System Identification Toolbox application

Identification is the definition of the relationship between output and input signals at a qualitative level.

For identification, use the System Identification Toolbox. Build a model of an asimulink.

Fig.1.1.1. Scheme for identification.

With the Ident command, go to the System Identification Toolbox.

Fig.1.1.2. System Identification Toolbox.

Import data to System Identification Toolbox:

Fig.1.1.3. Data imports

We obtain the coefficients of the transfer function:

Fig.1.1.4. Results of identification

K \u003d 44.994 T \u003d 9,0905

1.2 Approximation using CURVE FITTING TOOLBOX

Approximation or approximation is a method that allows you to investigate the numerical characteristics and properties of the object, reducing the task to the study of simpler or more convenient objects.

For approximation, we use the Curve Fitting Toolbox package. Letter in Simulink without a delay link.

Fig.1.2.1. Scheme for approximation.

Using the CFTool command, go to the Curve Fitting Toolbox. On the axis x, we choose the time, and on the axis of the output values. Describe the object object A-B * EXP (-C * X). We obtain the coefficients a, b and p.

Fig.1.2.2. The results of the approximation.

K \u003d (A + B) / 2 \u003d 45 T \u003d

1.3 Approximation by elementary links (graphic method)

Fig.1.3.1. Graphic method

Determine the delay time. To determine to, we spend direct from the well-minded value to the ordinate axis. To determine the time constant, we carry out a tangent to the curve before the intersection with a direct established value, from the intersection point we carry out perpendicular to the abscissa axis, we submit the retardation time from the resulting value.

K \u003d 45 T \u003d 47

1.4 Comparison of transitional functions

To compare three methods, we calculate the error of each method, we will find the sum of the squares of errors, we will find the dispersion. To do this, we build a model in Simulink and substitute the received parameters.

Fig.1.4.1. Comparison of transitional functions.

Three methods obtained the parameters of the transfer function of the object of the study. The criterion for estimating the resulting mathematical model of the object is the error dispersion and for this indicator the best results are marked in the approximation method using the Curve Fitting Tool. Further, for the mathematical model of the object we accept: w \u003d 45 / (1 / 0.022222 + 1) * E ^ (- 22,5p).

2. Selection of the law of regulation

We produce a choice of regulator from the ratio

Since, choose a PI regulator.

3. Synthesis SAU one-dimensional object

3.1 Calculation of SAU by Tsigler-Nikols

The cyglera-nicols method is based on the Nyquist criteria. The essence of the method is to find such a proportional regulator, which displays a closed system to the border of stability, and finding the operating frequency.

For this transfer function, we find the phase-frequency characteristic and will stand it.

We define the operating frequency as an abscissa of the intersection point of the FCH S.Obovaya frequency is 0.082.

Fig. 3.1.1 Finding the operating frequency

Calculate the parameters of the PI regulator settings. Rapidate the KD coefficient:

From the value obtained, we calculate the proportionality coefficient:

Calculate the time of isoorom:

We find the attitude:

Fig. 3.1.2 System Reaction over Control Channel on Step-Function

Fig. 3.1.3 System reaction on the perturbation channel on stepped function

Fig. 3.1.4 Reaction of the system over the perturbation channel per pulse function

Fig. 3.1.5 System Reaction over Pulse Function Control Channel

Calculate the degrees of attenuation by the formula:

We find the average attenuation degree of 0.93 and compare with a true value of 0.85.

3.2 Calculation of SAU by extended frequency characteristics method

This method is fully based on the use of the modified Nyquist criterion (E. Dudnikov's criterion), which says: if the open system is stable and its extended amplitude-phase characteristic passes through the coordinates [-1, j0], then the closed system will not only be stable, But will have some stability reserves determined by the degree of oscillativity.

- (3.2.1) extended response of an open system;

- (3.2.2) Expanded FFC Open System.

For a PI regulator, extended frequency characteristics have the form:

Calculation in MathCAD environment:

for sh \u003d 0.85 m \u003d 0.302

We will calculate the PI regulator settings in the MathCAD environment:

Let us turn to the area of \u200b\u200bextended frequency characteristics of the object. To do this, we will replace:

We turn to the region of extended frequency characteristics of the regulator:

Extended amplitude-frequency response of the regulator:

Extended phase-frequency response of the regulator:

After some transformations of equation (3.2.6) we get:

Build a schedule:

Fig.3.2.1 Settings parameters using the advanced frequency characteristics method

From the graph, we calculate the maximum value of KP / TU on the first twist and the corresponding value of the Kyrgyz Republic:

KP \u003d 0.00565 KP / TU \u003d 0.00034

We investigate the system reaction to type signals across control channels and perturbations.

Transitional control channel:

Fig. 3.2.2 System Reaction on Control Channel on Step Function

Transitional function on the perturbation channel:

Fig. 3.2.3 Reaction of the system on the perturbation channel to the stepped function

Pulse transitional function on the perturbation channel:

Fig. 3.2.4 Reaction of the system on the perturbation channel per pulse function

Pulse transitional function control:

Fig. 3.2.5 Reaction of the system on the control channel to the pulse function

Calculate the degrees of attenuation:

For the transition function on the control channel

For a transitional function on the perturbation channel

For a pulse transitional function on the perturbation channel

For a pulse transitional function on the control channel

We find the average value of the degree of attenuation of 0.98 and compare with the true value of 0.85.

The method of extended frequency characteristics and the method of cygler-nicols were calculated parameters of the PI regulator settings, degrees of attenuation. The average value of the degree of attenuation exceeds a true value of 9.41% obtained by the method of cygler-nicols. The average value of the attenuation obtained by the method of extended frequency characteristics exceeded the true by 15.29%. It follows from this that it is better to use the values \u200b\u200bobtained by the cygler-nicols method.

4. Synthesis of automatic control systems multidimensional object

4.1 Synthesis Cascade Management Systems

Cascade systems are used to automate objects with large inertia over the regulatory channel if you can choose a less inertial coordinate with respect to the most dangerous perturbation and use the same regulatory effect for it as for the main output of the object.

Fig. 4.1.1 Cascade Management System

In this case, the regulatory system includes two regulators - the main (external) regulator, which serves to stabilize the main output of the object Y, and auxiliary (internal) regulator designed to regulate the auxiliary coordinate Y1. The task for the auxiliary regulator is the output signal of the main regulator.

The calculation of the cascade ASR implies the definition of settings of the main and auxiliary regulators for the specified dynamic characteristics of the object in the main and auxiliary channels. Since the settings of the main and auxiliary regulators are interconnected, their calculation leads to the method of iterations.

At each step of iterations, the given single-circuit ACR is calculated, in which one of the regulators relates conventionally to an equivalent object. The equivalent object for the main regulator is a sequential connection of a closed auxiliary circuit and the main regulatory channel; The transfer function is equal to:

(4.1.1.)

The equivalent object for auxiliary regulator is a parallel connection of the auxiliary channel and the basic open system. Its gear ratio has the form:

(4.1.2.)

Depending on the first step of iteration, two methods for calculating cascading ACR distinguish:

1st method. Calculation starts from the main regulator. The method is used in cases where the inertia of the auxiliary channel is much smaller than the main one.

In the first step, assumes that the operating frequency of the main circuit is much less than the auxiliary. Then:

(4.1.3.)

Thus, in the first approximation, the setting of the main regulator does not depend on the settings of the auxiliary regulator and are located on WE0OSN (P).

At the second step, calculate the settings of the auxiliary regulator for the equivalent object.

In the case of approximate calculations, limited to the first two steps. With accurate calculations, they are continued until the adjustment settings found in two consecutive iterations will not coincide with the specified accuracy.

2nd method. Calculation starts with auxiliary regulator. In the first step, it is assumed that the external regulator is disabled, i.e.:

Thus, in the first approximation, the setting of the auxiliary regulator is found along a single-circuit ACR for the adjusting channel. In the second step, calculate the settings of the main regulator using the transmission function of the equivalent object WE1OSN (P), taking into account the settings of the auxiliary regulator. To clarify the settings of the auxiliary controller, the calculation is carried out by a gearing function in which the found settings of the main regulator are substituted. Calculations are carried out until the auxiliary regulator settings found in two consecutive iterations do not coincide with the specified accuracy.

Calculate the parameters of the auxiliary PI regulator:

Fig.4.1.2. Speed \u200b\u200breaction by control channel

Fig.4.1.3. Reaction to stepped impact on the perturbation channel

Fig.4.1.4. Impulse reaction by control channel

Fig.4.1.5. The reaction to the impulse effect on the perturbation channel

The covariant system task and invariant indignation. The main quality criterion is performed. The second quality criterion in the form of regulation time is not performed. A dynamic error criterion is performed.

4.2 Synthesis of the combined control system

There is a case when the object is attached to the object that can be measured, but a single-circuit management system is proposed, and the so-called combination system, which is a combination of two principles - the principle of feedback and the principle of compensation of perturbations.

It is proposed to intercept the perturbation before their impact on the object and using the auxiliary regulator to compensate them.

Fig.4.2.1. Combined control system

Apply to the diagram shown in Fig. 4.2.1, the condition of the invariance of the output value of Y relative to the disturbing effect Yv:

The principle of invariance to indignation: the system is invariant to perturbation, its transfer function through the control channel must be zero. Then the gear ratio of the compensator will be recorded:

(4.2.2.)

Calculate the PI regulator in the MathCAD of the regulator using standard Newton Binomial forms:

Step impact on the control channel:

Fig.4.2.2. Speed \u200b\u200breaction by control channel

Step impact on the perturbation channel:

Fig.4.2.3. Reaction to stepped impact on the perturbation channel

Pulse impact on the control channel:

Fig.4.2.4. Impulse reaction by control channel

Pulse effect on the perturbation channel:

Fig.4.2.5. The reaction to the impulse effect on the perturbation channel

The covariant system task and invariant indignation. Quality criteria in the form of regulation time is not performed. The criterion of dynamic error is not performed. The system is invariant in static, but non-invariant in dynamics due to the inert properties included in it elements.

4.3 Synthesis of the autonomous control system

When managing multidimensional objects, we often encounter the following picture:

Fig. 4.3.1 Control object with two input and two output variables

X1, X2 - Control Variables

Y1, Y2 - Controlled Variables

U1, U2 - Direct links

P1, P2 - cross-links.

If, for the output variable Y1, select the X2 variable as an adjusting variable, then due to the cross-channels, the control variable X2 will influence through the transfer function W21 to the Y1 variable, and the control variable x1 will affect W12 to Y2. These circumstances significantly complicate the calculation of this kind of system.

The task of calculation is greatly simplified if the system impose additional requirements - the requirements of the autonomy of regulatory channels. The autonomy of regulatory channels can be carried out by introducing additional links between the input variables, this kind of device is called compensators.

Fig. 4.3.2 A two-dimensional control system

As a result of the introduction of compensators, new regulatory variables appeared, which affect the source variables, taking into account compensating effects.

Calculate the transfer functions of compensators:

Calculate the parameters for setting PI regulators using standard Newton binomine forms.

Calculate the first PI regulator in MathCAD:

Calculate the second Pi-regulator in Mathcad:

Transitional function on the first control channel:

Fig. 4.3.3. System Reaction on Step Exposure

Transitional function on the second control channel:

Fig. 4.3.4. System Reaction on Step Exposure

The covariant system task and invariant indignation. The main quality criterion is performed. A second quality criterion is performed in the form of time.

Conclusion

In the first job of work, methods used to identify the function specified table were considered. Three methods were considered: Identification method using System Identification Toolbox, the approximation method using the Package Fitting Toolbox and an approximation method by elementary links. According to the results of approximation, the most adequate model was selected. This turned out to be a model obtained by approximation using the Curve Fitting Tool.

Then the law of regulation was determined and the settings of the PI regulator were calculated by two methods: the method of extended frequency characteristics and the method of cygler-nicols. When comparing the degrees of attenuation, it was determined that it was better to use the values \u200b\u200bobtained by Tsigler-Nicols.

The fourth center of the course work was in the modeling of systems. We spent the synthesis of control systems of a multidimensional object. For these systems, perturbation compensators were calculated, as well as PI regulators, for the calculation of which the standard Binomial forms of Newton were used. Reactions of systems were obtained on typical input effects.

List of sources used

Automatic control theory: Textbook for universities / V. Ya. Rothach. - 5th ed., Pererab. and add. - M.: Publishing House MEI, 2008. - 396 p., Il.

Modal control and observing devices / N.T. Body. - M.: "Mechanical Engineering", 1976. - 184 p.

Consultation center MATLAB [Electronic resource] // MATLAB.EXPONENTA, 2001-2014. URL: http://matlab.exponenta.ru. Date of handling: 03/12/2016.

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Until now, we mainly studied the task of analyzing the SAU, when the mathematical model of a closed saau was considered specified, and it was necessary to determine the quality of its operation: stability, accuracy of playing the input signal, etc.

An important, and more complex is the synthesis task when the mathematical model of the managed object is considered specified (and may be measuring and actuating devices). It is required to select the Sau structure, the law of control and the numeric values \u200b\u200bof the parameters of the regulator, which determine the desired quality of SAU.

With the tasks of the synthesis, we have already met. Synthesis of SAU can be carried out using stability criteria, D-partition, root humanographic methods.

Synthesis of one-dimensional single-circuit SAU with a single OOS with the help of a boiled system

This method uses a close relationship between the transition function of a closed SAU with a stepped effect and the real part of the frequency response of a closed saau.

Here . (one)

So According to the frequency response of the open system, you can determine the frequency characteristics of the closed system and vice versa. There are nomograms connecting these frequency characteristics.

By we can estimate the transition process (see (1)). Thus, knowing, we can estimate the transition process in the system.

Solve the task of SAU synthesis in frequency characteristics is more convenient when frequency characteristics are built in a logarithmic scale.

In a logarithmic scale on the axis of ordents w. postponed B. dB.

An increase in this ratio is 10 times complies with increasing

The abscissa axis is postponed in a logarithmic scale.

Decade - change in frequency 10 times.

The main advantage of building frequency characteristics in a logarithmic scale is that they can be built approximately, almost without computing.

Take inertial link . Its gear ratio

Ahh. Frequency, i.e. - Frequency conjugation.

With approximate construction of Lacha:

1) in neglect and, and dB

2) in neglect 1 and and in a logarithmic scale

Determine the slope at:

Consequently, the system of HCH in a logarithmic scale can be declining the characteristics to replace the straight - 20db / dec. The greatest error will be at the bend point ().

Integrating link.

At.

Consider first for example The principle of constructing approximate lacc (FCH is calculated precisely by formulas).

The approach of the construction of Lacha is that in the frequency response in members:

1) when the member is neglected and the link is considered as an amplifying;

2) when they neglect 1 and consider them as an integrating link with frequency response, the slope of whose characteristics - 20 dB / decand with the magnitude of the amplitude is equal 20LGK.

Frequency where - called frequency of conjugation.

We define the frequency of the pairing, where ()

What will turn into account of the assumptions made:

Put on the pairing frequency frequency axis.

Building start with an integrating link: I put on the frequency 20LGK \u003d 20LG100 \u003d 40DB and carry a line with a slope -20db / dec. At the frequency "Connect" another integrating link - the slope has become -40DB / Dec.

Two differentiating links are "connected" at the frequency. At one differentiating link slope + 20 dB / Dec, two integrating links will be the slope + 40 DB / Dec,consequently, the resulting tilt is -40DB / Dec + 40DB / Dec \u003d 0 dB / dec.

The phase-frequency characteristic is calculated.

			1ZV 2pc
0,2
0,8
∞

With the help of lach and FFH it is easy to establish the stability of a closed system.

According to the stability criterion of Nyquist, a closed SAU is stable if an AFCH opening system has the form (as astatic system):

At the amplitude frequency is 1 and therefore - the phase resistance margin.

When the phase is equal, then the amplitude resistance margin.

For Sau stability, it is necessary to

Synthesis of SAU with Lacha

it is carried out as follows:

SAU represent

The object and known elements of the regulator are included, for example, measuring, executive devices.

A corrective device that needs to be defined in the synthesis process.

Then the transmission function of the open system

Here is the gear ratio function of SAU, the dynamics of which satisfies the requirements for the designed system.

Then in a logarithmic scale

For minimally-phase sau, the type of lach is fully determined by the transition process and should not be considered a phase-frequency characteristic.

Minimal phase links (systems) are such that the roots of the numerator and the denominator are located in the left half-plane. Thus, the transfer function of the minimum phase system should not have zeros and poles in the left half-plane.

It can be written by the gear ratio of the corrective link. In this case, it will look:

The literature provides tables connecting the view from

And with the corresponding schemes of corrective devices that implement these. The above can be implemented in the form of the following corrective chain:

Here and we know.

We define the graphics and ,.

From here we find.

We define on schedule.

I define from here.

I define from here.

I define from here.

I define from here.

I define from here.

By defining the parameters of the corrective link, we introduce it into the system and simulate SAU, we obtain the transition process. If it does not suit - change the parameters of the link.

Requirements to .

The desired lacc open system is based on the general requirements for the system:

1. Accuracy (determines the gain),

2. The procedure for Aestamism,

3. Transition time,

4. Overregulation.

1. Must cross the frequency axis at a point that provides the specified transition time

And can be different:

Located from the nomograms that determine the dependence here - overshoot.

For example,

2. In order for sau to be stable, there should be an axis of frequencies with a slope - 20 dB / dec.

3. For the provision of the specified

4. The subsecient part of the characteristic should be done as wide as possible. The larger the range, the closer the process to the exponential.

The mid-frequency part basically determines the quality of the transition process.

The low-frequency part determines the accuracy of the control process.

There is another way to determine the endpoints of the central segment:

The phase stability margin at the point at the LFCH should be no less

Module resistance supply (amplitude) at the point L 2. Selected depending on the overalling:

Conjugation of the central segment of Lacha with a low-frequency part is made by a straight - 40 dB / dec or - 60 dB / dec.

The high-frequency part is not to complicate the corrective device, select similar source lacc.

After construction, you need to check the phase resistance supply. (on the )

Unfortunately, this synthesis method does not guarantee the required quality of the transition process.

The procedure for calculations during SAU synthesis with consistent

corrective device

1. It is built by the stronger part of the SAU (without corrective

road).

2. The desired lacc is built as specified quality requirements.

3. The corresponding LFCH is built.

4. The stocks of amplitude and phase resistance are determined.

5. By subtracting from finding a stroke corrective device.

6. Only choose its technical analogue.

7. If the technical analog is different, it is necessary to adjust the technical counterpart.

If a good result is obtained, the solution of the synthesis problem ends. If the result does not satisfy another analogue.

Synthesis SAU by root humanographers

The quality of the projected SAU from the point of view of speed and stability reserve can be characterized by the location of the root of the numerator and the denominator of the transfer function of the closed system.

Knowing roots, you can depict their location on the complex plane. Roots can be determined by calculating standard programs.

The more - the degree of sustainability, and the less - the degree of oscillativity, the better the quality of the Sau.

When a smooth change in the value of any parameter of the roots will be moved on the roots plane, the thumbnails of a certain curve, which is called the root trajectory or root year. By building the trajectory of all the roots, you can choose such a value of the variable parameters that correspond to the best location of the roots.

Let there be a gear ratio of a closed system

The coefficients of the numerator and the denominator are defined through the object parameters, the regulator, corrective devices. If you need to select the value of any parameter, then it is necessary to take some constant values \u200b\u200bfor all other parameters, and for the desired parameter, set various numeric values. For each specified value of the variable parameter, it is necessary to calculate the values \u200b\u200bof the root of the numerator and the denominator and build the root paths for which the value of the parameter is selected, which provides the best location of the roots.

Synthesis using standard transition processes

(standard coefficients method)

A particular method of using this method is the Vysnegrad's Chart for third-order systems.

Standard transition processes are constructed in a normalized form with a single input effect on dimensionless time, where

Synthesis of linear SAU by allocating the boundaries of stability and boundaries of a given degree of stability

Having allocate the method D-splitting Stability area, we must select a working point (defined by system parameters) within this area. However, different points will correspond to the different distribution of the roots of the characteristic equation, and, consequently, the different nature of the transition process. I would like to have a good transition process.

It is known that the duration of the transition process is determined by the closest to the imaginary axis.

If we specify the required transition time, then we can define. If the roots are located to the left, the duration of the transition will be less than the specified. .

If in equation (3) parameters, in the plane of which we want to construct the boundary of a given degree of stability, included in the characteristic equation linearly independently, then to the equation (3) you can apply the method considered earlier Driving. The selected boundary will be a line of a given sustainability.

Under the synthesis, the construction, creation, design, setting up the optimal system in relation to its parameters is understood. Therefore, designers are engaged in synthesis, the creators of SAR. When operating already created systems, for example, serially issued, it can only be about adjusting the parameters when the system outputs from the required modes for one reason or another.

Methods of synthesis

1. When creating an SAU of the necessary destination, first of all care that it performs its control and regulation functions with a given accuracy, had the optimal component of the element base (amplifiers, regulators, converters, engines, sensors, etc. ) So that it provides the necessary power, speed, moments of movement, was simple, reliable, convenient to operate and economical.

At this stage, the questions of the speakers can take into account only in the rough approximation, for example, do not choose the elements of knowingly unstable, with large time constants, resonant, etc.

2. Questions to ensure static characteristics, the accuracy of working out asked teams and high technical and economic indicators are for technological processes and the economy central and to solve the most difficult. Therefore, despite the fact that without the good quality of the dynamic modes of SAU will not be commissioned, the synthesis of its structure to ensure the required modes is carried out at the second stage, when the functional circuit, the composition of the elements and the system parameters are pre-installed. You cannot combine any effect on both steps.

In general, the Sau designed at the first stage is usually a multi-constructive structure with a complex transfer function, the analysis of which gives unsatisfactory results on the quality of transient processes. Therefore, it must be simplified to the desired characteristics and adjust.

Synthesis of SAU of the required quality

The synthesis of the system should be carried out by changing the structure to meet the necessary requirements. The characteristics of the system that meet the requirements are called the desired characteristics in contrast to the disposable, which has a source non-optimal system.

The basis for the construction of the desired characteristics is the required indicators of the system: stability, speed, accuracy, etc. Since the largest distribution was the logarithmic frequency characteristics, then consider the synthesis of SAU on the desired lacch and LFCH.

1. Construction of the desired characteristics start from the mid-frequency area characterizing the stability, speed and form of the transition process of the system. Its position is determined by the frequency of the cut S.Zh. (Fig.1.8.1).

The cutoff frequency is determined by the required transition process TPP and allowable redistribution:

Fig.2.

• 2. After the point C, the average-frequency asymptotus of the desired characteristics with an inclination of 20 dB / Dec are carried out (Fig.1.8.1.).
• 3. Find a low-frequency component from 2.

Usually defined by the quality of the system in the speed of DSK and at accelerating the DUSK.

We find frequency

The intersection of this asymptotes with the mid-frequency limits it to the left on the mating frequency.

4. The mating frequency 3 is chosen so that 3/2 \u003d 0.75 or Lg 3-Lg 2 \u003d 0.7ds, providing stability conditions.

In this condition, relations are taken into account:

which can also be used to limit the mid-frequency asymptotes.

If there are no obvious restrictions, then 2 and 3 are selected from the conditions (Fig.1.8.1, b)

L2 \u003d (616) DBLC (C) \u003d - (616) dB (1.8.4)

The increase in the section 3 - 2 is inappropriate.

5. Find a low-frequency component with 1. The speed of the speed is determined by the gain coefficient

DSK \u003d KSK. (1.8.5)

We postpone on the axis of the frequencies of KSK, we carry out asymptotes with a slope of 20 dB / decline through this point and end at intersection with the second asymptota. The intersection point is the low-frequency component C 1.

6. We check the phase stability margin

the phase at the cut frequency C should not exceed - with a warranty 45.

7. We check the fulfillment of the conditions for non-paying the desired lach in the forbidden zone (Fig.1.8.1, a).

and lk \u003d 20lgksk, (1.8.7)

where KSK \u003d is the gain of the open system or speed factor.