I am publishing the first chapter of lectures on the theory of automatic control, after which your life will never be the same.

Lectures on the course "Management of Technical Systems", reads Kozlov Oleg Stepanovich at the Department of "Nuclear Reactors and Power Plants", Faculty of Power Engineering, Moscow State Technical University. N.E. Bauman. For which he is very grateful.

These lectures are only being prepared for publication in the form of a book, and since there are TAU specialists, students and those simply interested in the subject, any criticism is welcome.

1. Basic concepts of the theory of control of technical systems

1.1. Goals, principles of management, types of management systems, basic definitions, examples

The development and improvement of industrial production (energy, transport, mechanical engineering, space technology, etc.) requires a continuous increase in the productivity of machines and units, improving product quality, reducing costs and, especially in nuclear energy, a sharp increase in safety (nuclear, radiation, etc. .e) operation of nuclear power plants and nuclear installations.

Realization of the goals set is impossible without the introduction of modern control systems, including both automated (with the participation of a human operator) and automatic (without the participation of a human operator) control systems (CS).

Definition: Management is such an organization of a particular technological process that ensures the achievement of the goal.

Management Theory is a branch of modern science and technology. It is based (based) on both fundamental (general scientific) disciplines (for example, mathematics, physics, chemistry, etc.) and applied disciplines (electronics, microprocessor technology, programming, etc.).

Any control process (automatic) consists of the following main stages (elements):

• obtaining information about the control task;
• obtaining information about the result of management;
• analysis of the received information;
• implementation of the decision (impact on the control object).

To implement the Management Process, the management system (CS) must have:

• sources of information about the control task;
• sources of information about the results of management (various sensors, measuring devices, detectors, etc.);
• devices for analyzing the received information and developing a solution;
• executive devices affecting the Control Object, containing: regulator, motors, amplifying-converting devices, etc.

Definition: If the control system (CS) contains all the above parts, then it is closed.

Definition: The control of a technical object using information about the results of control is called the feedback principle.

Schematically, such a control system can be represented as:

Rice. 1.1.1 - Structure of the control system (CS)

If the control system (CS) has a block diagram, the form of which corresponds to Fig. 1.1.1, and functions (works) without the participation of a person (operator), then it is called automatic control system (ACS).

If the control system functions with the participation of a person (operator), then it is called automated control system.

If the Control provides a given law of object change in time, regardless of the control results, then such control is performed in an open loop, and the control itself is called program management.

Open-loop systems include industrial machines (conveyor lines, rotary lines, etc.), machine tools with numerical control (CNC): see an example in fig. 1.1.2.

Fig.1.1.2 - Example of program control

The master device can be, for example, a “copier”.

Since in this example there are no sensors (meters) that control the part being manufactured, if, for example, the cutter was installed incorrectly or broke, then the goal (manufacturing of the part) cannot be achieved (realized). Typically, in systems of this type, output control is required, which will only record the deviation of the size and shape of the part from the desired one.

Automatic control systems are divided into 3 types:

• automatic control systems (ACS);
• automatic control systems (ACS);
• tracking systems (SS).

ATS and SS are subsets of ACS ==> .

Definition: An automatic control system that ensures the constancy of any physical quantity (group of quantities) in the control object is called an automatic control system (ACS).

Automatic control systems (ACS) are the most common type of automatic control systems.

The world's first automatic regulator (18th century) is Watt's regulator. This scheme (see Fig. 1.1.3) was implemented by Watt in England to maintain a constant speed of rotation of the steam engine wheel and, accordingly, to maintain a constant speed of rotation (motion) of the transmission pulley (belt).

In this scheme sensitive elements (measuring sensors) are “weights” (spheres). "Loads" (spheres) also "force" the rocker and then the valve to move. Therefore, this system can be attributed to the direct control system, and the regulator - to direct acting regulator, since it simultaneously performs the functions of both a "meter" and a "regulator".

In direct acting regulators additional source energy is not required to move the regulator.

Rice. 1.1.3 - Scheme of automatic Watt regulator

In indirect control systems, the presence (presence) of an amplifier (for example, power), an additional actuator, containing, for example, an electric motor, a servomotor, a hydraulic drive, etc., is necessary.

An example of an ACS (automatic control system), in the full sense of this definition, can be a control system that ensures the launch of a rocket into orbit, where the controlled variable can be, for example, the angle between the axis of the rocket and the normal to the Earth ==> see fig. 1.1.4.a and fig. 1.1.4.b

Rice. 1.1.4(a)

Rice. 1.1.4 (b)

1.2. Structure of control systems: simple and multidimensional systems

In the theory of Technical Systems Management, any system is usually divided into a set of links connected into network structures. In the simplest case, the system contains one link, to the input of which an input action (input) is applied, at the input a response of the system (output) is obtained.

In the theory of Control of Technical Systems, 2 main ways of representing the links of control systems are used:

- in the variables "input-output";

— in state variables (see sections 6…7 for more details).

Representation in variables “input-output” is usually used to describe relatively simple systems that have one “input” (one control action) and one “output” (one controlled variable, see figure 1.2.1).

Rice. 1.2.1 - Schematic representation of a simple control system

Typically, such a description is used for technically simple ACS (automatic control systems).

Recently, the representation in state variables has become widespread, especially for technically complex systems, including multidimensional automatic control systems. On fig. 1.2.2 shows a schematic representation of a multidimensional automatic control system, where u1(t)…um(t) — control actions (control vector), y1(t)…yp(t) — ACS adjustable parameters (output vector).

Rice. 1.2.2 - Schematic representation of a multidimensional control system

Let us consider in more detail the ACS structure, represented in the “input-output” variables and having one input (input or setting, or control action) and one output (output action or controlled (or controlled) variable).

Let us assume that the block diagram of such an ACS consists of a certain number of elements (links). By grouping the links according to the functional principle (what the links do), the block diagram of the ACS can be reduced to the following typical form:

Rice. 1.2.3 - Block diagram of the automatic control system

Symbol ε(t) or variable ε(t) denotes the mismatch (error) at the output of the comparing device, which can “work” in the mode of both simple comparative arithmetic operations (most often subtraction, less often addition), and more complex comparative operations (procedures).

As y1(t) = y(t)*k1Where k1 is the gain, then ==>
ε(t) = x(t) - y1(t) = x(t) - k1*y(t)

The task of the control system is (if it is stable) to “work” for the destruction of the mismatch (error) ε(t), i.e. ==> ε(t) → 0.

It should be noted that the control system is affected by both external influences (controlling, disturbing, interference) and internal interference. A hindrance differs from an impact by the stochasticity (randomness) of its existence, while the impact is almost always determined.

To denote the control (setting action), we will use either x(t)or u(t).

1.3. Basic laws of management

If we return to the last figure (structural diagram of the ACS in Fig. 1.2.3), then it is necessary to “decipher” the role played by the amplifying-converting device (what functions it performs).

If the amplifying-converting device (ACD) performs only amplification (or attenuation) of the mismatch signal ε(t), namely: Where is the coefficient of proportionality (in the particular case = Const), then such a control mode of a closed ACS is called a mode proportional control (P-control).

If the UPA generates an output signal ε1(t) proportional to the error ε(t) and the integral of ε(t), i.e. , then this control mode is called proportionally integrating (PI control). ==> Where b is the coefficient of proportionality (in the particular case b = Const).

Typically, PI control is used to improve the accuracy of control (regulation).

If the UPA generates an output signal ε1(t) proportional to the error ε(t) and its derivative, then this mode is called proportionally differentiating (PD control): ==>

Usually, the use of PD control increases the speed of the ACS

If the UPA generates an output signal ε1(t) proportional to the error ε(t), its derivative, and the integral of the error ==> , then such a mode is called then such a control mode is called proportional-integral-differential control mode (PID control).

PID control can often provide “good” control accuracy with “good” speed

1.4. Classification of automatic control systems

1.4.1. Classification by type of mathematical description

According to the type of mathematical description (equations of dynamics and statics), automatic control systems (ACS) are divided into linear и non-linear systems (ACS or ACS).

Each "subclass" (linear and non-linear) is subdivided into a number of "subclasses". For example, linear ACS (SAR) have differences in the type of mathematical description.
Since this semester will consider the dynamic properties of only linear automatic control (regulation) systems, we will give a classification according to the type of mathematical description for linear automatic control systems (ACS):

1) Linear automatic control systems described in “input-output” variables by ordinary differential equations (ODE) with permanent coefficients:

where x(t) – input action; y(t) – output action (adjustable value).

If we use the operator ("compact") form of writing a linear ODE, then equation (1.4.1) can be represented in the following form:

Where, p = d/dt is the differentiation operator; L(p), N(p) are the corresponding linear differential operators, which are equal to:

2) Linear automatic control systems described by linear ordinary differential equations (ODEs) with variables (in time) coefficients:

In the general case, such systems can also be attributed to the class of non-linear ACS (SAS).

3) Linear automatic control systems described by linear difference equations:

where f(…) is a linear function of arguments; k = 1, 2, 3… - whole numbers; Δt – quantization interval (sampling interval).

Equation (1.4.4) can be represented in a "compact" form:

Typically, such a description of linear ACS (SAR) is used in digital control systems (using a computer).

4) Linear automatic control systems with delay:

where L(p), N(p) — linear differential operators; τ is the lag time or lag constant.

If the operators L(p) и N(p) degenerate (L(p) = 1; N(p) = 1), then equation (1.4.6) corresponds to the mathematical description of the dynamics of the ideal delay link:

and a graphic illustration of its properties is shown in Fig. 1.4.1

Rice. 1.4.1 - Graphs of the input and output of the ideal delay link

5) Linear automatic control systems described by linear differential equations in partial derivatives. Often such self-propelled guns are called distributed control systems. ==> An "abstract" example of such a description:

The system of equations (1.4.7) describes the dynamics of a linearly distributed ACS, i.e. the controlled value depends not only on time, but also on one spatial coordinate.
If the control system is a "spatial" object, then ==>

where depends on time and spatial coordinates determined by the radius vector

6) ACS described systems ODEs, or systems of difference equations, or systems of partial differential equations ==> and so on...

A similar classification can be proposed for non-linear ACS (SAR) ...

For linear systems, the following requirements are met:

• linearity of the static characteristics of the ACS;
• linearity of the equation of dynamics, i.e. variables in the equation of dynamics are only in linear combination.

A static characteristic is the dependence of the output on the magnitude of the input action in the steady state (when all transient processes are damped).

For systems described by linear ordinary differential equations with constant coefficients, the static characteristic is obtained from the equation of dynamics (1.4.1) by equating to zero all non-stationary terms ==>

Figure 1.4.2 shows examples of linear and non-linear static characteristics of automatic control (regulation) systems.

Rice. 1.4.2 - Examples of static linear and non-linear characteristics

The nonlinearity of terms containing time derivatives in the equations of dynamics can arise when using nonlinear mathematical operations (*, /, , , sin, ln, etc.). For example, considering the equation of dynamics of some "abstract" ACS

note that in this equation with a linear static characteristic the second and third terms (dynamic terms) on the left side of the equation are non-linear, so the ACS described by such an equation is non-linear in dynamic Plan.

1.4.2. Classification by the nature of the transmitted signals

According to the nature of the transmitted signals, automatic control (or regulation) systems are divided into:

• continuous systems (systems of continuous action);
• relay systems (systems of relay action);
• discrete action systems (pulse and digital).

system continuous action is called such an ACS, in each of the links of which continuous change in input signal over time corresponds continuous change of the output signal, while the law of change of the output signal can be arbitrary. For the ACS to be continuous, it is necessary that the static characteristics of all links were continuous.

Rice. 1.4.3 - Example of a continuous system

system relay action is called ACS, in which at least in one link with a continuous change in the input value, the output value at some moments of the control process changes “jump” depending on the value of the input signal. The static characteristic of such a link has break points or fracture with break.

Rice. 1.4.4 - Examples of relay static characteristics

system discrete action is called a system in which at least one link with a continuous change in the input value, the output value has type of individual impulsesappearing after a certain period of time.

The link that converts a continuous signal into a discrete signal is called a pulse. A similar type of transmitted signals takes place in an automatic control system with a computer or a controller.

The following methods (algorithms) for converting a continuous input signal into a pulsed output signal are most often implemented:

• pulse amplitude modulation (AIM);
• pulse width modulation (PWM).

On fig. 1.4.5 is a graphical illustration of the pulse amplitude modulation (AIM) algorithm. At the top of Fig. time dependence is presented x(t) - signal at the entrance into the impulse. Output signal of the pulse block (link) y(t) is a sequence of rectangular pulses appearing with permanent quantization period Δt (see the lower part of the figure). The pulse duration is the same and equal to Δ. The amplitude of the pulse at the output of the block is proportional to the corresponding value of the continuous signal x(t) at the input of this block.

Rice. 1.4.5 - Implementation of pulse-amplitude modulation

This method of pulse modulation was very common in the electronic measuring equipment of control and protection systems (CPS) of nuclear power plants (NPP) in the 70s ... 80s of the last century.

On fig. 1.4.6 is a graphical illustration of the pulse-width modulation (PWM) algorithm. At the top of Fig. 1.14 shows the time dependence x(t) – signal at the input to the impulse link. Output signal of the pulse block (link) y(t) - a sequence of rectangular pulses that appear with a constant quantization period Δt (see bottom of figure 1.14). The amplitude of all impulses is the same. Pulse duration Δt at the output of the block is proportional to the corresponding value of the continuous signal x(t) at the input of the impulse block.

Rice. 1.4.6 - Implementation of pulse-width modulation

This method of pulse modulation is currently the most common in the electronic measuring equipment of control and protection systems (CPS) of nuclear power plants (NPP) and ACS of other technical systems.

Concluding this subsection, it should be noted that if the characteristic time constants in other parts of the ACS (SAR) significantly more Δt (by orders of magnitude), then the impulse system can be considered a continuous automatic control system (when using both AIM and PWM).

1.4.3. Classification by the nature of management

By the nature of the control processes, automatic control systems are divided into the following types:

• deterministic ACS, in which the input signal can be uniquely associated with the output signal (and vice versa);
• stochastic ACS (statistical, probabilistic), in which the ACS “responds” to a given input signal random (stochastic) output signal.

The output stochastic signal is characterized by:

• distribution law;
• mathematical expectation (average value);
• dispersion (standard deviation).

The stochastic nature of the control process is usually observed in essentially non-linear ACS both from the point of view of the static characteristic, and from the point of view (even to a greater extent) of the nonlinearity of the dynamic terms in the equations of dynamics.

Rice. 1.4.7 - Distribution of the output value of the stochastic ACS

In addition to the above main types of classification of control systems, there are other classifications. For example, classification can be carried out according to the control method and be based on interaction with the external environment and the possibility of adapting the ACS to changes in environmental parameters. Systems fall into two broad classes:

1) Ordinary (non-self-adjusting) control systems without adaptation; these systems belong to the category of simple ones that do not change their structure in the process of control. They are the most developed and widely used. Ordinary control systems are divided into three subclasses: open-loop, closed-loop, and combined control systems.

2) Self-tuning (adaptive) control systems. In these systems, when external conditions or characteristics of the controlled object change, an automatic (not predetermined) change in the parameters of the control device occurs due to changes in the CS coefficients, the CS structure, or even the introduction of new elements.

Another example of classification: on a hierarchical basis (single-level, two-level, multi-level).

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