Automatic control theory

The theory of automatic control (TAU) is a scientific discipline that studies the processes of automatic control of objects of different physical nature. At the same time, using mathematical tools, the properties of automatic control systems are identified and recommendations for their design are developed.

It is an integral part of technical cybernetics and is intended for the development of general principles of automatic control, as well as methods of analysis (functioning research) and synthesis (choice of parameters) of automatic control systems (ACS) for technical objects.

For this theory, only the character ^{[1] of} signal transformations by control objects is important .

History

For the first time, information about automata appeared at the beginning of our era in the works of Heron of Alexandria “Pneumatics” and “Mechanics”, where automata created by Geron himself and his teacher Ctesibius were described : pneumatic automatic machine for opening the doors of the temple, water organ, automatic machine for selling holy water, etc. The ideas of Heron were far ahead of their age and did not find application in his era.

In the Middle Ages, imitative “android” mechanics developed significantly, when the mechanical designers created a number of automata imitating individual human actions, and to reinforce the impression, the inventors gave the automata an external resemblance to a person and called them “ androids ”, that is, humanlike. At present, such devices are called robots , in contrast to the automatic control devices widely used in all spheres of human activity, which are called automata.

In the XIII century, the German philosopher-scholastic and alchemist Albert von Bolstadt built a robot for opening and closing doors.

Very interesting androids were created in the XVII-XVIII centuries. In the 18th century, Swiss watchmakers Pierre Droz and his son Henri created a mechanical scribe, a mechanical artist, and others. The beautiful automaton theater was created in the 18th century. Russian self-taught mechanic Kulibin . His theater, which is stored in the Hermitage , is located in the “egg figure clock”.

In its embryonic form, many of the provisions of the theory of automatic control are contained in the General Theory of (linear) Regulators, which was developed mainly in the years 1868-1876 in the works of Maxwell and Vyshnegradsky . The fundamental works of Vyshnegradsky are: "On the general theory of regulators", "On the regulators of indirect action." In these works, one can find the origins of modern engineering methods for studying the stability and quality of regulation.

The decisive influence on the development of national research methodology of the theory of automatic control was played by the work of the outstanding Soviet mathematician Andrei Markov (junior) , the founder of the Soviet constructivist school of mathematics, the author of the theory of algorithms and mathematical logic. These studies have been applied in the scientific and practical activities of Academician Lebedev on military topics - torpedo control and instrumentation and stability of large energy systems .

By the beginning of the 20th century and in its first decade, the theory of automatic control was formed as a general scientific discipline with a number of applied sections.

Basic Concepts

Automatics is a branch of science and technology, covering the theory and practice of automatic control, as well as the principles for constructing automatic systems and the technical means forming them.

The control object (OA) is a device, a physical process or a set of processes that must be managed to obtain the desired result. The interaction with the OS is performed by submitting a control input to its conditional input (which corrects the processes occurring at the OS), and an output parameter is obtained (which is a process-consequence).

Control - the impact (signal) applied to the input of the control object and ensuring that the processes in the control object are such that they will achieve the specified control goal at its output.

The goal is the desired flow of processes in the control object and obtaining the desired change in the parameter at its output.

Objects:

managed
unmanaged

Automatic control system (ACS) includes a control object and a control device.

Control device - a set of devices with the help of which the control inputs of the control object are controlled.

Regulation - a special case of control, the purpose of which is to maintain at a given level one or more outputs of the control object.

Regulator - converts the regulation error ε (t) into a control action applied to the control object.

The specifying effect g (t) - determines the required law of regulation of the output value.

Regulation error ε (t) = g (t) - y (t), the difference between the desired value of the controlled variable and its current value. If ε (t) is nonzero, then this signal is fed to the input of the regulator, which forms such a regulating effect, so that eventually with time ε (t) = 0.

The disturbing effect f (t) is the process at the input of the control object, which is a control hindrance.

Automatic control systems:

Open:

software control system. The control unit generates a control action without receiving information about the state of the system on the basis of any signs, a temporary program (simplicity and increased reliability, low quality control);
SU on indignation. The control system generates a control action based on information on the magnitude of the disturbing effect on the system.

Closed: The control unit generates a control action based on the measured information on the state of the object on the selected parameter.
Combined system: The control unit generates a control action based on information about the parameters of the object and on the basis of information of the disturbing action.

Functional Diagrams

Typical SAU scheme

Functional diagram of the element - the scheme of the system of automatic regulation and control, based on the function that this element performs.

Output signals are parameters that characterize the state of the control object and are essential for the control process.

System outputs are system points at which output signals can be observed in the form of certain physical quantities.

The system inputs are system points at which external influences are applied.

Input Signals:

interference - signals that are not related to the sources of information about the tasks and results of management.
useful - signals associated with the sources of information about the tasks and results of management.

Systems:

one-dimensional - systems with one input and one output.
multidimensional - systems with multiple inputs and outputs.

SAU control principles

Feedback is a connection in which the actual value of the output variable and the specified value of the controlled variable are fed to the input of the regulator.

hard - such an OS, at which a regulator receives a signal proportional to the output signal of the object at any time.
flexible - such an OS in which not only a signal proportional to the output signal of the object, but also a signal proportional to the derivative of the output variable arrives at the controller input.

Control by the principle of deviation of a controlled variable - feedback forms a closed loop. The controlled object is subjected to an impact proportional to the sum (difference) between the output variable and the specified value, so that this sum (difference) decreases.

Control according to the principle of compensation of disturbances - a signal proportional to the disturbing influence is sent to the regulator input. There is no relationship between the control action and the result of this action on the object.

Control by the principle of combined regulation - is used at the same time regulation of perturbation and deviation, which ensures the highest accuracy of control.

The principle of deviation of a controlled variable in TAU
The principle of compensation of disturbances in TAU
The principle of combined regulation in TAU

SAU Classification

By the nature of management:

control systems
regulatory systems

By the nature of the action:

continuous systems
discrete action systems
relay action systems

By the degree of use of information about the state of the control object:

OS management
control without OS

According to the degree of use of information about the parameters and structure of the control object:

adaptive
non-adaptive
search
non-searching
with identification
variable structure

According to the degree of transformation of coordinates in the ACS:

deterministic
stochastic (with random effects)

By the form of a mathematical model of coordinate transformation:

linear
nonlinear (relay, logic, etc.)

By type of control actions:

analog
discrete (discontinuous, pulsed, digital)

According to the degree of human participation:

tame
automatic
automated (man in control)

According to the law of changing the output variable:

stabilizing : the prescribed value of the output variable is unchanged.
program : the output variable is changed according to a specific, predetermined program.
tracking : the prescribed value of the output variable depends on the value of the unknown variable in advance at the input of the automatic system.

By the number of controlled and regulated variables:

one-dimensional : if the object has only one controllable value;
multidimensional: if the object has a relatively large number of controllable quantities and the corresponding number of control actions.

According to the degree of self-adjustment, adaptation, optimization and intelligence:

extreme
self-adjusting
intellectual

By the impact of the sensitive (measuring) element on the regulator:

direct control systems
indirect control systems

Intelligent Power

ISAU is a system that allows you to conduct training, adaptation or customization by memorizing and analyzing information about the behavior of an object, its SU and external influences. A feature of these systems is the availability of a database of the logical inference engine, an explanation subsystem, etc.

Knowledge Base - formalized rules in the form of logical formulas, tables, etc. The MIS is used to manage poorly formalized or complex technical objects.

The class of MIS corresponds to

The presence of SU interactions with the real external world using information channels of communication.
The openness of the system - is needed to replenish and acquire knowledge.
The presence of mechanisms for predicting changes in the environment of the system.
Inaccuracy of the OS information can be compensated by increasing the intellectualization of the control algorithm.
Preservation of operation when the connection is broken.

If an ISU satisfies all 5 features, then it is intellectual in a “big”, otherwise in a “small” sense.

Mathematical models of linear ACS

Deterministic

$W_{0}(p)={\frac {A(p)}{B(p)}}$ ${\ displaystyle W_ {0} (p) = {\ frac {A (p)} {B (p)}}$

$W_{0}(p)={\frac {K_{0}}{T_{0}p}}e^{-p\tau }$ ${\ displaystyle W_ {0} (p) = {\ frac {K_ {0}} {T_ {0} p}} e ^ {- p \ tau}}$

Statistical

Statistical features are characterized by a set of statistical parameters and distribution functions. For their research methods of mathematical statistics are used.

Adaptive

Adaptive use for the description of the control object deterministic stochastic methods.

Types of impacts. Transitional, weighting, transfer functions

A single step function is a special mathematical function whose value is zero for negative arguments and one for positive arguments. It is a natural simplest effect on the control object. In mathematics, it is expressed by a Heaviside single function.
The unit impulse function is a derivative of the unit step function. It characterizes an impulse of infinitely large amplitude that flows over an infinitely small period of time. Geometrical meaning - the area bounded by this function is 1.
It is used in a number of cases when the process of determining dynamic characteristics in the simplest way due to exceeding the value of the output value for a given value is impossible. ^[2]
The transient function is the response of the system to a single step signal.
The weight function is the response of the system to a single impulse.
The transfer function is the ratio of the output Laplace transform to the input Laplace transform with zero initial conditions and zero external perturbations.

Link Transfer Function

Serial connection

W _e (p) = W ₁ (p) W ₂ (p) ... W _n (p) = $\prod _{i=1}^{n}W_{i}$ ${\ displaystyle \ prod _ {i = 1} ^ {n} W_ {i}}$ (p)

Parallel connection

W _e (p) = W ₁ (p) + W ₂ (p) + ... + W _n (p) = $\sum _{i=1}^{n}W_{i}$ ${\ displaystyle \ sum _ {i = 1} ^ {n} W_ {i}}$ (p)

The transfer function of a closed system

W _OC (p) - equation describing the feedback circuit
W (p) - the equation describing the link
G (p) - equation describing the input action
U _OC (p) - equation describing the output signal of the feedback link
ΔU (p) is an equation describing the sum (difference) G (p) and U _OC (p)
Y (p) - equation describing the output signal of the system

$f(n)=\left\{{\begin{matrix}Y(p)=W(p)\Delta U(p)\\U_{OC}(p)=W_{OC}(p)Y(p)\\\Delta U(p)=G(p)\mp U_{OC}(p)\end{matrix}}\right.$ ${\ displaystyle f (n) = \ left \ {{\ begin {matrix} Y (p) = W (p) \ Delta U (p) \\ U_ {OC} (p) = W_ {OC} (p) Y (p) \\\ Delta U (p) = G (p) \ mp U_ {OC} (p) \ end {matrix}} \ right.}$

Solving this system of equations, we obtain the following results:

$Y(p)=W(p)(G(p)\mp W_{OC}(p)Y(p))$ ${\ displaystyle Y (p) = W (p) (G (p) \ mp W_ {OC} (p) Y (p))}$

$Y(p)\pm W(p)W_{OC}(p)Y(p)=W(p)G(p)$ ${\ displaystyle Y (p) \ pm W (p) W_ {OC} (p) Y (p) = W (p) G (p)}$

$Y(p)={{W(p)G(p)} \over {1\pm W(p)W_{OC}(p)}}$ ${\ displaystyle Y (p) = {{W (p) G (p)} \ over {1 \ pm W (p) W_ {OC} (p)}}}$

$W_{\ni }(p)={Y(p) \over G(p)}={W(p) \over 1\pm W(p)W_{OC}(p)}$ ${\ displaystyle W _ {\ ni} (p) = {Y (p) \ over G (p)} = {W (p) \ over 1 \ pm W (p) W_ {OC} (p)}}$

Getting the transfer function in the state space

The system in the state space is given in the form:

$\left\{{\begin{matrix}{\dot {x}}(t)=A\cdot x(t)+B\cdot u(t)\\y(t)=C\cdot x(t)+D\cdot u(t)\end{matrix}}\right.$ ${\ displaystyle \ left \ {{\ begin {matrix} {\ dot {x}} (t) = A \ cdot x (t) + B \ cdot u (t) \\ y (t) = C \ cdot x (t) + D \ cdot u (t) \ end {matrix}} \ right.}$

The system has m inputs u (t), l outputs y (t), n states x (t), n> = max (m, l), A, B, C, D are numerical matrices of the corresponding dimension nxn, nxm, lxn lxm ..

Let I be the unit matrix of dimension nxn, then:

pI X (p) - AX (p) = BU (p)

(pI - A) X (p) = BU (p)

x (0) = 0

X (p) = Wxu (p) U (p); Wxu (p) = (pI - A) ^ {- 1) B

Y (p) = Wyu (p) U (p); Wyu (p) = C (pI - A) ^ (- 1) B + D

Linearization of systems and links

Let the ACS is regulated and described by a nonlinear equation

$F(x,{\dot {x}},y,{\dot {y}},{\ddot {y}},...,f,{\dot {f}},{\ddot {f}})=0$ ${\ displaystyle F (x, {\ dot {x}}, y, {\ dot {y}}, {\ ddot {y}}, ..., f, {\ dot {f}}, {\ ddot {f}}) = 0}$

Moreover, the nonlinearity is insignificant, that is, this function can be expanded into a Taylor series in the vicinity of a stationary point, for example, with an external perturbation f = 0 .

The equation of this link in the steady state is as follows:

$F(x^{0},0,y^{0},0,0)=0,x_{k}^{0},y_{k}^{0}$ ${\ displaystyle F (x ^ {0}, 0, y ^ {0}, 0,0) = 0, x_ {k} ^ {0}, y_ {k} ^ {0}}$ , starting points, derivatives are absent.

Then, decomposing a nonlinear function in a Taylor series, we get:

$F(x^{0},y^{0})+\left({\frac {\partial F}{\partial x}}\right)^{0}\Delta x+\left({\frac {\partial F}{\partial {\dot {x}}}}\right)^{0}\Delta {\dot {x}}+\left({\frac {\partial F}{\partial y}}\right)^{0}\Delta y+\left({\frac {\partial F}{\partial {\dot {y}}}}\right)^{0}\Delta {\dot {y}}+\left({\frac {\partial F}{\partial {\ddot {y}}}}\right)^{0}\Delta {\ddot {y}}+R_{n}=0,R_{n},$ ${\ displaystyle F (x ^ {0}, y ^ {0}) + \ left ({\ frac {\ partial F} {\ partial x}} \ right) ^ {0} \ Delta x + \ left ({\ frac {\ partial F} {\ partial {\ dot {x}}}} \ right) ^ {0} \ Delta {\ dot {x}} + \ left ({\ frac {\ partial F} {\ partial y }} \ right) ^ {0} \ Delta y + \ left ({\ frac {\ partial F} {\ partial {\ dot {y}}}} \ right) ^ {0} \ Delta {\ dot {y} } + \ left ({\ frac {\ partial F} {\ partial {\ ddot {y}}}} \ right) ^ {0} \ Delta {\ ddot {y}} + R_ {n} = 0, R_ {n},}$ - residual member

$F(x,y)^{0}+\left(-b_{1}\right)\Delta x+\left(-b_{0}\right)\Delta {\dot {x}}+\left(a_{2}\right)\Delta y+\left(a_{1}\right)\Delta {\dot {y}}+\left(a_{0}\right)\Delta {\ddot {y}}+R_{n}=0$ ${\ displaystyle F (x, y) ^ {0} + \ left (-b_ {1} \ right) \ Delta x + \ left (-b_ {0} \ right) \ Delta {\ dot {x}} + \ left (a_ {2} \ right) \ Delta y + \ left (a_ {1} \ right) \ Delta {\ dot {y}} + \ left (a_ {0} \ right) \ Delta {\ ddot {y} } + R_ {n} = 0}$

$\left\{{\begin{matrix}\Delta x\rightarrow 0\\\Delta y\rightarrow 0\end{matrix}}\right.\Rightarrow R_{n}\rightarrow 0$ ${\ displaystyle \ left \ {{\ begin {matrix} \ Delta x \ rightarrow 0 \\\ Delta y \ rightarrow 0 \ end {matrix}} \ right. \ Rightarrow R_ {n} \ rightarrow 0}$

$a_{0}{\frac {d^{2}y}{dt^{2}}}+a_{1}{\frac {dy}{dt}}+a_{2}y=b_{0}{\frac {dx}{dt}}+b_{1}x$ ${\ displaystyle a_ {0} {\ frac {d ^ {2} y} {dt ^ {2}}} + a_ {1} {\ frac {dy} {dt}} + a_ {2} y = b_ { 0} {\ frac {dx} {dt}} + b_ {1} x}$

From non-linear recording moved to linear. Let us turn to the operator equation:

$(a_{0}p^{2}+a_{1}p+a_{2})y=(b_{0}p+b_{1})x$ ${\ displaystyle (a_ {0} p ^ {2} + a_ {1} p + a_ {2}) y = (b_ {0} p + b_ {1}) x}$

$F()\rightarrow F(\Delta x,\Delta y)\rightarrow LE\rightarrow OE$ ${\ displaystyle F () \ rightarrow F (\ Delta x, \ Delta y) \ rightarrow LE \ rightarrow OE}$

Controllability, observability of ACS

The ACS is controllable (fully controllable) if it can be transferred from any initial state x ₀ (t) to another arbitrary state x ₁ (t) at an arbitrary point in time by applying a piecewise continuous action U (t) ∈ [t ₀ ; t ₁ ].

САУ наблюдаема (полностью наблюдаема), если все переменные состояния x(t) можно определить по выходному (измеряемому) воздействию y(t).

Устойчивость линейных систем

Устойчивость — свойство САУ возвращаться в заданный или близкий к нему установившийся режим после какого-либо возмущения. Устойчивая САУ — система, в которой переходные процессы являются затухающими.

$(a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n})y=(b_{0}p^{m}+b_{1}p^{m-1}+...+b_{m})g$ $(a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n})y=(b_{0}p^{m}+b_{1}p^{m-1}+...+b_{m})g$ — операторная форма записи линеаризированного уравнения.

y(t) = y _уст (t)+y _п = y _вын (t)+y _св

y _уст (y _вын ) — частное решение линеаризированного уравнения.

y _п (y _св ) — общее решение линеаризированного уравнения как однородного дифференциального уравнения, то есть $D(p)=(a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n})y=0$ $D(p)=(a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n})y=0$

САУ устойчива, если переходные процессы у _n (t), вызываемые любыми возмущениями, будут затухающими с течением времени, то есть $y_{n}(t)\rightarrow 0$ $y_{n}(t)\rightarrow 0$ at $t\rightarrow {\mathcal {1}}$ $t\rightarrow {\mathcal {1}}$

Решая дифференциальное уравнение в общем случае, получим комплексные корни p _i , p _i+1 = ±α _i ± jβ _i

Каждой паре комплексно-сопряженных корней соответствует следующая составляющая уравнения переходного процесса:

$c_{i}e^{(\alpha _{i}+j\beta _{i})t}+c_{i+1}e^{(\alpha _{i}-j\beta _{i})t}=\alpha _{i}(c_{i}e^{j\beta _{i}t}+c_{i+1}e^{-j\beta _{i}t})=Ae^{\alpha _{i}t}\sin {(\beta _{i}t+\varphi _{i})}$ $c_{i}e^{(\alpha _{i}+j\beta _{i})t}+c_{i+1}e^{(\alpha _{i}-j\beta _{i})t}=\alpha _{i}(c_{i}e^{j\beta _{i}t}+c_{i+1}e^{-j\beta _{i}t})=Ae^{\alpha _{i}t}\sin {(\beta _{i}t+\varphi _{i})}$ where $A={\sqrt {c_{i}^{2}+c_{i+1}^{2}}}$ $A={\sqrt {c_{i}^{2}+c_{i+1}^{2}}}$ , $\operatorname {tg} {\varphi _{i}}={c_{i}+c_{i+1} \over c_{i}-c_{i+1}}$ $\operatorname {tg} {\varphi _{i}}={c_{i}+c_{i+1} \over c_{i}-c_{i+1}}$

Из полученных результатов видно, что:

при ∀α _i <0 выполняется условие устойчивости, то есть переходный процесс с течением времени стремится к у _уст (теорема Ляпунова 1);
при ∃α _i >0, выполняется условие неустойчивости (теорема Ляпунова 2), то есть $Ae^{\alpha _{i}t}\sin {(\beta _{i}t+\varphi _{i})}\rightarrow {\mathcal {1}}$ $Ae^{\alpha _{i}t}\sin {(\beta _{i}t+\varphi _{i})}\rightarrow {\mathcal {1}}$ , что приводит к расходящимся колебаниям;
при ∃α _i =0 и ¬∃α _i >0 $Ae^{\alpha _{i}t}\sin {(\beta _{i}t+\varphi _{i})}=const$ $Ae^{\alpha _{i}t}\sin {(\beta _{i}t+\varphi _{i})}=const$ , что приводит к незатухающим синусоидальным колебаниям системы (система на границе устойчивости) (теорема Ляпунова 3).

Критерии устойчивости

Критерий Рауса

Для определения устойчивости системы строятся таблицы вида:

Coefficients	Strings	столбец 1	столбец 2	столбец 3
	one	$C_{11}=a_{0}=T_{1}T_{2}T_{3}$ $C_{11}=a_{0}=T_{1}T_{2}T_{3}$	$C_{12}=a_{2}=T_{1}+T_{2}+T_{3}$ $C_{12}=a_{2}=T_{1}+T_{2}+T_{3}$	$C_{13}=a_{4}$ $C_{13}=a_{4}$
	2	$C_{21}=a_{1}=T_{1}T_{2}+T_{2}T_{3}+T_{1}+T_{3}$ $C_{21}=a_{1}=T_{1}T_{2}+T_{2}T_{3}+T_{1}+T_{3}$	$C_{22}=a_{3}=1+k$ $C_{22}=a_{3}=1+k$	$C_{23}=a_{5}$ $C_{23}=a_{5}$
$r_{3}={\frac {C_{11}}{C_{21}}}$ $r_{3}={\frac {C_{11}}{C_{21}}}$	3	$C_{31}=C_{12}-r_{3}C_{22}$ $C_{31}=C_{12}-r_{3}C_{22}$	$C_{32}=C_{13}-r_{3}C_{23}$ $C_{32}=C_{13}-r_{3}C_{23}$	$C_{33}=C_{14}-r_{3}C_{24}$ $C_{33}=C_{14}-r_{3}C_{24}$
$r_{4}={\frac {C_{21}}{C_{31}}}$ $r_{4}={\frac {C_{21}}{C_{31}}}$	four	$C_{41}=C_{22}-r_{4}C_{32}$ $C_{41}=C_{22}-r_{4}C_{32}$	$C_{42}=C_{23}-r_{4}C_{33}$ $C_{42}=C_{23}-r_{4}C_{33}$	$C_{43}=C_{24}-r_{4}C_{34}$ $C_{43}=C_{24}-r_{4}C_{34}$

Для устойчивости системы необходимо, чтобы все элементы первого столбца имели положительные значения; если в первом столбце присутствуют отрицательные элементы — система неустойчива; если хотя бы один элемент равен нулю, а остальные положительны, то система на границе устойчивости.

Критерий Гурвица

$D(p)=a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n}$ $D(p)=a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n}$

$\Delta _{n}=a_{n}\cdot \Delta _{n-1}={\begin{vmatrix}a_{1}&a_{3}&a_{5}&...&0\\a_{0}&a_{2}&a_{4}&...&0\\0&a_{1}&a_{3}&...&0\\...&...&...&...&...\\0&...&...&...&a_{n}\end{vmatrix}}$ $\Delta _{n}=a_{n}\cdot \Delta _{n-1}={\begin{vmatrix}a_{1}&a_{3}&a_{5}&...&0\\a_{0}&a_{2}&a_{4}&...&0\\0&a_{1}&a_{3}&...&0\\...&...&...&...&...\\0&...&...&...&a_{n}\end{vmatrix}}$ — определитель Гурвица

Теорема : для устойчивости замкнутой САУ необходимо и достаточно, чтобы определитель Гурвица и все его миноры были положительны при $a_{0}>0.$ $a_{0}>0.$

Критерий Михайлова

$D(p)=a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n}$ $D(p)=a_{0}p^{n}+a_{1}p^{n-1}+...+a_{n}$

Заменим $p=j\omega$ $p=j\omega$ where ω is the angular frequency of the oscillations corresponding to the pure imaginary root of the characteristic characteristic polynomial.

$D(j\omega )=X(\omega )+jY(\omega )=A(\omega )e^{j\psi (\omega )}$ ${\ displaystyle D (j \ omega) = X (\ omega) + jY (\ omega) = A (\ omega) e ^ {j \ psi (\ omega)}}$

$X(\omega )=a_{n}-a_{n-2}\omega ^{2}+...$ ${\ displaystyle X (\ omega) = a_ {n} -a_ {n-2} \ omega ^ {2} + ...}$

$Y(\omega )=a_{n-1}\omega -a_{n-3}\omega ^{3}+...$ ${\ displaystyle Y (\ omega) = a_ {n-1} \ omega -a_ {n-3} \ omega ^ {3} + ...}$

Criterion : for the stability of a linear system of the nth order, it is necessary and sufficient that the Mikhailov curve constructed in the coordinates $X(\omega ),Y(\omega )$ ${\ displaystyle X (\ omega), Y (\ omega)}$ , passed sequentially through n quadrants.

$D(p)=a_{0}(p-p_{1})(p-p_{2})...(p-p_{n})$ ${\ displaystyle D (p) = a_ {0} (p-p_ {1}) (p-p_ {2}) ... (p-p_ {n})}$

$p=j\omega \Rightarrow D(j\omega )=a_{0}(j\omega -p_{1})(j\omega -p_{2})...(j\omega -p_{n})$ ${\ displaystyle p = j \ omega \ Rightarrow D (j \ omega) = a_ {0} (j \ omega -p_ {1}) (j \ omega -p_ {2}) ... (j \ omega -p_ {n})}$

Consider the relationship between the Mikhailov curve and the signs of its roots (α> 0 and β> 0)

1) The root of the characteristic equation is a negative real number $p_{1}=-\alpha _{1}$ ${\ displaystyle p_ {1} = - \ alpha _ {1}}$

Corresponding root factor $(\alpha _{1}+j\omega )$ ${\ displaystyle (\ alpha _ {1} + j \ omega)}$

$\left\{{\begin{matrix}\omega \rightarrow +{\mathcal {1}}\\p_{1}=-\alpha _{1}\end{matrix}}\right.\Rightarrow \psi \rightarrow {\frac {\pi }{2}}$ ${\ displaystyle \ left \ {{\ begin {matrix} \ omega \ rightarrow + {\ mathcal {1}} \\ p_ {1} = - \ alpha _ {1} \ end {matrix}} \ right. \ Rightarrow \ psi \ rightarrow {\ frac {\ pi} {2}}}$

2) The root of the characteristic equation is a positive real number $p_{1}=+\alpha _{1}$ ${\ displaystyle p_ {1} = + \ alpha _ {1}}$

Corresponding root factor $(\alpha _{1}-j\omega )$ ${\ displaystyle (\ alpha _ {1} -j \ omega)}$

$\left\{{\begin{matrix}\omega \rightarrow +{\mathcal {1}}\\p_{1}=+\alpha _{1}\end{matrix}}\right.\Rightarrow \psi \rightarrow -{\frac {\pi }{2}}$ ${\ displaystyle \ left \ {{\ begin {matrix} \ omega \ rightarrow + {\ mathcal {1}} \\ p_ {1} = + \ alpha _ {1} \ end {matrix}} \ right. \ Rightarrow \ psi \ rightarrow - {\ frac {\ pi} {2}}}$

3) The root of the characteristic equation is a complex pair of numbers with a negative real part. $p_{2,3}=-\alpha _{1}\pm j\beta$ ${\ displaystyle p_ {2,3} = - \ alpha _ {1} \ pm j \ beta}$

Corresponding root factor $(j\omega +\alpha _{1}-j\beta )(j\omega +\alpha _{1}+j\beta )$ ${\ displaystyle (j \ omega + \ alpha _ {1} -j \ beta) (j \ omega + \ alpha _ {1} + j \ beta)}$

$\left\{{\begin{matrix}\omega \rightarrow +{\mathcal {1}}\\p_{2}=-\alpha _{1}+j\beta \\p_{3}=-\alpha _{1}-j\beta \end{matrix}}\right.\Rightarrow \left\{{\begin{matrix}\psi _{2}\rightarrow +{\frac {\pi }{2}}-\gamma \\\psi _{3}\rightarrow +{\frac {\pi }{2}}+\gamma \end{matrix}}\right.\Rightarrow \psi \rightarrow +2{\frac {\pi }{2}}=+\pi$ ${\ displaystyle \ left \ {{\ begin {matrix} \ omega \ rightarrow + {\ mathcal {1}} \\ p_ {2} = - \ alpha _ {1} + j \ beta \ p_ {3} = - \ alpha _ {1} -j \ beta \ end {matrix}} \ right. \ Rightarrow \ left \ {{\ begin {matrix} \ psi _ {2} \ rightarrow + {\ frac {\ pi} {2 }} - \ gamma \\\ psi _ {3} \ rightarrow + {\ frac {\ pi} {2}} + \ gamma \ end {matrix}} \ right. \ Rightarrow \ psi \ rightarrow +2 {\ frac {\ pi} {2}} = + \ pi}$ where $\gamma =\operatorname {arctg} {\frac {\beta }{\alpha }}$ ${\ displaystyle \ gamma = \ operatorname {arctg} {\ frac {\ beta} {\ alpha}}}$

4) The root of the characteristic equation is a complex pair of numbers with a positive real part. $p_{2,3}=+\alpha _{1}\pm j\beta$ ${\ displaystyle p_ {2,3} = + \ alpha _ {1} \ pm j \ beta}$

Corresponding root factor $(j\omega -\alpha _{1}-j\beta )(j\omega -\alpha _{1}+j\beta )$ ${\ displaystyle (j \ omega - \ alpha _ {1} -j \ beta) (j \ omega - \ alpha _ {1} + j \ beta)}$

$\left\{{\begin{matrix}\omega \rightarrow +{\mathcal {1}}\\p_{2}=+\alpha _{1}+j\beta \\p_{3}=+\alpha _{1}-j\beta \end{matrix}}\right.\Rightarrow \left\{{\begin{matrix}\psi _{2}\rightarrow -{\frac {\pi }{2}}+\gamma \\\psi _{3}\rightarrow -{\frac {\pi }{2}}-\gamma \end{matrix}}\right.\Rightarrow \psi \rightarrow -2{\frac {\pi }{2}}=-\pi$ ${\ displaystyle \ left \ {{\ begin {matrix} \ omega \ rightarrow + {\ mathcal {1}} \\ p_ {2} = + \ alpha _ {1} + j \ beta \ p_ {3} = + \ alpha _ {1} -j \ beta \ end {matrix}} \ right. \ Rightarrow \ left \ {{\ begin {matrix} \ psi _ {2} \ rightarrow - {\ frac {\ pi} {2 }} + \ gamma \\\ psi _ {3} \ rightarrow - {\ frac {\ pi} {2}} - \ gamma \ end {matrix}} \ right. \ Rightarrow \ psi \ rightarrow -2 {\ frac {\ pi} {2}} = - \ pi}$ where $\gamma =\operatorname {arctg} {\frac {\beta }{\alpha }}$ ${\ displaystyle \ gamma = \ operatorname {arctg} {\ frac {\ beta} {\ alpha}}}$

Nyquist criterion

Nyquist criterion is a graph-analytical criterion. Its characteristic feature is that the conclusion about the stability or instability of a closed system is made depending on the type of amplitude-phase or logarithmic frequency characteristics of an open-loop system.

Let an open system be represented as a polynomial $W(p)={\frac {R(p)}{Q(p)}}={\frac {b_{0}p^{n}+b_{1}p^{n-1}+...+b^{n}}{a_{0}p^{m}+a_{1}p^{m-1}+...+a^{m}}}$ ${\ displaystyle W (p) = {\ frac {R (p)} {Q (p)} = {\ frac {b_ {0} p ^ {n} + b_ {1} p ^ {n-1} + ... + b ^ {n}} {a_ {0} p ^ {m} + a_ {1} p ^ {m-1} + ... + a ^ {m}}}$

then we do the substitution $p=j\omega$ ${\ displaystyle p = j \ omega}$ and get: $W(j\omega )={\frac {R(j\omega )}{Q(j\omega )}}(*)=X(\omega )+jY(\omega )=A(\omega )e^{j\psi (\omega )}$ ${\ displaystyle W (j \ omega) = {\ frac {R (j \ omega)} {Q (j \ omega)}} (*) = X (\ omega) + jY (\ omega) = A (\ omega ) e ^ {j \ psi (\ omega)}}$

For a more convenient construction of the hodograph for n> 2, we will reduce the equation (*) to the “standard” form: $W(j\omega )={\frac {K(1+j\omega \tau _{1})(1+j\omega \tau _{2})[(1-\tau _{3}^{2}\omega ^{2})+2j\xi _{3}\tau _{3}\omega ]...}{(j\omega )'[(1-T_{2}^{2}\omega ^{2})+2j\xi _{2}T_{2}\omega ](-1+j\omega T_{3})...}}$ ${\ displaystyle W (j \ omega) = {\ frac {K (1 + j \ omega \ tau _ {1}) (1 + j \ omega \ tau _ {2}) [(1- \ tau _ {3 } ^ {2} \ omega ^ {2}) + 2j \ xi _ {3} \ tau _ {3} \ omega] ...} {(j \ omega) '[(1-T_ {2} ^ { 2} \ omega ^ {2}) + 2j \ xi _ {2} T_ {2} \ omega] (- 1 + j \ omega T_ {3}) ...}}}$

With this representation, the module A (ω) = | W (jω) | equal to the ratio of the modules of the numerator and the denominator, and the argument (phase) (ω) - the difference of their arguments. In turn, the module of the product of complex numbers is equal to the product of modules, and the argument is the sum of the arguments.

Modules and arguments corresponding to the transfer function multipliers:

Multiplier	$A(\omega )$ ${\ displaystyle A (\ omega)}$	$\psi (\omega )$ ${\ displaystyle \ psi (\ omega)}$
k	k	0
p	ω	${\frac {\pi }{2}}$ ${\ displaystyle {\ frac {\ pi} {2}}}$
$p^{2}$ ${\ displaystyle p ^ {2}}$	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$\pi$ ${\ displaystyle \ pi}$
$Tp+1$ ${\ displaystyle Tp + 1}$	${\sqrt {1+T^{2}\omega ^{2}}}$ ${\ displaystyle {\ sqrt {1 + T ^ {2} \ omega ^ {2}}}}$	$\operatorname {arctg} \omega T$ ${\ displaystyle \ operatorname {arctg} \ omega T}$
$Tp-1$ ${\ displaystyle Tp-1}$	${\sqrt {1+T^{2}\omega ^{2}}}$ ${\ displaystyle {\ sqrt {1 + T ^ {2} \ omega ^ {2}}}}$	$\pi -\operatorname {arctg} \omega T$ ${\ displaystyle \ pi - \ operatorname {arctg} \ omega T}$
$1-Tp$ ${\ displaystyle 1-Tp}$	${\sqrt {1+T^{2}\omega ^{2}}}$ ${\ displaystyle {\ sqrt {1 + T ^ {2} \ omega ^ {2}}}}$	$-\operatorname {arctg} \omega T$ ${\ displaystyle - \ operatorname {arctg} \ omega T}$
$T^{2}p^{2}+1$ ${\ displaystyle T ^ {2} p ^ {2} +1}$	$\left\|1-T^{2}\omega ^{2}\right\|$ ${\ displaystyle \ left \| 1-T ^ {2} \ omega ^ {2} \ right \|}$	${\begin{matrix}0,&\omega <{\frac {1}{T}}\\\pi ,&\omega >{\frac {1}{T}}\end{matrix}}$ ${\ displaystyle {\ begin {matrix} 0, & \ omega <{\ \ frac {1} {T}} \\\ pi, & \ omega> {\ frac {1} {T}} \ end {matrix}} }$
$T^{2}p^{2}+2\xi Tp+1$ ${\ displaystyle T ^ {2} p ^ {2} +2 \ xi Tp + 1}$	${\sqrt {(1-T^{2}\omega ^{2})^{2}+4\xi ^{2}T^{2}\omega ^{2}}}$ ${\ displaystyle {\ sqrt {(1-T ^ {2} \ omega ^ {2}) ^ {2} +4 \ xi ^ {2} T ^ {2} \ omega ^ {2}}}}$	${\begin{matrix}\operatorname {arctg} {\frac {2\xi \omega T}{1-T^{2}\omega ^{2}}},&\omega <{\frac {1}{T}}\\\pi +\operatorname {arctg} {\frac {2\xi \omega T}{1-T^{2}\omega ^{2}}},&\omega \geqslant {\frac {1}{T}}\end{matrix}}$ ${\ displaystyle {\ begin {matrix} \ operatorname {arctg} {\ frac {2 \ xi \ omega T} {1-T ^ {2} \ omega ^ {2}}}, & \ omega <{\ frac { 1} {T}} \\ pi + \ operatorname {arctg} {\ frac {2 \ xi \ omega T} {1-T ^ {2} \ omega ^ {2}}}, & \ omega \ geqslant { \ frac {1} {T}} \ end {matrix}}}$

Then build the hodograph for the auxiliary function $W_{1}(j\omega )=1+W(j\omega )$ ${\ displaystyle W_ {1} (j \ omega) = 1 + W (j \ omega)}$ why we will change $\omega [0;{\mathcal {1}})$ ${\ displaystyle \ omega [0; {\ mathcal {1}})}$

With $\omega =0,\quad W_{1}(j\omega )=K+1$ ${\ displaystyle \ omega = 0, \ quad W_ {1} (j \ omega) = K + 1}$ while $\omega ={\mathcal {1}},\quad W_{1}(j\omega )=1$ ${\ displaystyle \ omega = {\ mathcal {1}}, \ quad W_ {1} (j \ omega) = 1}$ (since n <m and $W(j\omega )=0$ ${\ displaystyle W (j \ omega) = 0}$ )

To determine the resulting angle of rotation, we find the difference of the arguments of the numerator $\psi _{1}$ ${\ displaystyle \ psi _ {1}}$ and denominator $\psi _{2}$ ${\ displaystyle \ psi _ {2}}$

The polynomial of the numerator of the auxiliary function has the same degree as the polynomial of its denominator, whence it follows $\psi _{1}=\psi _{2}$ ${\ displaystyle \ psi _ {1} = \ psi _ {2}}$ therefore, the resulting angle of rotation of the auxiliary function is 0. This means that for the stability of a closed system, the hodograph of the vector of the auxiliary function should not cover the origin, $W(j\omega )$ ${\ displaystyle W (j \ omega)}$ respectively a point with coordinates $(-1;j0)$ ${\ displaystyle (-1; j0)}$

SAU stability margin

Under operating conditions, the system parameters for one reason or another may vary within certain limits (aging, temperature fluctuations, etc.). These fluctuations in parameters can lead to a loss of stability of the system if it is operating near the stability limit. Therefore, they strive to design the system so that it works away from the stability boundary. The extent of this removal is called the sustainability margin.

The need for sustainability is determined by the following conditions:

Dropping nonlinear terms with linearization.
The coefficients included in the equation describing the ACS are determined with an error.
Stability studies for typical systems under typical conditions.

Criteria

Routh's criterion: in order to model the stability margin, it is necessary that the elements of the first column be greater than some fixed value ε> 0, called the factor of stability.
The Hurwitz criterion: the stability margin is determined similarly to the Routh stability margin, only ε characterizes the value of the Hurwitz determinant.
Mikhailov's criterion: a circle of nonzero radius with the center at the point O (0; 0) fits. The stock is determined by the radius of this circle. The system is unstable when the Mikhailov criterion is violated or when the Mikhailov curve intersects with a circle.
Nyquist criterion : the point (-1; j0) is critical here, therefore, a restricted area is built around this point, the radius of which will represent the factor of stability.

Comparative characteristics of sustainability criteria

The Nyquist frequency criterion is applicable mainly when it is difficult to obtain phase characteristics experimentally. However, the calculation of AFC, especially frequency, is more complicated than the construction of Mikhailov curves. In addition, the location of the AFCh does not give a direct answer to the question: is the system stable, that is, additional research is needed on the stability of the system in the open state.

Mikhailov's criterion is applied to systems of any order, unlike the Routh criterion. Using the Nyquist frequency criterion and the Mikhailov criterion, the characteristic curves can be constructed gradually, taking into account the influence of each link, which gives the criteria clarity and solves the problem of choosing system parameters from the stability condition.

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↑ Rotach V.Ya. The theory of automatic control. - 2nd, pererab. and additional .. - Moscow: MEI, 2004. - p. 3-15. - 400 s. - ISBN 5-7046-0924-4 .
↑ A.V. Andryushin, V.R.Sabanin, N.I. Smirnov. Management and Innovation in Heat Power Engineering. - M: MEI, 2011. - p. 15. - 392 p. - ISBN 978-5-38300539-2

[1] Rotach V.Ya. The theory of automatic control. - 2nd, pererab. and additional .. - Moscow: MEI, 2004. - p. 3-15. - 400 s. - ISBN 5-7046-0924-4 .

[2] A.V. Andryushin, V.R.Sabanin, N.I. Smirnov. Management and Innovation in Heat Power Engineering. - M: MEI, 2011. - p. 15. - 392 p. - ISBN 978-5-38300539-2