home Heating installation How to turn an unstable system into a stable one. Determination of the stability of automatic control systems for industrial robots

How to turn an unstable system into a stable one. Determination of the stability of automatic control systems for industrial robots

This section discusses the most important characteristics of the quality of managed systems. These characteristics are system stability, accuracy and noise immunity.

The concept of stability refers to the situation when the input signals of the system are equal to zero, i.e. there are no external influences. At the same time, a properly constructed system should be in a state of equilibrium (rest) or gradually approach this state. In unstable systems, even with zero input signals, natural oscillations occur and, as a result, unacceptably large errors.

The concept of accuracy is related to the quality of operation of controlled systems with varying input signals. In properly designed control systems, the mismatch between the given control law g(t) and the output signal x(t) should be small.

Finally, to characterize the effect of interference on control systems, the dispersion or standard deviation of the error component due to interference is used.

The concept of sustainability

One of the first questions that arise in the study and design of linear control systems is the question of their stability. The linear system is called sustainable if, when it is removed by external influences from a state of equilibrium (rest), it returns to it after the cessation of external influences. If, after the termination of external influence, the system does not return to a state of equilibrium, then it is unstable. For the normal functioning of the control system, it is necessary that it be stable, otherwise large errors will occur in it.

The determination of stability is usually carried out on initial stage creation of a control system. This is due to two reasons. First, stability analysis is quite simple. Second, unstable systems can be corrected, i.e. converted to stable by adding special corrective links.

Stability Analysis Using Algebraic Criteria

The stability of a system is related to the nature of its natural oscillations. To clarify this, suppose that the system is described by the differential equation

or, after the Laplace transform,

where g(p) is the input action.

The stable system returns to the state of rest if the input action g(p) 0 . Thus, for a stable system, the solution of the homogeneous differential equation must tend to zero as t tends to infinity.

If the roots p1, p2, ... , pn of the characteristic equation are found, then the solution of the homogeneous equation will be written as .

In what cases is the system stable?

Assume that pk = ak is a real root.

The term ck corresponds to it. At ak< 0 это слагаемое будет стремиться к нулю, если t стремится к бесконечности. Если же ak >0, then x(t) as t goes to infinity; . Finally, in the case when ak = 0, the term under consideration does not change even as t tends to infinity,

Let us now assume that is the complex root of the characteristic equation. Note that in this case also will be the root of the characteristic equation. Two complex conjugate roots will correspond to terms of the form , .

Moreover, if ak< 0, то в системе имеются затухающие колебания. При ak >0 - oscillations of increasing amplitude, and at ak = 0 - oscillations of constant amplitude сk.

Thus, the system is stable if the real parts of all roots of the characteristic equation are negative. If at least one root has a real part ak ³ 0, then the system is unstable. A system is said to be on the boundary of stability if at least one root of the characteristic equation has a zero real part, and the real parts of all other roots are negative.

This definition is well illustrated geometrically. Let us represent the roots of the characteristic equation as points on the complex plane (Fig. 15).

If all roots lie in the left half-plane of the complex variable, then the system is stable. If at least one root lies in the right half-plane of the complex variable, the system is unstable. If the roots are on the imaginary axis and in the left half-plane, then the system is said to be on the stability boundary.

Consider, as an example, a closed control system with one integrating link. In this case, H(p) = , , and the transfer function of the closed system

System output x(p) = W(p)g(p) or . Note that the characteristic equation p+k=0 is written by equating to zero the denominator of the transfer function of the closed control system. In this case, there is one root p1= -k< 0 и поэтому система управления всегда устойчива. Предположим теперь, что . Тогда . The characteristic equation p2 + + k = 0. Therefore, p1,2=. The system is on the edge of stability. It has undamped oscillations.

Stability Analysis Using Frequency Criteria

The main disadvantage of the considered algebraic approach to stability analysis is that in complex control systems it is difficult to establish a connection between the roots of the denominator рk, k=1, 2, ..., n, and the parameters of the elementary links that make up the control system. This leads to difficulties in correcting unstable systems. In order to simplify the analysis of stability, it is desirable to carry out this analysis on the transfer function H(p) of an open-loop control system.

In 1932, the American scientist Nyquist developed an effective method for analyzing the stability of feedback amplifiers. In 1938, the Soviet scientist A.V. Mikhailov generalized the Nyquist method to closed automatic control systems.

The Nyquist criterion is based on the construction of the hodograph of the transfer function H(jw) of an open-loop control system. Hodograph of the transfer function H(jw) is the curve drawn by the end of the vector H(jw) =|H(jw)|ejj(w) on the complex plane when the frequency w is measured from 0 to infinity.

The Nyquist stability criterion is most simply formulated: a closed control system is stable if the hodograph of the transfer function H(jw) of an open system does not cover a point with coordinates (-1, j0) on the complex plane. The figures show examples of hodographs of stable (Fig. 16, a) and unstable (Fig. 16, b) control systems.

If the hodograph passes through the point -1, then the system is said to be on the boundary of stability. In this case, at some frequency H(jw0)= -1 and the system may have undamped oscillations of the frequency w0. In unstable systems, the signal level x(t) will increase with time. In stable - decrease.

margin of stability

Another advantage of the criterion under consideration is the possibility of determining the stability margin of the control system. The stability margin is characterized by two indicators: gain stability margin and phase stability margin.

Gain Stability Margin is determined by the value g =1/|H(jw0)|, where w0 is the frequency at which (Fig. 17a). The stability margin g shows how many times the modulus of the transfer function of an open-loop control system must change (increase) in order for the closed-loop system to be on the stability boundary. The required stability margin depends on how much the transmission coefficient of the system can increase in the process of operation compared to the calculated one.

Phase stability margin is estimated by the value of the angle , where the frequency wсp , called cutoff frequency, is determined by the condition |H(jwcp)|=1 (Fig. 17, b).

The value of Dj shows how much the phase characteristic of an open-loop control system must change in order for the closed-loop system to be on the stability boundary. The phase margin is usually considered sufficient if
|Dj| ³ 30o.

Stability Analysis Using Logarithmic Frequency Responses

In many cases, an open-loop control system can be represented as a serial connection of n typical links with transfer functions . In this case, the transfer function of an open system is determined by the product . Logarithmic frequency response will be equal to the sum of the LAH of individual links:

Since the LAC of many elementary links can be approximated by segments of straight lines, the LAC of an open control system will also be represented as segments of straight lines with slopes to the frequency axis that are multiples of 20 decibels per decade.

Example. Let the transfer function of an open system have the following form

Such a system contains two integrators, a boosting link with a transfer function and an aperiodic link with a transfer function . Let us represent the LAH of individual links of such a system in the form of graphs in fig. 18, a. Summing up the presented graphs, we get the LAH of an open system (Fig. 18, b).

As follows from the figures, the construction of the total LAH is quite simple. It is only necessary to take into account the change in the slope of the LAH at the points and corresponding to the conjugate frequencies of the forcing and aperiodic links.

To check the stability conditions of a closed-loop automatic control system, it is necessary to plot the phase-frequency characteristic on the same logarithmic scale along the frequency axis . However, the experience of engineering calculations shows that a closed ACS, as a rule, is stable and has a margin of stability, if the LAH of an open system is near the frequency

The cutoff has a slope of -20 dB/dec. At the same time, the stability margin is the greater, the greater the length of this LAH section. It is usually believed that the length of the section with a slope of - 20 dB/dec should be at least 1 decade. There are stable ACS with LAH slope greater than -20 dB/dec, but for such systems, as a rule, the margin of stability is very small.

Assume that the ACS under study has a slope around the cutoff frequency greater than -20 dB/dec (Fig. 19)

Considering that when the ACS links are connected in series, their LAH are summed up, it is necessary to include in the ACS such a link that will ensure the stability of the system. In the case under consideration, such a link can be a link with a LAH shown in Fig. 20.

Indeed, after summing the LAC of the control system (Fig. 19) and an additional link, we obtain an LAC with a constant slope of 20 dB/dec at all frequencies, including

cutoff frequency. In the example under consideration, the transfer function of the additional corrective link is Hf(jw) =1+jwTf, with w1 = 1/Tf. The introduction of additional links to ensure the stability of control systems is called correction self-propelled guns, and the links themselves - corrective.

In this section, methods for studying one of the most important indicators of the quality of control systems, the stability of linear systems, were considered. The application of these methods to the analysis of specific systems is usually carried out as follows. First, the LAH of an open-loop control system is built. If the system is unstable, then corrective links are selected and introduced into it in such a way that the LAH slope at the cutoff frequency is -20 dB/dec and the necessary stability margin is provided. After that, the stability of the adjusted system is necessarily investigated using the Nyquist-Mikhailov criterion and the exact values of the stability margins for gain and phase are determined. If necessary, after that, the parameters of the control system are changed to ensure a given stability margin.

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Lecture #4

ACS stability

The property of a system to return to its original state after the removal of a perturbation is called stability.

Definition.

Curves 1 and 2 characterize a stable system, curves 3 and 4 characterize unstable systems.ε

Systems 5 and 6 on the edge of stability 5 - neutral system, 6 - oscillatory stability limit.

Let the SAC differential equation in operator form have the form

Then the solution of the differential equation (the motion of the system) consists of two parts Forced movement of the same kind as the input action.

In the absence of multiple roots where C i -integration constants determined from the initial conditions,

 1 ,  2 …,  n are the roots of the characteristic equation

The location of the roots of the characteristic

system equations on the complex plane

The roots of the characteristic equation do not depend either on the type of perturbation or on

initial conditions, and are determined only by the coefficients a 0 , а 1 , а 2 ,…,а n , that is, the parameters and structure of the system.

1 - the root is real, greater than zero;

2-root real, less than zero;

3-root is equal to zero;

4-two zero roots;

5-two complex conjugate roots whose real part is

positive;

6-two complex conjugate roots, the real part of which is negative;

7-two imaginary conjugate roots.

Stability Analysis Methods:

Straight lines (based on solving differential equations);
Indirect (stability criteria).

Theorems of A.M. Lyapunov.

Theorem 1.

Theorem 2.

Notes:

If there are two or more zero roots among the roots of the characteristic equation, then the system is unstable.
If one root is zero, and all the others are in the left half-plane, then the system is neutral.
If 2 roots are imaginary conjugate, and all the others are in the left half-plane, then the system is on the oscillatory boundary of stability.

ACS stability criteria.

The stability criterion is a rule that allows you to find out the stability of the system without calculating the roots of the characteristic equation.

In 1877 Rous installed:

1. Hurwitz stability criterion

The criterion was developed in 1895.

Let the characteristic equation of a closed system be defined: the equation is reduced to the form so that a0 >0.

We compose the main Hurwitz determinant according to the following rule:

the coefficients of the equation are written along the main diagonal, starting from the second to the last, the columns up from the diagonal are filled with coefficients with increasing indices, and the columns down from the diagonal are filled with coefficients with decreasing indices. In the absence of any coefficient in the equation and instead of coefficients with indices less than 0 and greater n write zero.

We single out the diagonal minors or the simplest determinants in the main Hurwitz determinant:

Formulation of the criterion.

For systems above the second order, in addition to the positivity of all coefficients of the characteristic equation, the following inequalities must be satisfied:

For third order systems:
For fourth order systems:
For systems of the fifth order:

For systems of the sixth order:

Example. A characteristic equation is given to investigate the stability of the system according to Hurwitz.

For sustainable systems, it is necessary

2. Routh criterion

The Routh criterion is used in the study of the stability of high-order systems.

The wording of the criterion:

Routh table.

Algorithm for filling the table: in the first and second lines, the coefficients of the equation with even and odd indices are written; the elements of the remaining rows are calculated according to the following rule:

The advantage of the criterion is that it is possible to study the stability of systems of any order.

2. Nyquist stability criterion

Argument principle

The basis of frequency methods is the argument principle.

Let's analyze the properties of a polynomial of the form:

Where  i - roots of the equation

On the complex plane, each root corresponds to a well-defined point. Geometrically, each rooti can be represented as a vector drawn from the origin to a point i : | i | - vector length, argi - the angle between the vector and the positive direction of the x-axis. Let us map D(p) into the Fourier space, then where j - i is an elementary vector.

The ends of the elementary vectors are on the imaginary axis.

Modulo vector, and argument (phase)

The direction of rotation of the vector counterclockwise is taken as POSITIVE. Then when changing from to each elementary vector ( j  -  i ) will rotate by an angle + , if  i lies in the left half-plane.

Let D ( )=0 have m roots in the right half-plane and n-m roots in the left, then with increasing from to the change in the argument of the vector D(j ) (rotation angle D(j ), equal to the sum of changes in the arguments of elementary vectors) will be

Argument principle:

The Nyquist criterion is based on the frequency characteristics of an open circuit of the ACS, since the stability of a closed system can be judged from the form of the frequency characteristics of an open circuit.

The Nyquist criterion has found wide application in engineering practice for the following reasons:

The stability of a system in a closed state is studied by the frequency transfer function of its open circuit, and this function most often consists of simple factors. The coefficients are the real parameters of the system, which makes it possible to choose them from the conditions of stability.
To study the stability, it is possible to use the experimentally obtained frequency characteristics of the most complex elements of the system (regulated object, executive body), which increases the accuracy of the results obtained.
Stability can be studied by LFC, the construction of which is not difficult.
It is convenient to determine the stability margins.

1. System stable in the open state

Let us introduce an auxiliary function and replace p  j  , then

According to the argument principle, the change in the argument D(j ) and D s (j  ) at 0<  <  equals Then that is the hodograph W 1 (j  ) must not span the origin.

To simplify the analysis and calculations, we will shift the origin of the radius vector from the origin to the point (-1, j 0), but instead of the helper function W 1 (j  ) we use the open-loop AFC W (j  ).

Formulation of Criterion #1

Examples.

Note that the difference between the number of positive and negative transitions of the AFC is to the left of the point (-1, j 0) is equal to zero.

2. A system with poles on the imaginary axis in the open state

To analyze the stability of the AFC systems, they are supplemented with a circle of infinitely large radius at 0 counterclockwise to a positive real semiaxis at zero poles, and in the case of purely imaginary roots - a semicircle clockwise at the point of discontinuity of the AFC.

Formulation of criterion #2

Unstable open circuit system

A more general case - the denominator of the transfer function of an open-loop system contains roots lying in the right half-plane. The appearance of instability of an open-loop system is caused by two reasons:

The consequence of the presence of unstable links;
A consequence of the loss of stability of links covered by positive or negative feedback.

X Although theoretically the whole system in a closed state can be stable in the presence of instability along the local feedback loop, in practice this case is undesirable and should be avoided, trying to use only stable local feedbacks. This is explained by the presence of undesirable properties, in particular, the appearance of conditional stability, which, with the nonlinearities usually present in the system, can in some modes lead to a loss of stability and the appearance of self-oscillations. Therefore, as a rule, when calculating the system, such local feedbacks are chosen that would be stable with the main feedback open..

Let the characteristic polynomial D(p ) of an open system has m roots with a positive real part.

Then

Auxiliary function when replacing p  j  according to the principle of argument for stable closed systems should have the following change in the argument when

Formulation of Criterion #3

The wording of Ya.Z. Tsypkina

Nyquist criterion for LFC

Note: the phase response of the LFC of astatic systems is supplemented by a monotone section + /2 at  0.

Example 1

Here m =0  the system is stable, but with decreasing k the system can be unstable, therefore such systems are called conditionally stable.

Example 2

20 lgk

1/T0

Here

For any k the system is unstable. Such systems are called structurally unstable.

Example 3

AFC covers a point with coordinates (-1, j 0) 1/2 times, hence the closed system is stable.

Example 4

at  0 AFC has a discontinuity, and therefore it must be supplemented with an arc of infinitely large radius from the negative real semiaxis.

In the area from -1 to - there is one positive transition and one and a half negative ones. The difference between positive and negative transitions is -1/2, and +1/2 is required for the stability of a closed system, since the characteristic polynomial of an open system has one positive root - the system is unstable.

Absolutely stableis called a system that remains stable with any decrease in the open circuit gain, otherwise the system is conditionally stable.

Systems that can be made stable by changing their parameters are calledstructurally stableotherwise, structurally unstable.

Sustainability margins

For normal operation, any ACS must be removed from the stability boundary and have a sufficient stability margin. This is necessary for the following reasons:

The equations of the ACS elements are, as a rule, idealized; when they are compiled, secondary factors are not taken into account;
When the equations are linearized, the approximation errors increase additionally;
The parameters of the elements are determined with some error;
The parameters of the same type of elements have a technological spread;
During operation, the parameters of the elements change due to aging.

In the practice of engineering calculations, the most widely used is the determination of the stability margin based on the NIEQUIST criterion, by removing the AFC of an open-loop system from the critical point with coordinates (-1, j 0), which is evaluated by two indicators: the phase stability margin and stability margin modulo (amplitude) H.

In order for the ACS to have stability margins of at least and H , the AFC of its open circuit, while satisfying the stability criterion, should not enter the part of the ring shaded in Fig. 1, where H is determined by the ratio

If stability is determined by the LFC of conditionally stable systems, then to ensure stability margins of at least and h is needed so that:

a) at h  L  - h the phase-frequency characteristic satisfied the inequalitiesθ > -180  +  or θ< -180  -  , i.e. did not enter the shaded area 1 in Fig. 2;

b) at -180  +   θ  -180  -  the amplitude-frequency characteristic satisfied the inequalities L< - h или L >h , i.e. did not enter the shaded areas 2" and 2"" in Fig. 2.

For an absolutely stable system, the stability margins and h are determined as shown in Fig. 3:

1. Phase margin

Margin modulo h =- L (ω -π ), where ω -π – frequency at which θ=-180˚ .

The required values of the stability margins depend on the ACS class and the requirements for the quality of regulation. Approximately should be =30  60  and h =6  20dB.

The minimum allowable stability margins in amplitude must be at least 6 dB (that is, the transfer coefficient of an open system is two times less than the critical one), and in phase at least 25 30 .

Stability of a System with a Pure Delay Link

If the open-loop AFC passes through the point (-1, j 0), then the system is on the verge of stability.

A system with a pure delay can be made stable if an inertialess link with a transfer coefficient less than 1 is included in the circuit. Other types of corrective devices are also possible.

Structurally stable and structurally unstable systems

One way to change the quality of a system (in terms of stability) is to change the open-loop ratio.

When changing k L ( ) will rise or fall. If a k increase, L ( ) rises and  cf will increase and the system will remain unstable. If a k decrease, the system can be made stable. This is one way to correct the system.

Systems that can be made stable by changing the parameters of the system are called STRUCTURALLY STABLE.

For these systems, there is a critical open-loop ratio. K crit. - this is such a transfer coefficient when the system is on the verge of stability.

There are STRUCTURALLY UNSTABLE systems - these are systems that cannot be made stable by changing the parameters of the system, but it is required to change the structure of the system for stability.

Example.

Consider three cases:

Let be

Then

Let's check the stability of the system.

Δ \u003d a 3 Δ 2\u003e 0.

To determine k rs.cr. equate to zero 2 .

Then

When at

The system under consideration is STRUCTURALLY STABLE, since it can be stabilized by changing the parameters of the links.

Let the same as in the first case.

Now there is no Static error on the control channel.

Hurwitz stability conditions:

Let  2 =0, then if then the system is unstable.

This system with 1st order astatism is STRUCTURALLY STABLE.

Let be

The system is always unstable. This system is STRUCTURALLY UNSTABLE.

LECTURE 7

In the previous lectures, the steady processes in the ACS were studied. We now turn to the consideration of transient processes. We begin to consider them with the concept of stability.

Any system must first of all be functional. It means that it should function normally under the action of various external disturbances. In other words, the system must work stably.

Sustainability - this property of the system to return to the original or close to it steady state after any exit from it as a result of any impact.

On fig. 7.1 shows typical transient curves in unstable (Fig. 7.1, a) and stable (Fig. 7.1, b) systems. If the system unstable, then any shock is sufficient for the divergent process of leaving the initial steady state to begin in it. This process can be aperiodic (curve 1 in Fig. 7.1, a) or oscillatory (curve 2 in Fig. 7.1, a).

An aperiodic diverging process can, for example, occur in an ACS if the polarity of the impact on the object is mistakenly switched in its control device, as a result of which the control unit will implement not negative, but positive feedback around the object. In this case, the CU will not eliminate the deviation at, but act in the opposite direction, causing its avalanche-like change.

An oscillatory divergent process can occur, for example, with an unlimited increase in the transmission coefficient of the system. As a result, the control unit will act too vigorously on the object, trying to eliminate the initially occurring deviations. at. In this case, with each successive return at to zero under the action of the control device curve at will cross the x-axis with increasing speed and the process as a whole will be divergent.

In the case of a stable system (Fig. 7.1, b), the transient process caused by some influence decays over time aperiodically (curve 1) or oscillatoryly (curve 2), and the system returns to the steady state again.

Thus, a stable system can also be defined as a system in which the transient processes are damped.

The above concept of stability defines steady state stability systems. However, the system can operate under conditions of continuously changing influences, when there is no steady state at all. Given these operating conditions, the following more general definition of sustainability can be given: the system is stable if its output value remains limited under the influence of perturbations of limited magnitude on the system.

It is easy to show that if the transient process in the system is damped, then the system will also satisfy the last definition.

A linear automatic control system is called stable if its output coordinate y(t) remains limited for any input actions x(t) and f(t) that are limited in absolute value. The stability of a linear system is determined by its characteristics and does not depend on the acting influences.

Thus, to determine the stability of a linear system, it is required to find the change in its controlled variable. The block diagram of the linear system is shown in Fig. 7.2, where W(s) is the transfer function of an open-loop system, which in general terms, as was defined in the second lecture, has the form:

Rice. 7.2. Structural diagram of a linear system

The transfer function of the closed system shown in fig. 7.2 is determined by the following formula

Substituting (7.1) into (7.2) and getting rid of fractions in the numerator and denominator of the transfer function of a closed system, we can represent it as follows:

The processes in the system (Fig. 7.2), as follows from (7.3), are described by a differential equation of the form

The solution of the linear inhomogeneous equation (7.4) in general form consists, as is known, of two components:

Here - a particular solution of the inhomogeneous equation (7.5) with the right side, describing the forced regime of the system, which is established after the end of the transition process; - general solution of the homogeneous equation

describing the transient process in the system.

As shown above, the system will be stable if the transients caused by any perturbations decay, i.e. if over time it tends to zero.

The solution of a homogeneous differential equation, as is known, has the form:

Here C i are the constants of integration determined by the initial conditions and the perturbation; s i are the roots of the characteristic equation

where the polynomial , called the characteristic one, is the left side of the equation (7.4) of the system dynamics.

It is known from the theory of complex variables that if the real part of the root s i is negative, then the term tends to zero as t ® ¥.

Thus, for the stability of the system necessary and sufficient , to all roots of the characteristic equation had negative real parts.

If we represent the roots of the characteristic equation of the system with points on the complex plane (Fig. 7.3), then the general condition for the stability of the linear system found above can also be formulated as follows: the condition for the stability of the system is the location of all the roots of the characteristic equation, i.e. poles of the transfer function of the system, in the left complex half-plane or, in short, they must all be left.

Rice. 7.3. Roots of the characteristic equation on the complex plane.

The presence of a root on the imaginary axis means that the system is on the boundary of stability. In this case, two cases are possible:

Root at the origin;

A pair of imaginary roots.

The zero root appears when the free term of the characteristic equation is equal to zero. In this case, the stability limit is called aperiodic ; the system is stable not with respect to the output signal, but with respect to its derivative: the output signal in the steady state has an arbitrary value. Such systems are called neutrally stable .

In the case when the characteristic equation has a pair of imaginary roots, the stability boundary is called vibrational , while in the transient there will be undamped harmonic oscillations.

If at least one of the roots has a positive real part, i.e. lies in the right half-plane of the complex plane of the roots of the characteristic equation, then the system is unstable.

To judge the stability of a system, it is practically not necessary to find the roots of its characteristic equation due to the fact that indirect signs have been developed that can be used to judge the signs of the real parts of these roots and thus the stability of the system without solving the characteristic equation itself. These indirect signs are called sustainability criteria.

There are three main stability criteria: the Routh-Hurwitz criterion, the Mikhailov criterion, and the Nyquist criterion. Let's consider them sequentially.

Sustainability called the property of the system to independently return to the state of equilibrium after an external input action brought it out of the state of equilibrium. Equilibrium is the state of the system when the controlled variable y(t) is constant and all its derivatives are equal to zero. The study of stability is one of the main tasks in the theory of automatic control.

As already noted, the control process is determined by the transition process: the law of change y(t) after changing x(t). The ACS transient process can be obtained by solving the ACS differential equation (1). This solution can be represented by the sum of two components, forced y to(t) and transitional y p(t):

y(t) = y to(t) + y p(t),

where y in(t) is determined by the properties of the system and the type of input action. The ACS will be stable if, over time, the transient component tends to zero:

One can unambiguously judge the stability of a system by the type of its transient process: a damped transient process (converging to a certain constant) corresponds to a stable system, divergent (going to infinity) - unstable.

EXAMPLES of transient processes of unstable ACS.

When studying the stability of ACS, the following tasks are solved:

Determining whether the ACS is stable for given parameters;

Determination of permissible changes in the parameters of the ACS without violating stability;

Search for the parameters and/or structure of the ACS under which it can become stable.

Lyapunov's theorem

necessary and sufficient stability condition linear ACS is formulated in the Lyapunov theorem:

If the characteristic equation of the ACS has all roots with a negative real part, then the system is stable;

If at least one root has a positive real part, then the ACS is unstable.

The characteristic equation of the ACS is written according to the form of the differential equation or the transfer function of the system. So, from equation (1) after the Laplace transform we have (see derivation (2)):

Polynomial on the left side of the equality of the form:

called characteristic. Setting the characteristic polynomial to zero gives characteristic equation system or link:

The roots of the characteristic equation, the number of which corresponds to the order of the characteristic equation of the ACS, can be real, complex and purely imaginary. They can be represented as points on the complex plane of quantity R. According to the theorem, for the stability of the system it is necessary and sufficient that all roots lie in the left half-plane. An example of one of the possible distributions in the complex plane of the roots of the characteristic equation sustainable ACS of the 5th order is shown in fig. 75.

If among the roots of the characteristic equation there is a zero root or a pair of conjugate purely imaginary roots located on the imaginary axis, the system is on the boundary of stability. Examples of possible distributions in the complex plane of the roots of the characteristic equation of the 5th order ACS, on the edge of stability are shown in fig. 77.

Systems that have one pair of imaginary roots can perform undamped oscillations (self-oscillations). Such systems are practically inoperable.

Rice. 77

Let us consider examples of stability assessment by the Lyapunov theorem and the connection between the assessment results and the ACS transient response.

Let the 3rd order ACS have a characteristic equation of the form:

On fig. 78 shows the result of solving this equation, obtained using the mathematical package Mathcad. The set of roots of the equation is presented in parentheses. As you can see, one of the roots of the equation turned out to be negative real number –3.55, and the other two are complex conjugate numbers with negative real part -0.525: (-0.525 - 0.657 j) and (–0.525 + 0.657 j).

Similarly, consider another ACS of the 3rd order, with a characteristic equation of the form:

On fig. 80 shows the result of solving this equation, obtained using the mathematical package Mathcad. The set of roots of the equation is presented in parentheses. As you can see, one of the roots of the equation turned out to be negative real number –7.2, and the other two are complex conjugate numbers with positive real part of 1.31: (1.31 + 4.64 j) and (1.31 - 4.64 j), i.e. the distribution of roots in the complex plane testifies, according to the Lyapunov theorem, to the instability of the ACS.

ACS sustainability criteria

To assess the stability, it is necessary to estimate the location of the roots of the characteristic equation of the system relative to the coordinate axes of the complex plane. This estimate can be made by directly solving the characteristic equation. But to determine the stability, it is not necessary to know the values of the roots of the characteristic equation, it is enough to check whether the real parts of all the roots are negative.

The rules that make it possible to investigate the stability of a system without directly finding the roots of the characteristic equation are called sustainability criteria.

At an early stage in the development of control theory, the problem of determining the stability of a polynomial without calculating its roots was topical, because characteristic equations of high orders were difficult to solve "by hand". Now it is easy to find the roots of the characteristic polynomial with the help of computer programs, but this approach does not allow one to study the stability theoretically, for example, to determine the boundaries of the stability regions of individual parameters of the ACS.

With the help of stability criteria, not only the fact of system stability is established, but also the influence of certain parameters and structural changes in the system on stability is assessed. Mathematically, all forms of stability criteria are equivalent, because they determine the conditions under which the roots of the characteristic equation fall into the left half-plane of the complex coordinate system.

6.2.1. Hurwitz criterion

The Hurwitz criterion refers to the algebraic criteria of stability, which allow you to determine whether the ACS is stable or not based on the results of algebraic operations on the coefficients of the characteristic equation.

Most of the real ACS are closed, i.e. have a common unit feedback and, accordingly, a transfer function of the form:

where W times(R) is the transfer function of an open ACS (without taking into account the general feedback).

Consider the derivation of the characteristic equation of a closed-loop ACS if the transfer function of the corresponding open-loop ACS is given. According to (17), the characteristic equation of the ACS is obtained by equating to zero the denominator of its transfer function, therefore, for a closed system we write:

However, the transfer function of an open system, according to (2), has the form:

therefore, the characteristic equation of a closed system can be written as:

A fraction is zero when its numerator is zero, therefore, the characteristic equation of a closed system can be written as the sum of the polynomials of the numerator and denominator of the open system transfer function, equating the resulting expression to zero:

(18)

Important! To apply the Hurwitz criterion, a special form of writing the characteristic equation is used, which differs from (16) by the reverse numbering of the polynomial coefficients:

The Hurwitz criterion uses a coefficient matrix of the characteristic equation with the size n´ n, composed as follows:

All coefficients of the characteristic equation are written along the main diagonal, starting from a 1 and ending a n;

Each row is supplemented with coefficients with increasing indices from left to right so that rows with even and odd indices alternate;

In the absence of a coefficient, and also if the index is less than 0 or greater n, in its place is written 0.

The result is a matrix, the first row of which contains the coefficients of equation (19) a 1 ,a 3 ,a 5 ,… (all with odd numbers) and zeros in place of the missing elements, the second line is the coefficients a 0 ,a 2 ,a 4 ,… (all with even numbers) and zeros in place of the missing elements. The third line is obtained by shifting the first line by one position to the right, the fourth by shifting the second line by one position to the right, and so on. For example, for ACS of the 5th order ( n= 5) this matrix has the form:

The Hurwitz criterion determines the necessary and sufficient condition for the stability of the ACS as follows: all roots of the SAC characteristic equation have negative real parts, if for a 0 > 0all n Hurwitz determinants of the coefficient matrix are positive.

The Hurwitz determinants are calculated as follows:

Under the condition that all coefficients of the characteristic equation are positive, it suffices to check only n– 1 first Hurwitz determinants without calculating the determinant for the full matrix. Under this condition, particular cases of the Hurwitz criterion for systems of low orders are obtained by expanding the determinants of the matrix of coefficients. So, as a result of the disclosure of determinants, for ACS of the first and second orders, the necessary and sufficient condition for stability is the proper positivity of all coefficients of the characteristic equation. For ACS of the 3rd order - the positivity of all coefficients and the condition of the form:

Using the Hurwitz criterion, we determine at what values of the static conversion coefficient of the controller k the system under consideration is stable. Let's write the transfer function of an open ACS:

Using (18), we write the characteristic equation of a closed ACS:

For that equation, according to the form (19), the coefficients, respectively, are:

If all coefficients of this 3rd order equation are positive, necessary condition stability is also the fulfillment of condition (20):

a 1× a 2 – a 0 × a 3 > 0,

Thus, the considered ACS will be stable if the value of the static conversion coefficient k satisfies the condition:

Let us consider examples of estimating the stability by the Hurwitz criterion of systems of the 3rd order studied earlier using the Lyapunov theorem (see Fig. 78 and Fig. 80). The matrix of Hurwitz coefficients for ACS of the 3rd order has the general form:

those. the Hurwitz matrices for the considered ACS are, respectively:

and
.

The characteristic equations of both ACS satisfy the criterion of positivity of all coefficients, therefore, to assess the stability by the Hurwitz criterion, it is sufficient to calculate and check for positivity n– 1 first Hurwitz determinants, i.e. for the 3rd order - the second determinant. The results of calculating the second determinants of the Hurwitz matrix for the systems under consideration (see Fig. 78 and Fig. 80), obtained using Mathcad, are shown in fig. 83– a and fig. 83– b respectively. As can be seen, the results of the Hurwitz stability assessment coincide with the previously obtained Lyapunov estimates and the results of constructing the transient characteristics of the considered ACS (see Fig. 79 and Fig. 81, respectively) - a positive determinant corresponds to a stable ACS, and a negative determinant - unstable.

The hodograph according to the formula (21) is calculated by changing the frequency w from 0 to +¥, and is built in the complex plane.

The Mikhailov criterion determines the necessary and sufficient condition for the stability of the ACS as follows: ACS is stable if, when the frequency changes from 0to +¥ hodograph of the Mikhailov vector A(j w ) starts on the positive part of the real axis and, without turning to zero, turning counterclockwise, passes sequentially n quadrants of the complex plane, where n is the order of the characteristic polynomial of the ACS.

For stable systems, the Mikhailov hodograph has a smooth spiral shape and, at w = 0, cuts off on the real axis in the positive direction a segment equal to the free term of the characteristic equation a 0 .

By the form of the Mikhailov hodograph, one can also determine the boundary state of the ACS stability: in the case of the stability boundary of the first type, i.e. the characteristic equation of the ACS has a zero root (see Fig. 77) there is no free term of the characteristic equation a 0 = 0 and the hodograph starts from the origin. At the stability boundary of the second type, i.e. the characteristic equation of the ACS has a pair of purely imaginary roots (see Fig. 77), the hodograph passes through the origin (turns to zero) at a certain non-zero value w, and this value is the frequency of undamped oscillations of the system.

Let us consider examples of estimating the stability by the Mikhailov criterion of systems of the 3rd order studied earlier using the Lyapunov theorem (see Fig. 78 and Fig. 80). The formulas for calculating the Mikhailov hodographs of these systems are, respectively:

Mikhailov's hodograph for the first ACS is shown in fig. 84. As you can see, its form satisfies all the conditions of the criterion:

The hodograph starts on the positive part of the real axis (cutting off at w = 0 on the real axis a segment equal to the free term of the characteristic equation a 0 = 3);

Does not vanish;

As the value of the frequency w increases, turning counterclockwise, it passes successively the first and second quadrants and in the third quadrant, as w ® ¥, goes to infinity.

It should be noted that for systems with a high order of the characteristic equation ( n= 5 or more) the counting of quadrants when checking the conditions of the Mikhailov criterion after the fourth one continues counterclockwise in the same order. That is, for example, for a stable ACS of the 5th order, the hodograph must sequentially pass four quadrants, return to the first (for the hodograph - in order of the fifth) and go to infinity in it. An example of Mikhailov's hodograph for a stable ACS of the 5th order with a formula for calculating the hodograph of the form:

shown in fig. 86. For the convenience of analysis, the initial section of the hodograph, obtained at low values of the frequency w, is shown as a separate fragment. It can be seen that the hodograph at w = 0 starts on the positive part of the real axis and, sequentially, counterclockwise, passing through five quadrants, goes to infinity in the fifth.

The Nyquist criterion for the amplitude-phase characteristic (APC) is formulated as follows: the closed system will be stable if the AFC of the corresponding open system does not cover the point with coordinates [–1, j0] when the frequency changes from 0 to.

Let us consider an arbitrary open ACS that does not contain integrating links. In this case, the value of the AFC for the frequency w = 0 is equal to the static conversion coefficient of the ACS:

W(j w) = W(j 0) = k.

In this case, if the degree of the numerator of the transfer function is less than the degree of the denominator, then the graph of the AFC, starting at a point with coordinates ( k, j 0) when the frequency changes from 0 to ¥ tends to the origin. On fig. 88– a shown AFC sustainable ACS - the graph does not cover the point with coordinates [–1, j 0], and in Fig. 88– b– unstable(the graph covers the point).

If there are integrating links in the ACS, then the AFC at w = 0 turns to infinity, i.e. the AFC graph in this case does not start on the real axis, but comes from infinity. In this case, to assess the stability by the Nyquist criterion, the contour includes not only the curve of the AFC plot, but also a part of a circle of infinite radius drawn clockwise from the real axis. Example sustainable ACS with AFC of this type is shown in fig. 90– a, unstable- in fig. 90– b.

Rice. 90

Consider an example of stability assessment by the Nyquist criterion for AFC using the example of a closed automatic control system, which corresponds to an open system with a transfer function of the form:

We write according to the given W times(p) AFC calculation formula:

and, changing the frequency w from 0 to +¥, we will plot the AFC of an open ACS using the mathematical package Mathcad (Fig. 91). For convenience of analysis, the section of the AFC in the region of the point [–1, j 0], obtained for large values of the frequency w, is shown in Fig. 91 separate fragments. The fragment clearly shows that the graph covers point [–1, j 0], hence the closed ACS is unstable.

Rice. 91

6.2.4. Nyquist criterion for LACH and LPCH

The Nyquist criterion for the logarithmic amplitude-frequency and phase-frequency characteristics is formulated as follows: A closed-loop system is stable if two conditions are met for the characteristics of the corresponding open-loop system:

- at a frequency equal to the cutoff frequency of the ACS w with module phase response less than 180 degrees: < 180° ;

- at a frequency equal to wp LAFC value is less than zero: L(wp)< 0.

As follows from the wording of the criterion, in order to check its conditions by the characteristics of an open ACS, it is initially necessary to determine two frequencies: the cutoff frequency w with and frequency w p . After that, for the found frequency values, the feasibility of both conditions of the criterion should be checked.

ACS cutoff frequency called the frequency at which the LAFC of the system crosses the frequency axis, that is L(w with) = 0. This frequency is also called unity gain frequency ACS, since the signal of this frequency at the output of the ACS has the same amplitude as at the input: A out = A in. For this case, the following is true:

Important! Do not confuse the concepts of the cutoff frequency of individual typical units of the ACS and the entire system as a whole. The definition of cutoff frequencies of typical links is considered in the "Notes" column. Applications 1.

Frequency w p ACS is the frequency at which the PFC of the ACS is equal to 180° with a "plus" sign or with a "minus" sign. If the PFC crosses the ±180 ordinate several times, then the condition is checked for the rightmost point.

Important! The characteristics under consideration are the cutoff frequencies w with and frequency w p are not available for every ACS. If the LAFC of the system does not cross the frequency axis at all, that is L(w) ¹ 0 for any values of w, then such a system has no cutoff frequency. Similarly, if the PFC of the system does not take the value of ±180° for any frequency values, then this ACS is not characterized by the parameter w p . In these cases, other criteria should be chosen to assess sustainability.

On fig. 92– a it is shown how to determine the frequencies w with and w p .

Rice. 92

EXAMPLES: 1) LACHH ACS without cutoff frequency w s; 2) LFCH ACS without frequency w p .

Let us check the feasibility of the conditions of the Nyquist criterion for the characteristics of an open ACS, shown in Fig. 92– a. Let us define graphically the quantities L(w p) and j(w with) as shown in Fig. 92– b. As seen, L(wp)< 0, а < 180°, i.e. both conditions of the Nyquist criterion are satisfied, therefore, the closed ACS corresponding to the considered open one is sustainable. From fig. 92– b we can also conclude that for the stability of the ACS according to the Nyquist criterion, it is sufficient that the condition w with < w p .

For the characteristics of an open ACS in Fig. 93– a L(wp) > 0, and > 180°, i.e. both conditions of the Nyquist criterion are not met, therefore, the closed ACS corresponding to the considered open one is unstable. From fig. 93– a we can also conclude that for the instability of the ACS according to the Nyquist criterion, it is sufficient that the condition w with> w p .

Rice. 93

For the characteristics of an open ACS, which corresponds to a closed system located on the edge of stability, L(wp) = 0 and = 180° w with= w p (see Fig. 93– b). For such a system, for a signal with frequency w with, i.e. with a frequency of unity gain, the phase shift of the output signal relative to the input is -180°. This suggests that after passing through the ACS, the signal magnitude changes sign, retaining the absolute value (energy), that is, undamped oscillations are established. AFC of such an ACS is shown in Fig. 89 .

Consider an example of stability assessment by the Nyquist criterion for LAFC and LPFC using the example of a closed automatic control system, which corresponds to an open system with a transfer function of the form:

Graphs of LAFC and LPFC of an open ACS, built using the mathematical package Mathcad according to formulas (11) and (12), are shown in fig. 94. As can be seen from the figure, the LAFC is zero for w with» 13.5 s -1 . The LPFC at a frequency w p » 5.7 s -1 changes sign - after j(w) reaches the value of –180° (the radius vector, turning clockwise, goes into the upper half-plane), the phase shift reading continues in the region of positive values . In this case, of the two conditions of the Nyquist criterion, only the second is formally violated: the value of the LAFC at the cutoff frequency is not negative ( L(wp) » 18 > 0). First condition ( < 180°) is formally fulfilled: » 130° < 180°. However, it should be understood that a 130° phase advance corresponds, when counted clockwise without sign change, to a lag of:

j(w with) = –360° + 130° = –230°,

therefore, the closed ACS is unstable. The same conclusion can be reached by comparing the values w with and w p: w with> w p . The assessment of the stability of this ACS according to the Nyquist criterion for AFC, performed at the end section 6.2.3 also showed a lack of sustainability.

Let us check the stability estimate according to the Nyquist criteria using the Lyapunov theorem. According to the given using formula (18), we write the characteristic equation of a closed ACS:

The solution of the characteristic equation of a closed ACS, obtained using the mathematical package Mathcad, has the form:

The set of roots of the equation is presented in parentheses. As you can see, one of the roots of the equation turned out to be negative real number –17.74, and the other two are complex conjugate numbers with positive the real part is 3.657. These roots are, respectively, (3.657 + 12.22 j) and (3.657–12.22 j). That. according to the Lyapunov theorem, closed ACS unstable, which is consistent with the results of the stability assessment obtained using both Nyquist criteria.

Rice. 94

ACS stability margins

Specifications devices that are part of the ACS change during operation, and, therefore, the constants of the ACS transfer function change over time. Investigator, it is not enough to design just a stable system; possessed reserves of sustainability. The margin determines the distance of the system from the stability boundary.

Amplitude stability margin D L the value in decibels is called, by which it is necessary to shift up the LAFC of an open ACS so as to bring the corresponding stable closed system to the stability limit. On fig. Fig. 95 shows the upward shift of the LAFC of a stable ACS, the initial characteristics of which were considered in the example of stability assessment using the Nyquist criterion (see Fig. 92– b).

where A(wp)< 1 – модуль АФХ на частоте w p .

Knowing D L, it is possible to determine the value of the coefficient of static transformation of an open ACS, at which the corresponding closed system will be on the stability boundary:

;

(23)

where k

Consider an example of determining the boundary value of the static conversion coefficient for an open ACS with a transfer function of the form:

LACHH and LPCHH of this ACS are shown in fig. 96. According to the performance graphs, it can be seen that the cutoff frequency of the ACS is w with» 50 s -1 , and the LPFC reaches –180° at the frequency w p » 100 s -1 and then changes sign. The amplitude stability margin for this ACS is
, therefore, according to formula (23):

When changing the static conversion coefficient of the ACS to a value equal to k gr, the LFC of the ACS will not change, but the LFC will shift upwards (see Fig. 96). As can be seen, with the found value k gr= 425.975 cutoff frequency of the open ACS w with 1 becomes equal to 100 s -1 , i.e. w with 1 = w p . So, in accordance with the Nyquist criterion for LAFC and LPFC, the closed system corresponding to the considered open ACS will indeed be on the stability boundary.

On fig. Fig. 97 shows the downward shift of the LFC of an open-loop ACS, the initial characteristics of which were considered in the example of stability assessment using the Nyquist criterion (see Fig. 92– b). As can be seen, the shift of the original LPFC parallel to itself down by Dj(w with) leads to a frequency shift w p of the open ACS to the left: for the new LPFC shown by the dotted line, the value of this frequency is w p1 = w with, which, according to the Nyquist criterion for LAFC and LPFC, indicates that the closed system is on the stability boundary. From fig. 97 that the quantity Dj(w with) can be defined as:

Recall that w with this is the unity gain frequency: a signal with this frequency at the output of the ACS has the same amplitude value as at the input. Therefore, the length of the radius vector drawn to the AFC point, which corresponds to w with, is equal to 1. This point can be found on the AFC graph at the intersection with a circle of unit radius (see Fig. 98).

From fig. 98 it is clearly seen that if the AFC graph of an open ACS is rotated by an angle equal to Dj(w with), then the graph will pass through the point [–1, j 0], which will bring the closed system to the stability limit according to the Nyquist criterion for AFC.

For the same AFC, consider the definition of the stability margin in amplitude. The frequency w p corresponds to a phase shift of ±180 °, therefore, the AFC point corresponding to this frequency can be found at the intersection of the graph with the real axis (Fig. 99). The AFC module, which determines the coefficient of attenuation of the signal amplitude with such a frequency at the ACS output, is equal to the length of the radius vector drawn from the origin to the corresponding point of the AFC. For AFC in Fig. 99 this value is equal to A(w p), and using it, using formula (22), one can calculate D L.

where k– coefficient of static transformation of the original open ACS .

Let us consider an example of determining the boundary value of the static conversion coefficient according to the AFC of an open ACS, for which earlier calculation k gr was performed according to logarithmic characteristics (see starting from formula (23) and up to Fig. 96). AFC of this ACS with the initial value k= 107 is shown in fig. 100. For the convenience of analyzing the graph in the region of the point [–1, j 0] its fragment is shown separately. As you can see, the ACS with the initial value k AFC modulus A(w p) » 0.25, therefore, according to formula (25):

Found value k gr= 428 coincides with a satisfactory accuracy with the result of the calculation according to the LAFC ( k gr= 425.975). The errors in the calculations are due to the approximate determination from the graphs D L and A(wp).

Rice. 100

As can be seen from fig. 100, when the ACS static conversion coefficient is changed to a value equal to k gr= 428, AFC ACS will pass through the point with coordinates [–1, j 0], which means that, in accordance with the Nyquist criterion for AFC, the closed system corresponding to the considered open ACS will indeed be on the stability boundary.

ACS stability margins by amplitude D L and phase Dj(w with), along with indicators determined by the transient response (see. chapter 2.3.2.), are the main indicators of the quality of management.

Literature

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2. Besekersky, V.A. Theory of automatic control systems / V.A. Besekersky, V.P. Popov. – M.: Nauka, 1975. – 766 p.

3. Andryushchenko, V.A. Theory of automatic control systems / V.A. Andryushchenko. - L.: LSU, 1990. - 256 p.

4. Klyuev, A.S. Design of automation systems technological processes: reference manual / A.S. Klyuev, B.V. Glazov and others - M .: Energoatomizdat, 1990. - 464 p.

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6. Fedorov, Yu.N. Handbook of an engineer for automated process control systems: design and development: study.-pract. allowance / Yu.N. Fedorov. – M.: Infra-Engineering, 2008. – 928 p.

7. Polyakov, K.Yu. The theory of automatic control for "dummies". K.Yu. Polyakov // Teaching, science and life [Electronic resource]. - 2009. - Access mode: http://kpolyakov.narod.ru/uni/teapot.htm. – Access date: 06/01/2011.

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9. Yakovlev, A.V. Motor speed stabilization system: laboratory work at the rate " Technical means SAU" /A.V. Yakovlev. - M .: MSTU im. N.E. Bauman, 2007. - 24 p.

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The stability considered above (together with the criteria for its determination) is not the only property of automatic control systems. Systems are characterized by: stability margin, areas of stability, attraction, quality of regulation and other characteristics. Let's consider some of them.

Structural stability (instability)

This is such a property of a closed system, in the presence of which it cannot be made stable under any changes in parameters.

Let be
. The Nyquist hodograph for this system is shown in Fig.A. The stability of this system is determined by the values of the parameters and
. The system under consideration is structurally stable.

Let be
. (Fig. B). Stability also depends on the parameters and . The system is structurally stable.

Let be
. In any case (for any values of the parameters), the system will be unstable. That is, the system is structurally unstable.

In a particular case, the transfer function has the form
. In this case, the corresponding characteristic equation of a closed system: . The principle of intermittency of roots and poles is violated. The system is unstable. Structurally unstable.

Transfer function system
- structurally unstable, since for a closed system, while the coefficients
,
,
,
, are all positive, but the condition implies that
, where
, or
. That is, the system is unstable.

System
also structurally stable. Here is the link
- quasi-aperiodic (statically unstable). Characteristic equation of a closed system. From where you can get two boundary conditions:
and
.

For single-loop systems, the following conditions take place (Meierov M.V.):

Let the single-loop system consist of:

- integrating links,

- unstable links,

- conservative links. Then, in the absence of differentiating links in the system, it will be structurally stable if

In the case of multi-loop systems, Meyer's relations must be applied to each loop included in the system.

margin of stability

The fact of detection of stability does not give confidence in the operability of the system.

Inaccuracies (errors) are possible, because:

the mathematical description of the system is idealized;

links are often linearized;

inaccuracy in determining parameters;

change in working conditions (in relation to the simulated ones).

Therefore, a margin of stability is needed.

When using the Hurwitz criterion, the margin is determined by the value of the penultimate minor:

If a
- there is no stability margin;
- there is stock.

The stability margin in the system characterizes the degree of stability.

The margin of stability and the degree of stability can be determined by the location of the roots of the characteristic equation and by the frequency characteristics of the system.

Similarly, one can determine the stability margin by the logarithmic characteristics L( ) and  ( ) , used in determining stability by the Nyquist criterion.

Stability area

In practice, designers of automatic control systems are interested in the space (region, limits, range) of parameters for which the system is stable. The set of parameter values for which the system has the property of stability is called the stability region of the system.

There are several methods for determining the areas of stability.

Based on the algebraic Hurwitz stability criterion;

D-partition method;

Root locus method.

Stability area according to Hurwitz is determined by using equalities under the Hurwitz conditions instead of inequalities. Most often, the determination of the boundary of the desired area can be made under the condition
. (See "Determining the Critical Gain"). From here, the dependence of the parameter of interest to us is determined from parameter . The resulting dependence() is the boundary of the system stability area.