Stability is the ability of the system to return to the nominal mode if it deviates from this mode for some reason.
Stability requirements are mandatory for all self-propelled guns.
A rigorous definition of stability was given by A.M. Lyapunov in his work " General task on the stability of movement "(late 19th century)
Let the dynamics of the system be described by the equation
y - output value
x- input value
y ( i ) , x ( j ) - derivatives.
Suppose that in this system there is a nominal mode of operation at n (t), which is uniquely determined by the nominal input action X n (t) and nominal initial conditions.
(2)
Since the nominal initial conditions (2) are difficult to maintain in practice, there are “deviated” initial conditions in the system.
(3)
For the nominal mode, the equation is true:
The rejected initial conditions correspond to the rejected mode.
For the deflected mode, the following equation is true:
(6)
We subtract equation (4) from equation (5), we obtain (7)
Let's introduce a definition.
Rated mode at n (t) stable according to Lyapunov, if for any deviated initial conditions (3) that differ sufficiently little from the nominal nominal initial conditions (2), for all t > 0, z(t) will be small.
If the nominal regime is stable according to Lyapunov and the limit 
, then the nominal mode is called asymptotically stable.
If there are initial conditions (3) arbitrarily little different from the nominal initial conditions (2), and at the same time 
becomes greater than some small, predetermined value, then the nominal mode at n (t)
called unstable.
From (7) it follows that the behavior z(t) completely independent of the type of input X n (t) .
Hence the conclusion follows: either in system (1) are asymptotically stable all nominal modes corresponding to different input X n (t), or they are all unstable.
Therefore, we can talk about the stability or instability of the system, and not of any one of its regimes.
This is an important conclusion that reduces the amount of research on ACS.
Unfortunately, it is valid only for linear self-propelled guns.
Necessary and sufficient conditions for the stability of linear self-propelled guns.
For the asymptotic stability of linear systems, it is necessary and sufficient that all the roots of the characteristic equation.
would have a negative real part.
It is known that the solution of a differential equation with constant coefficients
1. Let the roots be real.
At 
- and this is a deviation from the nominal mode.
2. If the roots are complex.
A necessary condition for stability.
For the asymptotic stability of system (1), (8), it is necessary that all the coefficients of the characteristic equation have the same sign.
Geometric interpretation of the stability condition
For the stability of the ACS, it is necessary and sufficient that the roots of the characteristic equation be located in the left half-plane complex plane roots.
ACS stability criteria.
These are artificial methods that allow, without finding the roots of the characteristic equation, to answer questions about the stability of the ACS, i.e. determine the signs of the real parts of the roots.
Two types of sustainability criteria:
one). Algebraic stability criterion (Hurwitz stability criterion).
Let a characteristic equation be given.
For the stability of the ACS, it is necessary and sufficient:
one). So that all the coefficients of the characteristic equation would have the same sign - 
(
system is unstable)
2). The main Hurwitz determinant, compiled according to a certain rule, and all its minor diagonals would have the sign of the coefficients - they would be greater than zero.
Rules for writing the main definition of Hurwitz.
one). All the coefficients of the characteristic equation are located along the main diagonal of the determinant in ascending order of indices, starting from a 1 .
2). Places in the determinant above the main diagonal are filled with the coefficients of the characteristic equation in ascending order of indices.
3). Places in the determinant under the main diagonal are filled with the coefficients of the characteristic equation in descending order of indices.
four). Places in the determinant where coefficients with indices greater than n and less zero, filled with zeros
Thus, the main Hurwitz determinant has the form:
A= 
>0
ACS is stable if
one). All coefficients of the characteristic equation are greater than zero ( 
0!)
,
,
….
2). The main Hurwitz determinant and all its diagonal minors > 0.
,
,
,
….
Consider examples.
1.
1.
2.
For the stability of the second-order ACS, a necessary and sufficient condition for stability is the positiveness of the coefficients of the characteristic equation.
1.
i=0…3
2.
A necessary and sufficient condition for the stability of third-order systems is the positivity of the coefficients and the product of the internal terms 
should be more artwork extreme members 
characteristic equation.
,
,
,
There is also the algebraic Routh criterion. This is the same Hurwitz criterion, but organized in such a way that it is convenient to use it to create programs for determining stability.
Vyshnegradsky stability criterion for third-order systems.
Vyshnegradsky I.A. proposed to depict the stability boundary on the so-called Vyshnegradskii parameter plane.
Let we have a characteristic equation of the third degree.
Let's transform it using substitution:
Then it will take the form:
A 1 andA 2 are called the Vyshnegradsky parameters (dimensionless quantities), in the plane of which the stability boundary is constructed.
Let us apply the Hurwitz stability criterion to the transformed equation
or A 1 A 2 > 1
On the edge of stability 
.
From here 
- equation on the stability boundary
The coefficients of the characteristic equation are used to determine BUT 1 and BUT 2 . If the point is below the hyperbola, the ACS is stable; if it is higher, it is unstable.
The automatic control system has inertia of various physical nature, which slow down the processes. A single jump, which is usually considered as an ACS test signal (Figure 1), can be expanded into a series:
Figure 1. Typical structure of the ACS
The presence of inertia causes a phase shift of the feedback signal 
relative to the input 
, and the phase shift depends both on the harmonic number and on the time constants. So for the aperiodic link of the 1st order, the phase shift is determined:
.
(2)
Figure 2. Phase shift at the ACS output
Since an infinite spectrum of harmonic components acts at the ACS input, then among them there is such a harmonic, the phase shift of which is equal to 
(Figure 2), i.e. The output signal will be in phase with the input.
Since the feedback is negative, at the input of the system it acts in phase with the input (dashed line in Figure 2), and the feedback signal acts at the moment when 
.
Let the amplitude of the harmonic component whose phase shift 
, is equal to 0.5, and the system transfer coefficient for this harmonic is greater than one, for example, equal to 2. Then the output signal after the first period 
, after the second period 
, after the third 
etc., i.e. the process is divergent (unstable) (Figure 3).
Figure 3. Harmonic transient
atk
>1.
If the system gain for a harmonic whose phase shift is 
, less than one, then the process will decay (the system is stable).
Thus, a closed system will be stable if its transmission coefficient for the harmonic component, the phase shift, which is equal to 
, less than unity.
If the transmission coefficient for the specified harmonic is equal to one, then the system is on the stability boundary and the output coordinate changes according to the harmonic law with a constant amplitude.
For the system (Figure 1), the output coordinate is determined by:
The reasons for the deviation of the ACS from the equilibrium position are the change in the input value 
and disturbing influences 
.
If a 
and 
those. there are no reasons for the deviation of the system from the equilibrium position, then 
.
If, in the absence of reasons for rejection 
,
denominator 
, then this means that the output coordinate 
can take any non-zero value, since in this case we have:
.
(4)
Consequently, undamped oscillations arise in the system under the condition:
.
(5)
Note that this condition is similar to the self-excitation condition of a Barkhausen feedback amplifier: self-excitation of the system occurs when as much voltage or other value is amplified as he (she) is removed through the feedback channel:
.
(6)
1.2 Determining the stability of automatic control systems
Any system automatic control(ACS) must be operational, i.e. function normally under the influence of disturbances of various kinds. The operability of the ACS is determined by its stability, which is one of the main dynamic characteristics of the system.
Stability is the property of a system to return to its original equilibrium position or a regime close to it after the end of the perturbation that caused the system to deviate from the equilibrium position. Unstable operation can occur in any ACS with feedback, while the system moves away from the equilibrium position.
If the system weight function is known ω(t) , then the linear system is stable if ω(t) remains limited for any input disturbances of limited magnitude:
,
(7)
where With - const.
Therefore, the stability of the system can be judged by the general solution of the linearized homogeneous differential equation of a closed ACS, since the stability does not depend on the type of the described disturbance. The system is stable if the transient component decays in time:
.
(8)
If a 
,
then the ACS is unstable.
If a 
tends neither to zero nor to infinity, then the system is on the boundary of stability.
Since the general solution of the differential equation depends on the form of the roots of the characteristic equation of the ACS, the stability can be determined without directly solving the homogeneous differential equation.
If the characteristic equation of a linear differential equation with constant ACS coefficients has the form
then his solution is:
,
(10)
where c- integration constants;
p t are the roots of the characteristic equation.
Therefore, the ACS is stable if
(11)
Thus, in order for the linear ACS to be stable, it is necessary and sufficient that the real parts of all the roots of the characteristic equation of the system be negative
R e p i < 0, (12)
a) for real roots p i < 0,
,
(12.a)
for real roots p i > 0;
;(12.b)
b) for complex roots of type p i =α± jβ at α< 0
, (12.v)
for complex roots p i =α± jβ at α> 0
,(12.g)
Therefore, the ACS is stable if all the roots of the characteristic equation (9) located in the left half-plane of the complex plane of the roots. The system is on the stability boundary if at least one real root or a pair of complex roots is on the imaginary axis. There are aperiodic and oscillatory boundaries of stability.
If at least one root of the characteristic equation of the ACS is equal to zero, then the system is on the aperiodic stability boundary. The characteristic equation in this case ( a n = 0) has the following form:
In this case, the system is stable with respect to the rate of change of the controlled variable, but with respect to the realized value, the system is neutral (neutrally stable system).
If the characteristic equation of the ACS has at least a pair of purely imaginary roots, then the system is on the border of oscillatory stability. In this case, undamped harmonic oscillations take place in the system.
Thus, to determine the stability of the ACS, one should solve the characteristic equation, i.e. find its roots. Finding the roots of the characteristic equation is possible because W 3 (p) is usually the ratio of two algebraic polynomials. However, such a direct method for determining the stability turns out to be very laborious, especially when n> 3. In addition, to determine stability, it is necessary to know only the signs of the roots and it is not necessary to know their meaning, i.e. the direct solution of the characteristic equation gives “extra information”. Therefore, to determine the stability, it is advisable to have indirect methods for determining the signs of the roots of the characteristic equation without solving it. These indirect methods for determining the signs of the roots of the characteristic equation without directly solving it are stability criteria.
ACS stability
Zeros and poles of the transfer function
The roots of the polynomial in the numerator of the transfer function are called zeros, and the roots of the polynomial in the denominator are poles transfer function. Poles at the same time roots of the characteristic equation, or characteristic numbers.
If the roots of the numerator and denominator of the transfer function lie in the left half-plane (while the roots of the numerator and denominator lie in the upper half-plane), then the link is called minimum-phase.
Correspondence to the left half-plane of the roots R the upper half-plane of the roots (Fig. 2.2.1) is explained by the fact that, or 
, i.e. a vector is obtained from a vector by rotating it by an angle clockwise. As a result, all vectors from the left half-plane come to the vectors in the upper half-plane.
Non-Minimum-Phase and Unstable Links
The links of the positional and differentiating types discussed above are referred to as stable links, or links with self-alignment.
Under self leveling is understood as the ability of a link to spontaneously come to a new steady value with a limited change in the input value or perturbing action. Usually the term self-alignment is used for links that are objects of regulation.
There are links in which a limited change in the input value does not cause the link to arrive at a new steady state, and the output value tends to increase indefinitely in time. These include, for example, links of an integrating type.
There are links in which this process is even more pronounced. This is due to the presence of positive real or complex roots with a positive real part in the characteristic equation (the denominator of the transfer function set to zero), as a result of which the link will belong to the category unstable links.
For example, in the case of the differential equation 
, we have the transfer function 
and a characteristic equation with a positive real root. This link has the same amplitude-frequency characteristic with an inertial link with a transfer function. But the phase-frequency characteristics of these links are the same. For the inertial link we have 
. For a link with a transfer function, we have
those. larger in absolute value.
In this regard, unstable links belong to the group non-minimum-phase links.
Non-minimum-phase links also include stable links that have real positive roots or complex roots with a positive real part in the numerator of the transfer function (corresponding to the right side of the differential equation).
For example, a link with a transfer function 
belongs to the group of non-minimum-phase links. The modulus of the frequency transfer function coincides with the modulus of the frequency transfer function of the link having the transfer function 
. But the phase shift of the first link is greater in absolute value:
The minimum phase links have smaller phase shifts compared to the corresponding links having the same amplitude frequency characteristics.
They say that the system stable or has self-alignment if, after removing the external perturbation, it returns to its original state.
Since the motion of a system in a free state is described by a homogeneous differential equation, the mathematical definition of a stable system can be formulated as follows:
The system is called asymptotically stable if the condition 
(2.9.1)
From the analysis of the general solution (1.2.10), the necessary and sufficient stability condition follows:
For the stability of the system, it is necessary and sufficient that all roots of the characteristic equation have strictly negative real parts, i.e. Rep i , I = 1…n. (2.9.2)
For clarity, the roots of the characteristic equation are usually depicted on the complex plane Fig. 2.9.1a. When the necessary and sufficient
Fig.8.12. Root plane
characteristic
equations A(p) = 0
OS - area of stability
th condition (2.9.2) all roots lie to the left of the imaginary axis, i.e. in the field of sustainability.
Therefore, condition (2.9.2) can be formulated as follows.
For stability it is necessary and sufficient that all roots of the characteristic equation are located in the left half-plane.
A rigorous general definition of stability, methods for studying the stability of nonlinear systems, and the possibility of extending the conclusion about the stability of a linearized system to the original nonlinear system given to the Russian scientist A.M. Lyapunov.
In practice, stability is often determined indirectly, using the so-called stability criteria without directly finding the roots of the characteristic equation. These include algebraic criteria: the Stodola condition, the Hurwitz and Mikhailov criteria, and the frequency Nyquist criterion. In this case, the Nyquist criterion makes it possible to determine the stability of a closed system by AFC or by the logarithmic characteristics of an open system.
Stodola condition
The condition was obtained by the Slovak mathematician Stodola at the end of the 19th century. It is interesting from a methodological point of view for understanding the conditions for the stability of a system.
We write the characteristic equation of the system in the form
D(p) = a 0 p n + a 1 p n- 1 +…a n = 0. (2.9.3)
According to Stodola, for stability it is necessary, but not enough, that when a 0 > 0, all other coefficients were strictly positive, i.e.,
a 1 > 0 ,..., a n > 0.
Need can be formed like this:
If the system is stable, then all roots of the characteristic equation have , i.e. are left.
The proof of necessity is elementary. According to Bezout's theorem, the characteristic polynomial can be represented as
Let , i.e. a real number, and 
are complex conjugate roots. Then
This shows that in the case of a polynomial with real coefficients, the complex roots are pairwise conjugate. Moreover, if , , then we have a product of polynomials with positive coefficients, which gives a polynomial only with positive coefficients.
Failure Stodola's condition is that the condition does not guarantee that all . This can be seen in a specific example by considering a polynomial of degree .
Note that in the case the Stodola condition is both necessary and sufficient. It follows from . If , then and to .
For from the analysis of the formula of the roots of the quadratic equation, the sufficiency of the condition also follows.
Two important corollaries follow from Stodola's condition.
1. If the condition is met, and the system is unstable, then the transient process has an oscillatory character. This follows from the fact that an equation with positive coefficients cannot have real positive roots. By definition, a root is a number that turns the characteristic polynomial to zero. No positive number can vanish a polynomial with positive coefficients, that is, be its root.
2. The positiveness of the coefficients of the characteristic polynomial (respectively, the fulfillment of the Stodola condition) is ensured in the case of negative feedback, i.e. in the case of an odd number of signal inversions in a closed loop. In this case, the characteristic polynomial. Otherwise, they had, and after reduction of similar ones, some coefficients could turn out to be negative.
Note that the negative feedback does not exclude the possibility of non-fulfillment of the Stodola condition. For example, if , and , then in the case of a single negative feedback . In this polynomial, the coefficient at is equal to zero. There are no negative coefficients, but, nevertheless, the condition is not satisfied, since it requires strict fulfillment of inequalities.
This is confirmed by the following example.
Example 2.9.1. Apply Stodola's condition to the scheme of fig. 2.9.2.
The transfer function of a single negative feedback system open in a circuit is equal to and the characteristic equation of a closed system is the sum of the numerator and denominator, i.e.
D(p) = p 2 + k 1 k 2 = 0.
Since there is no member with R in the first degree ( a 1 = 0), then the Stodola condition is not satisfied and the system is unstable. This system is structurally unstable, since for any values of the parameters k 1 and k 2 cannot be sustainable.
To make the system stable, it is necessary to introduce an additional link or a corrective link, i.e. change the structure of the system. Let's show this with examples. On fig. 2.9.3. a direct chain link is represented by successively connected links with transfer functions and . Parallel to the first introduction, an additional connection.
P 
the transfer function of a system open with respect to a single negative connection and the characteristic equation of a closed system, respectively, are equal to
,
Now the Stodola condition is satisfied for any 
. Since in the case of an equation of the second degree it is not only necessary, but also sufficient, the system is stable for any positive gains .
In Fig. 2.9.4, a boosting link is introduced into the circuit in series. The transfer function of a system open in a single negative connection in this case is equal to 
and the characteristic equation of a closed system is
Similarly to the previous one, the system is stable for any positive .
Rouss-Hurwitz stability criterion
Mathematicians Rauss (England) and Hurwitz (Switzerland) developed this criterion at about the same time. The difference was in the calculation algorithm. We will get acquainted with the criterion in Hurwitz's formulation.
According to Hurwitz, for stability it is necessary and sufficient that at a 0 > 0 Hurwitz determinant = n and all its major minors 1 , 2 ,..., n -1 were strictly positive, i.e.
(2.9.4)
The structure of the Hurwitz determinant is easy to remember, given that the coefficients are located along the main diagonal a 1 ,… ,a n, in the lines there are coefficients through one, if they are exhausted, then the free places are filled with zeros.
Example 2.9.2. Investigate for Hurwitz stability a system with a single negative feedback, in the forward circuit of which three inertial links are included and, therefore, the transfer function of an open-loop system has the form (2.9.5)
We write the characteristic equation of a closed system as the sum of the numerator and denominator (2.9.5):
Consequently,
The Hurwitz determinant and its minors have the form
taking into account a 0 > 0, the strict positivity of the Hurwitz determinant and minors (2.9.6) implies the Stodola condition and, in addition, the condition a 1 a 2 - a 0 a 3 > 0, which after substituting the values of the coefficients gives
(T 1 T 2 + T 1 T 3 +T 2 T 3 )(T 1 +T 2 +T 3 ) > T 1 T 2 T 3 (1+ k) . (2.9.7)
It can be seen from this that with increasing k the system can change from stable to unstable, since inequality (2.9.7) will no longer hold.
The transfer function of the system is erroneously equal to
According to the original finite value theorem, the steady-state error of processing a single step signal will be equal to 1/(1+ k). Therefore, a contradiction between stability and accuracy is revealed. To reduce the error, increase k, but this leads to a loss of stability.
The argument principle and Mikhailov's stability criterion
Mikhailov's criterion is based on the so-called argument principle.
Let us consider the characteristic polynomial of a closed system, which, according to Bezout's theorem, can be represented as
D(p) = a 0 p n + a 1 p n- 1 +…+a n = a 0 (p-p 1 )…(p - p n ).
Let's make a substitution p = j
D(j ) = a 0 (j ) n + a 1 (j ) n- 1 +…+a n = a 0 (j -p 1 )…(j -p n ) = X( )+jY( ).
For a specific value has a point on the complex plane given by the parametric equations
E 
if change
in the range from - to , then a Mikhailov curve will be drawn, i.e. a hodograph. Let's study the rotation of the vector D(j
)
when it changes
from - to , i.e. we find the increment of the vector argument (the argument is equal to the sum for the product of vectors): 
.
At = - difference vector, the beginning of which is at the point R i , and the end on the imaginary axis is directed vertically down. As you grow the end of the vector slides along the imaginary axis, and when = the vector is directed vertically upwards. If the root is left (Fig. 2.9.19a), then arg = + , and if the root is right, then arg=- .
If the characteristic equation has m right roots (respectively n-m left), then 
.
This is the principle of the argument. When isolating the real part X(
)
and imaginary Y(
)
we referred to X(
)
all terms containing j
to an even degree, and Y(
)
- to an odd degree. Therefore, the Mikhailov curve is symmetric about the real axis ( X(
)
- even, Y(
)
is an odd function). As a result, if you change
from 0 to +, then the increment of the argument will be half as much. For this reason, finally argument principle formulated as follows 
.
(2.9.29)
If the system is stable, i.e. m= 0, then we obtain the Mikhailov stability criterion.
According to Mikhailov, for stability it is necessary and sufficient that
,
(2.9.30)
that is, the Mikhailov curve must pass successively through n
Obviously, the application of the Mikhailov criterion does not require precise and detailed construction of the curve. It is important to establish how it goes around the origin of coordinates and whether the sequence of passage is violated n quarter counterclockwise.
Example 2.9.6. Apply the Mikhailov criterion to test the stability of the system shown in Figure 2.9.20.
Characteristic polynomial of a closed system at k 1 k 2 > 0 corresponds to a stable system, so the Stodola condition is satisfied, and for n = 1 is sufficient. You can directly find the root R 1 = - k 1 k 2 and make sure that the necessary and sufficient stability condition is satisfied. Therefore, the application of the Mikhailov criterion is illustrative. Assuming p= j , we get
D(j ) = X( )+ jY( ),
where X( ) = ; Y( ) = . (2.9.31)
According to the parametric equations (2.9.31), the Mikhailov hodograph was constructed in Fig. 2.9.21, from which it can be seen that when changing 0 to vector D(j ) rotates counterclockwise by + /2 , i.e. the system is stable.
Nyquist stability criterion
To 
As already noted, the Nyquist criterion occupies a special position among the stability criteria. This is a frequency criterion that allows you to determine the stability of a closed system from the frequency characteristics of an open one. In this case, it is assumed that the system is open in a single negative feedback circuit (Fig. 2.9.22).
One of the advantages of the Nyquist criterion is that the frequency characteristics of an open system can be obtained experimentally.
The derivation of the criterion is based on the use of the argument principle. The transfer function of an open system (along the single negative feedback circuit in Fig. 2.9.22) is equal to 
Consider . (2.9.32)
In the case of a real system with limited bandwidth, the power of the denominator of the open-loop transfer function P greater than the power of the numerator, i.e. n> . Therefore, the degrees of the characteristic polynomials of an open system and a closed system are the same and equal n. The transition from the AFC of an open system to the AFC according to (2.9.32) means an increase in the real part by 1, i.e. moving the origin to the point (-1, 0), as shown in Fig.2.9.23.
Let us now assume that the closed system is stable and the characteristic equation of the open system A(p) = 0 has m right roots. Then, in accordance with the argument principle (2.9.29), we obtain the necessary and sufficient condition for the Nyquist stability of a closed system
Those. for the stability of a closed system vector W 1 (j ) must do m/2 full turns counterclockwise, which is equivalent to rotating the vector W pa s (j ) relative to the critical point (-1.0).
In practice, as a rule, an open-loop system is stable, i.e. m= 0. In this case, the increment of the argument is zero, i.e. The AFC of an open system should not cover the critical point (-1.0).
Nyquist criterion for LAH and LPH
In practice, the logarithmic characteristics of an open system are more often used. Therefore, it is advisable to formulate the Nyquist criterion for determining the stability of a closed system with respect to them. The number of AFC rotations relative to the critical point (-1.0) and its coverage or non-coverage
depend on the number of positive and negative intersections of the interval (-, -1) of the real axis and, accordingly, intersections of the phase characteristic of the -180 ° line in the region L( ) 0 . Figure 2.9.24 shows the AFC and shows the signs of the intersections of the segment (-, -1) of the real axis.
Fair rule
where is the number of positive and negative intersections.
According to the AFC in Fig. 2.9.24c, the LAH and LPH shown in Fig. 2.9.25 are constructed, and positive and negative intersections are marked on the LPH. On the interval (-,-1) the modulus is greater than one, which corresponds to L( ) > 0. Therefore, the Nyquist criterion:
D 
For the stability of a closed system LPH of an open system in the region where L(
)
>
0 should have more positive -180° line crossings than negative ones.
If the open system is stable, then the number of positive and negative intersections of the -180° line phase characteristic in the region L( ) > 0 for the stability of a closed system should be the same or there should be no intersections.
Nyquist criterion for astatic system
It is especially necessary to consider the case of an astatic order system r with an open-loop transfer function equal to
.
In this case 
at 0, i.e., the amplitude-phase characteristic (AFC) of an open-loop system goes to infinity. Previously, we built the AFC when changing
from - to and it was a continuous curve closed at
= 0. Now it also closes at
= 0, but at infinity and it is not clear on which side of the real axis (on the left or on the right at infinity?).
Fig.2.9.19c illustrates that in this case there is an uncertainty in the calculation of the difference vector argument increment. It is now located all the time along the imaginary axis (coincides with j ). Only when passing through zero does the direction change (in this case, the rotation of the vector counterclockwise by or clockwise by -?), For definiteness, we conditionally assume that the root is left and the rounding of the origin occurs along an arc of an infinitely small radius counterclockwise (rotation by + ). Accordingly, in the neighborhood = 0 can be represented as
,
where = + when it changes from – 0 to + 0. The last expression shows that with such disclosure of uncertainty, the AFC rotates when from – 0 to + 0 by an angle - clockwise. Accordingly, the constructed AFC is necessary for = 0 complement the arc of infinity of radius at an angle , i.e. counterclockwise to the positive real semiaxis.
Stability margins modulo and phase
To guarantee stability with changes in system parameters, stability margins are introduced in modulus and phase, which are defined as follows.
Modulo stability margin shows how many times or by how many decibels it is permissible to increase or decrease the gain in order for the system to remain stable (to be on the border of stability). It is defined as min( L 3 , L 4) in Fig.2.9.25. Indeed, if you do not change the LPH, then when the LAH rises to L 4 cutoff frequency sr will move to the point 4 and the system will be on the boundary of stability. If you lower LAH to L 3 , then the cutoff frequency will shift to the left to the point 3 and the system will also be on the stability boundary. If we lower the LAH even lower, then in the area L( ) > 0 will remain only the negative intersection of the LPH line -180 °, i.e. according to the Nyquist criterion, the system will become unstable.
Phase stability margin shows how much it is permissible to increase the phase shift at a constant gain in order for the system to remain stable (turned out to be on the stability boundary). It is defined as an addition ( cf) down to -180°.
On practice L 12-20 dB, 20-30°.
10.1. The concept of structural stability. AFC of astatic self-propelled guns
ACS can be unstable for two reasons: inappropriate composition of dynamic links and inappropriate values of link parameters.
ACS that are unstable for the first reason are called structurally unstable. This means that by changing the parameters of the ACS it is impossible to achieve its stability, it is necessary to change its structure.
For example, if the ACS consists of any number of inertial and oscillatory links, it has the form shown in Fig.72. With an increase in the ACS gain K, each point of its AFC moves away from the origin of coordinates, while at a certain value K crit AFC will not cross the point ( -1, j0). With further increase K, ACS will be unstable. Conversely, when decreasing K in principle, it is possible to make such an ACS stable, therefore it is called structurally stable.
If the ACS is astatic, then when it opens, the characteristic equation can be represented as: pD 1 p(p) = 0, where n - astatism order, equal to the number of integrators connected in series. This equation has zero roots, so when 0 , the AFC tends to (Fig. 71c and 71d). For example, let W p (p) =, here = 1 , then the AFC of the open ACS:
W(j) = 
= P() + jQ().
Since the order of the denominator is greater than the order of the numerator, then
0
we have P() -,
Q()
-j. A similar AFC is shown in Fig. 73. 
Since the AFC suffers a break, it is difficult to say whether it covers the point (-1,j0). In this case, the following technique is used: if the AFC suffers a break, going to infinity at 0 , it is supplemented mentally with a semicircle of infinite radius, starting on the positive real semiaxis and continuing to the AFC in the negative direction. After that, the Nyquist criterion can be applied. As can be seen from the figure, the ACS, which has one integrating link, is structurally stable.
If the ACS has two integrating links (the order of astatism = 2 ), its AFC goes to infinity in the second quadrant (Fig. 74). For example, let W p (p) =, then AFC ACS:
W(j) = P() + jQ().
At 0 we have P() -, Q() + j. Such an ACS will not be stable for any parameter values, that is, it is structurally unstable.
A structurally unstable ACS can be made stable by including corrective links (for example, differentiating or forcing ones) or by changing the structure of the ACS, for example, using local feedback.
10.2. The concept of stability margin
Under operating conditions, the parameters of the system, for one reason or another, can change within certain limits (aging, temperature fluctuations, etc.). These fluctuations in parameters can lead to a loss of system stability if it operates near the stability boundary. Therefore, they strive to design ACS so that it works far from the stability limit. The degree of this removal is called margin of stability.
Modulo stability margin characterizes the removal of the AFC hodograph of an open ACS from the critical point in the direction of the real axis and is determined by the distance h from the critical point to the point where the hodograph crosses the abscissa axis (Fig. 75).
Phase stability margin characterizes the distance of the hodograph from the critical point along the arc of a circle of unit radius and is determined by the angle between the negative direction of the real semiaxis and the ray drawn from the origin to the point of intersection of the hodograph with the unit circle.
As already noted, with an increase in the transmission coefficient of an open ACS, the module of each point of the AFC increases, and at a certain value K = K cr The AFC will pass through the critical point (Fig. 76) and hit the stability boundary, and when K > K cr closed ACS will become unstable. However, in the case of “beak-shaped” AFCs (obtained due to the presence of internal feedbacks), not only an increase, but also a decrease K can lead to loss of stability of closed ACS (Fig. 77). In this case, the stability margin is determined by two segments h1 and h2 between the critical point and the AFC.
Usually, when creating ACS, they are set by the required stability margins h and , beyond which it should not go. These limits are set in the form of a sector drawn around the critical point, in which the AFC of an open ACS should not enter (Fig. 78).
10.3. LFC Stability Analysis
It is more convenient to assess the stability by the Nyquist criterion using the LFC of an open ACS. It is obvious that certain points of LAFC and LPFC will correspond to each point of the AFC.
Let the frequency characteristics of two open ACSs (1 and 2) be known, differing from each other only in the transmission coefficient K 1 2. Let the first ACS be stable in the closed state, the second one not (Fig. 79).
If a W 1 (p) is the transfer function of the first ACS, then the transfer function of the second ACS W 2 (p) = KW 1 (p), where K = K 2 /K 1. The second ACS can be represented as a serial chain of two links with transfer functions K (inertialess link) and W 1 (p), so the resulting LFC are built as the sum of the LFC of each of the links.
Therefore, the LACHH of the second self-propelled guns: L 2 () = 20lgK + L 1 (),
and LFCH: 2 () = 1 () .
The intersections of the AFC of the real axis correspond to the phase value = - . This corresponds to the point of intersection of the LPCH = - grid lines. In this case, as can be seen in the AFC, the amplitudes A 1 () 2 () > 1, which corresponds to the values L 1 () = 20lgA 1 () 2 () > 0.
Comparing the APFC and LPFC, we can conclude that the system in the closed state will be stable if the LPFC value = - will correspond to negative values of LACH and vice versa. Stability margins modulo h1 and h2 determined by the AFC correspond to the distances from the abscissa axis to the LAFC at points where = - , but on a logarithmic scale.
The singular points are the points of intersection of the AFC with the unit circle. Frequencies c1 and c2 at which this happens is called cutoff frequencies.
At the intersection points A() = 1 => L() = 0- LACH crosses the horizontal axis. If, at the cutoff frequency, the phase of the AFC c1> - (Fig. 79a curve 1), then the closed ACS is stable. In Fig. 79b, it looks like the intersection of the LFC of the horizontal axis corresponds to the point of the LFC located above the line = - . And vice versa for an unstable closed ACS (Fig. 79a curve 2) c2-, so when = c2 LFCH passes below the line = - . Corner 1 = c1 -(-) is the phase stability margin. This angle corresponds to the distance from the line = - to LFCH.
6.1. The concept of stability of automatic control systems
The ACS dynamics is characterized by a transient process that occurs in it under the influence of any disturbance (control action, interference, load changes, etc.). The type of transient process in the ACS depends both on the properties of the ACS itself and on the type of disturbance acting on it. Depending on the type of transient process in ACS, the following varieties are distinguished.
Sustainable ACS- a system that, with steady values of disturbing influences, after a certain period of time returns to a steady state of equilibrium.
Unstable ACS- a system that does not return to a steady state of equilibrium at steady-state values of perturbing actions. The deviation of the system from the state of equilibrium will either increase all the time, or continuously change in the form of undamped constant oscillations.
Graphs of curves of transient processes, typical for stable and unstable ACS, are presented in fig. 6.1. Obviously, a workable ACS must be stable.
| a) | Examples of stability and instability of a certain system can also be illustrated by the following examples (Fig. 6.2). On fig. 6.2a an example of an unstable system is given - at the slightest deviation of the ball from the initial stable position, it rolls down the slope of the surface and does not return to its original position; rice. 6.2b illustrates an example of a stable system, since with any deviation the ball will necessarily return to its original position; rice. 6.2c shows a system that is stable under some small disturbances. As soon as the perturbing action exceeds a certain value, the system loses stability. Such systems are called stable in the small and unstable in the large, since stability is related to the magnitude of the initial perturbing action. |
| b) | |
| Rice. 6.1. Types of transient curves in stable (a) and unstable (b) ACS: 1 – aperiodic transient; 2 - oscillatory transient |
An analysis of the performance or stability of a linear ACS can be carried out using its mathematical model. As shown earlier, a linear ACS can be described by differential equation (2.1). The solution of this differential equation in the general case has the form (2.3)
where is the free component of the solution of equation (2.1), which is determined by the initial conditions and properties of the considered ACS;
is the forced component of the solution of equation (2.1), determined by the perturbed influences and the properties of the considered ACS.
The stability of the ACS is characterized by the processes occurring within the ACS itself. These processes are determined by the form of the free component of the solution of equation (2.1). Therefore, in order for the ACS to be stable, the following condition must be met:
On the other hand, it can be represented in general as
where are the roots obtained by solving the characteristic equation (2.7). In table. 6.1 shows some varieties of transient processes in the ACS, depending on the type of roots of the characteristic equation (2.7).
Table 6.1
Varieties of transient processes in ACS depending on the type of roots
characteristic equation (2.7)
The end of the table. 6.1
| m- complex conjugate roots, the real part of which is negative: | oscillatory damped | sustainable | ||
| the roots are real, positive, while | aperiodic divergent | unstable | ||
| among the roots (item 1) is present m- complex conjugate roots, the real part of which is positive: | oscillatory divergent | unstable | ||
| among the roots (item 1) there is a pair of complex roots, the real part of which is equal to zero: | undamped oscillations | system on the verge of stability (purely theoretical case) |
To satisfy condition (6.1), it is necessary that each term in expression (6.2) for t®¥ would tend to zero. As follows from the analysis given in table. 6.1 examples of transient processes in the ACS, for this it is necessary that all the roots of the characteristic equation (2.7) be negative real or complex with a negative real part. If among the roots of the characteristic equation (2.7) there is at least one positive real root or a pair of conjugate complex roots with a positive real part, then the ACS under consideration will be unstable, since the term of equation (6.2) corresponding to this root, at t®¥ will increase indefinitely.
On fig. 6.3 and 6.4 are examples of the location of the roots of the characteristic equation of the ACS on the complex plane, corresponding to stable and unstable ACS. As follows from these examples, in order for the ACS to be stable, it is necessary that all the roots of the ACS characteristic equation be to the left of the imaginary axis.
To analyze the stability of the ACS by the form of the roots of its characteristic equation, it is required to find an analytical solution of the differential equation (2.1), which is a rather laborious task, and in some cases impossible. Therefore, in practice, sustainability criteria are widely used, which means the following.
Stability criterion- a set of features that allow you to have an idea about the signs of the roots of the characteristic equation without solving the equation itself. There are the following types of sustainability criteria:
− algebraic criteria of stability (criteria of Vyshnegradsky, Routh, Hurwitz). To analyze the stability of the ACS in this case, the coefficients of the characteristic equation of the system are used;
− frequency stability criteria (Nyquist, Mikhailov criteria). These stability criteria assume the application of the frequency characteristics of the system.
The use of this or that stability criterion makes it possible to judge the stability of the ACS more simply and efficiently than when solving the differential equation (2.1) that describes it. In addition, some stability criteria allow us to establish the cause of the ACS instability and outline ways to achieve system stability.
6.2. Algebraic Hurwitz stability criterion
This type of algebraic criterion is the most common in practice for studying the stability of ACS. The initial data for the study of stability in this case is the characteristic equation of a closed ACS
From the coefficients of the characteristic equation (6.3) a matrix (6.4) is formed, the dimension of which is equal to the order of the characteristic equation (6.3). Matrix (6.4) is compiled according to the following rule: the coefficients of the characteristic equation are written down sequentially along the main diagonal, starting from C1. The columns of the table, starting from the main diagonal, are filled up by increasing indices, down - by decreasing ones. All coefficients with indices below zero and above the degree of order of the characteristic equation n are replaced with zeros.
Hurwitz stability conditions: for the stability of the ACS with the characteristic equation (6.3), it is necessary and sufficient that all the coefficients of the characteristic equation (6.3) be positive, and also be positive n determinants composed of the coefficients of equation (6.3) based on matrix (6.4). To compile the determinant 1,2, ..., n th order, 1,2, …, n columns and rows. The examples below illustrate this rule.
Example 1. For ACS with a characteristic equation of the 2nd order:
matrix (6.4) can be written as
Determinants D1, D2, based on (6.6), have the form
C0, C1, C2 will be greater than zero, and the determinants (6.7) and (6.8) will also be positive.
Example 2 For ACS with 3rd order characteristic equation:
matrix (6.4) can be written as
Determinants D1 – D3, based on (6.10), have the form
According to the Hurwitz stability criterion, this system will be stable provided that the coefficients C0 – C3 will be greater than zero, and the determinant (6.12) will also be positive.
Example 3 For ACS with 4th order characteristic equation:
matrix (6.4) can be written as
Determinants D1 – D4, based on (6.15), have the form
According to the Hurwitz stability criterion, this system will be stable provided that the coefficients C0 – C4 will be greater than zero, and the determinants (6.16)–(6.19) will also be positive.
The algebraic Hurwitz criterion makes it possible to visually assess the influence of one or another parameter on the stability of the ACS as a whole. Let us assume that for the considered ACS, mathematical model which has the characteristic equation (6.3), it is necessary to investigate the influence of the parameter value C n for sustainability. To do this, by giving a number of valid values for C n, calculate n determinants composed of the coefficients of equation (6.3) based on matrix (6.4). Each of the determinants D i where i=0,..,n will be a function depending on the parameter C n, which can be represented as a graph (Fig. 6.5). Depicting functions on one graph D i (C n), where i=0,.., n, we determine on the x-axis the segment of change C n, during which all n determinants will be positive (in Fig. 6.5 this segment is marked with a bold line). Therefore, according to the Hurwitz criterion for the values C n that belong to the selected segment, the system will be stable. If after plotting the functions D i (C n), where i=0,.., n, on the x-axis it is impossible to select the segment of the change C n, during which all n determinants will be positive (Fig. 6.6), this means that by changing the value C n it is impossible to bring the ACS to a state of stability.
The application of the algebraic Hurwitz stability criterion assumes that the differential equation describing the ACS (6.3) is known and its coefficients are known quite accurately. In some cases, it is impossible to fulfill these conditions in practice. In addition, with an increase in the order of the characteristic equation of the ACS (6.3), the complexity of calculating the determinants compiled on the basis of the matrix (6.4) increases. Therefore, in practice, frequency stability criteria have also become widespread, which make it possible to assess the stability of the system, even if the differential equation (2.1) is unknown, and the experimental frequency characteristics of the ACS under consideration are available.
6.3. Frequency Nyquist Stability Criterion
Frequency stability criteria are now widely accepted. One of such criteria is the Nyquist criterion or the frequency amplitude-phase criterion. This type of criterion is a consequence of the Cauchy theorem. The proof of the validity of the Nyquist criterion is given in. The criterion under consideration makes it possible to judge the stability of a closed ACS by studying the AFC of this ACS in the open state, since this study easier to perform.
The initial data for studying the stability of the ACS using the Nyquist criterion is its AFC, which can be obtained either experimentally or using the well-known expression for the transfer function of an open-loop ACS (3.6) by replacing p=jw.
Nyquist stability conditions:
1) if the ACS is stable in the open state, then the amplitude-phase characteristic of this ACS, obtained by changing w from - ¥ to + ¥ j 0);
2) if the system is unstable in the open state and has k roots in the right half-plane, then the AFC of the ACS when changing w from - ¥ to + ¥ should cover k times a point on the complex plane with coordinates (–1, j 0). Vector Rotation Angle W(jw) must make up 2p k.
A closed ACS will be stable if, when changing w from 0 to + ¥ the difference between the number of positive and negative transitions of the AFC hodograph of an open system through a segment of the real axis (– ¥ , –1) will be equal to k/2, where k is the number of right roots of the characteristic equation of an open system. For the negative transition of the vector hodograph W(jw) its transition from the lower half-plane to the upper one is considered with increasing w. For the positive transition of the vector hodograph W(jw) its transition from the upper half-plane to the lower one is accepted with the same frequency change sequence.
With a negative sign for the complex frequency response, the above positions are determined by the point (+1, j 0).
The Nyquist criterion is also valid for the case when the polynomial C(p) in (3.6) ACS has a zero root, which corresponds to the AFC value equal to infinity. To study the stability of such ACS, it is necessary to mentally supplement the AFC hodograph with a circle of infinite radius and close the hodograph with a real semi-axis in the shortest direction. Next, check compliance with the Nyquist stability conditions and draw conclusions.
Examples of the AFC of stable and unstable ACS are shown in fig. 6.7, 6.8.
6.4. Logarithmic stability criterion
This stability criterion is an interpretation of the frequency Nyquist stability criterion in logarithmic form. Consider two AFC (Fig. 6.9) corresponding to an open ACS, while AFC (1) corresponds to an ACS that is unstable in the open state, AFC (2) - ACS that is stable in the open state. Let us introduce the characteristic points of the considered AFC: w 1s, w 2s are the points corresponding to the frequencies at which the amplitudes of the vectors W(jw) respectively, systems (1) and (2) become equal to one. This frequency is called the cutoff frequency. On the complex plane, this point corresponds to the point of intersection of the AFC with a circle of unit radius, the center of which is at the origin of coordinates (in Fig. 6.9 this circle is shown by a dotted line). The same point corresponds to the point of intersection of the LAFCH with the abscissa axis (Fig. 6.10); w 1 p, w 2 p are the points corresponding to the frequencies at which the phases of the vectors W(jw) respectively, systems (1) and (2) become equal to –180 O. On the complex plane, this point corresponds to the point of intersection of the AFC with the real negative semiaxis. The same point corresponds to the intersection point of the LPFC with the abscissa axis, provided that the LPFC and LPFC are shown on the same graph in the form shown in Fig. 6.10.
| Rice. 6.9. AFC ACS: 1 - unstable in the open state; 2 - stable in the open state | Rice. 6.10. LAFC and LPFC of unstable (1) and stable (2) ACS |
According to the Nyquist stability criterion, if the ACS is stable in the open state, then the amplitude-phase characteristic of this ACS, obtained by changing w from - ¥
to + ¥
, must not enclose a point on the complex plane with coordinates (–1, j 0). In other words, as follows from Fig. 6.9, the system will be stable if w p >w c, otherwise ( wp
If the number of points of intersection of the AFC and the negative real semiaxis on the segment (– ¥ , –1) when changing w from 0 to + ¥ more than one (Fig. 6.11), then, in order for the ACS to be stable in the closed state, it is necessary that the number of such points on the segment (- ¥ , –1) was even. In this case, the LPFC must cross an even number of times the x-axis in the segment from 0 to the cutoff frequency w c(Fig. 6.12).
For the stability of the ACS in the closed state, which are unstable in the open state and have k-roots lying to the right of the imaginary axis, the logarithmic criterion of stability can be formulated as follows: such ACS will be stable if the difference between the numbers of positive and negative transitions of the LPFC and negative transitions of the LPFC through the value of –180°, lying on the segment from 0 to w C, will be equal to k/2. Recall that the positive transition of the characteristic is taken to be its transition from the upper half-plane to the lower one with increasing w. The negative transition of the characteristic is taken as its transition from the lower half-plane to the upper one with the same frequency change sequence. Frequency characteristics of ACS, unstable in the open state and stable in the closed state, for which k=1 are shown in fig. 6.13, 6.14.
6.5. Mikhailov's frequency criterion for estimating stability
The initial data for studying the stability of the ACS using the Mikhailov criterion is the AFC of a closed system, which can be obtained using the characteristic polynomial of the closed ACS (3.35), which has the order n:
Mikhailov stability conditions: if the vector characterizing the closed ACS, when changing w from - ¥ to + ¥ describes in the positive direction (without changing direction) an angle equal to np(where n is the degree of the characteristic polynomial (6.20)), then such an ACS will be stable. Otherwise, the ACS will be unstable. The proof of this assertion is given in .
Since the hodograph of the curve of the transfer function vector of a closed ACS is symmetrical, it is allowed to confine ourselves to considering only its part corresponding to changes w from 0 to + ¥ . In this case, the angle described by the vector , when changing w from 0 to + ¥ will be halved.
On fig. 6.15, 6.16 examples of vector hodographs are given, corresponding to stable, unstable and neutral ACS (a system that is on the verge of stability).
6.6. Construction of ACS stability areas
The stability criteria considered above make it possible to determine whether the ACS under consideration is stable for given parameters or not. If the ACS is unstable, it is often necessary to look for an answer to the question: what is the cause of the instability, and to determine ways to eliminate it. In addition to assessing stability, in practice it often becomes necessary to determine ways to improve the dynamic performance of ACS. The listed tasks can be solved using the existing ACS stability criteria, however, they are most effectively solved by constructing the ACS stability and instability areas.
Let us assume that the ACS under consideration is unstable and, at the same time, it can be represented by a linear differential equation (2.1), the characteristic equation of which will have the following form (6.3):
Let us further assume that the coefficients C 0 -C n -1 given characteristic equation are given, and the coefficient C n may vary within the range C n (min)–C n (max). By setting a series of values for C n from the specified range, we find segments within this range, during which C n has such values at which the ACS will be stable (Fig. 6.17), i.e. all roots of the characteristic equation (6.21) will lie on the complex plane to the left of the imaginary axis. Boundary points of "segments of stability" correspond to the values C n, at which the ACS is on the verge of stability.
In equation (6.21), two or more coefficients can change. If two coefficients change in it (suppose it is From 0 and C n), then a study is made of the dependence of the stability of the ACS on the values of the coefficient
ents From 0 and C n by setting a number of values for these coefficients from some allowable ranges and checking the stability of the ACS for the selected values From 0 and C n. In this case, the stability regions will be some sections on the coordinate plane of the variable coefficients From 0 and C n(Fig. 6.18). The boundary of the stability of the system in this case will be the curve that limits the stability areas.
If in the characteristic equation three parameters change within certain allowable limits (for example, From 0, From 1 and C n), then when studying the dependence of the ACS stability on the values From 0, From 1 and C n the ACS stability region will be found, which will be a part of the space bounded by some complex surface (Fig. 6.19). This complex surface in this case will be the stability limit of the ACS.
Rice. 6.19. ACS stability area when changing three parameters
(From 0, From 1 and C n)
In the general case, if we assume that in the characteristic equation (6.21) all the coefficients included in it From 0-C n can change within certain acceptable limits, then the stability of the ACS can be considered as a logical function defined in some multidimensional space. At some points of this multidimensional space, this function will take the value "True" (the ACS is stable), at others - "False" (the ACS is unstable). Each point of such a space (the space of coefficients) will correspond to certain values From 0-C n, which are its coordinates. The hypersurface limiting the ACS stability region will be the boundary of the stability region in the considered coefficient space.
When determining the stability areas of the ACS, one stability area can be allocated, several stability areas can be allocated, or none can be allocated.