EntranceUse of l.a.ch.h. and phase frequency characteristics to analyze system stability
The Nyquist stability criterion was formulated and justified in 1932 by the American physicist H. Nyquist. The Nyquist stability criterion is most widely used in engineering practice for the following reasons:
- the stability of the system in a closed state is studied by the frequency transfer function of its open part W p (jw), and this function, most often, consists of simple factors. The coefficients are the real parameters of the system, which allows you to select them from the stability conditions;
- to study stability, you can use experimentally obtained frequency characteristics of the most complex elements of the system (control object, executive bodies), which increases the accuracy of the results obtained;
- the stability of the system can be studied using logarithmic frequency characteristics, the construction of which is not difficult;
- the stability margins of the system are determined quite simply;
- convenient to use for assessing the stability of an ATS with a delay.
The Nyquist stability criterion makes it possible to evaluate the stability of an ACS based on the AFC of its open-loop part. In this case, three cases of application of the Nyquist criterion are distinguished.
1. The open part of the ACS is stable.For the stability of a closed-loop system, it is necessary and sufficient that the AFC response of the open-loop part of the system (Nyquist hodograph) when changing frequencies w from 0 to +¥ did not cover the point with coordinates [-1, j 0]. In Fig. 4.6 shows the main possible situations:
1. - the closed system is absolutely stable;
2. - ATS is conditionally stable, i.e. stable only in a certain range of changes in the transmission coefficient k;
3. - ATS is on the border of stability;
4. - ATS is unstable.
Rice. 4.6. Nyquist hodographs when the open part of the ACS is stable
2. The open part of the ACS is on the stability boundary.In this case, the characteristic equation has zero or purely imaginary roots, and the remaining roots have negative real parts.
For the stability of a closed system, if the open-loop part of the system is on the stability boundary, it is necessary and sufficient that the AFC response of the open-loop part of the system when changing w from 0 to +¥, supplemented in the discontinuity area by an arc of infinitely large radius, did not cover the point with coordinates [-1, j 0]. In the presence of ν zero roots of the AFC response of the open-loop part of the system at w=0 by an arc of infinitely large radius moves from the positive real semi-axis by an angle of degrees clockwise, as shown in Fig. 4.7.
Rice. 4.7. Nyquist hodographs in the presence of zero roots
If there is a pair of purely imaginary roots w i =, then the AFC response at frequency w i an arc of infinitely large radius moves at an angle of 180° clockwise, which is reflected in Fig. 4.8.
Rice. 4.8. Nyquist hodograph in the presence of a pair of purely imaginary roots
3. The open-loop part of the system is unstable, i.e. the characteristic equation has l roots with positive real part. In this case, for the stability of a closed-loop system it is necessary and sufficient that when the frequency changes w from 0 to +¥ AFC of the open part of the ACS covered the point
[-1, j 0) l/2 times in the positive direction (counterclockwise).
With a complex shape of the Nyquist hodograph, it is more convenient to use another formulation of the Nyquist criterion, proposed by Ya.Z. Tsypkin using transition rules. Transition of the phase response response of the open-loop part of the system with increasing w the segment of the real axis from -1 to -¥ from top to bottom is considered positive (Fig. 4.9), and from bottom to top negative. If the AFC response begins in this segment at w=0 or ends at w=¥ , then it is considered that the AFC makes a half transition.
Rice. 4.9. Transitions of the Nyquist hodograph through the segment P( w) from -¥ to -1
The closed system is stable, if the difference between the number of positive and negative transitions of the Nyquist hodograph through a segment of the real axis from -1 to -¥ is equal to l/2, where l is the number of roots of the characteristic equation with a positive real part.
Task condition.
Using the Mikhailov and Nyquist stability criterion, determine the stability of a single-loop control system that has a transfer function of the form in the open state
Enter the values of K, a, b and c into the formula according to the option.
W(s) = 
,
(1)
Construct Mikhailov and Nyquist hodographs. Determine the cutoff frequency of the system.
Determine the critical value of the system gain.
Solution.
Problems of analysis and synthesis of control systems are solved using such a powerful mathematical apparatus as the operational calculus (Laplace transform). Problems of analysis and synthesis of control systems are solved using such a powerful mathematical apparatus as the operational calculus (Laplace transform). The general solution of the operator equation is the sum of terms determined by the values of the roots of the characteristic polynomial (polynomial):
D(s) = d s n d n ) .
Construction of Mikhailov's hodograph.
A) We write out the characteristic polynomial for the closed system described by equation (1)
D(s) = 50 + (25s+1)(0.1s+1)(0.01s+1) = 50+(625+50s+1)(0.001+0.11s+1) =0.625+68.85 +630.501+50.11s+51.
Roots of a polynomial D(s) may be: null; real (negative, positive); imaginary (always paired, conjugate) and complex conjugate.
B) Transform to the form s→ ωj
D()=0.625+68.85+630.501+50.11+51=0.625ω-68.85jω- 630.501ω+50.11jω+51
ω – signal frequency, j = (1) 1/2 – imaginary unit. J 4 =(-1) 4/2 =1, J 3 =(-1) 3/2 =-(1) 1/2 = - j, J 2 =(-1) 2/2 =-1, J =(-1) 1/2 = j,
C) Let us select the real and imaginary parts.
D= U()+jV(), where U() is the real part and V() is the imaginary part.
U(ω) =0.625ω-630.501ω+51
V(ω) =ω(50.11-68.85ω)
D) Let's build Mikhailov's hodograph.
Let's build Mikhailov's hodograph close to and away from zero; for this we'll build D(jw) as w changes from 0 to +∞. Let's find the intersection points U(w) and V(w) with axles. Let's solve the problem using Microsoft Excel.
We set the values of w in the range from 0 to 0.0001 to 0.1, and calculate them in the table. Excel values U(ω) and V(ω), D(ω); find the intersection points U(w) and V(w) with axles,
We set the values of w in the range from 0.1 to 20, and calculate them in the table. Excel values U(w) and V(w), D; find the intersection points U(w) and V(w) with axles.
Table 2.1 – Definition of the real and imaginary parts and the polynomial itself D()using Microsoft Excel
Rice. A, B, ..... Dependencies U(ω) and V(ω), D(ω) from ω
According to Fig. A, B, .....find the intersection points U(w) and V(w) with axles:
at ω = 0 U(ω)= …. And V(ω)= ……
Fig.1. Mikhailov's hodograph at ω = 0:000.1:0.1.
Fig.2. Mikhailov's hodograph at ω = 0.1:20
D) Conclusions about the stability of the system based on the hodograph.
Stability (as a concept) of any dynamic system is determined by its behavior after removing the external influence, i.e. its free movement under the influence of initial conditions. A system is stable if it returns to its original state of equilibrium after the signal (perturbation) that brought it out of this state ceases to act on the system. An unstable system does not return to its original state, but continuously moves away from it over time. To assess the stability of the system, it is necessary to study the free component of the solution to the dynamics equation, that is, the solution to the equation:.
D(s) = d s n d n )= 0.
Check the stability of the system using the Mikhailov criterion :
Mikhailov criterion: For a stable ASR, it is necessary and sufficient that the Mikhailov hodograph (see Fig. 1 and Fig. 2), starting at w = 0 on the positive real semi-axis, goes around successively in the positive direction (counterclockwise) as w increases from 0 to ∞ n quadrants, where n is the degree of the characteristic polynomial.
It is clear from the solution (see Fig. 1 and Fig. 2) that the hodograph satisfies the following criteria conditions: It starts on the positive real semi-axis at w = 0. The hodograph does not satisfy the following criterion conditions: it does not go around all 4 quadrants in the positive direction (degree of polynomial n=4) at ω.
We conclude that this open-loop system is not stable .
Construction of the Nyquist hodograph.
A) Let’s make a replacement in formula (1) s→ ωj
W(s) = 
=
,
B) Open the brackets and highlight the real and imaginary parts in the denominator
C) Multiply by the conjugate and select the real and imaginary parts
,
where U() is the real part and V() is the imaginary part.
D) Let's construct a Nyquist hodograph: - dependence of W() on .
Fig.3. Nyquist hodograph.
E) Let's check the stability of the system using the Nyquist criterion:
Nyquist criterion: In order for a system that is stable in the open state to be stable in the closed state, it is necessary that the Nyquist hodograph, when the frequency changes from zero to infinity, does not cover the point with coordinates (-1; j0).
It is clear from the solution (see Fig. 3) that the hodograph satisfies all the conditions of the criterion:
The hodograph changes its direction clockwise
The hodograph does not cover the point (-1; j0)
We conclude that this open-loop system is stable .
Determination of the critical value of the system gain.
A) In paragraph 2, the real and imaginary parts have already been distinguished
B) In order to find the critical value of the system gain, it is necessary to equate the imaginary part to zero and the real part to -1
C) Let us find from the second (2) equation
The numerator must be 0.
We accept that , then
C) Substitute into the first (1) equation and find
The critical value of the system gain.
Literature:
1.Methods of classical and modern theory of automatic control. Volume 1.
Analysis and statistical dynamics of automatic control systems. M: Ed. MSTU named after Bauman. 2000
2. Voronov A.A. Theory of automatic control. T. 1-3, M., Nauka, 1992
The left hodograph is obviously a hodograph sustainable system, does not cover the points , which is required according to the Nyquist criterion for the stability of a closed-loop system. Right hodograph – hodograph three-pole, a obviously unstable system bypasses the point three times counterclockwise, which is required according to the Nyquist criterion for the stability of a closed-loop system.
Comment.
The amplitude-phase characteristics of systems with real parameters - and only such are encountered in practice - are symmetrical about the real axis. Therefore, only half of the amplitude-phase characteristic corresponding to positive frequencies is usually considered. In this case, half-travels of the point are considered. The intersection of the segment () when the frequency increases from top to bottom (the phase increases) is considered an intersection, and from bottom to top is considered an intersection. If the amplitude-phase characteristic of an open-loop system begins on the segment (), then this will correspond to either an intersection, depending on whether the characteristic goes down or up as the frequency increases.
The number of intersections of the segment () can be calculated using logarithmic frequency characteristics. Let us clarify that these are the intersections that correspond to a phase when the magnitude of the amplitude characteristic is greater than one.
Determination of stability using logarithmic frequency characteristics.
To use the Mikhailov criterion, you need to construct a hodograph. Here is the characteristic polynomial of the closed system.
In the case of the Nyquist criterion, it is enough to know the transfer function of the open-loop system. In this case, there is no need to construct a hodograph. To determine Nyquist stability, it is enough to construct the logarithmic amplitude and phase frequency characteristics of an open-loop system.
The simplest construction is obtained when the transfer function of an open-loop system can be represented in the form
, then LAH 
,
The figure below corresponds to the transfer function
.
Here and 
built as functions.
The logarithmic frequency characteristics shown below correspond to the previously mentioned system with a transfer function (open-loop system)
.
On the left are the amplitude and phase frequency characteristics for the transfer function, on the right - for the transfer function, in the center - for the original transfer function (as calculated by the Les program, the “Integration” method).
The three poles of the function are shifted to the left (stable system). The phase characteristic, accordingly, has 0 level crossings. The three poles of the function are shifted to the right (unstable system). The phase characteristic, accordingly, has three half-level intersections in areas where the modulus of the transfer function is greater than unity.
In any case, the closed system is stable.
The central picture - the calculation in the absence of root movements, is the limit for the right picture, the course of the phase in the left picture is radically different. Where is the truth?
Examples from.
Let the transfer function of the open-loop system have the form:
.
An open-loop system is stable for any positive k And T. A closed system is also stable, as can be seen from the hodograph on the left in the figure.
When negative T the open-loop system is unstable - it has a plus in the right half-plane. The closed system is stable at , as can be seen from the hodograph in the center, and unstable at 
(hodograph on the right).
Let the transfer function of the open-loop system have the form ():
.
It has one pole on the imaginary axis. Consequently, for the stability of a closed-loop system, it is necessary that the number of intersections of the segment () of the real axis by the amplitude-phase characteristic of the open-loop system is equal (if we consider the hodograph only for positive frequencies).
This is the locus of the points that the end of the vector of the frequency transfer function describes when the frequency changes from -∞ to +∞. The size of the segment from the origin to each point of the hodograph shows how many times at a given frequency the output signal is greater than the input signal, and the phase shift between the signals is determined by the angle to the mentioned segment.
All other frequency dependencies are generated from the AFC:
- U(w) - even (for closed automatic control systems P(w));
- V(w) - odd;
- A(w) - even (frequency response);
- j(w) - odd (phase response);
- LACHH & LFCH - used most often.
Logarithmic frequency characteristics.
Logarithmic frequency characteristics (LFC) include a logarithmic amplitude characteristic (LAFC) and a logarithmic phase characteristic (LPFC) constructed separately on one plane. The construction of LFC & LFCH is carried out using the following expressions:
L(w) = 20 lg | W(j w)| = 20 lg A(w), [dB];
j(w) = arg( W(j w)), [rad].
Magnitude L(w) is expressed in decibels . Bel is a logarithmic unit corresponding to a tenfold increase in power. One Bel corresponds to an increase in power by 10 times, 2 Bels - by 100 times, 3 Bels - by 1000 times, etc. A decibel is equal to one tenth of a Bel.
Examples of AFC, AFC, PFC, LFC and LPFC for typical dynamic links are given in Table 2.
Table 2. Frequency characteristics of typical dynamic links.
Principles of automatic regulation
Based on the control principle, self-propelled guns can be divided into three groups:
- With regulation based on external influences - the Poncelet principle (used in open-loop self-propelled guns).
- With regulation by deviation - the Polzunov-Watt principle (used in closed self-propelled guns).
- With combined regulation. In this case, the ACS contains closed and open control loops.
Control principle based on external disturbance
The structure requires disturbance sensors. The system is described by the open-loop transfer function: x(t) = g(t) - f(t).
Advantages:
- It is possible to achieve complete invariance to certain perturbations.
- The problem of system stability does not arise, because no OS.
Flaws:
- A large number of disturbances requires a corresponding number of compensation channels.
- Changes in the parameters of the controlled object lead to errors in control.
- Can only be applied to objects whose characteristics are clearly known.
Deviation control principle
The system is described by the open-loop transfer function and the closure equation: x(t) = g(t) - y(t) W oc( t). The algorithm of the system is based on the desire to reduce the error x(t) to zero.
Advantages:
- OOS leads to a reduction in error, regardless of the factors that caused it (changes in the parameters of the controlled object or external conditions).
Flaws:
- In OS systems, there is a problem of stability.
- It is fundamentally impossible to achieve absolute invariance to disturbances in systems. The desire to achieve partial invariance (not with the first OS) leads to complication of the system and deterioration of stability.
Combined control
Combined control consists of a combination of two control principles based on deviation and external disturbance. Those. The control signal to the object is generated by two channels. The first channel is sensitive to the deviation of the controlled variable from the target. The second one generates a control action directly from a master or disturbing signal.
x(t) = g(t) - f(t) - y(t)Woc(t)
Advantages:
- The presence of OOS makes the system less sensitive to changes in the parameters of the controlled object.
- Adding reference-sensitive or disturbance-sensitive channel(s) does not affect the stability of the feedback loop.
Flaws:
- Channels that are sensitive to a task or to a disturbance usually contain differentiating links. Their practical implementation is difficult.
- Not all objects allow forcing.
ATS stability analysis
The concept of stability of a regulatory system is associated with its ability to return to a state of equilibrium after the disappearance of external forces that brought it out of this state. Stability is one of the main requirements for automatic systems.
The concept of stability can be extended to the case of ATS movement:
- undisturbed movement
- indignant movement.
The movement of any control system is described using differential equation, which generally describes 2 operating modes of the system:
Steady State Mode
Driving mode
In this case, the general solution in any system can be written as:
Forced the component is determined by the input influence on the input of the control system. The system reaches this state at the end of transient processes.
Transitional the component is determined by solving a homogeneous differential equation of the form:
Coefficients a 0 ,a 1 ,…a n include system parameters => changing any coefficient of the differential equation leads to a change in a number of system parameters.
Solution of a homogeneous differential equation
where are the integration constants, and are the roots of the characteristic equation of the following form:
The characteristic equation represents the denominator of the transfer function equal to zero.
The roots of the characteristic equation can be real, complex conjugate and complex, which is determined by the parameters of the system.
To assess the stability of systems, a number of sustainability criteria
All sustainability criteria are divided into 3 groups:
Root
- 
algebraic