Self control principle learning notes - system stability analysis (2) - loop analysis and Nyquist bode criterion

Self control principle learning notes
Self control principle learning notes column

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Loop analysis

3. Loop analysis

3.1 Basic idea of loop analysis ：

1. Transmission of sinusoidal signals with different frequencies in linear systems , The output is amplified 、 Narrowing or self-sustaining oscillation depends entirely on the frequency characteristics .

2. remember （-1,i0） Is the critical point . The phase angle crossover frequency is when the phase angle is -180° The frequency of time , At this time, if the amplitude is 1 Is self excitation condition .

3. simplify Nyquist The criterion ：

   If $|L(i{w}_{pc})>1|$, On the ring road $w_{pc}$ The signal is amplified , Closed loop system instability .

   As if $|L(i{w}_{pc})<1|$, On the ring road $w_{pc}$ The signal is attenuated , The closed-loop system is stable .

3.2 Performance index of stability （ Relatively stable ）

1. Gain margin

   ​
   $$\begin{gathered} g_m=\frac{1}{|L(i{w}_{pc})|} \\ \text{requirement :}{g}_m\ge 2 \end{gathered}$$
   

2. Phase margin
   $$\begin{gathered} {\phi }_m=180°+\angle L(i{w}_c) \\ {\phi }_m=30°\sim 60° \end{gathered}$$

3. Mold margin ( When there is resonance, we need to consider the mode margin )
   $$s_m=\underset{w}{\min }|L(iw)+1|=\frac{1}{\underset{w}{\max }|S(iw)|}=\frac{1}{M_s}$$
   

3 Margin calculation

3.3 Loop shaping

1. For open-loop stability 、 Closed loop unstable system , The critical point can be avoided by reducing the open-loop gain , Make the closed-loop system stable
2. The open-loop zero pole is introduced into the system through the controller , Change the shape of frequency characteristics , Bypass the critical point .

4.Nyquist The criterion

4.1 Relationship with argument principle

（1） The closed domain discussed defines the right half plane , The change of complex variable in the right half plane is equivalent to the frequency from negative infinity to positive infinity

（2） The path direction of the positive direction of the closed loop is changed to clockwise

（3） Limit the complex variable function discussed 1+L, Its zero is the pole of the closed-loop transfer function , Its pole is the pole of open-loop transmittal

（4） from 1+L To L That is to change the origin of the critical point to （-1,i0）

4.2 Rounding

4.3 Discrimination method

Let the open-loop transfer function $L(s)$ The number of poles of the right half complex plane is P,

1. Nyquist The curve revolves clockwise （-1,i0） The number of turns is $w_n$, Then the number of poles of the closed-loop system in the right half complex plane is $N=w_n+P$

2. The necessary and sufficient condition for the stability of the closed-loop system is N=0, namely Nyquist The curve surrounds （-1,i0） The number of counterclockwise turns of is P

3. Related instructions ：L（s） If RHP pole , The necessary condition for the stability of closed-loop system $|L(i{w}_{pc})|>1$, That is, you have to go around （-1,i0）.

4.4 Related examples

Number of open-loop poles	0	0	2
Number of circles clockwise	1	1	-2
Closed loop poles	0 Stable	0 Stable	0 Stable

2. n An integral link complements the circle $N\pi$

3.

The system adds open-loop poles , About to increase the phase angle of the system , So that it doesn't have to cross to the second quadrant , So as to achieve $w_n=0$.

There are pure imaginary poles ：

5. Stability and natural vibration analysis of nonlinear system

5.1 take Nyquist The criterion is applied to the determination of nonlinear systems

utilize $1+N(A)G(s)=0$, obtain $G(s)=-\frac{1}{N(A)}$, In the linear system Nyquist The criterion is transplanted to nonlinear systems , critical point -1 Become a borderline $-\frac{1}{N(A)}$

5.2 Necessary conditions for natural vibration

Negative inverse curve $–\frac{1}{N(A)}$ And $G(iw)$ There are intersections . At the intersection, it is in a critical stable state , Satisfy $N(A)\ast G(iw)=-1$ namely ：
$$\{\begin{array}{c}|N(A)|\cdot |G(iw)|=1\\ \angle N(A)+\angle G(iw)=-180°\end{array}$$

5.2 Steps for judging self-excited oscillation

1. On the same plane , Draw the negative inverse curve of open-loop transmittal and nonlinear link ;

2. According to the natural vibration analysis of nonlinear system, the stable region and unstable region are divided , Observe the relationship between the two curves , The surrounding area is unstable ;

3. If there is from along w Increasing direction , There are unstable regions to stable regions , Then there is a stable natural vibration point ; If there is natural vibration , Rely again 1+N(A)G(iω)=0 Write two equations to find A and ω.

   

6.Bode Figure stability analysis

6.1 And Nyquist Corresponding relation

Nyquist chart	Bode chart
Unit circle	Amplitude phase curve 0dB Line
Outside and inside the unit circle	Upper and lower sides of amplitude phase curve
Negative real axis	The phase angle is -180 Degree curve
$w_{pc}and{w}_c$ The correspondence is as follows	$w_{pc}and{w}_c$ The correspondence is as follows
Integral link nyquist Rounding $\nu \pi$	The phase frequency curve is complemented upward $\nu \frac{\pi }{2}$
Constant amplitude oscillation $\frac{1}{(s^2+w_n^2{)}^2}$, Amplitude phase semi closed curve （Nyquist Half of ） Need to be supplemented $\nu \pi$ from $w_n^{-}To{w}_n^{+}$	The phase frequency is compensated downward $\nu \pi$

6.2 Phase frequency characteristics - Number of crossings

limit ： Positive and negative crossing must be within the crossing frequency
Note that there may be half crossing after phase angle compensation ,-180 Upward is positive half crossing , Down is the negative half crossing

6.3 Calculate the relevant margin

7. Calculate the stability steps

1. Calculation $w_{pc}$： Calculate the contribution phase angle of each link , Avoid using will iw Replace the open-loop transfer function for simplification .