Self control principle learning notes
Self control principle learning notes column
Chapter one —— Dynamic model of feedback control system
Chapter two —— Stability analysis of control system
The third chapter —— Continuous time system performance analysis
Chapter four —— Calibration and synthesis of automatic control system
The fifth chapter —— Linear discrete system

Calibration and synthesis of automatic control system

1. Basic concepts of control system synthesis and correction

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1.1 Comprehensive introduction to the system

Give a certain design index in advance , This indicator is usually given in strict mathematical form , Then determine some control form or index based on the structure to get the desired system , Then find the controller that meets the predetermined index through analysis .

Is to seek the right number of tools and put forward the right indicators .

1.2 System calibration

Develop the expected performance indicators of the system , Based on these performance indicators and uncorrected system characteristics , Determine the structure of the correction , Through the comparison and calculation of indicators, the control parameters in the correction structure are determined .

The initial state of the time response corresponds to the high frequency characteristics of the frequency response , The final state of the time response corresponds to the low-frequency characteristics of the frequency response , The transient of the time response corresponds to the crossover region of the frequency response .

1.3 Rough correspondence between time response and frequency characteristics

The initial state of the time response corresponds to the high frequency characteristics of the frequency response , The final state of the time response corresponds to the low-frequency characteristics of the frequency response , The transient of the time response corresponds to the crossover region of the frequency response .

1.4 Index type

Performance indicators mainly include time domain indicators and frequency domain indicators , It is generally difficult to accurately establish the relationship between the two , At a glance , The initial state of the time response corresponds to the high frequency characteristics of the frequency response , The last state of the time response corresponds to the low-frequency characteristics of the frequency response , The transient of time response corresponds to the intermediate frequency characteristics of frequency response .

1. Static indicators ： Steady state error

2. Time domain dynamic index ： Adjust the time 、 Peak time 、 Overshoot

3. Frequency domain dynamic index ：

​ Open loop ： Crossover frequency 、 Phase margin

​ closed loop ： Resonant frequency 、 Resonance peak 、 Bandwidth frequency

1.5 Relationship between dynamic performance indicators

2. Loop shaping

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The idea of loop shaping ：

Closed loop performance index -> Desired open-loop frequency characteristics -> controller

2.1 Controller output requirements

Low frequency band ： Open loop gain $K$, System type $v\to$ Steady state error $e_{ss}$, Hope the shape is steep , high

If band ： Cut off frequency $w_c$, Phase margin $\gamma$, Dynamic performance ：$\sigma \%,t_s$, Hope the shape is slow , wide

High frequency band ： The ability of the system to resist high-frequency interference , High frequency roll off characteristics are expected , Hope the shape is low , steep

3. Series correction

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3.1 Series lead correction

$$G_c(s)=\frac{aTs+1}{Ts+1}(a>1)$$

a Usually take 4~10, The maximum lead angle of the primary lead network is 60 degree , The phase angle is ahead , The amplitude increases

The essence ： Using the phase angle advance characteristic of the lead network to improve the phase angle margin of the system , Put the controller Frequency at the maximum lead angle Just controlled in Crossing frequency It's about , In this way, the improved phase margin is the best .

Advance correction steps $(e_{ss}^{\ast },w_c^{\ast },{\gamma }^{\ast })$: An asterisk indicates the index requirements

1. Through the given steady-state error $e_{ss}^{\ast }=\frac{sR(s)}{1+L(S)}$ To calculate the K value
2. adopt $KP(iw)$, Calculate the crossing frequency at this time $w_{c0},{\gamma }_0$, Judge , Such as $w_{c0},{\gamma }_0$ Is it not enough The fruit is not enough , Then advance correction can be adopted
3. Calculation $C(s)$ The maximum lead angle to be provided ${\phi }_m={\gamma }^{\ast }-{\gamma }_0+\left(5^{\circ }\sim 10^{\circ }\right)$,
   $\left(5^{\circ }\sim 10^{\circ }\right)$ Used to compensate for the increase of crossing frequency , Cause the original $P(s)$ The loss of phase angle reduction provided at the corresponding crossing frequency . here a According to the formula $a=\frac{1+\sin {\phi }_m}{1-\sin {\phi }_m}$ To calculate the . meanwhile $C(s)$ The amplitude frequency characteristic provided at the maximum lead angle is $10\lg a$.
4. take $|KP(i{w}_c)|=-10\lg a$, That is to say $C(s)$ Frequency at the maximum lead angle , Calculate the corrected crossing frequency $w_c$, And then through the formula $\tau =\frac{1}{\sqrt{a}{w}_c}$ To calculate the $\tau$.
5. take $C(s)$ Plug in $P(s)$ Calculate whether the phase margin and crossing frequency of the corrected system meet the requirements

effect :

1. Keep the low band ： Meet the steady-state accuracy $e_{ss}$
2. Improve the intermediate frequency band ：$w_c\uparrow ,{\gamma }_0\uparrow$, Improved dynamic performance , The signal reproduction ability is improved , Improve the rapidity of the system
3. Raise the high band ： The ability to resist high-frequency interference decreases , Raise the amplitude frequency characteristic .

3.2 PD correction

wave filtering PD correction ：
$$C(s)=K_p(1+\frac{T_{d}s}{T_{d}s/N+1})$$

Compared with the ideal PD Adjust the , wave filtering PD Regulation is an additional inertia link in its controller .

In time $K_p=K,T_d=a\tau ,T_d/N=\tau ,N=a$, So it can be solved by the series advance correction algorithm .

No filtering PD correction （ Ideal filtering PD）:
$$C(s)=K_p(1+T_{d}s)$$

3.2 Series lag correction

$$G_c(s)=\frac{bTs+1}{Ts+1}(b<1)$$

The essence ： Using the high-frequency amplitude attenuation characteristics of the lag correction device , Reduce the transient gain and ride through frequency of the system , Excavate the reserves of the phase angle of the system itself

Lag correction steps :

1. According to the given steady-state error requirements , Calculate the static gain of the controller K

2. adopt $KP(iw)$ Of Bode chart , Calculate the crossover frequency and phase margin , When the crossing frequency is enough , When the phase margin is insufficient, the lag correction is generally used .

3. According to the phase margin , Determine the crossover frequency of the system after correction $w_c^{\ast }$
   $$180^{\circ }+\angle KP\left(\mathrm{i}\omega {}_{\mathrm{c}}^{\ast }\right)\ge {\phi }_{\mathrm{m}}^{\ast }+\delta ,\delta =5\sim 20^{\circ }$$
   Because for lag correction , We are mainly Excavate the phase angle reserve of the original system , Not through the maximum hysteresis angle of the controller , And the crossing frequency should be far away from the frequency corresponding to the maximum hysteresis angle , So here The phase angle provided by the controller is omitted .

4. Find out $KP(iw)$ At the new crossing frequency dB value , This value is offset with the medium and high frequency band of the controller . That is to say
   $$20\lg b=|KP(i{w}_c^{\ast })|$$

5. Then pass the order $T=(\frac{1}{5}\sim \frac{1}{10})w_c^{\ast }$

6. Determine the correction device , And further verify .

effect ：

1. Keep the low band ： Meet the steady-state accuracy $e_{ss}$
2. Lower the intermediate frequency band ： $w_c\downarrow ,\gamma \uparrow$ Lose quickness , Improve uniformity
3. Lower the high frequency band ： The ability to resist high-frequency interference is improved

3.3 PI correction

ratio $K_p$ No longer based on the steady-state error requirements of the system , It is used to offset the amplitude of the pull down .
$$C(s)=k_p(1+\frac{1}{T_{i}s})$$

3.4 Lag lead correction

The essence ： Comprehensive utilization of lagging network attenuation , The phase angle of lead network is advanced , Transform the open-loop characteristic and frequency characteristic , Improve system performance

Calibration steps :

1. Select the gain of the controller $K_c$, Use the given steady-state error requirements
2. $KP(iw)$ Of Bode chart , Determine whether lag lead correction is required
3. Select the lower limit of the expected crossing frequency , Determine the maximum lead angle that needs to be provided , Similar to the solution of lead correction
4. Introduce lag correction , bring $KP{C}_1$ stay $w_c$ The amplitude at is 1, Calculate the relevant parameters of lag correction

Be careful ：b<a The parameter setting of is not conducive to suppressing high-frequency noise .

3.5PID correction

Similar to the previous PD and PI correction , Note that this time $K_p$ effect Consistent with standard lag lead correction , Although the design methods are different .