Principle of control system based on feedback rate

Preface

Feedback control （feedback control） It is a very common control method , The basic idea is to use the sensor to multiply the actual value of the controlled quantity by the gain or not （ Unit feedback ） As feedback and reference input R(s) After subtraction, the deviation signal is obtained , The signal passes through the controller C(s)（controller） Then it is input into the original system as a control signal , The design of the controller is to make the output Y(s) Achieve the purpose of tracking reference input .

The basic form of feedback controller

The figure above shows two degrees of freedom （2-DOF） Basic form of controller .
Output function ：

Control input function ：control input


N:Numerator, molecular D:Denominator, The denominator
The relationship between various functions ：

System error

Steady state error of final value convergence ：

If it is a sinusoidal signal input ：

system stability

Internal stability of the system :
Characteristic equation DpDc+NpNc=0 All roots have negative real parts , And there is no unstable zero pole cancellation .

Routh criterion for system stability
Let's not talk about the calculation of Routh array , The basis for judging the stability of the system is that the number of numbers with positive real parts in the first column of the Routh array is equal to the number of sign changes .

Nai's criterion for system stability
Characteristic polynomial of open-loop system ：DgDh
If the open-loop characteristic polynomial of the system has p Zero point , At the origin q Zero point , So for the open loop Nai diagram of the system , When w from 0 When it comes to positive infinity , Its opposite （-1,j0） The angular variation of is （ppi+q(pi)/2） when , The system is stable after closed loop , Otherwise, it will be unstable after closed loop .

The relative stability of the system
There are differences between the nominal system model and the actual system , So you can't just keep the system stable , Leave a certain margin .
The traditional quantitative standards are amplitude margin and phase margin .
Better quantitative criteria are sensitivity peak and robust stability .

Significance of amplitude margin ：
Open loop transmittal L(s) The change of static gain will make the closed loop unstable , Therefore, a certain amplitude margin is required .
Significance of phase margin ：
Open loop transmittal L(s) The change of pure phase will make the closed loop unstable , Therefore, a certain phase margin is required .

Significance of sensitivity peak ：
Sensitivity peak frequency wp The meaning of is below this frequency , Naishi Tuli （-1,j0） The distance is the smallest .
|1+L(jw)| The smaller it is , The greater the sensitivity peak , The more unstable the system .
Therefore, the system may have large amplitude margin and phase margin , But it is still unstable .

The significance of robust stability ：
Because there is an error between the actual model and the nominal model , And the higher the frequency, the greater the error , The condition for the stability of the feedback system is ：

So robust stability is required T(s) In the high frequency band, the amplitude should be small .
On the other hand , If the modeling error is large , stay T(s) Very small ,S(s) It's big （ near 1） Under the circumstances , Closed loop feedback is almost ineffective .

Steady state error and system order

Analysis of the relationship between system performance and sensitivity function

1. Reduce noise interference ：
T(jw) for 0, But at the same time, it is impossible to track the input , Fortunately, the noise is generally high frequency , Input general low frequency , So make T(jw) As small as possible at high frequencies , Input the valid frequency band as large as possible ,S(jw)=1-T(jw) Will be small , This will also reduce the impact of interference .
2. Reduce interference
hope S(jw)=0, That is to say T(jw)=1, In the lower frequency band .
3. Tracking input
hope T(jw)=1,（w At low frequencies ）.
4. Reduce the influence of system modeling error （ Good robustness ）
hope T(jw) As small as possible , At high frequencies .