Contents
Canonical Form of Feedback Network
Sensitivity of some simple transfer functions
Steady State Error and coefficients
Mason's Gain Formula 

(sum of gain product of all individual loops)
(sum of gain product of all possible combination of 2 nontouching loops)
(sum of gain product of all possible combination of 3 nontouching loops)
(sum of gain product of all possible combination of 4 nontouching loops) …..
(sum of gain product of all individual loops not touching the forward path)
(sum of gain product of all possible combination of 2 nontouching loops not touching the forward path)
(sum of gain product of all possible combination of 3 nontouching loops not touching the forward path) …..
Gain of the forward path 




Sensitivity of a system with transfer function where is the parameter to which the system is sensitive is given by 

is generally complex so it can be represented as 

Sensitivity w.r.t to magnitude and phase 

Sensitivity overall 





Unity feedback System: Here closed loop transfer function can be expressed as a function of open loop transfer function.
Sensitivity of w.r.t magnitude of is given by:

This shows that a closed system is stable =1, even as open loop gain is infinity. 
Steady State Error 

This is direct application of final value theorem.

Position Error Coefficient 


Velocity Error Coefficient 


Acceleration Error Coefficient 


The type of open loop transfer function is important in determining error due to various signals
Type 
0 
1 
2 
3 
4 
Unit Step 





Ramp 





Parabolic 





Type 
0 
1 
2 
3 
4 
Unit Step 





Ramp 





Parabolic 





Gain Margin : Reciprocal of gain at phase crossover frequency.
Phase crossover frequency is the frequency at which phase is 

Phase Margin : Amount of phase that can be added at gain crossover frequency.
Grain crossover frequency is the frequency at which gain is unity 
Phase Margin = 
Resonant Peak : It is the maximum value of the magnitude of the closedloop transfer function. 

Resonant Frequency : It is the frequency at which resonant peak occurs. 

Cutoff rate : It is the frequency rate at which magnitude ratio changes after cutoff frequency 

Bandwidth : 

Delay time : Speed of response 

1. NP Gives measure of stability of closed loop systems from openloop transfer function.
2. NP is an extension of Polar plot.
3. Nyquist Plot is the mapping of Nyquist Path into function plane.
1. In control system analysis Nyquist Plot is drawn is for the function
2. is the denominator of the closed loop transfer function(CLTF) of a system.
3. The roots(or zeros) of determines the stability of CLTF.
4. For stability of CLTF, the roots of must not lie in the right half s plane.
If P is the number of poles of GH(s) inside the Nyquist Path and N is the number of encirclement of point by the Nyquist Plot then the CLTF is stable if is zero where is the number of zeros of that lie inside the Nyquist Path.
Mapping of into P(s) plane 

Plotting of a function of 

Analyticity of : is analytic at if is differentiable at 
If is analytic in the entire s plane then is an entire function.
For e.g. is an entire function whereas is not an entire function because it is not analytic at . 
If the angle between the curves and at is then the angle between the curves corresponding to and in the plane is also


when plotted for gives Polar plot. It is a special case of complex plot where .


· A Nyquist path encloses the entire right half of the splane. If any pole(s) or zero(s) lie on the axis then a detour is taken so as to exclude them from the path as shown in the figure below .
· Path components 2, 3 and 4 are common to any Nyquist Path but the number of appearances of path component 1 depends on the number of poles or zeros on the axis. · Nyquist Path is mapped into plane to obtain Nyquist Plot.



Nyquist Path 
Here Nyquist path is the entire RHS splane. A pole lies in the LHS splane.
This pole doesn't lie inside Nyquist Path.


Here Nyquist path is the entire RHS splane with detours around the poles on axis.
A pole and a zero lies in the RHS splane.



Nyquist Plot 

Nyquist Plot 










Lead Network 



Lag Network 
