Contents

Signal Flow Graph. 1

Mason’s Gain Formula. 1

Canonical Form of Feedback Network. 2

Sensitivity. 2

Sensitivity of some simple transfer functions. 2

Steady State Error and coefficients. 3

Error Coefficients and Type. 3

Steady state Error and Type. 3

Frequency -Domain Parameters. 4

Time Domain Parameters. 4

Nyquist Plot 4

Nyquist Plot (NP) 4

Plotting of complex functions. 5

What is Conformal MAPPING?. 6

Polar Plot 6

Nyquist Path. 6

Some Common Nyquist Plots. 7

Signal Flow Graph

Mason’s Gain Formula

Mason's Gain Formula

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

Canonical Form of Feedback Network

Sensitivity

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

Sensitivity of some simple transfer functions

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 and coefficients

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

Error Coefficients and Type

Type	0	1	2	3	4
Unit Step
Ramp
Parabolic

Steady state Error and Type

Type	0	1	2	3	4
Unit Step
Ramp
Parabolic

Frequency -Domain Parameters

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 closed-loop transfer function.
Resonant Frequency : It is the frequency at which resonant peak occurs.
Cut-off rate : It is the frequency rate at which magnitude ratio changes after cut-off frequency
Bandwidth :
Delay time : Speed of response

Time Domain Parameters

Nyquist Plot

Nyquist Plot (NP)

1. NP Gives measure of stability of closed loop systems from open-loop 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.

Plotting of complex functions

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 .

What is Conformal MAPPING?

If the angle between the curves   and  at  is  then the angle between the curves corresponding to  and  in the  plane is also

Polar Plot

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

Nyquist Path

· A Nyquist path encloses the entire right half of the s-plane. 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 s-plane. A pole lies in the LHS s-plane. This pole doesn't lie inside Nyquist Path.
Here Nyquist path is the entire RHS s-plane with detours around the poles on -axis. A pole and a zero lies in the RHS s-plane.

Some Common Nyquist Plots

	Nyquist Plot		Nyquist Plot
		Lead Network
		Lag Network