Summary : Procedure to plot root locus

Procedure   to plot Root locus

Step: 1  Determine the branch  number of loci.

Step: 2  Plot the poles and zeros on the S plane

Step: 3 Find real axis Loci .  

Step: 4   Find the number of Asymptotes and their angles, if any.

Step: 5  Determine the centre of Asymptotes  and draw.

Step: 6    Determine the break in/ away point , if any.

Step: 7   If the locus crosses the jw axis , determine the jw  cross over.

Step: 8  Calculate the angle of departure / angle of arrival  due to complex poles or zeros if any.

Sketch the root locus for the  system . (Brief discussion)

G(s)H(s) = K(s+4)(s+5)/[(s+3)(s+1)]

Steps

step 1:   m=2 (number of zeroes) , n=2 ( number of poles)

step 2:    n-m=0    ; No asymptotes //  no step required

Step 3:   Real axis loci      [ -3  -1]   and  [ -5   -4]

Step 4:                  1+G(s)H(s)=0

                          1+ K(s+4)(s+5)/[(s+3)(s+1)]=0

                           K=  -(s+1)(s+3)/(s+4)(s+5)

                                =  - s^2+4s+3/s^2+9s+20

                  dk/ds=  -   5s^2+34s+53/[(s^2+9s+20)]^2

              dk/ds=0    

               s=-2.42  and -4.38

    Plot

Note :   Root loci originate on the poles of G(s)H(s) (for K=0)

               and terminates on the zeros of G(s)H(s) (as K=∞).

Practice Question : Root Locus

Explain the term centroid and  asymptotes?

What is break away point?

Calculate the angle of asymptotes  and centroid for the system   having                                 G(s)= K(s+2)/[s(s+1)(s+4)

Find the  centroid and breakaway point for the system having   G(s) = K/[s(s+2)(s^2+6s+25)]

For G(s)H(s) = K/[s(s+1)(s+3)], determine the coordinates of valid break-away/break in points(s)

Sketch the root locus for  (i) G(s)H(s)= K/s(s+3)(s+5) and (ii) G(s)H(s) = K/[s(s+6)(s^2+4s+13)]

Explain the effect of addition of poles and zeros to a  second order control system. (root Locus)

For G(s) H(s)= K/s(s+4) Draw loot locus and determine the  value of K if the damping ratio is 0.707 .

How will you obtain the angle of departure and angle of arrival of root locii?

State the rule for obtaining the breakaway point  in the root locii.

Root Locus : Revision some points asked by students_5_fast exercise

Root Locus : Revision some points asked by students_4: intersection of root locus on Jw axis

Root Locus : Revision some points asked by students_3: break away/in point

Steps to determine the break away point:

1. Find the Characteristics Equation 1+G(s)H(s)=0 of the system.
2. Write  K in terms of  s.
3. Derive dK/ds  and Put dK/ds =0.
4. The roots of the equation dK/ds =0 are the break away points.
5. Note: If the value of K is positive for any roots of dK/ds=0, the roots   (are) valid break away /break in points.

In this example the value of K is positive  for S=-0.785. Hence this the break away point.In the video there is error in the last minutes. so correct by this.

Remaining points are ok.

Take Care

Jitender

Root Locus : Revision some points asked by students_2: Asimptotes and Centroid

Root Locus : Revision some points asked by students_1: real axis loci

There is no real root loci in between -2 and -1 since the sum of poles and zeroes is  even.  

Link for test : 06.05.2020

https://forms.gle/VEYeVy9NmYgVmXCP8

Note . complete Q no 1  , 4 and 5 in the google form and submit.

For QNo 2 and 3  solve this in your copy , Scan and send it in the class (google class room)

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Take Care 

Jitender

Two more examples: Electrical Network

All Students are required to complete "questions for practice" uploaded on 26.04.2020 in this blog. 

I am taking your class today at 2: pm on google class.  There we will discuss the problems related to Electrical network.

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Take care

Jitender

State Space representation : Electrical Network (Example 2)

Dear Students After seeing this video join  google class room   for practice 

State Space representation : Electrical Network

Time Response of the system: State Space Analysiss

Controllability and Observability: State Space Analysis

Parallel Decomposition Method: State space analysis

Question for practice: Compensating Network

1. 1.      What do you  mean by compensator?
2. 2.      Discuss lead/Lag/Lag-lead compensator? Sketch the bode plot and polar plot for Lead/Lag/  Lag-lead compensator network.
3. 3.      Sketch the bode plot  and Pole-Zero plot of lag-lead compensator.
4. 4.      What  are limitations of lead compensation?
5. 5.      Whar are the effects of  lag-lead compensator on the performance of the system.

Question for Practice : State Space Analysis

Decomposition : Cascaded

Decomposition Techniques: Example

Decomposition Technique: Direct decomposition

How to draw block diagram for the previous example

 After seeing this video join google class  at 2: 00 pm  we will discuss further problems


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Jitender Kumar

State space representation using Phase Variable in controllable conical form

Y(s)/U(s) = 3/[S^4+2S^2+3S+2)]

Questions for practice: Bode plot, Nyquist plot, Root Locus and M and N circles

Q1.  Sketch the bode plot and find the stability of the system for the following functions

1. G(s) = 50/(s+1)(s+2)
2. G(s) = 50/[s^2(s+1)(s+2)]

Q2. Define the following terms :  (i)  Bode Plot,  (ii) Gain Marging ,(iii) Phase margin

,       (iv) Gain cross over frequency and (v) Phase cross over frequency

Q 3. What is the requirement of gain margin and phase margin for a system to be          

          stable/unstable /    marginally stable?

Q4.  What information you get from Bode's plot.

Q5.  State advantages of Bode Plot

Q6 . Determine the angle and magnitude  of  G(s)H(s) = s(s+1)/[(s+1)(s+2)  at s =j5.

Q7. Determine the open loop transfer function from the bode plot shown below

Q8:  Determine the value of K for the open loop transfer function  G(s)H(s)=   K/[s(1+0.1s)(1+s)]          

        so that (a) the gain margin is 15 db and (b) phase margin is 60 degree.

Q9. Draw the Bode plot  for the transfer function  G(s)H(s) = 48(s+10)/[s(s+20)(s^2+2.4s+16)

Q10   Sketch the Bode plot for the transfer function  G(s)H(s)= 1000/[s(1+0.1s)(1+0.001s)] and

        determine (a) Gain cross over frequency

                         (b)   Phase Cross over frequency

                          (c)   G.M and P.M.

                         (d)  stability of the given system

Root Locus:
1.	Explain the term centroid and  asymptotes?
2.	What is break away point?
3.	Calculate the angle of asymptotes  and centroid for the system   having                                 G(s)= K(s+2)/[s(s+1)(s+4)
4.	Find the  centroid and breakaway point for the system having   G(s) = K/[s(s+2)(s^2+6s+25)]
5.	For G(s)H(s) = K/[s(s+1)(s+3)], determine the coordinates of valid break-away/break in points(s)
6.	Sketch the root locus for  (i) G(s)H(s)= K/s(s+3)(s+5) and (ii) G(s)H(s) = K/[s(s+6)(s^2+4s+13)]  (Manke)
7.	Explain the effect of addition of poles and zeros to a  second order control system. (root Locus)
8.	For G(s) H(s)= K/s(s+4) Draw loot locus and determine the  value of K if the damping ratio is 0.707 . (Manke)
9.	How will you obtain the angle of departure and angle of arrival of root locii?
10.	State the rule for obtaining the breakaway point  in the root locii.
Nyquist Plot
1.	What do you mean by Nyquist Criterion?
2.	For G(s)H(s)= K/[s(s+1)(s+2)], draw the nyquist plot and hence calculate the range of values of K for stability.
3.	Using Nyquist criterion  investigate the stability  of a unity feedback control system for G(sH(s) = 20/[s(s+2)(s+3)]
4.	Draw the Nyquist plot for the system G(s)H(s)=1/[s(s-2)] and  investigate the stability
5.	Using Nyquist criterion investigate the stability of a closed loop control system whose open loop transfer function is    K/[s^3(ST+1)]
M and N Circles
1.	What are M and N circles?
2.	Write short notes on M and N circles.
Note: Students  can download their study materials from the blog:                   bhaskaracharyablogspot.com. Three   links are given to you. You can download theses reference books online also.  If any doubt regarding these practice questions   you can contact me any time. Stay home  stay safe     and Take care to  you and your family

Control System Study Materials : these materials are available online also on google

Note : These books are available on  google .You can down load directly from there.

B C Kuo

I J Nagrath

Example (solve) : Nyquist plot

 In this example  N is not zero as said in above .  plz see this video also

 All students are required to see videos before joining the class so that i can clarify your doubts in class
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Take Care

Jitender

Stability : LHP and RHP

LHP=Left Half Plane
RHP=Right Half Plane
 Dear students if you have any doubt  you are required to clear from me by any mode i.e. either by mail/ call/classroom or through this blog 

Nyquist Path : Basics 2

Nyquist Plot Basics 1

Notice: Test

Dear Students 
 A test for the paper " Control System" will be held on 13.04.2020 at 2:00 PM on google classroom .
All are required to be there. Topics: Time domain Analysis, Error constants and coefficients.

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Jitender
BCAS

Starting : bode

More Questions from Bode Plot

Dear Students while preparing  notes   solve these questions for your reference

Q1.  Sketch the bode plot and find the stability of the system for the following functions:

1. G(s) = 50/(s+1)(s+2)
2. G(s) = 50/[s^2(s+1)(s+2)]

Q2. Define the following:  (i)  Bode Plot,  (ii) Gain Marging ,(iii) Phase margin

,    (iv) Gain cross over frequency and (v) Phase cross over frequency
Q 3. What is the requirement of gain margin and phase margin for a system to be stable/unstable /            marginally stable?

Q4.  What information you get from Bode's plot.

Q5.  State advantages of Bode Plot
Q6 . Determine the angle and magnitude  of  G(s)H(s) = s(s+1)/[(s+1)(s+2)  at s =j5.
Q7. Determine the open loop transfer function from the bode plot shown below

Q8:  Determine the value of K for the open loop transfer function  G(s)H(s)= K/[s(1+0.1s)(1+s)] so that (a) the gain margin is 15 db and (b) phase margin is 60 degree.

Q9. Draw the Bode plot  for the transfer function  G(s)H(s) = 48(s+10)/[s(s+20)(s^2+2.4s+16)

Q10   Sketch the Bode plot for the transfer function  G(s)H(s)= 1000/[s(1+0.1s)(1+0.001s)]

        Determine (a) Gain cross over frequency

                         (b)   Phase Cross over frequency

                          (c)   G.M and P.M.

                         (d)  stability of the given system

Semilog graph : Rough Idea to plot Bode

This video gives rough  idea to plot magnitude vs frequency. While  reducing its size for blog , Its quality is not good. So plz listen its audio carefully and if any doubt we will discuss in today's class (i.e. on 08.04.2020 from 2 to 4 PM)
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take Care

Jitender 

Vidoe to discuss example : BCAS_JK_7:51 80042020 manke

Dear Students as per your requirement regarding explanation of the example  (above from  book manke). I am uploading this video explaining your few doubts.
 Clarification: Note: Since this is type  1 order system  ( i.e in Den there is 's')   and in this example first corner frequency is   w=2 rad/sec . We can  plote  -20 db/dec for first order sytem till upto only w=2  rad/sec and we can start the plot  any where from any value , that should be less   than w= 2 rad /sec  .       In this example the initial  slope is started from w=1.

Bode Plot (Approximation): Rough idea : Magnitude Vs frequency

Note : In this video  when i am explaining 50 /[(s+1)(s+2)] . Since   to plot bode we need corner frequency  and for this  we have to make N and D as 1+ s. here   in D there is s+2  to make it as 1+s, we devide only (S+2)  by 2 and not ( s+1) which is already in standered  form . When we devide S+2 by 2  , to balance it we will divide 50 by 2 also.
When i am saying N and D divide by 2 means i am dividing only one  D term which is S+2 by 2 and N value 50  by 2 only.

  Take Care

Regarding Assignments

Dear Students   thanks for  sending  assignments. All of  you  are required to  send   remaining assignments on my  email ID  jitenderkumar83@gmail.com instead of my personal ID before or on the given date.


Also mention    "Name -Department- Roll No Sem"  in subject 

thanks for showing interest  

  Stay home , Stay safe

with best wishes
Jitender, BCAS, Dwarka

polar Plot: bcas_JK_polar

Dear students this is  a video regarding polar plot. In the last session of this video, I have equated the real part  Re (G(jw)) = 0 to determine the frequency (here w= sqrt(2))   at which the polar plot intersects the imaginary axis and then put this value in imaginary part to  determined the value of G(jw)  (in this case it is [(4*sqrt(2))/3]at the point of intersection. This video is taken by phone so it's not so clear but i have tried  to do my best with available of limit  number of resources at home.

Hope for  your healthy life.

Stay home, stay safe

Jitender Kumar

How to find G(s) from the Bode Plot Bcas_JK_G(s)

Dear Students if you have any doubts after seeing this video , plz clarify  or ask me to edit and resend again.

best wishes

Jitender

Resonant peak conditions : asked

Conditions  for resonance

Bode Plot: Questions for practice

All students are required to do the practice questions given today and yesterday till coming sunday
If any doubt  plz clarify . You are required to plz follow your reference books.

Q1. Sketch the bode plot and find the stability of the system for the following functions:

1. G(s) = 50/(s+1)(s+2)
2. G(s) = 50/[s^2(s+1)(s+2)]

Q2. Define the following:  (i)  Bode Plot,  (ii) Gain Marging ,(iii) Phase margin

,    (iv) Gain cross over frequency and (v) Phase cros over frequency

Q3  What information you get from Bode's plot.

Q4  State advantages of Bode Plot

Wishes for  your Healthy  life.

jitender, BCAS