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# Note for Control System - CS By NEERATI SIDDHIK

• Control System - CS
• Note
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If the numerator C(s) and Denominator R(s) are written as the product of   Linear factors,then the transfer function may be written as  C(s) R(s) =K (S−Z 1 )(S−Z 2 )..........(S−Z m ) (S−P 1 )(S−P 2 )..........(S−P n )   Where K is known as the gain (or) scale factor  By studying above transfer function we get the following results  1. When the variable S has value equal to Z 1 , Z 2 , ..........Z m then the  transfer function becomes zero. Hence the values Z 1 , Z 2 , ..........Z m are  called zeros of transfer function. It is denoted by a small circle.  2. When the variable S has value equal to P 1 , P 2 , ..........P n then the  transfer function becomes Infinity. Hence the values  P 1 , P 2 , ..........P n are called Poles of transfer function. It is denoted by  a small cross.  3. The number of poles at the origin gives type of the system  Transfer function of Non-feedback Control system   Consider the Non feedback system shown below   C(s)=R(s)G(s)  C(s) R(s) = G(s)   Transfer function of Feedback Control system  Consider the Feedback system shown below  N.SIDDHIK M.Tech Assistant Professor Department of EEE

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Where R(s) = Reference input    E(s) = Error Signal    B(s) = Feedback signal    C(s) = Controlled Output Signal  G(s) = C(s) Open loop (or) Feedforward path Transfer function  E(s) H (s) = B(s) Feedback Transfer function  C(s) From the figure we have  C(s)=E(s)G(s) (1)  E(s)=R(s)-B(s) (2)  B(s)=C(s)H(s) (3)  Put equation (3) in equation (2) we get  E(s)=R(s)-C(s)H(s) (4)  Put equation (4) in equation (1) we get  C(s)=[R(s)-C(s)H(s)]G(s)  C(s)=R(s)G(s)-C(s)H(s)G(s)  C(s)+C(s)H(s)G(s)=R(s)G(s)  C(s)[1+H(s)G(s)]=R(s)G(s)  N.SIDDHIK M.Tech Assistant Professor Department of EEE

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C(s) R(s) = G(s) 1+G(s)H(s)   This  negative  feedback  is  also  referred  as  degenerative  type  feedback.if  the  output  is  fed with positive feedback, then it is known as Regenerative  feedback.the transfer function of positive feedback Control system is  C(s) R(s) = G(s) 1−G(s)H(s)   Comparison between Open loop & Closed loop control system  Open loop system  ● ● ● ● ● ● ● Simple to construct  Feedback element is absent  No change in gain  Sensitivity of the system is unity  Response will be less when compared to feedback systems  No stability problem  Accuracy depends on the calibration of input  Closed loop system   ● Complicated to Construct  ● Feedback element is present   1 ● Gain will be reduced by a factor 1+G(s)H(s)   ● Sensitivity of the system is less than unity  ● Response will be faster when compared to Non-feedback systems  ● Stability may improve (or) may harmful to the system  ● Accuracy depends on feedback        N.SIDDHIK M.Tech Assistant Professor Department of EEE

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Effect of feedback     The  feedback  is  mainly  used  to  reduce  the  difference  between  reference  input  and  controlled  output.  This  difference  is  called  error.  There  are  some  parameters  of  the  system  which  have  a  tendency  to vary  under  some  conditions.  Such  parameters  affects  the  system  performance.  Hence  it  is  important  to look on the effect of parameters in  control system due to feedback.  1. a) Effect of feedback on Parameter variation  Open Loop System: C ​ onsider the Non feedback system shown  below    C(s)=R(s)G(s)  Suppose G(s) changes to [G(s) + ΔG(s)] due to parameter  variation,where ΔG(s) is very small. This corresponds to change in output  from C (s) to [C(s) + ΔC(s)]   C (s) + ΔC(s) = R(s) [G(s) + ΔG(s)]   From the above equation we can write effect of change in output  due to the parameter variation  ΔC(s) = R(s)ΔG(s)   b) Closed Loop System:  Consider the closed loop system transfer function  C(s) R(s) = G(s) 1+G(s)H(s)   Suppose G(s) changes to [G(s) + ΔG(s)] due to parameter  variation,where ΔG(s) is very small. This corresponds to change in output  from C (s) to [C(s) + ΔC(s)]   C (s) + ΔC(s) = get  [G(s)+ΔG(s)]R(s) 1+G(s)H(s)+ΔG(s)H(s)   The term ΔG(s)H(s) is negligible as compared to 1 + G(s)H(s) s ​ o we  N.SIDDHIK M.Tech Assistant Professor Department of EEE