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# Note for Control System - CS By NEERATI SIDDHIK

• Control System - CS
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Transfer function     In  modeling  a  control  system  we  need  a  function  called  Transfer  function.  The transfer function of a linear system is defined as “​The ratio  between  Laplace  Transform  of  output  to  the  Laplace  Transform  of  Input​”​ by keeping all initial conditions zero.  Consider  a  Linear  system having r(t) as an input and c(t) as output.  The behaviour of system is represented by differential equation  dn c(t) y n dtn + dn−1 c(t) y n−1 n−1 dt + ............... + y 1 dm r(t) xm dtm + dc(t) dt + y 0 c(t) =   dm−1 r(t) xm−1 m−1 dt + ............... + x1 dr(t) dt + x0 r(t)   Taking Laplace transform on both sides   [y n S n + y n−1 S n−1 + .......... + y 1 S + y 0 ]C(s) =   [xm S m + xm−1 S m−1 + .......... + x1 S + x0 ]R(s)   C(s) R(s) = xm S m +xm−1 S m−1 +..........+x1 S+x0 y n S n +y n−1 S n−1 +..........+y 1 S+y 0   By studying the above equation we get the following Conclusions  1. When the denominator polynomial of the transfer function equal  to zero, we get an equation which is known as Characteristic  equation equation of the system.  y n S n + y n−1 S n−1 + .......... + y 1 S + y 0 = 0   2. The highest power of S in the denominator of the transfer function  gives the order of the system  N.SIDDHIK M.Tech Assistant Professor Department of EEE

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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