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- 1. Feedback Control Systems (FCS) Lecture-2 Transfer Function and stability of LTI systems Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/ 1
- 2. Transfer Function • Transfer Function is the ratio of Laplace transform of the output to the Laplace transform of the input. Considering all initial conditions to zero. u(t) If Plant u(t ) U ( S ) y(t ) y(t) and Y (S ) • Where is the Laplace operator. 2
- 3. Transfer Function • Then the transfer function G(S) of the plant is given as G( S ) U(S) Y(S ) U(S ) G(S) Y(S) 3
- 4. Why Laplace Transform? • By use of Laplace transform we can convert many common functions into algebraic function of complex variable s. • For example sin t s Or e at 2 2 1 s a • Where s is a complex variable (complex frequency) and is given as s j 4
- 5. Laplace Transform of Derivatives • Not only common function can be converted into simple algebraic expressions but calculus operations can also be converted into algebraic expressions. • For example dx(t ) sX ( S ) x(0) dt 2 d x(t ) dt 2 2 s X ( S ) x( 0) dx( 0) dt 5
- 6. Laplace Transform of Derivatives • In general d x(t ) n dt n s X (S ) s n n 1 x( 0) x n 1 ( 0) • Where x(0) is the initial condition of the system. 6
- 7. Example: RC Circuit • u is the input voltage applied at t=0 • y is the capacitor voltage • If the capacitor is not already charged then y(0)=0. 7
- 8. Laplace Transform of Integrals x(t )dt 1 X (S ) s • The time domain integral becomes division by s in frequency domain. 8
- 9. Calculation of the Transfer Function • Consider the following ODE where y(t) is input of the system and x(t) is the output. • or A d 2 x(t ) dy(t ) C dt dt 2 Ax' ' (t ) dx(t ) B dt Cy ' (t ) Bx' (t ) • Taking the Laplace transform on either sides A[ s 2 X ( s ) sx( 0) x' ( 0)] C[ sY ( s ) y( 0)] B[ sX ( s ) x( 0)] 9
- 10. Calculation of the Transfer Function A[ s 2 X ( s ) sx( 0) x' ( 0)] C[ sY ( s ) y( 0)] B[ sX ( s ) x( 0)] • Considering Initial conditions to zero in order to find the transfer function of the system As 2 X ( s ) CsY ( s ) BsX ( s ) • Rearranging the above equation As 2 X ( s ) BsX ( s ) X ( s )[ As 2 X (s) Y (s) Bs ] Cs As 2 Bs CsY ( s ) CsY ( s ) C As B 10
- 11. Example 1. Find out the transfer function of the RC network shown in figure-1. Assume that the capacitor is not initially charged. Figure-1 2. u(t) and y(t) are the input and output respectively of a system defined by following ODE. Determine the Transfer Function. Assume there is no any energy stored in the system. 6u' ' (t ) 3u(t ) y(t )dt 3 y' ' ' (t ) y(t ) 11
- 12. Transfer Function • In general • Where x is the input of the system and y is the output of the system. 12
- 13. Transfer Function • When order of the denominator polynomial is greater than the numerator polynomial the transfer function is said to be ‘proper’. • Otherwise ‘improper’ 13
- 14. Transfer Function • Transfer function helps us to check – The stability of the system – Time domain and frequency domain characteristics of the system – Response of the system for any given input 14
- 15. Stability of Control System • There are several meanings of stability, in general there are two kinds of stability definitions in control system study. – Absolute Stability – Relative Stability 15
- 16. Stability of Control System • Roots of denominator polynomial of a transfer function are called ‘poles’. • And the roots of numerator polynomials of a transfer function are called ‘zeros’. 16
- 17. Stability of Control System • Poles of the system are represented by ‘x’ and zeros of the system are represented by ‘o’. • System order is always equal to number of poles of the transfer function. • Following transfer function represents nth order plant. 17
- 18. Stability of Control System • Poles is also defined as “it is the frequency at which system becomes infinite”. Hence the name pole where field is infinite. • And zero is the frequency at which system becomes 0. 18
- 19. Stability of Control System • Poles is also defined as “it is the frequency at which system becomes infinite”. • Like a magnetic pole or black hole. 19
- 20. Relation b/w poles and zeros and frequency response of the system • The relationship between poles and zeros and the frequency response of a system comes alive with this 3D pole-zero plot. Single pole system 20
- 21. Relation b/w poles and zeros and frequency response of the system • 3D pole-zero plot – System has 1 ‘zero’ and 2 ‘poles’. 21
- 22. Relation b/w poles and zeros and frequency response of the system 22
- 23. Example • Consider the Transfer function calculated in previous slides. G( s ) X (s) Y (s) C As B the denominator polynomial is As B 0 • The only pole of the system is s B A 23
- 24. Examples • Consider the following transfer functions. – Determine • • • • i) iii) Whether the transfer function is proper or improper Poles of the system zeros of the system Order of the system G( s ) G( s ) s 3 s( s 2) ( s 3) 2 s( s 2 10) ii) iv) G( s ) G( s ) s ( s 1)(s 2)(s 3) s 2 ( s 1) s( s 10) 24
- 25. Stability of Control Systems • The poles and zeros of the system are plotted in s-plane to check the stability of the system. j LHP Recall s RHP j s-plane 25
- 26. Stability of Control Systems • If all the poles of the system lie in left half plane the system is said to be Stable. • If any of the poles lie in right half plane the system is said to be unstable. • If pole(s) lie on imaginary axis the system is said to be marginally stable. j LHP RHP s-plane 26
- 27. Stability of Control Systems • For example G( s ) C , As B if A 1, B 3 and C 10 • Then the only pole of the system lie at pole 3 j LHP RHP X -3 s-plane 27
- 28. Examples • Consider the following transfer functions. i) iii) Determine whether the transfer function is proper or improper Calculate the Poles and zeros of the system Determine the order of the system Draw the pole-zero map Determine the Stability of the system G( s ) G( s ) s 3 s( s 2) ( s 3) 2 s( s 2 10) ii) iv) G( s ) G( s ) s ( s 1)(s 2)(s 3) s 2 ( s 1) s( s 10) 28
- 29. To download this lecture visit http://imtiazhussainkalwar.weebly.com/ END OF LECTURES-2 29

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