1. Shroff S.R. Rotary Institute of Chemical Technology Chemical Engineering Instrumentation and process Control

2. Root Locus for Stability Analysis  Group Member:- 1. Jayesh Kikani – 130990105011 2. Bhavik Mahant – 130990105012 3. Dhaval Mangukiya – 130990105013 4. Durgesh Mehta – 130990105015  Guided By:- Mr. Akshaysinh Magodara

3. Outline of Presentation  Basic Need  Introduction  Concept  Advantages & Disadvantages  Method  Examples  References

4. What do you mean by Stability for Linear System?  A control system is defined to be stable if the bounded input which is a function of time and has a definate value within fixed bound produces the bounded outputs.  A control system is defind to be unstable if it produces the unbounded or unlimited output response for every bounded input.  Bounded inputs are step and sinusoidal function.

5. Stability criteria for Linear system  The locations of the roots of the characteristic equation on the complex plane determine the stability of the control system. The criteria are given below:- 1.If the roots of the characteristic equation of a control system are in the right half of the complex plane, the control system is unstable and the roots have positive real parts.

6. 2. If the roots of the characteristic equation of a control system are in the left half of the complex plane the control system is stable and the roots have negative real parts. 3. If the roots of the characteristic equation of a control system are on the imaginary axis of the complex plane the system is just unstable.

7. Introduction to Root Locus Diagram  The root–locus diagram is used to test the stability of a control system.  It is a graphical method to determine the actual values of the roots for different values of gain or proportional sensitivity of the controller for the characteristic equation of the system.  It is used to study the transient response of the system subjected to the forcing function.

8.  The root-locus was introduced by W. R. Evans in 1948.  This method provides a clear indication of the effect of the necessary parameter adjustment with relatively small effort compared with other methods.  The important advantage of this method is that the roots of the characteristic equation of the system can be obtained directly.

9. Concept R(s) + C(s) -  The transfer function of forward path is G(s) = K G’(s) where K=Gain of the system K G’(s) H(s)

10.  The characteristic equation of the system is given by 1 + G(s) H(s) = 0 Or 1 + KG’(s) H(s)=0  The roots of the characteristic equation are dependent of upon the value of K which is varied from -∞ to +∞.  The root locus is obtained by joining all the such locations or root locus can be defined as the path of the closed-loop poles traced out in the complex plane.  If K is varied from -∞ to 0, the plot is called Direct root locus and if it is varied from 0 to +∞, the plot is called the inverse root locus.

11. What is a need of Root Locus method???  The Routh’s criteria is an algebric method and can be used for polynomial equation and it does not give the character of the response for different values of the gain of the controller.  Whereas the root locus method is a graphical technique which gives the values of the roots of the characteristic equation as one of the parameters the gain of the controller is continuously changing.

12. Advantages  The root locus can be used to describe qualitatively the performance of a system as various parameters are changed.  It gives a graphical representation of a system’s stability. Ranges of stability, ranges of instability, and the conditions that cause a system break into oscillation can also be seen from the root locus.  The general shape of the root locus gives an idea of the type of controller or compensator needed to meet the particular design criterion.

13. Disadvantages  Root locus is unable to deal with more than one variable at a time, and the difficulty with time delays.

14. Method of Plotting the Root Locus diagram 1)Determine the open-loop transfer function of the closed loop control system if it is G(s) which is the product of all the transfer function in the entire control system. 2)Determine the characteristic equation 1 + G(s) = 0 of the control system. 3) Determine the open-loop pole and open-loop zero from the open loop transfer function. The poles are indicated by cross X mark and zeros are indicated by 0 on the complex plane.

15.  The standard form of the open-loop transfer function can be given as, where, z 1,z 2,...z n is the open loop zero and p 1, p 2...,p n is the open loop poles.  Number of open loop zeros = a  Number of open loop poles = b  The value of s for which the open-loop transfer function becomes infinite known as open loop pole.  The value of s for which the open-loop transfer function becomes zero is known as open loop zero.

16. 4) The number of open-loop poles equal to the number of branches or loci. 5) Each branch begins at open loop pole and terminates at infinity. 6) If the sum of real poles and real zeroes is odd number then the real axis is the part of root locus diagram.

17. 7) Asymptotes or Asymptotic lines:- If the number of finites zeros(a), is less than the number of finite poles(b), then (b-a) branches of the root locus must end at zeros at infinity, travel along the straight lines called asymptotes or asymptotic lines of the root locus.  Center of Gravity or Centroid:- = b - a

18. 8) Angles of Asymptotes:- The angles made by asymptotic lines with the real axis is given by the following equation. where, n = 0,1,2,...,b-a-1 9) Determination of Break point:- Break away point:- Break away point is defined as the point of intersection of two branches radiating from the adjacent poles on the real axis and then leave the real axis following the asymptotic lines.

19. Break in point:- The break-in-point is defined as the point of intersection of two branches moving towards the adjacent zeros on the real axis and then enter the real axis.  Breakaway and break-in-point can be determined by the equation.

20. Important characteristics:- 1)If there are two adjacently placed poles on the real axis and the section of the real axis in between them is a part of the root locus, then there exists minimum one breakaway point in between adjacently placed poles. 2) If there are two adjacently placed zeros on the real axis and the section of the real axis in between them is a part of the root locus, then there exists minimum one break away point in between adjacently placed zeros.

21. 3) If there is a zero on the real axis and to the left of that zero there is no pole or zero existing on the real axis, and complete real axis to the left of this zero is a part of this root locus, then there exists minimum one breakaway point to the left of that zero. Break angles The root locus branches must approach or leave the break point on the real axis at an angle of +/- ( 180°/r), where r is the number of branches approaching or leaving the break point.

22. 10) Angle of Departure at a complex pole:- It is desired to determination the direction in which the locus leaves a complex pole or enters a complex zero. This information is needed to know moves towards the real axis or extends towards the asymptotes.  The angle at which a root locus branch leaves a complex pole in the s-plane is called the angle of departure. It is denoted by ɸ d. ɸ d = 180° - ɸ where ɸ = Σ ɸ p -Σ ɸ z

23. 11)Angle of arrival at a complex zero:- ɸ d = 180° - sum of all the angles subtended by phasors drawn to this zero from other zero + sum of the angles subtended by the phasors drawn to this zero from all the poles  The angle of departure and the angle of arrival need to be calculated only when there are complex poles and zeros. The angle of departure from a real open- loop pole and the angle of arrival of a real open-loop zero is always equal to 0° or 180°.

24. 12) Imaginary axis crossing point:- If the locus crosses the jω axis, the point of intersection of the root locus with the jω axis and the critical value of K can be determined by applying Routh-Hurwitz criterion to the characteristic equation of the system.  This allows knowing the stable operating condition of the system.

25. 13) Value of Gain margin from Root locus:- The ratio of the value of K at the point of intersection of the root locus with the jω axis to the design value of K of the system is called the value of the gain margin for the root locus of the system.  If the root locus does not intersect the jω axis, the gain margin is infinite.

26. Example  Sketch the Root Locus diagram. kckc H = 1 R (s) C (s) + - + +

27.  Let, Here, z 1 = 0  a=0, b = 3  p 1 = 0, p 2 = -2, p 3 =-4  Number of asymptotes:- b-a = 3  Angle:-

29. Characteristic equation:- s(s 2 + 6s + 8) + k c = 0 s 3 + 6s 2 +8s + k c = 0 18 6kckc kckc

30. -4 -2 0 - 0.8 4 k c =0 k c = 48

31. 2] The open loop transfer function of a control system is given as,  Compare above equation with,  Here K = 0.5 Kc, P 1 = 0, P 2 = -1, P 3 = -0.5 b = 3 Z 1 = -2 a = 1

32.  No. of Asymptptes:- b-a = 3-1 = 2  Centre of Gravity:- C.G. = (-1 -0.5 + 2)/ 2 C.G. = (0.5)/2 C.G. = 0.25

33.  Angle:-  For, n=0 θ = (π/2)  For, n=1 θ = (3π/2)  Break Away point:-

34.  2 s 3 + 7.5 s 2 + 6s + 1 = 0 Solving the equation, s = -0.2274 By using above root we can find the another two roots, which are -2.7117 & -0.8109.  Here, point -0.2274 is the break away point.  Using Routh criterion the value of gain kc of the controller can be determined for which the system becomes just stable.

35. The characteristic equation is given as, 1 + G(s) = 0 2s 3 + 3s 2 + s(1+kc) + 2kc = 0 By, generating the Routh array,

36. Row r 1 2 (1 + kc) r 2 3 2kc r 3 A 1 A 2 r 4 B 1 B 2 Where, A 1 = {[3(1+kc)/3](2kc))} / 3 A 1 = (3 – kc) / 3 A 2 = 0, same as B 1 = 2kc B 2 = 0

37. Row r 1 2 (1 + kc) m - 1 r 2 3 2kc m r 3 (3 – kc) / 3 0 m+1 r 4 2kc 0  System become just unstable when coefficients of m th will become zero.  (3 – kc) / 3 = 0 hence, kc = 3

38.  Location on imaginary line, A s 2 + B = 0 A = 3 and B = 6 3 s 2 + 6 = 0 s = +/- j root(2) s = +1.414 j or s = -1.414 j

39. 2j 1.414j 1j -2 -1 -0.5 1 2 -1j -1.414j -2j Asymptotic line unstable -0.2274

40. References  Industrial Instrumentation & process control, By R. P. Vyas  Process Systems Analysis & Control, By McGraw Hill Publication  Control System, By Hussain & Hussain URL:- www.sciencedirect.com