Root Locus Technique.pptx

Roots locus

1. Root Locus Technique

2. Knowing the locations of the closed loop poles is important because, It affects the transient response . It indicates whether the system is stable or not . An important property of the control system is the variation of these locations as the controller gain changes, therefore we need to quantify these variations. In this session we introduce the idea “the root locus” which as a graphical means of quantifying the variations in pole locations (but not the zeros). Introduction

3. Introduction Consider a closed loop system with unity feedback that uses a simple controller, Figure 1 : Closed-loop control system with a variable K

4. What is Root Locus?? For the simple single loop system shown in Figure 1, we have characteristic equation, Where ‘K’ is the variable parameter. A root locus is a plot in s-plane of all possible locations of closed-loop poles with some system parameter , usually controller gain varied from 0 to ∞. All the points on the root locus satisfies Equ(1) and Equ(1) can be rewritten in polar form as, Therefore it is necessary that   (1)

5. What is Root Locus?? For the simple single loop system shown in Figure 1, we have characteristic equation, Where ‘K’ is the variable parameter. A root locus is a plot in s-plane of all possible locations of closed-loop poles with some system parameter , usually controller gain varied from 0 to ∞. All the points on the root locus satisfies Equ(1) and Equ(1) can be rewritten in polar form as, Therefore it is necessary that   (1)

6. Continued…

7. Angle and Magnitude Condition The Angle Condition: The Magnitude Condition : In practice the angle condition is used to determine whether a point s lies on the root locus, and if it does, the magnitude condition is used to determine the gain K associated with that point, since Example : Consider system with , test a point for its existence on root locus. Also find corresponding K.  

8. Definition The root locus is a plot of the roots of the characteristic equation of the closed-loop system as a function of one system parameter varies , such as the gain of the open-loop transfer function. It is a method that determines how the poles move around the s-plane as we change one control parameter. This plot was introduced by Evans in 1948 and has been developed and used extensively in control engineering.

9. Difference between Root Locus & Routh-Hurwitz method The Root Locus Method ----- tells us the position of the poles in the s-plane for each value of a control parameter. The Routh -Hurwitz Method ---- could only tell us for could only tell us for which values of the control parameter the poles would be to the left of a given vertical axis in the s - plane.

10. Plotting roots of a characteristic equation-Example 1 Let us consider a position control servomechanism system. The plant consists of servomotor and load, controlled by controller power amplifier. The open loop transfer function Open loop poles,   K   R(s) C(s) Power Amplifier Servomotor + Load

11. Continued… The closed loop transfer function is given by, The characteristic equation is, The second order system under consideration is stable for positive values of ‘K’. The relative stability of the system depends upon the location of poles Values of the poles depends upon the system parameter ‘K’.   Gain of the open loop transfer function

12. At K=0 , (which are the open loop poles of the system) As ‘K’ increases , the roots move towards each other.  the roots are real and lie on the negative real axis of the s-plane between -2 & -1 and 0 to -1 , respectively At K=1 , . As ‘K’ is increased further , the roots breaks away from real axis and becomes complex conjugate and real part remains fixed at s=-1. , the roots complex and are given by,   Continued…

13. Continued… There are two branches A-C-E and B-C-D in the plot. Number root locus branches is equal to number of closed loop poles. Each root loci starts at open loop pole value at K=0 and terminates at zero as ‘K’ tends to infinity. For each value of ‘K’ gives one closed loop pole value. Note: Each point on the locus is closed loop root, so the locus is called root locus

14. Example 2 Consider Obtain the root locus using direct method. The characteristic equation is , Roots are   K -5 1 -0.1715 -5.828 5 -0.527 -9.472 . . . . . . ∞ -1 - ∞ K -5 1 -0.1715 -5.828 5 -0.527 -9.472 . . . . . . ∞ -1 - ∞

15. Continued…

16. Consider a second order multi-loop system with CLTF construct root locus. The characteristic equation , Which can be rewritten as Open loop poles and and open loop zero z = -5 Closed loop roots ,   Example 3

17. Continued…

18. Drawbacks of drawing root locus using direct method of substitution It is very difficult to plot the root locus for higher order systems by the method of substituting different values of ‘K’ in the roots of the characteristic equation. To simplify the construction of root locus for higher order systems certain rules are specified.

19. To draw a rough draft of root locus , some construction rules are devised. Let us discuss the rules, first stating the rule, then giving justification for its validity and citing an example. Rules for construction of root locus

20. Rules for construction of root locus Rule 1 Locate open loop poles and zeros on ‘s’ plane. The values of closed loop poles & zeros will be equal to open loop poles & zeros at K=0. Rule 1: Root locus are always symmetrical about the real axis . The root of the characteristic equation are either real or complex or combination of both.

21. Rule 2: The number of branches in the root locus Let G(s)H(s) = open loop transfer function of a given closed loop system , P = no of open loop poles & N = no of open loop zeros Note: The root locus branch always starts from open loop pole and ends on open loop zero. The number of root locus branches N is equal to the number of finite open loop poles  P  or the number of finite open loop zeros  Z , whichever is greater. If P > Z , N = P , no of branches that terminate at infinity is P-Z. If Z > P , N = Z , no of branches that originates from infinity is Z-P. Rules for construction of root locus Rule 2

22. Rule 3 : A point on the real axis lies on the root locus if the sum of number of open loop poles and open loop zeros , on the real axis , to the right hand side of that point is ODD. Example1 : , find on which sections of the real axis root locus exists   Rules for construction of root locus Rule 3

23. Example2 : Find the sections of real axis which belongs to the root locus.   Rules for construction of root locus Rule 4

24. Rule 4: It is a normal occurrence to have less finite zeros than finite poles, so system will have infinite zeros. For each infinite zero, the root locus will have a branch that travels to the infinite zero along a line called Asymptote. So there will be one asymptote for one infinite zero. But the question is where will these asymptotes located and what will be there angle of orientation?? Rules for construction of root locus Rule 4

25. Rules for construction of root locus Rule 4 Each asymptote is oriented at an angle from the positive real axis. The asymptote angles are designated θ,  So if there is one infinite zero, there is one asymptote and its asymptote angle is . If there are two infinite zeros, there will be two asymptotes with angles  and .  

26. Rules for construction of root locus Rule 4 Next question is where these asymptotes are located?? All the asymptotes intersect at a point on the real axis called centroid . The coordinates of this centroid can be calculated as, Centroid is always real , it may be located on negative or positive real axis. Note : Centroid may or may not be the part of the root locus.  

27. Rules for construction of root locus Rule 4 Example1: , calculate the angles of asymptotes and the centroid. Solution: The root locus of 1+G(s)H(s)=0, will consist of four root loci (branches) starting from open loop pole with K=0. One root will terminate on open loop zero s = -2 as Other three roots will terminate at infinity as along the asymptotes radiating from the centroid .  

28. Rules for construction of root locus Rule 4 Angles of the asymptotes , Here P-Z = 3, k= 0,1,2. Angles of asymptotes, are and . Centroid,  

29. Rules for construction of root locus Rule 4 Example1: , calculate the angles of asymptotes and the centroid.  

30. Rules for construction of root locus Rule 4 Conclusion from Rule 4: The angles of the asymptotes are fixed for fixed values of P – Z. The values of P & Z may be different but for particular values of P – Z, angles of asymptotes are fixed. P-Z Number of asymptotes required Angles of asymptotes - 1 1 180 2 2 90, 180 3 3 60,180,300 4 4 45,135,225,315

31. Rules for construction of root locus Rule 5 Rule 5: Breakaway point where multiple roots occur at same point , for a particular value of K Example 1: , in this case both roots & starts from open loop values and approach to common value -1 for K=1. From this breakaway point at s=-1, the roots becomes complex conjugate pairs as K increases towards infinity from 1. Such a point where two or more roots occur for a particular value of K is called breakaway point. The root locus branches will leave this breakaway point at an angle of , where ‘n’ is number of branches approaching breakaway point.  

32. Rules for construction of root locus Rule 5 Foer this example breakaway point is at s=-1. Two root locus branches approaches the breakaway point. Branch B-C-D depart away from breakaway point at 90 degree. Branch A-C-E depart away from breakaway point at -90 degree Note: the breakaway point will always be on root locus.

33. Rules for construction of root locus Rule 5 General predictions about existence of breakaway points If there are adjacently placed poles on the real axis and the real axis between them is part of root locus then minimum one breakaway point exists between the adjacent poles Example1 :  

34. Rules for construction of root locus Rule 5 2. If there are adjacently placed zeros on the real axis and the real axis between them is part of root locus then minimum one breakaway point exists between the adjacent zeros Example 2:  

35. Rules for construction of root locus Rule 5 3. If there is a zero on the real axis and to the left of that zero there is no zero or pole existing and complete real axis to the left of that zero is part of root locus then there exist breakaway point to the left of that zero Example 3:  

36. Rules for construction of root locus Rule 5 Determination of breakaway point: Construct the characteristic equation 1+G(s)H(s)=0. Write the characteristic equation as, K = f(s). Differentiate K = f(s) equation and equate it to zero Roots of the equation gives us breakaway points for range of K from -∞ to + ∞ So the valid breakaway point is obtained by substituting the point in equation of K = f(s), if value of K is positive that breakaway point is valid & if value of K is negative that breakaway point is invalid  

37. Example 1: For , determine the valid breakaway points.  

38. Rules for construction of root locus Rule 6 Rule 6: I ntersection of root loci with imaginary axis can be determined using Routh’s criteria. Segment of root loci can exist in right half ,this indicates the stability. The point at which root loci crosses the imaginary axis give stability limits. Basic Routh’s array gives this stability limit. Substituting the value of K margin obtained in auxiliary equation we get intersection point of root loci with imaginary axis.

39. Rules for construction of root locus Rule 6 Example 1:  

40. Rules for construction of root locus Rule 6 We obtain,

41. Rule 7: Angle of departure at complex conjugate poles and angle of arrival at complex conjugate zeros. A branch always leaves from an open loop pole. If the open loop pole is complex, we calculate angle of departure The angle of departure is given as,   Rules for construction of root locus Rule 7

42. Rules for construction of root locus Rule 7 Example: , calculate angles of departure at complex conjugate poles  

43. Rule 7: Angle of departure at complex conjugate poles and angle of arrival at complex conjugate zeros. If the open loop zero is complex, we calculate angle of arrival The angle of arrival is given as,   Rules for construction of root locus Rule 7

44. Rules for construction of root locus Rule 7 Example: , calculate arrival angles at complex conjugate zeros  

45. Example : Construction Root Locus without complex poles Draw the approximate root locus diagram for a closed loop system whose loop transfer function is given by, Comment on stability.  

46. Example : Construction Root Locus without complex poles Consider the feedback system with the characteristic equation, Sketch the root locus.  

47. Example : Construction Root Locus with complex poles Sketch the root locus for the system having