# Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax:

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1 Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax:

2 Introduction In this lesson, you will learn the following : The definition of a root locus How to sketch root locus How to use the root locus to find the poles of a closed loop system How to use root locus to describe the changes in transient response How to use root locus to design a parameter value to meet a desired transient response 2

3 Introduction What is Root Locus? Root Locus is graphical presentation of the closed-loop poles as the parameter is varied A powerful method of analysis and design for stability and transient response Ability to provide solution for system of order higher than two A graphic representation of a system s stability Why do we need to use Root Locus? We use Root Locus to analyze the transient qualitatively 3

4 Introduction E.g. we can use Root Locus to analyze qualitatively the effect of varying gain upon percent overshoot, settling time and peak time We can also use Root Locus to check the stability of feedback control system qualitatively The control system problem: The poles of the closed-loop transfer function are more difficult to find The closed-loop poles change with changes in system gain The poles of T(s) are not immediately known without factoring the denominator, and they are a function of K 4

5 Introduction Since the system transient response and stability are dependent upon the poles of T(s), we have no knowledge of the system performance unless we factor the denominator for specific values of K The Root Locus will be used to give us a vivid picture of the poles of T(s) as K varies 5

6 Introduction Complex Numbers and Their Representation as Vectors: Any complex number, j,described in Cartesian coordinates can be graphically represented by a vector, as shown in the following figure (a) The complex numbers also can be described in the polar form with magnitude M, and angle, as M If the complex number is substituting into a complex function, F(s), another complex number will result For example, if F(s)=(s+a), then substituting the complex number s j yields F() s ( a) j which is another complex number, as shown in figure (b) 6

7 Introduction F(s)=(s+a) Since F(s) has a zero at a, we ( s 7) have alternate representation s 5 j 2 7

8 Introduction Example: Evaluation of a complex function via vectors Magnitude and angle of F(s) at any point ( s 3 j4) Solution: s 3 j4 ( 3 j41) F( 3 j4) ( 3 j4)( 3 j4 2) ( )( ) 20 M ( )

9 Defining the Root Locus Consider security camera that can automatically follow a subject (auto tracking) Can get by 2 s 10s K 9

10 Defining the Root Locus (Pole plot from Table 8.1) (Representation of the paths of the closed-loop poles as the gain is varied ) Call Root Locus! 10

11 Properties of the Root Locus The root locus technique can be used to analyze and design the effect of a loop gain upon the system s transient response and stability The closed loop transfer function for a system with a gain, K, is (For control system of slide 5) From this equation, a pole, s, exists when the characteristic polynomial in the denominator becomes zero, or 11

12 Properties of the Root Locus Here, -1 represented in polar form as Alternatively, a value of s is a closed loop pole if and Let s demonstrate the relationship: For control system of slide 9, KG() s H () s s(s10) From Table 8.1, the closed-loop poles exist at and when the gain is 5 Substituting the pole at for s and 5 for K yields If these calculation is repeated for the point 5 j5, the angle equals an odd multiple of 180 K K K K KG() s H () s 180 s(s10) ( 5 j5)(5 j5) ( )( 5045 ) 50 KG() s H () s 1 (Thus, it is a point on the root locus!) 1 (2k 1)180 K 12

13 Properties of the Root Locus That is, 5 j5 is a point on the root locus for some value of gain Now we proceed to evaluate the value of gain K KG() s H () s 1 K For a point 2 j2, the angle does not equal () () K K K K KG s H s s(s10) ( 2 j2)(8 j2) ( 8135 )( 6814 ) Example: (Can confirm from Table 8.1!) Thus, it is not a point on the root locus, or is not a closed-loop pole for any gain A part of Root Locus for Gs () T() s 1 Gs ( ) 13

14 Properties of the Root Locus For this system, the open-loop transfer function is KG() s H () s K(s3)(s4) (s1)(s2) Consider the point 2 j3 Then, the angle does not equal as follows: Thus, this point is not a point on the root locus!! (That is, this point is not a closed-loop pole for any gain) 14

15 Sketching the Root Locus The following five basic rules allow us to sketch the root locus using minimal calculations Rule 1: Number of the branches Each closed loop pole moves as the gain is varied If we define a branch as the path that one pole traverses, then there will be one branch for each closed loop pole Our first rule defines the number of branches of the root locus: The number of the branches of the root locus are equal to the number of closed loop poles Rule 2: Symmetry Physically realizable system can not have the complex coefficients K in their transfer functions Thus, we conclude: The root locus is symmetrical about the real axis 15

16 Sketching the Root Locus Rule 3: Real-axis segment Using some mathematical properties of root locus plot, we get On the real axis, for K>0 the root locus exist to the left of an odd number of real axis, finite open loop poles and/or finite open loop zeros symmetrical 16

17 Sketching the Root Locus Rule 4: Starting and ending points Consider the closed loop transfer function As K approaches to zero, the closed loop poles approaches the poles of G(s)H(s) (Refer to Slide 10!) Similarly, as K approaches to infinity, the closed loop poles approaches the zeros of G(s)H(s) (Refer to Slide 10!) We conclude that The root locus begins at the finite and infinite poles of G(s)H(s) and ends at the finite and infinite zeros of G(s)H(s) 17

18 Sketching the Root Locus In this system, the root locus begins at the poles at -1 and -2 and ends at zeros at -3 and -4 Rule 5: Behavior at infinity A function can have infinite poles and zeros Every function of s has an equal number of poles and zeros For example, consider the function This function has two poles at -2 and -3 and one zeros at -1 Because of the number of poles and zeros must be equal, this function has a zero at infinity! (Ref. G(s)=1/s: G( ) 0+ a pole at s=0) a zero at s This rule tell us the following conclusion: (Complete Root Locus) 18

19 Sketching the Root Locus a The root locus approaches straight lines as asymptotes as the locus approaches infinity Further, the equation of the asymptotes is given by the real axis intercept,, and angle,, as follows: a where k = 0, ±1, ±2, ±3 and the angle is given in radians with respect to the positive extension of the real axis Example: Sketch the root locus for the system shown in figure a finite poles finite zeros (2k 1), a #finite poles #finite zeros #finite poles #finite zeros 19

20 Sketching the Root Locus Solution: Begin by calculating the asymptotes: If the values for k continued to increase, the angels would begin repeat The number of lines obtained equals the difference between the number of finite poles and the number of finite zeros Rule 4 states that the root locus begins at the open loop poles and ends at the open loop zeros 20

21 Sketching the Root Locus For this example, there are more open loop poles than open loop zeros Thus, there must be zeros at infinity The asymptotes tell us how we get to these zeros at infinity! This figure shows the complete root locus as well as the asymptotes that were just calculated Root locus with the 3 asymptotes 21

22 Sketching the Root Locus Notice that we have made use of all the rules learned so far The real axis segment lie to the left of an odd number of poles and/or zeros The locus starts at the open-loop poles and ends at the open-loop zeros For this example, there is only one open loop finite zero and three infinite zeros Rule 5, then, tell us that the three zeros at infinity are at the ends of the asymptotes Now, consider four additional rules for Refining the Sketch Once we sketch the root locus using the five rules discussed in the previous slides, we may want to accurately locate points on the root locus as well as find their associated gain 22

23 Refining the Sketch Rule 6: Real Axis Breakaway and Break-in Points Numerous root loci appear to break away from the real axis as the system poles move from the real axis to the complex plane At other times, the loci appear to return to the real axis as a pair of complex poles becomes real This is illustrated in the figure The breakaway point is found at the maximum point, and the break-in point is found at the minimum point (Variation of gain along the real axis ) 23

24 Refining the Sketch The figure shows a root locus leaving the real axis between -1 and -2 and returning to the real axis between +3 and +5 The point where the locus leaves the real axis, 1, is called the breakaway point, and the point where the locus returns to the real axis, 2, is called break-in point At the breakaway or break-in point, the branches of the root locus form an angle of 180/n with the real axis, where n is the number of closed loop poles arriving or departing from the single breakaway or break-in point on the real axis Thus for the poles shown in the figure the branches at the breakaway point form 90 angles with the real axis We now show how to find the breakaway and the break-in points There is a simple method for finding the points at which the root locus breaks away from and breaks into real axis The first method is to maximize and minimize the gain, K, using differential calculus 24

25 Refining the Sketch For all points on the root locus, because of the KG(s)H(s)= -1 as we pointed out before, K = -1 / G(s)H(s) For points along the real axis segment of the root locus where breakaway and break-in points could exist, s Hence along the real axis, K 1 1 GsHs () () G( ) H( ) s Hence if we differentiate this equation with respect to and set the derivative equal to zero, we can find the points of maximum and minimum gain and hence breakaway and break-in points dk 0,? d 25

26 Refining the Sketch Example: Find the breakaway and break-in points for the root locus of the figure using differential calculus Solution: Using the open loop poles and zeros, we present the open loop system whose root locus is shown in figure as follows But for all points along the root locus, KG(s)H(s)=-1, andalong the real axis,. Hence s Solving for K, we find 26

27 Refining the Sketch Differentiating K with respect to and setting the derivative equal to zero yields Solving for, we find =-1.45 (break-away point) and 3.82 (break-in point), which are the breakaway and break-in points (Data Sheet for breakaway and break-in points) 27

28 Refining the Sketch Rule 7: j -Axis Crossings j The -axis crossing is a point on the root locus that separates the stable operation of the system from the unstable operation The value of at the axis crossing yields the frequency of oscillation The gain at the j-axis crossing yields the maximum positive gain for system stability To find j -axis crossing, we can use the Routh-Hurwitz criterion as follows: Forcing a row of zeros in the Routh table will yield the gain; going back one row to the even polynomial equation and solving for the roots yields the frequency at the imaginary axis crossing 28

29 Refining the Sketch Example: For the system of the following figure, find the frequency and gain, K, for which the root locus crosses the imaginary axis For what range of K is the system stable? Solution: The closed loop transfer function for the system is Simplifying some of the entries by multiplying any row by a constant, we obtain the Routh array shown in table 29

30 Refining the Sketch A complete row of zeros yields the possibility for imaginary axis roots For positive value of gain only, the s 1 row can yield a row of zeros for K>0 Thus,, and K = 9.65 Forming the even polynomial by using the s 2 row with K =9.65 We obtain, and s is found to be ±j1.59 Thus, the root locus crosses the j -axis at ±j1.59 at a gain of 9.65 We conclude that the system is stable for 0 K

31 Refining the Sketch Rule 8: Angles of Departure and Arrival In order to sketch the root locus more accurately, we want to calculate the root locus departure angle from the complex poles and the arrival angle to the complex zeros If we assume a point on the root locus close to the point, we assume that all angles drawn from all other poles and zeros are drawn directly to the pole that is near the point Thus, the only unknown angle in the sum is the angle drawn from the pole that is close We can solve for this unknown angle, which is also the angle of departure from this complex pole Hence from the figure(a) (Angles of Departure) 31

32 Refining the Sketch (Angles of Departure) If we assume a point on the root locus close to a complex zero, the sum of angles drawn from all finite poles and zeros to this point is an odd multiple of 180 ( ) Except for the zero that is close to the point, we can assume all angles drawn from all other poles and zeros are drawn directly to the zero that is near the point 32

33 Refining the Sketch Thus, the only unknown angle in the sum is the angle drawn from the zeros that is close We can solve for this unknown angle, which is also the angle of arrival to this complex zero Hence from the figure(b) (Angles of Arrival) 33

34 Refining the Sketch Example: Given the unity feedback system of following figure, find the angle of departure from the complex poles and sketch the root locus Solution: Using the poles and zeros of G(s)= (s+2)/[(s+3)(s 2 +2s+2)] as plotted in the figure, we calculate the sum of angles drawn to a point close to the complex pole, -1+j1, in the second quadrant Thus, 34

35 Refining the Sketch 35

36 Refining the Sketch Rule 9: Plotting and calibration the Root locus We might want to know the exact coordinates of the root locus as it crosses the radial line representing an overshoot value Overshoot value can be represented by a radial line on the s- plane We learnt in Chapter 4, the value of zeta on the s-plane is cos Given percent overshoot value, we can calculate the zeta value ln(%os /100) 2 2 ln (%OS /100) Radial line representing overshoot value on the s-plane 36

37 Refining the Sketch Example: Draw a radial line on an s-plane that represents 20% overshoot Solution: Overshoot is represented by zeta (damping ratio) on the S-plane So, the first step is to find the value of zeta OS OS ln % /100 ln(20 /100) ln 2 % /100 2 ln 2 20 /100 Next step is to find the angle of the radial line cos cos 1 cos j 37

38 Refining the Sketch The point where our root locus intersect with the n percent overshoot radial line is the point when the gain value produces a transient response with n percent overshoot Our root locus intersect with the radial line. Meaning the gain at the intersection produces transient response with zeta = 0.45 A point on the radial line is on the root locus if the angular sum (zero angle pole angles) in reference to the point on the radial line add up to an odd multiple of 180, Odd multiple of 180 : (2k+1)180, k = 1, 2, 3, 4,. 180, 540, 900, 1260, 38

39 Refining the Sketch Then, we must then calculate the value of gain Refer to the previous root locus that we have calculated, we will find the exact coordinate as it crosses the radial line representing 20% overshoot

40 Refining the Sketch We can find the point on the radial line that crosses the root locus by selecting a point with radius value then add the angles of the zeros and subtract the angles of the poles Theory : Odd multiple of , 540, 900, 1260 point on the radial line is on the locus At a point s, 40

41 Refining the Sketch Example: Sketch the root locus for the system shown in figure and find the breakaway point on the real axis and the range of K within which the system is stable Solution: The root locus of the system is shown in following figure The real axis segment is found to be between -2 and -4, using Rule 3 The root locus starts at the open loop poles and ends at the open loop zeros, using Rule 4 The root locus crosses the -axis at ±j3.9, using Rule 7 j 41

42 Refining the Sketch The exact point and gain where the locus crosses the 0.45 damping ratio line: K=0.47 at , using Rule 9 To find breakaway point use the root locus algorithm to search the real axis between -2 and -4 for the point that yields maximum gain, using Rule 6 Naturally all points will have the sum of their angles equal to an odd multiple of 180 A maximum gain of is found to at the point Thus, the breakaway point is between the open loop poles on the real axis at Finally, the system is stable for K between 0 and 15 42

43 Root Locus Plot using matlab Example: Plot the root-locus of the following system Gs () 2 2 s s s s 2 ( s 2)( s 4) s 6s % matlab script 5 4 Root Locus clear all num = [1-4 20]; den = [1 6 8]; G = tf(num,den); rlocus(g); axis('equal'); Real Axis (seconds -1 ) 43

44 Homework Assignment #8 44

45 Homework Assignment #8 45

46 Homework Assignment #8 46

47 Homework Assignment #8 47

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