# Software Engineering 3DX3. Slides 8: Root Locus Techniques

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1 Software Engineering 3DX3 Slides 8: Root Locus Techniques Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on Control Systems Engineering by N. Nise. c 2006, 2007 R.J. Leduc 1

2 Introduction The root locus technique shows graphically how the closed-loop poles change as a system parameter is varied. Used to analyze and design systems for stability and transient response. Shows graphically the effect of varying the gain on things like percent overshoot, and settling time. Also shows graphically how stable a system is; shows ranges of stability, instability, and when system will start oscillating. c 2006, 2007 R.J. Leduc 2

3 The Control System Problem The poles of the open-loop transfer function are typically easy to find and do not depend on the gain, K. It is thus easy to determine stability and transient response for an open-loop system. Let G(s) = N G(s) D G (s) and H(s) = N H(s) D H (s). Figure 8.1. c 2006, 2007 R.J. Leduc 3

4 The Control System Problem - II Our closed transfer function is thus K N G(s) D T (s) = G (s) 1 + K N G(s) N H (s) D G (s) D H (s) (1) = KN G (s)d H (s) D G (s)d H (s) + KN G (s)n H (s) We thus see that we have to factor the denominator of T (s) to find the closed-loop poles, and they will be a function of K. (2) Figure 8.1(b). c 2006, 2007 R.J. Leduc 4

5 The Control System Problem - III For example, if G(s) = s + 1 s(s + 2) closed-loop transfer function is: and H(s) = s + 3 s + 4, our T (s) = K(s + 1)(s + 4) s(s + 2)(s + 4) + K(s + 1)(s + 3) (3) = K(s + 1)(s + 4) s 3 + (6 + K)s 2 + (8 + 4K)s + 3K (4) To find the poles, we would have to factor the polynomial for a specific value of K. The root-locus will give us a picture of how the poles will vary with K. c 2006, 2007 R.J. Leduc 5

6 Vector Representation of Complex Numbers Any complex number, σ + jω, can be represented as a vector. It can be represented in polar form with magnitude M, and an angle θ, as M θ. If F (s) is a complex function, setting s = σ + jω produces a complex number. For F (s) = (s + a), we would get (σ + a) + jω. Figure 8.2. c 2006, 2007 R.J. Leduc 6

7 Vector Representation of Complex Numbers - II If we note that function F (s) = (s + a) has a zero at s = a, we can alternately represent F (σ + jω) as originating at s = a, and terminating at σ + jω. To multiply and divide the polar form complex numbers, z 1 = M 1 θ 1 and z 2 = M 2 θ 2, we get z 1 z 2 = M 1 M 2 (θ 1 + θ 2 ) z 1 z 2 = M 1 M 2 (θ 1 θ 2 ) (5) c 2006, 2007 R.J. Leduc Figure

8 Polar Form and Transfer Functions For a transfer function, we have: G(s) = (s + z 1) (s + z m ) m (s + p 1 ) (s + p n ) = i=1 (s + z i) n i=1 (s + p i) = M G θ G where and M G = (6) m i=1 (s + z i) n i=1 (s + p i) = m i=1 M z i n i=1 M p i (7) θ G = Σzero angles Σpole angles (8) = Σ m i=1 (s + z i ) Σ n j=1 (s + p j ) (9) c 2006, 2007 R.J. Leduc 8

9 Polar Form and Transfer Functions eg. Use Equation 6 to evaluate F (s) = (s + 1) s(s + 2) at s = 3 + j4. Figure 8.3. c 2006, 2007 R.J. Leduc 9

10 Root Locus Introduction System below can automatically track subject wearing infrared sensors. Solving for the poles using the quadratic equation, we can create the table below for different values of K. Table 8.1. c 2006, 2007 R.J. Leduc Figure:

11 Root Locus Introduction - II We can plot the poles from Table 8.1. labelled by their corresponding gain. Table 8.1. Figure: 8.5 c 2006, 2007 R.J. Leduc 11

12 Root Locus Introduction - III We can go a step further, and replace the individual poles with their paths. We refer to this graphical representation of the path of the poles as we vary the gain, as the root locus. We will focus our discussion on K 0. For pole σ D + jω D, T s = 4 σ D, T p = π ω D, and ζ = σ D ω n. c 2006, 2007 R.J. Leduc Figure

13 Root Locus Properties For second-order systems, we can easily factor a system and draw the root locus. We do not want to have to factor for higher-order systems (5th, 10th etc.) for multiple values of K!. We will develop properties of the root locus that will allow us to rapidly sketch the root locus of higher-order systems. Consider the closed-loop transfer function below: A pole exists when T (s) = KG(s) 1 + KG(s)H(s) KG(s)H(s) = 1 = 1 (2k + 1)180 o k = 0, ±1, ±2,... (10) c 2006, 2007 R.J. Leduc 13

14 Root Locus Properties - II Equation 10 is equivalent to and KG(s)H(s) = 1 (11) KG(s)H(s) = (2k + 1)180 o (12) Equation 12 says that any s that makes the angle of KG(s)H(s) be an odd multiple of 180 o is a pole for some value of K. Given s above, the value of K that s is a pole of T (s) for is found from Equation 11 as follows: K = 1 G(s) H(s) (13) c 2006, 2007 R.J. Leduc 14

15 Root Locus Properties eg. For system below, consider s = 2 + j3 and s = 2 + j( 2/2). Figures 8.6 and 8.7. c 2006, 2007 R.J. Leduc 15

16 Sketching Root Locus Now give a set of rules so that we can quickly sketch a root locus, and then we can calculate exactly just those points of particular interest. 1. Number of branches: a branch is the path a single pole traverses. The number of branches thus equals the number of poles. 2. Symmetry: As complex poles occur in conjugate pairs, a root locus must be symmetric about the real axis. c 2006, 2007 R.J. Leduc 16 Figure 8.5.

17 Sketching Root Locus - II 3. Real-axis segments: For K > 0, the root locus only exists on the real axis to the left of an odd number of finite open-loop poles and/or zeros, that are also on the real axis. Why? By Equation 12, the angles must add up to an odd multiple of 180. A complex conjugate pair of open-loop zeros or poles will contribute zero to this angle. An open-loop pole or zero on the real axis, but to the left of the respective point, contributes zero to the angle. The number must be odd, so they add to an odd multiple of 180, not an even one. Figure 8.8. c 2006, 2007 R.J. Leduc 17

18 Sketching Root Locus - III 4. Starting and ending points: 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). Why? Consider the transfer function below T (s) = KN G (s)d H (s) D G (s)d H (s) + KN G (s)n H (s) The root locus begins at zero gain, thus for small K, our denominator is D G (s)d H (s) + ɛ (14) The root locus ends as K approaches infinity, thus our denominator becomes ɛ + KN G (s)n H (s) c 2006, 2007 R.J. Leduc 18

19 Infinite Poles and Zeros Consider the open-loop transfer function below K KG(s)H(s) = s(s + 1)(s + 2) (15) From point 4, we would expect our three poles to terminate at three zeros, but there are no finite zeros. A function can have an infinite zero if the function approaches zero as s approaches infinity. ie. G(s) = 1 s. A function can have an infinite pole if the function approaches infinity as s approaches infinity. ie. G(s) = s. When we include infinite poles and zeros, every function has an equal number of poles and zeros lim KG(s)H(s) = lim s s K s(s + 1)(s + 2) K s s s How do we locate where these zeros at infinity are so we can terminate our root locus? (16) c 2006, 2007 R.J. Leduc 19

20 Sketching Root Locus - IV 5. Behavior at Infinity: As the locus approaches infinity, it approaches straight lines as asymptotes. The asymptotes intersect the real-axis at σ a, and depart at angles θ a, as follows: Σfinite poles Σfinite zeros σ a = #finite poles #finite zeros (2k + 1)π θ a = #finite poles #finite zeros (17) (18) where k = 0, ±1, ±2, ±3, and the angle is in radians relative to the positive real axis. c 2006, 2007 R.J. Leduc 20

21 Sketching Root Locus eg. 1 Sketch the root locus for system below. Figure c 2006, 2007 R.J. Leduc 21

22 Real-axis Breakaway and Break-in Points Consider root locus below. We want to be able to calculate at what points on the real axis does the locus leave the real-axis (breakaway point), and at what point we return to the real-axis (break-in point). At breakaway/break-in points, the branches form an angle of 180 o /n with the real axis where n is number of poles converging on the point. c 2006, 2007 R.J. Leduc Figure

23 Real-axis Breakaway and Break-in Points - II Breakaway points occur at maximums in the gain for that part of the real-axis. Break-in points occur at minimums in the gain for that part of the real-axis. We can thus determine the breakaway and break-in points by setting s = σ, and setting the derivative of equation below equal to zero: 1 K = (19) G(σ)H(σ) c 2006, 2007 R.J. Leduc 23 Figure 8.13.

24 The jω-axis Crossings For systems like the one below, finding the jω-axis crossing is important as it is the value of the gain where the system goes from stable to unstable. Can use the Routh-Hurwitz criteria to find crossing: 1. Force a row of zeros to get gain 2. Determine polynomial for row above to get ω, the frequency of oscillation. Figure c 2006, 2007 R.J. Leduc 24

25 The jω-axis Crossing eg. For system below, find the frequency and gain for which the system crosses the jω-axis. Figures 8.11 and c 2006, 2007 R.J. Leduc 25

26 Angles of Departure and Arrival We can refine our sketch by determining at what angles we depart from complex poles, and arrive at complex zeros. Net angle from all open-loop poles and zeros to a point on root access must satisfy: Σzero angles Σpole angles = (2k + 1)180 o (20) To find angle θ 1, we choose a point ɛ on root locus near complex pole, and assume all angles except θ 1 are to the complex pole instead of ɛ. Can then use Equation 20 to solve for θ 1. c 2006, 2007 R.J. Leduc 26 Figure 8.15.

27 Angles of Departure and Arrival - II For example in Figure 8.15a, we can solve for θ 1 in equation below: θ 2 + θ 3 + θ 6 (θ 1 + θ 4 + θ 5 ) = (2k + 1)180 o (21) Similar approach can be used to find angle of arrival of complex zero in figure below. Simply solve for θ 2 in Equation 21. c 2006, 2007 R.J. Leduc Figure

28 Angles of Departure and Arrival eg. Find angle of departure for complex poles, and sketch root locus. Figures 8.16 and c 2006, 2007 R.J. Leduc 28

29 Plotting and Calibrating Root Locus Once sketched, we may wish to accurately locate certain points and their associated gain. For example, we may wish to determine the exact point the locus crosses the 0.45 damping ratio line in figure below. From Figure 4.17, we see that cos(θ) = adj hyp = ζω n = ζ. ω n We then use computer program to try sample radiuses, calculate the value of s at that point, and then test if point satisfies angle requirement. c 2006, 2007 R.J. Leduc Figures 4.17 and

30 Plotting and Calibrating Root Locus - II Once we have found our point we can use the equation below to solve for the required gain, K. 1 m K = G(s) H(s) = i=1 M p i n i=1 M (22) z i Uses labels in Figure 8.18, we would have for our example: K = ACDE B (23) Figures 4.17 and c 2006, 2007 R.J. Leduc 30

31 Transient Response Design via Gain Adjustment We want to be able to apply our transient response parameters and equations for second-order underdamped systems to our root locuses. These are only accurate for second-order systems with no finite zeros, or systems that can be approximated by them. In order that we can approximate higher-order systems as second-order systems, the higher-order closed-loop poles must be more than five times farther to the left than the two dominant poles. In order to approximate systems with zeros, the following must be true: 1. The closed-loop zeros near the two dominant closed-loop poles must be nearly canceled by higher-order poles near them. 2. Closed-loop zeros not cancelled, must be far away from the two dominant closed-loop poles. c 2006, 2007 R.J. Leduc 31

32 Transient Response Design via Gain Adjustment - II Let G(s) = N G(s) D G (s) and H(s) = N H(s) D H (s). We saw earlier, that our closed-loop transfer equals: T (s) = KN G (s)d H (s) D G (s)d H (s) + KN G (s)n H (s) (24) Figure c 2006, 2007 R.J. Leduc 32

33 Defining Parameters on Root Locus We have already seen that as ζ = cos θ, vectors from the origin are lines of constant damping ratio. As percent overshoot is solely a function of ζ, these lines are also lines of constant %OS. From diagram we can see that the real part of a pole is σ d = ζω n, and the imaginary part is ω d = ω n 1 ζ 2. As T s = 4 ζω n = 4 σ d, vertical lines have constant values of T s. Figure c 2006, 2007 R.J. Leduc 33

34 Defining Parameters on Root Locus - II π As T p = ω = π, horizontal lines thus have n 1 ζ 2 ω d constant peak time. We thus choose a line with the desired property, and test to find where it intersects our root locus. Figure c 2006, 2007 R.J. Leduc 34

35 Design Procedure For Higher-order Systems 1. Sketch root locus for system. 2. Assume system has no zeros and is second-order. Find gain that gives desired transient response. 3. Check that systems satisfies criteria to justify our approximation. 4. Simulate system to make sure transient response is acceptable. c 2006, 2007 R.J. Leduc 35

36 Third-order System Gain Design eg. For system below, design the value of gain, K, that will give 1.52% overshoot. Also estimate the settling time, peak time, and steady-state error. First step is to sketch the root locus below. We next assume system can be approximated by second-order system, and solve for ζ using ζ = q ln(%os/100). π 2 +ln 2 (%OS/100) c 2006, 2007 R.J. Leduc 36 Figures 8.21 and 8.22.

37 Third-order System Gain Design eg. - II This gives ζ = 0.8. Our angle is thus θ = cos 1 (0.8) = o. We then use root locus to search values along this line to see if they satisfy the angle requirement. The program finds three conjugate pairs on the locus and our ζ = 0.8 line. They are 0.87 ± j0.66, 1.19 ± j0.90, 4.6 ± j3.45 with respective gains of K = 7.36, 12.79, and We will use T p = π ω d, and T s = 4 σ d. Figures 8.21 and c 2006, 2007 R.J. Leduc 37

38 Third-order System Gain Design eg. - III For steady-state error, we have: K v = lim s 0 sg(s) = lim s 0 s K(s + 1.5) s(s + 1)(s + 10) = K(1.5) (1)(10) (25) To test to see if our approximation of a second-order system is valid, we calculate the location of the third pole for each value of K we found. The table below shows the results of our calculations. Table 8.4. c 2006, 2007 R.J. Leduc 38

39 Third-order System Gain Design eg. - IV We now simulate to see how good our result is: Figure c 2006, 2007 R.J. Leduc 39

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