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Chapter 5: Root Locus

Chapter 5: Root Locus. Nov. 13-15, 2012. Two Conditions for Plotting Root Locus. Given open-loop transfer function G k (s). Characteristic equation. Magnitude Condition and Argument Condition. Rules for Plotting Root Locus. Break-in point. Breakaway point.

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Chapter 5: Root Locus

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  1. Chapter 5: Root Locus Nov. 13-15, 2012

  2. Two Conditions for Plotting Root Locus Given open-loop transfer function Gk(s) Characteristic equation Magnitude Condition and Argument Condition

  3. Rules for Plotting Root Locus

  4. Break-in point Breakaway point Rule 6: Breakaway and Break-in Points on the Real Axis Use the following necessary condition

  5. =5.818 =0.172 -3 -2 -1 Example 5.3.1: Given the open-loop transfer function, please draw the root locus. 6.Breakaway and break-in points 1.Draw the open-loop poles and zeros 2. Two segments 4. Segments on real axis 3.Symmetry 5.Asymptote zero pole pole breakaway Break-in

  6. Example 5.3.2: j s1  0 s2 Given the open-loop transfer function please prove that the root locus in the complex plane is a circle. Conclusion: For the open-loop transfer function with one zero and two poles, the root locus of characteristic equation is probably a circle in the complex plane.

  7. -2 -1 5.Points across the imaginary axis 4. Breakaway and break-in points 2. Segments on real axis 3. Asymptotes • Open-loop poles and zeros Example 5.3.3: j1.414 K=6 -0.42 K=6 K=6 -j1.414

  8. Example 5.3.4: j3.16 =121 =121 -j3.16 484-4 =0 get =121 • Find poles and zeros 7.The point where the locus across the imaginary axis 6. Breakaway and break-in points 3. Symmetry 5. Asymptote 2. 4 segments 4. No segments on real axis Breakaway -1 2014/9/23 9

  9. Example 5.3.5 Please sketch the root locus with respect to K=[0,+∞).

  10. Extension of Root Locus Canonical form Conventional Root Locus Root locus gain • How to sketch the root locus for other parameters? • How to sketch if Gk(s)=1 • Parameter Root Locus • Zero-degree Root Locus

  11. 108o 90o 5.3.2 Parameter Root Locus Example 5.3.6: Ksis a ramp feedback gain, please sketch the root locus with respect to Ks=[0,+∞). 198o 1. Break-in and breakaway point 2. Angle of emergence

  12. For this kind of systems, the characteristic equations are like as: 5.3.2 Zero-degree Root Locus Example: Sketch the root loci of the system with the open loop transfer function: Analysis: Magnitude equation Argument equation

  13. Root locus by using the sketching rules with the following modification: • Real-axis: Left of even number of zeros or poles • Asymptote • Angles of emergence and entry For Kg varying from －∞→0 together with Kg=[0,+∞) simultaneously, the root loci are named the “complete root loci”.

  14. 5-4 Application of Root Locus Insert a zero

  15. Add a pole to the open-loop transfer function

  16. 5.4.1 The effects of Zeros and Poles Attracting effect Generally, adding an open zero in the left s-plane will lead the root loci to be bended to the left. The more closer to the imaginary axis the open zero is, the more prominent the effect on the system’s performance is. Repelling effect Generally, adding an open pole in the left s-plane will lead the root loci to be bended to the right. The more closer to the imaginary axis the open pole is, the more prominent the effect on the system’sperformance is.

  17. Example 5.4.1: How to enlarge the stability region? j0.2 ω1=0.17 Kc=0.778 ω1=0 Kc=0.278 60o -0.06 -0.027 ω3=0 Kc=0.278 -j0.2 ω2=-0.17 Kc=0.778

  18. 5.4.2 Performance Analysis Based on Root Locus Example 5.4.2: Given the open-loop transfer function please analyze the effect of open-loop gain K on the system performance. Calculate the dynamic performance criteria for K＝5.

  19. It is observed from the root locus that the system is stable for any K. For 0＜K＜0.5(0＜k＜1), there are two different negative real roots. For K=0.5(k=1), there are two same negative real roots. For K＞0.5(k＞1), there are a pair of conjugate complex poles. For K=5( k=10), the closed-loop poles are

  20. The criteria for transient performance can be given by Overshoot Peak time Settling time

  21. 1. We can get the information of the system’s stability by checking whether or not the root loci are on the left half s-plane with the system’s parameter varying. 2. We can get some information of the system’s steady-state error in terms of the number of the open-loop poles at the origin of the s-plane. 3. We can get some information of the system’s transient performance in terms of the tendencies of the root loci with the system’s parameter varying. 4. If the root loci in the left half s-plane move to somewhere far away from the imaginary axis with the system’s parameter varying, the system’s response decays more rapidly and the system is more stable, vice versa.

  22. Complement • Use Matlab to sketch root locus In Matlab: num=[1 7 12] sys=tf[num,den] den=[1 3 2] rlocus(num,den)

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