1. Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

2. Automatic Control Theory Excecies(16) 4 – 9, 11, 12, 14, 15

3. Summary Rule 5 The asymptotes of root-locus Rule 1 The starting and ending points of the root-locus Rule 2 Number of branches, symmetry and continuity of root-locus Rule 3 Root-locus on the real axis Rule 4 Sum of the roots Rule 6 The breakaway point of root-locus Rule 7 The points where the root-locus cross the imaginary axis Rule 8 The angle of departure and the angle of arrival

4. §4.2 General Rules for Constructing a Root-locus （ 18 ） Example 1 The system structure diagram is shown in figure Solution. (1) ② Asymptotes ： ① Root-locus on the real axis ： [-0.5, 1.75] （ 1 ） Sketch the root-locus when K * = 0→∞ ; （ 2 ） Analyze the stability versus variation rule of K *. ③ Angle of departure ： ④ Cross-point of imaginary axis ：

5. §4.2 General Rules for Constructing a Root-locus （ 19 ） Solution. (2) Analysis ： Open-loop stable ≠ Closed-loop stable Naegative feedback cannot always improve system performances Example 1 The system structure diagram is shown in figure （ 1 ） Sketch the root-locus when K * = 0→∞ ; （ 2 ） Analyze the stability versus variation rule of K *.

6. Automatic Control Theory （ Lecture 16) §4.1 The Basic Concept of Root-locus Method §4.2 General Rules for Constructing a Root-locus §4.3 Extension of Root-locus §4.4 System Analysis by Root-locus Method §4 Root-Locus Method

7. Automatic Control Theory （ Lecture 16) §4.3 Extension of Root-locus

8. §4.3 Extension of Root-locus （ 0 ） Example 2 The open-loop transfer function of a system is. Solution. (1) ② Asymptotes : ① The root-locus on real axis : [-∞,0] Sketch the root-locus a=0→∞. Determine the  (s) when x=1. ③ Breakaway point : Rewritten as ： ④ Intersection point with imaginary axis: §4.3.1 Parameter root-locus -- RL subjected to other parameters besides K * Construct “Substitutional open-loop transfer function ”

9. §4.3.1 Constant root-locus （ 1 ） Solution. (2) When ξ=1, a d =2/27 corresponding to the breakaway point d.

10. §4.3.1 Constant root-locus （ 2 ） Example 3 The open-loop transfer function of unity feedback system is. Solution I. ② Departure angle ： ① The root-locus on real axis[-∞,-587.7], [-27.7,0] Sketch the root-locus when T=0→∞. ④ Breakaway point ： Solving the roots: ③ Intersection point with imaginary axis: Rewritten as ：

11. §4.3.1 Constant root-locus （ 3 ） Solution II. ④ Arrival angle ： ① The root-locus on real axis ： [-∞,-587.7], [-27.7,0] ② Breakaway point ： ③ Intersection point with imaginary axis: Example 3 The open-loop transfer function of unity feedback system is. Sketch the root-locus when T=0→∞.

12. §4.3.1 Constant root-locus （ 4 ） Solution. ② Asymptotes ： ① Root-locus on the real axis ： [-∞, -2], [-1, 0] If the K * is determined., sketch root-locus with the vary of T. ③ Breakaway point ： ④ Intersection point with imaginary axis ： Example 4 The open-loop transfer function of unity feedback system is.

13. §4.3.1 Constant root-locus （ 5 ） 出射角： 渐近线： 出射角： 渐近线： Intersection point with imaginary axis ： Departure angle ：

14. §4.3 0° Root-locus （ 6 ） — Magnitude condition — A ngle condition §4.3.2 0 ° Root-locus —The root-locus of positive feedback systems

15. General Rule for sketching 0° Root-locus ★ Rule 5 The asymptotes of root-locus Rule 1 The starting and ending points of the root-locus Rule 2 Branch number, symmetry and continuity of root-locus ★ Rule 3 Root-locus on the real axis Rule 4 Sum of the root-locus Rule 6 The breakaway and break-in point of root-locus Rule 7 The points where the root-locus may cross the imaginary axis ★ Rule 8 Angle of departure Angle of arrival

16. §4.3 0° Root-locus （ 1 ） Example 5 Consider the system shown in the figure. Sketch the 0°and 180°root-locus when K * = 0→∞ Solution. ① The root-locus on real axis ： [-∞, -1] ② Departure angle ： ③ Breakaway point ： Rewritten as ： Thus ： (1) 180º root-locus (2) 0º root-locus [-1, ∞]

17. §4.3 0° Root-locus （ 2 ） Example 6 The open-loop transfer function of system is. Sketch the 0º and 180º root-locus Solution. ③ Asymptotes ： ① The root-locus on real axis ： [-3, -1] ② Departure angle ： (1) Sketch 180º root-locus

18. §4.3 0° Root-locus （ 3 ） Solution. ④ Asymptotes ： ① The root-locus on real axis ： [-∞,-3], [-1, ∞] ② Departure angle ： (2) Sketch 0º root-locus ③ Breakaway point ： Thus ：

19. §4.3 0° Root-locus （ 4 ） 0º root-locus Asymptotes ： Departure angle ： Dem o

20. Summary §4.3 Extension of Root-locus §4.3.1 Constant root-locus — Constructing substitutinal open-loop transfer function §4.3.2 0°Root-locus — T he 3 rules different from 180º root-locus.

21. Automatic Control Theory Exercise (15) 4 — 9,11,12,14,15

23. Automatic Control Theory The relativity of root-locus 利用 Matlab 说明