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2. 5.4Nyquist Stability Criterion Closed –loop TF Fig.5-35 In order to make the system stable,the characteristic roots must lie in the left of s-plane. Although open-loop TF poles and zeros may lie in the right of s-plane, only closed-loop characteristic roots lie in the left of s-plane,the system is definitely stable.

3. 1.Nyquist Stability Criterion is one connected with open loop frequency characteristic it is not necessary to solve closed loop poles by this method ，and widely used. 2.when TF is unknown,open loop frequency characteristic curve solved by analysis method/experimental method can all be used to stability analysis. 3.it is convenient to know its stability when some parameters change. 4.it can be applied to nonlinear system analysis. 5.even if there are some pure delay,this method can be used.

4. + G(s) - H(s) 5.4.1 additional function Open loop TF Closed loop TF Closed loop TF denominator

6. 1.the poles of F(s) is the poles of Open loop TF. the zeros of F(s) is the poles of closed loop TF. 2. the poles number of F(s) is equal to zeros number of F(s) . 3.the deviation between F(s) and Gk(s) is 1.

7. jω Jω+p -p -p 5.4.2 Nyquist stability criterion 1. First-order system If p>0,when ω is increases from 0 to ∞,＜(p+j ω ) is revolve 90º counterclockwise. If p>0,when ω is increases from 0 to ∞,＜(p+j ω ) is revolve 90º clockwise. Conclusion:when ω increases from 0 to ∞,＜(p+j ω ) revolves 90º counterclockwise,the first order system is stable.

8. 2. 2st-order system If Re(p>0),when ω increases from 0 to ∞,＜ (p+j ω ) :-φ→ 90º; ＜ (p*+j ω ) :φ→ 90º counterclockwise If Re(p>0),when ω s increases from 0 to ∞,＜ (p+j ω ) (p*+j ω ) revolves 2*90º counterclockwise. Conclusion:when ω is increases from 0 to ∞,＜(p+j ω ) (p*+j ω ) is revolve 2*90º counterclockwise,the 2-th order system is stable.

9. n. nst-order system jω+p1 p1 Conclusion:when ω is increases from 0 to ∞,phase increment is n*90º counterclockwise,the n-th order system is stable.

10. 5.4.3 Nyquist stability criterion 1. 0 type system 1)when open loop is stable,D(s) n roots are in left s plane, If closed loop is also stable ,DB(s) n roots are in left s plane,

11. Conclusion 1: when ω increases from 0 to ∞, F(j ω ) phase increment is 0º ,that is the trajectory of F(j ω ) does not surround the origin,the system is stable. because F(j ω )=1+Gk(j ω ) ,we can know when ω increases from 0 to ∞, the trajectory of Gk(j ω ) does not surround the (-1,j0) point, the system is stable.

12. 2)when open loop is unstable,D(s) (n-p) roots are in left s plane,p roots are in right s plane, If closed loop DB(s) z roots are in right s plane, when ω is increases from 0 to ∞, F(j ω ) curve surrounds the origin R circles,or Gk(j ω ) curve surrounds the (-1,j0) R circles, and z=p-2R

13. Conclusion 2: when open loop is unstable,its p roots are in right s plane,if when ω increases from 0 to ∞, the trajectory of Gk(j ω ) surrounds the (-1,j0) point R=p/2 times, the closed loop system is stable. Conclusion: given open loop system,its p roots are in right s plane. when ω increases from 0 to ∞, the trajectory of Gk(j ω ) surrounds the (-1,j0) point R times, the closed loop system has z roots are in right s plane. When z=0, the closed loop system is stable.

14. Exam from From

15. p = 0 ,R＝0 z=0 stable

16. exam -7.9 Crossing point to real axis:－7.9 (-1,j0) Nyquist criterion： P=0，R= -1, z= 2。 unstable。

17. exam Nyquist criterion： p=0，R=0，z=0 stable

18. exam r = 1 question：p＝? R = ?

19. 2. Open loop poles in imaginary axis From real axis, along a part circle with radius （ ），then from to along Axis.

20. To containing factor Of a open-loop TF when s Moves along a semi-circle with radius Curve will have When in s-plane , The phase of part-Circles with ∞ radius clockwise surrounding origin. For instance,TF:

21. No poles in right s plane，when K>0，the trajectory surrounds One time。So ,function Has 2 zeros in right s plane。The system is unstable。

22. Exam.5-4 a open-loop TF is： Check its stability When ：gain K is smaller  gain K is bigger 。 When K is small,stable When K is big,unstable

23. Exam.5-5 a open-loop TF is： Check its stability . Curve does not surround stable Curve passes closed poles in axis.

24. Curve surrounds clockwise one time，there are 2 closed loop poles in right 2 plane.the system is unstable.

25. some examples

26. Exam.5-6 a open-loop TF is： Check closed loop stability。 Fig.5-44

27. 1 1 pole in right s plane 2 Fig. 5-44 Nyquist diag. shows， Curve surrounds -1+j0 once clockwise. Polar plot 3 Fig.5-44 This shows 2 closed loop poles in right s plane,the system is unstable.

28. Exam.5-7 a open-loop TF is：。 GH plane Check closed loop stability。 asymptote Fig.5-45 Polar plot

29. Exam.5-8 a open-loop TF is： where Are positive。To make the system stable，how to select ？

32. ？crossing point to negative real axis expand set crossing point to negative real axis

35. 5.4.4 bode stability criterion

36. Exam. a open-loop TF is： Check its stability .