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1、1 Chapter 5 Frequency Domain Analysis of Automatic control systems 5.1 Basic Concepts of Frequency Characteristics 5.2 Logarithmic Frequency Characteristic Plot(Bode Plot)5.3 Polar Plot 5.4 Nyquist Stability Criterion(Part II)5.5 Stability Margins 5.6 Transient-state and Steady-state Performance Ana
2、lysis 2 sT sG sKsjjnkiim(1)()(1)11For Type I and Type II systems,it cannot be directly applied Nyquist stability criterion.How to reconstruct the Nyquist path?Open-loop transfer function Application of Nyquist Stability Criterion in Type I and Type II System jej 0 0j0 0 ej00 S-plane S-plane 3 sT sG
3、sKsjjnkiim(1)()(1)11Open-loop transfer function Positive imaginary axis：Right semicircle with infinite radius(clockwise):Negative imaginary axis:Right semicircle with infinitely small radius(counterclockwise):jej 0 0j0 0 ej00 22:sR ej，RsR eRj22,0,:，sj:0:0+sj,S-plane S-plane Application of Nyquist St
4、ability Criterion in Type I and Type II System 4 Part IV Nyquist path mapping：kGj()Part I and Part III：polar plot and its symmetrical part about the real axis Part ，sR eRGjkj22,()0 Type I system ssKsjjsniim(1)lim(1)1011The radius of the circle is ,the angle of the mapping curve is：0 G ssklim()0Gk(s)
5、-plane S-plane j 00+R ej sKslim00ReeKRjjlim0sRej1 1 +00+=Re Im:counterclockwise 22+:clockwise 225 Part IV Nyquist path mapping：kGj()Part I and Part III：polar plot and its symmetrical part about the real axis Part Type system The radius of the circle is ,the angle of the mapping curve is：counterclock
6、wise 22+：clockwise S-plane j 00+R ej 000+Gk(s)-plane，sR eRGjkj22,()0G ssklim()0 R eeKRjj()lim022 2+6 Example 1：The open-loop frequency characteristics of a type II system are known,and there is no pole in the right half S-plane,try the Nyquist criterion to judge the stability of the closed-loop syst
7、em.The Nyquist curve encloses(-1,j0)two times clockwise.N=2 First draw the complete Nyquist curve A：，ZNP2P 0The closed-loop system is unstable and has two poles in the right half S-plane.j00+R ej 0eRmI000+Gk(s)-plane S-plane 7 eRmI Example 2:The open-loop transfer function of the non-minimum-phase s
8、ystem is known as .Determine the stability of the closed-loop system.Find the number of right poles of the closed-loop when it is unstable.A：，PKTQ()()，PQ()0()0A ()()270，when ,0 when ,A ()0()180，AKT()1()2Magnitude-frequency characteristics（()90180arctan)270arctanTT Phase-frequency characteristics PKT
9、T22()1Real frequency characteristics（QKT22()1)Imaginary frequency characteristics 0 j00+R ej 00sTsG sKk(1)()1S-plane Gk(s)-plane 8 Example 2:The open-loop transfer function of the non-minimum phase system is known as .Determine the stability of the closed loop system.Find the number of right poles o
10、f the closed loop when it is unstable.A：sTsG sKk(1)()10 S-plane j00+R ej 00Gk(s)-plane sTssKKss(1)limlim-00If it is the minimum-phase system（ReeKRjjlim=-0-180)G ssklim()0When K0，P=1，N=1，Z=N+P=2,the closed-loop system is unstable,and there are two poles on the right half S-plane.180The radius of the
11、circle is ，and the angle of the mapping curve is non-minimum-phase system sRej：counterclockwise 220223:clockwise 9 mIeR15.05.0-1-5.1-2-5.005.0-1-5.1-2-Example 3：For ,determine the delay time when the system is critically stable.s ssG seks(1)(2)()A：Make the polar plot of respectively 0,0.8,2,4 Obviou
12、sly,as long as is slightly larger than 2s,the system will be unstable.4 20.8 0 When the delay is 0,the closed-loop system is stable.With the increase of,the stability becomes worse.When =2s,the system is on the edge of stability,and the polar plot crosses the negative real axis near the point(-1,j0).Gk(s)-plane Using analytical methods,we can solve for 2.0910 SUMMARY Application of Nyquist Stability Criterion in Type I and Type II Systems Application of Nyquist Stability Criterion in Systems with delay elements