线性系统的根轨迹分析法 (16).pdf

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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 I)5.5 Stability Margins 5.6 Transient-state and Steady-state Performance Anal

2、ysis 2 A complex function F(s)can be regarded as a mapping from the S-plane to the F-plane.spspspF sK szszsznm()().()()()().()1212zimi 1,2,.,pjnj 1,2,.,Argument Principle poles of F(s)zeros of F(s)C ClockwiseF s()-planeS-planeC Clockwises1F s()1jRe003 Argument Principle:If a closed curve C on the S-

3、plane contains P poles and Z zeros of F(s),and the closed curve does not pass through any zero or pole of F(s),so when s turns around the closed path C clockwise,the number of circles N of the closed curve which corresponds to F(s)surrounding the origin in the clockwise direction is NZP.CArgument Pr

4、inciple C ClockwiseF s()-planeS-planeC Clockwises1F s()1jImRe00spspspF sK szszsznm()().()()()().()12124 Example:F(s)is a rational function of s spspspF sK szszsznm()().()()()().()1212zimi 1,2,.,pjnj 1,2,.,Let F(s)be Only the angle of the vector changes-2,and the angle changes of the referenced vecto

5、rs from the other zero poles to the point s are all zero.F sszspijijmn()()()11By analogy,if the closed curve C encloses the Z zeros of F(s),the closed curve of the corresponding F-plane encloses the origin Z circles in a clockwise direction.C Only one zero of F(s)is included in the closed curve C.po

6、les of F(s)zero points of F(s)C clockwise-planeF s()-planeSC clockwisesF s()jImRe00z1z2p1p2F(s)s change of argument 2+1 +1 5 F sszspijijmn()()()11Only the angle of the vector changes-2,and the angle changes of the referenced vectors from the other zero poles to the point s are all zero.By analogy,if

7、 the closed curve C encloses P poles of F(s),the closed curve of the corresponding F-plane encloses the origin P circles in a counter clockwise direction.C Only one pole of F(s)is included in the closed curve C.spspspF sK szszsznm()().()()()().()1212C counter-clockwise-planeF s()-planeSC clockwisesF

8、 s()jImRe00p1z1p3p2F(s)s change of argument 2+1 +1 6 Only one pole of F(s)is included in the closed curve C.spspspF sK szszsznm()().()()()().()1212C counter-clockwiseF s()-planeS-planeC clockwisesF s()jImRe00p1z1p3p2Argument Principle:If a closed curve C on the S-plane contains P poles and Z zeros o

9、f F(s),and the closed curve does not pass through any zero or pole of F(s),so when s turns around the closed path C clockwise,the number of circles N of the closed curve ,which corresponds to F(s)surrounding the origin in the clockwise direction is:NZP.C+1 7 D sG sN s()()()11D sH sN s()()()22D s D s

10、G sG s H sN s N sk()()()()()()()1212N NDDsN D()121212 DDDDN NN NDD11212121212Nyquist stability criterion 1.Choose the complex function F(s)Poles of F(s)are the poles of open-loop transfer function.The transfer functions of forward channel:The transfer functions of feedback channel:The open-loop tran

11、sfer function:The closed-loop transfer function:Let a complex function be the characteristic polynomial of the system:NZPTimes that its mapping encloses the origin in the clockwise direction on the F-plane The number of poles of the open-loop transfer function in the closed curve of the S-plane.The

12、number of poles of the closed-loop transfer function in the S-plane closed curve.Given N and P,find Z Zeros of F(s)are the poles of the closed-loop transfer function.+)s(C)s(R)s(H)s(G1.Choose the complex function F(s)2.Determine the closed path on the S-plane 3.Obtain the relationship between F-plan

13、e and Gk(s)-plane F sG sk()1()8 1.Choose the complex function F(s)2.Determine the closed path on the S-plane 3.Obtain the relationship between F-plane and Gk(s)-plane j ej0C 2.Determine the closed path on the S-plane The positive imaginary axis:A semicircle with an infinite radius on the right half

14、S-plane:The negative imaginary axis:0+sj,22:sR ej，R，sj:0S-plane The closed curve is called the Nyquist path.The prerequisite is,the Nyquist path does not pass through the zeros and poles of F(s).9 plane-Foriginplane-F143211-2-3-4-ImRe543201-2-1.Choose the complex function F(s)2.Determine the closed

15、path on the S-plane 3.Obtain the relationship between F-plane and Gk(s)-plane ReImplane-)s(kGoriginoriginplane plane-kG5432101-2-43211-2-3-4-3.Obtain the relationship between F-plane and Gk(s)-plane F sGsk()1()Move the -plane one unit to the left to obtain the F-plane.G sk()Gk-plane(-1,j0)10 ImReRep

16、lane-)s(kGoriginplane-kG5432101-2-43211-2-3-4-When s changes along the positive imaginary axis,corresponding to the polar plot of the positive frequency part.Nyquist curve :0 When s changes on a semicircle with an infinite radius,its mapping is G sk()0 When s changes along the negative imaginary axi

17、s from infinity to the origin,corresponding to the part of the polar plot that is symmetric with the real axis.-:0Nyquist path mapping(Nyquist curve)sej0C jSplaneGenerally in ,the denominator order is higher than the numerator order,so when ,.G sk()sejG sk()011 Nyquist stability criterion:If the num

18、ber of poles of the open-loop transfer function in the right half S-plane is P,when w changes from-to+,and the times that the open-loop frequency characteristic curve encircles(-1，j0)is N(N0 is clockwise,N0,it is unstable.If the number of poles of the open-loop transfer function in the right half S-

19、plane is P,and the necessary and sufficient condition for the stability of the closed-loop system is:When changes from-to+,the open-loop frequency characteristic curve and its mirror image on the S-plane will enclose the(-1，j0)P times in a counterclockwise direction.For the stable open-loop system,P

20、=0,the sufficient and necessary condition for the stability of the closed-loop system is that the open-loop frequency characteristic curve and its mirror image do not surround(-1,j0).The number of poles of the unstable closed-loop system on the right half S-plane is:Z=N+P.12 When ，+0eRmIIntersection

21、 with imaginary axis:Intersection with real axis:and ，kG sKsss2()(2)(25)AK()4(5)422 22PK()(104)(104)(9)222222 ()arctan2arctan252，AKPKQ()/10()0()/10()0，APQ ()0()270()0()0QK()(9)(104)(9)222222Example:Suppose the transfer function of the open-loop system is:,and use the Nyquist criterion to judge the s

22、tability of the closed-loop system.When ，0Let ,Q()0then 03 PK(3)/26Let ,P()0then ，2.5QK(2.5)/(2.56.5)0K10K26 32.5+Magnitude-frequency characteristics:Phase-frequency characteristics:Real frequency characteristics:Imaginary frequency characteristics:13 0eRmI0eRmIeRmI4202-4-6-65432101-2-3-If the syste

23、m is to be stable,the Nyquist curve does not enclose(-1,j0),that is,P(3)=-K/26-1,then K26.When K=52,the Nyquist curve encircles(-1,j0)twice clockwise,N=2，Z=N+P=2，and closed loop system is unstable.When K-1 is to be satisfied,K-10 is required.So the condition for system stability is:-10 K 26 j(-1,0)PK26(3)K10 0 3+02 0+3P=0 kG sKsss2()(2)(25)Example:Suppose the transfer function of the open-loop system is:,and use the Nyquist criterion to judge the stability of the closed-loop system.(-1,0)j14 Summary Argument principle Nyquist stability criterion How to use Nyquist stability criterion