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1、Ch8 Stability in the Frequency Domain,Main content,Mapping contours in the s-planeThe Nyquist CriterionRelative Stability and Nyquist CriterionClosed-loop frequency responseSystem BandwidthExamples and SimulationSummary,Introduction,Feedback Control System,How to determine the stability?,Roots evalu
2、ation Routh-Hurwitz criterion(absolute/relative stability)Root locus Nyquist criterion(absolute/relative stability),8.1 Mapping contours in the s-plane,Contour mapConformal mapping,s-plane F(s)-plane,A contour mapping that retains the angles on the s-plane on the F(s)-plane,Figure 1 mapping F(s)=2s+
3、1,Figure 2 Mapping for F(s)=s/(s+2),Figure 3 Mapping for F(s)=s/(s+1/2),Principle of the argument,If a contour in the s-plane encircles Z zeros and P poles of F(s)and does not pass through any poles or zeros of F(s)and the traversal is in the clockwise direction along the contour,the corresponding c
4、ontour in F(s)-plane encircles the origin of F(s)-plane N=Z-P times in the clockwise direction.,The rules of contour,Clockwise traversal of a contour is assumed to be positive.Encirclement is defined by the rule:“Clockwise and eyes right”or“Counterclockwise and eyes left”,Figure 4 Evaluation of the
5、net angle of F(s)-plane2*Pi*N=2*Pi*Z-2*Pi*P,Figure 5 Evaluation of the net angle of F(s)-planeN=Z-P=3-1=2,Figure 6 Evaluation of the net angle of F(s)-planeN=Z-P=0-1=-1,8.2 The Nyquist Criterion,Figure 7 Single-loop feedback control system,The closed-loop characteristic equation:,Figure 8 Nyquist co
6、ntour in s-plane,The number of zeros of F(s)within the contour is:,Nyquist stability criterion,A feedback system is stable if and only if the contour in the L(s)-plane does not encircle the(-1,0)point when the number of poles of L(s)in the right-hand s-plane is zero(P=0)A feedback system is stable i
7、f and only if,for the contour,the number of counterclockwise encirclements of the(-1,0)point is equal to the number of poles of L(s)in the right-hand s-plane.,Application of Nyquist criterion,Example 1:system with 2 real polesExample 2:system with a pole at originExample 3:system with 3 polesExample
8、 4:system with 2 poles at originExample 5:system with a pole in RHPExample 6:system with a zero in RHP,Refer to P506-515,Figure 9 Nyquist contour and mapping for GH(s),Example 1,Example 2,Figure 10 Nyquist contour and mapping for GH(s),Figure 11 Nyquist contour and mapping for GH(s),Example 3,Figure
9、 12 Nyquist contour for GH(s)when(a)K=1,(b)K=2,(c)K=3,Example 4,Figure 13 Nyquist contour for GH(s),Example 6,Figure 10 Nyquist diagram and mapping for GH(j w)/K,稳定性分析的补充举例,开环传递函数不含积分环节,例1：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统稳定,例2：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统稳定,例3：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统不稳定,-1.59,开环传递函数含积分环节,需对开环幅相曲线作修正：从=0+处，逆时针补画90半径为无穷大的圆弧。,例4：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统不稳定,例5：系统开环传递函数为 试判定闭环系统的稳定性。,闭环系统不稳定,例6：系统开环传递函数为 试判定闭环系统的稳定性。,闭环系统不稳定,闭环系统稳定,Assignment,E9.7E9.24P9.2(resort to MATLAB)P9.4,
