控制工程(周川)a课件

Ch8 Stability in the Frequency Domain控制工程(周川)aMain contentMapping contours in the s-planeThe Nyquist CriterionRelative Stability and Nyquist CriterionClosed-loop frequency responseSystem BandwidthExamples and SimulationSummary控制工程(周川)aIntroductionFeedback Control System控制工程(周川)aHow to determine the stability?Roots evaluation Routh-Hurwitz criterion(absolute/relative stability)Root locus Nyquist criterion(absolute/relative stability)控制工程(周川)a8.1 Mapping contours in the s-planeContour mapConformal mapping s-plane F(s)-planeA contour mapping that retains the angles on the s-plane on the F(s)-plane 控制工程(周川)aFigure 1 mapping F(s)=2s+1控制工程(周川)aFigure 2 Mapping for F(s)=s/(s+2)控制工程(周川)aFigure 3 Mapping for F(s)=s/(s+1/2)控制工程(周川)aPrinciple of the argumentIf 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 contour in F(s)-plane encircles the origin of F(s)-plane N=Z-P times in the clockwise direction.控制工程(周川)aThe rules of contourClockwise traversal of a contour is assumed to be positive.Encirclement is defined by the rule:“Clockwise and eyes right”or “Counterclockwise and eyes left”控制工程(周川)aFigure 4 Evaluation of the net angle of F(s)-plane2*Pi*N=2*Pi*Z-2*Pi*P 控制工程(周川)aFigure 5 Evaluation of the net angle of F(s)-planeN=Z-P=3-1=2 控制工程(周川)a Figure 6 Evaluation of the net angle of F(s)-planeN=Z-P=0-1=-1 控制工程(周川)a8.2 The Nyquist Criterion Figure 7 Single-loop feedback control system控制工程(周川)aThe closed-loop characteristic equation:控制工程(周川)aFigure 8 Nyquist contour in s-plane控制工程(周川)aThe number of zeros of F(s)within the contour is:控制工程(周川)aNyquist stability criterionA 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 if 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.控制工程(周川)aApplication of Nyquist criterion Example 1:system with 2 real polesExample 2:system with a pole at originExample 3:system with 3 polesExample 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控制工程(周川)aFigure 9 Nyquist contour and mapping for GH(s)Example 1控制工程(周川)aExample 2Figure 10 Nyquist contour and mapping for GH(s)控制工程(周川)aFigure 11 Nyquist contour and mapping for GH(s)Example 3控制工程(周川)aFigure 12 Nyquist contour for GH(s)when (a)K=1,(b)K=2,(c)K=3控制工程(周川)aExample 4Figure 13 Nyquist contour for GH(s)控制工程(周川)aExample 6Figure 10 Nyquist diagram and mapping for GH(j w)/K控制工程(周川)a稳定性分析的补充举例稳定性分析的补充举例开环传递函数不含积分环节开环传递函数不含积分环节例例1 1：系统开环传递函数为系统开环传递函数为试判定闭环系统的稳定性。试判定闭环系统的稳定性。闭环系统稳定闭环系统稳定控制工程(周川)a例例2 2：系统开环传递函数为系统开环传递函数为试判定闭环系统的稳定性。试判定闭环系统的稳定性。闭环系统稳定闭环系统稳定控制工程(周川)a例例3 3：系统开环传递函数为系统开环传递函数为试判定闭环系统的稳定性。试判定闭环系统的稳定性。闭环系统不稳定闭环系统不稳定-1.59-1控制工程(周川)av开环传递函数含积分环节开环传递函数含积分环节需对开环幅相曲线作修正：需对开环幅相曲线作修正：从从=0=0+处，逆时针补画处，逆时针补画9090半径为无穷大的圆弧。半径为无穷大的圆弧。例例4 4：系统开环传递函数为系统开环传递函数为试判定闭环系统的稳定性。试判定闭环系统的稳定性。闭环系统不稳定闭环系统不稳定控制工程(周川)a例例5 5：系统开环传递函数为系统开环传递函数为 试判定闭环系统的稳定性。试判定闭环系统的稳定性。闭环系统不稳定闭环系统不稳定控制工程(周川)a例例6 6：系统开环传递函数为系统开环传递函数为 试判定闭环系统的稳定性。试判定闭环系统的稳定性。闭环系统闭环系统不稳定不稳定闭环系统闭环系统稳定稳定控制工程(周川)aAssignmentE9.7E9.24P9.2 (resort to MATLAB)P9.4控制工程(周川)a