自控课件自动控制理论

1、1Chapter 7 Stability in the Frequency Domain7.1 Introduction7.2 Mapping Contours in the s-plane7.3 Nyquist Stability Criterion7.4 Stability Margin of System 7.5 Dynamics performance of closed-loop from open-loop frequency characteristic7.6 Summary27.1 IntroductionDeveloped by H.Nyquist in 1932.Based

2、 on Cauchys theorem.3The frequency response can be obtained experimentally.It can be utilized to investigate the relative stability.4Where L(s) is a rational function of s.To ensure stability, it must be ascertained that all zeros of F(s) lie in the left-hand s-plane.Propose a mapping of the right-h

3、and s-plane in F(s)-plane.57.2 Mapping Contours in the s-planeA contour map is a contour in one plane mapped into another plane by a relation F(s).Example:6Cauchys theorem: If a contour s in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles and zeros of F(s) and t

4、he traversal is in the clockwise direction along the contour, the corresponding contour F in the F(s)-plane encircles the origin of the F(s)-plane N=Z-P times in the clockwise direction.7Another example:8The poles of F(s) are the poles of L(s).The zeros of F(s) are the characteristic roots of the sy

5、stem.7.3 Nyquist Stability Criterion9For a system to be stable, all the zeros of F(s) must lie in the left-hand s-plane.Choose a contour s in the s-plane that encloses the entire right-hand s-plane, the number of encirclements of the origin of the F(s)-plane is N=Z-P.Z: zeros in RHPP: poles in RHPSo

6、 the number of unstable poles of the system is Z=N+P10The contour F is known as the Nyquist diagram or ploar plot of F(s).As L(s)=F(s)-1, the number of encirclements of the origin in F(s)-plane es the number of encirclements of -1 point in L(s)-plane.L(s) is the open-loop transfer function.11Nyquist

7、 stability criterion1. A feedback system is stable if and only if the contour L 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).2. A feedback system is stable if and only if, for the contour L, the number of counter-clock

8、wise encirclements of the (-1, 0) point is equal to the number of poles of L(s) with positive real parts.12Example 7.1N=Z=0, so the system is stable.13Example 7.2 Assuming open loop transfer function is determine the stability of the system at K=20 and K=100.14We need to find the cross-over point an

9、d compare it with -1!15So the system is stable at K=20 and unstable at K=100.161200.3K=2017K=2018100K=1001911.48K=10020K=10021Example 7.322(a) The origin of the s-plane23is the polar plot of L(s).is mapped into the origin of the L(s)-plane.is symmetrical to the polar plot.24Note: 1.252.26 3. The con

10、clusion can be expanded to the system including delay unit. 4. If the contour L(jw) overpass the (-1, j0) point, that is one close-loop pole on the jw-axis, the system is critically stable.27 5. System with v poles at the origin The supplement curve must be draw. The small semicircular detour around

11、 the pole at the origin can be represented by setting 286. If the number of counter-clockwise encirclements is NP, then the closed-loop system is unstable with Z unstable poles, where Z=P-N. 29正负穿越正穿越：相角增加，逆时针负穿越：相角减少，顺时针极坐标图穿越点（-1，0）左边实轴的正负穿越次数之差等于极坐标图逆时针方向包围点（-1，0）的周数。Nyquist判据： Nyquist图穿越点（-1，0）左

12、边实轴的正负穿越次数之差应等于P。P：开环传递函数正实部极点数。30Example 7.4It is possible to encircle the -1 point.31At real axis, So the system is stable when 32Example 7.533So the system is stable when Tt.34Example 7.6 non-minimum phase system35Conclusion： Nyquist diagram encircles the 1 point one time in the direction of coun

13、ter-clockwise. N=1,P=1，Z=P-N=0, so the system is stable. The system is stable when K3. 36Example 7.7：The open-loop TF is Determine the changing range of K.The 1 point located on A or C, the system is stable. The 1 point located on B or D, the system is unstable.37We can get： 1 locus on A, K19.2, sta

14、ble1 locus on B, 19.2K334, unstable1 locus on C, 334K13200, unstableSo the changing range of K is 0 K19.2 and 334K0(PM1(0 dB) indicates a stable closed-loop system and the system will remain stable if the loop gain increase is less than GM. GM1(0 dB) indicates an unstable closed-loop system and a re

15、duction of loop gain at least GM is required for the system to e stable.420phase margin Gain margin Kg43Gain and Phase Margins on Bode plots.447.5 Dynamics performance of closed-loop from open-loop frequency characteristic1. System type and steady state error45System type Slope of the low frequency

16、asymptote type 0 0. type 1 -20dB/dec. type 2 -40dB/dec. 462. The specification of closed loop system in frequency domainThe second-order control system4748The peak value of magnitude of control system49Bandwidth of control system 503. The specification of open loop system in frequency domainGain crossover frequency51Phase margin524. The specification of high order system537.6 Summary In the frequency domain, Nyquists criterion can be used to determine the stability of a feedback control system. Nyquists criterion provides two relative stability measures: gain margin and phase margin