1、1,Chapter 7 Stability in the Frequency Domain,7.1 Introduction 7.2 Mapping Contours in the s-plane 7.3 Nyquist Stability Criterion 7.4 Stability Margin of System 7.5 Dynamics performance of closed-loop from open-loop frequency characteristic 7.6 Summary,2,7.1 Introduction,Developed by H.Nyquist in 1

2、932. Based on Cauchys theorem.,3,The frequency response can be obtained experimentally. It can be utilized to investigate the relative stability.,4,Where 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 mappi

3、ng of the right-hand s-plane in F(s)-plane.,5,7.2 Mapping Contours in the s-plane,A contour map is a contour in one plane mapped into another plane by a relation F(s). Example:,6,Cauchys 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

4、and zeros of F(s) and the 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.,7,Another example:,8,The poles of F(s) are the poles of L(s). The zeros of F(s) are the ch

5、aracteristic roots of the system.,7.3 Nyquist Stability Criterion,9,For 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.

6、 Z: zeros in RHP P: poles in RHP So the number of unstable poles of the system is Z=N+P,10,The 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 becomes the number of encirclements of -1 point in L(s)-plane. L(s)

7、is the open-loop transfer function.,11,Nyquist stability criterion 1. 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

8、, for the contour L, the number of counter-clockwise encirclements of the (-1, 0) point is equal to the number of poles of L(s) with positive real parts.,12,Example 7.1,N=Z=0, so the system is stable.,13,Example 7.2 Assuming open loop transfer function is,determine the stability of the system at K=2

9、0 and K=100.,14,We need to find the cross-over point and compare it with -1!,15,So the system is stable at K=20 and unstable at K=100.,16,1,20,0.3,K=20,17,K=20,18,100,K=100,19,1,1.48,K=100,20,K=100,21,Example 7.3,22,(a) The origin of the s-plane,23,is the polar plot of L(s).,is mapped into the origi

10、n of the L(s)-plane.,is symmetrical to the polar plot.,24,Note: 1.,25,2.,26,3. The conclusion 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 pole

11、s at the origin The supplement curve must be draw. The small semicircular detour around the pole at the origin can be represented by setting,28,6. 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,正負(fù)穿越,正穿越：相角增加，逆時(shí)針

12、負(fù)穿越：相角減少，順時(shí)針 極坐標(biāo)圖穿越點(diǎn)（-1，0）左邊實(shí)軸的正負(fù)穿越次數(shù)之差等于極坐標(biāo)圖逆時(shí)針?lè)较虬鼑c(diǎn)（-1，0）的周數(shù)。 Nyquist判據(jù)： Nyquist圖穿越點(diǎn)（-1，0）左邊實(shí)軸的正負(fù)穿越次數(shù)之差應(yīng)等于P。 P：開環(huán)傳遞函數(shù)正實(shí)部極點(diǎn)數(shù)。,30,Example 7.4,It is possible to encircle the -1 point.,31,At real axis,So the system is stable when,32,Example 7.5,33,So the system is stable when Tt.,34,Example 7.6 non-mi

13、nimum phase system,35,Conclusion： Nyquist diagram encircles the 1 point one time in the direction of counter-clockwise. N=1,P=1，Z=P-N=0, so the system is stable. The system is stable when K3.,36,Example 7.7：The open-loop TF is Determine the changing range of K.,The 1 point located on A or C, the sys

14、tem is stable. The 1 point located on B or D, the system is unstable.,37,We can get： 1 locus on A, K13200, unstable So the changing range of K is 0 K19.2 and 334K13200.,38,7.4 Stability Margin of System,The closeness of the L(jw) curve to “-1” is a measure of the relative stability of the system. Th

15、ere are two measures of relative stability - Gain Margin and Phase Margin.,39,Phase Margin (PM): Definition：the phase angle through which the L(jw) locus must be rotated so that the unity magnitude |L(jw)|=1 point will pass through the (-1, 0) point in the L(jw)-plane. wc : the gain crossover freque

16、ncy PM0(PM0) indicates a stable (an unstable) system.,40,Gain Margin(GM): Kg Definition：the reciprocal of the gain |L(jw)| at the frequency at which the phase angle reaches -180. wg: the phase crossover frequency,41,GM1(0 dB) indicates a stable closed-loop system and the system will remain stable if

17、 the loop gain increase is less than GM. GM1(0 dB) indicates an unstable closed-loop system and a reduction of loop gain at least GM is required for the system to become stable.,42,0,phase margin,Gain margin,Kg,43,Gain and Phase Margins on Bode plots.,44,7.5 Dynamics performance of closed-loop from

18、open-loop frequency characteristic,1. System type and steady state error,45,System type Slope of the low frequency asymptote type 0 0. type 1 -20dB/dec. type 2 -40dB/dec.,46,2. The specification of closed loop system in frequency domain,The second-order control system,47,48,The peak value of magnitude of control system,49,Bandwidth of control system,50,3. The specification of op