控制理论基础-自控a_ch7

1,Chapter 7 Stability in the Frequency Domain,7.1 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 Summary,2,7.1 Introduction,Developed by H.Nyquist in 1932.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 mapping 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 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 characteristic 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.Z: zeros in RHPP: poles in RHPSo 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) is the open-loop transfer function.,11,Nyquist stability criterion1. A feedback system is stable if and only if the contour L in the L(s)-plane does net 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-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=20 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 origin 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 poles 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,正负穿越,正穿越：相角增加负穿越：相角减少极坐标图穿越点（-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：极坐标图穿越点（-1，0）左边实轴的正负穿越次数之差应等于P/2。P：开环传递函数正实部极点数。,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-minimum 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 system is stable. The 1 point located on B or D, the system is unstable.,37,We can get： 1 locus on A, K13200, 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 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 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,Constant M circles,47,3. The second-order control system,48,49,The peak value of magnitude of control system,50,Bandwidth of control system,51,The specification of open loop system in frequency domain,Gain crossover frequency,52,Phase margin,53,4. The specification of high order system,54,7.6 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