Module 14 Nyquist Analysis and Relative stabilitv (I hours) Gain margin Phase margin
Module 14 Nyquist Analysis and Relative Stability (1 hours) • Gain Margin • Phase Margin
14. 1 Conditional Stability(条件稳定性) Example 1 (P272)F() K (2.+1)(3s+1 K Im OV(2O)2+1V(30)2+1 p=∠GH=-900-g-20-1g-30 when 9=-180 M=-K vhen=K<1(k<0.83) The system is stable Flg. 14.1 Nyquist diagrams for different values of K
14.1 Conditional Stability (条件稳定性） (2 1)(3 1) ( ) + + = s s s K F s (2 ) 1 (3 ) 1 2 2 + + = K M 90 2 3 −1 −1 = G H = − −t g −t g Example 1 （P272) when = −180 M K 5 6 6 1 = = 1( 0.83) 5 6 when K K The system is stable
I Re Unit circle Flg. 14.1 Nyquist diagrams for different values of K Fig. 14.8 Phase margin for stable and unstable systems K=0832 Fig. 14.2 Root locus for example system
Conclusion( p274: 1.-5.) 1. Write the open-loop transfer function in Bode form,and write down expressions for the magnitude and phase 2.Sketch the Nyquist diagram for an arbitrary value of gain to determine whether the critical point passes to the left or right of an observer moving along the frequency response curve in the direction of increasing frequency 3. USing computer method, if necessary, determine the frequency that makes the corresponding angle -180 4. Substitute this value of frequency into the magnitude equation and determine the corresponding magnitude 5. If the magnitude is less than unity and the critical point passes to the left, the system is stable; otherwise it is not
1. Write the open-loop transfer function in Bode form, and write down expressions for the magnitude and phase. 2. Sketch the Nyquist diagram for an arbitrary value of gain to determine whether the critical point passes to the left or right of an observer moving along the frequency response curve in the direction of increasing frequency. 3. Using computer method, if necessary, determine the frequency that makes the corresponding angle -180º. 4. Substitute this value of frequency into the magnitude equation and determine the corresponding magnitude. Conclusion ( P274: 1. ~ 5. ) 5. If the magnitude is less than unity and the critical point passes to the left, the system is stable; otherwise it is not
14.2 Gain and Phase Margins Ex. 1 Ex 2 Margin Small Great margin margin Great margin Incircle Flg. 14.1 Nyquist diagrams for different values Fig. 14.8 Phase margin for stable and unstable systems
14.2 Gain and Phase Margins Ex. 1 Ex. 2 Great margin Small margin No Margin ψ3 Great margin