《自动控制原理》课程教学资源（PPT课件讲稿）Module14 yquist Analysis and Relative stability（I hours）

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

Gain and Phase Margins: The closeness of the GH(jo)cure to-1 is a measure of the relative stability of the system. There are two numbers reflecting this measure Gain and Phase margin GM GM-gain margin ①三0 GM= GH(O) crossover freqency y-phase margin =PM=180°+∠GH(O,)GM=20lo 20logGHGo GHg

180 ( ) u  = PM = +GH j

Gain margin The increase in the system gain when phase 180 that will result in a marginally stable system with intersection of the-1+jo point on the Nyquist diagram

Gain margin • The increase in the system gain when phase = — 180ºthat will result in a marginally stable system with intersection of the -1+ j0 point on the Nyquist diagram

Phase margin The amount of phase shift of the ghg)at unity magnitude that will result in a marginally stable system with intersections of the -1+j0 point on the Nyquist diagram

Phase margin • The amount of phase shift of the GH(jω) at unity magnitude that will result in a marginally stable system with intersections of the - 1 + j0 point on the Nyquist diagram

Other Example 2 0 K K GH(S) GH(S) S(71s+1)(T2S+1) S(71S+1)(T2S+1)

Other Example ( 1)( 1) ( ) 1 + 2 + = s T s T s K GH s ( 1)( 1) ( ) 1 + 2 + = s T s T s K GH s

3 0 K K(71s+ GH(S) GH(S s(TS+l) s(2+D)(n>T2)

( ) ( 1) ( 1) ( ) 1 2 2 2 1 T T s T s K T s GH s  + + = ( 1) ( ) 2 + = s Ts K GH s