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1、Welcome any forms of ideas and advices! 0917-32723880917-3272388 examples some Exam.5-6 a open-loop TF is： Check closed loop stability。 Fig.5-44 Curve surrounds -1+j0 once clockwise.Polar plot 1 pole in right s plane Fig. 5-44 Nyquist diag. shows， This shows 2 closed loop poles in right s plane,the
2、 system is unstable. Fig.5-44 Exam.5-7 a open-loop TF is：。 Fig.5-45Polar plot Check closed loop stability。 asymptote GH plane Exam.5-8 a open-loop TF is： where Are positive。To make the system stable，how to select ？ set crossing point to negative real axis expand？crossing point to negative real axis
3、5.4.4 bode stability criterion Exam. a open-loop TF is： Check its stability . 5.5.1phase margin and amplitude margin In fig 5-46,K unstable; K Curve crossing with -1+j0; K stable。 The relative position of trajectory and -1+j0 can measure stability margin。In actual system, amplitude margin and phase
4、margin are usually used. 5.5relative stability Phase Margin Assure system gain cut-off frequency definition Phase Margin meaning delays more ，the system will be unstable. To a closed-loop stable system,if open loop frequency characteristic when ， Phase Margin is positive to make minimum phase system
5、 stable, Phase Margin must be positive。 ，Phase Margin is negative.when Positive Gain Margin Positive Phase Margin -1 1 Negative Gain Margin Negative Phase Margin -1 1 Stable System Unstable System Positive Gain Margin Positive Phase Margin Negative Gain Margin Negative Phase Margin Stable System Uns
6、table System 0 dB 0 dB Gain Margin h Assure Phase cross-over frequency definition Margin meaning to stable system, if the open loop gain increases more h times, the system will change into critically stable. If in db When expressing to db, Gain Margin is positive. ， positive Gain Margin (in decibel)
7、 means system stable, negative Gain Margin (in decibel) means system unstable 。 ifGain Margin is negative。 Positive Gain Margin Positive Phase Margin Negative Gain Margin Negative Phase Margin Stable System Unstable System 0 dB 0 dB another method to judge system stability In general,phase margin:30
8、0700 amplitude:515dB examK=5 and k=20 Check stability，solve stability margin. (1) 1 k=5 L(1) = 20lg5 = 14dB -20dB/dec k=20 L(1) = 20lg20 = 26dB -20dB/dec (2) 11 20dB/dec 210 20 dB/dec 14dB 26dB 3.16 K=5 -40 c1 1 14dB 0dB stable 66012.60 K=20 -40 c2 1 26dB 0dB unstable 77.4024.10 Solve g,have a try g
9、 3.2 exam 0.5c1 -40 -60 Exam 4.15 It is infinity, because their polar plots do not cross to negative real axis.Therefore, theoretically 1 or 2-th order system is definitely stable.Certainly, because math model is not accurate,they are not real 1 or 2-th order system.or so-called 1 or 2-th order syst
10、em may be unsteady. To stable minimum phase system, Gain Margin shows the value to which gain can increase until it is unstable. to unstable minimum phase system , Gain Margin shows the value to which gain can decrease until it is stable 。 How many is the Gain Margin of 1 or 2-th order system? Neith
11、er of them can show relative stability.For doing this, we must give both of them at the same time. 5.5.2 some notes on stable margin The Gain and phase Margin measure the relative of polar diag. To -1+ j0 point.They can be used as the design standard. For minimum phase system, when both of them are
12、positive,the system is just stable.The negative margin means the system unstable.Appropriate margin can prevent the influences from device parameter change. To get satisfied performance,the phase margin must be from . Gain margin must be6db。 Exam.5-9 a unit feedback system.its open loop TF K=1,solve
13、 2 margins。solve K to make 20logh=20dB， phase crossing frequencyGain margin at,open-loop log amplitude is Cut-off frequency Based on K=1,TF Phase marginGain crossing frequency from 20logh=20dB Check phase margin request。 According to ： Note when The 2 requests can be satisfied at the same time. Exam
14、.5-11 given a unit feedback system.its log frequency amplitude characteristic as fig. 5-50 (minimum system)。 solve its open loop TF.check its stabilitywhen the system is stable and ,solve steady-state error. from the fig. See log amplitude frequency characteristic -20dB/dec -20dB/dec -40dB/dec -40dB/dec 0.010.11 5 rad/s dB from the fig. when r(t)=t， 0 ,the system is stable.