线性系统的根轨迹分析法 (18).pdf

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1、1 Chapter 5 Frequency Domain Analysis of Automatic control systems 5.1 Basic Concepts of Frequency Characteristics 5.2 Logarithmic Frequency Characteristic Plot(Bode Plot)5.3 Polar Plot 5.4 Nyquist Stability Criterion 5.5 Stability Margins 5.6 Transient-state and Steady-state Performance Analysis 2

2、0plane-)s(kGImRe）0 j，1-（Open-loop transfer function of the system .Polar plot sT sT sG sKk(1)(1)()12 When ,and doesnt enclose(-1，j0),then the closed-loop system is stable.K K=1 When ,and passes(-1，j0),then the system is in critical steady-state.K K=2 When ,and encloses(-1，j0),then the closed-loop sy

3、stem is unstable.K K=3K1K2K30+3 Replane-)s(kGIm0The frequency corresponding to the polar plot that passes through the negative real axis(),and it is expressed by .o ()180gThe frequency corresponding to amplitude ,which is expressed by .A()1cPhase crossover frequency:Magnitude crossover frequency:The

4、 conditions for minimum-phase system stability are When ,o g()180Ag()1 When ,Ac()1o c()1800+gcAg()（1，j0）4 The reciprocal of the magnitude-frequency characteristic at the phase crossover frequency g.KAgg1()Magnitude stability margin cc()(180)180()Phase stability margin That is:The error between the p

5、hase-frequency characteristic at the magnitude crossover frequency c and 180.）0 j，1-（Replane-)s(kGIm00+gcAg()What is the physical meaning of the stability margin？That is:5 The amplitude stability margin is the multiple that the open-loop gain is allowed to increase when the closed-loop system reache

6、s critical stability.The physical meaning of the magnitude stability margin The physical meaning of the phase stability margin For a stable system,if the phase angle is reduced by degrees,the system is critically stable;if it is reduced again,it will become unstable.KAgg1()0plane-)s(kGImRe1-K1K20+gA

7、g()AKgg()cFor a stable system,if the amplitude increases Kg times,the system is in a critically stable state.If the increase is greater than the Kg times,the system will become unstable.The phase stability margin is the maximum phase lag that the open-loop frequency characteristic of the system allo

8、ws to increase at wc when the closed-loop system reaches critical stability.6 )()(L90-180-270-00dB line 2.The negative real axis on the polar plot corresponds to what on the logarithmic graph.1.The unit circle on the polar plot corresponds to what on the logarithmic coordinate diagram.The expression

9、 of stability margin on the Bode plot Replane-)s(kGIm0cgLg？-180 phase line？AKgg()1 LKAggg20lg20lg()cc()(180)180()Magnitude stability margin(gain stability margin)Phase stability margin Ag()1A20lg()07 )()(L90-180-270-0 and on the Bode plot cgFor the minimum-phase system,when the magnitude stability m

10、argin is ,the closed-loop system is stable.Otherwise it is unstable.However,and occur at the same time or not at the same time,so often only one stability margin is used to express the stability margin of the system.Commonly used phase margin.Lg0,0Lg00The expression of stability margin on the Bode p

11、lot cgLg8 Example:The open-loop transfer function of the unit feedback system is ,s ssG sKg(1)(10)()and separately determined phase margin when .,KKKggg330300It is required to be converted into the value when the open loop gain K=0.3,K=3 and K=30.When When When c0.288c1.583c5.1272.323.3 16K0.3K3K30s

12、 ssK(1)(0.11)KKg/10 c180()jjjK(1)(0.11)1cSolve:s ssG sKs(1)(10)()(Zero pole form)s ssG sKg(1)(0.11)()10(Time constant form)9 0010111.009-021-051-081-012-042-072-060402002-04-06-08-001-021-c72.30.2881 c165.123K0.3K3K30c23.31.583210 080010111.009-081-072-060402002-04-06-08-001-021-When the slope of L(

13、w)at wc is-20dB/dec,the system is stable.When the slope of L(w)at wc is-40dB/dec,the system may be stable or unstable.Even if it is stable,the phase margin is small.When the slope of L(w)at wc is-60dB/dec,the system is unstable.Unless the-60dB/dec section is very short,and the slope of the break lin

14、e connected at both ends of the section is greater than-40dB/dec;at this time,even if it is stable,the phase margin is also very small.is positive is positive is negative is negative 11 Summary Concept of stability margin.Definition of phase crossover frequency,magnitude crossover frequency,magnitude stability margin and phase stability margin.How to use stability margin to analyze system stability on Bode plot.