线性系统的根轨迹分析法 (19).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、If such a relationship exists in the frequency characteristics of the two systems:jj()()012h tht12()()the step response of the two systems will have such a relationship,The times the frequency characteristic of the system is broadened equals the times its step response is increased.0011.0101Bd/M0100

3、1-02-03-04-05-06-y4.12.118.06.04.02.00t41312111019876543210ss0.6112ss0.6112ss2.416162ss2.4161621.The inverse proportional relationship between frequency-scale and time-scale.3 2.The relationship between frequency characteristic and time-domain performance.The low-frequency region of the frequency re

4、sponse(the region where the frequency is much lower than magnitude crossover frequency)corresponds to the steady-state characteristics of the closed-loop system.The high-frequency region of the frequency response(the region where the frequency is much higher than magnitude crossover frequency)corres

5、ponds to the initial output response characteristic of the closed-loop system.The intermediate frequency region of the frequency characteristic(the region where the frequency is near magnitude crossover frequency)corresponds to the stability and transient-state performance of closed-loop system.0011

6、.0101Bd/L030201001-02-03-04-cthe low-frequency region the intermediate frequency region the high-frequency region The steady-state characteristics of the closed-loop system The stability and transient-state performance of closed-loop system.The initial output response characteristic of the closed-lo

7、op system 4 0mIeRme t sys 0 epyt+me t sysepyt+me t sysepyt+Use the frequency characteristic to analyze the steady-state performance of the system.1.use the open-loop frequency characteristic to analyze the steady-state performance of the system.use the polar plot to analyze the steady-state performa

8、nce of the system.5 met sysepyt )b(1)Bd()(Lmet sysepyt )c(1)Bd()(L04-04-02-02-04-04-02-1.use the open-loop frequency characteristic to analyze the steady-state performance of the system.use the polar plot to analyze the steady-state performance of the system.use the Bode plot to analyze the steady-s

9、tate performance of the system.calculate:the number of integral elements met sys 0 epyt )a(04-02-)Bd()(LKgl02Kgl02Use the frequency characteristic to analyze the steady-state performance of the system.6 04-04-02-When ,1LK(1)20lg,KL1020(1)jLKk()()20lg|20lg20 lgmet sysepyt )b()Bd()(Lmet sysepyt )c(1)B

10、d()(L04-04-02-02-04-04-02-When ，K is determined by the intersection of the and the horizontal axis.1L()0met sysepyt )b(1)Bd()(L1.use the open-loop frequency characteristic to analyze the steady-state performance of the system.use the polar plot to analyze the steady-state performance of the system.u

11、se the Bode plot to analyze the steady-state performance of the system.calculate:the number of integral elements the open-loop amplification factor K low-frequency asymptote:When ，0L0()0K020lg20 lg0K0met sys 0 epyt )a(04-02-)Bd()(LKgl02Kgl02Kgl02met sysepyt )a()Bd()(L02-04-04-02-，Use the frequency c

12、haracteristic to analyze the steady-state performance of the system.K0K07（1）M(0)=1 or M(0)1，M()=0；（2）For M()with a maximum value,we call this peak value Mp as the resonant peak.The corresponding frequency p is resonant frequency.(3)When M()decreases to 0.707M(0),the corresponding frequency b is call

13、ed bandwidth frequency.Bd/M45.335.225.115.00 The features of the closed-loop characteristic curve 2.use the closed-loop frequency characteristic to analyze the steady-state performance of the system.The zero frequency value M(0):the zero frequency value of the closed-loop magnitude-frequency charact

14、eristic tscc tsssM00()lim()lim()1lim()(0)The steady-state error:ssreM1(0)When 0,MK K(0)/(1)1ssreMK 1(0)1/(1)When 0,M(0)1ssreM 1(0)0For this unit step input of the unit feedback system,we can use M(0)to determine the steady-state error of the system.0.707M(0)b p Mp M(0)calculate the steady-state erro

15、r by the zero frequency value 8 The relationship between the transient-state performance index and frequency domain index for a typical second-order system The transfer function for a typical second-order open-loop system:s sG s(2)()nkn2Open-loop frequency characteristic:jjGj()(2)()nkn2Open-loop mag

16、nitude-frequency characteristic:A()(2)()nn22 22n2Open-loop phase-frequency characteristic:180arctan22()90arctannnMagnitude crossover frequency:412nc42substitute into j j()o 412()180arctan242cPhase angle margin:142arctan242Closed-loop transfer function:sss2()nn22n2Closed-loop frequency characteristic

17、:jjj()2()()nn22n2Closed-loop magnitude-frequency characteristic:M()(2)()nn2222n2When 012n12p2M2112pn1 224+4b2249 87654321)ged(0706050403020100800106pM04%02018.06.04.02.00,%Relationship between and%)ged(0806040200604020%0403020108.16.14.12.11pMMp,%Mp 100Relationship between and for underdamped second

18、-order system%Relationship between and Mp for second-order system%10 c st)ged(816141210186420080604020cb 42242(412)(2441 2)If both systems have the same phase margin,their relative stability(or the overshoot)is basically the same.However,the rapidity of the response(or the settling time)is inversely

19、 proportional to the magnitude crossover frequency c.The higher the magnitude crossover frequency c is,the shorter the settling time ts is.c0.63 b Relationship between ts c and for underdamped second-order system 11 Summary Relationship between frequency characteristic and time domain response.Use the frequency characteristics to analyze the steady-state performance of the system.Relationship between the transient-state performance index and the frequency domain index for a typical second-order system.