线性系统的根轨迹分析法 (14).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)-Part II 5.3 Polar Plot 5.4 Nyquist Stability Criterion 5.5 Stability Margins 5.6 Transient-state and Steady-state Performance An

2、alysis 2 Two methods to draw the Bode plot.Here is a transfer function of an open-loop system like this,ssssG ss5050(10.5)1()0.61()5(10.1)2e.g.1 How to manually draw the logarithmic frequency characteristic plot of a system.Drawing the exact logarithmic frequency characteristic plot by MATLAB.Drawin

3、g approximate logarithmic frequency characteristic plot manually.G sK()51sG s()12sG s10.5()13G ss()(1 0.1)4ssG s50501()0.61()1251)Proportional element 2)Integral element 3)Inertia element 4)Differential element 5)Oscillation element 3 001 05 01 2 1 2.0 1.002-01-0014102Logarithmic magnitude-frequency

4、 characteristic K20lg20lg514dBIt is a straight line with a slope of-20dB/dec through the(1,0)point.break frequency T0.5211T0.11011break frequency T5015011break frequency Step1.Drawing the Bode diagram of each typical elment.G sK()51G ss()1/2sG s10.5()13G ss()(1 0.1)4ssG s50501()0.61()1251）Proportion

5、al element 2）Integral element 3）Inertia element 4）Differential element 5）Oscillation element Transfer function 0 1.2 50 10.（1）（2）（3）（4）（5）G20lg,dB4 4050320001001011.01060504030201001-02-03-04-05-06-ced/Bd06-ced/Bd02-ced/Bd04-2)In the interval where the frequency is greater than 2 and less than 10.3)

6、In the interval where is greater than or equal to 10 and less than 50.4)In the high-frequency range,where is greater than or equal to 50.Step2.Add the sections of Bode plot of each typical element from low-frequency to high-frequency.1)Low-frequency range where is less than 2.L()/dBced/Bd02-ssssG ss

7、5050(10.5)1()0.61()5(10.1)25 0815310954054-09-531-081-522-072-（5）()（3）（4）（1）（2）10100100011.00954054-09-531-081-522-072-Drawing approximate logarithmic phase-frequency characteristics.G sK()51sGs()12sG s10.5()13Gss()(10.1)4ssG s50501()0.61()1251)Proportional element 2)Integral element 3)Inertia eleme

8、nt 4)Derivative element 5)Oscillation element Transfer function Step3.（4）（3）（5）（1）（2）1101000.1)()(6 dG sKsssesT sT sT siimkkkT skmjllllnjn()(1)(12)(1)(12)122122111221The steps of drawing the approximate Bode plot of an open-loop system directly.First of all,the transfer function should be written as

9、 time constant form.sassa saG sb sbsa sannnmmmmm().110111101aspspspG sbszszsznnmn()().()()()().()1212The second step is to find the 20lgK，K is known as open-loop gain in this expression.7 ced/Bd04-35.005030001001011.01060504030201001-02-ced/Bd02-ced/Bd04-ced/Bd04-ced/Bd02-.The third step is to deriv

10、e the break frequency of each typical element,and mark these break frequencies from low to high on the abscissa of the logarithmic coordinate system.Break frequency The fourth step is to determine the low-frequency asymptote.Its slope is The low-frequency asymptote passes the point(1,20lgK).123-20dB

11、/decL()/dB dG sKsssesT sT sT siimkkkT skmjllllnjn()(1)(12)(1)(12)122122111221Tii1/8 In actual drawing,you can write the total phase-frequency characteristics first,and then use the calculator every ten times to calculate a point and connect it with a smooth curve.The sixth step is after the drawing

12、of the Bode plot,if we want to improve the accuracy of the diagram,we can make corrections.Drawing approximate logarithmic phase-frequency characteristic plot.The slope of the highest high-frequency asymptote is The fifth step is to extend the determined low-frequency asymptote to the right,and chan

13、ge the slope of the line at each break frequency in turn,and the amount of the change in the slope of a straight line depends on the type of typical element encountered.nm20dB/dec.L()/dB3率率频频折折转转05030001001011.0060504030201001-02-ced/Bd0ced/Bd06-ced/Bd02-ced/Bd04-Break frequency 9 e.g.2 The known tr

14、ansfer function is as follows,G sss sss2()2000(1)(0.5)(14400)Solution.Step 1 G ssssss2()10(1)(21)(0.00250.0351)Five typical elements G s 1()102()1G ssG ss3()121G ss4()1ssssG s2020()(2 0.35)10.00250.035111()112225 Step 2 20lgK=20lg10=20dB,draw its Bode plot.10 0010402014215.02.01.00402002-04-ssssG ss

15、21(0.00250.0351)()(1)10112 Step3.Mark the break frequency.The break frequency is the reciprocal of the time constant NOTE.L()200 203120.51Tii1/11 0010402014215.02.01.00402002-04-ssssG ss21(0.00250.0351)()(1)10112Step 5.Extend the low frequency asymptote to the right.L()200 203120.51Step 4.Determine

16、low-frequency asymptote NOTE.Because in the first typical element,the break frequency of the inertial element is 0.5,which is lower than 1,so(=1，20)this point will be on the extension of the low-frequency asymptote.12 0010402014215.02.01.009-531-081-522-072-ssssG ss21(0.00250.0351)()(1)10112 1 0.002

17、5()90arctanarctan2arctan0.0352When When Step 6.Drawing phase-frequency characteristics.1 0.0025180arctan0.0352When 200 lim()900 lim()270 L().0.5 1 20 ()-109.4-110.4-181.4 13 When encountering the second-order dderivative element,the slope increases by 40 dB.Determine the number of open-loop gain,the

18、 integral element and the break frequency.TTikjlikjl,1111LK()20lg20 lg20 dB/decDetermine the low-frequency asymptote.The slope of high-frequency asymptote is-20(n-m)dB/dec.When encountering a first-order inertial element,the slope of this line goes down by 20dB.when encountering the oscillation elem

19、ent,the slop decreases by 40dB.After drawing the low-frequency asymptote,along the direction of increasing frequency,the slope of the line changes at each break frequency.,it is the first break line with the slop as,it goes through(1，20lgK).Summarize.The frequencies should be marked on the coordinat

20、e in an ascending order.When there is a first-order derivative element,the slop of the line increases by 20dB.14 e.g.3.Know the transfer functions of the five systems,draw the Bode plot,and summarize the characteristics of the minimum-phase system.G sT sT s121()11G sT sT s221()11()11321G sT sT s()11

21、421G sT sT s521()11G sT sT sesTAT1()1()12122TAT1()1()12222TAT1()1()12322TAT1()1()12422TAT1()1()12522 1()2()3()4()5()TTarctanarctan21TTarctanarctan21TTarctanarctan21TTarctanarctan21TTarctanarctan57.321TAATAAA1()()()1()()()()124522123Their magnitude-frequency characteristics are the same.Features.TT21

22、15 0815310954054-09-531-081-522-072-05-01-51-02-()11421G sT sT sG sT sT s121()11G sT sT s221()11()11321G sT sT s521()11G sT sT ses Features.The variation range of the phase-frequency characteristic is the smallest for the first and the fourth system.In the first system,the changing trend of the loga

23、rithmic magnitude-frequency characteristic is consistent with that of phase-frequency characteristic.11T21T210T1101T11T21T210T1101T)(L)()(3)(4)(2)(1)(516 If there is neither zero nor pole on the right half-open plane,and there is no pure delay element in a system,it is called a minimum-phase system.

24、Its corresponding transfer function is called the minimum-phase transfer function.On the contrary,if there is either a pole or a zero in the right half open S-plane,and there is a pure delay element in a system,its called a non-minimum-phase system.Its corresponding transfer function is called non-m

25、inimum-phase transfer function.In systems with the same magnitude-frequency characteristics,the phase shift of minimum-phase system is the smallest.And there is an internal relationship between the slope of the magnitude-frequency characteristic and the angle of the phase-frequency characteristic of

26、 the minimum-phase system.Definition Features.17 0010402014214.02.01.009-531-081-522-072-L()ced/Bd04-ced/Bd02-ced/Bd06-ced/Bd04-The features of the logarithmic magnitude-frequency characteristic of the minimum-phase system.20dB/decnm20()dB/dec0,()90the number of integral elements n m()90(-),A system

27、 that doesnt satisfy the above conditions is not the minimum-phase system.A system that satisfies the above conditions may not be the minimum-phase system.Logarithmic phase frequency characteristic.High-frequency asymptotic slope Low-frequency asymptotic slope 405038.00001001011.01060504030201001-02

28、-03-04-05-06-L()/dB30010501403100018.01.009-531-081-522-072-()18 Solution.1.Low-frequency slope is-40dB/dec:two integral elements.2.At w=0.8,the slope changed from-40dB/dec to-20dB/dec:a first-order differential element(s/0.8+1).3.At w=30,the slope is changed from-20dB/dec to-40dB/dec:inertia elemen

29、t 1/(s/30+1).4.At w=50,the slope is changed from-40dB/dec to-60dB/dec:inertia element (s/50+1).e.g.4 The approximate logarithmic magnitude-frequency characteristics of the known minimum-phase system are shown in the figure.Try to determine the transfer function of the system.L()/dB405038.00001001011

30、.01060504030201001-02-03-04-05-06-ced/Bd04-ced/Bd02-ced/Bd06-ced/Bd04-.L()/dB19 LK()20lg20lg 1(0.8)20lg20lg 1(30)20lg 1(50)2222G sssss2()3.2(10.81)(1301)(1501)405038.00001001011.01060504030201001-02-03-04-05-06-ced/Bd04-ced/Bd02-ced/Bd06-ced/Bd04-LLK(4)()20lg20lg0.820lg424=4，L()=0,at this time,the i

31、nfluence of the elements where the break frequency is greater than 4 can be ignored.sssG ss3050(1)(1)11()0.8 (1)12.KK2220lg20lg40.820lg420lg40.8 40K0.841K 3.2L()/dBK K?e.g.4 The approximate logarithmic magnitude-frequency characteristics of the known minimum-phase system are shown in the figure.Try to determine the transfer function of the system.20 Summary The methods and steps for drawing the approximate Bode plot of an open-loop system.Summarize the characteristics of the Bode plot of the minimum-phase system.