自动控制理论9

1、Lecture 13/14Frequency Response and Nyquist DiagramsNorth China Electric Power UniversitySun Hairong1Topics of this class1.Frequency response.2.Nyquist diagrams from transfer functions. (Reading Module 12)3.Nyquist stability criterion. (Reading Module 13)2The frequency response can be obtained in tw

2、o waysOne, considered a systems transfer function G(s), substituting the value s=j in the function, which may then be evaluated in terms of the magnitude and phase angle,1. Frequency response 3Two, considered a systems input is r(t)=Mr sin(t+1)And the steady output is Css(t)= Mc sin(t+2)The magnitud

3、e and phase angle can be obtained as,4Example 1 : RC circuit is known as following figure, determine its frequency response.Now we get the systems frequency response in two ways5 One, The systems transfer function is obtained Substituted the value s=j in the function,The magnitude and phase angle ma

4、y be evaluated6 Two, considered the systems input is ur (t)= sin tAnd the steady output have been gottenThe magnitude and phase angle can be obtained as, 7Obtained by transfer functionObtained by the input and the steady outputFrom the concept and the example above, we know that 8OneThere are three

5、parameters involved in the those equations, the independent variable frequency the magnitude M the phase angle .TwoThe magnitude M is the magnitude of the function, and also the ratio between the magnitude of the steady output and the magnitude of the input.The phase angle is the difference between

6、the input and the output.9ThreeThe frequency response can be expressed as In above example the frequency response can be expressed as 10Fourthe frequency response G(j) is called frequency characteristic, the magnitude M() is called magnitude-frequency characteristic, the phase () is called phase-fre

7、quency characteristic.11(Nyquist diagram & Bode diagram)2. Representation of frequency response In this class, we will learn whats the Nyquist diagram? Systems transfer function is The frequency response can be expressed asAssuming =1 , (which also mean that the frequency of the input signal is 1rad

8、/sec.), The frequency response is 12 The magnitude M and the phase angle , when =1 , can be shown in the following figure:Considering in different variable, when it changes from 0 to infinity133. Nyquist diagrams of the elementsAll common transfer functions can be divided into a set of simple elemen

9、ts. The procedure, therefore, is to take the element at a time and calculate the magnitude and the phase. And see what the conclusion is.14Gain elementThe integral elementFirst-order lag elementThe inverse of the first-order lag elementThe second-order lag element15Examples An open-loop system has a

10、 transfer function of the form , sketch the Nyquist diagram 16When we choose the positive frequency ,:0, the Nyquist stability criterion may be stated as follows 4.Nyquist stability criterion To investigate systems stabilityFor a system having P open-loop poles in the right-hand of the s plane to be

11、 stable, the open-loop frequency response must encircle the point (-1,0j) P/2 times in a counterclockwise direction.The point (-1,0j) is called critical point.17Some comments on Nyquist stability Comment #1The number of the systems closed-loop poles in the right-hand half plane can be obtained by th

12、e equation Z=P-2N Where Z=number of closed-loop poles in the right-hand half plane N=number of counterclockwise encirclements of the critical point. P= number of open-loop poles in the right-hand half plane. So, the Nyquist stability criterion may be written as: A system is stable if Z=0 18Comment #

13、2To determine N, the number of clockwise encirclements of the critical point. Where NCCW=the number of times the Nyquist path crosses the line (-，-1) in counterclockwise direction. NCW=the number of times the Nyquist path crosses the line (-，-1) in clockwise direction. 19Comment #3If the system has integral elements, that is to say the system has zero open-loop poles.Assuming the open-loop transfer function is written aswhen , G H(j)，the Nyquist path is discontinuous. How to determine i