1、PART2 Control TheoryUNIT3The Frequency Response Methods:Nyquist DiagramsIntroductionThere are times when it is necessary or advantageous to work in the frequency domain rather than in the Laplace domain of the root locus.For system analysis, the root locus method requires a transfer function,which m
2、ay be difficult or even impossible to obtain for certain components,subsystems.In many of these cases,the frequency response can be determined experimentally for sinusoidal test inputs of known frequency and amplitude.The nature of the input also influences the choice of techniques to be used for sy
3、stem analysis and design.Many command inputs merely instruct asystem, to move from one steady-state condition to a second steady-state condition.This type of input can be described adequately by suitable steps in position,velocity,and acceleration,and the Laplace domain is appropriate for this purpo
4、se.If,however,the interval between such step inputs is decreased so that the system never has time to reach the corresponding steady state,the step representation and Laplace domain are no longer adequate.Such rapidly varying command inputs (or disturbance) may be periodic（adj.周期性的）,random（adj.随机性的）
5、,or a combination thereof.The wind loading of a tracking radar antenna,for example,results from a mean velocity component that varies with plus superimposed random gusts. If the frequency distribution of these inputs can be calculated, measured, or even estimated, the frequency response can be used
6、to determine their effects upon the system output. The frequency response is a steady-state response .Although some information can be obtained about the transient response, it is only approximate and is subject to misinterpretation(n.曲解，误议).The Frequency Transfer FunctionIt is necessary to develop（
7、v.导出，引入） an input-output relationship that can be used in the frequency domain, i.e., a frequency transfer function. Consider a linear system with a known transfer function G(s) and apply the sinusoidal inputr (t)=sint or R(s)=where is the amplitude and the input or forcing frequency（强制频率） .The tran
8、sformed output isC(s)=G(s)The partial fraction expansion（部分分式展开式）of C(s) yieldsC(s)= +Where -,-, are the roots of the characteristic equation of the transfer function.The inverse transform isc(t)=+ +.where the first two terms represent an undamped oscillation resulting from the sinusoidal input, and
9、 the transient response. If the system is stable, the transient response will disappear with time, leaving as the steady-state response+The coefficients and are evaluated by the Heaviside expansion theorem as ;With these values for and,Eq.(2-3B-1) becomesSince they are complex functions,;Where the a
10、ngle is the argument of and is equal to arctg(ImG/ReG).Eq.(2-3B-2) can now be written asSince the bracketed（v.加括号） terms are equal to ,the steady-state response can be written as where From these equations we see that sinusoidal input to a linear stable system produces a steady-state response that i
11、s also sinusoidal, having the same frequency as the input but displaced through a phase angle and having an amplitude that may be different. This steady-state sinusoidal response is called the frequency response of the system. Since the phase angle is the angle associated with the complex function a
12、nd the amplitude ratio （c0/r0） is the magnitude of , knowledge of specifies the steady-state input-output relationship in the frequency domain. is called the frequency transfer function and can be obtained from the transfer function G(s) by replacing the Laplace variable s by j0 . Consequently, if c
13、an be determined from experimental data, G(s) can also be found by replacing j0 by s.For a given system, the frequency response is completely specified if the amplitude ratio and phase angle are known for the rage of input frequencies from 0 to + radians per unit time. Consider the stable first-orde
14、r system of Fig. 2-3B-1 with a transfer function G(s) =1/(s+1), the frequency transfer function is , where can be arbitrary(n.任意的) frequency. The amplitude ratio is and the phase angle isAs input frequency is increased from 0to +, we can draw the plot of M and , and a polar plot（极坐标图） that traces th
15、e tip（n.顶端） of the vector representing the frequency transfer function. Polar plots and M and versus (prep对.) plots are used to represent different types of complex functions in the frequency domain. Notice that the constant term in each factor is set equal to unity when working in the frequency dom
16、ain for convenience, whereas in the Laplace domain the coefficient of the highest power of s is set equal to unity.The Nyquist Stability CriterionIn the frequency domain, the theory of residues（余数定理） can be used to detect any roots in the right half of a plane. As with the root locus method, the cha
17、racteristic function in the form 1+ KZ(s)/P(s) is used, where again the function KZ (s)/P(s) may or may not be the open-loop transfer function. To develop the Nyquist criterion , the characteristic function itself is written as a ratio of polynomials so thatD(s)=1+Comparing the identities（n.一致性，等式）o
18、f Eq.(2-3A-2), we see that ,are the roots of the characteristic equation and that ,are the poles of both the characteristic function and KZ(s)/P(s).Poles and roots at the origin have been omitted（v.省略）in the interests of simplicity（n.简单）. In many cases, however, it is difficult to factor the denomin
19、ator polynomial of lose-loop transfer function D(s) to find the location of poles in the s-plane.To prove stability for D(s), it is necessary and sufficient to show that no zeros (for the closed-loop transfer function is poles) are inside the right half of the s-plane. We introduce the Nyquist conto
20、ur（n.轮廓，外形） D shown in Fig.2-3B-2, which encloses（v.围绕） the entire right half of the s-plane. D consists of the imaginary axis from to and a semicircle （n.半圆形）of radius（n.半径） R. In principle, stability analysis is based on plotting 1+KZ(s)/P(s) in a complex plane as s travels once clockwise around t
21、he closed contour D. The factors (s+) and (s+) are vectors from and to s, and for any value of s the magnitude and phase of 1+KZ(s)/P(s) can be determined graphically by measuring the vector lengths and angles in Fig.2-3B-2, if the were known.Note that on the imaginary axis . The plot of 1+KZ(s)/P(s
22、) for s traveling up the imaginary axis from to is effect just the polar plot of the frequency response function 1+KZ()/P(). Hence frequency response function indicated in Fig.2-3B-3 by measurement from the pole-zero pattern.Fig.2-3B-2 shows that if s moves once clockwise around D, vectors (s+) and
23、(s+) rotate 360 clockwise for each pole and zero inside D, and undergo（v.经历） no net（n.净值：adj.净值的） rotation for poles and zeros outside D. If the vector (s+) in the numerator rotates 360 clockwise, this will contribute a 360 clockwise rotation of the vector 1+KZ(s)/P(s) in the complex plane in which
24、it is plotted. If vector (s+p) in the denominator rotates 360 clockwise, this will contribute a 360 counter clockwise revolution（n.旋转） of 1+KZ(s)/P(s) has R zeros and P poles inside the Nyquist contour D ,a plot of 1+KZ(s)/P(s) as s travels once clockwise around D will encircle（v.环绕） the origin of t
25、he complex plane in which it is plotted N=R-P times in clockwise direction.The encirclements of a plot of 1+KZ(s)/P(s) around the origin equal the encirclements of a plots of KZ(s)/P(s) about the -1 point is equal to the number of poles of KZ(s)/P(s) inside the right-half plane, called open-loop uns
26、table poles.In the marginal case where Z(s)/P(s) has poles on the imaginary axis, these will be excluded from the Nyquist contour by semicircular indentations(n.缺口) of infinitesimal（adj.无限小的） radius around them. This is shown in Fig.2-3B-4 for the common case of a pole at the origin. The plot of KZ(
27、s)/P(s) as s travels once around D is called a Nyquist diagram and is needed to use the criterion.Gain Margin and Phase MarginMost practical systems are not open-loop unstable, so that stability requires zero encirclements of the -1 point. To determine this, it is in fact not necessary to plot the c
28、omplete Nyquist diagram; the polar plot, for increasing from 0+ to +, is sufficient.Simplified Nyquist criterion: If KZ(s)/P(s) does not have poles in the right-half s-plane, the closed-loop system is stable if and only if the -1 point lies to the left of the polar plot when moving along this plot i
29、n the direction of increasing .For example, the polar plot of a loop gain function shown in Fig.2-3B-5 indicates a stable system. If the curve passes through -1, the system is on the verge of instability. For adequate relative stability it is reasonable that the curve should not come too close to th
30、e -1 point .Gain margin and phase margin are two common design criteria, which specify the distance of a selected point of the polar plot to -1. Both are defined in Fig.2-3B-5: 1. Gain margin =1/OC.2. Phase margin plus the phase angle of KZ(s)/P(s) at the crossover frequency at which =1. It is also the negative phase shift (i.e., clockwise rotation) of KZ(s)/P(s) which will make the curve pass through -1. Each specifies the distance to -1 of only one point on the curve, so misleading indications(导致错误的读数) are possible. Phase margin is used very extensively in practice.
