四川大学自动化专业英语Chapterthree

1、Chapter three Transfer Performance and The S-Plane 3.1 Introduction In this chapter the transient behavior of system described by given transfer function is considered, whether these describe a single block or have been obtained by block diagram reduction. The purpose is to establish correlations an

2、d to specify requirements that ensure satisfactory performance. The use of feedback to satisfy such requirements will be considered later. 3.1 Introduction The correlations between transfer functions and response characteristics are developed in terms of the positions of the system poles and zeros i

3、n the s-plane. These are powerful concepts which will also be used in the design of feedback to improve unsatisfactory behavior. Inverse transformation of C(s)=G(s)R(s) was used in Section 1.7 to find the response c(t) to an input r(t), for the case where the roots of the denominators of C(s) are re

4、al and distinct. The cases of repeated roots and complex conjugate pairs will now also be considered, and the s-plane will provide the basis for a graphical alternative to the analytical method of calculating the residues. The alternative enhances insight and is often preferred for more complicated

5、transforms. 3.1 Introduction3.2 The S-Plane, Pole-Zero Patterns , and Residue Calculation In the input-output relation C(s)=G(s)R(s), each of C,G ,and R is in general a ratio of polynomials in s. Definitions: Zeros of C,G and R are the roots of the their numerator polynomials. Poles of C,G and R are

6、 the roots of the their denominator polynomials. System zeros and system poles are those of the system transfer function G(s). System characteristics polynomial is the name often used for the denominator polynomial of G(S). System characteristic equation identifies the result if the system character

7、istics polynomial is equated to zero. Evidently, its roots are the system poles. 3.2 The S-Plane, Pole-Zero Patterns , and Residue Calculation Since the polynomials have real coefficients, poles and zeros are either real or occur in complex conjugate pairs. They are values of s and can be plotted on

8、 a complex plane called the s-plane. Because s=+j, the real axis of the plane is the -axis, and frequencies are plotted along the imaginary -axis. 3.2 The S-Plane, Pole-Zero Patterns , and Residue Calculation It is important to note the angles in the s-plane have great significance, and therefore th

9、e scales on both axes must be identical. Figure 3.1(a) shows this plane and the system poles and zeros of the transfer function3.2 The S-Plane, Pole-Zero Patterns , and Residue Calculation )3)(1()2(1)3/4()3/1 (15 . 032)(2sssKSSSKsG This is the system pole-zero pattern. The potential power of represe

10、nting all dynamic characteristics by the positions of a number of points in this manner may be appreciated. For a unit step input R(s)=1/s, the pole-zero pattern of R(s) consists of just a pole at the origin if the s-plane. Since C(s)=G(s)R(s), the pole-zeros pattern of C(s) in Fig.3.1(b) is simply

11、the superposition of those of G(s) and of R(s). 3.2 The S-Plane, Pole-Zero Patterns , and Residue Calculation To derive this important alternative to the analytical method in Section 1.7, it is recalled that in the partial fraction expansion 3.2 The S-Plane, Pole-Zero Patterns , and Residue Calculat

12、ion )3)(1()2()()()(ssssKSRSGsC33121)3)(1()2()(sKsKsKssssKsC Of the output transform, the residues are310|) 1()2(3|)3()2(2|)3)(1()2(1ssssssKKsssKKsssKK3.3 Transient Response, Including Repeated and Complex Poles transient response are calculated for the cases of repeated real poles, a complex conjuga

13、tes pair of poles, and combinations of distinct real poles and complex conjugate pairs. Repeated Real poles Suppose that C(s) has m poles at pl )()()()()(2nmlpspspssZsCnmmpsnKmpsKmpsKmpsKpsKsC11)()(2)(1)(21111mipssCdsdiKiplsmlii,.,1| )()!1(111Then it can be shown that the partial fraction expansion

14、must be written as followsThe residues Km+1,Km+n-1 are found as before, but for the repeated root the equation is3.3 Transient Response, Including Repeated and Complex Poles A Complex Conjugate Pair of Poles Distinct Real Poles and Complex Conjugate Pairs For distinct poles, whether real or complex,

15、 the partial fraction expansion of C(S)=G(S)R(S) and the corresponding solution c(t) are 3.3 Transient Response, Including Repeated and Complex Poles)exp(.)exp()exp()(21)(221121tpKtpKtpKtcpsKnpsKpsKsCnnn All residues can be found by graphic rule. Those corresponding to real poles will be real. For c

16、omplex conjugate pairs, if pi and -pi+1 from such a pair, Ki and Ki+1 are also complex conjugate, as in (3.17), because all poles and zeros are real or occur in pairs which are complex conjugate.3.3 Transient Response, Including Repeated and Complex Poles3.4 Simple Lag: First-Order Systems(练习) Chapt

17、er 2 has shown that this system, with its pole-zero pattern, is very common. For a step input R(s)=1/s Hence the transient response is The first term is the forced solution, due to the input, and the second the transient solution, due to the system pole. Fig.3.4 shows the transient as well as c(t).

18、The transient is seen to be a decaying exponential. If it takes long to decay, the system response is slow, so the speed of decay is of key important. The commonly used measure of this speed of decay is the time constant. 12, 11/121)(KKTsKsKsCTtetc/1)( The time constant is the time in second for the

19、 decaying exponential transient to be reduced to e-1=0.368 of its initial value. Since e-t/T=e-1 when t=T, it is seen that The time constant for a simple lag 1/(Ts+1) is T seconds. This is, in fact, the reason a simple lag transfer function is often written in this form, The coefficient of s then im

20、mediately indicates the speed of decay. It takes 4T seconds for the transient to decay to 1.8% of its initial value.3.4 Simple Lag: First-Order Systems Stability- If -1/T is positive, the pole lies in the right half of the s-plane. The transient e-t/T then grows instead of dacays as t increases. The

21、 system is unstable and useless. Hence the most important rule for design: For system stability, the system pole(s) must lies in the left half of the s-plane.3.4 Simple Lag: First-Order Systems Speed of response- To speed up the response of the system(i.e., to reduce its time constant T), the pole -

22、1/T must be moved left. How such movement is to be achieved is a problem of design, considered later. Example 2.8.1 and comparison of Example 3.2.1 and 3.3.2 suggest, however, the use of feedback around the simple lag plant, together with an adjustable gain.3.4 Simple Lag: First-Order SystemsThis ve

23、ry common transfer function can always be reduced to the standard form: Where wn= undamped natural frequency =damping ratio The signification of these paramteres will be dicussed. For a unit step input R(s)=1/s, the transform of the output is 3.5 Quadratic Lag: Second-order Systems2222)(nnnsssG)2()(

24、222nnnssssC For 1 overdamped; =1, critically damped 1 these are on the negative real axis, on both sides of wn. For =1, both poles coincide at wn. For 1, when the poles are real and distinct, the transient is a sum of two decaying exponential, each with its own time constant. nnnS2222, 1)1()(3.5 Qua

25、dratic Lag: Second-order Systems As for the simple lag, it is again important to determine the correlation between dynamic behavior and the pole positions in the S-plane in Fig.3.5 Absolute stability The real part of the pole position is n. If it is positive, then from (3.30) for c(t), the transient

26、 will grow instead of decay due to the e-nt. Hence, for stability the poles lies in the left half of the s-plane. Relative stabilityTo avoid excessive overshoot and unduly oscillatory behavior, damping ratio must be adequate, since =cos, the angle may not be close to 90. Settling timeTime constant a

27、nd settling time are reduced by increasing the real part of the pole position Frequency of the natural oscillation The frequency, also called the resonant frequency or damped natural frequency, equals the imaginary part of pole position. Undamped natural frequency n This equals the distance of the p

28、oles to the origin. Moving the poles out radially(i.e. with constant) reduces settling time, peak time, and rise time while the percentage overshoot remains constant. Speed of responseFor a constant real part, this is improved by increasing the imaginary part until is reduced to a permissible level,

29、 thus reducing peak time and rise time. A secondorder system with _ larger than one is referred to as _ systemA . damping ratio /underdamping B. damping ratio/ ovredampingC. damped frequency/ underdamping D. damped frequency/overdamping For a overdamping second-order system, its characteristic equat

30、ion has two _ and _ polesA. equal /complex B. unequal / complex C. equal/real D. unequal /real As damping ratio is _ the system overshoot _A. decreased/ increases B. increased /increaseC. increased / decreases D. decreased / decreases 3.7 Routh-Hurwitz Stability CriterionThe Routh -Hurwitz stability

31、 criterion is a method for determining from the coefficients of the characteristic equation how many system poles are in the right-half s-plane or on the imaginary axisOf the Routh array below, the first two rows are produced by arranging the coefficient in the order indicated by the arrows. Each of

32、 the remaining entries bi,ci is found from the two rows preceding it according to the a pattern that can recognized from the following equations Calculations in each row are continued until only zero elements remain. In each of the last two rows the second and following elements are zeros. It can be

33、 shown that elements in any row can be multiplied by an arbitrary positive constant without affect the results below. This can be useful to simplify the arithmetic. For large n, computer algorithm can be written, based on (3.38) The Rouh-Hurwitz criterion states: A necessary and sufficient condition

34、 for stability is that there be no changes of sign in the elements of the first column of the arrayThe number of these sign changes is equal to the number of roots in the right-half s-plane.If the first element in a row is zero, it is replaced by a very small number , the sign changes when -0 are co

35、unted after completing the array. If all elements in a row are zero, the system has poles in the right half plane or on the imaginary axis. 3.8 Effect of system Zeros Much attention has been given to system poles and the correlations between their positions in the s-plane and the nature of the trans

36、ient response. A natural question is therefore what correlation exist between the response and the positions of the system zeros. The unit step response of a system with transfer function: Will be studied to examine the effect of adding a zero to an underdamped quadratic lag. Example 3.8.2 Find the

37、unit step response of the system The pole-zero pattern of C(S)=G(S)R(S) is whoen in Fig.3.11 .the residue are So the unit step response is It is seen that if a=1, so if the zros coincides with, or cancels, the pole at -1, the transient corresponding to that pole is zero. Similarly, if a=4, the zeo c

38、ancels the pole at -4, making the corresponding transient zero. )4)(1(4)(ssasasGaaKaaKK3423) 1(4211 So the unit step response is It is seen that if a=1, so if the zros coincides with, or cancels, the pole at -1, the transient corresponding to that pole is zero. Similarly, if a=4, the zero cancels th

39、e pole at -4, making the corresponding transient zero. The following remark generalize the implication of this example for the significance of the zeros in the system pole-zero pattern:tteaaeaatc4343) 1(41)(A residue Ki at pole pi corresponds to a transient term Kiexp(-pit), so the significance of t

40、he residue is that its magnitude is the initial size of the transient corresponding to the pole.If a zero is close to a pole, the residue at the pole tend to be small, because of a short vector in the numerator, so that the corresponding transient is probably small.If the zero coincides with the pol

41、e, it cancels it, and the transient term is zero.Thus the significance of zeros is that they affect the residues at the poles, and hence the sizes of the corresponding transients. A “slow” pole (close to the imaginary axis, with a large time constant) or a highly oscillatory pair (small damping ratio) may be acceptable if nearby zeros make the corresponding transient small Because pole