系统稳定性与零极点关系

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zero S-plane 2 (8) × 12 Figure 1: The pole-zero plot for a typical third-order systen with one real pole and a conplex conjugate pole pair, and a single real zero 1.1 The Pole-Zero plot A system is characterized by its poles and zeros in the sense that they allow reconstruction of the input/output differential equation. In general, the poles and zeros of a transfer function may be complex, and the system dynamics may be represented graphically by plotting their locations on the complex -planc, whose axes rcprcscnt the rcal and imaginary parts of thc complex variable s Such plots are known as pole-zero plots. It is usual to mark a zero location by a circle (o) and a pole location a cross(x). The location of the poles and zeros provide qualitative insights into the response characteristics of a system. Many computer programs are available to determine the poles and zeros of a system from either the transfer function or the system state equations 8. Figure 1 is an cxamplc of a polc-zcro plot for a third-order systcm with a singlc rcal zero, a rcal pole and a complex conjugate pole pair, that is H(s)= (3s+6) (s3+3s2+7s+5)(s-(-1)(s-(-1-2j)(s-(-1+2j) 1.2 System Poles and the Homogeneous Response Because the transfer function completely represents a system differential equation, its poles and zeros effectively define the system response. In particular the system poles directly define the components in the homogeneous response. The unforced response of a linear SIso system to a set of initial conditions is Wn(t)-∑ where the constants Ci are determined from the given set of initial conditions and the exponents A i are the roots of the characteristic equation or the system eigenvalues. The characteristic equation D(s) 1 ...+a0=0 and its roots are the system poles, that is Ai= Pi, leading to the following important relationship × () stable region unstable region r gure 2: The specification of the form of components of the homogeneous response from the system le locations on the pole-zero plot The transfer function poles are the roots of the characteristic equation, and also the eigenvalues of the system A matrix The homogeneous response may therefore be written n(t)=∑ The location of the poles in the s-plane therefore define the n components in the homogeneous response as described below 1. A real pole pi =-o in the left-half of the s-plane defines an exponentially decaying component Ceot, in the homogeneous response. The rate of the decay is determined by the pole location; poles far from the origin in the left-half plane correspond to components that decay rapidly, while poles near the origin correspond to slowly decaying components 2. A pole at the origin pi =0 defines a component that is constant in amplitude and defined by the initial conditions 3. A real pole in the right-half plane corresponds to an exponentially increasing component Ceo in the homogeneous response; thus defining the system to be unstable. 4. A complex conjugate pole pair ot jw in the left-half of the s-plane combine to generate a response component that is a decaying sinusoid of the form Ae-ot sin(at +o) where A and p are determined by the initial conditions. The rate of decay is specified by o; the frequency of oscillation is determined by w 5. An imaginary pole pair, that is a pole pair lying on the imaginary axis, +jw generates an oscillatory component with a constant amplitude determined by the initial conditions 6. A complex pole pair in the right half planc gcncrates an exponentially incrcasing componcnt These results are summarized in Fig. 2 ■ Example Comment on the expected form of the response of a system with a pole-zero plot shown in Fig. 3 to an arbitrary set of initial conditions ×/2 pla 36 ×一 Figure 3: Pole-zero plot of a fourth-order system with two rea. I and two complex conjugate poles Solution: The system has four poles and no zeros. The two real poles correspond to decaying exponential terms Cle- 3t and C2e-0. IL, and the complex conjugate pole pair introduce an oscillatory component Ae-t sin(2t+o), so that the total homogeneous response Is un(t)=C1e-3t+C2e-0It+Ae-t sin(2t +o) 12 Although the relative strengths of these components in any given situation is determined by the set of initial conditions, the following general observations may be made The term e -3t with a time-constant T of 0.33 seconds, decays rapidly and is significant only for approximately 4r or 1. 33seconds 2. The response has an oscillatory component Ae -t sin(2t+o) defined by the com plcx conjugate pair, and exhibits somc overshoot. The oscillation will decay in approximately four seconds because of the e t damping term 3. The tcrm c-O.It, with a timc-constant T=10 seconds, persists for approximately 40 seconds. It is therefore the dominant long term response component in the overall homogeneous response A S decreasing 5>0 S-plane 9(5) 0 s decreasi Figure 4: Definition of the parameters wn and s for an underdamped, second-order system from the complex conjugate pole locations The pole locations of the classical second-order homogeneous system +2(n,+2y=0, (13 described in Section 9. 3 are given by p1,p (14) If s2 l, corresponding to an overdamped system, the two poles are real and lie in the left-half plane. For an underdamped system, 0<s<l, the poles form a complex conjugate pair, P1,2= 土jn1 (15) and are located in the left-half plane, as shown in Fig. 1. From this figure it can be seen that the poles lie at a distance wn from the origin, and at an angle +cos(S from the negative real axis The poles for an underdamped second-order system therefore lie on a semi-circle with a radius defined by wn, at an angle defined by the value of the damping ratio s 1.3 System Stability The stability of a linear system may be determined directly from its transfer function. An nth order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases, or ∑Cep where the pi arc the systcm poles. In a stable system all componcnts of thc homogeneous responsc must decay to zero as time increases. If any pole has a positive real part there is a component in the output that increases without bound, causing the system to be unstable. In order for a lincar systcm to bc stablc, all of its poles must havc ncgative rcal parts that is they must a. lie within the left-half of the -plane. An "unstable?" pole, lying in the right half of the s-plane, generates a component in the system homogeneous response that increases without bound from any finite initial conditions. A system having one or inore poles lying On the iimaginary axis of the s-plane has n1on-decaying oscillatory componcnts in its homogencous responsc, and is dcfincd to bc marginally stablc 2 Geometric evaluation of the transfer function The transfer function may be evaluated for any value of s-g+ jw, and in general. when s is complex the function H(s) itself is complex. It is common to express the complex value of the transfer function in polar form as a magnitude and an angle H(s)=H(s)ejo(s) with a magnitude H(s)) and an angle (s) given by H(s)=VR{H()2+9{H (s) tanI/SII(Sy 9{H(8)} where rf is the real operator, and S is the imaginary operator. If the numerator and denomi- nator polynomials arc factored into torms(s pi) and(s- 2i) as in Eq(2) h(s=K (s-p1)(s-p2 )(s-pn) each of the factors in the numerator and denominator is a complex quantity, and may be interpreted as a vector in the s-plane, originating from the point zi or pi and directed to the point s at which the function is to be evaluated. Each of these vectors may be written in polar form in terms of a magnitude and an angle, for example for a pole pi=oi twi, the magnitude and angle of the vector to the point s= o +w are (a-0)2+(-)2, tan (22) s shown in Fig 5a. Because the magnitude of the product of two complex quantities is the product of the individual magnitudes, and the angle of the product is the sum of the component angles (Appendix B), the magnitude and angle of the complete transfer function may then be written m =1(-) ∠H(s)-∑(s-2)-∑∠(-p (2 The magnitudc of cach of the componcnt vectors in the numerator and do tor is thc dista of the point s from the pole or zero on the s-plane. Therefore if the vector from the pole pi to the point s on a pole-zero plot has a length i and an angle Di from the horizontal, and the vector from s-plane As(s) s-plane 沉(8) 0 ∠(s-P2) 日 2 H()|=K P2 X P2 ∠(s)=$1-01-02 Figure 5:(a)Definition of s-plane geometric relationships in polar form,(b)Geometric evaluation of the transfer function from the pole-zero plot the zero zi to the point s has a length ri and an angle i, as shown in Fig. 5b, the value of the transfer function at the point s is II(s) F 71·..7 25) q1·q H(s)=(1+…+om)-(61+…+n) The transfer function at any value of s may therefore be determined geometrically from the pole-zero plot, except for the overallgain'"factor K. The magnitude of the transfer function is proportional to the product of the geometric distances on the s-plane from each zero to the point s divided by the product of the distances from each pole to the point. The angle of the transfer function is the sum of the angles of the vectors associated with the zeros minus the sum of the angles of the vectors associated with the poles ■ Example A second-order system has a pair of complex conjugate poles as=-2+j3 and a single zero at the origin of the s-plane. Find the transfer function and use the pole-zero plot to evaluate the transfer function at S=0+j5 Solution: From the problcm dcscription H F (-2-j3)(-(-2-j3) 2+4s+13 (27) The pole-zero plot is shown in Fig. 6. From the figure the transfer function is V0-(-2)2+(5-3)2(0-(-2)2+(5-(-3)2 28 5 S-p × 3 (s) P2× 3 Figure 6: The pole-zero plot for a second order system with a zero at the origin ∠H(s)=tan-1(5/0)-tan-1(2/2)-tan-1(8/2) 20 3 Frequency Response and the Pole-Zero Plot The frequency response may be written in terms of the system poles and zeros by substituting jw for s directly into the factored form of the transfer function H()=h(1u-2)(0=2).-2m=1)(1-2m) (u-pi)(jw-p2).(w-pn 1)(jw-pn) Because the frequency response is the transfer function evaluated on the imaginary axis of the S-plane, that is when s= jw, the graphical method for evaluating the transfer function described bove may be applicd directly to the frcqucncy responsc. Each of the vectors from the n systcm poles to a test point s= jw has a magnitude and an angle +(u-c2) 31) s- pi tan (32) as shown in Fig. 7a, with similar expressions for the vectors from the m zeros. The magnitude and phase angle of the complete frequency response may then be written in terms of the magnitudes and angles of these component vectors (j) pi ∠H(0)=∑∠(-21)-∑∠(ju-P) i=1 A√a S-plane S-plane (@, -p,) 62 D2x P2 Figure 7: Definition of the vector quantities used in defining the frequency response function from the pole-zero plot. In (a) the vector from a pole (or zero)is defined, in(b) the vectors from all poles and zeros in a typical system are shown. As dcfincd above, if the vcctor from the pole pi to the point s= jw has lcngth gi and an angle B, rom the horizontal, and the vector from the zero xi to the point jw has a length ri and an angle Pi, as shown in Fig. 7b, the value of the frequency response at the point jw is H(w) K ∠H(j)=(1+.+φm)-(1+…+bn) (36) The graphical method can be very useful for deriving a qua itative picture of a system frequency response. For example. consider the sinusoidal response of a first-order system with a pole on the real axis at s=-1/T as shown in Fig. Sa, and its Bode plots in Fig. 8b. Even though the gain constant K cannot be determined Iron the pole-zero plot, Che following observations may be made directly by noting the behavior of the magnitude and angle of the vector from the pole to the imaginary axis as the input frequency is varied: 1. At low frequencies the gain approaches a finite value, and the phase angle has a small but finite lag 2. As the input frequency is increased the gain decreases(because the length of the vector incrcascs), and the phasc lag also incrcascs(thc angle of the vector bccomcs largcr 3. At very high input frequencies the gain approaches zero, and the phase angle approaches T /2 As a second example consider a second-order system, with the damping ratio chosen so that the pair of complex conjugate poles are located close to the imaginary axis as shown in Fig. 9a. In this case there are a pair of vectors connecting the two poles to the imaginary axis, and the following conclusions may be drawn by noting how the lengths and angles of the vectors change as the test requency moves up the imaginary axis 1. At low frequencies there is a finite(but undetermined gain and a small but finite phase lag associated with the system. 2. As the input frequency is increased and the test point on the imaginary axis approaches the pole, one of the vectors(associated with the pole in the second quadrant) decreases in length

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