Lecture 07: Poles and Zeros

Transcription

1 Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto s a ratoal fucto the complex varable s = σ + jω, that s H ( bms a s m b a s m m s... b s b... a s a It s ofte coveet to factor the polyomals the umerator ad deomator, ad to wrte the trasfer fucto terms of those factors 0 0 () H ( N( D( K ( s z ( s p )( s )( s z) ( s z p ) ( s p m )( s zm) )( s p ) () H ( N( D( K m j k ( s ( s z p j j ) ) (3) where the umerator ad deomator polyomals, N( ad D(, have real coeffcets defed by the system s dfferetal equato ad K = b m /a. As wrtte Eq. () the z s are the roots of the equato. N( = 0, (3) ad are defed to be the system zeros, ad the p s are the roots of the equato D( = 0, (4) ad are defed to be the system poles. I Eq. () the factors the umerator ad deomator are wrtte so that whe s = z the umerator N( = 0 ad the trasfer fucto vashes, that s lm H( = 0. s z ad smlarly whe s = p the deomator polyomal D( = 0 ad the value of the trasfer fucto becomes ubouded, lm H( =. s p All of the coeffcets of polyomals N( ad D( are real, therefore the poles ad zeros must be ether purely real, or appear complex cojugate pars. I geeral for the poles, ether p = σ, or else p, p + = σ ± jω. The exstece of a sgle complex pole wthout a correspodg cojugate pole would geerate complex coeffcets the polyomal D(. Smlarly, the system zeros are ether real or appear complex cojugate pars. Example: A lear system s descrbed by the dfferetal equato. Fd the system poles ad zeros.

2 d y dt 5 dy dt 6y du dt Example: A system has a par of complex cojugate poles p, p = - ± j, a sgle real zero z = -4, ad a ga factor K=3. Fd the dfferetal equato represetg the system. Fgure : The pole-zero plot for a typcal thrd-order system wth oe real pole ad a complex cojugate pole par, ad a sgle real zero.

3 Pole-zero plots: A system s characterzed by ts poles ad zeros the sese that they allow recostructo of the put/output dfferetal equato. I geeral, the poles ad zeros of a trasfer fucto may be complex, ad the system dyamcs may be represeted graphcally by plottg ther locatos o the complex s-plae, whose axes represet the real ad magary parts of the complex varable s. Such plots are kow as pole-zero plots. It s usual to mark a zero locato by a crcle ( ) ad a pole locato a cross ( ). The locato of the poles ad zeros provde qualtatve sghts to the respose characterstcs of a system. Fgure s a example of a pole-zero plot for a thrd-order system wth a sgle real zero, a real pole ad a complex cojugate pole par, that s; System poles ad the homogeeous respose: Because the trasfer fucto completely represets a system dfferetal equato, ts poles ad zeros effectvely defe the system respose. I partcular the system poles drectly defe the compoets the homogeeous respose. The uforced respose of a lear SISO system to a set of tal codtos s y h ( t) where the costats C are determed from the gve set of tal codtos ad the expoets λ are the roots of the characterstc equato or the system egevalues. The characterstc equato s C e t D( s a s... as a0 ad ts roots are the system poles, that s λ = p, leadg to the followg mportat relatoshp: 0 Fgure : The specfcato of the form of compoets of the homogeeous respose from

4 the system pole locatos o the pole-zero plot. The trasfer fucto poles are the roots of the characterstc equato, ad also the egevalues of the system A matrx. The homogeeous respose may therefore be wrtte y h ( t) The locato of the poles the s-plae therefore defe the compoets the homogeeous respose as descrbed below:. A real pole p = σ the left-half of the s-plae defes a expoetally decayg compoet, Ce σt, the homogeeous respose. The rate of the decay s determed by the pole locato; poles far from the org the left-half plae correspod to compoets that decay rapdly, whle poles ear the org correspod to slowly decayg compoets.. A pole at the org p = 0 defes a compoet that s costat ampltude ad defed by the tal codtos. 3. A real pole the rght-half plae correspods to a expoetally creasg compoet Ce σt the homogeeous respose; thus defg the system to be ustable. 4. A complex cojugate pole par σ ± jω the left-half of the s-plae combe to geerate a respose compoet that s a decayg susod of the form Ae σt s (ωt + υ) where A ad υ are determed by the tal codtos. The rate of decay s specfed by σ; the frequecy of oscllato s determed by ω. 5. A magary pole par, that s a pole par lyg o the magary axs, ±jω geerates a oscllatory compoet wth a costat ampltude determed by the tal codtos. 6. A complex pole par the rght half plae geerates a expoetally creasg compoet. These results are summarzed Fg.. C e The pole locatos of the classcal secod-order homogeeous system d y dy dt dt p t y 0 Roots are: p, If ζ, correspodg to a overdamped system, the two poles are real ad le the lefthalf plae. For a uderdamped system, 0 ζ <, the poles form a complex cojugate par, p, j ad are located the left-half plae, as show Fg. 4. From ths fgure t ca be see that the poles le at a dstace ω from the org, ad at a agle ± cos (ζ) from the egatve real axs. The poles for a uderdamped secod-order system therefore le o a sem-crcle wth a radus defed by ω, at a agle defed by the value of the dampg rato ζ.

5 Fgure 4: Defto of the parameters ω ad ζ for a uderdamped, secod-order system from the complex cojugate pole locatos. Fgure 3: Pole-zero plot of a fourth-order system wth two real ad two complex cojugate poles. Example: Commet o the expected form of the respose of a system wth a pole-zero plot show Fg. 3 to a arbtrary set of tal codtos. Soluto: The system has four poles ad o zeros. The two real poles correspod to decayg expoetal terms C e 3t ad C e 0.t, ad the complex cojugate pole par troduce a oscllatory compoet Ae t s (t + υ), so that the total homogeeous respose s y h (t) = C e 3t + C e 0.t + Ae t s (t + υ) Although the relatve stregths of these compoets ay gve stuato s determed by the set of tal codtos, the followg geeral observatos may be made:. The term e 3t, wth a tme-costat τ of 0.33 secods, decays rapdly ad s sgfcat oly for approxmately 4τ or.33secods.. The respose has a oscllatory compoet Ae t s(t + υ) defed by the complex cojugate par, ad exhbts some overshoot. The oscllato wll decay approxmately four secods because of the e t dampg term. 3. The term e 0.t, wth a tme-costat τ = 0 secods, perssts for approxmately 40 secods. It s therefore the domat log term respose compoet the overall homogeeous respose. System stablty The stablty of a lear system may be determed drectly from ts trasfer fucto. A th order lear system s asymptotcally stable oly f all of the compoets the homogeeous respose from a fte set of tal codtos decay to zero as tme creases, or

6 lm t C e p t 0 where the p are the system poles. I a stable system all compoets of the homogeeous respose must decay to zero as tme creases. If ay pole has a postve real part there s a compoet the output that creases wthout boud, causg the system to be ustable. I order for a lear system to be stable, all of ts poles must have egatve real parts that are they must all le wth the left-half of the s-plae. A ustable pole, lyg the rght half of the s-plae, geerates a compoet the system homogeeous respose that creases wthout boud from ay fte tal codtos. A system havg oe or more poles lyg o the magary axs of the s-plae has o-decayg oscllatory compoets ts homogeeous respose, ad s defed to be margally stable. Q. Commet o stablty for the dfferet locato of poles. Q. For the followg system, fd the poles ad zeros. Also establsh the dfferetal equato for the system. R( + 0 s(0.5s ) C( s