Second-Order LTI Systems

Transcription

1 Egieerig Scieces Systems Secod-Order LTI Systems First order LTI systems with costat, step, or zero iputs have simple expoetial resposes that we ca characterize just with a costat. Secod order systems are cosiderably more complicated, but are just as importat, ad are more iterestig. We wo't fid a super-simple shortcut for fidig solutios the Laplace trasform is usually the way to go. But we will gai a lot of isight. Secod order system Cosider a typical secod-order LTI system, which we might write as & y ( + k y& ( + k y( = b(, where b( is some iput fuctio. (Not all secod order LTI systems have exactly this same form this is just a commo example.). Solutio by Laplace trasform substitutig yields s Y sy() y& () + k sy ( k y() + k Y ( = ( B( which we solve for Y( Y ( s + k s + k ) = sy() + y& () + k y() + B( ( Y ( = sy() + y& () + ky() + B( ( s + k s + k ) The solutio to this depeds o what the iput b( is, as well as the iitial coditios. But i ay case we ca re-write the deomiator i factored form: sy() + y& () + ky() + B( Y ( = ( s + a)( s + b) where -a ad -b are the roots of the deomiator polyomial, also called the "characteristic equatio". The roots -a ad -b are also called the poles of Y(; the values of s where Y( goes to ifiity. d Order Systems Hadout Page

2 Egieerig Scieces Systems Some of the commo possibilities for Y( are give i table etries -4. (If the iput is costat, zero, or a step, Y( will i fact be oe of these, or a combiatio of them, depedig o iitial coditios.) We observe that all of the correspodig solutios ca be described by = C + Ae at + Be bt where A, B, ad C are costats. Thus, the solutios are like the first-order solutio, but with two expoetial terms, istead of just oe. We will ow study the behavior as a fuctio of the values of the roots (pole, -a ad -b. We ca divide the possibilities ito three cases: two distict real poles, two equal real poles, or a complex cojugate pair of poles. Case : Distict Real Roots The respose usually looks like a first-order expoetial respose. I fact, if A ad B have the same sig ad the poles are close to each other, it ca be early idistiguishable from a expoetial respose. However, careful examiatio ca ofte show differeces that are ot immediately visible. I Fig. we see a example of a secod order respose, =.5e t / +.5e t. At first it looks like a simple expoetial decay. It looks like the costat is, because the respose is dow to 37% there. But a simple first-order expoetial would be dow to 5% at three costats, ad at t = 3, it is oly dow to about %. So the geeral shape is similar, but it is a differet shape..9 d-order respose st-order respose t Fig.. First vs. secod-order resposes. However, it ca also look quite differet. The resposes with differet parameter values, = + e t e 4t ad =.4e t /4 +.6e 4t are show i Fig.. The first gives a respose with "overshoot." The secod gives a respose with a distict kee; the trasitio from the rapid decay of the last term to the d Order Systems Hadout Page

3 Egieerig Scieces Systems slow decay of the first. I homework two you solved a secod-order system with ode45 ad saw a respose very similar to this curve with a kee Case : Two Real Roots, exactly equal t This case is of little practical importace, because i practice you ca't ever get the roots to match exactly. Thus, it wo't be discussed further here. If you ever eed to aalyze this case you ca use table etries 34 ad 35. Case 3: Two Complex Cojugate Roots = C + Ae at + Be bt with b = a*. Defie: σ Re(a) = Re(b) = Re( p ) = Re( p ) () ad: ω d Im(a) = Im(b) () such that { a, b} = σ ± jω d Now = C + Ae σt jω d t + Be σt + jω d t = C + e σt Ae jω d t + Be jω d ( t ) We ca apply Euler's formula (e jx = cos x + jsi x ) to rewrite this as ( ) (3) = C + e σt (A + B)cosω d t + (B A) j si ω d t The solutio (3) is a little discocertig, because it looks like the solutio could be complex. If it was a solutio for a positio of a object, a complex value would't be physically meaigful. For example, what if B = -A, so the cosie term was zero, ad we had oly the imagiary sie term!? Actually, B ad A are complex, ad are related to a ad b, so it is't so simple to see what happes, but it turs out that with a ad b complex cojugates, (B-A) is always pure imagiary or zero, so the sie term always becomes real after all. (Try a example from the table to see this.) d Order Systems Hadout Page 3

4 Egieerig Scieces Systems Thus, we ca write the solutio as or = C + e σt ( Dcosω d t + Esi ω d = C + A e σ t cos( ω d t + φ) (4) I these expressios, C, D, E, A ad φ are real costats that we have't explicitly foud yet, ad σ ad ω d are real costats that ca be determied from the poles (see eqs () ad ()). Further Study of Case 3: Two Complex Cojugate Roots From (4), we see that the respose comprises a decayig siusoidal oscillatio. The frequecy of this decayig or damped oscillatio is ω d radias per secod, or ω d /(π ) Hz. The decay rate is σ. These two umbers correspod to the horizotal ad vertical positios of the poles o the complex plae. Sice these are the values of s that make Y( go to ifiity, we usually refer to this complex plae as the s-plae. Let's look at how the values o the s-plae ad the resposes correspod. O the ext pages are some series of plots of pole positios ad the correspodig respose (for the sake of cocreteess, these are all for zero iitial states, ad a step ipu. The first set is for costat ω d of.5 rad/sec, ad varyig σ. You ca observe a costat rig frequecy with varyig decay rate, util it gets to the poit where the oscillatio is damped so quickly that the oscillatio is't apparet, ad the rig frequecy ca't be see s = -. ± j s = -.3 ± j s = -.7 ± j s = -.5 ± j s = -.9 ± j Fig. 3. Costat rig frequecy with varyig decay rate. d Order Systems Hadout Page 4

5 Egieerig Scieces Systems Our ext series shows varyig rig frequecy with costat decay rate. Note the costat evelope, the peak amplitude of the rigig, which ca be described by a expoetial decay s = -. ± j s = -. ± j s = -. ± j Fig. 3. Costat decay rate with varyig rig frequecy. d Order Systems Hadout Page 5

6 Egieerig Scieces Systems Secod-Order LTI Systems Part II I the first part of this hadout we discussed describig secod-order LTI systems with complex cojugate poles i terms of σ ad ω d, the egative of the real part ad the imagiary part of the poles. However, aother, more commoly used descriptio ca be motivated by the three plots below, showig respose of the system equatios i the precedig sectio of this hadout with a step iput. For these plots we have added a curve for aother respose, due to some iitial coditios, i order to hit at the diversity of resposes possible with the same pole positios s = -.3 ± j.4; ζ =.6, ω = s = -.6 ± j.8; ζ =.6, ω = s = -. ± j.6; ζ =.6, ω = resposes resposes resposes PO=.95, t r =3.69, t s =.5, ω d = PO=.95, t r =.88, t s =6.6, ω d = PO=.95, t r =.95, t s =3.9, ω d = These resposes look similar i shape, ad are i fact idetical, except for the scale, ad the magitude of the iitial coditio respose. We see that the poles all lie alog a lie extedig out from the origi. We observe that the shape of the respose curve is determied ot be the absolute magitude of σ or ω d, but by their ratio. Thus, we'd like to express the characteristics of the system i terms of a uitless dampig ratio rather tha i terms of σ. If we use two umbers to describe a system, a dampig ratio ad oe other umber, we would like the other umber to be idepedet of the dampig i the system. With referece to actual physical systems, we ote that whe the elemet providig the dampig (e.g., a resistor i a electrical circuit, or frictio i a mechaical system) is chaged, ot oly do σ ad the dampig ratio chage, but ω d chages as well. This is because i the expressio Y ( = sy() + y& () + ky() + B( ( s + k s + k ) k reflects the degree of dampig, ad its value affects both the real ad imagiary parts of the roots. So the actual damped oscillatio frequecy ω d is ot idepedet of dampig. d Order Systems Hadout Page 6

7 Egieerig Scieces Systems Thus, we choose istead the atural frequecy, ω, defied as the frequecy that the system would oscillate if it were ot damped, ad we defie the dampig ratio as ζ = σ. ω This logic may or may ot be covicig, but what makes it practical is that we ca the rewrite the system equatio as && y( + ζω y& ( + ω y( = ω u( which allows us to easily relate ω ad ζ to k ad k or to the actual physical parameters i the system of iterest. Note that u( is ot the uit step fuctio, but rather is just a rescaled iput fuctio, u( = b( ω Similar to before, the solutio by the Laplace trasform may be foud., The solutios are iflueced by the roots of the polyomial s +ζω s + ω. s +ζω s + ω = s + ζω + ζ ω (completig the square ) = s + ζω + j ζ ω s + ζω j ζ ω (factorig ) We ca see i this that ω d = ω ζ. It is also useful to ote that the meaig of ω o the s- plae becomes the distace betwee the pole positio ad the origi. Thus, lie of costat ω are circles aroud the origi at a radius ω. Our previous three cases may ow be described i terms of the value of ζ: Uderdamped case, ζ < : complex cojugate roots Critically damped case, ζ = : idetical real roots Overdamped case, ζ > : distict real roots d Order Systems Hadout Page 7

8 Egieerig Scieces Systems 3. Solutios for a particular example: Step respose (zero iitial coditios, U( = / Ys = ω s s +ζω s + ω 3. Overdamped case (ζ > ) Real, egative roots, s = ζ± ζ ω The formulatio i terms of ad ω ad ζ is of most value i recogizig this case; for aalyzig it it is best to use the table etries that are i terms of the egatives of the roots, such as #, usig a = ( ζ + ζ ) ω, b = ( ) a b = ζ ω. The, ζ ζ ω. ab = ω, 3. Critically damped case (ζ = ) Rarely importat, but for referece, the result is: Idetical real, egative roots s = ω By table etry 33, ωt [ ( + ω t e ] y( = u( ) d Order Systems Hadout Page 8

9 Egieerig Scieces Systems 3.3 Uderdamped case (ζ < ) Complex cojugate roots Usig the table etry 7a, s = ζω ± j ζ ω The followig properties of the step respose ca be calculated oce you kow ζ ad ω : 3.3. Rise, t r : Time for the step respose to rise from % to 9% of its fial value..8 t r for resposes with moderate overshoot. ω 3.3. Settlig, t s : Time for respose to settle withi ±% of its fial value. A decayig expoetial reaches % i approximately five costats. To be precise, solve exp( ζω t s ) =., to obtai: t s = 4.6 ζω Peak, t p : Time util the respose hits its maximum overshoot. Differetiate with respect to t, obtaiig (after some algebra), ω exp( ζω si(ω d = The sie fuctio is zero for ω d t = kπ, k= iteger. Odd values of k correspod to peaks i the step respose. The first such peak is at k=. It follows that π π t p = = ω d ζ ω Peak overshoot, M p : The peak value of the step respose, y(t p ), is (+M p ) y, where y is the fial value. M p is frequetly expressed as a percetage. Kowig t p = π/ω d, we ca fid y(t p ) = + M p = + exp πζ, so ζ M p = exp πζ ζ ζ for ζ.6.6 d Order Systems Hadout Page 9