Today we re going to talk about time delays, affectionately referred to as e -st in the Laplace domain. Why e -st? Recall: time.
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1 6.302 Feedba Systems Reitation 23: ime Delays oday we re going to tal about time delays, affetionately referred to as e -st in the Laplae domain. Why e -st? Reall: time h(t) delay h(t-) Laplae transform: L{h(t-)} = - h(t-)e -st dτ Let t' = t - dt' = dt L {. } = - h(t') e -s(t ' + ) dt' = e -s h(t') e -st ' dt' - = e -s H(s) Pure delays are tough to deal with in ontrol systems. (Remember the ar with the delay in the steering olumn?) Let s see if we an understand why with our newfound nowledge of feedba theory. CLASS EXERCISE Suppose you had an unompensated system with loop transmission L(s) = 04 e -s ( 0 4. ( π 2 s. 00 (( ) What is the rossover frequeny ω? 2) What is the phase at rossover? 3) What would you do to arrange for 90 of phase ω? Page Cite as: Joel Dawson, ourse materials for Feedba Systems, Spring MI OpenCourseWare ( Massahusetts Institute of ehnology. Downloaded on [DD Month YYYY].
2 6.302 Feedba Systems Reitation 23: ime Delays Delays are tough. hey unload tons of negative phase without doing us the ourtesy of dropping the magnitude. If we add a pole to lower the magnitude, we pi up -90 of phase. If we add a zero to inrease the phase, the magnitude goes up. How do we deal with this in pratie? We don t. Very often, we opt for dominant pole ompensation and just mae sure ω ours before the delay ontributes signifiant phase shift. In our example, where would ω have to our in order to get 45 of pm? We ve already got our dominant pole, an integrator, whih gives us -90 of phase. We must figure out where the delay gives only -45 (- π ) of phase: 4 -ω π ( ω. 00 = ( π 4 π π. = ω = 50 rps So mae L(s) equal to: L(s) = 50 e ( 0 s -s ( π 2 (( Page 2 Cite as: Joel Dawson, ourse materials for Feedba Systems, Spring MI OpenCourseWare ( Massahusetts Institute of ehnology. Downloaded on [DD Month YYYY].
3 6.302 Feedba Systems Reitation 23: ime Delays How else an we loo at delays? It turns out that we an loo at them as a very, very large olletion of poles. Suppose we have the following system: x(s) Σ e -s Y(s) - Y(s) x(s) = e -s + e -s o figure out where the poles are, we rely on the harateristi equation + e -s = 0 e -s = - e -s = e j(2n + ) π (n = { -3, -2, -, 0,, 2, 3, }) Pole-zero diagram: ln [ e -s = e j(2n + )π ] ln( - s) = j(2n +)π -s = - ln + j(2n+)π s = ln - j(2n + ) π jω Page 3 3π/ π/ -π/ σ = / ln σ Cite as: Joel Dawson, ourse materials for Feedba Systems, Spring MI OpenCourseWare ( Massahusetts Institute of ehnology. Downloaded on [DD Month YYYY].
4 6.302 Feedba Systems Reitation 23: ime Delays Note we an mae this system stable by hoosing suh that these poles are in the LHP: ln < 0 ln < 0 < system is rossed over for all frequenies. Being familiar with delay helps us to model a physial situation that we have all witnessed: aousti feedba. r o A(s) mirophone amplifier speaer A(s): ypially a low-pass funtion of some sort. For now, though, let s say that it provides a frequeny-independent gain. Delay: Sound travels at a speed of v meters/se, and the mirophone is r meters away from the o o speaer. he time delay is thus: r o D = v o Moreover, sound amplitude deays in an inverse square manner with distane his is all we need to do our modeling. Page 4 Cite as: Joel Dawson, ourse materials for Feedba Systems, Spring MI OpenCourseWare ( Massahusetts Institute of ehnology. Downloaded on [DD Month YYYY].
5 6.302 Feedba Systems Reitation 23: ime Delays amp normal voie input + Σ mi input speaer input + positive sign e -sd We an use Nyquist to figure out when things will go nuts. L(s) = e -sd jω Im{L(s)} σ - Re{L(s)} D ontour number of irles!! Z = N + P Z = N p = 0 So, we get an infinite number of enirlements of the - point when Page 5 Cite as: Joel Dawson, ourse materials for Feedba Systems, Spring MI OpenCourseWare ( Massahusetts Institute of ehnology. Downloaded on [DD Month YYYY].
6 6.302 Feedba Systems Reitation 23: ime Delays his tells us what we already now by experiene. If we don t want the howling, we should: ) Lower turn down the volume 2) Raise r move the mirophone away from the speaer o It turns out that dynamis in the amplifier don t hange the qualitative results. Suppose A(s) was a single pole amplifier: A(s) = τs + So that L(s) is now e -sd τs +. Nyquist gives jω Im{L(s)} A σ A - (mirror image not sethed) Re{L(s)} Start here beause positive feedba Still, to avoid enirlements we either lower the volume or move the mirophone away from the speaer. Page 6 Cite as: Joel Dawson, ourse materials for Feedba Systems, Spring MI OpenCourseWare ( Massahusetts Institute of ehnology. Downloaded on [DD Month YYYY].
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