6.081 March Lecture 6
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1 Review exampe : Aut Zeda s ba accout growig from some iitia baace y [ ] y[.05y[ y[ ].05y[ Natura frequecy is.05 soutio grows as.05 Review exampe : Your robot from ab d d right d eft ] δtθ[ θ[ ] θ[ δtk Kafter some agebrak ] ] atura frequecies δt K were compex, so system osciates ustaby aguar veocity gai d Kd outside uit circe From differece equatios to iputoutput systems Rather tha a sige differece equatio, thi about trasformig a sequece of iputs to a sequece of outputs y[. This modes ots of situatios x [ y[
2 Exampe : Aut Zeda redux Aut Zeda does t die, but cotiues to mae deposits ad withdrawas: x [ y[ ba y [ ].05y[ Exampe : A more geera robot cotro program Rather tha taig aguar veocity x, to be Kd, et it be set by your program i some other way x [ robot ] δtθ[ θ[ ] θ[ δt Your cotro agorithm{measuremets} Exampe 3: oud processig Low-pass fiter y[ y[ b ] b ] b0 b ] b ] weighted average of 5 poits aroud
![/ Liear shift-ivariat systems Geera form of the trasformatios we re deaig with M a y[ ]](/docs-images/95/124867011/images/3-1.jpg "N 0 b ] Thi of this as a way of trasformig oe sequece ito aother x [ y[ ystem Combiig")
3 Exampe 4: D image processig Nowa, R. 005, Juy. Image Restoratio Basics. Retrieved from the Coexios Web site: Liear shift-ivariat systems Geera form of the trasformatios we re deaig with M a y[ ] N 0 b ] Thi of this as a way of trasformig oe sequece ito aother x [ y[ ystem Combiig systems w[ y[ compositio cascade y[ parae sum 3
4 Combiig sequeces Additio: y [ x [ x [ caig: hift: y [ y [ ] Framewor for abstractio sequeces systems primitives Meas of combiatio Meas of abstractio additio scaig shift cascade parae sum ow do we mae combiatios easier to wor with? Meas of capturig commo patters The big idea Ivet a way to mode sequeces ad systems i terms of somethig that modes the meas of combiatio as ordiary agebra This ets us aaye systems usig ordiary agebra A method for doig this is caed the Z- trasform 4
5 5 The Z-trasform of a sequece Let the be the coefficiets of a power series i a variabe caed. The resutig fuctio of is the Z-trasform, writte This is the biatera Z-trasform x ] [ x the coefficiet of ]is [ that Note Exampe L L Exampe 3
6 Qui What is the Z-trasform of Why the Z-trasform is ice additio ad scaig trasform to additio ad scaig Why the Z-trasform rocs hiftig trasforms to mutipicatio by : 6
7 7 Proof Qui What is the Z-trasform of What does this mea for systems? b Y a N M x b y a 0 0 ] [ ] [ ystem [ x ] y[ a b Y For the trasforms, we have
8 Wecome to the frequecy domai x [ y[ ystem Y Y is caed the system fuctio The system fuctio is a ratioa fuctio ratio of poyomias The poes of the system fuctio roots of the deomiator determie whether the system is stabe Aut Zeda s ba Ba Y y[ ].05y[ Y.05Y Y Ba.05 Qui y [ ] 3y[ ] y[ ] 3 What's the system fuctio? 8
9 9 What happes with system fuctios whe we combie systems? w[ y[ compositio cascade y[ parae sum ystem fuctio of a sum y[ Y ystem fuctio of a cascade w[ y[ Y W Y
10 Exampe : A more geera robot cotro program Rather tha taig aguar veocity x, to be Kd, et it be set by your program i some other way x [ robot ] δtθ[ θ[ ] θ[ δt Your cotro agorithm{measuremets} Exampe: Aayig the robot cotro program i the frequecy domai Θ D Robot δt ] δtθ[ D Θ δt θ[ ] θ[ δt Θ δt δt δt D ] ] δt Framewor for abstractio sequeces systems primitives Meas of combiatio Meas of abstractio additio scaig shift Z-trasform cascade parae sum ystem fuctio Meas of capturig commo patters 0
11 Exampe cotiued Let be the differece d right [-d eft [. Let be the robot s aguar veocity x [ robot ] δtθ[ θ[ ] θ[ δt Your cotro agorithm{measuremets} D Robot Robot D δt ere s the method you used i ab ast wee d [ ] K e[ K desired I.e., set to be some costat K times the error, where the error is the differece betwee what we wat ad what we have. Let s use frequecy-domai methods to redo the same aaysis we did previousy. Boc diagram for compete robot cotro system
12 Negative feedbac cofiguratio P Q P Q Bac s formua - Compute system fuctio for overa robot cotro system with feedbac where t K K D D Robot desired δ Compute system fuctio for overa robot cotro system with feedbac cot t K t K t K t K D D desired δ δ δ δ
13 Compute system fuctio for overa robot cotro system with feedbac cot K δt D Ddesired K δt K δt K δt ± - K δ t Natura frequecies are the roots of the deomiator This is ustabe, just as we ew at the begiig of the ecture. o what s the poit of goig through this? Now we have a way of aayig what happes with other cotro aws We ca repace K by a more eaborate cotro aw, for exampe Ke[ Ke[ ] Redo the aaysis with K K E END 3
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