Lecture 1 Overview. course mechanics. outline & topics. what is a linear dynamical system? why study linear systems? some examples

Transcription

1 EE263 Auumn 27-8 Sephen Boyd Lecure 1 Overview course mechanics ouline & opics wha is a linear dynamical sysem? why sudy linear sysems? some examples 1 1

2 Course mechanics all class info, lecures, homeworks, announcemens on class web page: course requiremens: weekly homework akehome miderm exam (dae TBD) akehome final exam (dae TBD) Overview 1 2

3 Prerequisies exposure o linear algebra (e.g., Mah 13) exposure o Laplace ransform, differenial equaions no needed, bu migh increase appreciaion: conrol sysems circuis & sysems dynamics Overview 1 3

4 Major opics & ouline linear algebra & applicaions auonomous linear dynamical sysems linear dynamical sysems wih inpus & oupus basic quadraic conrol & esimaion Overview 1 4

5 Linear dynamical sysem coninuous-ime linear dynamical sysem (CT LDS) has he form dx d = A()x() + B()u(), y() = C()x() + D()u() where: R denoes ime x() R n is he sae (vecor) u() R m is he inpu or conrol y() R p is he oupu Overview 1 5

6 A() R n n is he dynamics marix B() R n m is he inpu marix C() R p n is he oupu or sensor marix D() R p m is he feedhrough marix for ligher appearance, equaions are ofen wrien ẋ = Ax + Bu, y = Cx + Du CT LDS is a firs order vecor differenial equaion also called sae equaions, or m-inpu, n-sae, p-oupu LDS Overview 1 6

7 Some LDS erminology mos linear sysems encounered are ime-invarian: A, B, C, D are consan, i.e., don depend on when here is no inpu u (hence, no B or D) sysem is called auonomous very ofen here is no feedhrough, i.e., D = when u() and y() are scalar, sysem is called single-inpu, single-oupu (SISO); when inpu & oupu signal dimensions are more han one, MIMO Overview 1 7

8 Discree-ime linear dynamical sysem discree-ime linear dynamical sysem (DT LDS) has he form x( + 1) = A()x() + B()u(), y() = C()x() + D()u() where Z = {, ±1, ±2,...} (vecor) signals x, u, y are sequences DT LDS is a firs order vecor recursion Overview 1 8

9 Why sudy linear sysems? applicaions arise in many areas, e.g. auomaic conrol sysems signal processing communicaions economics, finance circui analysis, simulaion, design mechanical and civil engineering aeronauics navigaion, guidance Overview 1 9

10 Usefulness of LDS depends on availabiliy of compuing power, which is large & increasing exponenially used for analysis & design implemenaion, embedded in real-ime sysems like DSP, was a specialized opic & echnology 3 years ago Overview 1 1

11 Origins and hisory pars of LDS heory can be raced o 19h cenury builds on classical circuis & sysems (192s on) (ransfer funcions... ) bu wih more emphasis on linear algebra firs engineering applicaion: aerospace, 196s ransiioned from specialized opic o ubiquious in 198s (jus like digial signal processing, informaion heory,... ) Overview 1 11

12 Nonlinear dynamical sysems many dynamical sysems are nonlinear (a fascinaing opic) so why sudy linear sysems? mos echniques for nonlinear sysems are based on linear mehods mehods for linear sysems ofen work unreasonably well, in pracice, for nonlinear sysems if you don undersand linear dynamical sysems you cerainly can undersand nonlinear dynamical sysems Overview 1 12

13 Examples (ideas only, no deails) le s consider a specific sysem ẋ = Ax, y = Cx wih x() R 16, y() R (a 16-sae single-oupu sysem ) model of a lighly damped mechanical sysem, bu i doesn maer Overview 1 13

14 ypical oupu: y y oupu waveform is very complicaed; looks almos random and unpredicable we ll see ha such a soluion can be decomposed ino much simpler (modal) componens Overview 1 14

15 (idea probably familiar from poles ) Overview 1 15

16 Inpu design add wo inpus, wo oupus o sysem: ẋ = Ax + Bu, y = Cx, x() = where B R 16 2, C R 2 16 (same A as before) problem: find appropriae u : R + R 2 so ha y() y des = (1, 2) simple approach: consider saic condiions (u, x, y consan): ẋ = = Ax + Bu saic, y = y des = Cx solve for u o ge: u saic = ( CA B ) ydes = [ ] Overview 1 16

17 le s apply u = u saic and jus wai for hings o sele: u1 u y1 1.5 y akes abou 15 sec for y() o converge o y des Overview 1 17

18 using very clever inpu waveforms (EE263) we can do much beer, e.g..2 u u y1.5 y here y converges exacly in 5 sec Overview 1 18

19 in fac by using larger inpus we do sill beer, e.g. 5 u1 u y y here we have (exac) convergence in 2 sec Overview 1 19

20 in his course we ll sudy how o synhesize or design such inpus he radeoff beween size of u and convergence ime Overview 1 2

21 Esimaion / filering u w y H(s) A/D signal u is piecewise consan (period 1sec) filered by 2nd-order sysem H(s), sep response s() A/D runs a 1Hz, wih 3-bi quanizer Overview 1 21

22 1 s() u() w() y() problem: esimae original signal u, given quanized, filered signal y Overview 1 22

23 simple approach: ignore quanizaion design equalizer G(s) for H(s) (i.e., GH 1) approximae u as G(s)y... yields errible resuls Overview 1 23

24 formulae as esimaion problem (EE263)... 1 u() (solid) and û() (doed) RMS error.3, well below quanizaion error (!) Overview 1 24