×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

## Control Systems

by: Princess Rolfson

42

0

29

# Control Systems ME 451

Princess Rolfson
MSU
GPA 3.57

George Zhu

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
George Zhu
TYPE
Class Notes
PAGES
29
WORDS
KARMA
25 ?

## Popular in Mechanical Engineering

This 29 page Class Notes was uploaded by Princess Rolfson on Saturday September 19, 2015. The Class Notes belongs to ME 451 at Michigan State University taught by George Zhu in Fall. Since its upload, it has received 42 views. For similar materials see /class/207525/me-451-michigan-state-university in Mechanical Engineering at Michigan State University.

×

## Reviews for Control Systems

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/19/15
Math Review Complex Numbers 1 Complex number ordered pair of two real numbers sxjye C where xye R andjgxTl Conjugate 2 s x jy Addition s1x1jy1 S2 2x2 jyz S1S2 x1x2jy1 yz Multiplication M2 we y1y2 New xzyl ss 2 x2 y2 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Math Review Complex Numbers 2 Euler s identity 1390 j6 13949 j6 6 6 COS jSi11 Where cos zl sin zi 2 2j Polar form s x jy 2 re where x rcos y rsin6 Magnitude I lxz y2 11m 1 7 Phase 9 tan x 7 r S1 rle gl S2 rze gz S1S2 Iqrze 6162 9 s x Re S2 r2 S1 ejwl e 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 2 Math Review Logarithm The logarithm of x to the base 9 is denoted by 10gb x The logarithm of 1000 to the base 10 is 3 Le log10 1000 3 note that 103 21000 log10 10 1 and log101 0 Properties 191 x and logbxy ylogbx logbxy 2 10gb x 10gb y x logb 2 10gb x logb y Math Review Matrix Operations Aa11 a12 and B 2b11 b12 a21 a22 921 922 Determinant det A allazz anal2 Multiplication AB a11b11 61121921 61111912 61121922 1211911 1221921 1211912 1221922 6122 an Inverse a a A 1 21 11 det A 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 4 Laplace Transform Definition The Laplace transform LT of ft is Lft no fray ch 39 To sdomain ft t domain I L LT replace ODEs as linear inputoutput maps solve ODEs w constant coefficients 1111 Re 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Laplace Transform Example 1 Exam le 1 u LftFs Ifte dt je dr f 01 QC 0 1 0 0 e st 1 S 0 LftL1 0ll S s s ft6t1im tSA LfrFs Inna 12 J39Jte dt A gt0 O 0 A 0 lim iemdt AaOO A limiAzl AaOA A t LftL5t1 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 6 Laplace Transform Example 2 Example 2 ftt tZO 00 st 00 t 1 z LftFs ne drjresdr utv esj 0 0 S Jauvdt Ja uvdt Jauvdt a uvdtJ ua th 0 at 0a 0 at 0a 0 at 16 1 1eS 00 0 i2i2 s 0 s 0 s s erm Lrt 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 7 Laplace Transform Property 1 Linearity Laf t bgt aLf 0 bLgt aFS bGS Proof Laft 198 0 3161f t bg te dt a01fte dt be e ch Fs 05 aFS bGS Laplace Transform Property 2 s Shiftinq property If Lft Fs then Le ft Fs a Where Fs a 1irr1 Fs Proof Le ft Jemfte dt o f te dt 2 373 a 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 9 Laplace Transform Property 3 t Shiftinq property If L f t F s and ut is unit step function then Lft aut 61 e asFs Proof Lft aut a Tfa aut cue 1t TfU aut ae dl Tf7u7e m d7 e sTf7e d7 T t a e sFs 1 261 A t a t a 0 tlta 1 i 7 a t 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 10 Laplace Transform Property 4 Transforms of Derivatives a HLUVHF meLU HMWQlt mfm4g j Proof m m a Lfr Ifte dt Lia 61 u 6 v ft 80w an 80w au i J76 galCh JVCH Jar Vdf 6quot at 6 fte 3 s ne quotd sFs f0 0 175 LM0H Forw m fmgt Week 12 Math aaaaaaaaaaaaaaaaaaaaaaaaa ation Laplace Transform Property 5 Transforms of Inteqrals If Lft Fs then L Ifz39dz39 im 0 S Proof Let gt f7dz39 then go ft and g0 0 Using transforms of derivatives F S M 0 L t sLgt 80 505 J Gs Therefore Gs L jfmdofns Laplace Transform Example 3 Linearity ft 1 2t Lft L1 2t L1 2w l 2 S S S s Shiftinq properties If gt e nd Gs Lgt Gs L62 1L1Hs2 l 1 Ss gts 2 s 2 Therefore Lezt 5 2 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 13 Laplace Transform Example 4 t Shiftinq properties 6 8 Find 05 Lgt 3 where gt 2t 6ut 3 3 6 9 6 Leta3 ftt andft at 3Then 05 Lgt L2t 3ut 3 2Lt 3ut 3 26 3S i2 5 Therefore GS 32638 s 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 14 Laplace Transform Example 5 Calculating Lsin wt using L 1 s2Lft sf0 e Let ft sin wt f0 0 flt0gtw ft wcos wt f0 w2 Sin wt 2 2 Thus L w Nsm wt s Lsm wt 5 f 0 f 0 ft f0 039 wquot w2Lsin wt 2 52Lsin wt w 52 w2LSin wt 2 w Therefore Lsin wt 2 2 2 S W 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 15 Laplace Transform LT Table to remember Description ft Fs L f t Unit impulse 5m 1 Unit step 1 l S Unit ramp t i2 S nth order ramp tn Sn1 Exponential ear 1 S a Sine Sin wt W s2 w2 Cosine 003 wt S 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 16 Laplace Transform LT Properties Linearity Laft 1980 aLft bLg 0 s shifting properties Le ft F s a Lft SIS Cl t shifting properties Llf t 6W0 61 6 SF S e asLff Transforms of derivatives Lifm sFsf0 Lftfs2FsSf0f0 Transforms of integrals Lil mm Ems farm Convolution L 1FltsgtGltsgt Home rgtdr lgltofltr odr 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 17 Inverse Laplace Transform Given Fs Lft Find ft L 1Fs FS s domain f0 t domain Aggroaeh 1 Make F s quotlook likequot these transforms of common functions 2 Obtain L 1Fs term by term using tables properties etc 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 18 Inverse Laplace Transform Example 1 s1 S3S2 6S YS SH 05 S3S2 6S J S2 s3 Given Ys Find yt L1Ys V We Will discuss it next 1L 1iiL 11 iL 1 1 6 S 10 5 2 15 53 1 3 2t 2 3t 6 e 6 10 15 W 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 19 Inverse Laplace Transform PFE Case 1 PFE Partial Fraction Expansions A way to expand general rational functions into forms that appears in the LT table Case 1 real and distinct roots YSS 1 B ss 2s3 s Find A Band C 3 2 s3 gt s1As 2s3Bss3Css 2 s0 1A 23 A 1 i s 2 3 B25 10 2C 3 5 2 Ci 2009 Spring ME451 GGZ gtB s 3 Week 12 Math Review and Laplace Transfo m t39 Page 20 Inverse Laplace Transform PFE Case 1 Case 1 real and distinct roots cont d 16 310 215 1 3 2t 2 3t gt yt e e S 3 2 33 6 10 15 Alternative approach Ys s1 As 2s3BSS3CSS 2 AS2AS 6ABS23BSCS2 2CS ABCs2A3B 2Cs 6A Wehave s 0ABC 1 s 1 A 38 2C 3 equations with 3 unknowns so 1 6A 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 21 Inverse Laplace Transform PFE Case 2 Case 2 real and repeated roots s 3 A B C YS s23 32 322 s23 gt s3As22Bs2C Let 52 2 gt C21 1 12As2B as Lets 2 gt 821 1 022A gt A20 as 1 1 322 s23 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Ys Page 22 Inverse Laplace Transform PFE Case 2 Case 2 real and repeated roots cont d 1 1 1 1 2 3 2 3 S S S ss2 S t2 Ltss2 L ss2 2 t2 W mm m e 2 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 23 Inverse Laplace Transform PFE Case 3 Case 3 complex coniuqate roots s a i bi 1 1 Ys 2S S 2gtRootss 1i2i s 2s5 s1 2 Ys Lcos 2tss1 Me cos 2t S 239 ss1 Therefore yt 6 cos 2t 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 24 Inverse Laplace Transform Example 2 21 FUD s A 5 C DsE 2 2 2 ss1s 4 s s s1 s 2 s21Ass1s24Bs1s24Cs2s24Ds Es2s1 30 gt 14BgtB14 s 1 gt 25CgtC25 s2i gt 32iDE 42i1 4E 4D2iDE gt 316D 4ED Egt D E 320 8 2sAs1s24ss242s2s1Bs242ss1 s C2ss242s3Ds2s1DsE2ss1s2 S0 3 04A4B 3 A B 14 Fsl112 1 3 s1 1 1 2 3 1 gt t t e t Cos 21 sin2t 4S2 5s1 20s222 4 4 5 20 2 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 25 Laplace Transform Convolution If L1FS ft and L1GS 80 L 1FsgtGsgt IfrgtTdt jgmfo odr Examgle 3 1 3 38 38 YS 2 2 2 2 2 2 s 1s 9 s 1 s 9 s 1 s 9 WW 3 1 t sin z sin 31 y 8 8 Fs Gs ft L1Fs sin t and gt L1Gs sin 3t yt f239gt z39dz39 sin rsin 3t 239dz39 1 sm t sm 3t 8 8 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transfo mati Page 26 Laplace Transform Solving ODEs ODE in L AE in t domain s domain Direct l Solve for solution HS N 2009 Spring ME451 GGZ Week 12 Math Review and Laplace Transformation Page 27 Modeling Part of Control Design Process n p ut vCr I Controller i Output gt 4 Implementation Controller 139 MOdelmg v I Mathematical model 3 DesignSynthesis 2 Analysis Derive mathematical models for Electrical Systems Electrical systems Kirchhoff s voltage amp current laws Mechanical systems Mechanical systems Electromechanical system Newton s laws 2009 Spring ME451 GGZ Week 34 Modeling Physical Systems

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Jennifer McGill UCSF Med School

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over \$500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

Jim McGreen Ohio University

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com