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# Control Systems ME 451

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This 23 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 Jay Pil Choi in Fall. Since its upload, it has received 55 views. For similar materials see /class/207526/me-451-michigan-state-university in Mechanical Engineering at Michigan State University.

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Date Created: 09/19/15

ME451 Control Systems Lecture 7 Linearization time delays Dr Jongeun Choi Department of Mechanical Engineering FaH2008 Michigan State University Course roadmap Modeling Analysis Design 7 Laplace transform Vi Transfer function W Models for systems Yelectrical Wmechanical electromechanical J Block diagrams El Time response Transient Steady state Frequency response Bode plot Stability RouthHurwitz Nyquist Design specs Root locus Frequency domain PID amp Leadlag Design examples Linearization f Matab simulations amp laboratories FaH2008 What is a linear system A system having Principle of Superposition U100 gt y1t um a 9205 Va1a2 E R A nonlinear system does not satisfy the principle of superposition FaH2008 Linear systems Easier to understand and obtain solutions Linear ordinary differential equations ODEs Homogeneous solution and particular solution Transient solution and steady state solution Solution caused by initial values and forced solution Add many simple solutions to get more complex ones use superposition Easy to check the Stability of stationary states Laplace Transform FaH2008 Why linearization Real systems are inherently nonlinear Linear systems do not exist Ex ftKxt vtRit TF models are only for linear timeinvariant LTI systems Many control analysisdesign techniques are available for linear systems Nonlinear systems are difficult to deal with mathematically Often we linearize nonlinear systems before analysis and design How Fall 2008 5 How to linearize it Nonlinearity can be approximated by a linear function for small deviations 61 around an operating point 5130 Use a Taylor series expansion fo M A Linear approximation 613 New coordinate f fl Nonlinear functio Operating point 5130 1 Fa 2008 Old coordinate 6 Linearization Nonlinear system 2i 2 fm u Let uO be a nominal input and let the resultant state be x0 Perturbation u u0 6u I Resultant perturb 2130 Taylor series expansion are u 2960 M 8m uzuo f7u JimMo WWW 23 5 HOT 6 luu u T FaH2008 7 Linearization oont 8fvuo 58fmuo an 821 UZUQ 813 uzuo m39olSw39 f oa uo notice that m o f130 no henoe 8fwul0 68fvu0 6 913 QLZZIq 913 QLZZZQD M fo7uo 6 513 96 gt FaH2008 8 Linearization of a pendulum model I Motion of the pendulum gSin6t i ut O 7 r L mLL m9 f0 u Linearize it at 6 0 7r I Find MO New coordinates 9 60 66 W 69 uu06uO6u FaH2008 9 Linearization of a pendulum model cont Taylor series expansion of f9u at 6 Tl39 u O 8f07u 09 33 af0u au 33 FaH2008 10 Partialfraction Expansion Text page 637641 Fs is rational m S n a realizable condition ddt is not realizable bmsml l b15bo snan13 1a13a0 178 ll 21 TI393 A 03 I 5 0 39 2 i m 3 k1 i l 8 p2 3 191 ME451 807 34 Coverup Method 1 k2 kn ms 8291 sp2 spn 1 1 Elks pz Kinks Pi 8 Pz F8 Q kiquot Q T 0 11 vr i lki 8 P1F58pi Check the repeated root for the partialfraction expansion page 638 W451 807 35 99 54 1163 2551 M5 1H2 26 l 28 Roufh y I g 2 L e 1 J 5 5 Z 3 Z 1 3 64 A 5 O 3 J39 quotL 3 7 z J 3 0m 2 0 v x 9 3 o J 139 H1 3 Jquot 39 I J 2 Z 6 lt 6 I 6 e QCQC N t NM 00m Am 91 lt4 3 1 p 1 Q 0 Jr v V 0 1 N0 V99 M W s vfo K V WWM39W J 9 c A zquot poQog f3 6 MP TWO fmmem k1 j 2 W fmz ez id gt 1 19M a7 git 361 21quot H k1 Row NW3 7 S2 1 if3 9 9 5 3 1 db 3 4L 5 kl i 2k7 M41 1 quot3L3 015 k1 gquot3kfyi 3 RV OSMKIL 9wny 61 9180 p nj i lt33 lltz70 MC Kf fh l7o f x j 2 rx lt1 k 61 7 Root 20015 TMN I W III wegammaquot fudy m g 39oi 0 WWWquot 2 K 0V 439 oa e t o n ofr m Co a Aw pair5 6094 fooa ChmmeZ E7uv e39l39b id39 KG9SO j L 01964 ap0 TF 460115471516 gm W M L g 5 2 s n 39 ope 40 m5 L FlJwf 0pm 6000 F0105 4 2 J llt As 7 whm Kpo 9 CL po ea 0 00 law gt CL Iboes m an S 42 h Doegew ME451 Control Systems Lecture 2 Laplace transform Dr Jongeun Choi Department of Mechanical Engineering FaH2008 Michigan State University Course roadmap Modeling Analysis Design Laplace transform Transfer function Models for systems electrical mechanical electromechanical Block diagrams Linearization Time response Transient Steady state Frequency response Bode plot Stability RouthHurwitz Nyquist Design specs Root locus Frequency domain PID amp Leadlag Design examples t Matab simulations amp laboratories FaH2008 Laplace transform One of most important math tools in the course Definition For a function ft ft0 for tltO t Fs We denote Laplace transform of ft by F Fall 2008 Examples of Laplace transform Unit step function ft t 0 p90 1 igt Fs O 1 e stdt 7 quotA 1 a le sil 7 Memorize this s 0 s Unit ramp function ft ttgt0 0 AM 1 1 l VW 1 W est est W i W est 2 Igt Fs JO to dt 8 to O S39lO c dt 2 CI Integration by parts Fall 2008 Integration by parts d39 quotquot gtquot r r quotI 1quot r r r r Iquot 7 ut Ut U tUtutv t zbb u39uldt ulvdt mldt m u vdt uv dt oo EX testdt Let ut t 15 2l 390 10 5 0 St 1 00 1 F 1 A100 13900 1 V l I TI M39n39f H II 7r739w39 H I I ll 7P7SI IIT39 7 I M39quot lquot l w I I I v I 4 quot 3 J0 L 8 J0 J0 S 8 FaH2008 5 Examples of Laplace transform cont d ft Width o Unit impulse function Heightinf f 0 H Area 1 M 00 39 5 lt 4L 0 5 0 g t o gt Fs 6te5tdt 6 5390 1 Memorize this ft Exponential function 1 tfeat 20 t i 0 t lt 0 o 00 70 is 1 7 sa 0C 1 Igt FsuO 6 he fdt Sae t0 Sa Fall 2008 Examples of Laplace transform cont d Sine function Sinwt 2 2 3 w Memorize these Cosine function Coswt 2 8 2 s l W Remark Instead of computing Laplace transform for each function andor memorizing complicated Laplace transform use the Laplace transform table Fall 2008 7 Laplace transform table Table 81 in Appedix B of the textbook Tina anctinn Laplace Tranafnrln 1 A Linnttantp lll tim tn poaititre mega Iain prap Ef gf Slntt prapert aftf tan 1 H39 1 at can n Inverse Laplace Transform Fall 2008 Properties of Laplace transform 1 Linearity Proof Mama a2f2t jgkalm a2f2lttgtwdt p l st v so I l 73139 041 mm itH12 n I2U at IU JU W F15 F2S Fall 2008 Properties of Laplace transform 2 Time delay Proof 5 ft Tu5t T 00 7 5t 5 ft Te dt 0 T 0 fT STTdT 6 5TF5 45 Ex e 0 5t 4ust 4 6 In 39 aTUJ 7T5 Fall 2008 tdomain s domain 10 rerties f Lalace transfrm 3 Differentia tin L mm sFltsgt f0 Proof 11m O uxStew jaw853 O fte 8tdt 8Fs f0 EX COS 20 311 cos 2t 1 828 4 1 82 4 2 sin 2t t domain sdomain f gt m ft t Fs 39 8178 M 8 gt I FaH2008 11 rerties f Lalace transfrm 4 Interatin Otf 39 i r F3 S PFOOf fjde f0 f f7d7 em 1 f739d739 6 8t l O ft 3tdt 8 tdomain sdomain mm 5 ft 1 gtF FaH2008 12 Properties of Laplace transform 5 Final value theorem m 733 m sFs if all the poles osts are in 7900 5H0 the left half plane LHP Ex 5 5 148 3s2s2 39gtt39i Qoftls2s2E Poles osts are in LHP so final value thm applies EX Fs74 igt lim ftlt39im 48 0 52 4 two 39o 52 4 Some poles of sFs are not in LHP 80 final value thm does NOT apply FaH2008 13 Properties of Laplace transform 6 Initial value theorem Iim ft Iim sFs ifthe limits exist tHO H00 Remark In this theorem it does not matter if pole location is in LHS or not EX F1Igt 39 t F 0 8 532 S 2 t39gtr0n sggos 5 Ex39 o 4 gt lim Hm 3F30 PKG tHO i J l 3400 FaH2008 14 ME451 SO7 Transfer Function Laplace Transform of Unit Impulse Response of the System Input signal A x I Output signal u 11t 0 IrTu t T fox I1T5quotTIT 5C h139 quot quotdr def Transfer Function 00 hTc ST IT es39 0 IHs 2 Hsgtquot1 Take 8 jw 26 aim Hj39w6jwt coswt j sinwt ME451 07 Frequency Response A I Input ACOSwl e jt 87 Z We know x jd 39 j Complex numbers HUw Ale H jw Ale 5 Magnitude zw j 39l 6 1ULQ 2 JJC05wt 1 Phase Shift ME451 07 23 ME451 SO7 Frequency Response coswl yt Mcoswt How Me a jy 1m Hjw M lHjw 79 M Fwd Imlumw ll LReiHowm m Re Holly The Laplace Transform Appendix B Laplace transform converts a calculus problem the linear differential equation to an algebra problem How to Use it Take the Laplace transform of a linear differential equation Solve the algebra problem Take the Inverse Laplace transform to obtain the solution to the original differential equation M cmmw f nae cu O L 39N I def Inverse Laplace transform Jr ft 51Fs 6 WFse f ds 7U Jaij39xg ME451307 ME451 07 The Laplace Transform Appendix B Laplace Transform of a function ft f1i 26 jo f1 mom rgtd7 f1t f2t F18F28 ME451 07 Properties of Laplace Transforms page 641643 Linearity aft 69001 ft M 903 M Q 11 D Time Delay 0 JP0 fT 5TAdT 0 i A Wgt1 Nonrational function 27 ME451 07 ME451 SO7 Properties of Laplace Transforms Shi in Frequency Differentiation nlh pw 4 v 1 paw quot i ul 1 u LULJ JU UL eStm 30 7 ft7sestdt guru 7 I39n U l v MEASISW 28 Properties of Laplace Transforms Differentiation ll1t in time domain cgts in Laplace domain f s tf f0 mm orf Hm U J OhlJJ l V 2 t S f Sf0 f0 ntegrationquotlt in time domainltgt1s in Laplace domain 139 1 1 c I ftdt 2 7175 J 5 r LI l U0 MEASI SEI7 ME451 SO7 Laplace Transform of Impulse and Unit Step Impulse rrxnn 00 MADst 1 IA UUkx MD J JU UnitStep 1 4 n utlt b1 1 1 O t lt 0 gt t O 00 t 1t e 5 dt O ME451307 30 Unit Ramp W 1W7 2 0 t O 1 714 l L t I t6 7 JO 54 ME451 07 ME451 SO7 Exponential Function 00 s at eat 0 eateistdt 6 S a 1 Resiagt0 S CL ME451 07 Jwt BAPA n I 4 LUDKLUL LDI lulu j 1 044 3 5371391 I U I W L J s v39m e2 l 2 J L I 4 c coswt j 4 Sinwt UJ sinwt 82 ME451 07

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