Tutorial_8 solutions (1)

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Problem 9 I I I 3 i II Hu C EB AB I 4 rank C I s n i The system is uncontrollable Transfer fruitier from htt to X Lt c I o Gls CC SI A B i o Is 3 I i s I a El desists i a SIE s 3 k l s t I ss 2 s s l 5 13 S l 5 14 s12.5 s l 5 2.5 1 5 s l 1 S 1 An eigenvalue will only appear as a pole of thetransfer faction if its both controllable and observable X 2 5 didn't appear as a pole since it's uncontrollable mum too M y o l o X C E B AB A'B rank C 2 s n The system is not controllable Since A is an upper triangular matrix its eigenvalues appear on its diagonal cnet.es d l t 1 Gls 161 A B Io i od Is II to S I 0 0 IT 5 O n l

a G To 1 o Eg to set 03 S 1 Problem 3 x p I b y o 1 I X C E B AB A 733 it L rankle 2 a n The system is uncontrollable Since the A matrix is upper triangular d I I I Gls CC S1 A B co is It E I e is III To I I I s 1 Problem 4 x f x 191 u y E I I X O Ea L S1 Det G 3 14 I f 0 O has full rank The system is observable

Problem 5 X Ek 113 fly x 5123 f u Ek yl K 5 I x Ek o Kal II ul Det O 5 2 9 2 z f O i 0 has feel rank The system is observable

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Tutorial 7 Exercise 4 Revisited A B ii tt iI Itxi.Iii I t t y I o l o 1 contd input disturbance input Open logs system analysis a C T B AB A'B it i If rankle 3 n i The system is controllable State feedback design a Settling time a 5 s Mp Overshoot s 15 Zero steady state troukiy error to constant reference Mp exp o 15 E log o 15 T 11 log D 2 0 36472 E I 0.36 71 to 3647 0.5169 For Mp 15 we require 0.5169 choose 0.55 t s 4 2 criterion Wn 4 In 5 4 W n 5X o 55 Wn 1.4545 choose Wn l 5 We choose the third eigenvalue to be faster than the 2nd order ch characteristics polynomial Dcu d X 12 0.55 1.5 1 I 52 X 5 X 11 65 1 2 5 X 15 X 6.65 It to 5 11.25

Bass Cura forada 1 Is the system in CCF No 2 Oh characteristi polynomial Doc LX det Lite A aetfF III 3,1 Ht Tht Dat I EL E Lat Ati d 1 5 E X HD K 13 41 IX 314 Rtd 35 3 914 914 IX P 4h2 1 4.75 X t 1.5 d 4 Lz 475 Lz 5 3 Transformation matches e it I I e I il's I't 4 CL characteristic polynomial Dcu X X't 6.6512 t lo 5h 111.25 I 6.65 22 10.5 Iz 11.25 5 state feedback gain inCCF I di d Iz dz Tz 43 6.65 4 lo 5 4.75 Il T l 5 2.65 5 75 9.75 6 Transform back to the original x coordinates iii EEE's a I IN E 2.65 l 75 317 b Toachieve zero steady state trucking error To o 1 To D NC SI Aa B Aa A BK NcCSI At BK B To o Nc A 113k l B I N I d At Bk I d At Bk 13 20 I 04174g g t f f 2.65 1.75 31 to Hsi N 15

State feedbackdesign with integral action a Augment the system Aa 178 Ba Aa fi If Ba I o Bass Cura formula 2 Oh characteristic polynauli Ooh d X 14 2 14.75A 1 5 X 114 4 3 4.75A 11.5 X d 4 L 4.75 Ls 5 Ly O 2 Transformationmatrices Ea 1 Ba Aa Ba Ad Ba Aas Ba FIE III we www.aei eater eat i't I 3 CL characteristic polynomial Dominant mode is given by Rt 1.65A t 2 with S4 0.55 Wn L 5 Weneed to choose two more poles faster complex poles with larger damping ratio so they won'taffect the dominant mode too much E 0.95 Wnz 3 Deux Lit 11.65 2 A d 15 7 9 114 7.35 3 20.665 127675 X t 20.25 I 7.35 22 20.665 2 3 27.675 Ty 20.25 4 State feedback gain in CCF Tsa EE d Iz 22 2 3 23 Ix dx 735 4 20.605 4.75 21.67J I 5 20.25 0 3.35 15.905 26.175 20 5 Transform back to the original coordinates ka Tsa Ea Cai E3 35 15.06 11.23 27 K 3.35 15.06 11.23 Kz 27

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I f 1 I 1,11 i i 3 Hi a We only need one thermocouple to implement the observer We want to choose tee thermocouple such that the system is observable from that output Thermocouple X C Il O o Xz I O l O Xs Cg E O o I Off And IIE 9 rank 3 n V Ga Iaaf Ig Ig Ig rank G 2 an X Os Aa Iya hen 3 n V i We can either use x or Xz as the output to implement the observer b Goal in state feedback design regulate the temperature at Xa from Tutorial 7 Weneed one thermocouple to measure Xz to implement state feedback with integral action sincewe need to feedback the output that we are trying to regulate in the integral action If we only feedback an estimate of Xz the integrator will make tie estimate of Xz rather than the actual Xz To track the reference Since the system is not observable from Xa we need a 2nd thermocouple to measure or X 3 when implementing the observer Use Xz and Xs i.e C O O I Observer eigenvalues eig LA Lc We need the observer eigenvalues to be faster than the state feedback eigenulues A LC T AT I I A dual B dual Kdwal here Adual AT A Banal CT µ K dual LT

Bass Guru ferula Is the system in CCF No Oh characteristic polynoul Dam det XI A aa LI.IE III 4 344 71 2 Ei Il E cat att ht 144 1312 HK Xt X 13N 114 tech 1312 9 4 2 25 X if 3 2 3 1 1yd t E EX 314 D t 4 2 4.75 1 5 X L 4 Ls 475 Ls 5 Transformation matrices Cd B dual AdudBdual AetudBdure I ATE AT I 05 191 441 eat to I I l Yz 5 2 O O I Row1 t Roos 0 l 5 2 o Z O Row2 t 54 Row 3 O O 1 4 O O I K O O 2 I oo o p go g g Rm't knows 4 0 0 I 4 O O it it T I l i iii iiiEi I 4 O f j p od Rowe are i i'E O O I i O O I

CL characteristic polynomial Observer 3rd order Butterworth with cutoff freq 7 rad Is MATLABcommand Tuum den butter n Wn s n Wu 7 Duck X t 14K 98 X t 343 I 14 2 2 98 Is 343 Idwal in CCF i Kunal TT d Iz da Iz LD 14 4 98 4.75 343 1.5 Elo 93.25 341 5 Transform back to the original x coordinates me E 3 40 53 us 81 31 4 I 906.5 136.5 o 906.5 L Kafue 13,60 5

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