H.Ulaş Akova 1557263
ME 534 – Computer Control of Machines Mini-Project 3
Fall 2012
!"#$ &' "( !"# %&'()*#& *+(,%-.() '&# /&.+0#1 ). ') %. 1#%#&2-(# %"# .3#&'44 %&'()*#& *+(,%-.( .* %"# 04'(% #')-45 ') )".6( -( %"# *.44.6-(/ *-/+
()
( )
( )
( )
( )
8-/+&# 97 :4.,; <-'/&'2 .* %"# =.(%&.4 >5)%#2
() = () = ()() = ( ) () = () = () =
"#$ %&&$' ())* +,& -$ .'%//$& ,0
+ ( + ) ( )/( ) () () = 1 + ()() = 1+( )/( + ( +)+ ) ) + ) () = + ( + ) ) + +
1%&,((2 /#$ 3" *')+$004*(,&/5 /',&06$' 67&+/%)& )6 /#$ 020/$8 +,& -$ .'%//$& ,0
() = ()()() () = + ( + ) ) + + 9&0$'/%&: /#$ :%;$& &78$'%+,( ;,(7$0 %&/) /#$ ,-);$ $<*'$00%)& 2%$(=0 /)
() = 1 4.5×10 + 1.244.8 × 10 +125.4424 Then, the discrete time model of the system can be determined via Matlab’s built>%& built>%& 67&+/%)& !"#$ & -2 0*$+%62%&: , 0,8*(%&: /%8$ )6
= 10 ' "#$ 0+'%*/ 70$= /) =$/$'8%&$ /#$ =%0+'$/$ /%8$ 8)=$(
)6 /#$ 020/$8 +,& -$ 0$$& %& ?**$&=%<@
Θ() = 0.003568 + 3.4 × 10 − 2.8 × 10 () − − 8.5 × 10−1.5×10 1/9
ME 534 – Computer Control of Machines Mini-Project 3
H.Ulaş Akova 1557263
Fall 2012
)( ?#% +) *-&)% 4..; '% %"# /-3#( %#,"(-,'4 )0#,)@
= 4 ≤ 1 → = ≥ 4
≤ 0.10 → ≥ 0.733 → ≥ 0.592 → ≤ 53.7 = exp− !"# &..% 4.,+) .* %"# )5)%#2 6-%" ' AB,.(%&.44#& -) /-3#( -( %"# *.44.6-(/ *-/+ C% -) )##( %"'% 6-%" ' AB,.(%&.44#& %"# )0#,-*-,'%-.() ,'( D# )'%-)*-#1E '(1 %"+) %"# )%#0 &#)0.()# .* %"# )5)%#2 *.&
() = = 11.5 -) '4). /-3#( D#4.67 4
8
Root Locus
x 10
0.72
0.58
0.4
Step Response
0.2
1
0.83 6
0.9
SettlingTime: 0.9525 0.91
0.8
0.96
0.7
4
) s d n o c e s ( 1 s i x A y r a n i g a m I
1 -
2 0.99
0
0.6
8e+04
6e+04
4e+04
e d u t i l 0.5 p m A
2e+04
0.99
0.4
-2 0.96
0.3
-4 0.91
0.2
-6
0.1
0.83 0.72 -8 -1
-0.8
0.58 -0.6
0.4 -0.4
0.2 -0.2
0 0
0.2
0.4
0.6
0.8
1
0
0.5
1
5
Real Axis (seconds -1)
1.5
2
2.5
Time (seconds)
x 10
8-/+&# F7 G..% ?.,+) '(1 >%#0 G#)0.()# 6-%" ' A B ,.(%&.44#& H.6#3#&E %"# )5)%#2 &#I+-&#) 4'&/# ,.(%&.4) 1+&-(/ -%) .0#&'%-.(7 J( %"# .%"#& "'(1 ' 4'/ ,.20#()'%.& -) /&'0"-,'445 %+(#1 +)-(/ >C>J !..4 .* K'%4'D ')
() = 8 . . ). ') %. )'%-)*5
%"# )'2# )0#,-*-,'%-.()7 !"# 0#&*.&2'(,# .* %"# ,.(%&.44#& ,'( D# )##( -( 8-/+&# L7 Step Response 1.4
1.2
1
Rise Time: 0.6710 SettlingTime: Settling Time: 0.9613 0.9613 Settling Min: 0.9021 Settling Max: 1.0150 Overshoot: 1.4994 Undershoot: 0 Peak: 1.0150 PeakTime: 1.3936
e 0.8 d u t i l p m A 0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Time (seconds)
8-/+&# L7 =4.)#1B4..0 G#)0.()# 6-%" ?'/ =.(%&.44#& C( %"# *.44.6-(/ *-/+&# %"# ,.(%&.44#& .+%0+%) '&# /-3#( *.& %"# A B =.(%&.44#& '(1 4'/ ,.20#()'%.&7
2/9
ME 534 – Computer Control of Machines Mini-Project 3
H.Ulaş Akova 1557263
Fall 2012
Controller Output 12 M_Pcont M_lag 10
8
e d u t i l p m A
6
4
2
0
-2
0
0.5
1
1.5
2
2.5
3
Time (seconds)
8-/+&# M7 =.(%&.4) G#I+- D5 %"# =.(%&.44#&) !"# #I+-3'4#(% 1-),&#%#B%-2# ,.(%&.44#& *&.2 -%) ,.(%-(+.+) ,.+(%#&0'&% -) .D%'-(#1 +)-(/ 2'00-(/ %#,"(-I+#)7 8-&)%E %"# )'204-(/ 0#&-.1 -) %. D# )#4#,%#17 >-(,# 4#%
= 10 7
= = 67 > 10 E
N#O%E %"# 4.,'%-.() .* P#&.) '(1 0.4#) '(1 <= /'-( *.& %"# 1-),&#%#B%-2# ,.(%&.44#& '&# 1#%#&2-(#1 ') *.44.6)@
() = 8 0.13+1 → () = 8 0.25+1 → = −7.69, = −4, lim .(.) .) = 0.926, = ( (.) .) = 0.961 () = −− → = .( 1 − 0.92 .926 = 1.9 ( ) lim = → 1−0.961 . H#(,#E () = 1.9 .7 *( G#2#2D#& %"# -204-,'%-.( .* %"# /-3#( )0#,-*-,'%-.() ')
≥ 0.592, ≥ 4 → ≤ = = 0.9608 !"# &..% 4.,+) .* %"# +(,.20#()'%#1 )5)%#2 6-%" %"# 'D.3# &#I+-#(%) -20.)#1 -) /-3#( -( 8-/+&# Q7'7 C% -) ,4#'&45 )##( %"'% ' AB,.(%&.44#& ,'( )'%-)*5 %"# /-3#( &#I+-#(%) D5 0&.0#&45 '1R+)%-(/ %"# /'-( 3'4+#7 !"#&#*.&#E %"# 1.2-('(% 0.4# .* %"# )5)%#2 -) 04',#1 +)-(/ %"# Matlab’s SISO Toolbox and the location of the dominant closed loop pole can be se#( se#( -( 8-/+&# Q7D7 !"# ,.(%&.44#& /'-( -) )#% %.
= 12.
8-/+&# Q7 'S<#)-/( G#/-.(E DS =4.)#1B4..0 A.4# ?.,'%-.( 6-%" AB,.(%&.44#& 3/9
ME 534 – Computer Control of Machines Mini-Project 3
H.Ulaş Akova 1557263
Fall 2012
+( C( .&1#& %. )-2+4'%# %"# 15('2-, &#)0.()# .* %"# )5)%#2) 6-%" %"# ,.(%&.44#&) 1#)-/(#1 -( %"# 0-.+) 0'&%) ' >-2+4-(; 2.1#4 .* %"# )5)%#2 -) ,.()%-%+%#1 ') %"# .(# 0&#)#(%#1 -( 8-/+&# 97 !"# >-2+4-(; 2.1#4 .* %"# ,.(%&.4 )5)%#2 ,'( D# )##( -( 8-/+&# T7
8-/+&# T7 >-2+4-(; K.1#4 .* %"# =.(%&.4 >5)%#2 !"# 0.)-%-.( &#)0.()# .* %"# )5)%#2 6-%" %"#)# %6. 1-**#&#(% )5)%#2) -) /-3#( -( 8-/+&# U7 !"# ,.(%&.44#& .+%0+%) '&# '4). 04.%%#1 '(1 /-3#( -( 8-/+&# V7 Step Response 1.2 Design by Emulation Classical Discrete Time Controller 1
0.8
] s n a i d a R [ n o i t i s o P
0.6
0.4
0.2
0
-0.2
0
0. 5
1
1. 5
2
2. 5 Time [Second]
3
3. 5
8-/+&# U7 >%#0 G#)0.()# – G#)0.()# – W(/+4'& W(/+4'& A.)-%-.(
4/9
4
4. 5
5
ME 534 – Computer Control of Machines Mini-Project 3
H.Ulaş Akova 1557263
Fall 2012
Step Response 12 Design by Emulation Direct Discrete Time Controller 10
8 ] s t l o V [ M , t u p t u O r e l l o r t n o C
6
4
2
0
-2
0
0. 5
1
1. 5
2
2.5 Time [Second]
3
3. 5
4
4.5
5
8-/+&# V7 =.(%&.4) <#2'(1#1 D5 %"# =.(%&.44#&) ,( W ,.(%&.44#& , .(%&.44#& -) %. D# 1#)-/(#1 3-' 1-&#,% 1#)-/( %#,"(-I+#7 !. D#/-( 6-%"E ' ,4.)#1 4..0 %&'()*#& *+(,%-.( -) %. D# 1#%#&2-(#1 ). %"'% %"# 1#)- %&',;-(/ 0#&*.&2'(,# -) .D%'-(#17 ?#% +) )#4#,% %"# ,4.)#1 4..0 0.4# 4.,'%-.() -()-1# %"# '44.6'D4# 1#)-/( &#/-.( /-3#( -( 8-/+&# Q7'7 ')
, = 0.93±0.05 0.005 0.005 () = ( − 0.95 = )(−0.95−0.05)) −1. .95 + 0.05 .05)(−0.95−0.05 − 1.9 + 0.905 C% -) -20.&%'(% %. -(3#)%-/'%# %"# -(3#&)# 0&.,#)) 2.1#4E 6"-," -) -(3#&)# .* %"# 04'(% 2.1#4 1#%#&2-(#1 -( 0'&% 'E
1 = − − 8.5 × 10−1.5×10 () 0.003568 + 3.4 × 10−2.8×10 C% -) .D)#&3#1 %"'% %"# -(3#&)# 04'(% 2.1#4 "') '44 .* -%) 0.4#) '(1 P#&.) -()-1# %"# +(-% ,-&,4#7 !"#(E %"# ,.(%&.44#& %&'()*#& *+(,%-.( -) 1#%#&2-(#1 ') *.44.6)
) () = 1() 1 −(()
() −0.0145 + 0.014 −0.004525 −3 −6.84×10 0. 0 05 0.0 1402 02 − 3 . 8 5 3 × 1 0 = 0.003568 − 0.0135 0.01355 5 + 0.0193 0.01931 1 − 0.0122 0.01222 2 +0.002894 +2 + 2.783 × 10−2.278×10 !"# 0.4#) .* %"# ,.(%&.44#& %&'()*#& *+(,%-.( '&# /-3#( -( .&1#& %. )".6 %"'% %"# ,.(%&.44#& 1#)-/(#1 -) ,')+'4E
= 0,0, = 0.999,, = 0.95± 95 ± 0.0 0.05, 5, = −0.001, = 0.9 !"# &#)0.()# .* %"# )5)%#2 -) /-3#( -( 8-/+&# X7
5/9
ME 534 – Computer Control of Machines Mini-Project 3
H.Ulaş Akova 1557263
Fall 2012
Step Response 1.4 Direct Design 1.2
X: 0.576 Y: 1.05
1 X: 0.9044 Y: 1.01
] s n a i d 0.8 a R [ n o i t 0.6 i s o P
0.4
0.2
0
0
0.5
1
1.5
2 2.5 3 Time [Second]
3.5
4
4. 5
5
8-/+&# X7 A.)-%-.( G#)0.()# .* %"# >5)%#2 C( %"-) 2#%".1E *-(1-(/ ' )+-%'D4# ,4.)#1 4..0 %&'()*#& *+(,%-.( 6-%" 1#)- ,.22'(1 %&',;-(/ 0#&*.&2'(,# -) "-/"45 ,"'44#(/-(/7 !"#&#*.&#E .(# 2+)% D# &#'4-)%-, 6"-4# )#4#,%-(/
). %"'%
%"# 4-2-%) .* %"# #I+-02#(% -( %"# ,.(%&.4 )5)%#2 -) (.% &#',"#17 W(.%"#& 1-**-,+4%5 -) '%#1 %. %"# 1#)-/( #I+'%-.( 6"-," -(,4+1#) -(3#&)# 04'(% 2.1#47 !"# ,.(%&.44#& -) %. D# ,')+'47 !"+)E -* %"# 0&.,#)) -(,4+1#) P#&.) .+%)-1# %"# +(-% ,-&,4# -% )".+41 D#
7 K.&#.3#& %"# )%'D-4-%5 .* %"# 1#)-/(#1 )5)%#2 -) '(.%"#& -))+# %"'% 2+)% D# %';#( -(%. ',,.+(%7 8.& %"'%E 1 − ()E '(1 () 2+)% D# 0&.0#&45 "'(14#1 D5 )#4#,%-(/ ' 0&.0#& '1R+)%#17 Y9Z -( !"# 15('2-,) .* %"# )5)%#2 -) )-2+4'%#1 +)-(/ '( .([.** ,.(%&.44#&7 !. -204#2#(% %"# controller ‘Relay’ block of the Simulink is utilized. A differential gap of
0.2
-)
)#4#,%#17
8-/+&# 9\7 J([.** =.(%&.44#& !"# .+%0+% .* %"# )5)%#2 '(1 %"# ,.(%&.44#& .+%0+% -) /-3#( -( 8-/+&# 99 '(1 8-/+&# 9FE &#)0#,%-3#457
.&/ 8&'(;4-(E ]787E A.6#44E ^7<7E _.&;2'(E K7E <-/-%'4 =.(%&.4 .* <5('2-, >5)%#2)E !"-&1 `1-%-.(7
6/9
ME 534 – Computer Control of Machines Mini-Project 3
H.Ulaş Akova 1557263
Fall 2012
Position Output 1 0.9 0.8 0.7 ] s n 0.6 a i d a R [ 0.5 n o i t i s 0.4 o P
0.3 0.2 0.1 0
0
0.5
1
1.5
2 2.5 3 Time [Second]
3.5
4
4. 5
5
8-/+&# 997 A.)-%-.( .* %"# C(#&%-' YG'1-'()Z Controller Output 15
10 ] s t l o V [ M , t u p t u O r e l l o r t n o C
5
0
-5
-10
-15
0
0. 5
1
1.5
2 2. 5 3 Time [Second]
3.5
4
4. 5
5
8-/+&# 9F7 =.(%&.44#& J+%0+%E K Ya.4%)Z !"# )%&'%#/5 -) (.% )+-%'D4# *.& %"-) '004-,'%-.(7 G#2#2D#& %"'% '*%#& %"# 1#)- )%#'15 )%'%# 3'4+# -) &#',"#1 %"# .%"#& ,.(%&.44#&) 1#)-/(#1 -( 0-.+) )#,%-.() 1. (.% 2';# +)# .* '(5 ,.(%&.4 b#(#&/5S7 H.6#3#&E .([.** ,.(%&.44#& 1#2'(1) #(#&/5 *&.2 %"# 0.6#& #4#2#(% #3#( -* %"# 1#)- )%'%# -) &#',"#17 K.&#.3#&E %"# *&#I+#(% )6-%,"-(/ .* %"# ,.(%&.44#& -) '(.%"#& 0&.D4#2 -( %#&2) .* %"# #O0#,%#1 4-*# .* %"# )5)%#2 ,.20.(#(%)7
7/9
H.Ulaş Akova 1557263
ME 534 – Computer Control of Machines Mini-Project 3
Fall 2012
011,2+34 !"#$ &
! #$ %&' ( #)*) +,-./01 & 02/3, 322 3224 4 0204 ! +3,1 5 ! +678)032 +3,39/1/,8 -: 16/ ;781/9 <3 = >?5%/&4 ! @,931A,/ B*CA013*0/ D9EFGDEF H3 = '4 ! @,931A,/ H/8)813*0/ DI69F J = >?&/(&4 ! $KA)L32/*1 B*/,1)3 DMNO9PGF Q = >?R/(&4 ! $KA)L32/*1 S)80-A8 T39U)*N DV9WX,3CW8/0YF Z1 = >?&%4 ! [-,KA/ \-*813*1 DV9W@F Z/ = >?&%4 ! ]30M $#^ 0-*813*1 DSWX,3CW8/0YF VN = &G4 ! _/3,Q-` H/CA01)-* H31)Z3 = '4 ! +-a/, @9U2):)/, _3)* DSWSF Z8 = >?b%4 ! ;/*8-, _3)* DSW,3CF ! [/06*)032 ;U/0):)031)-*8 -: 16/ 0-*1,-22/C 8781/9 [8 = 5/(&4 ! ;39U2)*N [)9/ D8F 18 = 54 ! ;/112)*N [)9/c G! /,,-, Q3*C #U = >?54 ! #3`)9A9 IL/,86--1c DU/,0/*1F !"#$ "
! +3,1 3 ! \-*1)*A-A8 [)9/ [,3*8:/, :A*01)-*8 _5 = 1:XZ3Y4 _G = 1:XZ1OVNcD<3OJ 1:XZ1OVNcD<3OJ X<3OQdH3OJY H3OQFY4 _& = 1:XZ/OVNY4 _' = 1:X5cD5 >FY4 _% = 1:XZ8Y4 878\ = _5O://CQ30MX_Gc_&c(5YO_'O_%4 _5O://CQ30MX_Gc_&c(5YO_'O_%4 ! IU/*2--U [,3*8:/, ^A*01)-* ! T)80,/1/ 1)9/ 9-C/2c 1,3*8:/, :A*01)-* 878T = 0GCX878\c[8cef-6e 0GCX878\c[8cef-6eYWZ84 YWZ84 !"#$ )
! +3,1 Q ( T/8)N* Q7 $9A231)-* _g02 = ://CQ30MX55?%O878\c5Y4 ://CQ30MX55?%O878\c5Y4 :)NA,/4 81/UX_g02Y4 ;g+0-*1 = 81/U)*:-X_g02Y4 81/U)*:-X_g02Y4 ! H--1 <-0A8 -: 16/ A*0-9U3*831/C ;781/9 :)NA,/4 ,2-0A8X878\Y4 N,)C -* -*4 4 `2)9X5/%OD(5 5FY4 72)9X5/'OD(h hFY4 ! <3N \-9U/*831-, _0 = 1:XhOD>?5& 5FcD>?G% 5FY4 _gI< = _0O878\4 _g\< = ://CQ30MX_gIi>?>5i5>>c5>5i5i5/&F4 ! ,2-0A8X_gI 5>FY4 8/9
H.Ulaş Akova 1557263
ME 534 – Computer Control of Machines Mini-Project 3
Fall 2012
! \-9U3,/ j8/ -: \-*1,-2X$*/, \-*1,-2X$*/,N7Y N7Y #g+0-*1 = 55?%OX5 ( _g02Y4 #g23N = _0OX5 ( _g\
! U3,1 0 ,2-0A8X878TY4
fN,)C4
!"#$ +
! +3,1 C B*L+23*1 = 5W878T4 ! B*L/,8/ U23*1 9-C/2 ^C, = 1:X>?>>%cD5 (5?k >?k>%Fc[8c eS3,)3Q2/e eS3,)3Q2/ec cefe efeY4 Y4 ! T/8),/C 02-8/C 2--U 1,3*8:/, :A*01)-* _0 = B*L+23*1O^C,WX5(^C,Y4 B*L+23*1O^C,WX5(^C,Y4 ! \-*1,-22/, 1,3*8:/, :A*01)-*
9/9
