!"#$%&$'(%)*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%%
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1 !"#$%&$'(%)*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%% -%.'-/01-2%345%*$&"$/% 1').)*+6$7)8+9#:)'#+;<+=//8+ C)"#+)"-+?&2/'+D)3$*&-+:2?+:?$*#+*(E/+&?+>?("+$%F+!"#$%&'#()*+,*-./#,+&"#.*.(*0%&1%/%)* + EECS, UC Berkeley 4:)'5=).+G/&'/)&+ H?A<+IJ8+KLIM8+N/'E/*/-8+,;+
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4 )*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%+,/"&/'+ J*+,#+%)8%-%=&)E$/8)"+H)&C=8"/+%<-/"&$/8#)<%";%<=/8=$%</$H9;0<$>>E$%/$8$-/9#% $0<#-8)8%;&%)&C=8"/)-:%-<<:)9->;&84%K;<)986%! A-/C'-/$%-&C%8;Q'-/$%-/9#)"$9"=/$8%! A)B#:+%C+&-0)9%&$"';/S$C%8+8"$08%! 1;9-:)M->;&%-&C%:;9->;&H-'-/$%8$/E)9$8%! #T<6UU)9+<#+4;/B% icyphy,/-2-:%?=t-l%vc'-/c%1$$l%w:2$/";%.-&b);&e-&&)hx)&9$"$::)l%.-&y)"%.$8#)-% 4
5 )*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%+,/"&/'+ J*+,#+%)8%-%=&)E$/8)"+H)&C=8"/+%<-/"&$/8#)<%";%<=/8=$%</$H9;0<$>>E$%/$8$-/9#% $0<#-8)8%;&%)&C=8"/)-:%-<<:)9->;&84%K;<)986%! A-/C'-/$%-&C%8;Q'-/$%-/9#)"$9"=/$8%! A)B#:+%C+&-0)9%&$"';/S$C%8+8"$08%! 1;9-:)M->;&%-&C%:;9->;&H-'-/$%8$/E)9$8%! #T<6UU)9+<#+4;/B% icyphy,/-2-:%?=t-l%vc'-/c%1$$l%w:2$/";%.-&b);&e-&&)hx)&9$"$::)l%.-&y)"%.$8#)-% 6
6 *-<$*;C$% *-<$*;C$L%;/)B)&-::+%C$E$:;<$C%)&% K$//-.'-/0L%')::%2$%"-S$&%;E$/%2+% )*+,#+%'#$&%K$//-.'-/0%$&C8%)&% 35]^4% Lee, Berkeley 7
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11 *%?V%89#$C=:$/%7%?=S"-<$%7% c-e-.9/)<"%!%7%,>c$8n(% b&;"%8;%,>&+%,"-/b$"%!b,,(%,"/=:+%,>&+%,"-/b$"%!,,,(% Lee, Berkeley 12
12 )*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%+,/"&/'+ J*+,#+%)8%-%=&)E$/8)"+H)&C=8"/+%<-/"&$/8#)<%";%<=/8=$%</$H9;0<$>>E$%/$8$-/9#% $0<#-8)8%;&%)&C=8"/)-:%-<<:)9->;&84%K;<)986%! A-/C'-/$%-&C%8;Q'-/$%-/9#)"$9"=/$8%! A)B#:+%C+&-0)9%&$"';/S$C%8+8"$08%! 1;9-:)M->;&%-&C%:;9->;&H-'-/$%8$/E)9$8%! #T<6UU)9+<#+4;/B% icyphy,/-2-:%?=t-l%vc'-/c%1$$l%w:2$/";%.-&b);&e-&&)hx)&9$"$::)l%.-&y)"%.$8#)-% 13
13 @;/%*;0<:$a%.+8"$0%?+&-0)98% 9;0<;&$&"84%K#$%R;*%C$h&$86%! G#-"%)8%-%9;0<;&$&"N%! A;'%-/$%9;0<;&$&"8%$a$9="$CN%! A;'%C;%9;0<;&$&"8%9;00=&)9-"$N%
14 *+,#+.)0% *+,#+.)0% A$"$/;B$&$;=8%R;C$:)&B%-&C%.)0=:->;&% 9+2$/H<#+8)9-:%8+8"$084%K#$%8)0=:-";/%</;E)C$8% % *+,#+.)0%8=<<;/"8%"#$%@;::;')&B%R;C$:8%;@%*;0<="->;&6%!?)89/$"$%VE$&"%8)0=:->;&%! O=-&>M$CH."-"$%.+8"$08%!O..(%%8)0=:->;&%! *;&>&=;=8%>0$%!F=&B$Hb=T-(%8)0=:->;&%!?)89/$"$%>0$%8)0=:->;&%! R;C-:%R;C$:8%! I=&9>;&-:%R;9S=<%J&"$/@-9$%!IRJ(%! W:B$2/-)9%:;;<%8;:E$/8% #T<6UU9+<#+8)04;/B% icyphy Research Consortium 15
15 *+,#+.)06%R)a$C%C)89/$"$U9;&>&=;=8% 0;C$:)&B%i%8)0=:->;&% Actual Points Computed 10 <;*$;21=1;21;'#(* 854;212* >+.;9$%,54'$.;?*.9* position time 7,-'./$012*4+'.5* #$8545(*-,$'48#1*9.5* 7:"*-$/,#4'$.;* A==15*%54=&*$-*B&4'*$-* +./=,'126*C.B15*%54=&* $-*$'-*-1/4;'$+-6* 10 Interpolated Values!"#$%&'#()*+,-'./$012* 3$-,4#*12$'.56* position time icyphy Research Consortium 16
16 WCE-&9$C%*;&>&=;=8%.)0=:->;&% 10 O=-&>M$CH8"-"$%.+8"$08% J&%-%9:-88)9-:%e?V%8)0=:-";/L%-%8"$<H8)M$%9;&"/;:%-:B;/)"#0%C$"$/0)&$8%8-0<:$%>0$8L%-&C%-% 8"-"$%#-8%)"8%;'&%8-0<:$%>0$8L%-&C%8-0<:$8%-/$%</;9$88$C%=8)&B%-%?V%8)0=:->;&%$&B)&$%)&% $-9#%8"-"$%-&C%</;C=9)&B%8-0<:$8%;&:+%'#$&%"#$%E-:=$%9#-&B$8%2+%-%</$HC$"$/0)&$C% Chattering ";:$/-&9$L%9-::$C%"#$%P=-&"=04% But it doesn t quite work. Suppose the input 10 is instead Simple Continuous 8 0 t < 1 6 Then we get this: u(t) = t Quantized State time x q Simple Quantized-State System ẋ(t) =u(t) Quantize the state as follows x(t). Edward A. Lee (UC Berkeley) Quantized-State Systems Spring ẋ(t) =u(t) q(t) u(t) = 0 t < 1 10 t 1 icyphy Research Consortium 17 where Let q(t) =bx(t)c. 0 t < 1 u(t) = 10 t 1
17 A+2/)C%*;8)0=:->;&6%IRJ%,-<$/%9)/9=:-"$C% -0;&B%IRJ%-9>E)8"8L% -&C%8=20)T$C%-&C% <=2:)8#$C%)&%A.**% 35]f4% icyphy Research Consortium 18
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19 )*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%+,/"&/'+ J*+,#+%)8%-%=&)E$/8)"+H)&C=8"/+%<-/"&$/8#)<%";%<=/8=$%</$H9;0<$>>E$%/$8$-/9#% $0<#-8)8%;&%)&C=8"/)-:%-<<:)9->;&84%K;<)986%! A-/C'-/$%-&C%8;Q'-/$%-/9#)"$9"=/$8%! A)B#:+%C+&-0)9%&$"';/S$C%8+8"$08%! 1;9-:)M->;&%-&C%:;9->;&H-'-/$%8$/E)9$8%! #T<6UU)9+<#+4;/B% icyphy,/-2-:%?=t-l%vc'-/c%1$$l%w:2$/";%.-&b);&e-&&)hx)&9$"$::)l%.-&y)"%.$8#)-% 20
20 ..K6%.$9=/$%.'-/0%K;;:S)"% J;KL%=8-2:$%2+%-<<:)9->;&%C$E$:;<$/8%')"#%0;C$8"%8$9=/)"+%8S)::84% W<</;-9#6%%! e<$&h8;=/9$%:;9-:%-="#;/)m->;&%$&>"+%;$&2%-8%-% -="#;/)M->;&%8$/E)9$%!"#"$ %&$'()*$!"#$%"&'((&)*"+,- Accessor ma;s$=&%b)0n% +,&-(..$ /(-(01(*$!"#"$!"#$% &'()'(% 12+"' 12+"'!"#$%"$#& '($)*+,&-$#./%01& 12+"' 12+"'!"#$%&' -:"$/&->E$8%!</;";9;:8L%9/+<";B/-<#)9%-:B;/)"#08L% W="#;/)M->;&%V&>>$8`L%A4%b)0L%W4% G-8)9$SL%[4%R$#&$L%-&C%V4%W4%1$$L% I)*:;=C%o]l% 21
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22 )*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%+,/"&/'+ J*+,#+%)8%-%=&)E$/8)"+H)&C=8"/+%<-/"&$/8#)<%";%<=/8=$%</$H9;0<$>>E$%/$8$-/9#% $0<#-8)8%;&%)&C=8"/)-:%-<<:)9->;&84%K;<)986%! A-/C'-/$%-&C%8;Q'-/$%-/9#)"$9"=/$8%! A)B#:+%C+&-0)9%&$"';/S$C%8+8"$08%! 1;9-:)M->;&%-&C%:;9->;&H-'-/$%8$/E)9$8%! #T<6UU)9+<#+4;/B% icyphy,/-2-:%?=t-l%vc'-/c%1$$l%w:2$/";%.-&b);&e-&&)hx)&9$"$::)l%.-&y)"%.$8#)-% 23
23 -&C%e<>0)M->;&%K;;:S)"% mwss-+-n%! W9";/H;/)$&"$C% ";;:S)"% ";% 2=):C% 0;C$:8% ;&%8"/$-0)&B%C-"-%! 9;&8"/-)&$C%;<>0)M->;&%
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25 )*+,#+%!"#$%&'()*+,-./'012-%(3)*+4-%&/5%+,/"&/'+ J*+,#+%)8%-%=&)E$/8)"+H)&C=8"/+%<-/"&$/8#)<%";%<=/8=$%</$H9;0<$>>E$%/$8$-/9#% $0<#-8)8%;&%)&C=8"/)-:%-<<:)9->;&84%K;<)986%! A-/C'-/$%-&C%8;Q'-/$%-/9#)"$9"=/$8%! A)B#:+%C+&-0)9%&$"';/S$C%8+8"$08%! 1;9-:)M->;&%-&C%:;9->;&H-'-/$%8$/E)9$8%! #T<6UU)9+<#+4;/B% icyphy,/-2-:%?=t-l%vc'-/c%1$$l%w:2$/";%.-&b);&e-&&)hx)&9$"$::)l%.-&y)"%.$8#)-% 26
26 .<->-:%e&";:;b)$8%-&c%.$0-&>9%1;9-:)m->;&% mg$2$/n% W<<:)9->;&86%V&#-&9$%W99=/-9+L%.$9=/)"+L%-&C%,/)E-9+%)&%.'-/0% 1;9-:)M->;&% 1$E$/-B$%;&";:;B)9-:%0-<<)&B%";%"/-&8:-"$% 8$0-&>9%:;9-:)M->;&% 0-<<)&B%"$9#&)P=$8L`%4?S:)'/+T$)*(&-+)"#+;%%$')"3/L%355q4%! F$:-"$%8$0-&>9%:;9-:)M->;&%";%0-"#$0->9-:% TerraSwarm % Research 8"/=9"=/$8%% Center % 27
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