Transcript
6. DISEÑO: DOMINIO - TIEMPO 6.1 6.2 6.3 6.4
CANCELACIÓN DE POLOS CONTROLADOR P.I. CONTROLADOR PD CONTROLADOR PID
6.1 METODO DE CANCELACION DE POLOS Y CEROS METODO ANALITICO
Ejemplo 6.1 G P (s)
1
s ( s 2)
;
Especificaciones t S 2 sg ,
0.5 !
z
Con%&ola'o& GC ( z ) k z
a( )alla& el polo 'ominan%e
z e ( n j d )T ( T )
z e( sT )
ze
n
;
( j T )
e
d
s n j d
T"#.2s$
z
" Ma$ni%*' polo
z e ( n T )
z d T tS 2
n
4
+Tole&ancia 2,(
n
4 4 4 rad sg tS 0.5 2
n 4 rad
sg
d n 1 2 4 1 0.5 2 3.46 rad
d 3.46 rad z z
e(
sg
sg
0.540.2 )
0.67
z
3.46 * 0.2
0.692 rad
z 39.6 º
Polo 'ominan%e 0.6739.6º p 0 .51 j 0 .42
El polo 'ominan%e es en la-o ce&&a'o. S
2 2 31.42 rad sg T 0. 2
S 31.42 rad # muestras
ciclo
sg
S d
31.42 3.46
9
Si se 'esea ms m*es%&as! se 'e/e /aja& el %iempo 'e m*es%&eo. /( Con'iciones 'e n$*lo
G la G c G z 1 G p (s) s ( s 2)
G p s
G ( z ) (1 z 1 ) z
G ( z ) c 2 d (Gp , Ti , ' zoh ' ) +MATLA0( 0.017 ( z 0 .87 ) G( z) ( z 1)( z 0.67 ) z Gc ( z ) k z ( z ) 0.017 ( z 0.87 ) Gla ( z ) k ( z )( z 1)( z 0.67 )
M%o'o 'e cancelacin 'e polos 0.67
Gla ( z ) k
(0.017)( z 0.87) ( z )( z 1)
El polo en - " 1 no se 'e/e cancela&! p*es%o 3*e 4ace 3*e e ss 0 . k Siemp&e 3*e %en$amos en el 'enomina'o& z 1 ! es *n in%e$&a'o&. z 1
Si se cancela se p&esen%a lo si$*ien%e
0.42
tan 1
2
17.1º 0.87 0.51 0.42 180º tan 1 139.4º 1 0.51
1 2 180º 17 º 1 139.4 º 180 º 1 57.6º 0.42
tan 57.6 º
0.25
0.51
c( Con'icin 'e ma$ni%*' G la
1
k (0.017)( z 0.87) ( z 1)( z 0.25)
1
Po& MAT LA0 n*m " 5 #.#1 #. #178 9: 'en " 5 1 ;1.2< #.2< 9: Z 0.51 j 0.42 : = " a/s+pol>?al+'en!-(.@pol>?al+n*m!-(( = " 12.88 Con%&ola'o& Gc ( z ) 12.7 z 0.67 Compensa'o& en a'elan%o PD Pasa Al%o z 0.25
Gla Gc G z Glc feedback (Gla , [1],1) : +MATLA0( p pole(Glc )
0.5
tS 2 n=[1]; d=[1 2 0]; gp=tf(n,d); gz=c2d(gp,0.2,'zoh')
zpk(gz) nc=13*[1 -0.67]; dc=[1 -0.25]; gc=tf(nc,dc,0.2) zpk(gc) n1=[0.0175 0.01539]; d1=[1 -1.67 0.67]; g1=tf(n1,d1,0.2) g!=g1*gc zpk(g!) gc=f""d#!ck(g!,1) po"(gc) $t"p(gc)
POR MAT-LAB CANCELACIÓN DE POLOS %************************************************************ % p($) & = +24($10)($2) ++ / + + % % + + + t$ = 0.02 $"g, = 0.001 $"g, %&p = 0.5% %************************************************************* c"!8 ! ho" d:$p(' '); d:$p('******************************************************') d:$p('+ + +& + +'); d:$p('+ + + & + +&'); d:$p('+ &++ + + '); d:$p('******************************************************') % % % Gp( s )
2 s ( s 2)( s 10)
Gc ( z ) k *
z alfa z beta
% d:$p(' '); t$=:np?t(':g:t" " t:"po d" "$t!#"c::"nto > t$($"g) = '); % t$ = 0.02; d:$p(' '); =:np?t(':g:t" " t:"po d" ?"$t8"o @@ ($"g)> '); % =0.001; d:$p(' ');
&p=:np?t(':g:t" !A:o $o#8":p?$o > &p = '); % &p = 0.5%; c:t!= $oB"(&p = 100*"Ap(-c:t!*3.1C164$D8t(1-c:t!E2))') c:t! = "B!(c:t!) % c:t! = 0.F6 d:$p(' '); d:$p(' '); % (!) c!c?o d" poo do:n!nt" "n todo " $:$t"! !zo c"88!do Gn=C4(c:t!*t$); % f8"c?"nc:! n!t?8! p!8! 2% d" to"8!nc:! Gd=Gn*$D8t(1-(c:t!E2)); % <8"c?"nc:! !o8t:g?!d! G$=(2*p:)4; % <8"c?"nc:! d" ?"$t8"o ?"$t8!$=G$4Gd; % &?"$t8!$ po8 c:co % ?"$t8!$4c:co = 53 % !gn:t?d d" poo do:n!nt" &!goo="Ap(((-2*p:*c:t!)4$D8t(1-(c:t!E2))).*(Gd4G$)); ngoo8!d=(2*p:).*(Gd4G$); % !ng?o d" poo do:n!nt" "n 8!d:!n"$ ngoog8!d=(ngoo8!d*1F0)4(p:); % !ng?o d" poo do:n!nt" "n g8!do$ zpo"=&!goo*"Ap(H*ngoo8!d); poo do:n!nt" d:$p('+ B!o8 d" oo o:n!nt" "$>%') d:$p(zpo") % poo do:n!nt" > 0.F130 0.0969: %************************************************************* % (#) :$c8"t:z!8 ! p!nt! % p($) = 24($10)($2) gp$ = zpk([],[-10 -2],2); gp$=tf(gp$); gpztf=c2d(gp$,,'IJ'); % % + J+ + + + !f! / % + & + / %************************************************************ % J!!8 !f!
!f!=pgpz(1) d:$p(' '); d:$p('+ B!o8 d" f! "$> ') d:$p(!f!) % !f! = 0.99F %****************************************************** % () J #"t! + /
% ! f?nc:Kn d" t8!n$f"8"nc:! "n !zo !#:"8to t:"n" % ?n c"8o "n z = -0.996 L do$ poo$ "n z = 0.99 L z = #"t! % t"t!1 - f:1 - f:2 = -1F0M t"t!1N8!d=!t!n(:!g(zpo").4(!#$(zgpz)8"!(zpo"))); t"t!1Ng8!d=(t"t!1N8!d*1F0)4p:; f:1N8!d= p:-!t!n(:!g(zpo")4(pgpz(2)-8"!(zpo"))); f:1Ng8!d=(f:1N8!d*1F0)4p:; f:2Ng8!d=t"t!1Ng8!d1F0-f:1Ng8!d; f:2N8!d=t"t!1N8!dp:-f:1N8!d; % t"t!1(g8!do$) = 3.1M % f:1(g8!do$) = 151.3M % f:2(g8!do$) = 31.7M #"t!=8"!(zpo")-:!g(zpo")4t!n(f:2N8!d); % #"t! = 0.656C d:$p(' '); d:$p('+ B!o8 d" O"t! "$> ') d:$p(#"t!) %******************************************************* % J!!8 k po8 cond:c:Kn d" !gn:t?d % k*!#$(c(z)*(z)) = 1 "n " poo do:n!nt"
gcz=zpk(!f!,#"t!,1,); !f! L #"t! g!=gcz*gpz; % !(z)=c(z)*p(z)
%
c(z)
con
% % Gla( z )
9.9601e 007 ( z 0.998)( z 0.996) ( z 0.6564)( z 0.99)( z 0.998)
% % g! no !p!8"c" $:p:f:c!d!, no $" c!nc"! " poo con " % c"8o "ntonc"$ $" d"#" h!c"8 [z1,p1,k1]=zpkd!t!(g!,'B') :=f:nd(p1==!f!); p1(:)=[ ] :=f:nd(z1==!f!); z1(:)=[ ]; g!= zpk(z1,p1,k1,) g!Ntf=tf(g!); % onB:"8to ! &od"o tf [n?,d"n]=tfd!t!(g!Ntf,'B'); % +At8!:go o$ B!o8"$ g!1=poL2$L(n?,'z')4poL2$L(d"n,'z'); z=zpo"; !g = !#$("B!(g!1)); k=14!g; % k = 2.0593"00C d:$p(' '); d:$p('+ B!o8 d" k "$> ') d:$p(k) % () c(z) = ') gczz=zpk(!f!,#"t!,k,) % %
Gc ( z )
20593.3092( z 0.998) ( z 0.6564)
% % (+) +/+ / g!z=k*g!; gc=f""d#!ck(g!z,1); g8:d $t"p(gc) g8:d % op8o#!8 " poo do:n!nt" p=po"(gc); d:$p(' ');
d:$p('o$ poo$ "n !zo "88!do d" :$t"! $on> ') d:$p(p) % oB"8$hoot "n po8c"nt!H" &p = "Ap(-c:t!*Gn*p:4Gd)*100 % &p = 0.5%
6.2 DISEÑO DE UN CONTROLADOR P.I EJEMPLO 6.2 Gc ( z ) k
z z 1
G p (s)
10 ( s 1)(s 2)
Especificaciones T"#.1s$
M p 20% t S 5sg kp ?
ki ?
G PI k p k i
G PI (k p k i
T z 1 2 z 1 T 2
z )
k i T 2k p k i T 2k p z 1
k k p k i T : k i T 2k p k iT 2k p 2 G ( z ) c 2 dm (G p , T , ' zoh ' )
G ( z ) 0.045
+MATLA0(
( z 0.90) ( z 0.9)( z 0.81)
a( Polo 'ominan%e
M 20% p
t S 5sg T 0.1sg
Mp e
1 2
tS 5
0 .2
4
0.45
n 1.78 rad sg
0.45 n
d n 1 2 1.78 1 0.452 1.6 rad
d 1.66 rad z z
e(
sg
0.451.780.1)
0 .92
z 1.6 0.1
0.16 rad
z 9 .2 º
Polo 'ominan%e 0.92 9.2º p 0.9 j 0.15
/( Con'icin 'e n $*lo
sg
0.15
tan 1
1 tan
0.9 0.9 1 0.15
4.7 º
59º
1 0.9
2 180º tan 1 0.15 123.7 º 1 0.9
1 2 178º 180º
c( Con'icin 'e ma$ni%*' G la
1
Gla G c G z k
( z ) 0.045( z 0.90 ) ( z 1)( z 0.90)( z 0.81)
Cancelacin 'e polos 0.90 Gla k
0.045( z 0.90) ( z 1)( z 0.81)
Se cancela el polo 3*e es%e ms ce&ca 'e la *ni'a' pa&a 3*e 4a>a menos &ies$o 'e ines%a/ili'a'. k (0 .045 )( z 0 .90 ) ( z 1)( z 0 .81)
1
z 0.9 j 0.15
= " #. k k p ki
T
2
k iT 2k p k iT 2k p
0.77 k p 0 .05 k i
0 .9
+1(
0.1k i 2k p 0.1k i 2 k p
+2(
De +1( > +2( o/%enemos
kp
> ki
k p 0.73
k i 0.77
CONTROLADOR PI POR MAT-LAB c"!8 ! ho" d:$p(' '); d:$p('*****************************************************') d:$p('+ + +& + +'); d:$p('+ + + & + +&'); d:$p('+ + / '); d:$p('*****************************************************') % % 10
% Gp( s ) % %
( s 1)( s 2)
k kp ki *
T
Gpi ( z ) k * alfa
2
z alfa z 1
ki * T
2 * kp
ki * T
2 * kp
% % kp=on$t!nt" 8opo8c:on! % k:=on$t!nt" nt"g8! %*********************************************************** =:np?t(':g:t" " t:"po d" ?"$t8"o @@ ($"g)> '); % =0.1; % () +I gp$=zpk([],[-1 -2],10); gpz=c2d(gp$,,'IJ'); % !nt! "n " o:n:o I % %
Gp( z )
0.04528( z 0.9048) ( z 0.9048)( z 0.8187 )
% %************************************************************ % ! p!nt! d:$c8"t:z!d! no t:"n" poo "n z =1 , o $"! "$$ %("88o8 d" "$t!do "$t!c:on!8:o) no "$ c"8o, po8 t!nto " % cont8o!do8 p: d"#" !d:c:on!8 "$t" poo L c!nc"!8 " poo % !$ c"8c!no ! z=1 d" ! p!nt! !d:c:on!ndo " co88"$pond:"nt"
% c"8o ""B!ndo !$: " !8g"n d" "$t!#::d!d d" $:$t"! L % "B:t!ndo " :nc8""nto d" o8d"n %************************************************************ pgpz=po"(gpz); % oo$ "n p(z) zgpz=z"8o(gpz); % I"8o$ "n p(z) n="ngth(pgpz); % n = 2 % (O) J < po8 Ptodo d" !nc"!c:Kn d" oo$ L "8o$ % + poo !$ c"8c!no ! 1 $" c!nc"! con " c"8o d" % cont8o!do8 (!f!) poo$= $o8t( pgpz) % poo$ = 0.F1F7, 0.90CF % !f! "$ " poo !Lo8 !f! = poo$(n) % !f! = 0.90CF % % % %
() p: p: k =
') d:$p(kp); d:$p('on$t!nt" nt"g8!> ') d:$p(k:); d:$p('cont8o!do8 d:g:t!> ') d:$p!L(gp:) % %
Gpi ( z )
0.42085( z 0.9048) ( z 1)
% % "$p?"$t! ! "$c!Kn $t"p(gc)
6.3 CONTROLADOR PD: METODO ANALITICO (z ) Gc ( z ) k z
G PD k p k d
G PD
z 1 Tz
k p Tz k d z k d Tz
kd z k pT k d k p T k d T z
z (k pT k d ) k d Tz
k
k pT k d T
k
d : k T k p d
Ejemplo 1
G p ( s)
( s 1)(s 2)
:
Especificaciones T"#.1 M p 16% : t s 2 sg n d
M p e
4
: t S n
z e ( nT )
z d T
Sol*cin tS 2
4
n
n 2
2 d
d 3.46 rad sg
M p 0.16 e z
exp( 2 0.1)
z
0.8187
0.8187
z 3.46 0.1 0.346rad z 19.82 º
Polo 'ominan%e p
z
z
p
0.77
j 0.28
G p s
G ( z ) (1 z 1 ) z
G ( z ) c 2 dm (G p , T , ' zoh ' )
G( z)
+MATLA0(
0.0045( z 0.9048)
( z 0.9048)( z 0.8187) G la G c G z
k(z ) G( z) z 0.0045k ( z )( z 0.9048) Gla z ( z 0.9048)( z 0.8187)
Gla
Cancelacin 'e polos 0.9048
Gla
0.0045k ( z 0.9048)
z ( z 0.8187)
1 1 2 180 º 0.28 1 tan 1 9.49º 0.9048 0.77 0.28 1 tan 1 19.98º 0.77 0.8187 0.77 2 90º tan 1 99.86º 0.28
9.49 º 19 .98 º 99 .86 º 110 .35 º +No c*mple la con'icin 'e n$*lo(
Como no se c*mple la con'icin 'e n$*lo! el polo - " #.8#7 no se p*e'e cancela&! po& lo c*al p&oce'emos a /*sca&lo.
1 2 1 2 3 180 º
3 90º tan 1
0.9048 0.77 0.28
115.7 º
2 1 1 2 3 180 º
2 9.49 19.98º 99.86º 115.7 º 180º
2 46º tan 2 tan 46º
k ( z 0.5) G z
0.28 0.77
0.5
c
Con'icin 'e ma$ni%*' G la
1
0.0045 k ( z 0.5)( z 0.9048 ) z ( z 0.9048 )( z 0.8187 )
1
z 0.7 j 0.28
k 24.3 Gc ( z ) k
24.3( z 0.5)
k pT k d T
z k
d : k T k p d
k 24.3 : 0.5 : T 0.1 0.1k p k d 24.3 2.43 0.1k p k d 0.1 kd k 0.5 0.5 d k d 1.215 0.1k p k d 2.43
kp
2.43 k d 0.1
2.43 1.215 0.1
CONTROLADOR
k p 12.15
PD
POR MAT-LAB
c"!8 ! ho" d:$p(' '); d:$p('*****************************************************') d:$p('+ + +& + +'); d:$p('+ + + & + +&'); d:$p('+ + / '); d:$p('*****************************************************')
% % % Gp( s )
1 ( s 1)( s 2)
Gpd ( z ) k *
z alfa z
% %
k
kp * T kd
alfa
T
kd kp * T kd
% % kp=on$t!nt" 8opo8c:on! % kd=on$t!nt" "8:B!t:B! %************************************************************* d:$p(' '); t$=:np?t(':g:t" " t:"po d" +$t!#"c::"nto @t$@ ($"g)> '); % t$=2; d:$p(' '); p=:np?t(':g:t" " o8c"nt!H" d" o#8"p?$o @&p@ (%)> '); % p=16.3; p=p4100; d:$p(' '); =:np?t(':g:t" " t:"po d" ?"$t8"o @@ ($"g)> '); % =0.1; d:$p(' '); d:$p(' '); % () ++& + &+ + +& + I + % t$ = C4$:g! 2% to"8!nc:! $:g!=C4t$; % &p = "Ap(-$:g!*p:4Gd) Gd = "B!($oB"('p="Ap(-$:g!*p:4Gd)')) % poo$ d" $:$t"! cont:n?o $1=-$:g!H*Gd; $2=conH($1); G$=(2*p:)4; % <8"c?"nc:! d" &?"$t8"o % n?"8o d" ?"$t8!$ po8 c:co ?"$t8!$=G$4Gd; zpo"="Ap(*$1); d:$p(' '); d:$p('+ B!o8 d" oo o:n!nt" "$> ') d:$p(zpo") % &+ = 0.7701 0.2779: % (O) -0.90CF (t"t!1) poo$ d" ! p!nt! > [0.90CF (f:3) 0.F1F7 (f:2)] c"8o$ d" cont8o!do8 > !f! (t"t!2) poo$ d" cont8o!do8 > 0 (f:1) poo do:n!nt" > 0.7701 0.2779: t"t!1 t"t!2 -(f:1f:2f:3) = -1F0M t"t!(n) --T ng?o I"8o 8"$p"ct:Bo
t"t!1N8!d=!t!n(:!g(zpo").4(!#$(zgpz)8"!(zpo"))); t"t!1Ng8!d=(t"t!1N8!d*1F0)4p:; f:1N8!d=!ng"(zpo"); f:1Ng8!d=(f:1N8!d*1F0)4p:; f:2N8!d=p: - !t!n(:!g(zpo").4(!#$(pgpz(1,1))-8"!(zpo"))); f:2Ng8!d=(f:2N8!d*1F0)4p:; f:3N8!d=p: - !t!n(:!g(zpo").4(!#$(pgpz(2,1))-8"!(zpo"))); f:3Ng8!d=(f:3N8!d*1F0)4p:; t"t!2Ng8!d=-1F0-t"t!1Ng8!df:1Ng8!df:2Ng8!df:3Ng8!d; t"t!2N8!d=(t"t!2Ng8!d*p:)41F0; % t"t!1=9.C221M, f:1=19.FC56M, f:2=115.F610M, f:3=99.922FM
!f!=8"!(zpo")-(:!g(zpo")4t!n(t"t!2N8!d)); d:$p(' '); d:$p('+ B!o8 d" f! "$> ') d:$p(!f!) % !f! = 0.5036 % () + &/ !#$(pz*cz) = 1 gcz=zpk(!f!,0,1,); g!=gcz*gpz; g!Ntf=tf(g!); [n?,d"n]=tfd!t!(g!Ntf,'B'); g!1=poL2$L(n?,'z')4poL2$L(d"n,'z'); z=zpo"; !g = !#$("B!(g!1)); k=14!g; d:$p(' '); d:$p('+ B!o8 d" k "$> ') d:$p(k) % k = 2C.105F gczz=zpk(!f!,0,k,); % %
Gpi( z )
24.1058( z 0.5036) (z)
% % (+) +/+ / g!z=gczz*gpz; gc=f""d#!ck(g!z,1); g8:d $t"p(gc) p=po"(gc); d:$p(' '); d:$p('o$ poo$ "n !zo "88!do d" :$t"! $on> ') d:$p(p) % o$ poo$ "n !zo "88!do d" :$t"! $on> % [ 0.07C2 0.7701 0.2779: 0.7701 - 0.2779:] % Kt"$" D?" " poo$ 0.07C2 "$t! c"8c! ! o8:g"n L po8 o % t!nto no "$ do:n!nt"
6.4 CONTROLADOR PID:
METODO ANALITICO GPID k p k i G PID
T z 1
2 z 1 z 2 z B k z ( z 1)
Ejemplo
kd z 1 T
z
G p ( s)
10 ( s 1)(s 2)
:
T
0.1sg
Especificacin k ! 5 cons%an%e 'e ?eloci'a' (1 z 1 )Gla k ! l! T z 1 G z c 2 dm (G p , T , ' zoh ' )
Gz
0.0452 ( z 0.9048) ( z 0.9048)( z 0.8187)
T z 1
2 z 1
Gla G ( z ) k p k i
k!
1
k d z 1 T
z
0.0452 ( z 0.9048)
( z 0.9048)( z 0.8187 ) * k i T l! z 1
T * ( z 1) 2
0.0452 (1.9048) 0. 1 k ! 10 * ki * 2 2 (0.0952)(0.1813) k ! 0.988k i 5 0.988 k i k i 5.06
0.045 ( z 0.9048) k ( z 2 z B )
G ( z 0.9048)( z 0.8187) z ( z 1) la
B
k i T 2 2k p T 4k d k i T 2 2k p T 2k d 2k d
k i T 2k p T 2k d 2
Cancelacin 'e polos > ce&os z 2 z B ( z 0.9048)( z 0.8187) z 2 z B z 2 1.1724 z 0.741
k i T 2 2k pT 4 k d k i T 2 2 k p T 2k d
1.724
2k d
B k i T 2 2k pT 2k d 0.741 k d 0.4995 : k p 1 .4525 : k i 5.06
0.1 z 1 0.4995 z 1 0.1 z 2 z 1 z 1 z 1 1.4525 0.253 4.995 z 1 z 1.425 z ( z 1) 0.253 z ( z 1) 4.995 ( z 1) 2 z ( z 1)
G PID 1.4525 5.06 G PID G PID
CONTROLADOR
PID POR MAT-LAB
c"!8 ! ho" d:$p(' '); d:$p('******************************************************') d:$p('+ + +& + +'); d:$p('+ + + & + +&'); d:$p('+ + / '); d:$p('******************************************************') % % % Gp( s )
10
Gpid ( z ) k *
( s 1)( s 2)
z2 a * z b z * ( z 1)
% % k
ki"T$2 2"kp"T 2"kd 2"T
% % a
ki"T$ 2 2"kp"T 4"kd ki"T$ 2
2"kp"T
2"kd
2"kd
b
ki"T$ 2
2"kp"T
2"kd
% % kp=on$t!nt" 8opo8c:on! % kd=on$t!nt" "8:B!t:B! % k:=on$t!nt" nt"g8! %****************************************************** d:$p(' '); kB=:np?t(':g:t" ! on$t!nt" d" "88o8 d" "oc:d!d @kB@> '); % kB=5; d:$p(' '); =:np?t(':g:t" " t:"po d" ?"$t8"o @@ ($"g)> '); % =0.1; % +88o8 "n "$t!do "$t!c:on!8:o "$$=14kB; % "$$ = 0.2 no "$ n?o $" d"#" co88"g:8 con ?n % ()+I gp$=zpk([],[-1 -2],10); gpz=c2d(gp$,,'IJ'); pgpz=po"(gpz); zgpz=z"8o(gpz);
% !nt! "n " o:n:o I % oo$ "n p(z) % I"8o$ "n p(z)
% (O) J k: + + &+ + + kB $L$ k: kd kp z gpzNtf=tf(gpz);
[n?1,d"n1]=tfd!t!(gpzNtf,'B'); n?2=poL2$L(n?1,'z'); d"n2=poL2$L(d"n1,'z'); gz=n?24d"n2; gp:dN1=kp k:*(42)*((z1)4(z-1)) (kd4)*((z-1)4z); g!N1=(gz*gp:dN1); :=::t(((z-1)4z)*(g!N14),z,1); % = :, "ntonc"$ A kB = :-kB; k: = $oB"(A) k:="B!(k:) % () J kp, kd + + $L$ kp kd [kd,kp]=$oB"('2*kd4(k:*E22*kp*2*kd)=d"n1(3)','(k:*E22*kp*-C*kd)4(k:*E22*kp*2*kd) = d"n1(2)','kp','kd') kp = "B!(kp) kd = "B!(kd) % kp = 1.C125, kd = 0.C295 d:$p(' '); d:$p('on$t!nt" 8opo8c:on!> '),d:$p(kp); d:$p('on$t!nt" '),d:$p(kd); d:$p('on$t!nt" "8:B!t:B!> nt"g8!> '),d:$p(k:); % () 0.36FF 0.31F6: $t"p(gc) % t$ = 0.6 $"g, &p = 3.F %
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