UV Automatique - ASI ( 1 TD7 : Système en boucle fermée

UV Automatique - ASI ( 1 TD7 : Système en boucle fermée

"!$#&%'(*)*+-,/..-0214365798:.";<.>=?,@>.A (*'0B)*CD+>(E."FD.G7H3I=?=?.-7H)*.-5J=K(
L&M-N*OQPSRTPSNU
VW PSX WZY RT[]\SOQN^ W_ OQXP`N YY ^ Y [NbacX W P
dRTX W [NbdQOQe WZY acN*Ofd
H(s) =
1000
s(s + 10)2
H(s) =
1000
s(s + 10)2
HgihjN YkY dQ\SlmNnN Y dilmR Y [Ze WZY ^ W e YQY NSOQo>R YQY N*lpN W drqsONSdXH^ZOt^ W RudQeHRuOQNve?o9N*Pb^ W PSXHOQONEP
dQNS^]O _ OQX _ XHOdRTX WZW NSwxy[Nnz>e{R W
gZ|X WZW NSOrwTN Y P~}]*lUenacX W P
dQRuX W]W N*wJ[^ YfkY d\SlmN e YY NSOQo-R€g
 gikh‚e{wƒP`^ZwuN*OrwTepoKe{wTNS^]Ot[]^„z>e{R W k … ^]RJe YY ^]OQN†eH^ YkY dQ\SlmNb^ W N‡lUe{OQzHNb[N _ }Ze Y Nbˆ>‰ ◦ g
Š gr‹JeP`X WIY RTz W NnN Y d^ WŒY RTz W eHwJ*P~}ZNSwTX W ^ W RudQeHRuOQNHgh‚e{wƒP`^]wTNSOw€ NSOQOQNS^]ON W OQSz9RulmN _ NSOQlUe W N W dN W dOQN wƒeP`X WZY Ruz W N N`d†wTe
Y XHOdRTNb[]^ YkY dQ\SlmNHg]Žt _ X W [OQN‡qpwTeplmSlmN ^]N Y dRTX WY RwTemPSX WZY Ruz W N†N Y dt^ W N‡O~e{l _ N‡[N _ N W dQNUHg
ˆZg VW [ Y RuOQN‡e?oHX9RuOlUe{R W dN W e W dr^ W e YQY N*Oo-R YY NS… lmN W d‚ON Yf_ N*P`dQe W drwTN Y P`X W [RudRTX WZYrY ^]RToKe W dN Y
‘ NSNSOQOQOON*N*^]^]OOrY[d~NbeKddRO~… e?^]“ W N e{W z9^]NiwuwT” NHW ’ RuεN p(∞) = 0 ’ g
‘x•XH^]OtPSN†a:e{RTON9’X W eH[?–—X9R W dte{ε^p(∞)
P`X9OOQN*=P`d5%
N*^]O _ OX _ X9OfdQRuX W]W N*w6x•’]^ W PSXHOQONEP
dQNS^]OrqpONSdQeHOQ[„[N _ }Ze Y N9g
|X W]W N*OrwuN W XH^]o9N*e{^ Y P~}ZSlUevacX W P`dRTX W]W N*w[N‡w˜ e YY NSOQo-R YQY N*lmN W d*g
h‚e{wƒP`^ZwuN*OrwuN Y_ e{O~e{lm\`dQON Y [^"P`XHOQOQN*P
dQNS^]OEg
L&M-N*OQPSRTPSN 
VW PSX WZY RT[]\SOQN†[N W XH^]o9N*e{^wuN YkY dQ\SlmN‡[NbacX W P
dRTX W [NbdQOQe WZY acN*Ofd
VW [  Y RTON†OQ*e{wTR Y N*Oj^ W e YQY NSOQo>R YQY N*lpN W djqpONSdXH^ZO‚^ W R™d~e{RTON _ NSOQlpNSdfd~e W dt[N Y e{dR Y a:eHRuOQNiwTN‡PSe{}ZRuN*O[]NbP~}Ie{OQzHN Y‚Y ^]RuoKe W dbš
‘ OQ[  _ _ X e WZYQYY N*NblmX NY P`W RTdrwuwƒNeKW„dQXH› RTOX9N^ZN P`WwTNb› acX9NSOQ^ZlmP`wTSNbN acNSOQlmSN9’
‘ dN*l _ZY [N‡OQ _ X WZY Nbqm‰>œ[N‡žZg Ÿ Y D’ BF ≤ 5% ’
‘ NSOQOQNS^]O Y dQeKdQR ^]N W ^]wTwuN9g
‘
Hgx&OQX _ X Y NSO^ W … P`X9OOQN*P`dN*^]O*g|X W]W NSOrwTN Y P~}]*lUe acX W P
dQRuX W]W NSw[^ YfkY d\SlmN‡e YQY NSOQo>R˜g
W [RƒPSe{dRTX WZY‚_ XH^]OwTN†P~}ZXHRuM[^PSXHOQONEP
dN*^]O†š … ^Ze W [X W› XH^IP`wTNwTN YkY dQ\SlmN Y e WZY P`XHOQOQN*P
dQNS^]OE’HwƒevO _ X WZY N†R W [RƒP`RTNSwTwuN
N W ON*zHR Y dQO*N†ep^ W [ _ e YQY N*lmN W dt[N‡ŸHž>œg
 gr¡ _ e{OdRTO[^pPSeH}]RuN*O[NP~}ZeHOz9N Y ’S[`dQNSOQlmR W N*OwƒetlUeHOz9N4[N _ }Ze Y N … ^]N[XHRudDe?o9XHRTOwTN YkY dQ\SlmNjPSXHOQORTzH¢NSdDwƒe _ ^Zw Y eKdQRuX W
[N PSXH^ _ ^ZONbqUžH[]£
P`X9OOQN Y_ X W []e W dQNHg
Š gr|`dN*OlmR W NSOrwƒemlUe{OQωzHNbc0[N _ }Ze Y N … ^ e{^]O~e{RudwuN YkY dQ\SlmN W X W P`X9OOQRuz9†qmPSN`dfdQN _ ^]w Y e{dRTX W g
ˆZgL W [E[^]RTON‡wTN Y_ eHOQeHlm\`dOQN Y [^"PSXHOQONEP
dN*^]OEg
‰g VW P`X WIY Rƒ[\SOQN … ^ZNUwuN Yk-Y d\*lmNme YY NSOQo-R Y ^ › R™dn[N Yb_ N*OfdQ^]O › eKdQRuX WZY†W X{dQSN Y d gDhjN Yb_ NSOd^ZO › e{dRTX WZY eHzHR YY N W d Y ^ZO‡wTe
dO~e WIY lmR YY RuX W [N‡wƒemP`X9lplUe W [N‡e{^ YfkY d\*lpN PSXHlmlmNbwuNblmX W dOQNbwTev”Zz9^]OQNbPSR™¤[N YY XH^ Y g
%
d
yc
+
−
ε
+
C(s)
+
H(s)
y
Correcteur
Avance de phase
¥¦c§j¨D ‘ e YY NSOQo-R YQY N*lmN W de?o9N*P†P`X9OOQN*P`dNS^ZOjxy©ªe?o?e W P`Nb[N _ }Ie Y N
« e>¬„|iX W]W NSOrw˜ N`M _ OQN YQY RTX W [N
«:­ O~e WZY acXHOQlmSNb[N ‹Je _ wTe9P`Nb[N‡w˜ N*OOQNS^]O ¬‚N W acX W P`dRTX W [N
N`d
g
« › ¬Gh‚e{wƒP`^]wTNSOw˜ N*OOQNS^]OrN W O*zHRTE(s)
_ NSOd^ZOf¤
lpN _ NSOQlUe W N W d Y RwƒepPSX WZY RTz W NbN Y dt^ W *P~}ZNSwTX εW N`d Y RX W P`X WZY Rƒ[\*YONc(s)
^… ]NbwTe D(s)
› eKdQRuX W N Y dPSX WZY dQe W dQNHg®^]NPSX WZY dQe{dN`¤˜df¤<X Wi¯ hjXHlmlmN W djOQSlmE[RuN*O‚qnP`N _ OX › wu\*lmN ¯ x&OX _ X Y N*O4^ W P`XHOQOQN*P
dQNS^]O
qpe9[?–—XHR W [OQN‡e{^P`X9OOQN*P`dNS^ZOqme?oKe W P`Nb[N _ }Ze Y N _ OQ*PS*[N W dEg
« PE¬„|iSdNSOQlmR W NSODwuN Y_ eHOQeHlp\SdOQN Y [NjPSNjP`X9OOQN*P`dN*^]O Y e WZY lpX[Ru”ZN*OwTN Y_ N*OfacX9OlUe W P`N Y [^vP*e{}]RTNSOD[NjP~}Ie{OQzHN Y R W RudRƒe{w˜g

°p±s²²³b° ´ ±sµ "!$#&%'(*)*+-,/..-0$1j3I5D7H8€.";˜.>=K,/@-.A (S'0B)EC+-(E."F.G793Z=?=K.-7{)E.>5=K(
L&M-N*OQPSRTPSNU
¶ ­ [^ YfkY d\*lpN
H(s) =
1000
s(s + 10)2
ŽtS*PSORud^ZON‡[NbwƒeU¶ ­
H(s) =
10
s 1+
s 2
10
‹JN fY kY d\*lpN‡ep^ W z9eHR W N W o-RudN YQY N kv = 10 g
Hgt·P~}]*lUenacX W P
dQRuX W]W N*wJ[^ YfkY d\SlmN e YY NSOQo-RDš _ e Y [ N [Ru¸sP`^]wud _ e{OdRƒP`^]wTRu\*ON9g
 gr¹GeHOz9N†[N _ }Ze Y N‡[N‡ˆ>‰ ◦
VW oHNS^]die?oHXHRTO‚^ W N‡lUe{OQzHNb[N _ }Ze Y N‡[N‡ˆ-‰ ◦ N W —– X9^Ze W d Y ^ZOrwuN‡z>e{R W k [^"P`X9OOQN*P`dNS^ZO‚x•g
¶ ­ NW £ V
HBO (s) =
10k
s 1+
s 2
10
L¢M _ wTXHRudQe{dRTX W [N‡w€ R W acXHOQlUeKdRTX WY ^]OrwTeplUe{OQzHN‡[N _ }Ie Y N
mϕ
= π + ϕBO (ωC0 )
ω π
C0
mϕ = π − − 2arctan
2
10
ω π
π
C0
mϕ =
=
− 2arctan
2
10
4
ω π
C0
⇒ 2arctan
=
10
4
π ⇒ ωC0 = 10tan
8
i¡ º»š ωC0 = 4.14rad/s g
|`dN*OlmR W eKdQRuX W []N k
‹Je _ ^Zw Y eKdQRuX W ωC0 oH*ORu”ZNbwTepOQNSwƒeKdQRuX W
⇒
10k
ωC0 1 +
2
ωC0
100
⇒k
|HBO (jωC0 )| = 1
= 1
=
ωC0 1 +
10
2
ωC0
100
i¡ º»š k = 0.48
Š gL&OQON*^]O Y dQeKdQR … ^]N
‹N YkY dQ\SlmN‡N Y dt[N P`wƒe YQY Np « ^ W R W d*zHO~eKdN*^]O
¬ ⇒ εp(∞) = 0
L&OQON*^]Ot[N†dO~e?“ W eHzHN†XH^"[Nbo-R™dQN YQY N
‹N YkY dQ\SlmN‡N Y dt[N P`wƒe YQY Np « ^ W R W d*zHO~eKdN*^]O
¬ ⇒ N*OOQNS^ZO[]N†o-RudN YY N†” W RTN‡lmeHR YW X WW ^]wuwTNHg
1 e?o9N*P
εv (∞) =
Kv = lim sHBO (s) = 10k
s→0
Kv
ˆZgt· k-W d}]\ Y N‡[^"PSXHOQONEP
dN*^]OrqpONSdQe{O~[[]N _ }Ze Y N
¶ ­ []^PSXHOQONEP
dQNS^]OŽrx
1 + T s e?oHN*P
C(s) = b
b>1
1 + bT s
Š
¶ ­ N W £ V [^ YfkY d\SlmN e?o9N*P†P`XHOQOQN*P
dQNS^]O Y x¼NSdiŽrx
HBOC (s) = kb
1 + Ts
10
1 + bT s s 1 + s 2
10
‹N‡Žrx½[]XHRudre{–—XH^dQNSOrN W› e YY N Y acOQ … ^]N W P`N Y wuNbz>e{R WW EP`N YY e{RTON _ XH^]Ote?o9XHRTO‚wƒe _ OQ*P`R Y RuX W [ Y RTO*NHg
ºtX9^]oHN*wuwTNbN`M _ OQN YQY RTX W [N‡w˜ N*OOQNS^]Or[N†dQOQeK“ W eHzHN
1 e?o9N*P
εv (∞) =
Kv = lim sHBOC (s) = 10kb
s→0
Kv
h‚e{wƒP`^Zw[N‡wTepoKe{wTNS^ZOt[N b
εv (∞) =
1
5
=
10kb
100
2
⇒b=
k
¡iº»š b ≈ 4 g
h‚e{wƒP`^Zw[N
x•XH^]O W N _ e TY lmX[Ru”ZNSOrwƒeplmeHOz9N†[]N _ }Ze Y N _ OQ*P`E[NSlmlmN W drOQSzHwTSN9’Ruwa:e{^diP~}]X9R Y RTO
T
dN*w … ]^ NUš
ωc0
10
1
≤
⇒T ≥
T
10
ωc0
i¡ º»š T = 10
ω
L&M-N*OQPSRTPSN 
¶ ­ [^ YfkY d\*lpN
c0
⇒ T = 2.4s
H(s) =
1000
s(s + 10)2
H(s) =
10
s 2
s(1 + 10
)
g
ŽtS*PSORud^ZON‡[NbwƒeU¶ ­
¾ eHR W N W o-RudN YY N kv = 10 g
Hgihj}]X9R™M[ ^ W PSXHOQONEP
dN*^]O
‹N Yk-Y d\*lmNN Y dr[N‡P`wƒe YQY Nv N*OOQNS^]O Y dQeKdQR … ^]NdX9^K–—XH^ZO Y‚W ^]wTwuN†N W w€ e ›IY N W P`N†[N _ N*OfdQ^]O › eKdRTX WIY ⇒ w˜ e9P
dRTX W R W dS¤
zHO~e{wTN† W  N Y d _ e YW *PSN YQY eHRuOQNH⇒g
L W £‚¶ Y e WZY P`X9OOQN*P`dNS^ZO*’X W e"^ W [ _ e YQY N*lmN W dv[N„ŸHž>œg&hjN*PSR Y Ruz W Ru”ZN … ^]NwTeŒlmeHOz9Ns[N _ }Ze Y N[^ YfkY d\*lpN
R W R™dQRTeHwZN Y d _ NSdRudNHg>x•XH^]Ojw€ e{^ZzHlmN W dN*O*’HRuwZa:e{^d‚^ W Nie9P
dRTX W []SOQRuo9SNt|v’>[ XH¿U^ W P`X9OOQN*P`dNS^ZO&q e?oKe W PSNt[N _ }Ie Y Nt¡tx•g
 gr¡ W e{w k-Y N‡[^"PSeH}]RuN*Ot[N‡P~}Ze{OQzHN Y
ξBF
DBF% ≤ 5% ⇒ ξBF = 0.7
3
= 0.7 ⇒ ωn,BF tr5 % = 3 ⇒ ωn,BF =
tr5 %
⇒ ωn,BF = 5rad/s
¶]X9Olv^]wTN Y [ e _]_ OX?MRTlme{dRTX W
Žr _ X WZY NbX Y PSRuwTwTe{dX9RuOQN†[] Y RuOQSN Y ^ _]_ X Y N†^ W PSXHl _ XHOdNSlmN W dt[^  N†X9OQ[]ONb[XHlmR W e W d _ XH^]OrwTN YfkY d\*lpNbN W £¶‚g
mϕ = 100ξBF = 70◦
ωc0 = ωn,BF = 5rad/s
ˆ
Š gr¹GeHOz9N†[N _ }Ze Y N‡[^ YfkY d\SlmN _ XH^]Orwƒepo?eHwuN*^]Ot[N
ωc0
_ OEP`*[]N W dN
mϕ
= π + ϕ(ωC0 )
ω π
C0
mϕ = π − − 2arctan
2
10
ω π
C0
mϕ =
− 2arctan
2
10
mϕ = 37◦
¡iº»š
ŽrN*lmeHO … ^]NšwƒeoKe{wTNS^]O‡P`Ru¤[N YQY ^ Y N Y d wTe„lUe{OQzHNm[N _ }Ze Y N … ^ e{^]O~e{RudbwuN Y kY dQ\SlmN Y R Y eacO … ^ZN W PSNm[]NsP`XH^ _ ^]OQNmq
žH[Z£N Y d _ wTe9P`*NtN W ωc0 = 5rad/s g>L&wTwuNN Y d‚[RuÀBSOQN W dNi[]Niwƒe lUe{OQzHNt[N _ }Ze Y Nie9P
dQ^]NSwTNi[^ YkY dQ\SlmN … ^]R6N Y dj[N  ◦
qpwTe _ ^ Y w Y e{dRTX W ωc0 = 6.8rad/s « X › dN W ^]N Y XH^ Y lme{dwƒe › ¬`g
ˆZgt· -k W d}]\ Y N‡[^"PSXHOQONEP
dN*^]O
¶ ­ []^PSXHOQONEP
dQNS^]O
C(s) = Kc
e?oHNEP
1 + aT s
1 + Ts
a>1
Á&ÂKÃuÄSÅÃDÆ{ÅvÇ]Â{ÈÂKÉUÊ`˘ÈÌpÂ
‹ N YfkY d\*lpNve _]_ XHOdN*OQeHR™di^ W NnlUe{OQzHNv[N _ }Ie Y Nv[N Š>Í ◦ gIx•X9^]O†e?o9XHRTOrwTelUe{OQzHNn[N _ }Ze Y Nn[]N Í ž ◦ ’6RTwJa:eH^d … ^]NvwuN
P`X9OOQN*P`dN*^]Ore _]_ XHOdNb^ W N _ }Ie Y N Y ^ _]_ wu*lpN W dQe{RTOQNb[]N ∆mϕ = 33◦ g
ϕc,max
= ∆mϕ = arcsin
⇒a =
a−1
a+1
1 + sin(∆mϕ )
1 − sin(∆mϕ )
Á&ÂKÃuÄSÅÃDÆ{ÅvÇ]Â{ÈÂKÉUÊ`˘ÈÌÎ
‹& e?oKe W PSN†[]N _ }Ze Y NblUeKMRulUeHwuNbN Y dte _]_ uw R … ^]SN qpwƒe _ ^]w Y eKdQRuX W
ωc,max
=
⇒T
=
ωC0
1
√ = ωC0
T a
1
√
ωC0 a
Á&ÂKÃuÄSÅÃDÆ{ÅsÏ9Â{ÐÒÑ
‹Nbz>e{R W Kc _ N*OlmKN`cdi[]N _ wTe9P`N*O‚w˜ eKMN V []£½qmwƒe _ ^]w Y eKdQRuX W
ωC0
gV O
|HBOC (jωC0 )| = 1 ⇒ |C(jωC0 )H(jωC0 )| = 1
√
|C(jωC0 )| = Kc a
√
|HBOC (jωC0 )| = 1 ⇒ Kc a|H(jωC0 )| = 1
√
10
⇒ Kc a
2 ) = 1
ωC0
ωC0 (1 + 100
·e9P~}Ze W d … ]^ N
’X W N W []*[^]Rudnš
ω2 )
¡iº»š
’
’
C0
ωC0 (1 + 100
√
⇒ Kc =
10 a
a = 3.4 T = 0.108s K2 = 0.33
‰gr¡ YQY NSOQo>R YQY N*lpN W de?o9N*P _ N*OfdQ^]O › eKdRTX WIY
‘ L¢M _ ON YY RTX W [Nbw€ NSOQON*^]O E(s)
·-XHOdRTN‡e YQY N*Oo-RTN
Y (s) =
’ Y X9R™d‡š
g
d
H(s)
C(s)H(s)
Yc (s) +
D(s)
1 + C(s)H(s)
1 + C(s)H(s)
VW N W []*[^]Rudtw˜ NSM _ ON YY RuX W [N‡w˜ NSOQON*^]O
E(s) = Yc (s) − Y (s)
1
H(s)
E(s) =
Yc (s) −
D(s)
1 + C(s)H(s)
1 + C(s)H(s)
ωC0
o9SOQR™”ZNbwƒepON*wTe{dRTX W
‰
‘ &L­ O}]OQSNSX9^ZOOr\*lpN W N‡O[*NbzHRTwƒlpepN o?eH_ wuNSN*OQ^]lUOe ” W W N eHW wuNd _ XH^]O
ε(∞) =
ε(∞) =
D(s) =
1
s
N`d
Yc (s) =
1
s
lim sE(s)
s→0
lim
s→0
1
H(s)
−
1 + C(s)H(s) 1 + C(s)H(s)
h‚e{wƒP`^]wTX WZY wƒepwuRTlmR™dQN _ X9^]OtP~}Ze9P`^ W [N Y dNSOQlmN Y
­ *N OlmNm
ε1 (∞) =
ε1 (∞) =
1
1 + C(s)H(s)
1
lim
1+aT
s
s→0 1 + Kc
1+T s
lim
s→0
10
s 2
s(1+ 10
)
ε1 (∞) = 0
­ N*OlmN 
ε2 (∞) =
ε2 (∞) =
ε2 (∞) =
H(s)
1 + C(s)H(s)
10
10
1 + aT s
lim −
1 + Kc
s 2
s→0
1 + Ts s 1 + s 2
s 1 + 10
10
−1
Kc
lim −
s→0
hjX W PSwu^ Y RTX W š6w€ NSOQON*^]O Y dQe{dR ^]N W  N Y d _ e YW ^ZwuwTNnN WG_ OQ Y N W PSNn[N _ NSOd^ZO › e{dRTX WZY gZx•X9^]O*wuRTlmR W NSObP`N`ddNvwTe _ N*Of¤
d^]O › eKdQRuX W PSX WZY dQe W dNH’X W R W … dOQX[^]Rudt^ W `P XHOQOQN*P
dQNS^]O‚x& ‚N W_ wT^ Y [N‡w˜ ¡tx•g
‘ · -k W dQ}]\ Y N‡[^"P`X9OOQN*P`dNS^ZO‚x4 ¶ ­ [^x& C(s) = Kc2
1 + Ti s
Ti s
‹NsP`X9OOQN*P`dN*^]O‡x& bN Y dnOQSzHwTU[NUa:e9Ó`X W q"PSX WZY NSOQoHN*O†wTN Yn_ N*OfacX9OlUe W P`N Yb_ OEP`E[N W dN Y X › dN W ^]N Y e?oHNEPpw€ ¡tx•g VW
P~}]XHR Y Rud Ti [N†a:eHÓ`X W qpe?o9XHRTO
¡iº»š
ωc0
10
1
≤
⇒ Ti ≥
Ti
10
ωc0
Ti = ω10c0 ⇒ Ti = 2
Yg