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MANUAL CONTROL AND STABILIZATION OF AN INVERTED PENDULUM
Copyright (c) 2005 IFAC. All rights reserved 16th Triennial World Congress, Prague, Czech Republic
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1 v(mc Jp + mp Jp0 + m2p l2 sin2 θ) Jp i −m2 gl2 sin θ cos θ + Jp mp lθ˙ 2 sin θ
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0 p
p
x1 = θ u=
x2 =
s
Jp ˙ θ mp gl
τ=
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mp gl t Jp
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x3 =
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mp l x˙ Jp g
p
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È ÿsÓ
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1
1
00 0
1
' ·Pº&µ''¹Pº + ~ ! G#"¿Ã/'µ&$"Ãc `à P1T sy`RWllW1lll `1l c s l-s`RWyRWlZW 1`lsr; sy`RWll`TZWT11R`s1lRyR`#T`>WühN `El1>1~`s`1RW`ls+y¶ W>` R`s1llG~lW`~Ë`1 ``s`rl;^N\`1l`N -sc l¡R1R`WRrR1W N -lW1s#`é`1£ sy`RWllW1lllÄ`rËER1`§ ¾klR` ZWRl1l' sy`RW`lW1s6R1 jr`1s`1 ``s` ;=`1 `c`R¿/1RW1sy/R `1<WRlWl+ P1<R];¶Wlc`1 h -WlyR< s1`RR1`N&`1 sy`RWllW1lllR G`1]VWll Pl1wlPll^-wWR`1#R1W`1~ NWyR`W >11w^R1®`c`RúÈ WÓ l~`1süh NP``>TZWll Pl1 ``s11s`RWR1`s`lW u≥u u &È % Ó −u ≤ u ≤ u sat (u) = u P1§ NWyR`WllW>ll?R−u ¦^u`>≤¥−u>s1. Wr-1>1l1 ~WR`1>Nl È `cZs1V ¸ÑÌsÌWÍWÓ P>`¥`1± sc° ``sls1ll¦RZs |x | ≤ ~s NWl 1lWR+ù1W W#1l`>`NTR1l%1R;WRsªl#sl^Ný-l; <W ~lWPRPh¾1`>`lWrR `> WRP1N |x | ≤ π
ÿ &¼' P `º&µ®Ã©º ù¡Ð©º&P `º&µ ·W>`l1`1wlyW`R->11l1WT WRl 6s1` ¸ +§R1?-WRl`lWÂ^%`1? s`§- s1 `> W>Wlw;6R1<-11>> θ l >1WR<`x1<1l`RW1 N ZRW R1E1l^sT-scPR\R1E Ny`N^]s`P^]`> ->11l1© m R1¡#WR©^#`>¡-N111l1 W1 l`#W\y ^l1N``l¥ ÚÚ~`>©>Ws-sly J ùk>``>kl m 11s`TR1P#WR=;+`1T s` F `> Zs` <W N`l1®s`1< s`]W1 g `1PW NN`WRlW 11 `&WR cl^P1~Éc1s`lW1;>#W`ls';>R1~lyW`R ->11l1 # ¶ P`lR` p
p
0
c
0
u0
0
0
Jp θ¨ − mp l¨ x cos θ − mp gl sin θ = 0 ¨ −mp lθ cos θ + (mc + mp )¨ x − mp lθ˙ 2 sin θ = F.
È ¸ÑÓ
1
0
0
π 2
1
1125
ËÊ {Â[]H;fD{ |x| < π GZsl;P®Z`s R1#s1Wlc`ll3È'R`! s!W1éùkk° `1R1 ÿ sÓ `1s'Ë]R1W N l`s`lW¦lTls`WN'`>W .- l+l-s`R> RT1;^]T sy`RWllW1lll<`NP1l j Wll P`1%-1>1l1 `¦s©¶l `1#1s`lsy`s 1lW1PR1`s1WR1'1l^sW&`R1PR1Ww`>Ts1Wl<l `R``l R«` −α ≤ x ≤ α kP1l`Éc1l`NPR1WR1 W NN`WRlW±^<`1¶111l1 l `0 / ly`l¦s`s ``Pl11¶111l1 s~`RV`W¿R1&s1Wl α P11N`WW1slyRll\È 'R` W s1 ùk>`1R1 ÿ sÓ WlWP`1®VWll Pl1 RlWRs#¶~N α s1 u È ÒsÓ 2u α = π − arctan u + arccos p G'l'`W#P1sP W1c`Rc`1l`lW`1>+W'uR1E−R1slR W NN`WRlW u = 4/3 W1RWl1¦ZW`1lWRWR +VÚ; Ãc#WlP;Wl1 ; α RÉc1l`ls`WN α W NN`WαRl=W+π 1 61Á`1? sy`RWllW1lllÆ`¥~ 6R`¥W1R`W `1s`1r-W1>1WR¥^ Wc``sls1llG¦`slWésc `1RW1s£`1¶Wly x = α x = 0 G±`1 NW` ; c`>&W NN`WRlW#lw-s`l`lW<ZW W1α >10sWRW¥ZW |x| < π/2 ¼&`l1Æ`>|x|¡>>π/2 `s Éc1WRsÈ sÓ W>3#W` h1l1£`>¦1WRWN` C R `1-W1>1WR© W>1l`lW1 ~ W>`Wl¦`1 ZWll Pl1 h¾11`R`lWÄZsR11>- -W1>1WR x = f (x ) ;/`> WyR`sls1llGRWlW r Ø2 +
4
3
2
x2
1
0
−1
1
−2
−3
−4
−3
−2
−1
0
x1 0
1
2
3
ùl1 ÿ ' W11>W`l^1R1w WyR`sls1llG<`NWlW1+Zs l Z1l>l1NwW> Zs |x| < π/2 l1s`1l>^P1P l` lw`1;|x¡|`1≤Pα11=`R2.35 W1l Éc1lll1`ls1©`>s``;P®`> ÂR1>R Rs ^/`1W RW 6l+6 G%R1l< s` u = 3 (±) 6`R®>W`l ER1W&\ZW' W>``sy W Nl`s`lW R1`l&s«Éc1lll1`l1 W x = arctan u W1 u (ý1¥R1\W N l`s`lWÄ1sE`1 s1`RWy * x = 0 sl1 u c`>&Éc1WRs^#WRsE^`1&-N111l1 s%¶ ly`NW`s`rR sW
0
1
0
0
0 1
0 0 2
0
1
0
1 2 x = − cos x + u0 sin x1 + C 2 2 1 2 − cos x1 cos x01 + sin x1 sin x01 x2 = +C 2 cos x0 cos (x1 + x01 ) =− + C. cos x01
0 2 0
È WÓ
2
+
2
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P1E sy`RWllW1lll R®'`Ry`lslWlWry±È sÓ = ¾k>s``11N`Wll<~Pl¶ s1`l1NTG~ s`N
fα+ (x1 ) =
r
−2
cos (x1 + x01 ) cos x01
cos (x1 − x01 )
cos x01 r cos (x1 + x01 ) −2 0
π ≤ x1 ≤ α 2
+ C1
+ C2
1
Ø2
−
π π < x1 < 2 2
Ø2 LÊ qý[]H;fDq, |x| < π/2 354'6 P 1 ` P1lE NW` WR`R-W1?R¦`1#RR1WRlW§P1R%`> ->11l1 lT1N^N®sl;~%R\1WR&`1E1WRNWyRW C = 2 cos (α + x ) cos x 1lW>`1RW1sr`1E1lWWN> Gr`1l' s`;+`>¶W11k° WRl§^R1ª WyR`sls1llGÆRÄl§WlWÁc `> C = 2 cos (π/2 − x ) − 2 cos (π/2 + x ) + C `Rüh RW`l]`>`W>W`>11R`W>Éc1lll1`l (x , 0) cos x cos x W> (−x , 0) +l1NW`ls`lWrs`s11«`1 Éc1ll1Rl +x ) cos (−π/2 − x ) −2 +C `> P®`1s&`1rs``W>1l+P1``ühN `s`l s` C = 2 cos (−π/2 cos x x `>R`W>RWl1Rs1<^È WÓ `1RW1s%R1Éc1ll1Rl1 P1 E;~'¶W11>W`%;w`1 Wc``cos sls1llG`slW+ P1l<sl^N C = 1/ cos x W>%`1 ¾k>`R`lW1 k l ~ > 61rWlÉc1s`lW«È ÍWÓ k Gùls1` ÿ f (x ) `1< sy`RWllW1lll&`NWlWlER1< NW`P^ u = 3 s1 r WlW yÈ ÒsÓ l `1;P 1s`1\ 1RW>ùlW1R α `1;P sy`RWllW1lllRWls1&ZW |x | ≤ π +µ&WR N f (x ) = `1s¥`1ýRWlW WR P±Zs§ls`WN¥ s>¥; u r P1RlE; R1 sy`RWllW1lll `&>-1>®s`1 Zs `>1>-E¶W11>W`¦;P`1\ sy`RWllW1lll%Rh° ;`W Rx1`>s`PlW1 lyRl` u``sc`W>R\1WW\ R1l<&`-1R Wly`RR`l1sl<ls`1sl1lWG Wls+ W>` ^Ry#`R^^`>'l ~~-W>11WRl `'Ë=R1`Es` > `N``Rl `lW>'W x `>l`RWw^/`1>1-T¶W11>W`W1>1 È ÍWÓ P Ãc ~! ! 7- Gµ®Å ·Pº'µ&'¹< º *]¹ f (x ) = −f (−x ). ùlW1R ÿ `1;PR1 sy`RWllW1lll#RWlW©VW `1l ' 8 6R``R±R ~s`1`1 1RlW£;# NWyR`Wll s`PR u = 3 l#`sl# N1`WN 1W>l1R`W h¢yl1©;TRhZR1 r W#\W1>PZs`1 +
cos x1
−α ≤ x1 ≤ −
+ C3
π 2
0 1
1
0 1
0 1
2
0 1
3
0 1
2
+ π/2
cos x01
,
−
2
1
0
π ≤ x1 ≤ x01 2
1
1 − cos(x1 − x01 ) cos x01
0
π x01 ≤ x1 ≤ 2
,
0
1
1
− π/2
1
2
0 1
1
−
0 1
0 1
− α
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1
0 1
0 1
0 1
x01 )
0 1
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+ π/2
1
0
1126
8 4
6 3
4 2 2
x2
x2
1
0
0
−2 −1
−4 −2
−6 −3 −8
−3
−2
x1
−1
0
1
2
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1
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1 v(mc Jp + mp Jp0 + m2p l2 sin2 θ) Jp i −m2 gl2 sin θ cos θ + Jp mp lθ˙ 2 sin θ
P1` J lR1w#W\y¶;>l1R`l';>R1~-1>1l1 PR`R- ¥`ýR« Ny`§^\#W`NTR1Ä NWyR`W >11£``1éRc``N l£``s1GVW`#ÂRÆR1éW ° l`s`lW?^®`> s` ]RWR1\R1WéR1s `l1 ZWR F µ&W`l ¦`1sR1v«R`W>GZs`s`lW¡ Wý- 1W1EWlW1sl lR1E`RW``1s `l1 N J ≤ m l Gy`Rc11 N>`11WRWlNW`lW1 p
0 p
p
x1 = θ u=
x2 =
s
Jp ˙ θ mp gl
τ=
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mp gl t Jp
v g
x3 =
s
mp l x˙ Jp g
p
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È ÿsÓ
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1
1
00 0
1
' ·Pº&µ''¹Pº + ~ ! G#"¿Ã/'µ&$"Ãc `à P1T sy`RWllW1lll `1l c s l-s`RWyRWlZW 1`lsr; sy`RWll`TZWT11R`s1lRyR`#T`>WühN `El1>1~`s`1RW`ls+y¶ W>` R`s1llG~lW`~Ë`1 ``s`rl;^N\`1l`N -sc l¡R1R`WRrR1W N -lW1s#`é`1£ sy`RWllW1lllÄ`rËER1`§ ¾klR` ZWRl1l' sy`RW`lW1s6R1 jr`1s`1 ``s` ;=`1 `c`R¿/1RW1sy/R `1<WRlWl+ P1<R];¶Wlc`1 h -WlyR< s1`RR1`N&`1 sy`RWllW1lllR G`1]VWll Pl1wlPll^-wWR`1#R1W`1~ NWyR`W >11w^R1®`c`RúÈ WÓ l~`1süh NP``>TZWll Pl1 ``s11s`RWR1`s`lW u≥u u &È % Ó −u ≤ u ≤ u sat (u) = u P1§ NWyR`WllW>ll?R−u ¦^u`>≤¥−u>s1. Wr-1>1l1 ~WR`1>Nl È `cZs1V ¸ÑÌsÌWÍWÓ P>`¥`1± sc° ``sls1ll¦RZs |x | ≤ ~s NWl 1lWR+ù1W W#1l`>`NTR1l%1R;WRsªl#sl^Ný-l; <W ~lWPRPh¾1`>`lWrR `> WRP1N |x | ≤ π
ÿ &¼' P `º&µ®Ã©º ù¡Ð©º&P `º&µ ·W>`l1`1wlyW`R->11l1WT WRl 6s1` ¸ +§R1?-WRl`lWÂ^%`1? s`§- s1 `> W>Wlw;6R1<-11>> θ l >1WR<`x1<1l`RW1 N ZRW R1E1l^sT-scPR\R1E Ny`N^]s`P^]`> ->11l1© m R1¡#WR©^#`>¡-N111l1 W1 l`#W\y ^l1N``l¥ ÚÚ~`>©>Ws-sly J ùk>``>kl m 11s`TR1P#WR=;+`1T s` F `> Zs` <W N`l1®s`1< s`]W1 g `1PW NN`WRlW 11 `&WR cl^P1~Éc1s`lW1;>#W`ls';>R1~lyW`R ->11l1 # ¶ P`lR` p
p
0
c
0
u0
0
0
Jp θ¨ − mp l¨ x cos θ − mp gl sin θ = 0 ¨ −mp lθ cos θ + (mc + mp )¨ x − mp lθ˙ 2 sin θ = F.
È ¸ÑÓ
1
0
0
π 2
1
1125
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4
3
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x2
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x1 0
1
2
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0
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1 2 x = − cos x + u0 sin x1 + C 2 2 1 2 − cos x1 cos x01 + sin x1 sin x01 x2 = +C 2 cos x0 cos (x1 + x01 ) =− + C. cos x01
0 2 0
È WÓ
2
+
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r
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cos (x1 − x01 )
cos x01 r cos (x1 + x01 ) −2 0
π ≤ x1 ≤ α 2
+ C1
+ C2
1
Ø2
−
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cos x1
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1
+ π/2
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2
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1
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−2
0
2
4
0
2
4
6
8
10
6
8
10
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1
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2
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1
2
3
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4
0
1
2
3
4
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1
2
t
3
4
10
6
8
4
6
2
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x˙
12
4
0 −2
2
−4
0
−6
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0
t
1
2
3
4
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p
2
− π/2
c
p
p
2
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2
1129