Lección 10 ejemplos forma controlador.pdf

Lineales

1. Example ˙x1=x2,˙x2=−x1+ε(1−x
2
1
)x2+u, y=x1, ε >0
˙y= ˙x1=x2
¨y= ˙x2=−x1+ε(1−x
2
1
)x2+u
Relativedegree
= 2
over
R
2
Example ˙x1=x2,˙x2=−x1+ε(1−x
2
1
)x2+u, y=x2, ε >0
˙y= ˙x2=−x1+ε(1−x
2
1
)x2+u
Relativedegree
= 1
over
R
2
– p. 5/17

2. Example ˙x1=x2,˙x2=−x1+ε(1−x
2
1
)x2+u, y=x1+x
2
2
, ε >0
˙y=x2+2x2[−x1+ε(1−x
2
1
)x2+u]
Relativedegree
= 1
over
{x26= 0}
Example:
Field-controlledDCmotor
˙x1=−ax1+u,˙x2=−bx2+k−cx1x3,˙x3=θx1x2, y=x3
a
,
b
,
c
,
k
,and
θ
arepositiveconstants
˙y= ˙x3=θx1x2
¨y=θx1˙x2+θ˙x1x2= ()+θx2u
Relativedegree
= 2
over
{x26= 0}
– p. 6/17

3. Example
˙x1=x2,˙x2=−x1+ε(1−x
2
1
)x2+u, y=x2
˙y= ˙x2=−x1+ε(1−x
2
1
)x2+u⇒ρ= 1
y(t)≡0⇒x2(t)≡0⇒˙x1= 0
Non-minimumphase
– p. 14/17

4. Example
˙x1=−x1+
2+x
2
3
1+x
2
3
u,˙x2=x3,˙x3=x1x3+u, y=x2
˙y= ˙x2=x3
¨y= ˙x3=x1x3+u⇒ρ= 2
γ=LgLfh(x) = 1, α=−
L
2
f
h(x)
LgLfh(x)
=−x1x3
Z
∗
={x2=x3= 0}
u=u
∗
(x) = 0⇒˙x1=−x1
Minimumphase
– p. 15/17

5. Find
φ(x)
suchthat
φ(0) = 0,
∂φ
∂x
g(x) =
h
∂φ
∂x
1
,
∂φ
∂x
2
,
∂φ
∂x
3
i



2+x
2
3
1+x
2
3
0
1


= 0
and
T(x) =
h
φ(x)x2x3
i
T
isadiffeomorphism
∂φ
∂x1

2+x
2
3
1+x
2
3
+
∂φ
∂x3
= 0
φ(x) =−x1+x3+tan
−1
x3
– p. 16/17

6. T(x) =
h
−x1+x3+tan
−1
x3, x2, x3
i
T
isaglobaldiffeomorphism
η=−x1+x3+tan
−1
x3, ξ1=x2, ξ2=x3
˙η=
−
−η+ξ2+tan
−1
ξ2


1+
2+ξ
2
2
1+ξ
2
2
ξ2
!
˙
ξ1=ξ2
˙
ξ2=
−
−η+ξ2+tan
−1
ξ2

ξ2+u
y=ξ1
– p. 17/17

7. Denition:
Anonlinearsystemisinthecontrollerformif
˙x=Ax+Bγ(x)[u−α(x)]
where(A,B)iscontrollableandγ(x)isanonsingular
u=α(x)+γ
−1
(x)v⇒˙x=Ax+Bv
Then-dimensionalsingle-input(SI)system
˙x=f(x)+g(x)u
canbetransformedintothecontrollerformif∃h(x)s.t.
˙x=f(x)+g(x)u, y=h(x)
hasrelativedegreen.
Why?
– p. 2/18

8. Transformthesystemintothenormalform
˙z=Acz+Bcγ(z)[u−α(z)], y=Ccz
Ontheotherhand,ifthereisachangeofvariables
ζ=S(x)thattransformstheSIsystem
˙x=f(x)+g(x)u
intothecontrollerform
˙
ζ=Aζ+Bγ(ζ)[u−α(ζ)]
thenthereisafunctionh(x)suchthatthesystem
˙x=f(x)+g(x)u, y=h(x)
hasrelativedegreen.
Why?
– p. 3/18

9. Foranycontrollablepair(A,B),wecanndanonsingular
matrixMthattransforms(A,B)intoacontrollable
canonicalform:
MAM
−1
=Ac+Bcλ
T
, MB=Bc
z=Mζ=MS(x)
def
=T(x)
˙z=Acz+Bcγ(∆)[u−α(∆)]
h(x) =T1(x)
– p. 4/18

10. Insummary,then-dimensionalSIsystem
˙x=f(x)+g(x)u
istransformableintothecontrollerformifandonlyif∃h(x)
suchthat
˙x=f(x)+g(x)u, y=h(x)
hasrelativedegreen
Searchforasmoothfunctionh(x)suchthat
LgL
i−1
f
h(x) = 0, i= 1,2,...,n−1,andLgL
n−1
f
h(x)6= 0
T(x) =
h
h(x), Lfh(x),∆∆∆L
n−1
f
h(x)
i
– p. 5/18

11. TheLieBracket:
Fortwovectoreldsfandg,theLie
bracket[f,g]isathirdvectorelddenedby
[f,g](x) =
∂g
∂x
f(x)−
∂f
∂x
g(x)
Notation:
ad
0
f
g(x) =g(x), adfg(x) = [f,g](x)
ad
k
f
g(x) = [f,ad
k−1
f
g](x), k≥1
Properties:
[f,g] =−[g,f] Forconstantvectoreldsfandg,[f,g] = 0
– p. 6/18

12. Example
f=
"
x2
−sinx1−x2
#
, g=
"
0
x1
#
[f,g] =
"
0 0
1 0
# "
x2
−sinx1−x2
#
−
"
0 1
−cosx1−1
# "
0
x1
#
adfg= [f,g] =
"
−x1
x1+x2
#
– p. 7/18

13. f=
"
x2
−sinx1−x2
#
, adfg=
"
−x1
x1+x2
#
ad
2
f
g= [f,adfg] =
"
−1 0
1 1
# "
x2
−sinx1−x2
#
−
"
0 1
−cosx1−1
# "
−x1
x1+x2
#
=
"
−x1−2x2
x1+x2−sinx1−x1cosx1
#
– p. 8/18

14. Distribution:
Forvectoreldsf1,f2,...,fkonD⊂R
n
,let
∆(x) = span{f1(x),f2(x),...,fk(x)}
Thecollectionofallvectorspaces∆(x)forx∈Discalled
adistributionandreferredtoby
∆ = span{f1,f2,...,fk}
Ifdim(∆(x)) =kforallx∈D,wesaythat∆isa
nonsingulardistributiononD,generatedbyf1,...,fk
Adistribution∆is
involutive
if
g1∈∆ andg2∈∆⇒[g1,g2]∈∆
– p. 9/18

15. Lemma:
If∆isanonsingulardistribution,generatedby
f1,...,fk,thenitisinvolutiveifandonlyif
[fi,fj]∈∆,∀1≤i,j≤k
Example:
D=R
3
;∆ = span{f1,f2}
f1=



2x2
1
0


, f2=



1
0
x2


,dim(∆(x)) = 2,∀x∈D
[f1,f2] =
∂f2
∂x
f1−
∂f1
∂x
f2=



0
0
1



– p. 10/18

16. rank[f1(x),f2(x),[f1,f2](x)] =
rank



2x21 0
1 0 0
0x21


= 3,∀x∈D
∆isnotinvolutive
– p. 11/18

17. Example:
D={x∈R
3
|x
2
1
+x
2
3
6= 0};∆ = span{f1,f2}
f1=



2x3
−1
0


, f2=



−x1
−2x2
x3


,dim(∆(x)) = 2,∀x∈D
[f1,f2] =
∂f2
∂x
f1−
∂f1
∂x
f2=



−4x3
2
0



rank



2x3−x1−4x3
−1−2x22
0x30


= 2,∀x∈D
∆isinvolutive
– p. 12/18

18. Theorem:
Then-dimensionalSIsystem
˙x=f(x)+g(x)u
istransformableintothecontrollerform
ifandonlyif
thereis
adomainD0suchthat
rank[g(x),adfg(x),...,ad
n−1
f
g(x)] =n,∀x∈D0
and
span{g,adfg,...,ad
n−2
f
g}isinvolutiveinD0
– p. 13/18

19. Example
˙x=
"
asinx2
−x
2
1
#
+
"
0
1
#
u
adfg= [f,g] =−
∂f
∂x
g=
"
−acosx2
0
#
[g(x),adfg(x)] =
"
0−acosx2
1 0
#
rank[g(x),adfg(x)] = 2,∀xsuchthatcosx26= 0
span{g}isinvolutive
FindhsuchthatLgh(x) = 0,andLgLfh(x)6= 0
– p. 14/18

20. ∂h ∂x
g=
∂h
∂x2
= 0⇒hisindependentofx2
Lfh(x) =
∂h
∂x1
asinx2
LgLfh(x) =
∂(Lfh)
∂x
g=
∂(Lfh)
∂x2
=
∂h
∂x1
acosx2
LgLfh(x)6= 0inD0={x∈R
2
|cosx26= 0}if
∂h
∂x1
6= 0
Takeh(x) =x1⇒T(x) =
"
h
Lfh
#
=
"
x1
asinx2
#
– p. 15/18

21. Example(Field-ControlledDCMotor)
˙x=



−ax1
−bx2+k−cx1x3
θx1x2


+



1
0
0


u
adfg=



a
cx3
−θx2


;ad
2
f
g=



a
2
(a+b)cx3
(b−a)θx2−θk



[g(x),adfg(x),ad
2
f
g(x)] =



1a a
2
0cx3(a+b)cx3
0−θx2(b−a)θx2−θk



– p. 16/18

22. det[∆] =cθ(−k+2bx2)x3
rank[∆] = 3forx26=k/2bandx36= 0
span{g,adfg}isinvolutiveif[g,adfg]∈span{g,adfg}
[g,adfg] =
∂(adfg)
∂x
g=



0 0 0
0 0c
0−θ0






1
0
0


=



0
0
0



⇒span{g,adfg}isinvolutive
D0={x∈R
3
|x2>
k
2b
andx3>0}
FindhsuchthatLgh(x) =LgLfh(x) = 0;LgL
2
f
h(x)6= 0
– p. 17/18

23. x
∗
= [0,k/b,ω0]
T
, h(x
∗
) = 0
∂h
∂x
g=
∂h
∂x1
= 0⇒hisindependentofx1
Lfh(x) =
∂h
∂x2
[−bx2+k−cx1x3]+
∂h
∂x3
θx1x2
[∂(Lfh)/∂x]g= 0⇒cx3
∂h
∂x2
=θx2
∂h
∂x3
h=c1[θx
2
2
+cx
2
3
]+c2, LgL
2
f
h(x) =−2c1cθ(k−2bx2)x3
h(x
∗
) =c1[θ(k/b)
2
+cω
2
0
]+c2
c1= 1, c2=−θ(k/b)
2
−cω
2
0
– p. 18/18