f 1-revised
TRANSCRIPT
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.
a(
.
.
b(
.
backlash
.
c(
.
.
d(
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.
.
backlash.
.
(adaptive)
Bang-Bang
.
.LTI
x A x Bu= +&
LTI
.A)1
A)2
)(
3(
)
(
.)(
.)4
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0u)5 = BIBO
.
)6 .
.
2
2sin
sin
x d y dyy e x x y xdt dt
y x
= + + =
=
x:
y:
-1
v v v u+ =&
v:
u:
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.vu10
10 10
.
v
.vu
1101 ==+= SSS vvvu
2.31010010 ===+= SSS vvvu
sv:
v: .
:2
.
=)(vhi
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( )h vi
(0) 0 , (0) 0h h =
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1 2
2 1 1 2( )
x x
x x h x x
=
=
&
&
-3
(5)sp f
F ma
F F F my
=
=
&&
( )
(0) 0
spF g y
g
=
=
y, 1ay
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mgyg S)(.
mgK.((slipping friction
..K
.
F
.
(6) ( , ) 0my ky cy y y+ + + =&& & &
0
0 , /
0 , /
S
S
y
y y mg K
y y mg K
>
=
= >
&
&
&
( )
( , )
( )
k
S
mgSign y
y y Ky
mgSign y
=
&
&
&
( , )y y &
0y =
&
/Sy mg K
)6(
y&y&&..
.
1 2,x y x y= = &
1 2
x x=&
2 1 2 1 2
1( , )
k cx x x x x
m m m= &
(6) ( , ) 0my ky cy y y+ + + =&& & &
>
=
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-------------------------------------
0
0 , /
0 , /
S
S
y
y y mg K
y y mg K
>
=
= >
&
&
&
( )
( , )
( )
k
S
mgSign y
y y Ky
mgSign y
=
&
&
&
( , )y y & 0y /Sy mg K&= )6(
y&y&&..
.
1 2,x y x y= = &
1 2x x=&
2 1 2 1 21 ( , )k cx x x x x
m m m= &
)1
:
2x x= +&
=0(0)x x
& =(x=0)x x
)0 ) t
x t x e=
) )x t.
:
0
0 0
( )1
t
t
x ex t
x x e
=
+
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-------------------------------------
.
( )t .
.
:
limit cycles
.
.vander pol
22 ( 1) 0mx c x x kx+ + =& &
2
1x
.
x
( )x t
.
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)(
:
)1 .
.
)2
.
)(Bifurcations
.
)
(
.Bifurcation
3 0x x x+ + =&&
Duffing (undamped).
.
3
2
0 0
( ) 0
0 ,e e
x x
x x
x x
+ + =
+ =
= =
0ex .=
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0e
x = 3
( 0, , )e
x = e
.
0= Bifurcation Duffing
.
0e
x .=
Duffing x&
.
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xx&
.&x
.
0ex =
.
chaos
.
chaos
.
chaos
.
(chaotic (
.
50.1 6sinx x x t+ + =&& &
(0) 2x ((0= 3x &= (0) 2.01x ((0= 3.01x &=
.
5
.
.chaotic
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(turbulence)
.
.chaotic
Jump resonance
B.P OP-AMP . OP-
AMP
OP-AMP
Jump
resonance.
.
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)1
( , , )x f x u t=&
11
22
3
,
n
x f
x fx f
x f
= =
M M
)2
1
1[ , , ,..., , , ,...]
n n
n nd y d yh t y y u udt dt
= & &
1 2
12 3
2 13 4
1
111
1 2( , , ,..., , , ,...)
n
n nn nn
n n
x xx y
x xx y x
x x
d yx xx x
dtx h t x x x u u
=
= =
= = =
&
&& &
&M
M M
& &&
& &
)) )x t
)) )x t
)) )x t
1
sgn( )x x=&
.
2
1
2x =&
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,t x t= + =
3
21x x= +&
:
.
)u , ) 0x t =
) , , )x f x u t=&0
t0
.
0 0( , ) 0x t t t=
:
.
sin 0g
l + =&&
2
1 1
22 1 2 2
( , )
sin
f x t
x
x x g
x x x x l
= = = =
&& & &
14243
2
2 1 1
1
0( , ) 0 0 , sin 0
0sin
xg
x t x x x kglx
l
= = =
( , , ) ( , , ) 0x f x u t f x u t= =&
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.
0x0.
x A: x=&A0x =
.
.A
A
(1 4) (2 2) 0A = =
-1
-2
.
2
1 1 1 2 1
2 2 1 2 2
( , ) (0,0) 0,
( , ) (0,0) 0
x f x x f
x f x x f
= =
= =
&
&
: 1 2 2 4
0 ,
2 4 1 2
x x x
= = =
14243 144424443
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.
.
),()0,0(),( 21*
111
02
11
01
1211 XXffx
X
fx
X
fXXf
XX
+++
=
==
),()0,0(),( 21*
221
02
21
01
2212 XXffx
X
fx
X
fXXf
XX
++
+
=
==
=
=
0
00
2
1
X
XX
=11X
=22X
21f
1 1
1 21 1
2 2 2
1 2 0x
f f
x xx x
x xf f
x x=
=
&
&
:
f
.
.2x12
1 1 1 2
1 2
2 2 1 2
( ) ( , ) 0 1, 2
( ) i
x f x xf x x i
x f x x
= = =
=
&
&
1 1 1 2 2 2 ,x x x x x x= + = +
1 1 1 1 1 2 2 1 1 2
1 1 1 2
( ) ( , ) ( , )
( , )
x f x x x x g x x
x g x x
+ = + + =
=
&
2 2 2 1 1 2 2 2 1 2
2 2 1 2
, ( ) ( , ) ( , )
( , )
x x f x x x x g x x
x g x x
+ = + + =
=
&
1 2( , )i x x2x1 1 2( , )ig x x
.
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)1
)2
.
.
)3
.
))Phase Plane)1
.
.lyapunov
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describing functions
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