Lecture 3 System Modeling
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Transcript of Lecture 3 System Modeling
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7/31/2019 Lecture 3 System Modeling
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F = ma
F am
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k
x
b m b xF = ma
u (t) = M x (t)
U (s ) = Ms 2X (s )X (s )U (s )
=1
Ms 2
1s 2
x (t) u (t )
u bx = M x
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U (s ) bsX (s ) = Ms 2X (s )X (s )U (s )
=1
Ms 2 + bs
m 1
x 2 > x 1 x 2 > x 1 1
m 2
u + k(x 2 x 1) + b(x 2 x 1) = m 1 x 1k(x2 x 1) b(x 2 x 1) = m 2 x 2
X 1(s ) m 1s 2 + bs + k X 2(s ) [bs + k] = U (s )
X 1(s ) [bs + k] + X 2(s ) m 2s 2 + bs + k = 0
X 1 X 2 X 2
X 1(s ) =X 2(s ) [bs + k] + U (s )
[m 1s 2 + bs + k]
X 2(s ) [bs + k] + U (s )
[m 1s 2 + bs + k][bs + k] + X 2(s ) m 2s 2 + bs + k = 0
X 2(s )U (s )
=bs + k
m 1m 2s 4 + ( m 1 + m 2)bs3 + ( m 1 + m 2)ks 2
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M = I
=
I = ( )2
I =M
0
r 2dm
M = F l
M = F l mgr sin I = mr 2
=
M = I
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= U R
I 1 1 = UR + k (2 1) + b 2 1I 2 2 = k (2 1) b 2 1
F = m x
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F x = m xF y = m y
U H = M x
l sin x + l sin
H = md2
dt 2(x + l sin )
V mg = md2
dt 2(l cos )
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V l sin H l cos = I
ddt (sin ) = cos
d2
dt 2(sin ) =
ddt
(cos ) = sin + cos ddt
(cos ) = sin d2
dt 2(cos ) =
ddt
(sin ) = cos sin
U H = M x
H = m x ml sin ()2
+ ml cos
V mg = ml cos ()2
ml sin
V l sin H l cos = I
U = ( M + m )x ml sin ()2
+ ml cos
mgl sin ml 2 ml cos x = I mgl sin ml cos x = I + ml 2
sin ()2, cos , sin , cos x
v(t ) = Ri (t ) V (s ) = RI (s )
i = C dv (t )
dtI (s ) = CsV (s )
1sC
v(t) = Ldi (t )
dtV (s ) = LsI (s ) sL
v(t) = V si(t) = I s
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e0 = K (e2 e1) = K (e1 e2)
i+ = i = 0 e+ = e
e0 K = +
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E i E 2R 1 C 1sE 2
E 2 E oR 2
= 0
C 2s (E i E o ) +E 2 E o
R 2= 0
E 2(s ) E i (s ) E o (s )
E oE i
=R 1R 2C 1C 2s 2 + ( R 1 + R 2) C 2s + 1
R 1R 2C 1C 2s 2 + (( R 1 + R 2) C 2 + R 1C 1) s + 1
i1 i2 i3 = 0ei e
R 1 C ddt
(e eo ) e eo
R 2= 0
e = 0eiR 1
+ C de odt
+eoR 2
= 0
E iR 1
+ CsE o +E oR 2
= 0
E oE i = 1R 1 Cs + 1R 2= R 2R 1 11 + R 2Cs
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ei
T = ki.
ebm
eb = K b m
ei ebR + sL
= i
T = kiT bm = J m m m =
T J m s 2 + bs
eb = K b m eb = sK bm
J m
(s )ei (s )
=
1Ls + R k
1J m s 2 + bs
1 + sK b 1
Ls + R k 1
J m s 2 + bs
=k
s [J m Ls 2 + s (bL + RJ m ) + ( Rb K bk)]
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f (x 1 + x 2) = f (x 1) + f (x 2)
x 1 x 2
a n y(n ) + a n 1y(n 1) + + a 1y(1) + a 0y(t ) = bm u (m ) + bm 1u (m 1) + + b1u (1) + b0u (t)
(y1(t), u 1(t )) (y2(t ), u 2(t ))(y1(t ) + u 1(t), y2(t) + u 2(t ))
f (y, y, , y (n ) , u, u, , u (m ) ) = 0
f ( ) = a n y(n ) + a n 1y(n 1) + + a 1y(1) + a 0y(t ) bm u (m ) + bm 1u (m 1) + + b1u (1) + b0u (t )
f (.)
U = ( M + m )x ml sin ()2
+ ml cos mgl sin ml cos x = I + ml 2
f (, , ,x, x, x, u ) = 0
V R = i3R
V i = i3R + Ldidt
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y = f (x )
y = f (x)
= f (x) +df dx x = x
(x x ) +12!
d 2 f dx 2 x = x
(x x )2 + x
|x x |y= f (x ) +
df dx x = x
(x x )y = y f (x ) x = x x
y =df dx x = x
x
f (x ) x
y = f (x 1 , x 2 , u 1 , u 2)
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y = f (x 1 , x 2 , u 1 , u 2) +f x 1 x 1 = x 1 x 2 = x 2
u 1 = u 1 u 2 = u 2
(x 1 x 1)
+f
x 2 x 1 = x 1 x 2 = x 2u 1 = u 1 u 2 = u 2
(x 2
x 2) +
f
u 1 x 1 = x 1 x 2 = x 2u 1 = u 1 u 2 = u 2
(u 1
u 1)
+ +f u 2 x 1 = x 1 x 2 = x 2
u 1 = u 1 u 2 = u 2
(u 2 u 2)
x, x, u, u, x, x, x,u, u, u, x, 0, 0, u, 0, 0,
dxdt
= f (x (t ))
x = 0 f (x ) = 0
x (0) = x x (t) = x x (0) x x (t ) x (0, T )
dxdt f (x ) +
df dx x = x
(x x )
f (x ) = 0 x
x = x xddt
x =ddt
x
x =df dx x = x
x
d2
dt 2x (t ) = f (x (t ), x (t ), u (t))
u
f (x , 0, u ) = 0
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d2
dt 2x (t ) = f (x , 0, u ) +
f x x = x
x = 0u = u
(x (t ) x )
+ f x x = x
x = 0u = u
(x (t ) 0) + f x x = xx = 0
u = u
(x(t ) x )
f (x , 0, u ) = 0 x = x x u = u u
x =f x x = x
x = 0u = u
x +f x x = x
x = 0u = u
x +f x x = x
x = 0u = u
u
V i = i3
R + Ldidt
didt =
V iL
RL i
3
V = 1didt
= 0V iL
RL
i3 = 0
i =3 V R = 3 1R
f (i) = RL
i3
f (i) f (i ) + f i i = i (i i )
f i i = i
= 3RL
i2i = i
= 3L
3 R
didt
=V iL
+ f (i)
=V iL
+ f (i ) +f i i = i
(i i )
=V i
L+ f (i )
3
L
3 R (i
i )
V i = V i V i = i ii =
V iL
+V L
+ f (i )
=0
3L
3 Ri
=V iL
3L
3 Ri
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si (s ) =V i (s )
L 3L
3 Ri (s )i (s ) s +
3L
3 R = V i (s )L
i (s )V i (s )
=1L
s +3L
3 R
I = ul cos mgl sin bl
u = 0 = 0
mgl sin = 0
= 0
f (, u ) = ul cos mgl sin
f (, u ) = f (, u ) +f u = u
= 0
( )
+f u u = u
= 0
(u u )
f u = u
= 0
= ul sin mgl cos | u = u = 0
= mgl
f u u = u
= 0
= l cos | u = u = 0
= l
I = f (, u ) mgl ( ) + l(u u ) bl
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f (, u ) = 0 = u = u u
I = mgl + lu bl
s 2 I (s ) = mgl (s ) + lu (s ) sbl (s )(s ) s 2I + sbl + mgl = lu (s )
(s )u (s )
=l
s 2I + sbl + mgl
