matlab bode plot

8/13/2019 Matlab Bode Plot

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y(x) = xy.


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t

x

y(1) = 1

y(x)

y

y(x) x = 2 x 2

y

x

y

y

y(x) + 8y(x) + 2y(x) = cos(x); y(0) = 0, y(0) = 1.


3/16

x(t) = x(t) + 2y(t) z(t)

y(t) = x(t) +z(t)

z(t) = 4x(t) 4y(t) +5z(t).

C= 2C1 + 1/2C3

x

t

x(0) = 1 y(0) = 2 z(0) = 3


4/16


5/16

x

y0 = 1 y(0) = 1

x

y

y1(x) = y(x) y2(x) =y(x)

y1(x) =y2(x)y2(x) = 8y2(x) 2y1(x) + cos(x)

y1 y2

y(2)


6/16

x

y1 y2 y y y(m, n) m n y y1 y2

x

y1 y(:, 1) y1 y2

dx

dt = x+y

dy

dt = y xz

dzdt = bz+xy br,

= 10 b = 8/3 r = 28 x(0) = 8 y(0) = 8 z(0) = 27


7/16

20 15 10 5 0 5 10 15 2030

20

10

0

10

20

30

The Lorenz Strange attractor

x(t)

y(t)

dy

dx=f(x, y).

f(x, y) y(x) (x, y)

y(x) f(x, y) (x, y)

(x, y) x a b y c d

(1, 0) (1, 1)(1, 2) (1.5, 0) (1.5, 1) (1.5, 2) (2, 0) (2, 1) (2, 2) (a, b)

v = (x, y) (0, 0) o

v = (x, y) yx

(x, y) dydx

y(x) = x +sin(y)

dy

dx=

f(x, y)

1 .


8/16

3 2 1 0 1 2 3 42.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

y(x) = x+ sin(y)

dy = f(x, y) = x+ sin(y)

dx= 1

3 2 1 0 1 2 3 42.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

y(x) =x+ sin(y)


9/16

dy

dx=x2 y.

4 3 2 1 0 1 2 3 43

2

1

0

1

2

3

4

y(x) =x2 y

dxdt

=f(x, y)dy

dt =g(x, y).

t

dy

dt

dx

dt

dy

dx=

dydtdxdt

= g(x, y)

f(x, y),

dy

dx=

g(x, y)

f(x, y).


10/16

y

x

x(t)

t

y(t)

y(t) = x(t)

dx

dt

= y

dy

dt = g

l sin(x),

g = 9.81 2

l

dy

dx=

glsin(x)

y ,

y= 0

y

y

x sin(x)

y= 0

x

t

x(t)

dxdt

= y y >0 x(t) y


11/16

6 4 2 0 2 4 615

10

5

0

5

10

15

(x, y)

x

y

dy

dx

(0, 0) (0, 0) (, 0) (, 0)

x

x(t) = ax(t) bx(t)y(t)

y(t) = ry(t) +cx(t)y(t),

x(t) y(t)

dy

dx=ry+cxy

ax bxy ,

a = r = 2 b = c = 1 x = 0 y = 2

(a, b)

(a,b)


12/16

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

f(t) = t2

F(s) = 11+s

b

y(t)

y(0) = 4.06

t= .95 b

d2y

dt2 = g b dy

dt, y(0) = 4.06, y(0) = 0,


13/16

y(.95) = 0 y(.95) b

b

b

V

t

z

t= .95

y(0) =.39

y(2.13) = 0

d2y

dt2 = g b

dy

dt, y(0) =.39, y(2.13) = 0,

t = 0 t = 2.13

y(0) y(2.13)

b


14/16

y

y

t

v

x

x(t) t y(t) = 0

y(t) = g by(t); y(0) =.18; y(0) =vsin

x(t) = bx(t); x(0) = 0; x(0) =vcos

y(0) =.18

v

z1= y z2 = x z3 = y z4 = x

z1 = z3

z

2 = z4

z3 = g bz3

z4 = bz4

y(t) = 0 y(0) = 0 y(t) = 0


15/16

z(1) = 0 y(t) = 0 y(t) = 1

y

z0

t

z

te z ze


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matlab bode plot

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