03.06.2013 Views

### Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

π <br />

<br />

<br />

<br />

<br />

<br />

i = √ −1 <br />

π = 3.1415927 . . . <br />

e = 2.718281 . . . <br />

2.20D − 16

π <br />

<br />

<br />

<br />

<br />

<br />

i = √ −1 <br />

π = 3.1415927 . . . <br />

e = 2.718281 . . . <br />

2.20D − 16

π <br />

<br />

<br />

<br />

<br />

<br />

i = √ −1 <br />

π = 3.1415927 . . . <br />

e = 2.718281 . . . <br />

2.20D − 16

π <br />

<br />

<br />

<br />

<br />

<br />

i = √ −1 <br />

π = 3.1415927 . . . <br />

e = 2.718281 . . . <br />

2.20D − 16

3 × 3 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

• <br />

• <br />

A <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a 1 × 6

1 − 3i <br />

<br />

<br />

<br />

<br />

<br />

<br />

n<br />

m

1 <br />

1 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

A 3 × 3

A <br />

A <br />

<br />

<br />

<br />

b l × 2 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

: <br />

<br />

<br />

<br />

<br />

<br />

A 3 × 2 <br />

<br />

1

1 <br />

<br />

<br />

<br />

<br />

2 × 2

∗.

∧<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

A <br />

e A =<br />

∞<br />

i=0<br />

A i<br />

i!

exp(A) <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

A \ B <br />

<br />

AB −1

Ax = b <br />

A <br />

A<br />

<br />

<br />

xb = A<br />

<br />

<br />

<br />

A I x<br />

| <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

A <br />

B aij bij

cij = aij ·bij <br />

<br />

.∗ ./ . <br />

<br />

A n A <br />

A <br />

A n <br />

c A <br />

c. A c aij<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

D <br />

<br />

D

%i<br />

<br />

<br />

<br />

<br />

<br />

<br />

T <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

[] <br />

<br />

<br />

l × 1

−1 2 <br />

<br />

<br />

<br />

<br />

s

s

∗ /

=

=

=

=

F (s) =<br />

<br />

3(s + 2)<br />

(s + 3) ∗ (s + 1)<br />

F (s) = 1.5 1.5<br />

+<br />

s + 1 s + 3

1 + s<br />

F (s) =<br />

5 + s<br />

<br />

z − 1<br />

s =<br />

z + 1<br />

<br />

F (z) = F (s)| s= z−1<br />

z+1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

= z<br />

2 + 3z

1 + s<br />

F (s) =<br />

5 + s<br />

<br />

z − 1<br />

s =<br />

z + 1<br />

<br />

F (z) = F (s)| s= z−1<br />

z+1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

= z<br />

2 + 3z

1 + s<br />

F (s) =<br />

5 + s<br />

<br />

z − 1<br />

s =<br />

z + 1<br />

<br />

F (z) = F (s)| s= z−1<br />

z+1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

= z<br />

2 + 3z

1 + s<br />

F (s) =<br />

5 + s<br />

<br />

z − 1<br />

s =<br />

z + 1<br />

<br />

F (z) = F (s)| s= z−1<br />

z+1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

= z<br />

2 + 3z

1 + s<br />

F (s) =<br />

5 + s<br />

<br />

z − 1<br />

s =<br />

z + 1<br />

<br />

F (z) = F (s)| s= z−1<br />

z+1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

= z<br />

2 + 3z

z

A

A <br />

<br />

<br />

A <br />

<br />

Ax = b A <br />

<br />

Ā = [Ab]<br />

<br />

Ā1 A <br />

b <br />

<br />

Ā1 = [Aib1]<br />

A1 b1 <br />

AX = B <br />

B

A <br />

<br />

<br />

A <br />

<br />

Ax = b A <br />

<br />

Ā = [Ab]<br />

<br />

Ā1 A <br />

b <br />

<br />

Ā1 = [Aib1]<br />

A1 b1 <br />

AX = B <br />

B

A <br />

<br />

<br />

A <br />

<br />

Ax = b A <br />

<br />

Ā = [Ab]<br />

<br />

Ā1 A <br />

b <br />

<br />

Ā1 = [Aib1]<br />

A1 b1 <br />

AX = B <br />

B

A <br />

<br />

<br />

A <br />

<br />

Ax = b A <br />

<br />

Ā = [Ab]<br />

<br />

Ā1 A <br />

b <br />

<br />

Ā1 = [Aib1]<br />

A1 b1 <br />

AX = B <br />

B

∼ <br />

<br />

== <br />

< <br />

> <br />

≤ <br />

≥ <br />

∼=

∼ <br />

<br />

== <br />

< <br />

> <br />

≤ <br />

≥ <br />

∼=

∼ <br />

<br />

== <br />

< <br />

> <br />

≤ <br />

≥ <br />

∼=

∼ <br />

<br />

== <br />

< <br />

> <br />

≤ <br />

≥ <br />

∼=

∼ <br />

<br />

== <br />

< <br />

> <br />

≤ <br />

≥ <br />

∼=

T ∈ 33(R)

T −1 T = I<br />

I <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

T <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

• <br />

<br />

<br />

[−l, 3]<br />

0.1

[−l, 3]<br />

0.1

[0, 1] <br />

<br />

<br />

• <br />

· + × ⊕ ♦ △ ▽ ♣ ○

· + × ⊕ ♦ △ ▽ ♣ ○

· + × ⊕ ♦ △ ▽ ♣ ○

· + × ⊕ ♦ △ ▽ ♣ ○

· + × ⊕ ♦ △ ▽ ♣ ○

· + × ⊕ ♦ △ ▽ ♣ ○

x = f(x, y)(3.1) <br />

<br />

f(x, y) = cos(x)cos(y) (x, y) ∈ [0, 2π] × [0, 2π](3.2) <br />

z(i, j) =<br />

f(x(i), y(j))

• <br />

<br />

∗ <br />

∗ <br />

• <br />

<br />

∗ <br />

∗ <br />

• <br />

<br />

∗ <br />

∗ <br />

• <br />

<br />

∗ <br />

∗ <br />

• <br />

<br />

∗ <br />

∗ <br />

• <br />

<br />

∗ <br />

∗ <br />

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

. . . <br />

<br />

<br />

<br />

<br />

<br />

640 × 480

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

• <br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

˙x = Ax + Bu<br />

y = Cx + Du<br />

<br />

−5 −1<br />

A =<br />

6 0<br />

<br />

−1<br />

B =<br />

1<br />

C = −1 0 <br />

D = 0<br />

x(0) = [0 0] T

˙x = Ax + Bu<br />

y = Cx + Du<br />

<br />

−5 −1<br />

A =<br />

6 0<br />

<br />

−1<br />

B =<br />

1<br />

C = −1 0 <br />

D = 0<br />

x(0) = [0 0] T

10 −16 <br />

<br />

<br />

<br />

<br />

<br />

<br />

F 1 s = 2

A, B.C.D

A A

k <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

k = 3

k <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

k = 3

x = 0.5 <br />

<br />

<br />

<br />

1 + kF (s) = 0<br />

F (s) <br />

<br />

F (s) =<br />

s + 10<br />

(s + 2)(s + 3)

x = 0.5 <br />

<br />

<br />

<br />

1 + kF (s) = 0<br />

F (s) <br />

<br />

F (s) =<br />

s + 10<br />

(s + 2)(s + 3)

P 1 =<br />

<br />

s + 1<br />

s(s + 0.1)<br />

P 2 = s + 10<br />

s(s + 1)<br />

P 3 = s + 100<br />

s(s + 10)<br />

P 4 = s + 1000<br />

s(s + 100)

z s <br />

<br />

<br />

<br />

<br />

k <br />

<br />

<br />

<br />

<br />

<br />

<br />

1 + kF (s) = 0 k <br />

<br />

z s

F (jω) <br />

<br />

<br />

<br />

<br />

(−90 ◦ , 0) <br />

<br />

<br />

<br />

<br />

<br />

<br />

F (jω) =<br />

10 + jω<br />

jω(jω + 3)<br />

<br />

<br />

XdB = 20 log 10(X)

10 −3 , 10 3 <br />

<br />

<br />

z = e jω<br />

θ ∈ ]−π, π] θ ∈ [0, π] <br />

<br />

z = e 2πjωT<br />

ω <br />

Hz ω ∈ 10 −3 , 0.5 <br />

T = 1 <br />

θ = π ω = 0.5

F (jω) =<br />

20(10 + jω)<br />

(jω + 3)(jω + 5)<br />

<br />

<br />

<br />

4 × 2π ≈ 25rad/s <br />

F (jω) = 106 + jω<br />

10 5 + jω

F (jω) = 106 + jω<br />

10 5 + jω

F (jω) =<br />

1<br />

(jω + 3)<br />

ω = 3

s = 2<br />

T<br />

· z − 1<br />

z + 1<br />

T <br />

<br />

<br />

Sc T fp <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

u

sl u <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

sl u <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

sl u <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

sl u <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

sl u <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

sl u <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

• <br />

F (s) =<br />

1<br />

s 2 + s + 1<br />

<br />

F (s)

¡

• l <br />

• m <br />

• µ <br />

<br />

• ϑ <br />

• <br />

• τg <br />

<br />

<br />

µ ¨ ϑ = τg<br />

µ = ml 2<br />

τg = mgl sin(ϑ)<br />

ϑ<br />

l<br />

m<br />

mg

ϑ g <br />

¨ϑ + g<br />

sin(ϑ) = 0.<br />

l<br />

<br />

<br />

<br />

˙x = f(x, u)<br />

(5.1)<br />

y = g(x, u)<br />

<br />

u<br />

g<br />

f(x, u)<br />

x0<br />

˙x x<br />

g(x, u)<br />

<br />

˙ϑ<br />

− g<br />

l sin(ϑ)<br />

˙ϑ<br />

¨ϑ<br />

˙ϑ0<br />

<br />

<br />

y = ϑ u = g <br />

<br />

ϑ<br />

x = ˙ϑ<br />

<br />

⎧ <br />

⎨ ˙ϑ<br />

¨ϑ<br />

⎩<br />

y<br />

=<br />

=<br />

<br />

ϑ<br />

˙ϑ<br />

− g<br />

l sin(ϑ)<br />

<br />

ϑ0<br />

ϑ<br />

˙ϑ<br />

ϑ<br />

y<br />

y<br />

(5.2)<br />

<br />

x1 = ϑ x2 = ˙ ϑ <br />

<br />

<br />

˙x1 = x2

g<br />

f<br />

˙ϑ<br />

− g<br />

l sin(ϑ)<br />

˙ϑ<br />

¨ϑ<br />

ϑ0<br />

<br />

˙ϑ0<br />

<br />

<br />

<br />

<br />

<br />

m = 1 l = 1 <br />

g = ˆg = 9.81 2 .<br />

g<br />

− g<br />

l sin(ϑ)<br />

¨ϑ<br />

˙ϑ0<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

• <br />

<br />

• <br />

<br />

<br />

ϑ<br />

˙ϑ<br />

˙ϑ<br />

ϑ0<br />

<br />

ϑ<br />

ϑ<br />

y

• <br />

<br />

<br />

<br />

−(g/l) sin(ϑ) <br />

g ϑ <br />

g ϑ <br />

<br />

<br />

− g<br />

sin(ϑ) → −u1/l ∗ sin(u2),<br />

l<br />

l <br />

<br />

<br />

<br />

<br />

<br />

45 ◦ 45 ◦ <br />

<br />

45 ◦

• <br />

<br />

<br />

<br />

−(g/l) sin(ϑ) <br />

g ϑ <br />

g ϑ <br />

<br />

<br />

− g<br />

sin(ϑ) → −u1/l ∗ sin(u2),<br />

l<br />

l <br />

<br />

<br />

<br />

<br />

<br />

45 ◦ 45 ◦ <br />

<br />

45 ◦

˙ ϑ = 0 ◦ /s ϑ = π/4 <br />

<br />

<br />

= 0 <br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

−1 <br />

+1

f(ˆx, û) = 0<br />

<br />

k ∈ Nk <br />

ˆx1 = ˆ ϑ = kπ ˆx2 =ˆ˙x2 = 0 û = ˆg<br />

<br />

δϑ = ϑ − ˆ ϑ δ ˙ ϑ = ˙ ϑ −ˆ˙ϑ = ˙ ϑ<br />

<br />

⎧ <br />

⎨ ˙ϑ<br />

¨ϑ<br />

⎩<br />

<br />

0<br />

=<br />

−<br />

1<br />

ˆg<br />

y<br />

l cos(ϑ)<br />

= δϑ<br />

<br />

δϑ<br />

0 ˙ϑ<br />

=<br />

˙ϑ<br />

<br />

ˆϑ = 0 ˆ˙ϑ = 0 ˆg = 9.81 2<br />

− ˆg<br />

l cos( ˆ ϑ)δϑ<br />

<br />

⎧ <br />

⎨ ˙ϑ ˙ϑ<br />

¨ϑ<br />

=<br />

−<br />

⎩<br />

ˆg<br />

l ϑ<br />

<br />

y = ϑ<br />

<br />

(5.3)<br />

<br />

<br />

g<br />

− ˆg<br />

l<br />

¨ϑ<br />

˙ϑ0<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

˙ϑ<br />

ϑ0<br />

<br />

ϑ<br />

(5.4

− g<br />

sin(ϑ) → −u1/l ∗ sin(u2)<br />

l<br />

<br />

− ˆg<br />

ϑ<br />

l<br />

<br />

→ −u1/l ∗ u2<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

±45 ◦ <br />

<br />

5 ◦ <br />

<br />

5 ◦

• <br />

<br />

<br />

• <br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

• <br />

• <br />

• <br />

<br />

<br />

<br />

• <br />

• <br />

<br />

• <br />

<br />

• <br />

<br />

• <br />

<br />

<br />

<br />

• <br />

<br />

• <br />

<br />

• <br />

<br />

• <br />

<br />

• <br />

• <br />

<br />

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1

A > B<br />

A B <br />

A B <br />

<br />

<br />

<br />

<br />

1 × 1