東京大学hori.k.u-tokyo.ac.jp/lectures/Control_Book2.pdf · øæ1 $æ1):i7µmca a jhkÛ...

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m x =F ! x = F m

= f (3.1)

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x = x1

x2

(3.2)

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x (0) = 0

0

" x (T) = L

0

(3.3)

--opHI-zDlHK

$}^;

dx1

dt = x2 , dx2

dt = f (3.4), (3.5)

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dx

dt = f (x, u ) (3.6)

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dx

dt = f (x, u ) (3.7)

y = g (x, u ) (3.8)

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+,-ABK

$ILo@¿$OZúÅ�;

x1 = f1(x1, x2, , xn,u1, , um)

x2 = f2(x1, x2, , xn,u1, , um)

xn = fn(x1, x2, , xn,u1, , um)

(3.9)

y1 = g1(x1, x2, , xn,u1, , um)

y2 = g2(x1, x2, , xn,u1, , um)

yl = gl (x1, x2, , xn,u1, , um)

(3.10)

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A%&+,@qL�jAK^-p�;

x + a1x + a0x = u (3.11)

y = b1x + b0x (3.12)

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¤-GL�jA7DJHK

t#u÷ø%&./0

$ÌZ;'(�L�L:;I7%&./0@÷øM�ÉZ)ZAÝI-MGHK

x = Ax + Bu (3.13)

y = Cx + Du (3.14)

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JH-+pL�;

A = #f

#x , B =

#f

#u , C =

#g

#x , D =

#g

#u (3.15)

-OZ,oLHK-7m.@/0�MúÝ-;

A =

#f1

#x1

#f1

#x2

#f1

#xn

#f2

#x1

#f2

#xn

#fn-1

#xn

#fn

#x1

#fn

#xn-1

#fn

#xn

(3.16)

-�HK1;2;3Cå4DJHK

$÷ø%&./0;ST./07;¬®¯ã5677þ8:íî79 DJHK

= +x

Ax

Bu

.

n

n m

m

= +y C x D u

n m

lm

fghg$÷ø%&./07×®:=7þ8

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x(t) = Ax(t) + Bu(t) (3.17)

y(t) = Cx(t) + Du(t) (3.18)

@STS=+OGHK

sX(s) = AX(s) + BU(s) (3.19)

Y (s) = CX (s) + DU(s) (3.20)

0ÀghiÎÂj ;

(sI - A)X(s) = BU(s)

! X(s) = (sI - A)-1

BU(s)

(3.21)

0Àgh¤ÐÂM2ROZ;

Y(s) = C (sI - A)-1

BU(s) + DU(s)

= C (sI - A)-1

B + D U(s)

(3.22)

-�H�o;[\U,56:;

G(s) = C (sI - A)-1

B +D (3.23)

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$xy@±-²³GK

x1

x2

= 2 3

4 5

x1

x2

+ 1

0

u

y = 6 7 x1

x2

(3.24)

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G(s) = 6 7 s 0

0 s

- 2 3

4 5

-1

1

0

= 6s - 2

s 2 - 7s - 2

(3.25)

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fghÏ$�ó7Z[\ZÚ-Æ7;<=f

$I7°7%&+,:;

x = i $ T (3.26)

$RT:;

u = e TL T (3.27)

$ST:;

y = x = i $ T (3.28)

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7UV@úlîh�jÝ;

e -% $ = (R +sL) i

= Ri +L i

! i = - R

L i -

%

L $ + 1

L e

(3.29)

%i + TL = (Js + B)$

= J $ +B$

! $ = %

J i - B

J $ + 1

J TL

(3.30)

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i

$

= - R

L- %

L%

J- B

J

i

$ +

1

L0

0 1

J

e

TL

(3.31)

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I(s)

& (s)

= G1 1(s) G1 2(s)

G2 1(s) G2 2(s)

E(s)

TL(s) (3.32)

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G(s) = Y(s)

U(s) =

10(s+4)

s(s+1)(s+2)(3.33)

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U(s)

s(s+1)(s+2) =

Y(s)

10(s+4) = X(s) (3.34)

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U(s)

s(s+1)(s+2) = X(s) (3.35)

j ;

s 3X(s)+3s2X(s)+2sX(s) = U(s)

! sX3(s) = -2X2(s) -sX3(s) + U(s)

(3.36)

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sX1(s) = X2(s)

sX2(s) = X3(s)

(3.37)

(3.38)

-eAZAHKÊ.;

Y(s)

10(s+4) = X(s) (3.39)

�o:;

Y(s) = 40X(s) + 10sX(s)

= 40X1(s) + 10X2(s)

(3.40)

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x1 = x2

x2 = x3

x3 = - 2x2 - 3x3 +u

(3.41)

y = 40x1+10x2 (3.42)

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x =

0 1 0

0 0 1

0 -2 -3

x +

0

0

1

u (3.43)

y = 40 10 0 x (3.44)

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G(s) = Y(s)

U(s) =

bn-1sn-1 + bn-2s

n-2 + + b1s + b0

s n + an-1sn-1 + + a1s + a0

(3.45)

7P34wxøMjH%&./0©1:;

x =

0 1

0 1 0

0

0 1

-a2 -a1 -an-2 -an-1

x +

0

0

0

0

0

1

u (3.46)

y = b0 b1 b2 bn-1 x (3.47)

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x = Ax + bu (3.48)

y = c Tx (3.49)

t^DO;*RT*ST°-OZAHKu

MNOZ;

x = ATx + c Tu (3.50)

y = bTx (3.51)

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G(s) = Y(s)

U(s) =

10(s+4)

s(s+1)(s+2)

= 20s

- 30

s+1 + 10

s+2

(3.52)

®pM;

Y(s) = 20s

U(s) - 30

s+1U(s) + 10

s+2U(s)

' 20X1(s) - 30X2(s) + 10 X3(s)

(3.53)

IID;

X1(s) = U(s)/s

X2(s) = U(s)/(s+1)

X3(s) = U(s)/(s+2)

(3.54)

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x1 = u

x2 = -x2 + u

x3 = -2x3 + u

(3.55)

y = 20x1 - 30x2 + 10x3 (3.56)

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x =

0 0 0

0 -1 1

0 0 -2

x +

1

1

1

u

(3.57)

y = 20 -30 10 x (3.58)

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ABèO³´�øM�HK

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t*uÌn56

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x = Ax + Bu (3.59)

7$:;

x(t) = eAtx(0) + eA(t-()Bu(()d(

0

t

(3.60)

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56DJ ;´70DÐÑÌLHK

eAt = I + At + A2

2! + A

3

3! + = A

i

i! t

i!i=0

)

(3.61)

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H-;

sX - x(0)= AX + BU

! (sI - A ) X = x(0) + BU

! X = (sI - A ) -1

x(0) + (sI - A ) -1

BU (3.62)

$IL@sSTS=+OO^

x(t) = L-1 (sI - A )

-1 x(0) + L

-1 (sI - A )

-1BU (3.63)

:�Ô$A^0ÀghÏÐÂMÊÕGH:RDJHK

$0ÀghÏgÂ-0ÀghÏÐÂ@mÖGHI-Mj ;Ìn56:;

eAt = _-1¨ (sI - A )

-1(3.64)

DqpoLHI-z��HKx@±-²³GK

A = -1 1

0 -2

(3.65)

7-l;i-~@bôZEHK

(sI -A )-1

= s+1 -1

0 s+2

-1

= 1

(s+1)(s+2)

s+2 1

0 s+1

=

1

s+1

1

(s+1)(s+2)

0 1

s+2

(3.66)

! e At = _-1¨ (sI - A )

-1 =

e -t e -t -e -2t

0 e -2t

(3.67)

tµu×Ø�(34°7%&./0

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=>?@×Ø�(<=>?7%&./0D©1GH�¯zJHK²} ;

x(k+1) = F x(k) + G u(k) (3.68)

7jB�øDúl^A�ÅDJHK

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x(i+1) = eATx(i) + eA (i+1)T -( Bu(() d(

iT

(i+1)T

(3.69)

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x(i+1) = eATx(i) + eA (i+1)T -( B u(i)d(

iT

(i+1)T

= eATx(i) + eAt dt B u(i)

0

T

(3.70)

-�HKÄÉZ;×Ø�(<=>?7%&./0@qpH56:;

F = eAT = _-1¨ (sI - A )

-1 t =T (3.71)

G = eAT -I A -1

B (3.72)

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t#uP34É-P��É

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ÍÀ~Â§¨Ðî~�~zuvGH^ô7ïð�-ÐÑÌL;

Q =ñb Ab A2b ò An-1bó (3.74)

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lH^ô7ïð�-ÐÑÌL;

R =

c T

c TA

c TAn -1

(3.75)

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z��M�� ãGÝ;]Ê:¼Ü�(°DCåñDJH^ôDJHKu×Ø�(<=>?7%&./0

@;

x(k+1) = Ax(k) + bu(k) (3.76)

-úIBKÍÎ©¦ÀÐÂ�oøôZ;�(@ùÉZ%&+,7©@úlîOZAÝK

x(0) = x0

x(1) = Ax0 + bu(0)

x(2) = Ax(1)+bu(1) = A2x0+Abu(0) +bu(1)

x(3) = Ax(2)+bu(2) = A3x0+A

2bu(0)+Abu(1) +bu(2)

x (n) = Ax(n-1) +bu(n-1)

= An x 0+A

n-1bu(0)+A

n-2bu(1) + +Abu(n-2)+bu(n-1)

(3.77)

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ÐMNOZCu;¦À}Â&Ð¨ -�HR

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An-1

b , An-2

b , Ab , b

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t!uüÐÉ

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O^K%&./0©1MeAZ:;I7ïð:;

x = Ax + Bu (3.78)

MeAZ;Í&Ð-O^-l;

lim

t ") x(t) = 0 (3.79)

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�@NH.\`�Y��V0O

$%&'()*+,-

u = Fx +v (3.80)

Yr\=�\/ghij?%&3��4A

x = Ax +B(Fx +v )

= (A+BF)x + B v

= Ax+Bv

(3.81)

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IJY><`/HO=l\AB³9478ab´µ)?¶·:�wNH;oA¸7¶9v@>@uH./0¯

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zÇ|¾¿ÀM

$ÈÉ?=�9A{abghij

x = Ax +bu (3.82)

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.:�A�\/ghij?%&3��4A

x = Ax +b(fTx +v )

= ( A +b fT)x +bv

(3.84)

.:HO

$��AÊ9V<`��ËÊÌ?��zghij?¾|Y¤¥9¿À@uH].YÍ}0O

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=

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0

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(3.85)

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u = kTz +v

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+ v

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+ v

(3.86)

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T z + b

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z =

0 1

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z +

0

0

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"1 "2 "n-1 "n (3.90)

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Ø"#"UÛØ"#"

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èÛ��"è�µ

èîU"èîU��í��µ

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ï��ï (3.91)

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ai + ki = µi

! ki = µ i - ai (3.92)

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(3.93)

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-1

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(3.94)

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x = 0 1

2 3

x + 0

1

u (3.95)

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"I - A = " -1

-2 "-3

= "2 -3" -2 =0 (3.96)

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+,-4A

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x2

(3.97)

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" - "1 " - "2 = "2+3"+2 (3.98)

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0 1

2 + f1 3 + f2

(3.99)

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RSTUS$%&459@?:;

$/�A ¡?áHdghij

x = f(x, u) (3.101)

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0

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dt $ minu

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$Hdghij?%&3��4A

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(3.103)

(3.104)

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n

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PA + ATP - PBR

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u = -Kx + v = -R-1

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(3.114)

YÁG9sH%&'()*+,-4A

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P(sI-A) - (sI+AT)P + K

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I + K(-sI-A)-1

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T(-sI-A)

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TQC(sI-A)

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I + K(-j& I-A)-1

BTR I + K(j&I-A)

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x = Ax + b u (3.124)

y = c x (3.125)

9F/ÀDEFXA

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0

T

(3.126)

YÁG9sH%&'()*+,-4A

u = - k x , k = bTP (3.127)

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PA + ATP - Pb b

TP + c Tc = 0 (3.128)

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T(j& )QG(j& ) (3.129)

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D(j& ) 2

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(3.130)

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D(j&) 2

' 1

! 1 + L(j&) 2

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1

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(3.138)

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x1

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x2

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B2

u (3.141)

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(3.149)

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x =ATx + CTu (3.155)

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1+(W2T

"

< 1 (4.51)

W2T " < 1 (4.52)

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W1S + W2T < 1 (4.53)

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ý·¸/#$%&'¼Ì<=

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W1 + W2CP < 1+CP (4.54)

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W1

1+CP + W2

CP

1+CP < 1

* W1S + W2T < 1 (4.55)

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東京大学hori.k.u-tokyo.ac.jp/lectures/Control_Book2.pdf · øæ1 $æ1):i7µmca a jhkÛ...

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Transcript of 東京大学hori.k.u-tokyo.ac.jp/lectures/Control_Book2.pdf · øæ1 $æ1):i7µmca a jhkÛ...