制御工学第1 －平成xx年度－ 電気工学科 堀 洋一，...

制御工学第１　－平成ＸＸ年度－　　　　　　電気工学科　堀　洋一，古関隆章

講義時間： 月曜日 8:30~10:00講義予定日：別途通知教科書： 堀・大西：制御工学の基礎，電気電子情報通信基礎コース，丸善(株)参考書（制御工学第２の教科書）：

堀・大西：応用制御工学，電気電子情報通信基礎コース，丸善(株)

１．序論　１．１　　制御工学とは何か　１．２　　制御システムの例，分類　１．３　　閉ループ制御と開ループ制御　１．４　　制御工学の歴史

２．システム動特性の表現　２．１　　信号伝達と状態遷移　２．２　　動作点まわりの線形化　２．３　　線形システムの表現（２．４　　ラプラス変換法の基礎）　２．５　　ブロック図とその合成　２．６　　特性の計測法

３．制御システムの安定性　３．１　　線形システムの安定性　３．２　　ラウスの安定判別法　３．３　　ナイキストの安定判別法　３．４　　フィードバック系の安定度指標

４．フィードバック制御系の基本特性　４．１　　入力追従特性と外乱抑圧特性　４．２　　定常誤差と誤差係数　４．３　　２次系の過渡応答　４．４　　２次系の周波数応答　４．５　　高次系の代表根

５．線形フィードバック系の補償　５．１　　フィードバック制御系設計の基本指針　５．２　　直列補償とフィードバック補償（５．３　　ニコルズ線図を用いた制御系設計）　５．４　　根軌跡を用いた制御系設計　５．５　　直列補償とPID調節計　５．６　　フィードバック補償

６．制御工学第２への序章（古関）　　　古典的制御の復習と状態空間法との関係　　　（フィードバック制御と多項式法など）　　　状態空間におけるシステム序論　　　（連続量と離散量）

制御工学第１（教科書の一部，第２の内容も含む）

４．自動制御

４．１　自動制御の基礎

　４．１．１　制御系の基本構成

　４．１．２　制御系の分類と歴史

　４．１．３　閉ループ制御と開ループ制御

　４．１．４　システム動特性の表現

　　　　　　　　　　　　　（含 ラプラス変換法の基礎）

　４．１．５　制御系の安定性

４．２　フィードバック制御系の設計

　４．２．１　フィードバック制御系の基本特性

　４．２．２　２次系の性質

　４．２．３　フィードバック制御系設計の基本指針

　４．２．４　ニコルズ線図を用いた制御系設計

　４．２．５　根軌跡を用いた制御系設計

　４．２．６　PID調節計

　４．２．７　フィードバック補償

４．３　状態空間法

　４．３．１　状態変数と状態方程式

　４．３．２　状態方程式と伝達関数との相互変換

　４．３．３　状態方程式の基礎的性質

　４．３．４　状態フィードバック

　４．３．５　最適制御

　４．３．６　オブザーバ

４．４　ロバスト制御

　４．４．１　適応制御とロバスト制御

　４．４．２　Ｈ∞制御の考え方

　４．４．３　Ｈ∞制御問題の解とその意味

　４．４．４　μシンセシスとロバスト性能

（第４章になっているのはある事情のゆえであり，気にしないでください。）

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1+ j%T(1.64), (1.65)

,�Ä=

20 log G = 20 logA - 20 log 1+% 2T2

1/2(1.66)

arg G = - tan-1 %T (1.67)

ZkDQ

$Ö�׬|Z<B789G;E9_64Ô:%Õ,*+2AD9:;=Ù�2%Z<C8ÉÁ7%ZBDQzP;=

¾¸'L2ñ¤=CtR�[S<izC7jBwê7QjM<=ëÒ�[��1Z<=ÏU�*+aÿ>MN

�[*+<ív;ÿ>DjBaÖ�×,uÆB56¬PM67DKPZkDQ

$íî;=´«aç��;kDïëÒ�[��1Z<=1<DuZ<kDa|}~�Ö,r`Cj562�]r`

;CDQj,r`C56<5N5NEF]Dw,ZkDa=P¢ó7a�G;CD,Z=jjZ<ëÒ�[�

�1,23!G'L2ó`jB;5r`Q

|}~�Ö$j56

$z6í#AM1,Ö�׬|<|}~�r,r`;CDQ£Ô}~ääÕ,H�I<=%�JJ#,B¤=��ÉÁ7K#Ká

�=%�°LL#,B¤=K}f%���° K°M%���ZkDKP=-20 logT - 20 log%°BCDQÖ�׬|,Ù�<°ÉÁ7%°'

KP=jM<x¤a° ?��°,NàÊ;CDQÉÁ7}�¼<¼a}�¿;CDOB;�Pa��QRQjM2ݳÈbݳBS

TQ>9¼a�¿;CDBUä~��µÝÞ¶QRQjM28X��#BS-Z7DQ

|}~�r$##,23!G,¾¸*+

$|}~�r;J76�ëC*+<V¬,r`;CDa=jM2í«W¬,r`C>¬Z?@Z¤DQjM2d>

¬?@iBSTQÑ,{|a|}~�âZkDQ

|}~�â$##,23!G,>M¬?@

$BjÀZ=íî;=

G(s) = A (1+T2s)(1+T4s)

(1+T1s)(1+T3s)(1+T5s)(1.68)

CD­,23!G,89G*+<=

20log G(j% ) = 20logA + 20log1+% 2T22 1/2

+ 20log1+% 2T42 1/2

- 20log1+% 2T12 1/2

- 20log1+% 2T32 1/2

- 20log1+% 2T52 1/2

(1.69)

BCDQjM<=íî,>M¬?@Zk¢=9BhN=�}Á�

�Á¤Á�

â°,B¤;<|}~�ä,r`;CDQíÐ

íÐ,/h;<]Z;|}~�â,r`C?@aCzM6J¢=ÑMP2X�ÄEÅ6ÎÏ;ñ8jBaZ¤DQ

|}~�ä$>M¬?@

!"#"$%&'()*+,

%()*+,-./0123456789:&'(;<=>?@ABC01DEFG00D23*+,)HI

JKL3MNO3PQCR;/=>STFG

U#V*+,)WX

%YZ)[\;]^_\@YZDEF@)-`+abcdef3gh^>^ij@)-`k+abcdef1

=jGl1mn3opq6r2EFstuv-wmF1xyz+){|}~D{|�F��3[\-stu

v3_\-{|}~1��n+abcdeDEFG^�^3{|�~-_\1��n3z+}~Dq6r�{

|��n{|�~2kZ;�w^>=���3k+abcde1CFG

�����%����t

%�>3�����)����t)�����23��2C=1^>3

J! =mgl sin! +T (1.70)

D�m��FG�!�������)1�3

J! =mgl!+T (1.71)

1��D�F���b�e ¡^>3

Js 2! =mgl ! +T (1.72)

"G(s) = !

T = 1

Js 2-mgl = 1

J 1

s+mgl

J s-

mgl

J

= 1

(s+#)(s-#)(1.73)

1CFG

%!¢£¤¥¦§DEF��3[\§��)¨©ª,23§�e«9b¬­:C�§¢�®¥3¯°±¬­:C�n

§¢$�®¤¥�²$�¦CR-³[^>!-´µF01�D�FG^�^3!;23¶·¸#¹)º�_F)D0)(2*+

D2C=G0)»j;3z¼;`½¾¿H)ÀÁ)Â)ÃHÄ�Åil2¯)1�(2Æ*+fDEFG

%=i3!)ÇQÈ!É-Ê+^3}~)ÇQÈ!

É2Å;^l=U/i�3EFaËD�a��t-̵l=V1

�23

T = K1 ! - !0 + K2! (1.74)

1=j456789:&'-�13

J! = mgl ! + K1 ! -!0 + K2! (1.75)

»�3

J s2 - K2s - mgl - K1 ! = -K1!0 (1.76)

" !

!0

= -K1

J s 2 - K2s - mgl - K1

(1.77)

1CFG+ÎÏÐD23

!

!0

= K1

mgl + K1

(1.78)

1CÑ>3Ò�-ÓÀÔ����n3!2!

É;=��D@�=È1CF»j;ÕmFG

%^�^(�*+DEFlµ;23�¤����¦)ÀÁ)Â��T>Ö×ØÙ;EF¶B�EFGÂ23

s = K2 ± K2

2 + 4J mgl + K1

2J(1.79)

DEF��3Ò�ÚÉ3Ò

�ÚÉDEF01�Ûi^=G

%0)��=C�ªNC&')ÜÝ�C�>@Þ1C�ß�FG^�^3à�2!áà) âã�Ô��CÑ

l1�;3à��1ä]�å;¬­:-�æ�n»=31=jç�è)01-éÑ>=FêæDEFGÒ�áÒ

�

)ëìJCÈ2á2��íî1ïð^Cæ�nC�C=G

%ñò(D2ÀÁóº��EFQôõ;ö9÷�F»j;Ò�áÒ

�-øµ>@»=G^�^3ù�ñú;CÑl

ûA����tá3��;üH)Ô�C��t-*+;�lýF»j1�F13»�þ~C&'ÿI)ÜÝ-

¶B1�FG

UñV�!e)*+MNO

%�D;STl»j;3"e«#)$U½¾¿H)ÀÁóº�Uª,óº�V¢%1<=lª,���)ÂV

;3&1/D@¯)ÃHÄ-'/@)�EFU(×ØÙ;EFV1(2Æ*+DEFG

����)%456789:&'(

%����))456789:&'(;<=>3ÇQÈ��_\ÈiD)½¾¿H23

Y

R =

G(s)

1 + G(s)H(s)(1.80)

DEFGß�ß��*+�EF)2ÀÁ��²£¤¥¦,¤¥¦�¢É�)ÂDEFG

G(s)

1 + G(s)H(s) =

b0+b1s + b2s 2 + + bms m

a0+a1s + a2s 2 + + ans n(1.81)

1-.�F1�3�¤��)�¦)ÀÁ-`ª,óº�f1é=3ª,óº�)Â-`"e«#)$f1=jG

%Ô/C0123@^£¤¥¦�Æ*+DEÑ>@3à)Æ*+$2�¤��)�¦)Æ*+$12012z2^C=

1=j01DEFG3;=mn3Æ*+C"e«#£¤¥¦-456789:&';»Ñ>*+â�ýF01�4

�01;56DEFG

%�>3�!eU789¹:V)MNOD23ª,óº��

ans n + an-1sn-1+ +a1s + a0 = 0 (1.82)

;]^>3;<)=CH>U�!eH>1?@01;^»jV-/�FG

sn R11=an R12=an-2 R13=an-3 ...

sn-1 R21=an-1 R22=an-3 R23=an-4 ...

sn-2R3 1=

an-1an-2-an-3an

an-1

R3 2=an-1an-4-an-5an

an-1

R3 3=an-1an-6-an-7an

an-1 ...

sn-3

R4 1=R 3 1R2 2-R2 1R 3 2

R 3 1

R4 2=R 3 1R2 3-R2 1R 3 3

R 3 1%%...

...

...

s2 A ...

s1 A ...

s0 A

%i·BC��3B

CD�B

CE�D�B

CEF1=j»j;z/<�;GH-HT>I#J1�FGò;3B

CE���Kµ>3B

CE�D�B

CE

LD�BCEM1=j»j;HT>IñJ1�FGINJ;<2l���æ1O�ð3�PQ)J)zRÖ)HSDTF

1=jUðDEFG

%z¼;23Ri, 1 .... Ri, j

Ri+1, 1 .... Ri+1, j

-V=>37W²�D�X�-YF)DEF��3

Ri+2, j = Ri, 1 % Ri+1, j - Ri+1, 1 % Ri, j

Ri+1, 1

(1.83)

1Z�01@D�FG

%�>3(�*+DEF;23

U#VGHai ��T>[\]U¯;��n»=V

UñV�!eH> Ri1 ( i = 3, 4, ^ ) �_>¯

DE�n»=G7W�)\]� â�F1�3\] â){H�Æ*+Â)H;CF1=j01@`��>=FG

%;Q�3YaC�!e)*+MNODEF�3Ã2@jb^c�EFGª,óº�-d¤¥¦1Z�3à)òH

-C1�FGI#J1IñJDJÑ>=FÀe-3d¤¥¦¢dC¤¥¦²d

CE�¤¥¦1Z0jG�F13INJ;<fJ)

ïð23;<)»jCg+-'Ñ>=FG

sn Pn(s)

sn-1 Pn-1(s)

sn-2 Pn(s)-P

n-1(s)DTÑlEi� h%P

n-2(s)1�FG i2 a

ns

sn-3 Pn-1

(s)-Pn-2

(s)DTÑlEi� h%Pn-3

(s)1�FG i2 an-1

s

....

s1 P3(s)-P

2(s)DTÑlEi� h%P

1(s)1�FG i2 a

3s

s0 P2(s)-P

1(s)DTÑlEi� h%P

0(s)1�FG i2 a

2s

P1(s)2P

0(s)DT�/�>3 i2 a

1s 1CFG

%0)ïð2j6:k97)lmODEFGà^>ïð��l`Ei�f)?þò)GH��!eH>Ri,1DE

FG

%0�23�!e)�O23@1)óº�-=�/�)Bn;Àe^>=F01;CFGà)_ìo-pe�

F;2@j�0^c)ÜÝ�¶BDEF�3qrêæSTF13st�8uD/C�Ñl»jC"e«#;]

¨�FGà^>à)st)svá8uGHUfJ)iDEFaiDK��FGV;&1/D@w;CF@)�E

F1(2Æ*+1M+��F1=j01;CFG��;3svá8uGH)(>�&'()x�-KL�FG

�!e)�O23y;*+�Æ*+�-M+�FêæD2C�3@Ñ1z{=@)-|îD=F)DEFG

UNV4­}9~)*+MNO

%4­}9~U,9��W¹�V)�O2úsJ;�!e)�O1[zDEF01�Ü��>=FG00D2ïð�

Oêæ-`^><�G

Hn =

an-1 an-3 an-5 0

an an-2 an-4 an-6 0

0 an-1 an-3 an-5 0

0 an an-2 an-4 an-6 0

0

0 a5 a3 a1

0 a4 a2 a0

(1.84)

CF4­�59~J>�-Y�3

H1 = an-1 (1.85)

H2 = an-1 an-3

an an-2

(1.86)

H3 =

an-1 an-3 an-5

an an-2 an-4

0 an-1 an-3

(1.87)

������

CF�J>�U4­�59~J>�V-ïð�FG(�*+DEFlµ;23

U#VGHai ��T>¯U�!e)*+MN)1�1[�V

UñV4­�59~J>� HiUi=2, 3, ..., n-1V��T>¯

DCæ�nC�C=G

%C<37W��¢�,

WE�®,

WE�UW¢LD�FD����D�CE�V1=j¿G�EF01���;��µ��FG4­}9~)�O23

()òHC�!�~iDC�ïð56DEF�3à�;QÔ�CòH)(;CF1J>�)ïð���;CF

)D3��D2Ei�V=��C=»jDEFG

U!V�!e)�O)��

%i·3ª,����¥M� ²�L¥F� ²��¥L� ²�F¥��²�¥�²�M� ¢�É�)*+MN-^>�»jG�!eH>)?�)ûü

23

s5 1 2 1

s4 3 4 1

1C�3;��æF13

s3

3%2-1%4

3 = 2

3

3%1-1%5

3 = - 2

3

s2

2

3%4-3%(- 2

3)

2

3

= 7

2

3%5-3%0

2

3

= 5

s1

7%(- 2

3) - 2

3%5

7 = - 8

7

s0 5

%¥�)üD37M�¢E)®�Dw1CÑ>=FG\]) â237

F�¢���7

M�¢E)®�;CF1�137

M�¢E)®���

7��¢M1CF1�)ïñ{DEF��3û/)Æ*+Â-'/1=j�ß�FG

%ò;3�����)456789:&'(�*+DEFlµ)Ò)��-´µ>�»jG

�����%456789:&'(

Y

R = G

1+G(1.88)

»�3ª,óº�23

s4 + 4s3 + 5s2 + 2s + 2K = 0 (1.89)

1CFG�!eH>-YÑ>=�13

s4 1 5 2K

s3 4 2 0

s2

4%5-1%2

4 = 9

2 2K

s1

9

2%2-4%2K

9

2

= 2

9(9-8K)

0

s0 2K

%�Ñ>3(�*+DEFlµ;23

�Ò�%D���E)Ò��%»�3%Ú�Ò�Ú��®)�DEF01

���1CFGC<3Ò¢ÉD��®))��D23(2¥�X$U$�2ª���)��±HV1=jõ)Â-'Ñ>=

F2·DEFG��µ>�»G

U$V���e¬)*+MNO

%�!e)*+MNO23�­6b½¾¿H£®¤�²£,¦-ïð^3o ª,óº�-¡TF@)DEÑlG^

�^30)ïð2^n^nÔ Ø�DEÑl�3Cî��)¢+£6r��ª,óº�-´µF¶B�EFÃ

¤)¥¦;23ïð§ì�Æ56C01@bC�C=G

�!e)�O�����h��£®¤�²£,¦)$-¡TFUo JV

���e¬)�O��h��£,�êæ-¡TFU¨ JV

���LÉ%456789:&'(1*+MN

%0�;]^300DSTF���e¬U©ª«9W¥¹V)�O23.­6b½¾¿H£¤¥¦,¤¥¦êæ-¡T>3�

­6b½¾¿H� £¤¥¦®¤�²£¤¥¦,¤¥¦¦� )=t-¨ J;¬¢�F@)DEFGl1mn���LÉ)&'()*

+,23�²£¤¥¦,¤¥¦¢É)Â�_ÄÖ×ØU­,d®�­¸¯¹�,B°¯�d°BC¸V;E�n.±��FG^�^3�!e)*

+MNODáÑ>�l»j;30�-=í=íïð�F)2�C�Ø�DEFG@^£¤¥¦,¤¥¦êæ-¡T>

*+MN�D��nl=²î³´DEFG

µ���e¬)+p

%£¤¥¦,¤¥¦¢É)$-_ÄZ�ê^3¶·Q;$�EF¥¦20�-(;¸æC��3¥¹ØQ)���L�)=

CºÀ»-¼mFG

���L�%¥¹Ø;<æF¥)��^�

���L�%���e¬�½¾

%¥¹ØQD¶·-w��¯;å�Ñ>QÑ>=�¨230�-�±HX$ê1ÕÑ>»=GUl=>=)¿p

"e«#2�$hÀ)1�Þ)ä¨@`�C=U_\�Ác1CFVG���L�D$-Ô���F1Ât;þ^

>=F)�0�;]¨^>=FG

%¥¹ØQD���L�)Ä»D¥-Å��ýl1�3.­6b½¾¿H£¤¥¦,¤¥¦)½¾U���e¬½¾V�3

UE�D� ÉV){�-�{{|^3£¤¥¦,¤¥¦)$)jí3ÃHÄ�(×¹Ø;EF@)U/i�3Æ*+Â3l

ê^¥¢X$Q2)Æ�GCÇC����L�)½¾D@»æ>=FGV�ÈÉEÑl1�F13`�¢ÈDE�n�

­6b(�*+DEFGf0�����e¬)+pDEFG

%ä)"e«#D2È¢É)01�ó�3à)1�2��¢É��*+��1CFG0�-ÊË)���e¬)+p

1=jGB�F;3½¾�UE�DÉV-ÌiCæ�n»=)DEFG

%ò;ä;3�)�-1Ñ>½¾-Í=>�»jGÎy)lµ;,¤¥¦¢�1^><�G

���LL%¥¹ØQ)Ä»1à)R]

µÄ»Ï%¥¢X$U$в��h�²ÀV-³[^>3

G(j$ ) = K

j$ (j$+1)2 = - 2K

(1+$ 2)2

- j K(1-$ 2)

$ (1+$ 2)2

(1.90)

0�-Í�13���LF)»j;CFG

���LF%Ä»�Ñ;]�F½¾

µÄ»Ò%¥�¢�7�¸�EX&U&в'®��h�E'®�V-³[^>3

G(Re-j& ) = K

Re-j& (1+Re-j& ) (1.91)

1CF)D37hÉ1�F1%;ÓÔ�FG

µÄ»Õ%¥� ¢X$� U$ÐE�À�h� E�� VDE�30)½¾23$�ÃHGH)óº�)ÂDEF��3¶·3Ä

»Ï)¥¦1÷]Ö;CFG

µÄ»�×%¥�¢���¸�EX&U&ÐE'®��h�²�'®�V-³[�F13

G(r e -j& ) ( Kr

e -j& (1.92)

DEF)D30�2Ô�C×ØD3²'®�����E'®�;{F½¾1CFG

%Ò-Ô���F1¤E�Dɦ-­6b;|�á��CF�30�2Ù�Ú-Ô���F1(�Æ*+;C�á�

=1=jÎÝ1z2�FG

%;Q)01��31Î2£¤X$¦)�-¼m3$-ÉhÀ1^>¡T�nÓÀDEF01�ß�FG/i�3Ä

»Ïêæ-ÛÜ;^>3���LF)½¾êæ-¡T�n»=)DEFG

%Ý�;���LM)Þ/)¥¦;/=>3�­6b(�*+�Æ*+�-ST>�»jGà�2½¾)÷1)

ßt1t¤E�Dɦ)Ô�¿GDøi�3

à�*+3á�*+3â�Æ*+

1CFG

���LM%*+C¥¦3Æ*+C¥¦

%1Î)¿p"e«#23kZÔ�±H)ã];2ä¨^C=)D3x1îR)¥¦3£¤XÀ¦¢%;CFGz

�3£¤É¦;¿^>2û1<�E�3$¢É3/i�z+È)opJC[\-wml1�£¤É¦�YZ)È-1F

¥¦2���L�¤B¦3kZ)È-1F¥¦2¤ä¦)»jC½¾1CFGà�Æ�3+abcde3k+abcd

e;]¨�FG

¤B¦�+abcde%%%%%%%¤ä¦�k+abcde

���L�%£¤É¦�YZ3kZ)È-1F¥¦)���e¬½¾

UåV456789:&'()*+~PQ

µÙ�ÚæY1aÛæY

%0�iD3()ª,2ç_;�m��l1^>rè-��µ>�lG^�^3ä;2¢+^lª,;2¶

·éê�EF^3ª,§ì@z+12�ë�C=Gl1mn3�ìí)�ì~2uîuv� ßF1 â�F

^3ïJð)ñ)���23à)�~áþ~;»Ñ>òCÑ>�FG0)»jC01-¼mF134567

89:&'(23y;*+DEF1=jêæDC�3EF�~()ª,� ßÑ>@*+,-óßC=êæ)

æô-@Ñ>=F01�Ûi^=G

%0)»jC*+,)æô)z/)KLO1^>»�Ü��>=F)�3ò;`�aÛæY3Ù�ÚæYDE

FG

%=i3EFõõ"e«#)zö½¾¿HÒ£¤X$¦,¤X$¦)÷nØQ)½¾����L�)Ãõ)»j;CÑl1

^»jG0)¥¦23£¤¥¦,¤¥¦�*+CBnDEFU�Cßí$-¥)(×Ø;@lC=V�ë�3(2ø�

�;*+DEFG10ù�3ä;23¢+)éêáª,) â�EFlµ3Ò£¤X$¦,¤X$¦)½¾2ú9÷-

Í^l»jCûÌ;EFì-@Ñ>üý�F1¼mC�>2C�C=G^l�Ñ>3@1)½¾�3¤E�D� ɦ

)t)Ei���;EF13ä)½¾2¤E�D�ɦ-|�3¥¦;»Ñ>2(2Æ*+;CÑ>^ijG

���L�%456789:&'(1à) �

���L)%���e¬õ�Q)!þ3ÿ

þ

%0)»jC01-!Plµ;23Ò£¤X$¦,¤X$¦)½¾�3E�)t��EF�~;Q"�>=F01�¶B

DEFGà0D3*+,)æô)KL1^>3E���½¾iD)#"U���L))�V-1F01�¼m��

FG^�^30)#"-´µF;2�QD½¾1¤E�Dɦ)t-q$3à)%��?�;CFt-&�1=j

'Y�¶BDEF^3�±H¨©)x�)K`O3l1mn(6£õ�CRD2Îy;´µ��C=)t�E

FGà0D30)�)³ß�;3ß�ß�231Î3���L);`�»jCaÛæY!þ3Ù�ÚæYÿ

þ-3*+

,)æôU*+~V)KL1^>V=F)DEFG

%�Cßí3

!þ�¢�)É*²��+�Ò£¤X$¦,¤X$¦,

,Ò£¤X$¦,¤X$¦,¢��)t%-~. (1.93)

UÙ�Ú�É/0;CF�±H;<=>3aÛ�E1!þ<��F1

%*+Z�;¾�FGV

ÿþ�¢��É�°8ÿ,�®Ò£¤X$¦,¤X$¦,

Ò£¤X$¦,¤X$¦¢�E�)É*)t�-/0. �� (1.94)

UaÛ1���)É*;CF�±H;<=>3Ù�Ú�E1ÿþÔ�

%�CF1*+Z�;¾�FGV

1+Ë��FGÿþ;2�¤���F¦)»j;£"2­Ç3-V=FGà)p4230)»j;�F1(�*+)

1�2ÿþ2¯3Æ*+)1�2w3*+Z�;EF1�2í5jRÉ-/0.1CF��DEFG

%0)!þ-aÛæYU6:B¥¸�þB�ÿWCV3ÿ

þ-Ù�ÚæYUÿBWC�þB�ÿWCV1=jG

%7;STF»j;3Ò£¤X$¦,¤X$¦)½¾�E�;�8�;/�3z¼;()9~¨©2��J;C�3�±

Hª,2EF�±HDÔ�C:6:-@/G0)012()ª,1^>2;i^=@)D2C=Gà0D

456789:&'()Êï;ElÑ>230)aÛæY1Ù�ÚæY-<çCÈ;=@01�3()�ª,

-x;;�F1=jg+��@¶B;CÑ>�F)DEFG�Cßí3!þ3ÿ

þ2*+,)æô)KLDEF

�3à�1[>;3()�ª,)x?-`�z/)KLD@E�3à)g+D@ABDEFG

%Ù�ÚæY3aÛæY23(6£õ�U���L�V3Ù�ÚµaÛõ�U���FÉV��@3��;@�1F

01�D�FG

���L�%(6£õ�Q)!þD�ÿ

þ

���FÉ%Ù�ÚµaÛõ�Q)!þD�ÿ

þ

!"#$%&'()*+,-./01

!"#"2$%&'()*+,-./3456

$7789:%&'()*+,-./;<=>01?@ABCDEFGHI>J:34=>56BKLFMI

NFOPFQRS

T2UVWXY56IZ[\]56

$%&'()*+,-/^=B9:_`RabF:

c^deXY56 cZ[\]56

/fgILAhKiCJS

$jkl'/mn?oJ,-.9:pLqBrsFt?uVoJ7IBvEF7w?@ASDIxyz{|/q

m?z}|BsvAIoJI`:~}��/�/vAB:��EI�')'j�'���/�i���i�i�R

F�MsLvA8CJi:�BvEF9�/vAB�J��>��?oJ�iLL���w>LS�89�o

�JS7/vA>56?^deXY56I��S

$��:�I/pq/mni�i�:�/ ¡qmi�¢sFsMEDI`>£:7w?�IB¤o¥`:o

>¦�Z[\]56�§¨©ªH8CJS

~}��$jkl'/mn0«e/¬­BroJ®¯

~}�}$%&'()*+,-.

~}�°$^deXY56/±

~}�z$Z[\]56/±

$~}�}/,-.BQLF:^de²IZ[³�¨´WµM8/¶·¸¹?1ºsF»JI:

y = CP

1+CP r + 1

1+CP d (2.1)

I>�:Qv�:~}�°Qv¼~}�z/vA>56?½D¾DLILA¿ÀIsDHÁiCJ7Ii¦�JS

$778:,-ÂÃ?ÄÅÄÆIJÄÇĵÄI«Èowy:

e = 1

1+CP r - 1

1+CP d (2.2)

I>J�¨:�ÉoJI:^de®¯IZ[®¯I9ÊË6Ì?½EFLJvA8CJS

$778:ÍÎBÏwJ�ÐÑ�ÒÓÔÕÄ?Ön¸¹×ÑØÕI�¼:

S = 1

1+CP (2.3)

I«ÈoJI:^deXY56IZ[\]569×ÑØÕBvEF�ÙB¬ÚoJS7/vA>.9Û2ÜÝn

.ÞIßL:7/àá89âBOPDhK/569ÛÊËÞ7IB>JS

$s�sÛ#ÜÝn.Þ89:7/ãä9åæç�B018`JS

~}�è$#ÜÝn,-./�éê±

$~}�èBëo#ÜÝn,-.89:^de�¨/%&'(%ìl'(íîiïðoJS7/I`:

y = (C1+C2)P

1+C2P r + 1

1+C2P d (2.4)

YEF:ÂÃÄÅÄÆIJÄÇĵÄÄ9:

e = 1- C1P

1+C2P r - 1

1+C2P d (2.5)

I>JSÓ�:Ó

}BvEF:^deIZ[�¨´WM8/hK/569�>�ÜÝB¬x¨wJ7Ii¦�JS

$Ïð89Ôñò,-ó/vA>2ÜÝn/,-óiôõsDDNB:#ÜÝn,-9öw¨wFsMEFL

JSDIxy:%÷ø',-/ùR9:M©^deIúûe/Ã?I��/e�¬Úü/_ýB®ËF,-�

þ?ÿNJSs�s:Ã?IJI7!8:"#>$%?&FFsMA7IB>JS

$2ÜÝn.89:Z[\]6'?fgsvAIoJI:GÀ=B()*+B¾,J?->R>JSoJI^

de®¯9�R>�o��MsR>L./?0ËD�oJ/8:()*+ÚB912iCJI345wFL

JSs�s:#ÜÝn,-?�EFLwy:©EIMs>�þ?IJ7Ii8`J/8CJS

$778:~}�è/678*9BKLF:xF»vAS678*9;�¨´W<M8/¶·569:

y

! = - C2P

1+C2P (2.6)

8CJS��!=:7/_`5�>JPRý5RsDLS

$s�s:7/HÁ9:Z[\]56/fgHÁI9>?sFLJS@Ñ}�èÕ�¨9:Z[\]/DNB

�ÐÑ�ÒÓ}ÔÕ?AWBB�EFL`DLILAHÁiÉFIwJ/8CJi:@Ñ}�CÕI9DsF2B>EFL

J�¨:ã�I�ý5RoJ/9EF'8CJS

$s�s:678*99ùR/GH:IJ(KLMNBïðs:\]sDLZ[9OKLMNBCJ7Iiù

L�¨:^=?úÏoJKL¹MN?abwy:AMR01oJ7IiF'8CJS

T = C2P

1+C2P (2.7)

/7I?ÛPÖn¸¹ÞILASQÑØÕ9R)S�T«6/Ud8�CJS

$Ön¸¹×ÑØÕIPÖn¸¹QÑØÕ?I�Bý5RsDLILAHÁ9:VOoJWX,-/ªH>YZ/[I

K8C�ÛuHÖn\]ÚYZÞI�ywJS>Q:QÑØÕ92ÜÝn.89^de®¯I��^sFLJ7I

?_`sFQ7AS

T#U«aÂÃIÂÃb¹

c«aÂÃ

$Sc*de/VWBroJ\fe/ÂÃ/7I?Û«aÂÃÞILAS

e = r - y = 1

1+G(s) r (2.8)

8CJ�¨:²ÄÆÄ�ÐØTSc*dUgU/I`:hÄiÄX/I`/Å/e9:

es = lim

t "#e(t) = lim

s "0

s 1

1+G(s) 1s

= 1

1+G(0)(2.9)

8CJS

~}�C$Sc*dUgBroJ®¯

$�s:ÅØÆ{8CEDIoJI:�/j>7Iik7JDNB9:lÑ{ÕÆX8>RF9>¨>L�¨:lÑØÕ

9�ÐØ>Jmnoap?q>RI��KrM>RF9>¨>LSo>¦�:

G(s) = s (s + p1)(s+p2)

(2.10)

ILAs8>bwy>¨>LS7wtZ/GH9:lÑ{ÕÆuT"1/eUIsF:

es = 1

1+K (2.11)

>J«aÂÃ?0ËJS77B:u/7I?Û«a)*+ÞILAS@Ñ}��{Õ/vw9:xyBq>RI�2

KAiCwyzD5wJS�/A9d{+�T,-r|Ui½EFLF�}+�R'{i½EFLF�vLS

�s:d{+�i�EFL>RFs��«aÂÃ?BBsDL/8Cwy:}+�R'{B½D¾JGHiC

JS

$$$Ñ~ÕÄd{+�ixyBA?�K$$Ñ�ÕÄ}+�R'{ixyBA?�K

~}��$«aÂÃ?0Ë>L.

c�nÂÃ

$K�B:~}��BëovAB:��B�±sF_`R>J^deT{+d¸¹UBXY5¾JGH?:xv

AS;<=B9:��>�8CJi:�@�<B��?H¦¾J,-.iCJS7/GH/ÂÃÄÅ�Ä?�nÂÃ

Iv�i:

~}��${+dVWBroJ®¯

ev = lim

s "0

s 1

1+G(s) 1

s 2 = lim

s "0

1

1+G(s) 1s

(2.12)

8CJ�¨:t�/�K/GHi:x¨wJS

T2UÅ�ÄÄÆÄX� lÑØÕiÄ�ÐØÄ?rM>LS

T#UÅ�ÄÄÆÄ�Ðu� lÑØÕiÄ�ÐØÄ?�K�br=8LJSD�s:

K = lim

s "0

s G(s) (2.13)

T�UÅ�ÄÄÆÄ{� lÑØÕiÄ�ÐØÄ?hKt�r�

c(�/^deBroJ«aÂÃ

$^de?ùp@8��oJ7I?:xJI:²ÄIsF9Ä�ÐØTSc*dU:�ÐØ}T{+dU�>£/��'

+iCJS��B:lÑØÕir��ÐØ/¹?�IoJI:�}��/vA>¸b?-JS

�}�2$^de¸¹BroJ«aÂÃ

$89:L!L!>VWBrsF«aÂÃ?0Ë>RoJB9:}+�R'{B�ÐØ?DR5=½D¾FQL

F�wyvL89>L�:I�¦wJ���w>Li:YZ9�A��8>LS

$�ÐØiùR>wy>Jå£:T«6?�sR��oJvAB>J/8�ÐØ/ù�9 bDLs:(�VWB

roJXY56?¡RoJ7I95壪H8>LILA¢$�CEF:Îa9oaó?2£�bVwJ7I

iùLS

!"#"#$#�./6Ì

$Ïú/jSc¤BvRϦw:5M,M>Ï|/¥¦BªH>§¨?¡Do#�./©ª®¯Qv¼KL¹

®¯?:qs«sRÉFQ7AS

T2U#�./©ª®¯

~}�¬$­'®¯'�/°±,-.

$~}�¬/­'®¯'�/,-?±BI�:°±/Ug$²�¨úûe$M8/¶·¸¹?1ºoJI:

$

$*

=

K

s(1+Tms)

1+ K

s(1+Tms)

= K

Tms 2+s+K = K/Tm

s 2+ 1

Tm

s+ K

Tm

= % n2

s 2+2&%ns + % n2

(2.14)

$$T\V/p9:#�.³/¯´µ¶*·+¸?@EFLJSU

I>JS778:$²?Sc*dUgIoJI:$

²¹Ä�ÐØÄ8CJ�¨:$9:

$(s) = % n2

s 2+2&%ns + %n2

1s

= '(

(s-')(s-() 1s

=

1s

+ (

'-( 1s-'

- '

'-( 1

s-( (')()

1s

+ '

(s-')2

- 1s-'

('=()

(2.15)

I>�:��¸¹BºoDNB{d{S»¬¼sF:

$ (t) =

1 + (

'-( e't - '

'-( e(t (')()

1 + 't e't - e't ('=()

(2.16)

I>JS

$t�:¥/��'+?&BvEFGHabsvAS

T2U&½2 '¾(Ä>J¿/#úÀ TT«U

T#U&¹2 '¹( >J¿/ªÀ TT«U

T�UBÁ&Á2 ú¹Âi¿/ͧÃÄÀ TT«U

T!U&¹B ͧmŹ/À TT«12U

TÆU&ÁB ú¹ÂiÇ/ͧÃÄÀJÇ/ªÀJ#úÀ TET«U

$LAM8�>RT2UJT�U9U¹/ú¹Âi¿>/8T«8CJST!U9T«12BC�:TÆU9

ET«B>JMN8CJS

~}��{$T2UJTÆU/ÈÉ'SBQbJÀ/ʱ

$t�?MINJI~}��{BËRvAB:&?¬Ú5¾DGH/À/ÌÍ?-J7Ii8`JSt�B�Aq

s«sRÉFL7AS

$T2U&½2/GH

$À9hK/¿/úÀI>�:

', ( = -&% n ± &2- 1 % n (2.17)

Sc*d®¯9:

$(t) = 1 - & + &

2- 1

2 &2- 1

e - (& - &2-1)%nt + & - &

2- 1

2 &2- 1

e - (&+ &2-1)%nt (2.18)

I1º5wJS³$ÑhÕгh?1ºoJI:aBÇI>J7Ii¦�J�¨:7/��®¯9�ÎÏÐI>JS&

?_`RoJå£Ñ+Ò+¸iÓLFRJS5¨B:&iCJÔn_`R>JI:

$

$*

= % n2

s 2+2&%ns+%n2

" 1

1+2&

% n

s

(2.19)

I>�:åI=£���w.I»>sF�vR>JS

~}���$&½2/GH/��®¯

$T#U&¹2/GH

$À9:' = (Ä>J¿/ªÀI>JSo>¦�:

$$$', (ÄÆÄÄÇÄ%�$T#ªÀU ÄÄÄÄ(2.20)

Sc*d®¯9:

$ (t) = 1 - (%nt + 1)e -%nt (2.21)

$7/569ÛÕ2,/ÞIvyw:@`o�T�')'j�'�Ui>R���B\feB·oJDN:®

¯56/ÛvLÞ¯´µIsF�L¨wJSDIxyÖ]1>£Uë1ó/×/®¯9Õ2,/56B>Jv

AB}*µ)Ø/Ù5>£?0«sFLJS

~}��}$&¹2/GH/��®¯

$T�UBÁ&Á2/GH

$À9¿/ú¹Â?�KͧÃĹI>JS

', ( = -&% n ± j 1 - &2

% n (2.22)

Sc*d®¯9:

$(t) = 1 - e - & %nt cos %0t + & %n

% 0

sin % 0t

= 1 - e - & %nt cos %0t + &

1 - &2

sin % 0t

(2.23)

I>JSD�s:

% 0= 1 - &2% n (2.24)

$@Ñ}�}°Õ9ÚÛoJÇÜL?�¦sFLF:7/®¯9ÛÚÛ./ÞB>JS�/KL¹8CJ%{?Û.

/ÝKL¹ÞILAST%�9Þ"ÝKL¹ILASUG©%

{Á%

�I>Ji:«6=B9:ÚÛpTßà.8L

xyáâ>£UixãI>�:4ä./sDLKL¹v��/`BRR>EF�wi�B>JDN8CJIå

æ8`JS

$7/ÚÛ.//jç?�AqsÎPF»vAS~}��°/Äè�éÄè

}éÄè

°Ä/e9:

d$

dt = 1

1 - &2

e -& %nt sin%0t = 0 (2.25)

?¥LFÁN¨wJS7/¥

tn = n*% 0

(2.26)

9®¯Ls/êTA_yU/�ë?ìxJSê/(59:

An = e -

n*&

1 - &2 = e -

*&

1 - &2

n

= A1n

(2.27)

I>JS7wv�:./í9ÛîKïðIBe -

*&

1 - &2ñB>EFLRÞIßxF:&iB/I`B9ÚÛs>R

>JS7/`�8&/7I?ÛÚÛ«¹ÞILA/8CJS

~}��°$./ÚÛ/®¯Ls

T#U#�./KL¹®¯

$òn9#�./KL¹56?ÎPF»vASKL¹569:¶·¸¹/Ø?ó%8Q`�xF:

G(j% ) = % n2

(j%)2+2&%n(j%)+%n

2 = % n

2

% n2 - % 2 2

+ j 2&%n%(2.28)

8ìx¨w:.í9:

G(j%) = % n2

% n2 - % 2 2

+ 4&2% n

2% 2(2.29)

°ô9:

+G(j% ) = - tan-1

2& % n%

% n2 - % 2

(2.30)

I>JS%¹{89Ä õÄ lÑó%ÕÄ õÄiÄ �T{ö³÷øU:%¹X89Ä õÄ lÑó%ÕÄ õÄiÄ {TÇXö³÷øUI>J7I9où¦�

J�!AS89:úI/KL¹89£A>J�!AS

$T2U�Ç}&}ûB/I`:o>¦�:&ü�Ð 2¹{��{�/I`

$õÄlÑó%ÕÄõÄ9Ä%ÄBrsF�ÎÚqI>�:�/569t�/vAB>JS

~}��z$&ü�Ð 2/GH/KL¹®¯

$T#U�Ç}&}½B/I`:o>¦�:&Á�Ð 2¹{��{�/I`

$% = 1-2&2 % n , % p ÄBFÒ'+e

MP = 1

2 & 1 - &2

(2.31)

?½KvAB>JS

~}��è$&Á�Ð 2/GH/KL¹®¯

$7/vAB:Ò'+/CJI`9CJKL¹8VWTUgeUv��´Wi_`R>J7IB>JS5¨B

ýÔi_`R>JI./T«6iþ>¦wFLRS

$ÿ!8ÉDvAB:Û©ª®¯i./=�£A�Þ/"2e9&¹�8CJSMD:ÛKL¹®¯iýÔ?½K

�#�Þ/"2e9$¹�Ð 2Æ{��{�8C�:ãä9%>EFLJ/8_`5wDLSÄ

T�U(�./&�
$~}��C/vA>A/ʱ?�K.iCJI`:7/Sc*d®¯?ÎPvAI�ED¨:ÔnYZ89CJ

i:Å'B�(�L'�éÄ'

}Ä�b?ÎPwyvLS

~}��C$(�./AʱI&�
$DIxy:

G(s) = K

s+'1 s+'2 s+'3 (2.32)

ILA¶·¸¹?�K./Sc*d®¯?:xF»vAS��/DNB:~�éÄ~

}éÄ���éÄ~

�9oPF%>J)8>L

¹8CJIowy:Sc*d®¯/{d{S¬¼lÑØÕÐØ9�/vABÂaa¹B*+8`JS

G(s) = K0

s + K1

s+'1

+ K2

s+'2

+ (2.33)

$�/��®¯9:

y(t) = 1 + e - '1t + e - '2t + (2.34)

I>JS

$7/®¯BKLF�Aqs«sRs¨PF»vAS,-Èp/b¹u.Ä9*+«/?�LJI:

u.ÄÄÆ ÄÄÄÄ0T12A3)yUÄÄ0TØÄ4xy35/AU

ÄÄÄÄ0T12A35/AU0TØÄ4xy3)yU ÄÄ (2.35)

I>JT4689«sRåæsFL>LUSo>¦�u.9:Ø74�8/Ac)y/a8�¨1º8`J/8C

JS

$�±IsF:

G(s) = s+2

(s 2+s+1) (s+5) 5

2

= s+2

(s+1+j

2) (s+

1-j

2) (s+5)

5

2

(2.36)

?IJI:7/I`/A:)y9~}���/vABØ74�Ba8oJS

$Sc*d®¯µÑhÕ9

y(t) = 1 + K1e - 1+j

2 t+ K2e -

1-j

2 t + K3e -5t (2.37)

8CJS7/Äu�éÄu

}éÄu

°/9re?ÁNJB9~}���/�8:;?9��:@Ñ}�°èÕBsDiEF1ºowy

vLS7/GH9:

K1 = K2 = 1.6

1.4-4.4 1-5

2 . 0.65 (2.38)

K3 = 3

4.4-4.4 1-1

2 . 0.08 (2.39)

$o>¦�:u°/e9u

�:u

}B�PF©EIý5R>JS7w9u

}Br®oJAÇèi5/AB�Pxy�¨

©EI;wFLJDN8CJS5¨B:7/p9:ÅÇèh85/pB�P`¦NF�RÚÛoJS7/7I9~

}���/µÑhÕ/©ª®¯�¨�vR¦�JS@Ñ}�°CÕ8èÐÑØÒèÕ/p?<=sDGH/®¯I:Ç>>®¯

I/Ã9å=/¦©�8CJS

$�78:_,E?B./56?:xJI`9:xyB\��LA/»?:x5?<=sF�vLS7/vA

>:xyB\��LA/7I?:&�À:CJL9@ÊAT³AB.�~�hÄ ÄCADÅØUIv�S7/±89:ÇÑ�EóÕ

FG}i&�ÀB>JS%&'()*+,-.89:�K/ͧÃĹ/Hi&�ÀI>JvA015wJ7

IiùLS�sF5/A?<=oJI:./¶·¸¹9#�./�/B>JS¦w¦wi:7wM8#�./

®¯BKLF«sRIËD/Ý�77BCJS

~}���$G(s) = s+2

(s2+s+1) (s+5) 5

2 Ä/A:)y/a8

(a) G(s) = s+2

s2+s+1 1

2 ÄÄÄÄÄÄÄÄÄ(b) G(s) = s+2

(s 2+s+1) (s+5) 5

2

~}���$��®¯Ls/�J

!"#"$%&'()*+,-./0123456

%7789:;<-.=>?@=AB01CDEFGH23456BIJKL-.=9MNOPQ2RS8TUG

VW<XKY2Z[\8Y]ELMNOPQ89^_2`a<bc(d-.>&'()*+,-.ABefCDE

KL

MNOPQ%-.=2XKg7h

>iAjc(d-.gbc(d-.

%-.<9kjc(d-.>lmnopqllmprlostlqAgbc(d-.>rqlunvpqllmprlostlqAgwxK7g9y<Y

IJGL778k72z{2|}BE~����D�]`L

%bc(d-.~G9&'()*+,-.>�nnv��r�prlostlqA<9k������(���<�CD-.��

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 ¡<E�¢k72£¤B¥`¦§�K¨gE`7gwk&'()*+,-./201©2Y28xKL

%9ªH<«UD¬¨Z­®¢Z[ZE29kjc(d-.9¯°CYbc(d-.]±²UDEK³­89

ZEgE`7g8xKLMNON´2,µ(¶2·®¸H-.89k¹º2»¼½B1¾Ck¹º2·®w¿7[

ZE]`Z,µ(¶2À�Á��ÂÁ�2��(¶B1ÃCD¬�k7®BÄ�ÅCZw[-.�KL7®9

jc(d-.8x±kdÆÇ�È-.gE`7gYxKL~GkÉÊ2ËÌ8ÍÎB-.�Kg�kxKÆ(

�(¨[ÅGÍÎB1¾C&'()*+,�K�ÏBgKLg7hwk1¾¼®KÍÎ9�8<ÐUDC~U

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Æ(�(ÚÛ[®Ü³®KL7®9&'()&ÝÞ()-.8xKL72]`<k:;2-.89jc(dg

bc(dBx³ßDÜ`7gwàEL

MNON´%,µ(¶2·®¸H-.>jc(d-.A

ád�¶âãäåæçmèué2�ê2ëì

%4¤<ÔUDkd�¶â2ãäåæwíçmèuéî­��CGg�<k©2ëìwÅï<¥2]`<x[³®

K¨ðJD�]`L

%~°kjc(d-./89k

!Y

Y =

!Gp

Gp

(2.40)

gZ±kd�¶â��9©2~~kÅï��gZKL

%��kbc(d>&'()*+,A-./89k

!Y

Y = 1

1 + GcGpGF

!Gp

Gp

(2.41)

8x±k&'()*+,-.=<]UD��Fñò¼®K7gw³¨KL

ójc(d-.ô óbc(d-.ô

MNONP%d�¶â��2ëì

á���õö<]K��

%Ù<MNONNB÷EDk���õöwøù�Kg�©2ëìwÅï<¥2]`<ú®K¨B�D�]`L��

�õö9EFû¨2üý<X±k©2ëìYþC°ûÿZKL

%~°kjc(d-./89k

Y = GpGcN1 + GpD1 + D2 + GpGcU (2.42)

gZKL

%!P9-.Xï2��8x±"#$WwàEL%Eç

mçr9Æ(�&|�B'û7gwàF©2ëì9(¼EL

)P9*Ð=õö8xKL)

P<+&,+âwx±kç

mw-W|�BYUDEKg�Kg.&/È2Åï901C

DC~`L)N92¾��8xUDkÅï<345EDFKL

%��kbc(d>&'()*+,A-./89k

Y = 1

1 + GcGpGF

GpGcN1 + GpD1 + D2 - GpGcGFN2 + GpGcU

(2.43)

8xKLçr~G9ç

6w-W|�B'UDE®¢k+&,+â2]`Z&/+d72��)

P8p)N<]Kf9��B

:<�K7gw8�KL~GkXï��!P�k&'()*+,-.|;22¾õö!

´Y��F�<�KL

jc(d-. %%%bc(d-.

MNONN%���õö2ëì

>#A-.2=>>?mnpl�prlostlqA2W@

%-.2=9�ACD#g¬±~G9$g¬±xKLBCD9z{BEF<GA�K¯H9x~±ZEwk:

÷I8972JE9»H8xKL

áfK-.>tnLMq�sNloA

%tèséOrlous8xKY2BE`L7789kPQK9x~±ÁF�êCZE28kPQKRS|�9»H89

ZEL��TË�U>?VvW´Aw»HZ�U5QgZKLMNONX<Y�]`ZZ[\)-.w�]®G�UBY

�Y2gCD^[®DEKLdÆ,&-./w_`7®<ab�KL

MNONX%Z[\)p-.

ácd-.>st�r�NoLprlostlqeuntflprlostlqA

%PQKtèséw9<�¶gÈ<�ê�KY2BE`L-.2PQ9hgCDPQKcd|�B]F�K7g

>?VtWiA8x±kMNONi<Y�]`Z\Z)-.wjF¨[^[®DEKLk(l�mnoÈgYE`LC

¨Ckcd-.<¬EDY��TË9bp¯HZqP8x±krs89kÆ*&âk(l/gE`»HZWt

wxKL

MNONi%\Z)-.

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./gE`|}w~U8xK¨[kfK-.gcK-.BgF<GA�K¯H9ZEgY��KL

ádÆÇ�È-.>mtlLt�½½nvprlostlqA

%PQKwx[¨ªHf~UDEKü�k�Z³aktèséOLNfno8xKY2BE`LY`�7C�uz<E`

gkfH[®G��<dUDPQKB�±��Zw[-.PzBä�K.(�¶&-.2]`ZY2Y�~®

KL

%C¨CkPQKwx[¨ªHf~UDEKü�kY`�7C�2�EG-.Y~U8xKLGg�¢kPQ

K2��Kwx[¨ªH³¨K28©®B�÷CG���-.�kvªPQK<]UDvª�ÐB\�YF±

��`a<îyîy-.��w(¼FZUDEF��±�C-.�Z¥wx±k��Z-.�UBY�7gw

^[®DEKL

>$A01��>��3�A

%-.=B01�K<9��B��K¯HwxKL��9~Gk8�xwUG-./B���K3�8Yx

±kf97�g��7�<��KY2<W­K7gw8�KL

áf97�23�

%�f2Xï5�<�CD�WÕÖw��CGg�k¥2F[E2��w�K¨gE`5Q8x±k

lim

t "#

y(t) - r#

r# (2.44)

8f ¼®KLt¡9k5�Ktèsé2ÕÖ¡<¬­KK8x±kGg�¢k

r(s) = r#

s ,

r#

s 2 ,

r#

s 3 , (2.45)

Z¥g¢³CGg�2£æ8xKL

�7�23�

%MNON¤<Y�]`Z�uzZRS$¥9k¦aDw±ÕÖstk+(*(.§(â¨

fk©fÕÖs

ugE`$ª

2|}«B÷ED¢ú�K7gw8�KL¦aDw±ÕÖst9k-.«wk©2PQK2P´¬¨[Q´¬~8<

¨¨KÕÖ8xKL72]`Z­®¯¡Zf B�KB{9k°R±s89RS$¥w*�*�CG±õöB

�y8EG±CD²F<ÕÖ2°RBfH[®Z¨UG±k³´ÕÖ2xK/89k³´ÕÖ9A2æKgC

DµEk¦aDw±ÕÖ<�HGFZEgE`¶Ñ2GH8xKL

%+(*(.§(â¨f9·��¸«gYE³®k/2�f�BY�»HZ5Q8xKL

%©fÕÖsu9k·�w¹$¬<º~Kr»2Õ¼8¢³¼®kGg�¢¹N¬2�<fH[®KL½I�K]

`<k#Ù/89¨f8ps

u8ps

tp2Ö<9å£wxKL

MNON¤%-.�U2��

%7®[2&'()*+,-./|�2��B~gHDk¢<CD�]`L

¢NON%-./2�U5Q

%7®[2K<9P�wx±k�u<k¾]E¾-./2æK9¿w2g¬±8xKL

ÀmpOpPOPpÁpPO¤ >Â9POXF[EA

%½pOpP¤ÃÁpX´Ã >Â9N´ÃF[E8]EA

L½pOpÄpvÅpÁp¡pvÅ >Â9P´vÅF[E8]EA

¨fpOp´pÁpN¤¬ >Gg�¢N´¬g�KA

>!A3Æ?@g&'()*+,?@

%-./2|�ñò<÷E[®K?@HÇ>ȶâÆ(�©2Y2g�UDY]EA<9k-.�É<�CD

3Æ<ÊX�K3Æ?@gk&'()*+,�Ë<ÊX�K&'()*+,?@gwx±¥a[Y]F÷E[

®KL©2ÌÍzZ019½I�KgCDk7789¥2]`ZHÇwxK¨B~gHD¬7`L

MNONÄ%3Æ?@g&'()*+,?@

á3Æ?@

%%�?@

Gc(s) = 1+Ts

1+&Ts (0<&<1) (2.46)

%%×®?@

Gc(s) = 1+Ts

1+&Ts (& >1) (2.47)

MNONÏ%×®?@kÎ�?@25Ð

%¬¬ÑUÒ<E�¢k×®?@BÂ�KgL½B��F�K5Ðwx±kÎ�?@BÂ�Kg%

½B��F8�

KL:;<9ÓaB*Ð�®¢ÔÕ¶Y�êCDC~`¨[©`Ö×<9E¨ZEL

%%\Z)>$�ÐAðØ1

Gc(s) = KP + KI 1s

+ KDs

= KP (1 + 1

TIs + TDs )

(2.48)

gE`ãäåæBYûkE³¢Ù÷23Æ?@HÇ8xKLÚÛ9½I�KL

á&'()*+,?@

%%ÜW?@gm&�()-.wx±k!"#"Ý8ÚI�KL

ádµ&'c�

%PQK2½h<xKãäåæBX®G±k©2ãäåæBdÆÇ�È-.�K>&�Þ§(߶Ç�KAZ

¥2�ÏBgK7gwxKLGg�¢àáâ9k"�<]UD*ãä25�w�³K28k5�K>PQKA

<k"�<]UD�³K&'c�BX®KL

%72]`Z-./9#Ó{�-./2�å8xKLdµ&'c�9k��TËZ¥2bc(d|�<9æF

ëìBç`¼ZE>\YñòCZEA28kèéw¯H8xKL

!"#"!%nÈcoSMB÷EG-./01

%#$æêë8201ìgCDí¢zZY2wknÈcoSM>!Nrîlqupïî�tsA<]K018xKL

%ȶâÆ(�çrèué201B�K2wPz8xKwkðg~°kç

rèuégç

mèuéBðg~gH<CDkçèuég

CDñB��H]`L

pp

MNONò%ȶâÆ(�gd�¶â>ðg~gH<CD��KA

>iAfÀóôgf&óô

%¼DkõÕö&âSM2g7h8IJG]`<kbc(dãä|�

T(s) = G(s)

1+G(s) (2.49)

BEûY1Ã�K29��I÷Z28k�jc(d2ãäåæBðJDkbc(d2|�B�¾�K�gE`

��Zø®<dUD¿wùCCDE7`LxK#$æ'<¬EDk

çèú'épOpûpüpú? >jc(d|�A p (2.50)

>èú'épOpÀnpú& >bc(d|�A pp(2.51)

g�Kgk#$æ¾f�kãäåæýþc<YgÿEG1Ã�1¾<]±kû�?>Yahy#$æ'<]UD

�ê�KA9!"<#HK7gw8�KL

%$èNO¤´é8pèNO¤PéB$èNOiQé<íX�Kgk

M e j & = x + j y

1 + x + j y(2.52)

gZKL7®B¦EDk

x + M2

M2-1

2

+ y 2 = M

M2-1

2

(2.53)

x + 1

2

2

+ y - 1

2 1

tan &

2

= 1

4 tan

2 & + 1

tan2 &

(2.54)

B%KL$èNO¤XéB�fÀóô�k$èNO¤iéB�f&óô�gEEk7®[B&Fg¬]©MNONQ2]`<

ZKL

MNONQ%fÀóôgf&óô

>çèú'éOûüú?p2èû8p?éB��Kgp>èú'éOÀnú&pp2èÀ8p&éwÄHKgE`M8xKLA

%72SMD<kjc(d|�çèú'éOûüú?2óô>'B´¨[¡~8�ê¼ßKAB��kfÀóôkf&ó

ôg2'RBÄ(7g8kbc(d|�>èú'éOÀnú&2���(�w§H[®KkgE`7g<ZKL

>#A)ÈcoSM

%g7hw:;2jc(d|�9kMNONQ2]`<�®E<#H[®K7g9þZEL~GkÔÕ¶*B��

Kgk©2óô9+S¥<�ê�K28EaEa&�Z¬¼ZFD9Z[ZEL72�RB¦,CG2wnÈ

coSM8xKL

%jc(d|�2`ak�u<¾fC��E29·-.ç.gÓa�tLç8xh`L~Gbc(d|�9

>èú'éOÀnú&8xK¨[k.ç.g�tLçB��DÀg&w#H[®KZ[¢7®9/�8xKL©78k.ç.�

�tLç2Ç�&D<fÀóôkf&óôB&EG2wnÈcoSMZ28xKL�Z³aknÈcoSM9k

çèú0épO.ç.npúp�tLç >jc(d|�A (2.55)

B��Dk

%%>èú'éOÀnú& >bc(d|�A p (2.56)

BÄ(gE`|1<ZUDEKL

MNOX´%nÈcoSM

%72D<EUGyjc(d|�B2EDC~�¢k½¨[ÔÕ¶.ç.�Óa�tLçB��K29|�3SB

Dw45<6·À��Kî­8]EL�³Zakçèú'ép Op çrèú'ép ç

mèú'ép g�Kg�k-.=ç

rèú'é25Ð

9k

áÔÕ¶<åCD9k7'<��KRBDw<°[�î­

áÓa<åCD9k7'<��KRB45<°[�î­

gE`Ö×Zå£wxKL

%~Gk723Sw4�KfÀóô2À2K9Àwg±`Kr�K>8·9(,ÀmA8xK7gY³¨KLû

~±k#$æ|�29(,wÖ×<#H[®K28xKL

>$AnÈcoSM<]K01:

%�Î�?@�B:<CDknÈcoSM<]K01BÌÍz<�D�]`L

%GîCk-.=2ÔÕ¶*9d�¶â2�<�Hk

Gp(s) = 2K

s (s+1) (s+2) (2.57)

gCD¬FL

á~°kÁ�;�nfp<å�K��¨[*<PO¤pgZK>72ÚÛ9=>AL©78k*?PO¤2g�2ç

mèú'é2

óôBMNOXP<&ED�KL.ç.O´k�tLç?[Pò´Ã2R9bc(d/w@�f<ZK¾ABZR¾8k

çmèú'é2óô972R<sÿ­GFZELC¨Ckóô9'wP±s2RC872R<¨Z±sÿEDEKLó

ô<4�KfÀ3S2ÀBÄy8�Kg+9<��E¨[k72~~89+9<��Z#$æ|�29(,Àm

BYak*DPO¤gCZEgÀm9(¼F8�ZE7gw³¨KL

MNOXP%nÈcoSM<]K01

%C¨Ck*?PO¤gCG29kÁ�;�BÄϬ¿w<CGEgE`��<dUGGH8xK¨[7®¿D(

¼F98�ZELC¨Ck72~~89x~±<¾ABZR¾<sFkÓaEF%½�ÔÕ¶EFL

½w_gy¥Z

E7g<ZKL

á©78kÎ�?@HÇBÜ`7g<�KLÎ�?@HÇ

Gc(s) = 1+&T s

1+Ts(2.58)

2ÔÕ¶áÓa|�B¬¬ÑUÒ<2ED�KgkMNOXN2]`<ZKL

MNOXN%Î�?@HÇ2ÔÕ¶kÓa2#$æ|�

%778k'pOp'NpÁp'

Pp2GSVW9k�.ç.Bx~±��°k�tLçB5Ú��F°[��kû~±Óaî

­BÎHK�Î�?@�2|�gZUDEKL778k'?PV&>2±s8k¬]©k

�tLpçrèú'épOpi¤Ã[ps�o[PpèPV&éppppp pp(2.59)

2å£wxK7gBÜUDkMNOXP2�rYABZCR>'?P?PV&>±sA2xG±�B5Úi´Ã_¥°

[�7g<�KL$èNO¤QéBi´Ãg¬Fgks�o[PèPV&é?¤Ã]±&?P´Z28k>O´OPF[E<�Kgx

E7g<ZKL>5Úi´Ã_¥°[�gE`29kÀmOPOXF[E<CGEgE`H#¨[�DEKLA

%¿DB~gHKgkÎ�?@HÇ9k

Gc(s) = 1+s

1+ 0.1s(2.60)

gZK28kx[GHDóôB&ED�K>MNOXXAL

%�KgHCEóô9ÀmOPOX]±Y(¼EÀ

m2fÀóô<4�K28kY`þCæÍ2ÔÕ¶B��F�K7

gw8�KL©78kóôBD�<.&âCDÀmOPOX<4�K]`<�KLrIz<k*OPOÏÄgZ±kÁ�

��9��[®G��]±YEEY2<Z±k]E01w8�G7g<ZKL

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��Y��EL

!"#"U%VóôB÷EG-./01

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K�ì8xKL

jc(dãäåæçrèu8p*éç

mèuéB��k*B�ê¼ßk

bc(dãäåæpçrqlunvèu8*épOpç

rèu8p*éç

mèuéVcPüç

rèu8p*éç

mèuéd2 w¥2]`<�F¨B&F

MNOXi%Vóôì2�UDEK7g

>iAVóôì2���

%¼Dk:¤gCDk¿w2]`Z&'()*+,-./B��kÔÕ¶*B�¨CDkbc(d/2 >�

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mèuég

¬7`Lbc(d/2ãäåæ9k

T(s) = G(s)

1+G(s) , GîC%G(s) =

Z(s)

P(s)(2.61)

%¼DkE~ðJGE29k$èNOÄPé2WeB´g¬EG|��Y$

P(s) + K Z(s) = 0 (2.62)

2V8xKL

%*gE`�ê¼ßK«>�Z³a012�ÉgZUDEK���(�A9$èNOÄNé2]`Z¥8S¥z<�

~®®¢]FkGg�¢k

G(s) = K 1+2s

1+10s (2.63)

gE`]`Z×fZ�(&¿�<Yk

G(s) = 4 1+2s

1+Ks (2.64)

Z¥gE`¥8Y]EL(Ch$èNOÄié2]`<k-.=2 �gRB¥2]`<01C]`¨gE`£¤

2�wkú:<9]FxKL

>#AVóô2�Z

%7789k��[®GçèuéBYg<CDk-.=2ÔÕ¶*B01�KkYUgYÖ×Z�(&k�Z³

akPü*çèuéO´p2VóôBðJK7g<C]`L

%~°çèué2Weàq$\èuéBk

P(s) = Kp (s - pi)!i=1

m

(2.65)

>½p9çèué2 2æA

%Whàq$ièuéBk

Z(s) = Kz (s - zj)!j=1

l

(2.66)

>qp9çèué2gR2æA

g¬FL

%~Gk*pjpOpè*kpV*

mé*p>*pjl:AgCD¬FL

%½lq2g�k-.�É9NO<dÆ�(>ustNrsq?pmtlmntA8xKg�Ek½<q2g�dÆ�(>mtlmntA

8xKgE`LdÆ�(gE`29kºBz<éKwxKgE`Y2éK8xKLdÆ�(8ZEgkfmÜ

Ww¯H<Z±��KB2¾�K¯HwتKLEE¨�®¢nÐowS�<ZUDpZ7g<ZKLq2­

<øù�KºB.&/È2ãäåæ9dÆ�(8xKL

%$èNOÄNé<$èNOĤé8pèNOÄÄéBíX�Kgk$èNOÄÏéB%KL

(s - zj)!j=1

l

(s - pi)!i=1

m

= - 1

K'

(2.67)

%778k

s - pi = s - pi e j(i

s - zj = s - zj e j%j

(2.68)

2]`<k kgRgóôD2�RBrst,âcB uQ¢vB�KgkÙ2)û2wx>ÓawxkÔÕ

¶wxAw%[®KL

%áÓawx

%j"j = 1

l

- ( i"i = 1

m

= (2n+1)) (2.69)

%áÔÕ¶wx

(s - zj)!j=1

l

(s - pi)!i=1

m

= 1

K'

(2.70)

%72`akVóôB&FGH<÷EK�Z2_gy¥9Óawx¨[y¨®KLÔÕ¶wx9kóôD27

R8*pjpB#HK>�Z³a*B#HKAg�<î­Ü`L

MNOX¤% kgRgóôD2�RBrst,âc2 uQ¢v

%¿w<Vóô2�Z<ûEDIJDE7`L

%ó�Ziôóô24æ

%%*pj>~G9*AB��Kgk½�ûè½8pqép42óôw%[®KL

%ó�Z#ôzRgIR

%%*pjpOp´2g�k|��Y$2V9\èuépOp´p2Vp>çèué2 A<ZKL

%%*pjpOp¡2g�k|��Y$2V9ièuépOp´p2V>çèué2gRA<ZKL

%%û~±k*pjp{:W¡g�®¢kVóô9kçèué2� ¨[ÅDgR8IKL�

%%>èA�Zik#B��Kgk½lq8xKg�kIRgZKJ�gRw|±ZEL©2ü�>_

%%%%%gy¥72]`Zü�8xKwA9k½[qpª2³}~RBgRîg��KL�Z³aóô

%%%%%9³}~R<01�KLA

%ó�Z$ôóô2�É�

%%%Vóô9:^��8xKL 9:æ£æ2íæ�Y$2V8xK¨[k:V¨8��Çæ

%%>f�æY�(A8xKL©2��8xKVóô9:^�É8xKL

%ó�Z!ô:^D2óô

%%%:^D2GÖ8k©25�<mNpgk

úpw�18�æªxKg�k©2GÖ<9Vóôwø

%%ù�KL

%ó�ZUô�sS

%%%�sS9½p[pqp4øùCDk

pi"i=1

m

- zj"j=1

l

m - l , 0

pBzRgCk��

(2n+1))

m-l 2��7<�wK

%%3S�gZKL

%ó�Z�ô

%%%:^D2�®G#RBÅ0CGVóô9k*2�ÂggY<4sCkûE<9»Z±k¼

%%[<�Ç6I<ÅD·FL72]`ZkVóôw:^B�®KR9»Vwx¨[#H[®KL

%%%�Z³ak

\èuép*ièuéO´ p (2.71)

\pjèuéü*ipjèuéO´pp>D$Bu8ÜWCGY2A pp(2.72)

%%BW¦¼ßD¦­¢*guw#H[®KL

%ó�ZÝô

%%%Vóôw�^B]¸KR9k�f}�<xK2î¨[��&2�f[Aì¨[#H[®KL

>$AVóô<]K01:

%ðgû:¤B�UD�]`L

G(s) = 2

s3 + 3s 2 + 2s(2.73)

<ûEDÌÍz<VóôB&ED�]`L

%ó�Ziôóô24æpOp½�ûpèX8p´épOpX>4A

%ó�Z#ôóô9k

%%%%%%%zRW %>p´8p[P8p[NpA <z~±k

%%%%%%%IRWgR>¡8p¡8p¡pA <IxKL

%ó�ZUô�sS9k

� [�g Opp´p[pPp[pNpOp[X

 æ[gæ OppX

%%%%%%]±kè[P8´épB±kèNoüPé)VXpOp)VX8p)8p¤)VX2��BYû

%ó�Z�ôGÖó[P8p´ô2¥7¨8:^B�®KL»Vwx]±#HD�KL

|��Y$9k uèuüPéèuüNéüN*O´

ÜWCDk XuNüÄuüNO´

72z{BW¦¼ßD¦­¢ks =

-3 ± 3

3 pB%KL

%ó�ZÝô�^B]¸KR9k��&2�f[Aì8#HKL

%p\èuépüp*ièuépOpuXpüpXuNpüpNupüpN*pOp´

uX ppP pppppppN

uN ppX ppppppN*

uP ppÄ[N*

u´ ppN*

%%%72]`<��&2�ì<]Kgk�fwx9k*l´8pÄ[N*l´gZKLdUD�f}�9

%%*OX8xKL72g�k|��Y$9uXüXuNüNuüÄO´gZK¨[k 9uO¹ú�NpgZ±k/9

%%�f}�<xK7gw³¨KL

%¿D]±kÙ2VóôB%KL

MNOXÏ%Vóô2:

>!A-.=201

%¼DkVóôB&EGî­89\<YZ[ZE28xUDk-.=B01CZ­®¢Z[ZEL72ü�

Ykí¢V2���<]±k#Ù/2 �]~8B��®¢�W8xKL7789k´D*Di2�(&B�E

�KL72�T<01�K7gwàEGH8xKL)û2V9k

s = -*' n ± j ' n 1 - *2

(2.74)

8x±k&/+dRS

y(t) = 1 - e -*'nt

1 - *2

sin ' n 1 - *2

t + cos-1 * (2.75)

B&FgMNOXò2]`<ZKL

MNOXò%&/+dRS

á+(*(.§(â

%´D*Di2ü�k#Ù/2&/+dRS9¯°+(*(.§(âBتKLv?èséVvspOp´p]±k

t = )

' n 1 - *2

(2.76)

8r�KBYaki]±9�Å�W2��¼k�Z³a+(*(.§(â9k

Ov = e -

) *

1 - *2 (2.77)

8xKL7®BMY�®¢MNOXQ2]`<Z±k*g¨f2å£w³¨KL

MNOXQ%*g¨f2å£

% �]g�<fæ2å£9¥`ZUDEK2îh`L s = -*'n ± j 'n 1 - *2>´D*DP2ü�AB�Ç6

I<MY�KgMNOi´2]`<ZKL)û2 9'oB¯�g�K�#D<Ó]C>dUDk�2��¼wR

SÁ�<å£�KAk:^gZ���%w�<£æ<�R�KL

MNOi´%#Ù/2 �]

%¿DB~gHKgkw2¢B%KL

¢NOX%��gCD��K+(*(.§(â¨fg*p8p%2å£

ásu>©fÕÖA

%&/+dRS2��SB��D�Kgk

y(t) - y# + e - *'nt

1 - *2

+ $ (t , ts) (2.78)

gZKLswsuB��KgkrIK2$�¬�¿�<¬¼~±Y`)�g�<9ÅZEgE`7gB¢CDEK2

8ksu27gB©fÕÖg�sL

%$èNOÏòéB�¥CDk

*'nt , log 1$

+ 1

2 log 1

1 - *2

(2.79)

gZKL$pOp´O´NgCk´D*D´OÏÄpF[Eg�Kgk

*'nt , 3.9 + 1

2 log 1

1 - 0.762

= 4.3 (2.80)

gZK28k

*'n , 4.3

ts

(2.81)

Y�wP�gZKL*'o9 2:V2��¼B¢³�L©fÕÖ9©2 æB!¡CGY�8xKgE`7g

8xKL

ást>¦aD±ÕÖA

%°R¨[ w9Z®DEK�w¢EL~GkdÆ,&-.Z¥89k�7C+(*(.§(âwà�H2�

w¦aDw±ÕÖw¢E28£~®KL

á:^D2gR

%:^>¤AD2gR9+(*(.§(âB�¼ßK�¥<xKwk�^¨[�W�®DE®¢k_gy¥

ëì9ZEL5¯I2gR9@�fgRg�¢®K7gYx±k&/+dRSwkPQKwXï¼®G3½<

 �¥<RS�K>C¢[F�Kg��¥<ÔK28kRS2¦¥9x~±�³[ZEAgE`|ÿZúÉB

Y�7gwxKL7®B RSg�y8EKL-./<¬­KgR2�Z9x~±]F¦E¼®DEK³­8

9ZFkúùY§¨®D<xKL

!"#"$%&'()*+

%,-./0123456789:;<=4;>?@)*+2AB=>C:?DEFGHIJKLMNOP5QR

PS9Q<TUVWXYZ[\>TUVW?]^DE5_R`abcdefdghRi?DE?01234:

;<=4jklDEmnopqrZst3k?@5LuvwxLuKyz{|?}2NO>o5~={�Z7_N

Z)*+@�:�=4;kl}?}�@��U������:;k)*+_<R95��eR;3@�e�:

;k)*+o��NO�kl~?|5����?st227Z5�����:;k)*+oO<`=>��Z

RP5��N@���UTUV���:;k7?o5TU������ ¡U¢�G&£¤¥£¦§§¦¨©ª«

¬¤­®£¤©©ª£¯&¬M2AB=4�9:;<=4;>l

HIJKL%��°&'()*+

G±M&'()*+?./²³

%)*+?./01234?´tµ¶@5·¸rZ@¹?º°N»¼>l

Gc(s) = Kp + KI 1s

+ KDs = Kp 1 + 1

TIs + TDs (2.82)

%}?./²³�½>25)*+¾fo¿ÀZÁÂ_>ÃÄ5¿À?ÅÄZÁÂ_>ÃÄ5¿À?ÆÄZÁÂ

_>ÃÄ?ÇÈ?ÉNO>}2o`Q>l}?ÇÈ?²³�5Á²³5ÅIJ³5ÆIJ³2�Êl~3

45}?ÇÈ?²³�QË~RÌk)*+�5�9Ͳ³)*+5O>;@&'(G&£¤Î¤£®Ï¤­¦©Ð« '­®ª¥£¦©« ¦­Ñ«

(ª£ÏÒ¦®ÏÒª?ÓM)*+2�Êl

%Ôk5}?�a?ÕÈÔk@ÖÈ?²³?½�7È)*+o×Ì<=>l~?´tµ¶@{Ø?��ZR

>l

&«)*+ Gc(s) = Kp

'«)*+Gc(s) =

Kp

TI s

&'«)*+Gc(s) = Kp 1 + 1

TIs

&(«)*+ Gc(s) = Kp 1 + TDs

%²³?¶oÙR;Úi)*+?ÛÜ@ÝÞ2R>?N5ßZͲ³)*+�:;>�P5DEàáZâã4

äåR7?�:;>?oæçNO>l}?èé?êë@|ì_>lR[5)*+2;�Aíî�3R;N5Þ

ZDEF2AïN7�;l

%�45ðÎ5ñ

'«5ñ

(«@¹?��Z�B=>l

%%LòðίÁÂóGΣ¤Î¤£®Ï¤­¦©«¨¦­ÑM2;;ôÄõ»ö_>?o÷øNO>l}?�íî@)*+

%%%%?ÛÜQ<È\<=k�íîNO>l3Q35mnù@ú3�ðÎ�ûPü�îo8;?N5

%%%%pýN@ðÎGÁÂþ� M�:;>l

%««««ñ'«¯««ÅÄÿ!GÏ­®ª¥£¦®Ï¤­«®Ï§ªMÔk@"V�¡ÿ!G£ª#ª®«®Ï§ªM

%««««ñ(¯««ÆÄÿ!GÑÏ$$ª£ª­®Ï¦®Ï¤­«®Ï§ªMÔk@%¢¡ÿ!G£¦®ª«®Ï§ªM

%&'Z@ÆÄ?(@)*+N,ßZ-jR.2R>?N5)*+/�0�12_>k35

TDs

1+! s , k435%! " TD 10 (2.83)

2;�°���}2o8;lñ(ò!�ÆÄþ� 2AÊ}2oO>lð

(N@R;?N56�=k3l}=@5W

7�T8?9:.Zà345ÆIJ³Z��4®;y?,|Z<ã>DE=>?5Á²³Z�>DE=>?Ä

Zà_>Á2R�4;>l

G#M)*+?èé

%DEàáo@Ì<=>25i?��R)*+�:;>AjQoÔBCDZR>l7a�ïi?��REFN

75&'()*+�:;>?o./?GH2;�IQ<@JÔ3;?NO>o5DEàáZ��4@±²³R;

3@#²³3Q7kR;)*+N7OÔPGHoK`<R;}2oO>l~?2j@5&'()*+�:;>L

`PZ�PÝÞR)*+�:;>Ú�o5M:7ÙR94_ú35N�O¢P?¶oÙR;?N)*oQRN

O>l

%i?��Rèé�S�AjQ5DEàá?´tµ¶o°TIJUKVR;3@°TIJUwV?ºN·¸rZ»�Nj

>23k2j?èéêë�HIJKIZö_l89?TUVW?´tµ¶@}?��R·¸oWXNO>?N5H

IJKI?��Rèéêë?Y:Z.@-j;l

Gp(s) = e -Ds Kp

(1+T1s) (1+T2s)(2.84)

Gp(s) = e -Ds Kp

1+T1s (2.85)

HIJKI%&'()*+?èéêë

%}?HQ<`Q>��Z5&'()*+oGH�s[_>?@Á\r];^_NO�45±²³R;3#²³

?½?)*+N`ÄR}2o8;lÝÞZab�cÌ4[}�l

%d¯}?^_@5efÿ!(Z��4TUVW?ghoij�=>EFNO>lefÿ!?./@úBQ3

95Ͳ³_A4:;47GH@R;lÔk5}?EF�¢Tþ� �ùk>2lomnop3q_;?N5

oß¿À�r2_>Z@'²³os0ZR>l~=tÌ')*+ouv�=>l

%w¯}?^_@5dZÁx>2efÿ!?GHoÁ\ry�95±¹z=5#¹z=l234?gho{9

R>l~=tÌ5./od�PqPq_95'²³Z&²³�cÌ>2GHoQRPOo>l

%|¯}?^_N@5TUVW@Ú2ïi±¹z=?gh�7È?N5&²³?½N7ÁÂþ� �`Ä)9

2=B5noc�}R�}2R9oß¿À�y�935â~�c�Ok>}2oNj>l

%�¯}?^_N@5TUVW@#¹z=Z8Ùefÿ!oc`�kgh2R>l'²³?½N@â~o[~

95&²³?½N@ÁÂþ� �OÔPùk<=R;k3oß¿Ào-j9R>lÔk5&'²³N@â~�c

oOo<R;l}?EF@5&'(Ͳ³�Ãos0ZR>l

%�¯}?^_@5efÿ!?��@Ú2ïiR95TUVW@#¹z=l2×Ì<=>l~=tÌ5&(²

³5_R`a�½gh?./Z��45noc�~}R�}2R9`Ä);ÁÂþ� oæ<=>?N'²³?

s0h@y�;l

GÍM&'(N�O¢P?�o�

%;�kï)*+o@Ì<=>25~?N�O¢P�i�èo_>QoCDZR>lN�O¢Pèo?êë@

;�;�R�Z��4���=4;>o567�9�<=4;>?o��¢��¢G�Ϫ¥©ª£M2���0

G�Ï��¤©#M?���c�2��â~�NO>l

d���c�

%)*+�&²³4\23Gñ'�6-5ñ

(�y2_=B�;M5~?ÁÂþ� ð

Î�KÌ45DEl�no��

ZÔN7�4;9l}?8�NlZW7�T�f�@Ì>25¾f@Õo��N�²���_>l}?2j?

ÁÂþ� ?.�ð�5�²*��ñ

�2_>l_>25)*+?�N�O¢P@»IJK?��ZèB=>l

»IJK%���c�Z�>N�O¢Pèoêë

%

%}?î�@5DEàá?ghoÚ efÿ!2±¹z=N·¸Nj>235l?��â~?Sj_¡ðÒo

Iw¢£c2R>��ZN�O¢P�èï47?NO>l}?î�?YI@,¤DEàá?gh�¥3R92

75¦�¢§¨��DEl�È9Pùkk|N?©ª«HQ<5N�O¢P?.��3�>}2NO>l3Q

35¬­5l�;�kïno��ZÔN7�4;9?N5©ª�@DEàá?8�o-j9®�=43Ô�l

%3ko�45}?2j@DEàá@øß?¯Y�SÌR;}2o8;lk2ÌB5O>¬â�SR�p°T

UVWN?bcDElN@5}?��R©ª�S�2bco-j9K`>?N5¬â?�So±²�=43Ô

�l~}N5}?EF@©ª?'@¯Y��³_>Q5O>;@¯Y�SR�47~?!?¯Y«H@}´2

34ûPü�Q?;B=Q2¼µ>�ÌR;l}=o5}?î�?-jR¶INO>l

%·?¶I234Ok<=>?@53B3B5no��o@�jP3R;}2NO>ll�no��Z��4

;}�2_>25Õ¸Z-��²³2R�4DEàá?,¹º²³º»Z��43Ô�4;kP5¯³hZ¼

ÉoO�kP_>25Ú Õo��N�²_>��Râ~2RPoaNO>lDEF?þ� �KÌ47��

oK`>£cNs½343Ô���R}2@R;?Nno���o3>}2@6¾2¿3;}2o8;l

d��â~�

%��â~�@5DEàá?´tµ¶o·¸rZ°TIJUÀVO>;@°TIJUÁVN»�=>23k2j?N�

O¢Pèo�N5}?èoêë�»wJKZö_l

Gp(s) = e -Ds 1

1+T s (2.86)

Gp(s) = e -Ds 1

T s (2.87)

%��¢��¢2���0@5}?(2ñ�¥3>k3ZDEàá?��â~�Âo_>}2���23k?N

}?ÃoO>l}?EF75NjOo�kl?��â~?ÄÒoÚ Iw¢ZR>��N�O¢PoèB=4;

>l��â~?Âo@5-��NS�s0@R95Âo«HoÅÆZÇ7=R;£c?y��NS�kP5¶

�?Âo�ÈËF`¼4ÅÆ?���y�93kP_>Ri?XÉ�_>2�;l

»IJw%��â~�Z�>N�O¢Pèoêë

%}?��Z5)*+?N�O¢P?èíîZ@;�;�?î�o×Ì<=>l3Q35;B=?î�Z34

75Ê<Q?ºNDEàá?gh�¥3R;2ä:NjR;l2}�o5&'?p°XY5ËÌXYRi?T

UVWN@5}?��Rgh�¥3>k3?©ªoTUVW?¯Y8��®_k35&SNjR;}2o,ß

Z8;lÔk5DEàá?gh7ßZÕoN@R95ÿ!rKp�Í�}27O>?N5k2Ì;�kïgh

�Âo3475~?«Ho;È7=:Nj>2@Q¡<R;l

%~?k35}=<?N�O¢P@~=ÔN?ΪZêÏ;4äåZèo35DEl�È9Pùk5~?DE

l?DE8�?ÐÑ�ÒRo<5�!oäåZN�O¢P�©SÓ¿rZKÌ4;92;�ÔÕrRÖ�o�

92<=>l}?��R)×Z@ØÙ�0_>}27ÙR9R956·N@�X�X5¦ÚÛ¢unRi?Ö

Ü�2>}27ÙR9R;l

%Ôk5DEàáNO>TUVW?¯Y8��OÔP®_}2R9~?gh�Âo_>Ö�GÝW7Þßo�

2AB=>MZµ_>àáâs7ãïZR�=4;>l

dÃÄräå���æ �

%}}N5çèZ��4���=kÃÄräå���æ ��ZÈ;4ìA4[}�l~?é+êëZ@-K

ìíî;7?oO>l

%ÔB5ùÖZé+�=kDEl?ï�¢Tgh5ÈÔP9:.Q<¾f.ÔN?JÔ3;´tµ¶ðT#V

@5{Ø?��Z»��=>52×Ì>l

W(s) = 1

#0 + #1$s + #2$2s 2+

(2.88)

%}=�Äñl-»�2ò�l}}Z5#@â~?���»35ó¶?l-¦Ï@â~?º�»`_lçè?äå

�N@5

#0, #1, #2, #3, #4, = 1, 1, 0.5, 0.15, 0.03, (2.89)

2;�l-N@Ì<=>l}=�KÌ=Bô?â~+º2R>lHIJKvZö_��Z5çèäå�@Á\r9

õÈö¢¨¢Ý÷¢¡oÒ<=>k35k2ÌB5Uø�¡DERi?â~äå�Z@�ÌR;lGù?,�

N�¢Þ�³3k;2j5Õ�ù�ú34Q<oû�=@ü>lM3Q35TUVWDERiN@5}?9<

;ö¢¨¢Ý÷¢¡oO>îo5õùPÿ!oý;?NþÔ=>l

%ìíî;?@5°TIJUuV@« ¦yЫ ¦

LЫ ¦

I9<;ÔNoÈ0N5¦

v{ù?)¹?Ä?ó¶@4ï4ïÈ0NR9

R>2;�}2NO>l

%ßÿZ345DEàá7Äñl-N»�_>l

Gp(s) = b0 + b1s + b2s 2 +

a0 + a1s + a2s2 + = 1

a0' + a1

' s + a2' s2 +

(2.90)

%DEF7Ôk5

Gc(s) = + c-1

s + c0 + c1s + = 1

c0' + c1

' s + c2' s 2 +

(2.91)

2;���Z5¶!?N�O¢P���}2Z_>lÍ!4\���}2Z_=B&'(DEFZàâ_>l&'(

DEF@¼;";Í!£c?N�O¢PN»�Nj>?NO>Q<5äå�7DEàá7Í!£c?N�O¢

PN»�_=B`ÄNO>2×Ì>l

%{ùZ��45Äñl-»��=kðT#V2#ÎT#Voæ<=>?N5#

�T#V�þjR¹¶Zèí5

W(s) = Gc(s)Gp(s)

1 + Gc(s)Gp(s) (2.92)

ZL�35$3?Í(�%&34'%°234(9?NO>lG�)Í(3Q+*3R;?NÃÄräå�

���52;�lM

%}?+*@­+NO>o5k4,f2-�oO=B(9}2oNj45JJJJ«�.LЫ�

y�

LЫJJJJoæ<=>l

HIJKv%çèäå�?â~+º

d/0?8(°

%ùÖZé+�=kDEl?gho�ÈAjºZ@5çè?8(°{¾Z7;�;�R7?o×Ì<=

4;>l/0?8(°7~?12ÈNOP52W�¢Gðª##©ª£M?:ëº234�<=kº�343k7?N

O>l/0?î�@5+¶H�2�B=>k;xï5ÕrRDElé+�NO>o5}}N@~?ÕÃ�2P

Ok5mërR8(°?º4\�ab_>}2Z_>l

%;ÔÄñ8(°�5

««««« P(s) = 6ai si (2.93)

235¹°N%Ï�o7_>l

% i = ai2

ai-1ai+1 (2.94)

%}}N5/0?@Ì>8�;88(°2@5

% 1 = 2.5, % 2=%3=...=2 (2.95)

�9k_8(°NO>l}?º@-K[ Ìq_;lÔk5r¹?8(°@5:�V�G;ª##ª©M8(°Z·

9R>l

! = a1 a0 (2.96)

@%Zÿo¶2ò`=>l735lo±¹NO=B®@på?ÿo¶NO>l

%no<=@5

% i > 1.5% i* (2.97)

k435% i

*= 1% i+1

+ 1% i-1

N@Ì<=>l/08(°�!;L234>?rZ+*34½>25

P(s) = 1+s+0.5 s2+0.125 s3+0.0156 s4+0.001 s5 (2.98)

2R�4;>l°TIJuUV@r;¹¶Q<È0NOP5@Äoé+ÿZA93k;¹¶ÔN�ÌB�;l);

îÔNB,ZF`¼>}2@mWXN7O>35°TIJuÁV?no<=�Ì9k34;=B~=ÚiÈ0N@

R;l

!"#"C%¦�¢§¨��./

%¦�¢§¨��./@5DEl?�?¦�¢§¨���Z./���D�34l?gh�3E3��2;�

7?NO>l}?EF5DEàá?ghZ�P5Ôk5l?ÛÜZ��4;�;�R./��o×Ì<=>

o5�9:;<=>Â�ÖÈBQPZ¹Zö~�l

G±MÆÄFG

%HIJKK@5}=ÔNZ73B3B:;4jk�¢øHÛ?�U��HNO>l}?l?I¹Ã�I;k�¾

f!?gh@5

&

&*

=

A1A 2Km

s(1+Tms)

1 + A1A2Km

s(1+Tms)

= A1A 2Km

Tms2+ s + A1A2Km

=

A1A2Km

Tm

s 2 + 1

Tm

s + A1A2Km

Tm

= ' n2

s 2+2('ns+'n2

(2.99)

k435

' n = A1A2Km

Tm

(2.100)

( = 1

2

1

A 1A2Km

1

Tm

(2.101)

2R�4;>l

HIJKK%�¢øHÛ

%}}N5â~o�²rZR<R;��Z52Jo¶(�Õo.(yZèÊ25°TIJLyyVЫ TIJLyLVQ<5K

L*+¶@5

' n = 1

2(0Tm

(2.102)

l?�¢Tþ� @5

A1A2Km = 1

4(02Tm

(2.103)

25;B=7ñ§Z��4D��=43Ô�}2o`Q>l~}N5}?l?gh�3E_>67GHrRî�

@ñ§�r2_>}2NO�45}=�&GrZSR�?oÆÄFGNO>l

%}?î�@5MmrZ@5HIJKw?I¹?��Z¾f?ÆÄZÁÂ3k=>�2P434¦�¢§¨��_

>?NO>o5&'Z@�²HNZ,«3k�cs�HGP�O¢PM�:;>lP�O¢P?¾f�e@5

�O�c5_R`aÑ&«òÑ®ZÁÂ_>k3NO>l

HIJKw%ÆÄ./

%HIJKwNÆÄFG�:;kEF5P�F?¾fÒLQ<&ÔN?´tµ¶�¥34½��l

&(s)

v1(s) =

A2Km

s(1+Tms)

1 + A2KmKts

s(1+Tms)

= A

s(1+Tm's)

(2.104)

}}Z5

A = A2Km

1 + A2KmKt (2.105)

Tm '

= Tm

1 + A 2KmKt (2.106)

%_R`a5}?2j@ùÐQ<`Q>��Z5ÿo¶ñ§@LòTLQR

Ið§ð®VSZ2Ù_>ll?KL*+¶@

ùoP5y�Rä¢PZÒÌ>��ZR>`\NO>lþ� 72Ù_>o5}=@P�FL?þ� RL�-

j9_>}2Z�P./Nj>l

%°TIJLyÀV�½>25ñ§T« �;9<N7y�9Nj>��Zê`=>o5&'Z@�²H?ghoU�Z@

¹ºNR;}25~?·?mVNO>£c{ù@y�9NjR;l3Q35ÆÄFG@�¢øHÛ?gh3E

Z@ÝÞNLGRÖÜNO�45W9:;<=4jkl

%ä¢P?EF5&Zà3~?ÆÄNO>�O�c@P�O¢PN,¤æ<=>?N5&'û&?+Â.Q<

ÆÄZ��4¥3>s0@R;l~}N5P�O¢Pd¦�¢§¨��27AB=>lk456·@��Ý

G,gFM�:;kP�O¢P@X`=>YZZOP5L`PZ[°°?Ý\¦¡] �¢^oþïN:;<

=>l~?EF@5û&_`?ÆÄGÀÄM�+*_>s0oO>l

G#MaW2¢§DE2���¢DE�¢T

%;Ô5HIJKÀ?��Rj`34ÞbR±¹z=?àáoO>2_>l}=�P�F�:;4¦�¢§¨��

�Q\4q>25cT®VQ<dT®VÔN?´tµ¶@¹?��ZK`>l

y(s)

x(s) =

K1K

1+)s

1+ K1K

1+)s

= K'

1+)'s

(2.107)

k43

K' = K1K

1+K1K (2.108)

)' =

)

1+K1K (2.109)

%_R`a5¦�¢§¨��Z��4àá?ÿo¶@b<QZ2Ù_>}2o`Q>l

HIJKÀ%���¢DE�¢T?GH

%Ôkàá?þ� ðoKp35ðQ*ðZR�k2_>25àâ_>ðT?Kp@5°TwJUwVQ<5

*K' +

,K'

,K *K + K1

( 1+K1K)2

*K

- *K'

K'

+ 1

1+K1K *K

K

(2.110)

2R>l

%}?°Q<`Q>��Z5¦�¢§¨���S�25þ� ?Kpõ72Ù_>l}?��Rhe@_NZ

ìAk}2NO�45¦�¢§¨��?7È_f=kgg?ÕÈNO>l

%�45¦�¢§¨��Z��4ÿo¶qþ� ?Kpõo27Z2Ù_>he@5¦�¢§¨��l?gh

./Z:;>}2oNj>lk2ÌB5HIJKÀ?��Z5ÕÈ?Ãh?bcDE�bi?bc?)*Z��4

S�23��l}?2j5i@jQ<k�ø�4ÃhZg=lúo5m¢Px?=>Q<Ãh�nbcÔN?

ÿ!z=@-j;35i?g=îZ�P~?!?þ� @5ÿZ�PQRPKp_>l~?k3573}?Ô

ÔÃh?bcñy�o¾35pobcñq2?ÀZ�Pm¢P�²Q_2;�øß?¦�¢§¨��DE�S�

25nocZQRPrs�2�4DEl�é+3R94@R<R;lÔk57272m¢P²³=>Q<H?

ñyÔN?!ZQRP?z=oO>l}?��RÖÈ?mVQ<l?â~�c@QRPz9R�43Ô�l

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