制御工学第1 －平成xx年度－ 電気工学科 堀 洋一，...
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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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�[*+<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$¦)½¾�E�;�8�;/�3z¼;()9~¨©2��J;C�3�±
Hª,2EF�±HDÔ�C:6:-@/G0)012()ª,1^>2;i^=@)D2C=Gà0D
456789:&'()Êï;ElÑ>230)aÛæY1Ù�ÚæY-<çCÈ;=@01�3()�ª,
-x;;�F1=jg+��@¶B;CÑ>�F)DEFG�Cßí3!þ3ÿ
þ2*+,)æô)KLDEF
�3à�1[>;3()�ª,)x?-`�z/)KLD@E�3à)g+D@ABDEFG
%Ù�ÚæY3aÛæY23(6£õ�U���L�V3Ù�ÚµaÛõ�U���FÉV��@3��;@�1F
01�D�FG
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!"#$%&'()*+,-./01
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$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-.��
B��8�KgE`�RwxKL��k&'()*+,-./2��R9�f�B��C��E7g8xKL
¡<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
GÍ2ÑÒ8xK¨[kÓW2Æ(�(<ÔCG289ÕÖ×®wتKL©78k1¾CGÍÎ9kÙ2
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
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lim
t "#
y(t) - r#
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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
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DµEk¦aDw±ÕÖ<�HGFZEgE`¶Ñ2GH8xKL
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u8ps
tp2Ö<9å£wxKL
MNON¤%-.�U2��
%7®[2&'()*+,-./|�2��B~gHDk¢<CD�]`L
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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
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Gc(s) = KP + KI 1s
+ KDs
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(2.48)
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á&'()*+,?@
%%ÜW?@gm&�()-.wx±k!"#"Ý8ÚI�KL
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!"#"!%nÈcoSMB÷EG-./01
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T(s) = G(s)
1+G(s) (2.49)
BEûY1Ã�K29��I÷Z28k�jc(d2ãäåæBðJDkbc(d2|�B�¾�K�gE`
��Zø®<dUD¿wùCCDE7`LxK#$æ'<¬EDk
çèú'épOpûpüpú? >jc(d|�A p (2.50)
>èú'épOpÀnpú& >bc(d|�A pp(2.51)
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%$èNO¤´é8pèNO¤PéB$èNOiQé<íX�Kgk
M e j & = x + j y
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gZKL7®B¦EDk
x + M2
M2-1
2
+ y 2 = M
M2-1
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(2.53)
x + 1
2
2
+ y - 1
2 1
tan &
2
= 1
4 tan
2 & + 1
tan2 &
(2.54)
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ZKL
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çèú0épO.ç.npúp�tLç >jc(d|�A (2.55)
B��Dk
%%>èú'éOÀnú& >bc(d|�A p (2.56)
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Gp(s) = 2K
s (s+1) (s+2) (2.57)
gCD¬FL
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Gc(s) = 1+&T s
1+Ts(2.58)
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%¿DB~gHKgkÎ�?@HÇ9k
Gc(s) = 1+s
1+ 0.1s(2.60)
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mèuég
¬7`Lbc(d/2ãäåæ9k
T(s) = G(s)
1+G(s) , GîC%G(s) =
Z(s)
P(s)(2.61)
%¼DkE~ðJGE29k$èNOÄPé2WeB´g¬EG|��Y$
P(s) + K Z(s) = 0 (2.62)
2V8xKL
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~®®¢]FkGg�¢k
G(s) = K 1+2s
1+10s (2.63)
gE`]`Z×fZ�(&¿�<Yk
G(s) = 4 1+2s
1+Ks (2.64)
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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
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8xKgE`LdÆ�(gE`29kºBz<éKwxKgE`Y2éK8xKLdÆ�(8ZEgkfmÜ
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<øù�KºB.&/È2ãäåæ9dÆ�(8xKL
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(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)
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%%%�Z³ak
\èuép*ièuéO´ p (2.71)
\pjèuéü*ipjèuéO´pp>D$Bu8ÜWCGY2A pp(2.72)
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G(s) = 2
s3 + 3s 2 + 2s(2.73)
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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
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t = )
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(2.76)
8r�KBYaki]±9�Å�W2��¼k�Z³a+(*(.§(â9k
Ov = e -
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MNOXQ%*g¨f2å£
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%&/+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�
��æ �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
HIJKÁ%Ãh?bcDE
%~}N}?2j@5HIJKÁ?��Z5m¢PQ<Ãh�n?i?bcÔN?ÃÄ�2P¾35U¢a�R
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