Chpter Mthemticl Model of Sytem 重点掌握 : 微分方程传递函数系统结构图及信号流图梅逊公式
Min content Differentil Eqution of Phyicl Sytem. The Lplce Trnform nd Invere Trnform. The Trnfer function of Liner Sytem. Block Digrm nd Block Digrm Reduction. Signl-flow Grph nd Mon gin formul
Definition of Mthemticl model of ytem Mthemticl model:decription of the behvior of ytem uing mthemtic. 描述系统的输入 输出变量以及系统内部各个变量之间的数学表达式 3
Type of mthemticl model Differentil Eqution Trnfer Function 3 Frequency Repone 4 Stte Eqution 5 Difference Eqution 4
Differentil Eqution of Phyicl Sytem How to get the differentil eqution of phyicl ytem? The differentil eqution decribing the dynmic performnce of phyicl ytem re obtined by utilizing the phyicl lw of the proce. Step: 确定系统中各元件的输入 输出变量 Step: 按信号传递顺序列写微分方程 Step3: 化简 ( 线性化 消去中间变量, 写出输入 输出变量间的数学表达式 5
Differentil Eqution for Idel Element ( Electricl Reitnce i R U i U R 6
( Electricl Cpcitnce i C U i C du dt (3Electricl Inductnce i L U U L di dt (4 M block F v M F M dv dt
(5 Spring F x k x F k x x ( (6 Dmper v v F b( v v F b
exmple r(t Exmple :RLC circuit R L i(t di( t C c(t r ( t Ri( t L c( t i ( t C dc ( t dt dt d c( t dc( t LC RC c( t dt dt r( t 9
Exmple :m-pring-dmper b k by ky M y M y r(t r(t d y( t dy M b ky( t dt dt r( t
Liner Approximtion of Phyicl Sytem Wht i the liner ytem? A liner ytem tifie the propertie of uperpoition nd Homogeneity: (Principle of Superpoition. 满足叠加原理的系统称为线性系统 叠加原理又可分为可加性和齐次性
Principle of uperpoition Superpoition Property r r ytem y y ytem r ytem r y y Homogeneity Property r ytem y r ytem y
Exmple ( y ( (3 kx y kx y x b Doe not tify the homogeneity property Doe not tify the uperpoition property When x x 0 Δx nd y y Δ y 0 Eqution ( cn be rewritten y Δ y kx k Δ x 0 0 b We hve Δ y k Δ x or y kx
Lineriztion of Wek Nonliner Chrcteritic 4
Lineriztion uing Tylor erie expnion bout the operting point( Equilibrium Poition The output-input nonliner chrcteritic of yf(x i illutrted in the following figure: y df f (x dx x y0 Δy 0 Δy kδx y 0 A x0 x 0 Δx x 5
y f x df dt So we get: ( ( Δx Set Δyf(x-f(x 0, o we hve f x 0 x 0 Set Δ y df dx df dx x 0 x 0 Δ We get ΔykΔx x k y df f (x dx y0 Δy x0 Δy kδx y 0 A Or ykx x0 x 0 Δx x
The Lplce Trnform Definition If function of time,f(t,tify 0 f σ t ( t e dt < We hve the Lplce trnformtion for function f(t,i F ( f ( t e 0 t dt L{ f ( t } 7
The Lplce vrible cn be conidered to be differentil opertor o tht,we hve d dt t dt 0
Importnt Lplce Trnform Pir L ] ( [ t L n! n [ t ] n L[ e ] t L[in ωt] L[co ωt] ω ω ω 9
Invere Lplce Trnformtion Invere Lplce trnformtion cn be denoted f ( t invere Lplce trnformtion of F( L [ F( ] or f ( t c j F( e j c j π t d 0
( linerity Importnt Theorem of Lplce Trnform [ ( ] ( ( L ± ± k f ( t k f t k F k F ( differentition d L[ k f ( t ] k dt k F( f (0 f k ( k (0
(3 Shife in Time L [ ] T f ( t T ( t T e F( (4 Complex Shifting [ ] e f ( t F( L t ± (5Initil-Vlue Theorem lim t 0 f ( t lim F( (6 Finl-Vlue Theorem lim t f ( t lim F( 0 IF F( doe not hve pole on or to the right of the imginry xi in the -plne.
Solve the differentil eqution uing the Lplce trnform Exmple d y ( t dy M b ky ( t r ( t dt dt The Lplce trnform of the eqution i M [ Y( y(0 y (0] b[ Y( y(0] ky( R( when r ( t 0, y ( 0 y, 0 y ( 0 0 3
We cn get ( M b y0 Y ( M b k p( q( when, b/ M 3, y 0 k M / Then Y( become 3 Y ( ( ( The invere Lplce trnform of Eq.(i ( y ( t e t e t
Exmple d y dt 4 dy dt 3y When the initil condition re,, nd r( t y ( 0 y ( 0 0 r ( t The Lplce trnform,we obtin Y 4 ( ( 4 3 / / 6 3 / 3 o y ( t e t 6 e 3t 3
Exmple 3: Conider the function ( ( ( 3 G Clculte g(t. 3 3 ( ( ( k k G nwer 0,, 3 k k
The trnfer function of liner ytem definition Trnfer function: The rtio of the Lplce trnform of the output vrible to the Lplce trnform of the input vrible,with ll initil condition umed to be zero. 零初始条件下, 输出变量的拉氏变换与输入变量的拉氏变换之比 7
Exmple: RC electricl network R U r (t U ( t 0 C U U o r ( ( R / C / C RC U r ( U o ( G(
( ( ( ( t c dt t dc dt t c d dt t c d n n n n n 0 ( ( t r b dt t r d b dt t r d m m m m m 0 ( Conider the dynmic ytem repreented by the differentil eqution If the initil condition re ll zero,then the Lplce trnform of the Eq. yield ( ( 0 C n n n ( ( 0 R b b b m m m The trnfer function i 0 0 ( ( ( b b b G R C n n n m m m
Some Comment bout the Trnfer Function. The concept of trnfer function only pplie to the LTI ytem.. Trnfer function i only determined by the tructure nd prmeter of ytem. 3. Trnfer function i rtionl proper frction, nd there reltionhip of the order of the numertor nd denomintor i n the order of the denomintor m the order of the numertor n m 4. The invere Lplce trnform of trnfer function i the impule repone function of the ytem.
5. A certin trnfer function correpond certin portrit of zero- pole ditribution. et C( G ( R( N( D( Pole: The root of the denomintor polynomil D(. Zero:The root of the numertor polynomil N(. 6. The method of the trnfer function h ome limittion. (. It only pplie to the SISO ytem. (. It only cn reflect the reltionhip of input nd output. (3. It only cn nlyze the motion chrcteritic of zero initil condition.
The trnfer function of ome component. Potentiometer. θ mx θ U ( Θ U( K E K E θ mx 3
. Potentiometer Bridge. θ θ E K p θ K p θ U ( Θ U( Θ ( ΔΘ K p
3.Ger trin Ger rtio n N ω L θ L n ω N m θ m, θ L N N n θ m
4.rmture-controlled dc motor V (t R L ω,θ i f InertilJ Frictionb V b (t ( ( ( ( V I L R V b ( ( ( K K V b b b θ ω ( ( ( ( L R K V I b ω (t i (
Motor torque T ( K I ( m m ( where T ( T ( T ( m L T L ( :lod torque ( :diturbnce torque T d d (3 T L ( J θ ( b θ ( (4 ( 0 We cn obtin the trnfer function(with T d G( ( θ V ( [( R L K ( J m b K b K m ]
The block digrm of dc-motor V ( T L K b ω( ω( Jb R L I K m θ ( V (t T m R i (t L V b (t ω,θ T d ( V ( R Km L T m T m ω( Jb θ ( K b
Component of block digrm. Signl line: A line with rrow tht indicte the direction of ignl trnform. U (. Block: It expree the trnfer function. U ( C( G( 3. Derivtion point (meuring point. U ( U (
4. Synthei Point (Compring point. U ( ( R( U ± ± R ( Exmple:Poition lve ytem θr θc Z f J R L θ m Z U K U E b f
θ r ( K U ( U ( U ( K A ( U ( θ c ( E b ( L R I K m T m ( θ m ( J f K b θ m( /i θ c ( r ( K U ( ( E b L R I K m T m ( θ m ( J f θ c ( i K b
Block Digrm Trnformtion ( Combining block in ccde X X G X 3 X X 3 ( G ( X X ( ( 3 ( ( G ( Prllel Connection of Block X G ( G ( X X 3 ± ( G ( X 4 G ± X X 4 ( G ( G G X 4 G ( ± G ( X
(3Eliminting feedbck loop X X ± G H X X G GH
(4Moving umming point behind block X ± X G X 3 X G ± X 3 G X (5Moving umming point hed block X G ± X X 3 X ± G G X 3 X
(6 Moving pickoff point hed block X G X X G X X X G (7 Moving pickoff point behind block X G X X G X X X G
Block digrm reduction H3 R( _ G G G3 G4 C( H H
R( G( G( C ( C ( C( R( G G G _ C ( R( G G G _ C (
G R( G G C( G G G C R ( ( G G G G G G
Signl-Flow Grph Model Signl-flow grph: A digrm tht conit of node connected by everl directed brnche nd tht i grphicl repreenttion of et of liner reltion. brnch G( V f ( θ ( Node: The input nd output point or junction 48
Pth : A pth i brnch or continuou equence of brnche tht cn be trvered from one ignl(node to the nother ignl (node. Loop: A loop i cloed pth tht originte nd terminte on the me node R( G C ( H
Mon ignl-flow gin formul P k Δ k T :kth pth gin :determinnt of the pth :cofctor of the pth Pk Δ Δ P k k k Δ L n LmLq Lr LLt Ln LmLq Sum of ll different loop gin Sum of the gin product of ll combintion of nontouching loop L r L L t
R( H H3 L L G G G3 G 4 G 5 G8 G G 6 7 Y( GG G3G4 L3 L4 H6 H7 P P G 5G 6G 7G 8 L L G3H 3 6 6 GH L L 4 G7 H7 3 G H Δ ( L L L3 L4 ( L L3 LL 4 LL3 LL4 Δ ( L 3 4 L Δ ( L L Y ( P Δ P R ( Δ Δ
G 7 R( G 8 G G G 3 G4 G5 G6 Y ( H 4 H H H 3 P GG G3G4G5G 6 GG G7G6 L G G G G P P 3 GG G3G4G 8 3 4 5H L6 GG G3G4G5G 6H3 8 3 4 8H L 3 7 GG G7G 6H3 L G G G G G 5 6H L3 G8H L4 G7H G L G G L G 5 4H 4
Δ ( L L L3 L4 L5 L6 L7 L8 ( L L L L L 4 5 7 5 4 3L Δ L Δ Δ3 5 Y ( P P Δ P R( Δ 3
H G 0 R( E( G G G 3 C( H R( G 0 E( H G G G 3 C( - H
G 4 R( E( G G G 3 C( H H G 4 R( H G G 3 G C( - H
G 5 H 4 G G G3 G4 y y y3 y4 y5 y6 y7 H H y G3H H 4 G3H H y Δ y y 4 Δ ( 4 GG H 4 Δ G H G H GG G H H GG H H 3 H 3 3 3 4 3 G HH 4 G3H H 4 GG G3H 3H 4 GG 3HH H 4
Some function of MATLAB ( conv(p,q multiply polynomil p nd polynomil q ( plot(t,y plot vrible y veru vrible t. (3 root(p clculte root of p(0 (4 pole(y clculte pole of y (5 zero(y clculte zero of y (6 erie(y,y erie interconnection of y nd y. (7 feedbck(y,y,ign feedbck interconnection of y nd y,ignor -.