Models of physical processes

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1 Model o phcal procee ecre noe 5DD Janar 6 PPJ van den Boch EH4 ppjvdoch@enl onen Procee and Model Modellng Model Derenal eqaon Sae pace eqaon Dervng a odel SI n naloge aong deren phcal eld Dervng a odel n 4 ep 4 Sgn rle Model n deren phcal eld Elecrcal procee Mechancal procee Hdralc procee 4 Theral procee 5 Ma low coparen odel pharac knec 6 hecal reacon 4 nearaon o non-lnear ae eqaon Traner ncon aplace Tranoraon Relaon eween derenal eqaon ae pace odel and raner ncon Dervaon o a ae pace odel ro a derenal eqaon Dervaon o he raner ncon ro a ae pace odel Dervaon o he derenal eqaon ro a raner ncon Freqenc repone Reglaon Sac eedack loop Dnac eedack loop Sead ae error 4 Eerce

2 Procee and odel Thee condened lecre noe deal wh odel o phcal or eaple elecrcal echancal hdralc heral pharaknec and checal procee I rn o ha hee deren pe o procee ca ne repreened he ae knd o dnacal aheacal odel can e derved B ng derenal eqaon and ae pace odel he ae aheacal orlaon are ed Dervng hee aheacal odel ro a phcal proce called odellng a llraed n gre For analng he dnac ehavor o hee aheacal odel and coneqenl o he nderlng phcal procee he raner ncon nrodced I rn o ha he characerc o hee raner ncon can e nerpreed drecl o predc dnac ehavor or eaple ep repone Proce Model Modellng Fgre Modellng and anal nal

3 Modellng Model Model are ed o nderand and predc he ehavor o real-le procee oon a we can decre hee real-le procee cenl accrae wh he ad o a aheacal odel analng and nderandng hee procee ecoe ncreangl eaer When he odel even cenl accrae pole o ake predcon on re ehavor o he proce eg weaher orecang ch odel can e condered a or knowledge o real-le procee One o he o dcl ep o deerne he e ondar n he real-le proce wh nde he coponen and ncon o e odelled and ode oher par ha have o e negleced I he nde par oo all a ple odel rel vale o decre or predc he ehavor o he proce oo rerced or oo naccrae I he nde par oo large a ver cople odel can rel whch oo coplcaed o deal wh Boh aon are no aracve and have o e avoded So he odel hold conan onl hoe coponen or ncon ha are reall poran or or d coneqence wll e ha here are an deren odel or he ae proce each odel or own applcaon Moreover an odel wll alwa e aller and decre le deal han he real-le proce o Model Proce noher poran acor deal wh he e range I we wan o d phologcal phenoena he e cale o he procee can e ver a llecond or elecrcal acve o a llon ear dealng wh evolon o ro - o 6 econd a range o 9 eween alle and large e nerval Th oo large! Reaonale odel have a range o aal 4 larger range wll nrodce cople large calclaon e and no coparale rel In general procee ha are aer han he eleced e range o he odel are aed o have reached an eqlr o are aed o e conan Proce ha are lower ha he eleced e range o he odel are o low ha he are condered oo a conan n he e condered he odel Eaple In dng he change o he lng drng evolon he e range ha o e a lea ear or ore Then decrng n he ae odel alo he perodc reahng o econd oh nrealc and echncall alo pole Fa odel Slow odel e [ec] In general we can dngh eween wo deren knd o odel nael rcral or echanc odel ha have a cloe reelance wh he nderlng proce or eaple elecrc crc drawng o hdralc or echancal e Real phcal coponen and her nerconnecon can e dnghed The nernal varale are clearl vle Thee odel ncorporae all avalale knowledge The are qe cople wh an oee nknown paraeer nconal odel ha j relec he eernal ehavor o a proce aong np and op The nernal conrcon or operaon o le porance Man goal o repreen eernal ehavor The nernal procee can e hdden Eaple o nconal odel are aheacal odel ch a derenal eqaon ae pace odel raner ncon ple repone or her graphcal repreenaon a lock dagra Fnconal odel are ore arac and coneqenl ore generall applcale Maheacal odel can e decred eqaon For or prpoe we dngh eween algerac eqaon derenal eqaon and ae pace eqaon: In algerac eqaon here an nananeo relaon aong he varale or eaple he relaon eween orce and dplaceen o a prng: k he relaon eween prere derence p and low φ n a ppe: φ c p

4 In derenal eqaon he e ehavor or dnac o varale poran For eaple when applng an np o a dnac proce ake oe e eore he eec ecoe vle and reache a eec Soee dappear agan Sch procee are called dnac Eaple: drnkng one or an glae o eer rel n a gradall ncreae o alcohol n he lood and aer oe hor n a hang-over Man hor laer all eec wll dappear charged capac can e dcharged a reor ake oe e eore volage ha dropped elow a ceran vale Preng he ga pedal n a car wll rel n an ncreae o veloc no edael he nal peed wll e reached Derenal eqaon are approprae ool o decre dnacal phenoena n eaple o a derenal eqaon In general we call he np varale and he op varale Sae pace eqaon are alo ed o decre he e ehavor or dnac o procee Derenal eqaon can e rewren a ae pace eqaon and vce vera For eaple he prevo derenal eqaon eld he ollowng ae pace odel Bede he np and he op he nernal varale he ae nrodced n n-h order derenal eqaon wll eld a ae pace odel wh an n-denonal ae vecor and n r-order derenal eqaon one or each o he n eleen o Moreover here an algerac op eqaon o dene he op a a ncon o he ae and np g For lnear ncon and g he ae eqaon can e orlaed wh vecor and arce or eaple: Th ae eqaon epree he ae dnacal ehavor eween and a epreed he derenal eqaon Derenal eqaon In derenal eqaon no onl he vale o a varale poran alo rae o change epreed a: d d l Th rae o change o he e dervave o o he lope o when drawn a a ncon o e Bede h r-order dervave alo hgher-order dervave o can occr n a derenal eqaon ch a he econd-order hrd-order dervave or n-h order dervave n The order n o a derenal eqaon deerned he hghe dervave o n he eqaon For calclang a nqe olon o an n-he order derenal eqaon he vale o n nal condon have o e known o n- Derenal eqaon rn o o e valale aheacal decrpon or dng analng and nheng dnacal procee 4

5 Eaple: Fr-order derenal eqaon wh nal condon and her olon Derenal eqaon Inal condon Solon a a a ep-a; a /a {-ep-a} ep-a; Second-order derenal eqaon wh nal condon and olon - ep- ep- Sae pace eqaon In dnac odel we can dene a ae vecor ede he np and op Th ae an nernal varale o he odel and repreen n general he phcal er o a proce I we know he ae a e o and he vale o he np or hen oh and or are ll dened he ae pace eqaon: g Thee ae eqaon dene he r e dervave o he ae a an algerac ncon o ONY he ae el and he np The op dened a an algerac ncon o ONY he ae and he np When he odel o n-h order he ae wll e a vecor wh n eleen When he odel ha one np and one op oh and are calar ncon For oanng a nqe olon and o here ha o e an nal condon or Vecor ha n eleen When oh ncon and g are lnear n oh and lnear ae eqaon rel: B D Vecor and arce: Sae pace odel le vecor and arce Thee are nce an aracve coponen ro lnear algera o pl varale and eqaon Vecor have n eleen eher ordered a a row n or a a coln n T ar ha or eaple n row and coln The denon o a row vecor *n a coln vecor ha denon n* and h ar ha denon n* Eaple The ar eqaon repreen he ollowng eqaon when a -denonal and a - denonal coln vecor: a a a a a a The aocaed eqaon are: a a a a a a learl a ch horer noaon Soe ore ar characerc: - I wh I he n ar When B T T ro ranpoe he ollowng vald: a kl lk Dervng a odel SI n 5

6 When dealng wh phcal e a nece o ake care o approprae n or all varale and paraeer o coponen Mch conon and even error can e avoded proper age ade o hee n In an eqaon he coned eec o all n ha o e he ae or all ndvdal par ha are added or raced For eaple n he ollowng eqaon he hrd er no correc j lookng a he proper n o he ndvdal er: R R R R R R R R There an nernaonal cenc andard o n SI: See Inernaonale Unhappl he US le oher n le n ead o eer pond n ead o kg ec whch gh gve conderale conon The even ac SI n are: phcal varale ol n lengh l [] eer a [kg] klogra e [] econd elecrcal crren [] pere eperare T [K] Kelvn qan aer n [ol] ol lgh low I [cd] candela naloge aong deren phcal eld In everal phcal eld he ae pe o varale and coponen can e recogned Thee varale alwa occr a par The are eleced ch ha her prodc eld power [W] The neracon aong phcal coponen can e e nderood conderng he echange o power aong he Underandng one eld ake nderandng he oher eld ore ea Tale decre he analoge aong he eld elecrcal echancal hdralc acoc and heral e The r varale called eor or acro varale volage orce orqe prere eperare The econd one he low or hrogh varale crren veloc or low The prodc o eor and low eld he phcal qan power P [W] In hee ve phcal eld or applcaon area he ollowng generc coponen can e dnghed wh her nqe relaon eween her eor e and low varale: Reance R apac Indcance e R e or e e τ dτ τ e or e τ dτ τ For elecrcal e he eor he volage [V] and he low varale he crren [] Then hee generc eqaon ecoe: Elecrcal reance R [Ω] Elecrcal capac [F] Elecrcal ndcance [H] R or τ or τ τ τ dτ dτ Onl a reance R or dpae power and o hea p Th heral power can n general no e recovered and wll leave he e apac and ndcance and are er o he can ore power [W] a energ [J] [W] The aon o energ ored n elcdaed n ale The energ E [J] ored n a [F}: E E τ τ dτ E d E d E [ τ τ τ τ τ ] 6

7 In he ae wa he energ ored n an ndcance [H] eld: τ E E τ τ dτ E τ τ dτ E τ d E [ ] Power ener and leave a er who a lo When akng a odel each er nrodce a ae Becae we know conen o he er a e o and he vale o he np or hen oh and or are ll dened he ae pace eqaon The ner o ae eqal he ner o er In ale addonal varale wh her n are nrodced and dened or pecc applcaon area ch elecrcal echancal hdralc and heral varale For each o hee area he wo o poran ac varale are elcdaed n ale Thee varale are eleced ch ha her prodc eqal power Onl he heral area ha a deren nerpreaon Tale Bac varale Elecrcal Mechancal ranlaon Volage: [V] Force: [N] rren: [] Veloc: v [/] Mechancal roaon Hdralc/ acoc Theral Torqe: M or τ [N] nglar peed: ω [rad/] Prere: p [N/ ] Vole low: φ[ /] Teperare: T [K] Hea low: q [W] Tale naloge aong deren phcal eld Elecrcal Mechancal ranlaon Mechancal roaon Eor [V] [N] M or τ [N] Flow [] v [/] ω [rad/] Hdralc/ acoc p [N/ ] φ[ /] Theral T [K] q [W] Power P [W] Reance daper R apac prng Indcance a nera nerance Ber wh ae varale Energ ored n - er [J] Energ ored n - er [J] P P v P τ ω P p φ P q!! R [Ω] R [F] n o [H] [N/] v k [N/] k v k [kg] v a v v v [N/rad] τ ω k [N/rad] τ k ω τ k θ J [kg ] ω J τ τ J α J ω θ θ ω θ ω J τ R H [N/ 5 ] ln: p φ R H non-ln: p R H φ φ k p H [ 5 /N] p H φ φ φ n φ o H [kg/ 4 ] φ H p H p p φ H H φ φ H p ~ ~ R T [K/W] T q R T T [J/k] ½ ½k ½kθ ½ H p T T ½ ½v ½Jω ½ H φ T T q q q n q o T T T T q 7

8 Bede he coponen R and here are ore cople coponen ha ake care o power converon whn or eween wo eld or eaple ranorer oor and pp When hee coponen are deal here no power lo oneqenl he power a one de ha o e eqal o he power a he oher de Thee coponen are llraed n ale Tale Power converon On oh de he ae varale eor or low E/E ranorer α α P P Mr/Mr gear o τ α τ ω α ω P τ ω τ ω P M/H pp p p φ v P M v p p φ P H On oh de deren varale: E/Mr D-oor τ α α ω P E αωτ/α ωτ P M α E/M lnear oor/ lodpeaker α α α v P E αv/α v P M Mr/H cenrgal pp p p α ω τ α φ P H p p φ αω τ / α ω τ P M Dervng a odel n 4 ep a: Dene np Inp are eernal varale ha canno e nlenced : Dene ae Each er a ae : Dene ae eqaon Each ae varale ha own ae eqaon ale ' : Reove all varale n rgh hand de ecep or and 4: Dene op eqaon onl a ncon o and g Ue addonal eqaon or h reoval Ver gn o ae varale n eqaon Op are gven he prole orlaon or applcaon ae pace odel can alwa e ond ng he ollowng 4 ep: Dene np and deerne he er Each er ha a ae varale aocaed wh Dene h ae The ner o ndependen ae ecoe he order o he ae pace eqaon Each ae varale ha own ae eqaon ee ale Reove/elnae all varale ha are no np and no ae ro he rgh hand de Th goal can alwa e acheved ng addonal eqaon or eaple applng Krchho law coponen eqaon ec Ver gn o ae varale n rgh hand de 4 Dene he op eqaon a ncon o onl he np and he ae 4 Sgn rle The ae varale a he rgh hand de o he ae eqaon ha correpond wh he ae varale a he le de wll have WYS a gn Th rle a characerc o all phcal pave procee Reark: When an np ed ha depend on he ae varale h rle doe no appl The gn o he ae varale n he np hen nceran Eaple: echancal a-prng e eld: k or v k v k k v 8

9 Model n deren phcal eld Elecrcal procee * agn a volage /- and crren drecon o each coponen * Each capac eld a ae * Each ndcance eld a ae * Each jncon o wre : appl Krchho cren law: n j j ne crren n an connecon ero or * Each loop o coponen: appl Krchho volage law n j j ne volage along a loop n a crc ero or Eaple parallel connecon o a reor R [Ω] an ndcance [H] and a capacor [F] wh a crren orce I [] eld a odel wh np I and ae c [V] and crren []: I I R R I here a ere connecon o a volage orce U [V] wh a reor R [Ω] an ndcance [H] and a capacor [F] he aocaed odel ecoe wh np U and ae c [V] and crren []: U R U R Mechancal procee ranlaon: * gn a veloc v drecon and poon drecon o each a * Each a eld a ae or oh veloc v and poon * Search or all orce n one drecon or each a Take care o he gn The ne o hee orce eld he acceleraon n ha drecon I here are ore drecon n D and and n D repea procedre or ne drecon Soee he ae and acceleraon are negleced Onl veloc and poon are condered a relevan Then he ae eqaon redce o k or who prng conan roaon: gn an anglar veloc ω and an angle θ o each nera Each nera eld a ae or oh anglar veloc ω and angle θ Search or all orqe/oen n one drecon or each nera Take care o he gn/roaon drecon The ne o all hee orqe eld he acceleraon n ha drecon I here are ore drecon n D and and n D repea procedre or ne drecon Hdralc procee * gn a prere acro /- and low drecon o each coponen * Each hdralc capac H eld a ae p 9

10 * Each hdralc nerance H eld a ae φ The nonlnear hdralc reance R H odelled wh φ k p onl vald or pove vale o φ I he low goe n he oppoe drecon h reance ha o e odelled a: φ k gn p p The hdralc capac H [ 5 /N] dened a he rao o he change o vole V a an ncreae o he prere p: H V p [ 5 /N] For an open veel wh cro econ [ ] grav g [/ ] and ld wh pecc a ρ [kg/ ] and hegh o ld h []: V h and pρg h o H h ρ g h ρ g In ead o he prere p oee he hegh h led a ae varale: ρg p φ φ wh pρgh h eld H p ρgh ρg φ o h φ I he wall o a veel or e elac ha veel/e eh a coplance H The hdralc nerance H [kg/ 4 ] repreen he a o he ld o e acceleraed he prere derence The hdralc nerance o a ld n a ppe wh lengh l [] and cro econ [ ] ecoe: H l ρ [kg/ 4 ] Eaple veel wh cro econ [ ] conan a lqd wh hegh h [] There a low n φ and a low o φ o o he veel The o low leave he veel a he oo a nonlnear rercon k Inp: ae p The odel ecoe: ρg p φn φo φn k p φn k p H When he hegh eleced a ae he odel ecoe: h φn φ o H φn k ρgh φ p h φ k 4 Theral procee Each heral capac T eld a ae T Frcon and a reance nrodce loe whch are he np q [W] or a heral proce Th heral power canno e ranored who loe o echancal or elecrcal power Eaple Sppoe a a wh a eaol o q [W] n an ncaor wh heral capac T and reance o envronen R T and heang q h [W] Teperare envronen eqal T e [K] Wh np q h q and T e ae T o he ncaor he odel ecoe: T T T e qn qo q qh T T R T T E R T T T q h q

11 5 Ma low coparen odel pharac knec Relevan qane are a and a low o e epreed n [ol] or [kg] and [ol/] or [kg/] Ma he ae varale o a er ha receve and releae a Bac law: Preervaon o a n a coparen: n o When here are an coparen wh a low aong he he ollowng varale are o e dnghed: : a n coparen [ol] or [kg] j : a low ro j o [ol/] or [kg/] o : a lo/ecreon ro coparen [ol/] or [kg/] : eernal ppl n coparen [ol/] or [kg/] Ma echange eween wo coparen and envronen: n n o o The low j can ollow everal law: near oncenraon angr Mchael-Menen j k j * j j c j * j j k j *-σ * j j a* j / j k j *V j * j /V j Two coparen odel wh lnear low: k k k or k k k k k k k k k Eaple: Sppoe a veel wh a conan vole V [ ] and low [ /] n and o o The concenraon n he np low o oe aeral c [kg/ ] Then he a low no he veel *c [kg/] The ae he a [kg] n he veel The concenraon c v n he veel ecoe c v /V oneqenl he concenraon o he op low c o c v Inp: c and o ae: Now he ae eqaon ecoe: c o c o c o V V o c Eaple: Sppoe a veel wh a changng vole V h [ ] and low [ /] n and o k h The concenraon n he np low o oe aeral c [kg/ ] Then he a low no he veel *c [kg/] The ae are he a [kg] n he veel and he hegh h [] o he lqd The concenraon c v n he veel ecoe c v /V/h oneqenl he concenraon o he op low c o c v Inp: and c ae: and h Now he ae eqaon ecoe: c h c o c o c o o k h h h k o

12 Eaple: Two cell are conneced There an echange o a eween oh cell j k*c j - c wh c /V he concenraon n cell and here are addonal np and op [kg/] or each cell Inp: and ae: and Then he ae eqaon or he ae and ecoe k * V k * V V V V o c c V o In h eaple he echange o a eween he wo vole/er drven he derence n concenraon The larger h derence he large he a low wll e The addonal np and op are odelled a a low ndependen o concenraon I n a new prole he olow are dened a vole low [ /] hen he oded odel ecoe k * V V V k * V 6 hecal reacon V Sae varale: aon o aeral X: [X] ner o ol o aeral X: [ol] k B k [] [B] []: aon o ol [ol] k reacon conan [ol - - ] k reacon conan [ - ] Eqlr: V ] [ B ] [ ] k[ ]*[ B] k [ ] [ ] [ B ] [ ] o [ ]*[ ] [ ] [ ]*[ B] k k B k or conan [ ] k [ Sppoe a [] a [ol] [B] [ol] and [] c [ol] Dene [ol] a he ncreae o o alo he decreae o and B Then vald: [] a a - [B] - and [] c c There onl one ae varale ecae he aeral are ed ogeher he checal reacon eqaon Sae eqaon o h checal reacon: k a k c Becae here can e no negave aon o aeral here are ond or : a and -c oneqenl c na The eqlr or ond when So he eqlr e or and o or B and deerned he olon o k a k c oneqenl a e a e e e and c e c e e e e For he ore cople checal reacon ollow: k B D k

13 [ ] [ B ] [ ] [ D ] k[ ]*[ ]*[ B] k [ ]*[ D] ½ Sppoe a [] a [ol] [B] [ol] [] c [ol] and [D] d [ol] Dene [ol] a he ncreae o o alo he decreae o and B and ncreae o D Then vald: [] a a - [B] - and [] c c and [D] d d There onl one ae varale ecae he 4 aeral are ed ogeher he checal reacon eqaon Sae eqaon o h checal reacon: k a k c d Becae here can e no negave aon o aeral here are ond or : ½ a -c and d oneqenl nc d n½ a The eqlr or ond when So he eqlr e or and o or B and D deerned he olon o k a k c d oneqenl ae a e e e c e c e and d e d e Eaple lood veel ranpor nren along an organ The lood low [ /] and he concenraon o c [ol/ ] When loodveel pae he organ here an echange o he nren don wh don conan k d [ /] The vole o he lood veel cloe o he organ V [ ] and o he organ he vole eqal V o In he organ a checal reacon ake wh B o prodce wh reacon conan k [/ol] and k [/] e [B] [ol] and [] [ol] o e conan Inp c [B] and [] ae [ol] and o [ol] he ner o ol n he vole V and V o repecvel The odel ecoe o * c k d k * V d * V V o o o V V k [ ]*[ B] k o [ ] 4 nearaon o non-lnear ae eqaon For a nonlnear algerac ncon a lnear approaon can e calclaed n oe eleced operang pon o I wll e he angen d/d n ha operang pon { o } So he lnear approaon n { o } ecoe wh : Th approaon accrae when o or all vale o

14 Fgre Nonlnear ncon and lnear approaon on Eaple I 57 hen he lnear odel or 4 ecoe Th lnearaon o an algerac ncon can alo e appled or he lnearaon o nonlnear ae eqaon pplng he ollowng ep eld a lnear ae pace odel Dene operaon pon or non-lnear e g ang re/ead ae So Selec he re/ead ae/average vale o I n5 alclae he operang pon ro alclae he vale o n he operang pon: g neare he non-lnear ncon and g ng he Talor approaon e ha and o he odel vald n he neghorhood o and wh all varaon o and Oerve - and - Then we have he ollowng lnear odel n he neghorhood o : B In he ae wa he lneared op eqaon ecoe: 4

15 g g g g g Oerve g - g - and - hen we have he ollowng lnear odel n he neghorhood o : D g g Reark: paral dervave I he odel econd order hen oh { } T and { } T are vecor wh eleen Sppoe here onl one np The paral dervave o and g wh repec o and ecoe: g g g g g Eaple: 4 Operang pon: 4 and - Dervave o wh repec o ecoe and he dervave wh repec o ero oneqenl he ollowng lnear odel rel: : 4 whch an nale lnear odel - : 4 whch a ale lnear odel Sal can e hown lookng a he eqaon e or pove So wll dervave whch ple ha wll ecoe even larger and gger wll no rern o ero and o no o Th ehavor called nale e or pove Then dervave negave whch ple ha wll ecoe aller wll o ero and o o Th ehavor called ale eer eplanaon gven n chaper Eaple: Second-order e wh ae T ; now no he op a ae varale! Operang pon: o T -5 T or T and For he r operang pon -5 T wh egenvale - and - o ale 5 For he econd operang pon T : wh egenvale and o nale 5

16 Traner ncon ople ner The aplace ranoraon and he reqenc repone le cople ner cople ner c can e decred wh real a and agna par: c a j odl c and phae ϕ argc wh c a and ϕ argc aan/a o c c epjϕ j ndcae he agnar ner wh j j c c a a j c - c a - a j - c c a a - ja a c c epϕ ϕ c /c c / c epϕ - ϕ epc epaj epaepj epc epa argepc Reark: Maheahcan and Mala le he o ndcae he cople ner Here we e he j o ake a dncon eween crren and cople ner j aplace Tranoraon The aplace ranoraon agn or a real-valed varale a ncon o e a cople varale Y a ncon o he cople varale σω { } Y e d For oe ncon he aocaed vale o Y hown n he ale I Y ha een calclaed he aocaed vale o can alo e rereved ro h ale o Y o Y ple: nω ω/ ω ep: U / coω / ω rap: / ep-anω ω/a ω ep-a /a ep-acoω a/a ω ep-a /a When oh np and have een aplace ranored no U and Y he relaon een oh ecoe he raner ncon H wh YHU I rn o ha ch eaer o anale H han he derenal eqaon o and Ver aracve propere o he aplace ranoraon are he ollowng relaon: { } Y { } and Y ec pplng hee relaon o a derenal eqaon eld Y Y o 5Y 5 Y 4U 8U 6

17 U Y The raner ncon H o he e he relaon eween Y and U {HY/U} neglecng all nal condon So H o h derenal eqaon ecoe: H We call he roo o he denonaor o H pole and he roo o he neraor ero Once Y known can e calclaed ng paral racon epanon Fr he n roo pole o denonaor D o Y have o e calclaed and ne he n rede n D N Y wh no lple pole occr: l Y The e repone ecoe aer ranorng each eparae roo a an eponenal ncon: n e I here are par o cople pole -σ ± jω o σ ω ζω n ω n hee qadrac er can e ranored a par: ω σ ω ω σ ω σ σ ω σ c c Y n co e c e ω ω σ ω σ σ Eaple: a wh o and he n ep Then a a a a a U a Y o o o a a and a a a a /a{-ep-a} o ep-a or The nlence o everal paraeer σωζω n o he raner ncon H concernng cople roo o he denonaor pole can e lnked wh he paraeer p M p r o he ep repone o he e 7

18 paraeer ol nae nlence Degn real par -σ aole dapng al elng e real par < σ> large σ agnar par ω reqenc peak e p large ω angle ζσ/ω n ζ relave dapng overhoo M p large ζ eg ζ>7 odl ω n naral reqenc re e r large ω n The aole dapng σ ndcae he peed wh whch he repone ele For pove vale o σ o negave roo! he repone wll reach a conan vale For /σ ep-σep-5 Then ao whn 5% o nal vale For 5/σ ep-σep-57 oneqenl σ deerne he elng e o a repone The larger σ he aer he eer The agnar par ω deerne he reqenc n The hgher ω he hgher he reqenc and alo he aller he peak e p wll e The relave dapng ζ {ζnθ} deerne he overhoo M p o a ep repone For ζ θ here no dapng and he overhoo wll e % For ζ θ9 o here are no ocllaon no agnar pole and o no overhoo degn vale or eaple ζ7 whch eld ao 5% overhoo The naral reqenc ω n deerne he re e o a repone The hgher ω n he aer he repone and he aller he re e r 4 v 5 I M p ζ θ ω 8 v 5 ω n σ Re p Fgre Relaon eween paraeer o H and ep repone Inal vale heore Who ng he nvere aplace ranoraon pole o calclae he vale o j aer ndcaed wh drecl ro Y: l ly Fnal vale heore The nal vale or ead ae vale o can e calclaed or ale odel: l l Y Eaple: Sppoe H and a n ep o U o 4 8 Y hen and 8 5 MTB: np neraor polnoal: n [4 8]; 8

19 np denonaor polnoal: den [ 5 ]; roo o denonaor: roo rooden; egenvale o ar B egenvale egb; rede [ p k] reden den; : rede p: pole roo k conan degreendegreeden Mala calclae cople rede or cople pole Takng wo conjgaed pole ogeher a a qadrac er eld real coecen and o e preerred raner ncon h n den; ep repone ep h; draw ep repone plo aplace olc laplaceep- eld / nvere aplace olc laplace/ eld ep- Relaon aong derenal eqaon ae pace odel and raner ncon Dervaon o a ae pace odel ro a derenal eqaon Sppoe he ollowng derenal eqaon avalale or he decrpon o a proce: Then he aocaed raner ncon can e derved: U Y Inrodce he neredae varale Z and dngh eween he neraor N and he denonaor D o H Then we have } 5 { 5 Z Z N Y or U Z U U D Z U D N U H Y There are an pole o dene he ae One wa o elec T wh Then we have 5 5 Moreover he neraor e YNZ48Z or onng oh eqaon eld he ollowng ae pace odel: 9

20 8 4 5 The ae eqaon can alo e eleced or drawng a lock chee gre 4 a graphcal repreenaon o he aheacal eqvalen decrpon o derenal eqaon he raner ncon and he ae pace odel / / / 4 8 Z U Y Fgre 4 Block chee o a aheacal odel Reark: The coecen o he denonaor are vle wh a n gn! on he r row o he ar The coecen o he neraor are placed n he -ar Th ae pace odel j one o he an pole ae pace odel ro a derenal eqaon or raner ncon MTB [ B D] [4 8] [ 5 ]; D B Mala calclae a deren alo correc ae pace odel! Dervaon o he raner ncon ro a ae pace odel D B Eerce he aplace ranoraon on oh de and neglec nal condon : } { D U B I DU X Y BU I X B X X B or D B I H

21 MTB [n den] B D; Dervaon o he derenal eqaon ro a raner ncon I he raner ncon H avalale hen YHU For eaple: 4 8 Y U 5 Mlpl oh de wh he denonaor o H eld: Y Y 5Y Y 4U 8U Recogne ha n Y n he aplace doan correpond wh n he n-he dervave o n he e doan oneqenl he aocaed derenal eqaon Freqenc repone Sppoe Y HU Boh Y H U and are cople varale The aplace ranoraon allow he calclaon o ranen repone ch a he repone on a ep or an ple For e gong o nn onl he ead ae repone rean In he cople aplace operaor σjω σ repreen he ranen When dealng wh reqenc repone onl he ead ae repone o nere oneqenl σ dappear and he aplace operaor redce o jω or H jω Hjω The reqenc repone Hjω alo a cople ner wh odl H jω and phae arghjω When he np gnal o e H a ne wave ˆ n ω hen he op gnal ecoe ˆ n ω φ wh ˆ H jω ˆ and ϕ arg H jω So he odl H jω he gan o he proce a ncon o he reqenc learl h gan reqenc dependen When he degree o he neraor aller han he degree o he denonaor o here are ore pole han ero he odl H jω decreae o ero or ncreang vale o he reqenc Eaple: I H 4/4 hen Hjω 4/jω4 The odl o H eld Hjω 4/ ω 6 When he np gnal eqal 5n hen or ω he reqenc repone o Hjω: H 4/ 6 9 and argh aan/4 46 [rad] oneqenl he op ecoe 9*5n46 45*n46 MTB hn den; odeh

22 Reglaon Man phologcal procee are reglaed o garanee ha he phcal or checal condon o he od rean whn accepale ondare n pe o an nernal and eernal drance For eaple he od eperare he lood prere and he O concenraon n he lood are kep reaonal conan n pe o large varng acv o he od Th conan operang pon can e realed oe knd o eedack The vale o gnal are eared and are ed or akng correcon The ac dea o eedack llraed n gre g g Sac eedack loop Fgre 5 Sac reglaon The wo procee and g are ed ogeher ean o a eedack loop Boh procee deerne he operang pon nael g{ and o g{ } or { g } Eaple: The olon or operang pon ha o a oh eqaon oneqenl - - o and The operaon pon ae oh ncon and he cro-pon o oh graph and g When and/or are dred he wll rern o he operang pon deerned he ncon and g When hee ncon change he operang pon wll no e he ae h a For eaple change ro - no - he new operang pon ecoe 5 5 long a hee wo graph cro each oher o have a coon pon here wll e one or ore operang pon Sll havng ch a pon doe no garanee ha he operang pon ale I pole ha a all devaon ro he operang pon a rel n a decreae o h devaon o convergence o he operaon pon or an ncreae o dvergence The r aon called ale and he econd one nale For a phologcal proce ale operang pon are reqred Eaple pp hear ake care ha here a prere eween np and op Owng o nernal rcon he prere drop when he lood low ncreae Ppe arere ven have a reance when here a lood low The larger he low he larger he prere drop over hee ppe Sppoe P he prere derence over he pp and over he ppe and call F he low Then he pp can e odelled a PF and he ppe a FgP or Pg F The cro-pon o oh ncon eld he operaon pon P F ha ae oh he characerc o he

23 pp and ppe Sppoe: pp: P P - af or he pp and P F or he ppe Then he operang pon o h pp wh hee ppe wll e P F wh F P P a a P Boh pp and ppe deerne ogeher he vale o he operang pp There no caal ha arge ha he pp deerne he prere and he ppe he low or vce vera Boh do Dnac eedack loop In he prevo paragraph he ncon and g are aed o e ac ncon However hee ncon can alo e condered eng dnac eg repreened raner ncon or eaple X H Y and Y H X Then X H {H X} or X H H H and Y H H Now he al ll deerned he pole o he denonaor {- H H } I he all have negave real par h eedack loop ale Devaon ro he operang pon wll e parl ppreed and he e wll rern o operang pon Eaple: Pre - predaor relaonhp Ra and oe n he ae ecologcal area depend on each oher Th can e orlaed wh he ollowng odel wh r he ner o ra and he ner o oe Ra ncreae a no oo ch oe and cen ood c avalale Who ra e oe wll de d In h odel r repreen he proal ha a o ee a ra whch negave or he ra and pove or he oe I here are oo an ra and he ood ecoe a laon her growh wll e rerced Th acor odelled wh a qadrac er r : r a r r c r d er For ale vale o he coecen a he ale operang pon ede can e deerned: r d e ae cd e Feedack loop For echncal conrol e he ollowng conrol chee oen pleened a elcdaed n gre 6 I clearl recogne an np an op and a reerence r whch repreen he reqred vale o he op Moreover a drance d added whch relec drance and no-odelled par o he phcal e The proce ndcaed raner ncon H ha np and op conroller ha een added wh np e r - I calclae an approprae vale or he np o he proce Nearl alwa he conrol reqreen are aed ch ha ha o ollow reerence r a good a pole So he goal o keep e he error a all a pole or deren vale o r and d and ae alo or changng vale o H R - E U H D Y Fgre 6 Un eedack loop

24 Ung oe calcl we oan: Y D HU U E E R - Y Y D H[R - Y] or H Y R D H H and E R Y R D H H I we can garanee ha H >> he nlence o D wll e redced and Y R The onl reqreen we ake ha he conrolled e ale and ha he prodc o proce and conroller ver large The larger he eer wll r So a well-degned conroller can reale ha he op o a proce wll ollow he reerence vale r n pe o he preence o drance d and alo n pe o change n H The reqreen ha H >> o neher he odel H o he proce nor he drance d have o e accrael known Feedack a a echan or le o rvve Feedack one o he ore powerl characerc o le In pe o large varaon n phologcal procee H and large varaon n he envronen D proper eedack allow a cell an organ an organ ndvdal or even poplaon o anan ale vale or poran phologcal varale ch a concenraon eperare and prere In hee ore ale condon he cell organ organ or ndvdal can rvve and even proper Derence eween phologcal and echncal eedack loop R H D Y g Fgre 7 oparon eween phologcal and echncal eedack loop In a echncal eedack loop pcall can e recogned: o a enor o ene or eare he varale o e conrolled o a conroller wh a dedcaed ak o ne he error eween reerence and eared op o an acaor ha e he conroller gnal a np and nlence he proce eg a oor or a valve In general here onl one enor and one acaor In phologcal eedack loop here are everal procee ha ogeher deerne he ale operang pon o hee procee In an aon no clear reerence gnal can e oerved The ale operang pon ealhed he cro-pon o wo procee Reall poran conrol loop n he han od or eaple or reglang he lood prere have an enor an acaor and an conroller o anan al n an qe deren aon Even coplee alre o one or ore enor or acaor can e acceped ecae oher conrol loop wh deren enor and acaor wll replace he ncon o he roken one So eedack or conrol loop n pholog rn o o e ore ro and ore relale han echncal one Thee echncal olon have a hgher perorance or her dedcaed ak Ode ha ak or wh evere drance he wll copleel al Relal rn o o e poran or lve han perorance 4